LIBRARY I Michigan State University I}. This is to certify that the dissertation entitled W HO LES ALE PR I CE DETERM I NAT ION presented by Tzong-Rong Tsai has been accepted towards fulfillment of the requirements for Ph.D. Economics degree in @414»; n/lbuwu U Major professor {I Dan June 15, 1987 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 MSU ’ LlBRARlES —,—. RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. ML) Q JALJf) WHOLESALE PRICE DETERMINATION BY Tzons-Rons T831 A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the decree of DOCTOR OF PHILOSOPHY Department of Economics 1987 0 Copyright by Tzons-Rong Tsai 1987 ABSTRACT WHOLESALE PRICE DETERMINATION BY Tzong-Rong Tsai The study of wholesale price determination is important because there are many industries characterized by retailers and wholesalers that both.have significant bargaining power. Firms' bargaining power arises from two sources: their fewness in number and their ability to differentiate the product being sold. To model the fewness of firms on both sides. bargaining theory provides an adequate analytical framework, while for the product differentiation issue, spatial economics proVides some useful techniques. This dissertation is one of the first to integrate the above tools for the analysis of wholesale price determination. Throughout this dissertation. we assume that the retailers are identical to one another. and so are the wholesalers. Also. we assume that the demand for the wholesaler's product is a derived demand. For the bargaining solution, we use the one advocated by Nash [1950]. [1953]. Binmore [1982] and HcLennan [1982]. For the bargaining process. we adopt the one used by Rubinstein Tzong-Rong Tsai [i982] and Davidson [1985]. To incorporate spatial factors into the analysis. we use a model similar to that of Novshek [1980]. Our maJor findings are as follows. First of all. when there is one wholesaler at the wholesale level. with a linear demand function. the equilibrium wholesale price is invariant to the number of retailers if spatial factors are not incorporated into the analysis. Secondly. in our spatial bargaining model. the equilibrium retail price is not equal to but higher than the retailer's marginal cost. (i.e. the wholesale price). even in the Bertrand type price- setting games. Thirdly. in the two-by-two pure bargaining game. the bargaining outcome is very sensitive to the bargaining process. Fourthly. in a game where the wholesaler cooperate With the retailers to maximize their sum of profits. the equilibrium.wholesale price is equal to the wholesaler's marginal cost if the retailers are local monopolists; it will be higher than the wholesaler's marginal 6081’. if the retailers compete With one another. ACKNOWLEDGEMENTS I am.glad to have this opportunity expressing my sincere thanks to these individuals for their friendship. support. and encouragement. First of all. I want to thank Professor Jack Meyer. Chairman of my dissertation committee. for his many-hour adrice during the past two years. His insightful comments and suggestions are also most highly appreciated. Special thanks go to Professor Carl DaVidson. for his enthusiastic encouragement and ever-lasting help since the first day I entered Michigan State University. Sincere appreciation is also given to Professor Paul Segerstrom for his valuable conInents resulted from his careful reading this dissertation. I also want to thank Professor Lawrence S. Martin for his valuable suggestions and inspiration. Also. I want to express my sincere thank to Professor Anthony Y. C. too for his kindness and help in past years. Most importantly. I owe my deepest thanks to my wife. BihrShiow Chen. for her continuing support and never-ending encouragement. Without her unselfish love and considerate help (as well as the tedious typing work). I could not finish my graduate study. Also. I would like to thank my son. Jay. who brightened my life with his Joy and love. Finally. I want to thank my parents for their love and care since my childhood. TABLE OF CONTENTS Page LIST OF TABLES .......... ..... .......................... vii CHAPTER I. IRTRODUCTIOH ....... ........... ..... ............. 1 Background of the Problem . ...... .... ...... ... 1 Methodology and Purpose ................... ... 7 structure OOOOOIOOOOOOOOOOOOOOOOOO 00000000 0000 9 Footnote .......... ..... ................ ...... 15 II. LITERATURE REVIEW . .............................. 16 Introduction ..... ................... . 16 Review of Literature on the Theory of Bargaining 00000000 000000000 .0000. 00000000000 .00... 18 Review of Literature on the Theory of Spatial Economics 00000000000000.0000.0000000000000 29 Conclusion ... ......... ....................... Footnotes ................. .......... ......... 45 III. WHOLESALE PRICE DETERMINATION (I) - PURE BARGAINING CASES - .................... 49 Introduction ....... .. .......... .......... 49 Bilateral Monopoly Case ........... . ...... . 50 Monopoly Wholesaler versus Duopoly Retailers .................................... ...... 56 Monopsony Retailer versus Duopoly Wholesalers ........................................... 58 Duopoly Wholesalers versus Duopoly Retailers (I) - Every Wholesaler provides goods to one and only one retailer .................... ... 76 Duopoly Wholesalers versus Duopoly Retailers (II) - Wholesalers take turns bargaining With all the retailers ......... ...... . ..... .. 82 Duopoly Wholesalers versus Duopoly Retailers (III) - Retailers take turns bargaining with all the wholesalers ... ..... ..................... 94 Conclusions ... ............ . ....... .. ..... .... 102 Footnotes .. ......... . ........................ 105 vi CHAPTER Page IV. WHOLESALE PRICE DETERMINATION (II) - BARGAINING CASES WITH SPATIAL FACTORS - .... 109 Introduction ................................. 109 Assumptions and Notation for the Linear Demand CaseOOOOOOOO 0 000000 000000 0000 111 The Analysis of the Linear Demand Case ....... 114 Assumptions and Notation for the Completely Inelastic Demand Case ...... ..... ....... 127 The Analysis of the Completely Inelastic Demand case 00000 0 0000000000000000000000000 O 00000 128 The Long- -Run equilibrium of the Retailing Level ........................................... 137 Conclusions ....................... . .......... 145 Footnotes ..... ...... . ........................ 151 V. WHOLESALE PRICE DETERMINATION (III) - COOPERATIVE CASES WITH SIDE PAYMENTS - ..... 154 IntroauctionioootOooo 000000 00000000 154 Assumptions and Notation for the Linear Demand Case ..................................... 156 The Analysis of the Linear Demand Case .. ..... 159 Assumptions and Notation for the Completely Inelastic Demand Case . ........ ............ 176 The Analysis of the Completely Inelastic Demand caseOOOOOOI 00000 0 0000000000000 00.000.00.00 17? Conclusions .................................. 167 Footnotes ......... ...... ..... . ............... 191 VI. CONCLUSIONS AND EXTENSIONS ............. . ........ 193 APPENDIX A Conditions for Different Types of competitive Situations at the Retailing Level ......... ...... 198 APPENDIX B The Long-run Equilibrium at the Retailing Level in the Cooperative Game .... ....... ...... ........ ... 207 BIBLOGRAPNY ........................ . ......................... 217 V11 LIST OF TABLES TABLE Page I. Comparative Static Results for Bargaining Cases II. Comparative Static Results for Cooperative Cases OOOOOIOOOOOOOOO 000000 0000 000000 000.000.0000... 189 V111 .--Q 11 or re hr Chapter One: Introduction I. Background Of the Problem In traditional economic theory. with perfect competition in all markets. every good is sold at its marginal cost. The distinction between intermediate goods and final product is thus of little importance. The study of wholesale price determination (or the price detenmination of any intermediate good) is. therefore. unnecessary if the assumptions of perfect competition closely approximate the reality of the economy. However. perfect competition actually characterizes a small fraction of the markets in the real world. For example. one of the most important assumptions in perfect competition. - that there are many firms in every market. with no firm having market power - is obviously unrealistic in many industries. In our real world. the auto industry. the computer industry. the breakfast cereal industry. etc. are characterized by a small number of large firms. The equilibrium price is therefore by no means equal to the firm’s marginal cost. Furthermore. another important assumption of perfect competition - goods are homogeneous in every industry - is not realistic. either. Even though it is possible that. in some industries. goods produced by different firms are homogeneous in every physical aspect. consumers may treat them as differentiated goods due to the differences in firm’s location. brand image. etc. mix 3:01 [19' amc' A We Oler The economic reality of a small number of firms in many industries encouraged the study of oligopoly theory. Economists. like Cournot [1836]. Chamberlin [i942]. Friedman [1977]. Shubik [1960]. etc.. have studied.the interactions among firms in oligopolistic markets. In their analyses. strategic considerations play a critical role as firms have a variety of complex strategies to choose from. For a firm. these strategic considerations include: What price or quantity should be set to deter entry; how to punish other firmm for deviating from.collusive behaVior; When the firm should exit from the market. etc. Ow1ng to the striking developments made by economists in this field. oligopoly theory has become one of the major streams in microeconomics over 938?. twenty years. Oligopoly theory. which.deals with the problem of a few sellers versus numerous buyers. assumes that producers produce and distribute goods directly to consumers without the help of retailers. That is. the vertical market relationship between the Wholesalers (i.e. the producers) and the retailers has been ignored. This is not a serious problem if the retailers are competitive buyers to the wholesalers. Under this situation. standard oligopoly theory can be used to solve the problem of Wholesale price determination without any trouble. However. economists. such as Porter [1976]. have shown empirically that the retailers are not competitive buyers in most industries. [19 1nd 301 Furthermore. Porter [1976] emphasized that. from two sources. the retailers may have bargaining power over the wholesale price. The first one is the 'fewness' of retailers within a local area. The second one is their ability to differentiate goods. By providing product information. after-sales services or even by different locations. the retailers make consumers feel that their commodities are not homogeneous. Product differentiation proVides the retailers with market power at the retailing level and bargaining power in the Wholesale price 401’. erminat 1 on. With a small number of firms on both.sides of the market. wholesale price is likely to be determined by bargaining between agents. Most economists before Nash [1950] regarded the solutions of bargaining problems as indetermdnate. Nash [1950]. however. prov1ded a definite solution to the bargaining problem by using the axiomatic approach. In his [1951. 1953] articles. he provided a different approach.- the noncooperative game-theoretic approach.or the strategic approach. The strategic approach has the advantage of revealing the connection between the bargaining process and.the resulting outcome. However. in his [1953] model. the 'Nash.Demand Game” had a baSIc difficulty that any Pareto efficient point is a possible Nash equilibrium. So. there are infinite number of equilibria in.this model. However. a model with a multitude of equilibria is clearly of limited interest. There were SC 10 CO 9‘1 99 some unsuccessful tries to refine the Nash equilibrium prior to Selten [1975]. Selten [1965. 1973. 1975] introduced the concept of ”perfect equilibrium“ to refine the Nash equilibrium concept. A perfect equilibrium is a Nash equilibrium that satisfies some additional properties to rule out the Nash equilibria supported by non-credible threats. Selten called this concept of perfect equilibrium " subgame perfection" . Rubinstein [1962). based on the concept of (subgame) perfect equilibrium. constructed an example of a two-person bargaining model with an iterated bargaining process under complete information. He demonstrated that. in his example. the bargaining equilibrium is unique. This was a break- through In the theory of bargaining. NcLenhan [i982] extended Rubinstein'swork by demonstrating that this result holds in a much more general setting. In particular. he showed that the bargaining outcome by the differential approximation is the Nash solution of the static bargaining problem given the initial feasible set and the threat point. Binmore [1962] refined Rubinstein's results by considering examples in which the bargaining process involves random moves or in which the "pie" does not shrink over time. Davidson [1965] constructed a multi-unit bargaining model for the analysis of union-management bargaining. Davidson's [1965] work provided us with a basic framework for the 811812318 Of the multi-unit bargaining between the whol 8831 ers and retai I ers. Besides the eminent research in oligopoly theory and bargaining theory. spatial economics also provides a new frontier for the study of market imperfections. Harket imperfection prov1des firms with market power. which results in an equilibrium outcome that is different from perfect competition. Greenhut. Hwang. and Ohta [1975] have shown that the perfectly competitive paradigm is inappropriate for the explanation of pricing behavior in many ”real life” markets. Again. Greenhut. Norman and Hung [1987] claim that the very concept of spatial economics requires rejection of many principles derived in the theory of perfect competition. In spatial economics. the shape of the market. the location of firms (and/or consumers). the pricing policies adopted by the firms. and transportation costs. all play critical roles. The spatial approach to the economics of imperfect competition has been applied to many aspects in microeconomics. especially to the analysis of product differentiation. (See. for example. Salop [1979] and Novshek [i960] . ) Most of the prominent researchers in spatial economics. however. have assumed that the wholesalers (producers) are the retailers. The buyers. in their analyses. do not have any ability to influence price determination except indirectly through spatial factors such as distance and transport cost. 30. 1t 18 implied that the buyers are the consumers of the final products and the producers are the retailers in most spatial economics models. Their emphases on the location of firms/consumers and the transport cost help us to understand.further the relationship between the retailers and the consumers as well as the retail price determination. Nevertheless. a complete picture of Wholesale price determination can not be satisfactorily found in the literature of spatial economic models . This is because most spatial models ignore the vertical market relationship between the Wholesalers and.the retailers in which.the fewness of firms on both sides plays an important P010 in the price determination. The pure bargaining theory may not serve satisfactorily as the sole tool for the Wholesale price determination. either. The pure bargaining theory is indeed a useful analytical tool for the studies of price determination between agents with.small number on both.sides of the market. Spatial factors. however. are usually not incorporated into these studies. Spatial factors. in fact. are important for retailers to differentiate their goods and thus allow retailers to gain market power at the retailing level and bargaining power in the Wholesale price negotiation. Therefore. a study of Wholesale price determination needs both pure bargaining theory to serve as the basic framework and spatial economics to emphasize retailers' strength in the bargaining. Studies on this topic are important because they help us to understand the (en: real bet hat: of I devc IL 3°1u11 R‘- "uh b general facts of the Wholesale price determination in our real world. Furthermore. with some modifications. it could be used to analyze many other cases where the traders of both sides of a vertical market relationship have some sort of bargaining power. This dissertation. therefore. is devoted to the study of the wholesale price determination. II. Nethodology and Purpose In this dissertation. a general model of the wholesale price determination is constructed to analyze the situations where perfect competition does not prevail at either the wholesale or the retailing level. As it was stated before. this research is an application based on the findings from the fields 0f bargaining theory and spatial economics. In this dissertation. the major participants of the Wholesale price bargaining game are the wholesalers and the retailers. while the role of consumers is not significant. Firms in the same category are assumed to be homogeneous. Also. it is assumed that firms behave noncooperatively and. except for some special cases. the goal of every firm is to maximize its own profits. Retailers compete in the final market. and wholesalers compete to sell their goods to the retailers (although the number of wholesalers is limited to one in some chapters of this dissertation). The bargaining solution concept used is the Nash solution which was pioneered by Nash [1950. 1953). 1 factor retail retail even 1 Ignore simpll the It (In no Itcles ueda‘ ind Va; the r91 the Cor Mall: Rice 1; maximiz In most cases of this thesis. the effects of spatial factors are embodied in consumer's demand function in the retailing level. When spatial factors are considered in the retailing level. each retailer has some sort of market power even though the number of the retailers may be large. We ignore spatial factors at the wholesale level not only for simplicity1 but also for the fact that in many industries the wholesalers have market power by the nature of fewness. (In most chapters of this dissertation. the number of wholesaler is limited to one.) The models of spatial firms used at the retailing level are similar to those in Capozza and Van Order [1978] and Novshek [1980]. It is assumed that the retailer charges a mill price to all consumers and that the consumers pay the transport cost for the goods. The retailers negotiate with the wholesalers on the wholesale price before they set an optimal retail price for profit maximization. In this dissertation. the major concern is the Wholesale price in the bargaining equilibrium under different market structures. Novshek [1980] has shown that at the retailing level there are three different market structures depending on the spatial and cost conditions. That is. in the final product market. there may be a monopoly retailer. several retailers Which are local monopolists. or several retailers which are not local monopolists. Therefore. the Wholesale price level COUId be sffer mics. profit canly price a on the m. 3 II Fill". are the m1 en discuss econ-om the 3m 'llh 5p different under these different situations. Once the Wholesale price is determined. the retail price. and the profit levels of the Wholesaler and the retailer can be easily calculated. Beyond the discussion on the equilibrium price and profit levels. the comparative static predictions on the impacts Of parameter changes are 3180 studied. III. The Structure In this dissertation. Chapter Two contains a literature review. In the bargaining theory. articles to be reviewed are those w1th.game-theoretic approaches to the bargaining problems While articles with other approaches will be discussed more concisely. In the field of spatial economics. articles to be reviewed are those which focus on the study of the oligopolistic firms' pricing strategies With spatial considerations. Chapter Three begins the study of the bargaining between the Wholesalers and the retailers over the wholesale price. In this chapter. we study the pure bargaining cases without consideration of spatial factors. We limit the number of firms to be no greater than two on each side. This gives rise to a bargaining problem over the wholesale price. There are six cases studied in this chapter: 1L bilateral monopoly; 2). monopoly Wholesaler versus duopoly retailers; 3).:monopsony retailer versus duopoly Wholesalers; 4). anPOIY WDOICSEIBPS versus anPOIY reta only reta both coul coul Prev with 10 retailers where each wholesaler provides goods to one and only one retailer; 5). duopoly Wholesalers versus duopoly retailers With the wholesalers taking turns bargaining With both retailers at each time period. where each wholesaler could provide goods to both retailers and each retailer could buy goods from both wholesalers; 6). same as the previous case except that retailers take turns bargaining With both “0188319?! at each time PQPIOd. Some of the results in Chapter Three are quite interesting. Due to the assumption that the demand for the Wholesaler's product is a derived demand from the demand for the retailers' goods. the bargaining equilibrium Wholesale price in case 2 (monopoly Wholesaler versus duopoly retailers) is just equal to that in case 1 (bilateral monopoly case). This implies that the monopoly wholesaler can hardly improve its bargaining position through the increase in the number of retailers in the wholesale price bargaining when the demand for its product is a derived demand. The equilibrium Wholesale price in case 3 - the monopsony retailer versus duopoly wholesalers - is equal to the wholesaler's marginal cost because the monopsony retailer can threat each wholesaler not to buy anything from it if the threatened one does not lower its wholesale price. The equilibrium Wholesale price in the pair-wise bargaining between the wholesalers and the retailers in case 4 is lower than that in cases 1 and 2. This implies that the Incl case the reta equa that out: the . prev. Itole Ihole taxi: lhole Unlla W10 71th ‘uPPe 9“ b; mike] Ir 11 wholesalers compete With each other indirectly. Besides. in case 5 - duopoly Wholesalers versus duopoly retailers With the W taking turns bargaining with both the retailers - the bargaining equilibrium wholesale price is equal to Wholesaler's marginal cost. which is the same as that in case 3. The intuition for this seemingly strange outcome is that the retailers. at each time period. may use the agreed wholesale price With the other wholesaler in the previous period to threat the current wholesaler for a lower Wholesale price. Contrarily. in case 6 - the duopoly wholesalers versus duopoly retailers with the retailers taking turns bargaining with both wholesalers - the Wholesale price will be so high as if it is determined unilaterally by the wholesalers because. at each time period. the wholesalers may use the agreed Wholesale price with the other retailer in the previous period to threat the current retailer for a higher wholesale price. Therefore. the bargaining equilibrium wholesale price in case 6 is higher than that in any other case. The bargaining equilibrium wholesale price in case 1 and case 2 is next to that in case 6. The bargaining equilibrium wholesale price in case 4 is higher than that in case 3 and case 5 but lower than that in case 1 and case 2. This comparison shows that the bargaining equilibrium outcome is affected by the number of players on each side and by the bargaining structure (especially when there are two players on both sides). In Chapter Four and Chapter Five. we incorporate proce to re price. he pa: (mm the as m cm Ilium the mo. c°°Per. 11mg“l the m Challter on beer ’QYNent QlaPier In 3"?“th 12 spatial factors at the retailing level. Throughout these chapters. the number of wholesalers is limited to one because we have found that when there are two or more firms on both sides. the bargaining results Will be unpredictable unless we impose some restrictions on the bargaining procedures. We assume that consumers pay the transport cost to get the good from the retailer with the lowest delivered price. Also. by assuming that the same wholesale price will be paid by all retailers. the number of retailers can be any finite number. The major difference between Chapters Four and Five is the assumption about the goal of firms. In Chapter Four. as in Chapter Three. we assume that the goal of each firm is to maximize its own profits: but in Chapter Five we assume that the monopoly wholesaler and all the retailers are willing to cooperate to maximize the sum of profits of all firms. although it is not necessary that the retailers cooperate at the retailing level. The sum of profits of all firms in Chapter Five is no less than that in Chapter Four. so firms on both sides are willing to cooperate if there is a side- payment scheme which guarantees everyone better-off than in Chapter Four. In Chapter Four. With the assumption that all firms want to maximize their own profits. bargaining over the Wholesale price is necessary. Using a similar bargaining structure 38 that “SEC in Chapter Three and With the N381) crlte equll value retal retal relat the rl facto: whole large there Spill; Price mile: 'hole °f Pr: 'hole. 18 a I mal Vertl. p’06u. divis‘ do no: if lhe Dame, 13 criterion for the bargaining solution. we solve the equilibrium price and profit levels. These equilibrium values depend on the types of competitive situations at the retailing level. The competitive situation among the retailers basically depends on the number of retailers relative to the size of the market. If the number is small. the retailers are local monopolists. In this case. spatial factors have no effect on the bargaining equilibrium wholesale price. When the number of the retailers is so large that the retailers are not local monopolists or when there is only one retailer serving the entire market. spatial factors affect the bargaining equilibrium wholesale price. At the end of this chapter. we also study the long- run equilibrium at the retailing level. In Chapter Five. With the assumption that the Wholesaler and all retailers cooperate to maximize their sum of profits. the equilibrium Wholesale price is equal to the Wholesaler's marginal cost of production When the retailer is a monopolist serv1ng the entire market or all the retailers are local monopolists. This phenomenon is like a vertically integrated firm charging the marginal cost of production to the sales division from the manufacturing division. However. since the wholesaler and the retailers do not vertically integrate. the wholesaler Will run a loss if there is no side-payment scheme provided. The side- payment scheme 18 the one WhiCh guarantees the wholesaler a prof11 then 1 are me 1: no Incree 1: a1: Furthe Profit the be case 1 than I the 31 1‘1» PPOfit level no less than if it bargains With the retailers. When the number Of retailers is so large that the retailers are not local monopolists. the equilibrium wholesale price is no longer the Wholesaler's marginal cost but an increasing function Of it. The equilibrium Wholesale price is also an increasing function of the number of retailers. Furthermore. under this situation the retailers earn less profits while the Wholesaler earns higher profits than in the bargaining game. So. the side-payment scheme for this case is the- one Which guarantees the retailers no worse-off than if they bargain With the Wholesaler. The formula of the side-payment schemes are presented in this chapter. Also. the long-run equilibrium is studied. In Chapter Six. we conclude the research With a sumary of findings from preVious chapters. Also. for the purpose of further studies we have listed many possible extensions at the end of this chapter. such as studies of dynamic games. incomplete information. different criteria of the bargaining solution. different bargaining structures. etc. 1. 15 Footnote of Chapter One There is a conmon problem in spatial economic research that the symbolic computations are very complicated WhiCh sometimes cause the implications Of the model to be inconclusive. To handle this problem. Beckmann [1976] simplified the analysis by standardizing his model through some transformations Of variables. while Novshek [1960] plugged some numeric figures into his model. Incl: dev0' frel. Chapter Two: Literature Review I. Introduction Having searched for previous studies on the topic of Wholesale price determination. we find only a few references devoted to this topic. Our search covered several academic fields discussed below. At first. we expected to find many studies of Wholesale price determination in the field of marketing. This was due to fine fact that fine field of marketing studies the pricing. advertising and distribution of goods from the manufacturers to the final consumers. However. although.there are many studies on the Wholesale distribution channels‘. there are only a few on the Wholesale price determination.a Also. there are many studies on the retail price levels of certain specific goods or on the retail price determination for goods distributed through certain special types of sales outlets. However. most of the studies on the Wholesale or retail price determination are based on survey results rather than on economic models. Therefore. few useful references were obtained from.the source book Pricing Policies and Strategies: An Annotated Bibliography published by the American Narketing Association (edited by Lund. Nonroe and Choudhury [1981]). It seems that scholars in the field of marketing are more interested in the specific. real-world case-studies than the generalized. abstract model-building. 16 perfc orzar Inole seeru inter have mole [1976 Po'er indus 1 Ice OVer f"he litre: ”We. beta.) serV1< 17 As the market structure. market conduct. and market performance are the major foci of the field of industrial organization. 3 we hoped to find some studies on the Wholesale price determination from this field. However. it seems that most economists in this field do not have much interest in the Wholesale price determination although some have displayed interest in the relationship between the wholesalers and the retailers. (See. for example. Porter [1976].) Porter [1976] studied the relative bargaining power of the Wholesalers and the retailers in different industries. He showed that the fewness of the retailers in a local market provides the retailers with bargaining power over the wholesale price determination. Furthermore. beside fewness. he emphasized the fact that the ability to differentiate goods gives the retailers another source of bargaining power. For example. in the computer industry. better pre-sales consulting advice and/or better after-sales service provided by a computer dealer make its product more valuable than the product sold by other dealers. Porter [1976]. therefore. concluded that the retailers in the industries of non-convenience goods (such as computers. automobiles) have higher bargaining power than the retailers in the industries of convenience goods (such as groceries). However. Porter [1976] did not precisely describe how the wholesale prices are determined. He only explained why some of the Wholesale prices are relatively higher or lower. based on the economic data available. whole node} ”131 analy issue This . harea re re: Sectlr Sectlr Sectlt II. 33 more ( ioum Strata bOIOre chaPac [1950] leflnl “10m inc 3 . Frayed 1‘ unit 18 Porter's [1976] work does shed light on the study of Wholesale price determination in the following ways. To model the fewness of firms at both the wholesale and the retailing levels. the theory of bargaining provides an analytical framework. while for the product differentiation issue. spatial economics provides us with useful techniques. This dissertation is. therefore. a combination of both bargaining theory and spatial economics. In this chapter. we review the literature on the theory of bargaining in Section II. and the literature on spatial economics in Section III. Finally. we present a short conclusion in Section IV. II. Review Of Literature on the Theory Of Bargaining Theories of bargaining before Nash [1950] could do no more than specify a range in Which an agreement may be found.4 As the outcome of bargaining depends on the strategic interaction of the bargainers. many economists before Nash [1950] concluded that bargaining is characterized byithe indeterminancy of its outcome. Nash [1950) was (perhaps) the first economist who offered a definite solution for the bargaining problem. With the axioms of expected utility theory developed by Von Neumann and Morgenstern [1944]. Nash added three more axioms5 and proved that the resulting solution to the bargaining problem is unique and is characterized DY the maximum Of the product of 1151 re 30! ha 10 m. I: ha C0 0!" C0 D0 DC Fe 11: as CC 19 Of gains from.agreement. A bargaining model Of this type is usually called an Xi matic mode 0 b ainin because it relies heav1ly on the properties Of the axioms. It is sometimes called a cooperative model since it models the bargaining process as a cooperative game. In another paper. Nash [1951] presented a new approach to the theory of bargaining - a noncooperative game- theoretic approach or strategic approach. Also. in his [1953] article. Nash pointed out the fact that the bargaining process that determines the outcome of any cooperative game can be represented by a bargaining game having the nature of a noncooperative game in extensive form or in normal form. Furthermore. the solution of this cooperative game can be defined in terms of the equilibrium points of that noncooperative bargaining game. 6 The noncooperative game-theoretic approach has the advantage of revealing the connection between the bargaining process and the resulting outcome. It also may appear more realistic. as one can check directly how well the proposed process corresponds to the actual bargaining. However. it has the disadvantage of failing to explain how the bargaining procedure was obtained. 7 Furthermore. in the Nash demand game there is a basic difficulty that this noncooperative game allows a multitude of equilibria.8 The multiplicity of Nash equilibria does not yield plausible descriptions of bargaining behavior. A bargaining analysis under which any 20 outcome is possible is clearly 0f limited interest. Nash then suggested that the one corresponding to the prediction Of his axiomatic model had distinguishing characteristics. such as the minimax and maximin properties. 9 Nash's axiomatic approach has the advantage of obtaining a unique solution regardless of the specific extensive form used to model the bargaining process. Therefore. it has been developed widely by many economists. In contrast. there has been much less successful development in the strategic approach. For example. Harsanyi [1956] attempted to construct a noncooperative model of an iterated bargaining process. In his model. the bargainers begin by making presumably incompatible offers. The bargaining proceeds as agents make concessions. Concessions occur to increase the value of the product of utilities (measured incrementally from the disagreement point). This concession process accordingly leads toward the Nash bargaining solution. There are. however. several difficulties in Harsanyi's model. It fails to provide an explicit account of time structure to the bargaining process. and fails to explain the magnitude of the concessions made by the players. The progress in the strategic approach to the theory Of bargaining regained its momentum due in large to the developments in the general theory Of noncooperative games. 1° One of the developments. originating in the work of m it: 1m Pr Tl: co as be Fr Pl 21 of Selten [1965. 1973. 1975] on ”perfect equilibria". offers a technique for reducing the multiplicity of Rash equilibria. By investigating the perfect equilibria in a two-person mnltiperiod bargaining game under complete information. Rubinstein [1982] prOVided a breakthrough flhat the sequential equilibriul in his node! was unique. (We will discuss his paper latter.) Another development. originating in the work of Harsanyi [1967. i968a. 1968b]. extends the theory of bargaining to include games of incomplete information. this development allows more realistic modeling of bargaining situations in which.a bargainer holds private information. Economists. such as Fudenberg and Tirole [1983]. Cramton [1984). and Rubinstein [1965). have constructed bargaining models with.different features in the nature of incomplete information. the probability distributions. etc. Since the bargaining models studied in this thesis are models of complete information. we will focus on the literature with.models of complete information only. Among them. we briefly describe the work of Rubinstein [1962] first. Rubinstein [1982) constructed a two-person mmltiperiod bargaining game under complete information. He assumed that two players have to reach an agreement on the partition of a pie size 1. Each has to make. in turn. a proposal as to how the pie should be divided. After one player has made an offer. the other must decide either to accept it. or to reject it and continue the bargaining.11 A share reJect bareai 2) max '3'; a;- to any 1 (Plat t. m dlscom 8mm the m: “7 PM 91111112 bu -; this b; in and by "Name 3113 Sn L" us may hav “413.0. in. Play” i b): 22 At time 0. Player 1 proposes that he receives some share s. Player 2 may accept the offer by replying "I" or reject it by replying “R“. Acceptance of the offer ends the bargaining. After rejection. the rejecting player (Player 2) makes a counter-offer to which Player 1 replies "Y" or "ll“; and so on. There are no rules which bind the players to any previous offers they have made. The payoff to Player 1 (Player 2) equals his share of the pie as agreed at time t. multiplied by 91*- (resp. 62‘). where 61. ea represent discount factors. and 91. ea < i. A strategy for Player 1 specifies his proposal/reply at each point. as a function of the history of the game up to that point. It is clear that any partition of the pie can be supported by a flash equilibrium. 1‘3 However. the partition of the pie supported by a ”perfect equilibrium" is unique in this model. We show this by backward induction in the following paragraphs. 13 With the bargaining process and discount factors given and by assuming the original game begins at time t : o. the subgame begins with an offer made by Player 1 at time t = a. This subgame has the same structure as the original game. Let us denote l! as the undiscounted optimal share Player 1 may have in any perfect equilibrium of this game. (The undiscounted optimal share Player 2 may obtain then is (1 - 11)). At time t = 1. Player 1's decision to accept or reject player 2's period 1 offer (s1) will accordingly be governed by: 23 (2.1) accept s1. if s1 2 61H. reject s1. if s1 < 61H. For Player 2. an offer s1 > 913 in time t = 1 is clearly suboptimal since a lower offer will be accepted by Player 1 and will provide Player 2 with.a higher payoff. Player 2's choice is accordingly to offer 81 = 91M or 81 < 91H. If 81 : 61H. the offer is accepted and Player 2's payoff (discounted back.to t = O) is (2.2) 63(1 - 61H) If s1 < 61H. Player 1 rejects the offer. and Player 2': payoff discounted back to t : O is at most (2.3) 923(1 - 3) As 0 < 91.92 < 1. it is clear that 93(1 - e1n) > 923(1 - a). Therefore. Player 2's optimal offer at t = 1 is (2.4) 31' = 911‘! We now consider the decision problem.at t = 0. It is known that if an agreement is reached at t = 1. equilibrium strategies involve Player 2 offering Gin and Player 1 accepting it. An argument. analogous to that constructed for t : 1. then reveals Player 2 will accept Player 1’s offer of so at t : o if (2.5) 1 - so 2 92(1 - Gin) and reject it otherwise. Player 1 will not offer an so ‘which satisfies equation (2.5) with.strict inequality. since an offer which equalizes both.sides of equation (2.5) would also be accepted and would yield a higher payoff to him. 80, Player 1 offers an so such that (2.6) 1 The pa: make: a 13 re): 0) is (2.7) e the (a: Player Player the val 1:. (2.8) 1 It 1; . mm; (3.9) l (3.10) mm. Mane, there ; “Nan be all, “”13, Wages: “M: 24 (2.6) so = 1 - 92(1 - 91K). The payoff to Player 1 is 1 - 92(1 - 9111). If Player 1 makes an offer of so Which is greater than this. the offer is rejected. and Player 1's payoff (discounted back to t = 0) is (2.7) eiau. The game facing Player 1 at t = o is identical to the game Player 1 faces at t = 2. So. the value of the payoff to Player 1 from the game's continuation at t = 2 must equal the value at t : 0. Therefore. the undiscounted optimal share to Player 1 is the larger of (2.6) and (2.7). That is. (2.5) u : max[1 - 93(1 - 61H). eiau] It is clear that ll > 9131! since a, < 1. Therefore (2.8) yields (2.9) u = 1 - 93(1 - 9.x). or (2.10) n = (1 - 92)/(1 - e192). Equation (2. 10) shows that the perfect equilibrium is unigge and determined by the discount factors. The more impatient a player is. (of which the discount factor is smaller). the smaller his share of the pie is. 1“ Also. there is an advantage to the player who starts the bargaining when a, : ea = e. 15 However. this advantage may be eliminated either by "tossing a coin"16 to see who starts. or by assuming that the time lapse between successive offers is exceedingly small. (approaching zero). 17 Furthermore. the above analysis shows that the 25 unique perfect equilibrium agreement Will be reached at the initial period. These results from the work of Rubinstein [1982] provide a sophisticated theory of bargaining. Attention accordingly turns to the sensitivity of the results to the specification of the bargaining model. Binmore [1982] refined Rubinstein’s results and considered. in particular. examples in which the bargaining process involves random moves or in which the “pie" does not shrink steadily over time. HcLennan [1982] defined the bargaining between two agents as ”a class of noncooperative games in which moves that make proposals available are distinguished from moves with physical effect. '15 Furthermore. with the following assumptions: 1) there is a single threat-point that does not change with time. ii) the transmission and acceptance of proposals are rapid and also symetrical. and iii) the proportion of the feasible set lying to the northeast of the threat-point shrinks radially. HcLennan [1962] predicted that the bargaining outcome by the differential approximation is the flash solution of the static bargaining problem given the initial feasible set and the threat point. The purpose of most articles reviewed above is to solve the abstract bargaining problems. The major concern of those articles is to prove the existence and uniqueness of the bargaining solution. Also. the bargaining problems in those studies are two-person bargaining games. Davidson [198 labo: anal: and : diff) modei relat unior are 1 vorke Inde; PEPra 15 us under First Union 'ales '381 millm ”41m. '38es ratio; a “N 26 [1965] applied the theory of bargaining to the analysis of labor-management bargaining. In addition. he extended the analysis from two-person bargaining to multi-unit bargaining and studied the outcome of collective bargaining under different bargaining structures. We briefly describe his mode 1 as below. Davidson [1985) constructed a model to investigate the relationship between bargaining structure and wage in unionized oligopolistic industries. He assumed that there are two different bargaining structures. In one. the workers of each firm are represented by separate and independent unions. In the other. an industry-wide union represents all the workers in the industry. His model then is used to compare the outcome of collective bargaining under these t'O different bargaining structures. The basic assumptions of this model are as below. First of all. it is assumed that there are two identical unionized firms in the market. Each firm bargains over wages with the union representing its workers. Once the wage is set. the firm chooses output and employment to maximize profits. The union attempts to maximize a utility function which is an increasing function of employment and wages of its members. Furthermore. complete information and rational agents are assumed to rule out the possibility of a strike19 in equilibrium. 27 In his analysis. Davidson [1985] showed that when the workers at firm i manage to secure a higher wage for themselves. a positive externality is created that the size of the pie to be divided by firm j and its workers is increased. (Here. i = 1. 2. J = 1. 2. and i t J.) 'lherefore. when the unions bargain independently. they ignore this externality and settle for too low a wage. An industry-wide union. which may internalize this externality and has a lower cost of striking one firm (than a local union). leads to higher wages. lower profits for the firms. and higher prices for the consumers. In the case of collective bargaining with independent unions. Davidson [1985] used a bargaining process similar to that used by Rubinstein [1962]. in which the firm and its union exchange their offers. With the assumptions that the time between periods is arbitrarily small and that production occurs only when both firms have come to terms with their workers or when one firm settles with its union and the other union decides to leave the bargaining table and strike. Dav1dson [1985] showed that for any discount factor 9 E (o. 1) there exists a unique perfect equilibrium outcome. Also. as e approaches 1. the solution converges to the Nash cooperative bargaining solution. In addition. this Perfect equilibrium wage vector will be proposed and accepted in the initial period. In the case of collective bargaining with the industry- ride 1 proce playe) perfec are p) Furtnc result benef) thesis retail tree Also, toms °f the tom: Halon, 'holes c°11ec ha'e e: CASe; ‘ Our Fe: 1n“la: 28 wide union. Davidson [1985] proposed a similar bargaining process while in this bargaining game there are three players rather than two players. Again. he showed that the perfect equilibrium wage vector is unique and these wages are proposed and accepted in the initial period. Furthermore. he proved that the industry-wide bargaining results in higher wages throughout the industry. which benefits workers and reduces profits Of DOth firms. Davidson’s [1985) work initiated the research of this thesis. The relationship between the wholesalers and the retailers in this thesis is similar to the relationship between the unions and the firms in Davidson’s [1965] work. Also. there are some cases in Chapter Three of this thesis corresponding to those in his work. For example. the case of the monopoly wholesaler versus duopoly retailers corresponds to his collective bargaining with industry-wide union. The pair-wise bargaining between the duopoly wholesalers and the duopoly retailers corresponds to his collective bargaining with independent unions. Although we have extended our research to many other pure bargaining cases in Chapter Three. and incorporated spatial factors to our research in Chapters Four and Five. we owe him for his initial studies. 29 III. Review Of Literature on the Theory Of Spatial Economics Porter [1976] painted out. retailers’ bargaining power emerges not only from their fewness in number but also from their ability to differentiate product differentiation. Retailers' ability to differentiate goods can come from many different sources. such as their location. their counseling service before sales. their after-sale maintenance service. etc. Among these. the difference in their location is often modeled. Furthermore. most other sources of product differentiation such as different levels of service may be modeled as different ”location" on a scale of serv1ce levels. Therefore. spatial economics which emphasizes the location of firms relative to location of consumers. may be used to study product differentiation. The studies on spatial economics can be traced back to. at least the classic work by Hotelling [1929]. Hotelling analyzed the behavior of two firms selling a homogeneous product locating on a finite one-dimensional geographical market (which he called a "main street"). He assumed that the consumer demand is completely inelastic (and without any “outside good“ as assumed by Salop [1979]). That is. "one unit of the comodity is consumed in each unit of time in each unit of length of line. '30 Having stated the reasons why the duopolists would not cooperate but act 11(311-cooperatively.a1 Hotelling asserted that successive location moves would lead the duopolists to an equilibrium 30 in which.both.firms are located at the center Of the market and set identical prices. The results presented by Hotelling [1929] stimulated many economists to extend his study mainly in two ways. The first one is to apply the concept of "distance” in space as the "differences" in quality or characteristics of the good to analyze the general problem of product differentiation. Among them. Boulding [1966] applied Hotelling's result that firms are located at the center of the market to explain ”why all the dime stores are usually clustered together. . . . why certain towns attract large numbers of firms of one kind; . . . It is the principle which can be carried over into other 'distances' than spatial differences. '33 He named this principle the "Principle of Minimum Differentiation". Hany subsequent researchers. however. have shown that this result is very sensitive to the assumptions made by Hotelling. The relaxation of some of them can lead to an equilibrium with firms being scattered over the market. Eaton and Lipsey [1976] investigated cases of one-dimensional markets and two-dimensional markets with different assumptions on the number of firms. the conjectural variation. and the population density. They found that "minimum differentiation is a property only of those in which firms pursue a strategy of zero conjectural Variation and where the number of firms is restricted to two. "33 The principle of minimum differentiation. thus. is a special case rather than “a principle of the utmost 31 generality. '34 They also showed that the equilibrium might not be unique but multiple. or even non-existing. depending on the as sumpt i ons . The other type of studies after Hotelling [1929] studied how the firms behave when the spatial factors are embodied in the analysis. Lerner and Singer [1937] extended Hotelling's study but modified Hotelling's assumption of complete inelastic demand with a so-cal led ”rectangular" demand function. That is. a consumer purchases one. and only one. unit of the coumodity if the delivered price is lower than a given upper limit. (which they called the ”demand price"35) and none in the opposite case. With the ”demand price" and the length of market given. they discussed the relationship between transport cost and the maximum distance for which a firm faces a positive demand. With a special assumption on the firms interaction. 35 they investigated the equilibrium price when the number of firm is given. Furthermore. they studied the clustering of firms when the number of firms increases. They showed that when the number of firms is greater than three. the location- equilibrium is usually not unique and firms in general do not cluster at the center Of market. Hoover [1937] looked at the other side of the spatial effect on firm’s behaVior. He assumed that the marginal cost of production is constant and transport cost per unit Of distance is identical between any two locations. He also 32 assumed that all the buyers have identical demand schedules of constant as well as equal elasticity. at their respective locations. Furthermore. he assumed that firms charge delivered prices rather than mill prices to their customers. With the assumptions mentioned above. he showed that to maximize profits a monopolistic firm will charge discriminatory prices to customers within his market area. Price discrimination is against more distant buyers. Also. he showed that if the elasticity of demand curves decreases rapidly as price falls. (For example. as in any "straight line" demand curve). then the price discrimination is against nearer buyers . Smithies [1941a] studied the relationship between the shape of demand curve and the optimal price policy for a spatial monopolist. He assumed that the freight rate is uniform per unit of distance as did Hoover [1937]. Furthermore. he assumed that there is a monopolist in the market who can ”fix a separate price at every point in the market. '37 (That is. the monopolist can exercise price discrimination against some of its customers.) Based on these assumptions. Smithies [1941a] studied the optimal price policy for this monopolist to maximize its profits. He concluded that the monopolist "will sell f. o. b. mill if the logarithmic demand curve is linear or concave. and will absorb freight if the logarithmic demand curve is convex. '35 In another paper. Smithies [1941b] followed the 33 tradition of Hotelling [1929] by adopting all the assumptions of Hotelling [1929] except for the assumption on consumer demand. Smithies [1941b] assumed that the consumer demand is price elastic rather than completely inelastic. When the demand is price elastic. the gains from.moving towards the rival will be offset by the loss incurred in the hinterland. Therefore. it is not necessary that at equilibrium the duopolists cluster at the center of market (as Hotelling [1929] asserted). Instead. Smithies [1941b] pointed.out the equilibrium locations of the firms depend on the conjectural hypotheses made by the competitors as to each.other’s behavior. Each competitor's strategy depends on his estimate of his rival’s reactions in respect to both of price and location. Smithies considered three cases of the conjectural variation. (full quasi-cooperation. quasi- cooperative as to prices and competitive as to locations. and full competition).29 and discussed the conditions for equilibrium in those cases. The articles reviewed above are important in spatial economics not only because they are pioneering works in this field but also because they introduced different directions for further studies. For example. the pricing policies which.were first studied by Hoover [1937] and Smuthies [1941a] has many extensions by economists like Dewey [1966]. Greenhut and Pfouts [1967]. Beckmann [1976]. Beckmann and Ingene [i976]. Gronberg and Meyer [i981b]. etc. Also. as 34 was shown by Hotelling [1929] and Smithies [1941b]. different assumptions on the consumer demand can lead to different results on the spatial equilibrium of firms clustering. This caused many economists like Losch.[1964]. Gannon [1971]. Greenhut. Hwang and Ohta [1976] to investigate the shape of spatial demand function. Furthermore. the effect of spatial competition on location and/or price first studied by Hotelling [1929]. Lerner and Singer [1937]. and Smithies [1941b]. has been further analyzed by many economists like Beckmann [1972]. Eaton [1972]. Eaton and Lipsey [1976]. Capozza and Van Order [1978]. Salop [1979]. Hovshek [1980] and Gronberg and Meyer [1981a]. etc. Among the huge volumes of studies in spatial economics. those studies by Capozza and Van Order [1978]. Salop [1979]. and Novshek [1980] have a close relationship with.the models in this dissertation. We review them below in more detail. Capozza and Van Order [1978] claimed there are two essential distinguishing features of spatial competition. The first one is transportation cost. Transport cost gives spatial firm its monopoly power over customers close to it. The second essential feature is that average cost curve must be downward sloping over some range. This assures the advantage of concentrating production at specific locations. The average cost curve is downward sloping either because of positive fixed costs or because Of economies Of scale in 35 production. Capozza and Van Order [1978] further enumerated a set of criteria that equilibria in a reasonable model of spatial competition should satisfy. Those are: (1) As transport costs approach.zero. price should approach marginal cost; (2) As fixed costs approach zero. price should approach marginal cost; (3) As costs (fixed. marginal. or transportation) rise. price should rise; (4) As demand density rises. price should fall in the long-run; (6) As more firms enter the industry. price should fall.30 Capozza and Van Order [1978] constructed a model of spatial competition to investigate the extent to which spatial price theory replicates the results of classical price theory When there is free entry. Based on different assumptions about the conjectural variation concerning the response of competing firms. there were two models31 studied. The first one was so-called "Loschian Hodel” in which each firm assumes that its market area is fixed and sets price like a monopolist within its market area. That is. in Loschian model. the conjectural variation for price (dP'/dP) is equal to one. (Here. P and P' represent the prices charged by a representative firm and its competitor respectively.) The other one which.they named as SHC (Spatial Honopolistic Competition) model or H-S (Hotelling- Smithies) model. is the model in which zero conjectural variation Of price prevails. To have a complete picture Of Capozza and Van Order’s 36 [1978] model. we list their assumptions as below. A1). There is a single commodity that can be produced in two- dimensional space with the same cost function. A2). The cost function has constant marginal cost and (positive) fixed costs. A3). Transport cost per mile is identical between any two points. All). Potential consumers occupy a homogeneous unbounded plain at uniform density D. AS). All consumers are identical and have a demand curve that is linear in delivered price. A6). Firms charge a mill price to their customers. Also. firms continue to enter until profits for all firms are driven to zero. A7). All market areas are circular. A8). There are three different types of competitive situations - Loschian model. H—S (or SHC) model. and 8-0 (Greenhut-Ohta) model. (The implications of the first two models have been discussed before. while the last one is. by nature. a model with the conjectural variation of dP'ldP = -1.) The major concern of Capozza and Van Order [1978]. however. was on the first two models. Capozza and Van Order [1978]. according to the assumptions mentioned above. proved that Loschian model did not satisfy any of the criteria stated before. That is. when transport costs or fixed costs approach zero. the equilibrium price of the Loschian model does not approach marginal cost. Also. when the fixed costs. or transport costs rise. the equilibrium price of the Loschian model falls rather than rises. Contrarily. the normal case of the 811C model satisfies every criterion. That is. as transport 01 CI 9‘1 Do '1 oni be; no; 291- Spa ‘3) 3? costs or fixed costs approach zero. the equilibrium price of the normal case of SHC model approaches marginal cost; when the costs (fixed. marginal. or transportation) rise. the equilibrium price rises. However. the comparative static predictions of the perverse case of the SHC model. (in this case the transport costs are a large portion of the delivered price or I'firms are so spread out that individual firms are almost regional monopolists").3a are qualitatively the same as those in the Loschian model. Capozza and Van Order [1978] assumed the long-run equilibrium is characterized by zero-profits to all firms. This is correct if firms are infinitely divisible. However. with positive fixed costs assumed and with an integer number of firms required. it is not always true that. after entries/exits. every firm earns zero profits in the long-run equilibrium. It is more likely that active firms earn positive profits in the long-run equilibrium. Varian [1984] made this same point. 33 Therefore. the long-run equilibrium with free entry and exit is better to be described as the one “at which active firms earn non-negative profit. and the best a potential entrant could do by entering is to earn a nonpositive profit'34 as was claimed by Novshek [1980]. With the assumption that firms compete in price With zero conjectural variation. Salop [1979] Studied a model Of spatial competition in WhiCh a second comodity is explicitly treated. He modeled a market in the spirit Of 38 Hotelling [1929]. with the market being a circle. The circle has unit circumference. with a total of L consumers located evenly around.it. Each consumer buys one unit of the differentiated product from only one seller. or he buys none. That is. a consumer located at 11 will buy a unit of commodity from.a firm located at 1' if the following condition is satisfied. (2.11) main: [U - cpl - 1*) - P1] 2. s" where U is the maximum.utility the consumer may have when the distance and price are zero; c is the constant transport cost per unit of distance; '11 - 1" is the distance between l1 and l”; P1 is the price paid by the consumer; s“ is the surplus of utility from the homog ene ous o ther 8 00d. Equation (2.11) may be rewritten as (2.12) max [v - c|l - 1" - P ] l O 1 i i where v = U - s” > O is the effective reservation price. Salop [1979] pointed out the perceived demand curve for a single representative firm is a function of other firms' prices and locations. Three regions of the representative firm/s demand curve may be distinguished: the ”monopoly." "competitive." and "super competitive" regions.35 The “monopoly" region consists Of those prices in WhiCl‘l the fmn’s surplu. (he (10! comes mo (0 firm. prices firm a flrm': (2.13 (ha. ”Ari 39 firm’s entire market consists of consumers for whom the surplus of no other firm’s comodity exceeds the surplus of the homogeneous outside good. The “competitive" region is composed of those prices in which customers are attracted who would otherwise purchase comodities from some other firm. The "super-competitive" region consists of those prices in which all the customers of the closest neighboring firm are captured. The maximum distance of one side of the representative firm's monopoly region is (2. 13) xi!) : (v - P)/c. So. the demand in this region is (2. 14) q“ = 2L(v - P)/c. The demand for firm i in the competitive region is (2. 15) qc : L(P’ + c/n - P) where P is the price charged by the representative firm: P' is the price charged by the nearest competitors located at distance 1/n; n is the number 0f firms. Salop [1979] defined the ”symetric zero profit equilibrium” (SZPE) as ”a price and a number of firms such that every equally spaced Hash price setter's maximum profit price choice earns zero profits. '35 By assuming a constant marginal cost (m) and fixed cost (F). Salop [1979] presented the monopoly equilibrium as (2.161 (2.17} and t! (2.15 (2.19 mm (hang the f in Cc size Rum, 11m fall. r188 EVep Howe 9%} "m: We 31p 0f 40 (2.16) Pm m + c/(2nm). and (1/2)(2cL/F)1/a and the competitive equilibrium as (2. 18) Po = m + c/no. and (2.19) no (ct/F)1/8. Furthermore. Salop [1979] demonstrated that if equilibrium lies at the kink. the effects of parameter changes are perverse. In the short-run. prices are rigid in the face of small cost changes. In the long-run. increases in costs lower equilibrium prices. Increases in the market size raise prices. 37 If equilibrium does not lie at the kink. equilibrium price rises and equilibrium number of firms falls as fixed costs (F) rise or market size (L) falls. When the transport cost (c) rises. equilibrium price rises and equilibrium number Of firms rises too. 38 The above model constructed by Salop [1979] satisfies every criterion set by Capozza and Van Order [1978]. However. it raises a similar question for the zero-profit equilibrium as that in Capozza and Van Order [1978]. The equilibrium number of firms shown in equations (2. 17) and (Z. 19) is not likely to be an integer for most situations. 39 Hovshek [1980] pointed out the comon problems of previous spatial models where location and price are strategic variables. For those models with the assumptions of zero conjectural variation (ZCV). Novshek [1980] showed Cl f1 41 that "when firms have constant marginal costs. no equilibrium at which ZCV is relevant ever exists. "*0 For those models assuming zero profits in equilibrium. he showed that there might be nonexistence of equilibrium because the number Of active firms must be an integer. Therefore. be constructed a model with the assumption of “modified zero conjectural variation". Hodified ZCV assumes that "firms believe the strategies of other firms are fixed unless a strategy change for the firm undercuts some other active firm (at firm's location) in Which case the affected firm will respond by lowering its price. '41 The shape of the market in his model is similar to that in Salop [1979] which is a circle with circumference L. All firms are assumed to have identical cost functions consisting of a fixed cost F plus a constant marginal cost times output. By measuring all prices as net of marginal cost. without loss of generality. he assumed the marginal cost is zero. Also. in this model. firms choose location and price to maximize their profits. Consumers are uniformly distributed on the circle with density A. Each consumer only purchases from firms with the lowest delivered price. with consumers paying transportation costs. The transport costs are measured by the unit cost. c. times the distance. All consumers possess identical linear demand functions which are functions of delivered price. With the assumptions above. Novshek [1980] proved that for all positive integers n. there is an n-firm.equilibrium (Iltn (that narke situa servl: case . prlce Popula 11(ua4 EQUIIJ e(mill Purim 3bcL/( relat) Tue e; We the Dc equlli temp)1 112 (without free entry and exit). When n 1 maxti. 3bcL/4a]. (that is. when the number of firms is small relative to the market). all firms are local monopolists. There are two situations studied. The first one is a single monopolist serving the entire market. The equilibrium price for this case is P” = (a/2b) - (cL/8). That is. the equilibrium price is a function of all parameters except for the population density and the fixed costs. For the other situation. where some consumers may not be served. the equilibrium price is P" : (1/3)(a/b). That is. the equilibrium price is not affected by any spatial factor. Furthermore. Novshek [1980] proved that when n > maxll. 3bcL/4a]. (that is. when the number of firms is large relative to the market). firms are not local monopolists. The equilibrium is unique and symnetric. The equilibrium price is. again. a function of every parameter except for the population density and the fixed costs. This equilibrium price is. however. expressed by a lengthy and complicated combination Of the parameters in the model. Hovshek [1980] also demonstrated that when fixed costs are sufficiently small relative to other market parameters. a free entry equilibrium exists. This equilibrium is ”metric and approximately competitive (i. e. the equilibrium price is approximately equal to the marginal cost). Spatial models in Chapters Four and Five of this thesis are 1 Rover level Secor price assun betve DOSll both cases funct and V compa PTlCe tOplc the k In Ch :luqy bilae baPEa nOt O abll) ham. 43 are based on the framework provided by Novshek [1980]. However. there are some modifications. First. we have two levels of firms - the wholesaler and the retailers. Secondly. the retailer's marginal cost (i. e. the wholesale price) is not an arbitrary number (which Novshek [1980] assumed to be zero) but is determined by the negotiations between the wholesaler and the retailers and must be positive. For the spatial models in this thesis. we study both the linear demand and the completely inelastic demand cases. In those cases with a completely inelastic demand function. we follow the way of Salop [1979]. We owe Capozza and Van Order [1978] for their emphases on the studies of comparative statics. IV. Conclusion Having reviewed the articles related to the wholesale price determination. we find that more can be done on this topic. With the theory of bargaining. we are equipped with the knowledge for analyzing the behavior of bargainers. So. in Chapter Three. we apply the pure bargaining theory to study the wholesale price determination from the simplest bilateral monopoly case to some sophisticated two-by-two bargaining cases. As the retailers' bargaining power arises not only from their fewness in number but also from their ability to differentiate product. we combine the theory of bargaining and spatial economics in Chapters Four and Fl tr 44 Five to study the wholesale price determination. This treatment not only enlarges the applications of bargaining theory but also brings spatial economics to a new frontier where vertical market-relationship plays an important role. 45 Footnotes Of Chapter TWO For example. see Ertel. K.A. [1978]. The only article related to this topic. which we could find. is ”The Influence of Hanufacturers' Price Policies upon Price Determination by Wholesalers.“ by Alton [1967]. However. the study by Alton [1957] was based on a business survey rather than on an economic model. For example. see Caves [1982]. For example. the concept of "core" by Edgeworth [1881). See Hash.[1960]. p. 166 and p.157. They are quoted as below: "(1). An individual offered two possible anticipations can decide which is preferable or that they are equally desirable. (2). The ordering thus produced is transitive: if A is better than B and B is better than C then A is better than C. (3). Any probability combination of equally desirable states is just as desirable as either. (4). If A. B. and C. are as in assumption (2). then there is a probability combination of A and C which is just as desirable as C. This amounts to an assumption of continuity. (6). If 0 1 p 1 1 and A and B are equally desirable. then pA + (1 - p)C and p3 + (1 - p)C are equally desirable. Also. if A and B are equally desirable. A may be substituted for B in any desirability ordering relationship satisfied by B. Let u. and ua be utility functions for the two individuals. Let c(S) represent the solution point in a set S which.is compact and convex and includes the origin. We assume: (6). If a is a point in S such that there exists another point B in S with the property u1(B) > u1(a) and ua(B) > ug(d). then a ¢ c(S). (7). If the set T contains the set S and c(T) is in S. then c(T) = c(S)- We say that a set S is symetric if there exist utility 10. 11. 12. 13. 14. 46 operators u, and ug such that when (a. b) is contained in S. (b. a) is also contained in S: that is. such that the graph becomes symmetrical with respect to the line u. : uz. (8). If S is symmetric and u. and.ua display this. then c(S) is a point of the form (a. a). that is. a point on the line u. : ua." The first five are from Von Neumann and Horgenstern [1944] while the last three are his additions. The interpretation and proof of Nash [1950] model may be found in Friedman [1986]. pp.154-159. See Hash.[1953]. p.129 and Harsanyi [1982]. p.51. See Chatterjee [1986]. p.170 and p.187. See Rash.[1953]. p.131. See Hash.[1953]. P.136. See Roth [1985]. p.3. See Rubinstein [1962]. p.97. See Rubinstein [1982]. p.101 for the proof. For simplicity. we omit the case of fixed bargaining cost in Rubinstein [1982]. Part of this proof is indebted to Chatterjee [1986] and Sutton [1986]. See Sutton [1986]. p. 711 and Chatterjee [1986]. p.177. However. this statement is not correct in a strict sense because it ignores the advantage to the player who starts the bargaining. For example. if 61 = 1/3 and 92 = 1/2. then H = 3/5 > 1/2. Player 1 gets a bigger share of the pie since H > 1/2. although he is less patient as e. < 92. In his original literature. Rubinstein [1982] did not explicitly make such a Statement. Instead. by 15. 16. 17. 16. 19. 20. 21. 22. 23. 24. as. 26. 4? assuming that 61 = ea = 9. he mentioned the advantage to the player who starts the bargaining. See Rubinstein [1982]. p. 99 and p. 108: Sutton [1986]. p.711. Our example in note 14 further shows that it is possible that a less patient player earns more when he starts first. See Binmore [1980]. See Sutton [1986]. p.711. and Binmore [1982]. p.29. See HcLennan [1982]. This is quoted from the abstract of his article. Davidson and Cheung [1987] constructed a model of incomplete information to investigate the relationship between the bargaining structure and strike activity. See Hotelling [1929]. p.45. See Hotelling [1929]. p. 49. See Boulding [1966]. p. 4841-. See Eaton and Lipsey [1975]. p.46. See Boulding [1966]. p. 484. See Lerner and Singer [1937]. p.148. See Lerner and Singer [1937]. p.161. For the convenience of the readers. we quote this assumption as below: '. . . each seller assumes that his competitor will not respond to an encroachment on his customers. even when the encroachment deprives him of nearly half his customers: but that he will react to the loss of the whole of his trade when undercut. and that this reaction will take the form of mov1ng to the position most profitable on the assumption that the undercutter's position is fixed. " 27. 28. 29. 3°. 31. 32. 33. 34. 36. 36. 37. 36. 39. 4o. *1. 46 See Smithies [1941a]. p.64. Again. see Smithies [1941a]. p.64. For detailed definitions and implications of these three cases. see Smuthies [1941b]. pp.427-428. See Capozza and Van Order [1978]. p.897. Although.they also mentioned the model which was first presented by Greenhut and Ohta [1973]. they did not study this model in detail. See Capozza and Van Order [1978]. p.903. The so-called perverse case of the sac model is similar to the case of local monopolists in Hovshek [1980] and in this dissertation. See Varian [1984]. p.69. See Novshek [1980]. p.317. The following definitions are quoted from Salop [1979]. p.143. The only change is that we replace the word “brand” by the word "firmP to maintain the consistency in this paragraph. See Salop [1979]. p.145. See Salop [1979]. p.141 and pp.149-150. See Salop [1979]. p.149. Salop [1979] had mentioned this problem. See Novshek [1960]. p.313. See Novshek [1980]. p.315. Chapter Three: Wholesale Price Determination (I) - pure bargaining cases - I. Introduction Hav1ng reviewed the literature in the previous chapter. we begin our study Of wholesale price determination in this chapter. As the first step in this research. we concentrate our attention on the pure bargaining between the retailers and the wholesalers.1 So. spatial factors are not incorporated into the analysis of this chapter. For simplicity. we assume that the consumers’ demand for the retailers’ goods is a downward-sloping linear function. The retailers’ demand for the wholesalers’ product is assmmed to be a derived demand from.the consumers’ demand for the retailer’s goods. The bargaining structure used in this chapter is simular to that used in Rubinstein [1982] and Davidson [1985]. We use the bargaining solution proposed by Nash [1950]. Throughout this chapter we assume that complete cooperation between the wholesalers and the retailers is prohibited. Side-payments from agents of one level (e.g. the wholesalers) to agents of another level (e.g. the retailers) are allowed only to insure that every agent earns non-negative profits in the equilibrium. There are many cases studied in this chapter. For most of the cases. the bargaining equilibrium is unique. In Section II. we study the bilateral monopoly case. In 49 50 section III. we study the case where a monopoly wholesaler deals With duopoly retailers. While in Section IV. we study the case where a monopsony retailer deals With duopoly wholesalers. After that. we study the cases where there are duopoly firms at both levels. The bargaining outcome Will be unpredictable if all the retailers bargain With all the wholesalers simultaneously. ‘We hence impose restrictions on the bargaining structure to make the bargaining equilibrium predictable. Therefore. in Section V, we assume that every wholesaler provides its goods to one and only one retailer and every retailer buys goods from one and only one wholesaler. Also. in Section VI. we assume that the wholesalers take turns bargaining with all the retailers at each time period. In Section VII. we assume that the retailers take turns bargaining with all the wholesalers at each.time period. Finally. we summarize the results in Section VIII. II. Bilatepa; Monopoly Case A. Assppptipps apd Rotation (A1) A single homogeneous 800d. (A2) There are two firms ‘ one wholesaler and one retailer - in the market. The wholesaler is the manufacturer. (A3) The (inverse) consumer demand is a linear function shown as P = a - bx where P is the retail price: 51 X is the quantity demanded by consumers. a/b l X l 0: a and b are positive constants. (A4) The retailer’s profit function is "r (P " P')X "' fr (a - bx - Pw)x - fr where w, is the profit level of the retailer. wrao; P' is the wholesale price which.is the marginal cost of the retailer. a 2 P, l 0; fr is the fixed costs of the retailer. (A5) The demand for the wholesaler’s product. 0. is a derived demand from the consumers’ demand for the retailer’s good. In this case. 0 = X. (A6) The wholesaler’s profit function is “m (P' ‘ C)Q " fm (P' - c)x - fm where rm is the profit level of the wholesaler. um i o; c is the constant marginal cost of production. a l c l 0: fm is the fixed costs of the wholesaler. £m_z 0. (A7) The goal of every firm is to maximize its own profits. 62 (A8) Complete information is assumed in this analysis. (A9) Clearly. the retailer prefers a low P' while the (A10) wholesaler prefers P' to be as high as possible. ‘We therefore assume that the agents determine P' through a bargaining process. A bargaining equilibrium is achieved if the bargaining solution satisfies the Hash.criterion advocated by Nash [1950]. [1953]. Binmore [i982] and HcLennan [1982]. That is. Pw' is the bargaining equilibrium if P,” maximizes O : (um - umG)(wr - wrd) where "m4 is the threat-point payoff to the wholesaler if there is no agreement: «rd is the threat-point payoff to the retailer if there is no agreement. The bargaining process is as below. The players on both.sides take turns submitting their offers about P'. If an offer is accepted by the other side. the bargaining ends and production (and sales) begins. If an offer is rejected. the rejecting side makes a counter-offer to see if it would be accepted. This process of exchanging offers continues until an agreement is reached. Assume that both sides discount the future gains and loss with time. and that the time lapse is relatively short between offers submitted. The bargaining equilibrium price-offer Would be 53 proposed and accepted in the initial period if it ex1sts. no matter which side proposes such an offer. This bargaining process is similar to that used in Rubinstein [1983) and DaVidson [1985]. B. The Analysis For any given P'. the retailer will maximize its profits by choosing an optimal sales level. x”. with (3.1) x" = (1/2)(a - Pw)/b. then the equilibrium retail price Will be (3.2) P” = (1/2)(a + Pw)- So. the profit levels Will be (3.3) fir” (1/4)(a - Pw)2/b - fr (3.4) um” (1/2)(Pw - c)(a - P')/b - fm. Using the bargaining structure stated in assumption (A10). With assumptions (A8) and (A9). both players know that they will run a loss of their own fixed costs if there is no agreement between them. This means the threat-point payoff to the retailer is wrd = -fr and the threat-point payoff to the wholesaler is mm? = -fm. According to the flash criterion. the bargaining equilibrium wholesale price is one which maximizes (3.5) 0 ("mi _ wmd)(1rr' - 11rd.) (um. + in) ("r' + fr) (1/8)(1/b2)(a - P')3(P' - c). The solution which maximizes Q in equation (3.5) is 54 (3.6) P," = (1/4)(a + 3c). Therefore. With the bargaining equilibrium wholesale price given in equation (3. 5), We know that (3.7) X" = Q” = (3/6) (a - c)/b (3.8) p" = (1/8)(5a + 3c) (3.9) w,” = (9/64)(a - c)3/b - fr (3.10) Wm” : (3/32)(a - c)a/b - fm. Equation (3. 6) shows that the equilibrium wholesale price is a function of both the consumer’s evaluation and the marginal cost of production. Other factors, such as the slope of the demand curve and the fixed costs. do not affect the equilibrium wholesale price. The same result applies to the equilibrium retail price. according to equation (3.6). Equations (3.9) and (3.10) show that. when the demand for the wholesaler’s product is a derived demand. the wholesaler earns a smaller profit than the retailer does in the bilateral monopoly case if fn = fr. The reason is as below. 2 When the retailer and the wholesaler bargain over the wholesale price. PW. only. as the demand for the wholesaler’s product is aoderived demand. the retailer chooses the quantity. x“. which gives the retailer a strategic advantage in bargaining. (If they bargain over (P'. X) combination. then the retailer would have no choice over P” a - bit". 11' would be split in Nash solution given frzfmzo.) 55 The comparative static predictions for this case are (3.11) OP"/6a 1/4 c 5/8 = aP*/6a (3.12) or,*/oc 3/4 > 3/5 = 6P'/3c (3.13) our'/6a (9/32)(a - c)/b > (3/16)(a - c)/b = owmf/oa (3.14) owr'/Ob = -(9/64)(a - c)2/b2 < -(3/32)(a - c)2/b2 Gulf/8b (3. 15) dur'IOC -(9/32)(a - c)/b < -(3/16)(a - C)/b : dum‘/OC (3.16) owr'/0fr = -1 < o = awmf/Of, (3.17) our'/6fm_= o > -1 : awmf/ofm. Equation (3. 11) shows that an increase in the consumer’s evaluation on the 800d. (that is. an outward shift Of t_h_e demand curve). causes the equilibrium wholesale and retail prices to rise With the retail price rising more. Also. equation (3. 13) shows that an increase in the consumer’s evaluation also increases bOth players’ profit levels. Equations (3.6) and (3.8) show that the slope of the demand curve has no impact on the equilibrium wholesale and retail prices. However. equation (3.14) shows that the steeper the demand curve the lower the equilibrium profit levels. The impact on the profit levels comes from its negative effect on the quantities demanded as (BX/ab < 0. Equation (3. 12) shows that an increase in the marginal cost of production causes the equilibrium wholesale and 56 retail prices to rise. It also shows that the rise in the retail price is less than the rise in the wholesale price While the rise in the wholesale price is less than the rise in the marginal cost Of production. Therefore. an increase in the marginal cost Of production causes the retailer’s profit margin and the wholesaler’s profit margin to fall. In addition. the rise in the prices cause the quantities demanded to fall. SO. as shown in equation (3.15). an increase in the marginal cost Of production causes the profit levels to fall. Furthermore. the fixed costs have no effect on the equilibrium wholesale or retail price. Each firm’s fixed costs have a negative effect on its own profits only. according to equation (3.16) and (3.17). III. HOhOEle Wholesaler versus Duopoly Retailers In this section. We assume that there are tWO firms at the retailing level. We study two cases. In the first case. We assume that the retailers cooperate in the wholesale price bargaining. In the second case. We assume that the retailers act non-cooperatively in every aspect. Case 1. Duopoly retailers compete with each other but cooperate in the wholesale ppice bargaining.3 A. Assumptions and Notation Assumptions (A1) and (A7) to (A10) are the same as 57 those used in the preVious section. We therefore list only those With modifications as below. (A2) (A3) (A4) (A5) (A6) There is one wholesaler at the wholesale level and two symmetric retailers at the retailing level. Using the Cournot-Hash behaVioral assumption. the retailers compete at the retailing level. Also. we assume the retailers cooperate in the wholesale price bargaining to maximize their sum of profits. up. fr : #1 + Va. The (inverse) consumer demand. which is linear. is P a - bx a - b(X1 + X2) x1 is the quantity sold by retailer 1: X1 2 0 for all i =1. 2. The profit function for each retailer is 1'1 (P "' P“)X1 " fr [a - b(X1 + x2) - ijxl - fr for all i = 1. 2: where W1 is the profit level of retailer i. i = 1. 2: The demand for the wholesaler’s product. 0. is a derived demand from consumers’ demand for the retailers’ goods. In this case. 0 : x = x1 + X3. The wholesaler’s profit function is 56 "m : (PW ‘ C)Q " fm 3 (PW ‘ C)x ‘ fm (Pw - c)(X1 . x3) - fm. (All notation here is the same as before.) B. The Analysis For any given PW. each retailer Will maximize its profits by choosing an optimal sales level as (3.16) x1" = (1/2)(a/b - xJ* - Pwlb) foralli=1.2.j:1.2.andi¢j. By Substituting x * into x . we get J i (3.19) X1” = (1/3) (a - P')/b for all i = 1. 2. So. (3.20) x“ : x1” + x2” : (2/3)(a — P')/b and (3.21) P" (1/3)(a + 22,). Therefore. the profit level for each retailer is (3.22) «1* : (1/9)(a - Pw)a/b - f, for all 1 = 1. 2 and the wholesaler’s profit level is (3. 23) Wm” '3 (2/3) (a ' Pw) (PW " C)/b ‘ fm. With assumption (A2) - both.retailers cooperate in the wholesale price bargaining to maximize their sum of profits. 7r. - the sum Of the retailers’ profits. fir. is (3.24) w,“ u, + ”2 (2/9)(a - P.)3/b - 2fr. Also. With assumptions (A.8) and (A.9). the threat- Point payoff to each side is «pa = -2fr and wmd : —fm. 59 According to the Nash criterion. the bargaining equilibrium wholesale price is the one WhiCh maximizes (3.25) o 2 (Wm‘ - um9)(wr* - wrd) = (4/27)(1/b2)(Pw — c)(a - Pw)3 The solution which maximizes O in equation (3.25) is (3.26) P,” = (1/4)(a + 3c). With.the bargaining equilibrium wholesale price given in equation (3.26). We know that (3.27) X1“ = (1/4)(a - c)/b for all i = l. 2 (3.28) x“ xi! + x2! (1/2)(a - c)/b (3.29) P' = (1/2)(a + c) (3.30) a.” = (1/16)(a - c)3/b - fr for all 1 = 1. 2 (3.31) wr' : a,” + we” = (1/6)(a - c)3/b - 2fr (3.32) wm' = (1/8)(a - c)a/b - fm. It is interesting that the bargaining equilibrium wholesale pgice in this case is the same as that in the bilateral monopoly cupg. When the retailers cooperate in the wholesale price bargaining. they look like a monopsony buyer. Although they compete at the retailing level. which causes the retail price to fall for any given Pw. the equilibrium wholesale price is the same as that in the bilateral monopoly case. To explain why the equilibrium wholesale price is 6O unchanged. let us look at the algebraic computation. For any PW given. the increase in the number of retailers in this case endows the wholesaler with 4/3 times of its previous profits. This is because competition between the retailers leads to an increase in aggregate output. which benefits the wholesaler. On the other hand. the sum of the retailers’ profits becomes 8/9 of their previous amount.4 This is because the competition between the retailers causes the retail price to fall for any given PW. The objective function. 0. is therefore 32/27 times of the previous value. for any wholesale price given. Friedman [1986] pointed out. as one of Nash’s axioms. the bargaining solution is not affected by any positive affine transformations.s Therefore. the bargaining equilibrium solutions for the wholesale price are the same in these two cases because the only difference between them is in the constant terms of the objective functions. The same phenomenon happens in other types of consumer demand function.6 Thg uuplication of the phenomenon above is as follows. When the demand for the wholesaler’s product is a derived demand. in the wholesale price bargaining. the increase in the number of retailers does not affect the wholesale price. In other words. the increase in the number of retailers does not improve the bargaining position of the wholesaler. However. the wholesaler’s profits increase due to the fact that the retailers compete With each other at the retailing 61 level. which.causes the aggregate output to increase. That is. the wholesaler's profit level is higher than that in the previous case according to equations (3.10) and (3.32). Also. from equations (3.31) and (3.32). we Know that the wholesaler's profit level is equal to the sum of the retailers' profits in this case if fm = afr. The comparative static predictions. for all i 1. 2. (3.33) owi*/oa (1/8)(a - C)/b < (i/4)(a - C)/b owr'/Oa owm*/Oa (3.34) ov1'/ab -(1/16)(a - c)3/ba > -(i/6)(a - c)2/b3 owr‘/°b : OWm'/°b (3.3s) aw1*/oc -(1/e)(a - c)/b > -(i/4)(a - c)/b = our*/ac cum. / 6C (3.36) me'lafr O > “i : °w1./°fr ) awr”/Ofr : -2 (3.37) ow1*/0fm curl/aim 3 0 > ’1 3 OWm'/°fmo The wholesale price. in this case. is the same as that in the previous case. So. the comparative static predictions and their implications regarding P.” are the same in both.cases. The impacts of the parameters (a. b. and c) on the retail price in this case are smaller than those in the previous case. but the sign patterns of parameter changes on P” are the same in both cases. Their implications regarding P”. then. are the same for both cases. «U. be 62 The wholesaler's profit level is higher than any individual retailer’s profits. So. the impact of parameter changes is greater on the former than on the latter in the absolute value. This is shown in equations (3.33) to (3.35). Again. changes in the fixed costs of each.firm.have impacts on its own profits only according to equation (3.36) and (3.37). We have analyzed.the case where the duopoly retailers cooperate in the wholesale Price bargaining but compete at the retailing level. TWO natural extensions are to analyze the case where the retailers are completely noncooperative and to analyze the case where they are completely cooperative. The latter case can be reduced to the bilateral monopoly case. SO. we only analyze the case where all the retailers are completely noncooperative. Case 2. Duopoly retailers act noncooperativelz in every aspect. A. Asspgptions and Notation All the assumptions used in the preVious case Will be used in this case except for the following changes: (A6) The monopoly wholesaler's profit function is 12m = (1’1“ - c)x1 + (Paw - c)X3 - fm (A9) When bargaining occurs between the wholesaler and the retailers. the bargaining equilibrium solution is according to the Nash criterion advocated by (A10) 63 Nash [1950]. [1953]. Binmore [1962] and McLennan [1982]. That is. P1" is the bargaining equilibrium if P1" maximizes Q = (um - umd)(w1 - ald) for all i = 1. a where and is the threat-point payoff to the wholesaler if there is no agreement on P1" by the wholesaler and retailer i. but an agreement on PJ' by the wholesaler and retailer J: «14 is the threat-point payoff to retailer i if there is no agreement on P1w by the wholesaler and retailer i. but an agreement on PJ' by the wholesaler and retailer j. The bargaining process between the wholesaler and the retailers is as below. In the initial period. the wholesaler suggests a pair of wholesale prices (Pi'. Pg"). The retailers decide simultaneously whether to accept or reject the wholesale prices offered. If both wholesale prices are accepted. the negotiations end and the production and sales take place. If P3” is accepted and P1" is rejected. then negotiations between.the wholesaler and retailer 1 continue until either an agreement is reached or the wholesaler decides to withdraw from the bargaining table and makes retailer j the monopolist in the market. If both wholesale prices are rejected. then the retailers make b: 64 counter-offers simultaneously. The wholesaler may accept both.cffers. accept one and continue bargaining over the other. accept one offer and withdraw from bargaining with the other retailer. or reject both.wholesale-price offers. This process continues until negotiations are completed at both retailers. The bargaining process stated above is the same as that in Davidson [1985]. B. The Analzsis For any given P1”, retailer i will maximize its profits DY choosing an optimal sales level as (3.36) x1! = (1/2)(a - be - Pl')/b foralli:1.2;j 1.2.and1¢J. So. when both P1" and P3“ are given. the optimal output level for each retailer Will be (3.39) x1” : (1/3)(a - 2P1w + PJ')/b for all i = 1. 2; J : 1. 8 and 1 i J. The equilibrium retail price and profit levels. then. will be (3.40) (3.41) (3.42) P' = (i/3)(a + p," + Pa") 11' = (1/9)(a - 2P1w + PJ')3/b - fr for all i = 1. a; j : 1. a and i r J mm” = (1/3)(P1' - c)(a - 2P1w + PJW)/b + (1/3)(PJ' - c)(a - 2PJ' + P1W)/b - fm for all 1 = 1. 2; J : 1. 2 and i #:J. h the 1! amen has a] whole: fur } Ithh (3.43] be the (3. 44; (3. 45 (3. 45 Equll ‘3- 47 65 With.the bargaining process stated in assumption (A10). the threat-point payoff to retailer i is «14 : -fr if no agreement on P1" has been reached but an agreement on P3" has already been reached. The threat-point payoff to the wholesaler. in this situation. would be wmd : (PJW - c)xJm - fm. Here, me is the sales by the monopoly retailer j. which is (1/2)(a - PJ')/b when PJ' is given. So. (3.43) umd (PJ' - c)me»- rm (l/a)(PJw - c)(a - PJW)/b - fm. For any given PJW. the bargining equilibrium P1“ will be the one which.maximizes Q. where (3.44) o (wm' - and)(w1* - v.4) (i/54)(1/b2)(a - aplv + PJ')3(2P1V - PJ' - c). The solution which maximizes Q in equation (3.44) is (3.45) P1'-' = (1/6)(a + 4PJ' + 3c) for all i = l. 2; j = 1. 2 and i t j. when pr is 8 iven. By symmetry. for any P1“ given. (3.46) pr : (1/8)(a +4P1' + 3c) for all i = 1. 2: j : 1. 2. and i t J. when P1" is given. With assumption (A8) Of complete information. at equilibrium the pair Of equilibrium wholesale prices Will be (3.47) P1'-' : P2"' : Pw' : (a +3c)/4 molesa This 1m] success: wholesa. the con: art1fac1 molesal retailel demand. (God to the qua; PFICB hi I"Halley 66 It is interesting that the bargaining efiilibrium Lholesale price is eflal to th_at in the previous two cases. This implies that the monopoly wholesaler can not successfully threaten any retailer and Obtain a higher wholesale price. This result also holds in the cases where the consumer demand is not linear. 7 This phenomenon is an artifact of the assumptions that the demand for the wholesaler's product is a derived demand and that the retailers are symetric. When the demand is a derived demand. it might not be wise for the wholesaler to sell its good to only one retailer with a higher price. The loss in the quantity sold can hardly be offset by the gain in the price hike. Also. with the assumption of sylmletric retailers. if the wholesaler sells its goods to one retailer. it will sell its goods to the other retailer at the same wholesale price. in eQilibrium. Therefore. the bargaining equilibrium wholesale price will be the same for al 1 these cases. Sumarizing the studies above. we have reached the following conclusion. When the demand for the monomlz wholesaler's oduct is a derived deman with the.s tric retailersI the bargainipg eggilibrium wholesale price is always the same regardless of the number of retailers and regardless if they coograte or not. 5 With the equilibrium wholesale price shown in equation (3.47), we may calculate the equilibrium retail price and (3F? IE? '1 1e 67 the equilibrium profit levels as the following: (3.48) I”I = (1/2)(a + c) (3.49) «1* (i/io)(a - c)3/b - fr for all i = 1. a (3.50) am“ (1/8)(a - c)3/b - rm. Equations (3.46). (3.49) and (3.50) are the same as equations (3.29). (3.30) and (3.32) in the previous case. This implies that when the demand for the wholeggler:§ product is a derived demandI with the same number of retailersI all the eggilibrium results are the same. no matter if they cooperate or compete with.each other. All the previous cases have a common assumption that there is only one wholesaler at the wholesale level. We will investigate. beginning in the next section. the cases where there are two symmetric wholesalers in the wholesale level. 68 IV. Honopsonx Retailer versus Duopoly Wholegglerg In this section. we will discuss the situation where there is one retailer at the retailing level while there are two wholesalers at the wholesale level. A. Assgptions and Rotation Assumptions (A1). (A3). (A4). (A7) and (A8) are the same as those in the second case of the previous section. ‘We do not repeat them here. Instead. we list those assumption WhiCh are different as below. (A2) There is one retailer at the retailing level. At the wholesale level. there are two symmetric. noncooperative wholesalers.9 (A5) The demand for each wholesaler’s product by the monoposony retailer is a derived demand. The monopsony retailer purchases goods from the wholesaler with a lower wholesale price. Or. it purchases equal amount of goods from both wholesalers if they offer the same price. That is. 01 = o2 = X/2 if Pw1 = P"? = Pw: or. QJ o and 01 = x if Pwi < PwJ foralli:1. 8. J:1.2. andi¢j. Here. P'1 is the wholesale price for goods sold by wholesaler l; 01 is the retailer's demand for wholesaler i ' s 800d. 69 (A5) Each wholesaler’s profit function is «mi = (P, - c)(X/2) - fm if Pwi : PwJ = Pw; or. wmi = (Pwl - c)x - fm if Pwl < PwJ for all i = 1. 2. J : 1. 2 and i t J where ”m1 is the profit level of wholesaler 1. (A9) If the wholesale price. PW. preferred by the retailer is different from that preferred by the wholesalers. then there is a bargaining between them. The bargaining equilibrium is achieved if the bargaining solution satisfies the Nash criterion advocated by Nash [1950). [i953]. Binmore [1982] and McLennan [1982]. That is. Pwi is the bargaining equilibrium if Pwi maximizes Q : (wmé - «mlvd)(wr - urd) for all i : 1. 2 where wmlvd is the threat-pOint payoff to the wholesaler i if there is no agreement on Pwi by wholesaler i and the retailer but an agreement on PwJ by wholesaler j and the retailer; «rd is the threat-point payoff to the retailer if there is no agreement on Pwi by wholesaler i and the retailer. but an agreement on PwJ by wholesaler j and the retailer. (A10) When there occurs a bargaining between the retailer and the holesalers. the bargaining process Will be as below. In the initial period. (l 70 the retailer suggests a pair of wholesale prices (Pw1. Pwa). The wholesalers decide simultaneously whether to accept or reject the wholesale price offered. If both wholesale prices are accepted. the negotiations end and the production and sales take place. If P'J is accepted and PV1 is rejected. then negotiations between the retailer and wholesaler i continue until either an agreement is reached or the retailer decides to withdraw from.the bargaining table and makes wholesaler j the monopolist in the wholesale level. If both wholesale prices are rejected. then the wholesalers make counter-offers simultaneously. The retailer may accept both offers. accept one and continue bargaining over the other. accept one offer and withdraw from bargaining with the other wholesaler. or reject both wholesale-price offers. This process continues until negotiations are completed at both who l esal ers. B. The Analzsis To maximize profits. with the lowest wholesale price PW given. the monopsony retailer sets its sales at (3.51) x” : (l/a)(a - Pw)/b. According to assumption (AS). the retailer buys goods 71 only from the wholesaler with a lower price. This makes the other wholesaler run a loss of its fixed costs. So. with the assumptions of complete information and symmetric wholesalers. at equilibrium. both wholesalers will supply goods to the monopsony retailer at the same price level. That is. Pw1 : Pwa = Pw*. The amount supplied by each wholesaler will be (3.52) 01 = x*/a : (1/4)(a - Pw')/b for all i = 1. a. The retail price and the profit levels. then. will be (3.53) P“ : (1/2)(a + P") (3.54) w,” : (i/4)(a - Pw')3/b - fr (3.55) um1-* = (i/4)(Pw - c)(a - Pw)/b - fm for all i = l, 2. With the bargaining process stated in assumption (A10). the threat-point payoff to wholesaler i is wmivd = -fm if no agreement on Pwi has been reached but an agreement on PwJ has been reached. The threat-point payoff to the retailer is «,4 = (i/4)(a - PwJ)2/b - fr at this time. Also. this implies that all the price offers on Pwi are at least as large as the agreed PwJ at the threat-point. For any PwJ given. Pwi is the bargaining equilibrium wholesale price if it maximizes (3.56) o (wr* - urd)(wm1u* - «m1:d) (1/3a)(1/b3)(aa - Pwi - PwJ)(PwJ - Pwi) (Pwi - c)(a - Pwi). 72 The solution for maximizing Q in equation (3.66) is (3.57) 9,1 = 9,3 = p,“ = c which.implies (3.58) x! = (1/2)(a - c)/b (3.59) oi = o? = (1/4)(a - C)/b (3.60) P” = (1/2)(a + c) (3.61) a,” = (1/4)(a - c)3/b - fr (3.62) «m1-* = -fm for all i = 1. 2. Equation (3. 57) shows that when the retailerfi the monopgony buyer. with more than one wholesaler, the wholesale price will be bargained down to the wholesaler's marginal cost of production. This low price will cause the producers (wholesalers) to run a loss of their fixed costsio in the short-run equilibrium. This short-run equilibrium is fragile because the wholesalers' profit levels are the same as if they do not supply anything. However. we assume that they will stay in the market because both of them know that if they exit they lose their fixed costs but if they remain they have a chance of being the natural monopolist in the long-run. In the long-run. one of the two wholesalers will exit which makes the other a natural monopoly at the wholesale level. That is. in the long-run. there exists a bilateral monopoly situation for the wholesale price bargaining. Side-payments from the monopsony retailer to bOth retailers11 can make wholesalers earn non-negative profits m t] (he 1 not I Paym' £1er (3. 6: (3.6: (3.6' (3.6: (3.6' 'ho). life 23!; Ml marg fixe. Dam. ”ta. °°n31 °f t] ”011 73 in the short-run equilibrium. Side-payment transfers affect the players' profits with.a lump-sum change only. WhiCh do not affect the equilibrium prices. The simplest side- payment scheme is that the retailer pays each wholesaler its fixed costs. fm. The comparative Static predictions are (3.63) 6P*/oa = 1/2 > o (3.64) OP"/Oc 1 > 1/2 : 6P*/6c (3.65) 6w,*/oa (1/2)(a - c)/b > 0 (3.66) Ovr'IOb —(1/4)(a - c)3/l>a < o (3.67) ovr'/0c -(1/2)(a - c)/b < 0. The eggilibrium.wholesale pric . WhiCh is equal to the wholesaler's marginal cost of production. will not be affected by the changes of other parameters. The egpilibrium.retail price will be higher when the consumer's evaluation on the good is higher. or when the wholesaler’s marginal cost of production is higher. In this case. both wholesalers run a loss of their fixed costs in the short-run if there is not any side- payment from the retailer to the wholesalers. The monopsony retailer’s profit level is positively affected by the consumer’s evaluation and negatively affected by the slope of the (inverse) demand curve. the marginal cost of production and its own fixed costs. To assure that every wholesaler earns non-negative profits bargain that in (at the (3.68) (3.69) (3.70) (3.71) [3.72 (3.73 (3.7) 74 profits in the short-run equilibrium. we may solve this bargaining equilibrium wholesale price With a constraint that wmi l 0. (In a trivial sense. let "m1 : 0.) So. we get the bargaining equilibrium wholesale price as (3.68) P"' = (1/2){(a + c) - [(a + c)2 - 4(ac + 4b£m)1(1/3>3 WhiCh means (3.69) x" = (1/4)((a - c) + [(a + c)2 - 4(ac + 4bfm)](1/3))/b (3.70) P'” = (1/4)((3a + c) - [(a + c)2 - 4(ac + 4bfm)](i/2)] (3.71) Vr'” = (1/16)((a - c) + [(a + c)2 - 4(ac + 4bfm)](l/3)]2/b - fr (3.72) «mi-'* : m2.'* = o. The comparative static predictions then become (3.73) 6P'**/6a (1/a)(1 - (a - c)H(-1/3)) < (1/4)(3 - (a - c)H('1/3)) = OP"/6a (3.74) OPw**/6b 4fmn('1/3) > aimH(-1/2) = op'*/ob (3.75) OP'”'/6c (1/a)(1 + (a - c)H(-1/3)) > (1/4)(1 + (a - c)H(‘1/3)) 8P"/8c (3.76) 6Pw”'/6fm_= 4bH(-1/2) > abH(-1/3) = 6P**/6fm (3.77) ow,**/aa (1/8)((a - c) + H(1/3))(1 . (a - c)H('1/3))/b > 0 (3.78) owr*'/ab -(1/16){(a - c) + H(1/3))(16b£m + (a - c) + H‘l/all/ba < o (3.79) awr**/oc -(1/8){(a - c) + H(1/3)){1 + (3.80) E 7111 be evaluat steeper have no Profit PPOflt ‘006 1: lower ; is hlg‘ \he ‘3 the Wt SlUQY Same. 75 (a - c)H('1/3))/b < o (3.60) ourH/afm = -((a - c) + H(1/3))H(-1/2) Here. H : (a +c)a - 4(ac + 4bfm). Here. we find that the egpilibrium wholeggle and retail prices are positivelz affected by all parameters except for the retailer's fixed costs. That is. the equilibrium prices will be higher when there exists a higher consumer evaluation. a higher marginal cost of production. or a steeper demand curve. The retailer’s fixed costs. however. have no effect on the equilibrium prices. The wholeggler1§ profit level. in this case. is always zero. The retgéler;§ profit level is higher when the consumer’s evaluation on the good is higher. However. the retailer's profit level is lower if the demand curve is steeper or if any kind of cost is higher. All the cases studied before have not yet investigated the game in which two players on each side bargaining over the wholesale price. In the following sections. we will study three different cases of the two-by-two bargaining 76 V. Duopoly Wholesalers versus Duopoly Retailers (1) - Every wholesaler provides goods to one and only one retailer From.this section. we assume there are two firms on both sides. Beginning with the simplest case. we assume that every wholesaler provides goods to one and only one retailer and every retailer buys goods from one and only one wholesaler. So. there is no explicit or direct competition among the wholesalers. The indirect competition at the wholesale level is from the competition at the retailing level because the demand for wholesaler's product is a derived demand. A. Assumptions and Rotation Assumptions (A1). (A7) and (A8) are the same as before. Some of the other assumptions are similar to but with.some modifications from those in other sections. Therefore we list all the assumptions except for (A1). (A7) and (A8). (A2) There are two symmetric wholesalers at the wholesale level and two symmetric retailers at the retailing level. With Cournot-Nash behavioral assumption. the retailers compete with each other at the retailing level. Every wholesaler proVides goods to one and only one retailer and every retailer buys goods from one and only one wholesaler. (A3) (A4) (A5) (A6) (A9) 77 The (inverse) consumer demand is a linear function shown as P a - bx a - b(X1 + X2) x1 is the quantity sold by retailer i. i = 1. 2. Every retailer's profit function is shown as V1 : (P - P1')X1 - fr : [a - b(X1 + Xe) - P1'1x1 - fr for all i = 1. 2. The demand for each.wholesaler’s product by the retailer is a derived demand from consumers' demand for retailer’s goods. Q1 : X1 1 = 1. 2 where 01 is the demand for wholesaler i's product. Every wholesaler's profit function is shown as «m1 = (1’1" - c)oi - fm = (I31w - c)X1 - fm for all i : 1. 2. As the retailer prefers a low PW while the wholesaler prefers P' as high as possible. We therefore assume that the agents determine PW through a bargaining process. The negotiations are assumed to occur simultaneously and take the following form. In the initial time period. each wholesaler proposes a wholesale price that its corresponding retailer may accept or reject. If the retailer rejects the price offer. it makes a counter-offer in the next period. which the .00 [3'3 B. (A10) 76 corresponding wholesaler may accept or reject. This process of exchanging offers continues until an agreement is reached. To capture the notion that the time it takes to come to terms is small relative to the length of the contract. we assume that the time between periods is very small and.that the production and sales occur only when both retailers have come to terms with their wholesalers or when one retailer settles with.its wholesaler and the other wholesaler (or the other retailer) decides to leave the bargaining table. This bargaining process is the same in spirit as that used in Davidson [1986). The bargaining equilibrium solution is according to the Hash.criterion developed by Nash [1950]. [1953]. Binmore [1982] and HcLennan [1982]. That is. P1")' is the bargaining equilibrium solution if P1W)' maximizes 01 = (77...1 - wmi-d) (171 - 1714) for all i = 1. 2 where Wmi'd is the threat-point payoff to wholesaler i and «id is the threat-point payoff to retailer i if there is no agreement on P1" by wholesaler i and retailer i. but an agreement on P1” by wholesaler j and retailer j. for all i = 1. 2; J = 1. 2 and i ttj. The analysis 79 By the method of backward deduction. and using the same process as in section III. we know that when both.P1W and P1" are given. the optimal sales for retailer i will be (3.81) x1“ = (1/3)(a + PJ' - 2P1")/b for all i = 1. 3; J = 1, 2 and i t J. Also. according to assumptions (A3) to (A6). With P1" and PJ' given. the retail price and profit levels will be (3.82) P” : (1/3)(a + P," + P3“) (3.83) «1' : (1/9)(a + 93' - 2P1')3/b fr for all i = 1. 2; j = 1. 2 and i t j. (3.84) «m1-' = (1/3)(P1' - c)(a + PJ' - 2P1')/b - fm for all i = 1. 2: j : 1. 2 and i t j. With the bargaining structure stated above. the threat- point payoffs to retailer i and wholesaler i are «14 : -fr and «m4-d = -fm. separately. This happens when there is no P1” agreed by retailer i and wholesaler 1 while an agreement on PJ' has been reached by retailer J and wholesaler J. For any given PJ'. P1" is the bargaining equilibrium wholesale price if it maximizes (3.65) 0 (”mi .. “mi.d) (1'1 - 171d) (l/27)(P1' - c)(a + p1" - 291W)3/ba. The solution for Q maximization in equation (3.85) is (3.86) P1" : (1/2)(a + PJW + 6c) for all i = 1. 2; J = 1. 2 and 1 ¢ J. 80 Equation (3.86) implies that the equilibrium wholesale prices will be (3.87) P1"* : Paw)“ : 9".” = (1/7)(a + 6c) which further implies (3.88) x1" : (2/7)(a - c)/b for all i : i. 2 (3.89) P' = (1/7)(3a + 4c) (3.90) «1' = (4/49)(a - c)3/b - fr for all i : 1. 2 (3.91) «mi-* : (2/49)(a - c)2/b - fm for all i = 1. 2. Equation (3.87) shows that the equilibrium wholesale price in this case is higher than that in the monopsony- retailer versus duopoly-wholesalers case but lower than that in all other cases studied before. This implies that the wholesalers. in this case. compete with.each.other indirectly. which.causes the wholesale price lower. The equilibrium retail price. in this case. is lower than that in the previous cases. With the lower retail price. the consumers consume more than they do in the previous cases. Also. equations (3.90) and (3.91) show that an individual retailer earns more profits than an individual wholesaler The comparative static predictions show that. for all 1 = 1. 2. (3.92) OPVv'/6a : 1/7 < 3/7 : 8P*/8a (3.93) aPW-'/oc : 8/7 > 4/7 = aP*/ac (3.94) ox1*/aa : 2/7 > o (3.95) 0X1”/0c : -2/7 < o 81 (3.96) ox1*/8h -(2/7)(a - c)/ba < 0 (3.97) 071*/8a = (8/49)(a - c)/b > (4/49)(a - c)/b = awm1-*/aa (3.96) OW1'/Oc -(8/49)(a - C)/b < -(4/49)(a - c)/b awm4.*/ac (3.99) Ow1”/0b = -(4/49)(a - c)/ba < -(2/49)(a - c)/ba = 8wm1"/0b. The comparative static predictions shown in the equations above have the implications as below. The equilibrium prices will be higher if wholesaler's marginal cost of production and consumer's evaluation on the good are higher. Also. changes from.the cost side have a bigger impact on the wholesale price. However. changes from the demand side have a bigger impact on the retail price. The fixed costs of each firm affect its own profit level only. Furthermore. the impacts of the parameter changes on wholesaler’s profits are qualitatively the same as those on retailer's profits. This is because the demand for wholesaler's product is a derived demand. 82 VI. Duopoly Wholesalers with Duopoly Retailers (II) - Wholesalers take turns bargaining with all the retailers For a two-by-two bargaining game where all the wholesalers bargain With all the retailers simultaneously. the bargaining equilibrium outcome is unpredictable. The reason is stated below. When a retailer tries to threaten any wholesaler that it will not buy anything from.the threatened one unless the wholesale price is lowered. the threatened wholesaler has no fear because it may sell its product to another retailer. So. the retailer's threat is not credible. By the same logic. neither wholesaler can threaten any retailer. Therefore. it is necessary to impose some restrictions on the bargaining process to make it workable. In the following two sections. we impose a restriction on the bargaining procedure.12 The restriction is. at each time period. there is only one firm on either side bargaining with all firms on the other side. In this section. we assume that the wholesalers take turns bargaining with all the retailers at each time period. (In the next section we assume that the retailers take turns bargaining with all the wholesalers at each time period.) Also. we assume that any agreement made in the previous period is binding at the current period. We list the assumptions and notation as below. A. Asspgptions and Notation 63 Assumptions (A1). (A3). (A7) and (A6) are the same as those in the previous sections. Therefore. we list all assumptions except for (A1). (A3). (A7) and (A8). (A2) (A4) There are two symmetric wholesalers and two symmetric retailers in this industry. The wholesalers compete with each other at the wholesale level and so do the retailers at the retailing level. Every wholesaler may sell its goods to every retailer and every retailer may buy the goods from every wholesaler. Every retailer buys goods from the wholesaler with a lower wholesale price or it buys the same amount of goods from both wholesalers if P11 = P13 for all i = 1. 2. By the same analogue. every wholesaler supplies goods to the retailer with a higher wholesale price or it supplies equal amount of its product to both retailers when P." = Pa' for all w : 1. 2. Here. P1' is the wholesale price agreed by retailer i and Wholesaler w. The retailer's profit function is shown as 2 W1 : 81(P - P1')X1w - fr '2 Zita - b(X1 + X2) - lelxlw * fr '3 for all i = 1. 2 where X1' is retailer i’s demand for wholesaler w's products. W = 1, 2; 64 2 x1 = 8 x1“ is the goods sold by retailer 1. (A5) The demand for each.wholesaler's product is a derived (A5) (A9) demand from the consumer demand for retailers' 8000.3. That is. 2 0' = 181X1' for all w = 1. 2 where 0V is the demand for wholesaler w’s product from the retailers. The wholesaler's profit function is shown as 2 77' : 181(1’1' - c)x1w - fm for all w = 1. 2 where w' is the profit level of wholesaler w. At each time period. the wholesalers take turns bargaining with all the retailers. The bargaining process is as below. In the initial period. wholesaler w suggests a pair of wholesale prices (P1'. P3"). The retailers decide simultaneously whether to accept or reject the wholesale prices offered. If both wholesale prices are accepted. then it is the turn of another wholesaler bargaining with both retailers in the next period. If P1“ is accepted and P1" is rejected. then the negotiations between wholesaler w and retailer 1 continue until either an agreement is reached or the wholesaler decides to withdraw from the bargaining table and to supply its goods only to retailer j. If (A10) 65 both (P1". P3") are rejected. then the retailers make counter-offers simultaneously. Wholesaler w may accept both.offers. accept one and continue bargaining over the other. accept one offer and withdraw from bargaining With.the other retailer. or reject both retail-price offers. This process continues until negotiations are completed at both retailers. After wholesaler w has finished its bargaining With both retailers. wholesaler w’ begins its negotiations in the next period. The agreements made in any time period are binding in the next period. We are interested in analyzing the steady-state bargaining equilibrium in which.the wholesale price once determined will not be changed in the following time periods. Furthermore. we assume that the time lapse between periods is very small so the time it takes to come to the steady-state bargaining equilibrium is small relative to the length of time for production and sales. The bargaining equilibrium solution is according to the Nash criterion developed by Nash [1950]. [1953]. binmore [1982]. McLennan [1982]. That is. P1.t'v* is the steady-state bargaining equilibrium solution if P1.t")* maximizes 0t = ("t' ‘ 1rt"'dl(l'i.t ‘ "i.td) for all i = 1. 2; W = 1. 2 and for all t. where t is the notation of time. which may be any positive integer: 66 wtwvd is the threat-point payoff to wholesaler w and w1,td is the threat-point payoff to retailer i if there is no agreement on P1,." by retailer i and wholesaler w at time t. but an agreement on PJ.tW by retailer j and wholesaler w at time t and an agreement on P1,t-1" by retailer i and wholesaler w' at the previous period. t-l. B. The Analzsis With.the assumptions stated before. we may investigate the wholesale prices in the steady-state bargaining equilibrium by the following steps. First of all. let us assume that at the beginning of time period t. there exists a pair of wholesale prices (P1.t-1". P2.t-1")- These wholesale prices were agreed by the retailers and wholesaler w' in.the preVious period but they are not necessarily the steady-state equilibrium.pricesi3. So. the retailers and wholesaler w want to bargain for a new pair of wholesale price (P1.t'. P2.t')- The wholesale prices (P1.t-1"'. Pa.t-1") are binding. in time period t. to wholesaler w’and both retailers. When there is an agreement on PJ.tw by wholesaler w and retailer j but no agreement on P1,tw by wholesaler w and retailer i. the threat-point payoff to retailer i will be (3.100) w1.td : (Ptd - P1,.d)x1,td - fr for all i = 1, 2 where w1.td is the threat-point payoff to retailer 67 i. P1.t')d is the threat-point wholesale price by which retailer i may get goods from the other wholesaler. Here. P1,t')d = P1,t-1w'. P1,t_1V' is the wholesale price agreed by wholesaler w’ and retailer i in the previous period. x1,td is the threat-point quantity of goods sold by retailer i to the consumers. With the assumption that the wholesale price P1,t-1" is binding in this period and with equation (3.29) in Section III. we know that x1.td : (1/3)(a - 2P1.t-1" + PJ.tw)/b. Ptd is the threat-point retail price. Ptd = a - b(X1'td + x1..m) where XJ,tm is the quantity retail j could sell xJ.tm : (1/3)(a - 2PJ,tW + P1,.-1W’). So. Ptd a - b[(1/3)(a - 2P1’._1V’ + PJ.tw)/b + (1/3)(a - 2PJ,tW + p1,.-.w’)1 = (1/3)(3 * Pl't-1w' + PJ'tw). Therefore. the threat-point payoff to retail i is (3.101) w..td = (1/9)(a - 2P1’t-1" + PJ’tW)a/b - fr for all i : 1. 2. Also. the threat—point payoff to the wholesaler w is (3.102) «twid = (PJ.tW - c)XJ.tm - fm (1/3)(PJ.tw - c)(a - 2PJ,tW + P1,._1W’)/b 88 - fmo Furthermore. according to assumptions (A4) and (A6) and equations (3.41) and (3.42) in Section III. if P1.tw and PL..." are steady-state equilibrium Wholesale Prices. the retailer’s profits and the wholesaler’s profits Will be (3.103) «1,. = (1/9)(a - 2?..." + PJ.t')a/b - fr for all i = 1. 2 (3.104) Vt" : (lump...w - c)(a - 2P1'tw + PJ.t')/b + (1/3)(PJ..' - c)(a - 2PJ'tW + Pl'tW)/b - fm for all W = 1. 2. So. with PJ’t' given. the bargaining equilibrium P1,." is the one WhiCh maximizes Qt’ where (3.106) 0. (wt' - th-d)(w1,t - w1.td) = {[(1/3)(P1.t' - c)(a - 2P..." + pJ,.')/b . (1/3)(PJ,.w - c)(a - apJ,tW + p1,.')/bl - (1/3)(PJ.tV - c)(a - 3P1.t' + P1,.-.V')/b)l((1/9)(a - 291.." + P3,.")3/b - (1/9)(a - 2P1't-1" . p1..W)B/b). The first-order condition of maximizing at in equation (3.105) is (3.106) ant/6P... = 0. That is. ((1/3)(a - 4P1.tW + 2PJ.tW + c)l((2/9)(a - Pi.t' ‘ Pi.t-1" + Pj.t')(Pi.t-1" ' Pi.t')l - ((1/3)(P1.t' - c)(a - 2P1't' + PJ’t') + (1/3)(PJ.tw - c)(P1,.' - P1,.-1"))((4/9)(a - 2p..." + p1,.w)) = 0. 69 At the steady-state equilibrium. P1,..." = P1,t-1w' for all time periods. Equation (3. 106). then. is reduced to (3.107) (4/27)(P1,.V - c)(aP1.tW + PJ,.W)2 = 0 which yields the solution for P1..." as (3.108) P1,." = c for all i = 1. 2 and w = 1. 2. That is. at equilibrium. the wholesale price is equal to wholesaler’s marginal cost Of production. here are some important implications here. First. the wholesale price in the steady-state equilibrium Will be most dis-advantageous toward the wholesalers if the wholesalers take turns bargaining with.the retailers at each.time period. Furthermore. the equilibrium wholesale price is equal to that in the case where a monopsony retailer bargains 'iUh duopoly wholesalers. The intuition of the phenomenon above is not difficult. Let us discuss a similar case here. Assume that Taiwan and S. Korea are the only sources supplying car stereos while General Motor (GM) and Ford Motor are the only car makers. Taiwan and S. Korea compete with each other to sell their products to GM and Ford while GM and Ford compete With each other to sell their cars to the consumers. Let us further assume that all the players are noncooperative in the game of car-stereo price bargaining. Furthermore. we assume that the sellers (Taiwan and S. Korea) take turns bargaining with GM and Ford. Whenever one seller (say. Taiwan) has a 90 tentative agreement on the price with GM and Ford. GM and Ford will use this agreement to threaten S. Korea for a lower price: otherwise they will buy nothing from S. Korea. When 8. Korea yields to GM and Ford for an agreement with a lower price. GM and Ford then call Taiwan for a much lower price. Back and forth. the buyers may lower the price to a level where the sellers can barely survive. With.the equilibrium wholesale price that P1.tW : c. we may calculate the equilibrium quantity. profits. and the retail price as below: (3.109) x1” = 0*.” = (1/3)(a - c)/b for all i : 1. 2; w : 1. 2 (3.110) P” : (1/3)(a + 2c) (3.111) «1* : (1/9)(a - c)a/b - fr for all i = 1. 2 (3.112) w'l' : -fm for all w : 1. 2 When the steady-state equilibrium is achieved. there is no need to distinguish different time periods. 80. we suppress the notation of time. t. in equations (3.109) to (3.112). Equation (3.112) shows that every wholesaler will run a loss of its fixed costs at the steady-state equilibrium. So. this equilibrium can be hardly sustained if there is no side-payment scheme proVided. (Unless the wholesaler’s fixed costs are zero.) The simplest side-payment scheme is that the retailers 91 pay the wholesalers their fixed costs to make the wholesalers break-even14. This means the profit levels become (3.113) «1" = (1/9)(a - cla/b - fr - fm for all i = 1, 2 and‘ (3.114) w')*' : o for all w = 1. 2. The necessary condition for this side-payment scheme to be accepted by the retailers is that «1" l 0. This implies that the wholesaler's fixed costs should not be greater than wl‘ in equation (3.111% The comparative static predictions Will be (3.115) 8P'/8a : 1/3 > 0 (3.116) 8P*/8c = 2/3 > o (3.117) 8w1"/8a (2/9)(a - c)/b > o for all i = 1. 2 (3.118) 8w1*’/8c -(2/9)(a - c)/b < 0 for all i = 1. 2 (3.119) 8w1*’/8b -(2/9)(a - c)3/b2 < 0 for all i = l, 2. The equations above show that the wholesale price at the steady-state bargaining equilibrium is affected by the wholesaler's marginal cost only. The retail price is positively affected by both the Wholesaler’s marginal cost of production and the consumer’s evaluation on the good. Also. the higher the consumer's evaluation on the good. the higher the retailer's profits. Other factors such as the marginal cost and.fixed costs have negative effects on the retailer's profits. The wholesalers. in this case. always earn zero profits. 92 An alternative. to make the Wholesalers not run a loss. is to add the restriction of 7' l 0 in the bargaining problem. (TriVially. let 77" = 0.) This brings about the solution of the Wholesale price. the equilibrium retail price. as well as the profit levels as below: (3.120) P1'-" = (1/2)((a + c) - [(a + c)2 - 4(ac + shim)l(1/3)l for all i 1. 2; W : 1. 2 (3.121) x1*' = (1/6)((a - c) + [(a . c)a - 4(ac . 3bfm))(1/3)l/b for all i = 1. 2 (3.122) P'” = (1/3)((2a + c) - [(a + c)2 - 4(ac + 3bfm)] (1/8); (3.123) «1" = (1/36)((a - c) + [(a + c)2 - 4(ac + 3hfm))(1/3))3/b - fr for all i = l. 2 (3.124) u'-*" = o for all w = 1. 2. The solutions in equations (3.120) to (3.124) will converge to those in equations (3. 108) and (3. 109) to (3. 112) when the wholesaler’s fixed costs. fm. converge to zero. Furthermore. equation (3. 120) shows that the wholesale equilibrium price now is affected by all parameters except for the retailer's fixed costs. 80 is the retail price. The retailer's profit level is affected by all parameters while the wholesaler's profit level is always zero. If 171*" in equation (3. 123) is greater than 171" in equation (3.113). then lei'” in equation (3.120) is the stfieady- state equilibrium wholesale price. Otherwise, it is pi)" in equation (3.108). 93 The comparative static predictions are (3.125) 8P1'o*"/8a = (1/2)11 - (a - c)x('1/3)l < (1/3)(2 - (a - c)K(‘1/3)l = 8P*"/6a for all i : 1. 2: w : 1. 2 (3.126) 6P1"*"/0b = 3fm2(-1/3) > meK(‘1/3) = 8P*”/8b for all i = 1, 2; w = 1. 2 (3.127) 8P1V-*”/8c : (1/2)(l + (a - c)x('1/3)l > (1/3)(1 + (a - c)K(‘1/3)) : 8P*"/8c for all i : 1. 2; w = 1. 2 (3.128) 8P1'-*”/8£m = 3bx(-1/3) > 28x(-1/3) = 89*"/afm for all i = 1. 2; w : 1. 2 (3.129) 8u1*”/8a = (1/18)J(1 + (a - c)x('1/3)l/h > o for all 1 = 1. 2 (3.130) ow1*"/ah = -(1/36)(12bme(‘1/3) + Jl/ba < o for all i = 1. 2 (3.131) 6fl1*"/8c = -(1/18)J{1 + (a - c)H('1/3))/b < o for all i = 1. 2 (3.132) Owi'"/ofm = -(l/3)Jx(-1/3) < 0 for all i = 1. 2 where J : ((a - c) + K‘i/all. K = [(a + c)2 - 4(ac + 3bfm)]. The equations above show that the cost-side parameters. except for fr. have positive impacts on the equilibrium wholesale and retail prices. Their impacts are stronger on the wholesale price than on the retail price. Also. every cost-side parameter has a negative effect on retailer's profits. (Mote that the wholesaler's profit level is always 94 zero in this case.) Among the demand-side parameters. consumers' evaluation on the 800d has positive effects on both the equilibrium prices and the retailer's profit level. Its impact on the retail price is stronger than that on the wholesale price. Although the slope 0f demand curve has no impact on either price. the steeper the demand curve. the lower the retailer’s profits. VII. Duopoly Wholesalers with Duopoly Retailers (III) - Retailers take turns bargaining with all the wholesalers In this section. most assumptions are the same as those in the preVious section. The only change is that we assume the retailers take turns bargaining with all the wholesalers at each time period. It is expressed in assumptions (A9) and (A10). This small change results in a quite different equilibrium wholesale price. A. Asspgptions and Notation As was mentioned in last paragraph. there are only four assumptions different from those in the preVious section. We list them as below. (A9) At each time period. the retailers take turns bargaining with all the wholesalers. The bargaining process is as below. In the initial period. retailer 1 suggests a pair of wholesale prices (P11. 95 P13). The Wholesalers decide simultaneously whether to accept or reject the wholesale prices offered. If both wholesale prices are accepted. then it is the turn of retailer j bargaining with both wholesalers in the next period. If P1" is accepted and P1“ is rejected. then the negotiations between retailer i and wholesaler w continue until either an agreement is reached or the retailer decides to withdraw from the bargaining table and to buy goods only from wholesaler w'. If both (P11. P13) are rejected. then the wholesalers make counter-offers simultaneously. Retailer i may accept both offers. accept one and continue bargaining over the other. accept one offer and withdraw from.bargaining with the other wholesaler. or reject both wholesale-price offers. This process continues until negotiations are completed at both wholesalers. After retailer i has finished its negotiations with both wholesalers. retailer j begins its negotiations in the next period. The agreements made in any time period are binding in the next period. We are interested in analyzing the steady-state bargaining equilibrium in which.the wholesale price once determined will not be changed in the following time periods. Furthermore. we assume that the time lapse between periods is very small so the time it takes to come to the steady-state bargaining equilibrium is small B. 96 relative to the length Of the time for production and sales. (A10) The bargaining equilibrium solution is according to the Mash criterion developed by Mash [1950]. [1953]. binmore [1982]. McLennan [1982). That is. P1.twv* is the steady-state bargaining equilibrium solution if P1,tw-' maximizes 9t = Wt" ' fl‘t""‘1)(1"i,t ‘ "Ltdl for all w = 1. 2; and for all t. where t is the notation of time. which may be any positive integral. wtwld is the threat-point payoff to wholesaler w and wl’td is the threat-point payoff to retailer i if there is no agreement on Pl’tw by retailer i and wholesaler w at time t but an agreement on P1.tw' by retailer i and wholesaler w' at time t and an agreement on PJ,t-1V by wholesaler w and retailer j at the previous period. t-l. Analysis At the beginning of time period t. there exists a pair of wholesale prices (Pj,t-11- P3,.-13). This pair 0f ”101938.19 prices were agreed upon by the wholesalers and retailer J in the previous period, which are binding in the current period. This pair of wholesale prices need not 97 constitute a steady-state equilibrium. So. the wholesalers and the current retailer (retailer i) want to bargain for a new pair of wholesale prices (P1.t1. P1,t3). With.the assumptions stated before. when there is an agreement on P1.t" by retailer i and wholesaler w’. but no agreement on P1,." by retailer i and wholesaler w. the threat-point payoff to wholesaler w will be (3.133) utW-d = (pt':d - c)0twvd - fm for all i : 1. 2 and w : 1. 2 Where wt'vd is the threat-point payoff to wholesaler w at time t. Pt'vd is the wholesale price at the threat- point. Here. Pt'19 : PJ.t-1W. PJ.t_1' is the wholesale price agreed by retailer j and wholesaler w in the prev10us period. QtW.d is the quantity sold by wholesaler w to retailer j at the threat point. With the assumption that the agreement in the previous period is binding in this period. and by equation (3.52) in Section IV. we get (3. 134) 0)th = (1/4)(a - PJ.t_1")/b. 30. the threat-point payoff to wholesaler w is (3. 135) «Land : (1/4)(P3,t-1" - c)(a - P3,.-1")/b - fm; A180. the threat-pOint paYOff t0 retailer i is (3. 136) "Ltd : (Ptd - thW. d)x1.td - fr where wi'td is the threat-point payoff to retailer 96 i; Ptd is the retail price at the threat-point; P1.t'-d is the threat—point wholesale price for retailer i. which is the wholesale price agreed by retailer i and wholesaler w'. Pi.t"d = PLt": x1.td is the quantity which the retailer would like to sell to maximize its profits at the threat-point. With Gt'hd (in equation (3.134)) and P1. t" given. we may calculate X1. td and Ptd as below. (3.137) x1.td = (1/8)(3a + 91,.-." - 4P1..V’)/b and (3.138) Ptd a - b(x1,.d + 0.'-d) (1/8)(3a + PJ.t_1' + 4P1.tw'). SO, the threat-pOint payoff to retailer i is (3. 139) 171.61 (Pt‘1 - P1, t'" dlxi. t‘1 ' fr (1/64)(3a + PJ.t_1W - 4P...W’)3/h - fr. By assumption (A2). we know that when every P1.t" Proposed is at least as large as P1. t‘". and not higher than Pj,t-1‘2 the threat-point situation may occur. With the assunlption of symetry. at equilibrium. the wholesale prices will be the same to both wholesalers and retailer i will buy equal amount of goods from both wholesaler if PIA” : 3P1.t'”. Accompanied by assumptions (A4) and (A6) as well as equations (3. 54) and (3.55) in Section IV. the profit levels :for time wholesalers and the retailers at equilibrium will 99 be: (3.140) wtw = (1/4)(P1..w - c)(a - P1,.W)/b - fm for all w = 1. 2 (3.141) «1,. : (1/4)(a - P1.tw)3/b - f, for all i = 1. 2. Therefore. a bargaining equilibrium solution for the wholesale price P1,." is the one which maximizes (3.142) o. ("t' - th-d)(w1,t - «1,td) [(1/4)(P1,tw - c)(a - P1,tw)/b - (1/4)(PJ,t.1w - c)(a - P1,.-1')/bll(1/4)(a - P1,.W)2/b - (1/64)(3a + 93,.-." - 4P1.tw')3/b]. The first-order condition of maximizing at in equation (3.142) is (3.143) ant/8P." = 0 which implies (3.144) [(1/4)(a - 2?...“ + c)][(1/4)(a - P1,.')3 - (1/64)(3a + PJ.t-1V - 4P1.tw')3] - [(1/4)(P1.tw - c)(a - P1,.V) - (1/4)(PJ,.-.' - c) (a - PJ,.-.')l((l/2)(a - 9...")) = 0. At steady-state equilibrium. P1.tW-* = P1,.-." for all t. then equation (3. 144) may be reduced as (3.145) -(7/256)(a - 2?..." + c)(a - Pt")(a - spi,tw + 4P1,.W’) = 0. Therefore. for all w = 1. 2 with any PL." given. the solution of Pi.t' to maximize (2.. is (30146) Pl'twi' 2 (1/2) (a + c)’ 100 There are some important implications here. First. the wholesale price at the steady-state equilibrium is most dis- advantageous toward.the retailers if the retailers take turns bargaining with.all the wholesalers at each.time period. Furthermore. the equilibrium wholesale price. WhiCh is equal to (1/2) (a + c). is the same as that determined unilaterally by the Wholesalers. In this case. the current retailer at each time period is forced to agree upon a higher wholesale price (than the one agreed upon in the previous period). The intuition for this result is similar to that in the previous section. The retailer. at each time period. can not threaten any wholesaler by cutting off its purchases. Instead. the wholesalers can use the wholesale price agreed with another retailer in the previous period to threat the current retailer for a higher price. The wholesalers will repeat this practice until the wholesale price is high enough to maximize the wholesalers’ profits. With the equilibrium wholesale price shown in equation (3.146). we may calculate the equilibrium quantity. profits and the retail price as below. (3.147) x.“ : 0"." = (1/6)(a - c)/6 for all i : 1. 2; w = 1. 2 (3.148) P“ = (1/3)(2a + c) (3.149) 71* = (1/36)(a - c)2/b - f, for all i H .... N (3.150) 7*)“ = (1/12)(a - c)3/b - fm for all w H p N 101 When the steady-state equilibrium is achieved. there is no need to distinguish different time periods. Therefore. the notation of time. t. in equations (3.147) to (3.150) is suppressed. It is clear that. in this case. the Wholesaler earns a higher profits than the retailer does at steady-state equilibrium. Furthermore. the retail price is the highest among all the cases studied in this chapter because the wholesale price is the highest. Therefore. the quantity of goods consumed by the consumers is the smallest among all the cases. The consumers’ welfgpe in this case is worse than that in all otger cases. The comparative static predictions are as following: (3.151) 8P1V-*/8a (1/2) < (2/3) = 8P*/8a for all i 1. 2; w = 1. 2 (3.152) 6P1'v*/6c (1/2) > (1/3) = 6P*/6c for all i = 1. 2; w = 1. 2 (3.153) dfli'lda : (1/18)(a - C)/b < (1/6)(a - c)/b 8w'1'/8a for all i = i. 2; w : 1. 2 (3.154) 6w1'/6b -(1/36)(a - c)3/ba > -(1/12)(a - o)2 /lo2 : 8w'1'/8b for all i : 1. 2; w : 1. 2 (3.155) 8w1*/8c : -(1/18)(a - c)/b > -(1/6)(a - c)/b = 8wV1*/8c for all i = 1. 2; w = 1. 2 (3.156) 8w1*/8fr : -1 < o = (WWW/8fr for all i = 1. 2; W = 1. 2 (3.157) 8u1'/8fm : 0 > -1 8wV1*/8fm 102 for all i = 1. 2; w : 1. 2. Equations (3.151) and (3.152) show that the impacts from the demand side have a bigger effects on the retail price. However. the impacts from.the cost side have bigger effects on the Wholesale price. The fixed costs of each fimm affect its own profits with a dollar for dollar negative effect. Furthermore. every parameter (except for £1. and flu) has a greater influence on the wholesaler’s profits than on the retailer’s profits. VIII. Conclusions Having discussed carefully all the cases presented in this Chapter. we sumarize some major conclusions as below. i). When the demand for a monopoly Wholesaler’s product is a derived demand. With symetric retailers. the equilibrium wholesale price is always the same regardless Of the number of the retailersi5. ii). However. for the case of a monopsony retailer versus duopolistic wholesalers. the equilibrium.wholesale price is equal to the wholesaler’s marginal cost. This Will cause the wholesalers to run a loss of their own fixed costs in the short-run equilibrium. Side-payments from the retailer to the wholesalers Will make wholesalers earn non- negative profits. iii). For the case of duopoly wholesalers versus 103 duopoly retailers. the bargaining equilibrium is unpredictable if all the retailers bargain with all the wholesalers simultaneously. Therefore. we need to place some restrictions on the bargaining process. iv). If we modify the bargaining structure so that each wholesaler supplies goods to one and only one retailer and each retailer buys goods from one and only one wholesaler. then the bargaining equilibrium is unique. The equilibrium wholesale price is lower than that in the cases where there is only one wholesaler in the wholesale level. The reason is that the wholesalers now compete with each other indirectly. v). With a bargaining process in which the wholesalers take turns bargaining with all the retailers. the equilibrium wholesale price is driven to the Wholesaler’s marginal cost of production. This equilibrium wholesale price is equal to that in the case where a monopsony retailer bargains With duopoly wholesalers. vi). With a bargaining process in which the retailers take turns bargaining with all the wholesalers. the equilibrium.wholesale price is so high that it is the same as that determined unilaterally by the wholesalers. The equilibrium wholesale and retail prices in this case are higher than those in all the other cases studied in this chapter. Therefore. the quantity of goods consumed by the consumers is smaller than all other cases. The consumers’ 104 welfare is. then. worst in this case. In next chapter. we introduce spatial factors into the analysis. Also. for simplicity. the number of wholesalers is limited to one. We show that some of the conclusions that hold in this chapter do not hold when spatial factors are included. 105 Footnotes of Chapter Three It is not necessary that the retailers always bargain over the wholesale price with the wholesaler. Under some situations. with a well-designed side-payment scheme. the retailers and the wholesalers may be Willing to cooperate if everyone may be better-off through cooperation. we are indebted to Professor Segerstrom for initiating this explanation. This kind of business practices is not uncommon in the real world. For example. in some LDC countries. to increase the fiscal revenue. the government establishes a trade institution controlling the imports of some goods. This institution becomes the monopoly wholesaler in the domestic market. Domestic retailers usually form a trade union bargaining With this monopoly wholesaler over the wholesale price. Members of this trade union compete with each other at the retailing level. A historical example of this kind of business practice is the trade of apples in Taiwan before 1982. Taiwanese government had an institution named Central Trust of China (CTC) which monopolized the import of apples before 1982. The domestic retailers bid the quota to get apples from.CTC. However. they made a tacit agreement on the wholesale price in their trade union meeting. CTC had the right to reject any bid which was too low while the trade union would buy nothing if the price demanded by 106 CTC was too high. Only when a price was agreed by both sides. the import and sales would begin. When we say the Wholesaler’s profit is 4/3 times of its preVious level and the retailers’ profit level is 8/9 times of its previous level. we implicitly ignore the existence of fixed costs. The values of fixed costs. however. have no effect on the computation of the bargaining equilibrium. It is because they are always canceled out in the calculation of the objective function. 0. See Friedman [1986]. pp.155-156. This is the second condition for the Mash solution in Friedman [1986]. We are grateful to Professor Paul Segerstrom for showing us this explanation. We have tried some Other types Of demand function such as P : l/(X1 + X2). P: 1 - (x. +xa)3. and P : 1 - X1Xa. All of them yield the same equilibrium wholesale price no matter if the number of retailers is one or two. Furthermore. When the number of retailer is two. the equilibrium wholesale price is the same no matter if they cooperate or not in the wholesale price bargaining. See note 6. With the linear demand function. we calculate the equilibrium wholesale prices for the situations where the number of retailers is greater than two. The results 100 11. 12. 13. 107 show that for any finite number of retailersI with a sipgle monopply wholesalerll the eggilibrium wholesale price is always the same. However. due to the complication of calculation. we are not sure if this result holds for other types of demand function. If wholesalers cooperate With each other for the wholesale price bargaining. then the equilibrium wholesale price will be the same as that in the bilateral monopoly case. This is similar to the problem well known in the Bertrand price-setting games. In those games. the equilibrium price is equal to the marginal cost of production. Unless the fixed costs are zero. it can not be a long-run equilibrium. The retailer is more likely to make side-payments to both wholesalers than to only one wholesaler although it seemingly costs him more. The reason is that. with complete information. the retailer knows that if he makes side-payments to only one wholesaler. the other one will exit and it ends up With bilateral monopoly situation. The monopsony retailer earns less in bilateral monopoly situation than in this situation. This modification of bargaining process is suggested by Professor Carl Davidson. We are grateful for his kind help. If they are steady-state equilibrium wholesale prices, 14. 15. 108 then there is no need to search for the new wholesale prices. For simplicity. we assume that every wholesaler receives an amount Of its fixed COStS from a retailer. Strictly speaking. this statement holds when the consumer demand is linear. For more detailed information. please refer to notes 6 and a. Chapter Four: Wholesale Price Determination (II) - bargaining cases with spatial factors I . Introduction The pure bargaining cases studied in Chapter Three show that when there are two firms at both the wholesale and the retailing levels. the strategy space is very large. So. the bargaining equilibrium solution becomes unpredictable. This problem can be cured if we impose some restrictions on the bargaining process. However. with the restrictions on Che bargaining process considered. the solution then becomes most dis-advantageous toward either the wholesalers or the retailers, depending on the bargaining structure. To avoid this complication. in Chapters Four and Five. we assume there is only one wholesaler at the wholesale level. Also. we assume that all symmetric retailers pay the same wholesale price to the monopoly wholesaler. Furthermore, in these two chapters. we incorporate spatial factors into the analyses. Spatial factors are embodied at the retailing level as the consumers pay the transport cost to get the good from the retailer. With.spatial factors. the homogeneous good sold by different retailers becomes differentiated. Product differentiation prov1des the retailers with market power at the retailing level and bargaining power for the wholesale price determination. The major difference between Chapters Four and Five is the assumption about the goal of firms. In this chapter. as 109 110 in Chapter Three. we assume that the goal of each.firm.is to maximize its own profits. In Chapter Five. we assume that the monopoly wholesaler and all the retailers are willing to maximize their sum of profits. (However. we do not assume that the retailers also cooperate at the retailing level.) ‘With different assumptions on the goal of firms. the equilibrium wholesale price levels are significantly different in these two chapters. The demand for the good by each consumer. in general. is a decreasing function of the delivered price. A downward sloping linear demand function is a good example and will be used in these two chapters. Some other types of demand functions. such as the completely inelastic demand function. will be incorporated into the analysis. too. While dealing with cases with a completely inelastic demand function. we assume there exists an outside good so the equilibrium price will be finite.1 Competition among spatially distributed retailers depends on the number of firms relative to the market size. demand density. and other parameters. Novshek [1980] has shown that. under different relationships between the parameters of demand. cost. and spatial factors, there are three possible types of competitive situations at the retailing level. These include a single retailer who is a monopolist. several retailers which are local monopolists. and several retailers who are not local monopolists. 111 Therefore. in Chapters Four and Five. we will study each of these three cases for each type 0f demand functions. The structure of this chapter is as below. Section II contains the assumptions and notation for models with a linear demand function. Section III contains the analysis of those linear demand models divided into three cases. Section IV contains the assumptions and notation for models with a completely inelastic demand function. Section V contains the analysis of models with a completely inelastic demand function. Section VI contains a study of the long- run equilibrium. Finally. in Section VII. conclusions of this chapter are presented. II. Assumptions and Notation for the Linear Demand Case (A1) A single homogeneous good. (A2) A one dimensional market of length L shaped as the circumference of a circle. (A3) There are n symmetric retailers evenly located on the market; neither entry nor exit occurs. The distance between any two neighboring retailers is L/n. Novshek [1980] shows. With the appropriate conjecture concerning the response of competing firms. this can be an equilibrium position if firms choose price and location as their decision variables. (A4) Consumers are symmetric and uniformly distributed along the market With the density D. 112 (A5) Each retailer charges a mill price to all consumers. (A6) Each consumer has the demand function (A?) (A8) x a - b(P + tu) if a/b > (P + tu); : O otherWise. Where x is the quantity demanded by a consumer; P is the mill price charged by the retailer, P l O; t is the transport cost per mile regardless of direction; u is the distance between the customer and the retailer. u z 0. a. b are positive constants. One manufacturer produces good x. OWing to the legal restrictions. the wholesaler charges the same wholesale price. P". to all the retailers.2 Furthermore. we assume no shipping costs for delivering goods from this monopoly wholesaler (or manufacturer) to the retailers. or. equivalently. the costs are constant per unit and the same for each customer. This assumption is equivalent to assuming that the wholesaler is located at the center of the circular market. With constant shipping cost to each point on the circumference.3 The cost function for each retailer is cr = P'X + fr Where C, is the total cost to the retailer; P. is the wholesale price of good x. Which is the retailer’s marginal cost. PV, 2 0; 113 fr is the fixed cost of running a retailing store. fr 2 o; X is the amount 0f 800d. X 8016 DY a retailer. X n R R EDI x(P, u)du : 2DJ [a - b(P + tu)]du 0 O 2DR(a - DP - btR/Z). where R is the maximum distance of each side of the market area a retailer can reach. R S (1/2)(L/n). Also. R is a function of P as R : (i/a)(P’ - P + tL/n)/t. where P' is the mill price charged by a neighboring retailer. when the retailer is not a local monopolist. 0r. R : minIL/a. (a/b - P)/tl when the retailer is a local monopolist. (A9) The profit function for each retailer is wr=PX"Cr (P "' P')X " fr 2DR(P - Pw)(a - bP - btR/a) - fr 2 o With.P' given. the retailer chooses a retail price. P. to maximize its profits. LA1<3) The cost function for the monopoly wholesaler is Cm = C0 + fm Where Cm is the total cost of the monopolist; c is the constant marginal cost of production, (including constant shipping cost to all retailers). c z 0; fm is the fixed cost of production. fm z 0; (A11) (A12) (A13) LAlll) III. (ll) , 114 G is the quantity sold by the Wholesaler. which is a derived demand from the consumers' demand for the retailers’ goods. 0 = nX. The profit function for the monopoly wholesaler is anDR(P' - c)(a - DP - btR/a) - fm z o The goal of each firm is to maximize its profits. The wholesale price. P". is determined by the Nash bargaining solution where players in this bargaining game are the wholesaler and the retailers. The Nash bargaining solution. according to Nash [1950]. [1953]. Binmore [1982] and McLennan [1982]. is the PW Which maximizes n = ("r - dramm - am) ("r ‘0' fr) (17m 4‘ fm). Here, dr : -fr is the threat-point payoff to each retailer if no agreement is reached: dm = -fm is the threat-point payoff to each Wholesaler if no agreement is reached. complete information is assumed in this bargaining game. The Analysis of the Linear Demand Case cases of monopoly retailer(s) When n s maxii. 3th/[4(a - wa)]l.4 the number of 115 retailers is small relative to the market. With the situation that n = i z 3th/[4(a - wa)]. there is a single monopoly retailer serving the entire market. For other situations where 1 s n < 3th/[4(a - wa)]. there might be a single monopoly retailer or several local monopolists serving part of the market.5 Therefore. R : mintL/a. (a/b - P)/t] for the cases mentioned above. We will study all of them as below. Case 1. A single monopolist serves the entire market when n = 1 z 3th/[4(a - Pw)]. In this case R : L/a and P s a/b - tL/a. This is the case where the prevailing retail price is so low that the single monopoly retailer serves the entire market. This retailer is a monopolist to the consumers and a monopsonist to the manufacturer. Since we have assumed one manufacturer in this industry. this case becomes a situation of bilateral monopoly. Substitutine R : L/Z into (A6). we get (4w 1) X : bDL(a/b - tL/4 - P) > 0. For any given PW. to maximize up in assumption (A9). the retailers will choose (4- a) P" -.- (1/2)(a/b - tL/ll + PW) Which implies (4. 3) "r” (i/l-l)bDL(a/b - tL/4 - Pwa - fr (4. 4) 11m (1/2)bDL(a/b - tL/4 - Pw)(Pw - c) - fm (4. s) Q' = (1/43)u:>1>1.)?‘3(1=W - c)(a/b - tL/ll - Pwls' 116 All the equations above are functions of the wholesale price. P". If PW is determined unilaterally by the monopoly Wholesaler to maximize «m3. then it Will be Pw* : (1/2)(a/b - tL/4 + C). Also. if PW is determined unilaterally by the monopoly retailer to maximize vb”. then it Will be PW” = o. The Wholesale price levels preferred by both players are different. So. we have a bargaining situation between the Wholesaler and the retailer. The Hash bargaining equilibrium exists When Q’I is maximized. The first order condition of Q” maXimization is OQ'/0Pw : 0. Which implies (4.6) (a/b - tL/4 - P') - 3(Pw - c)(a/b - tL/4 - Pw)3 = o. The solution for equation (4.6) is (4.7) P.” = (i/4)(a/b - tL/4 + 3c). 80. (4.8) P” = (i/8)(5a/b - 5tL/4 + 3C) (4.9) u,” = (9/64)bDL(a/b - tL/4 - c)a - fr (4.10) am“ = (3/32)bDL(a/b - tL/4 - c)a - rm. Equations (4. 7) and (4. 8) show that every demamd-side parameter and every spatial factor. except for population density, D. affects the equilibrium wholesale and retail prices. As the monopoly retailer serves every consumer in the market. it adjusts the retail price to accomodate every change in the consumer demand function. To maximize its profits. the retailer bargains with the Wholesaler for a new wholesale Price when it makes adjustments on the retail 117 price. By the same logic. the monopoly wholesaler Will bargain with the retailer for a new wholesale price when there is a change in its marginal cost. The population density. D. enters the wholesaler and.the retailer’s profit function in the same manner. so it has no effect on the equilibrium prices. Also. fixed costs have no effect on the short-run equilibrium prices. Furthermore. as P” : (1/8)(5a/b - 5tL/4 + 3c) 2 a/b - tL/a. it is necessary that the marginal cost of production. c. satisfy c S a/b - (11/12)tL. From equations (4.7) and (4.8). we know that the slope of the consumer's demand. b. affects the equilibrium prices. This is different from cases in the pure bargaining games. In the pure bargaining games. the slope of demand curve has no effect on the equilibrium prices. Furthermore. equations (4.9) and (4.10) show that fir” : (3/a)wm* if fr : fm = O. The profit level for the monopsony buyer is higher than that for the monopoly seller in this bilateral monopoly situation. It is because the monopsony buyer. (fine retailer). has strategic advantage in the bargaining.6 As we assume that the demand for the wholesaler’s product is a derived demand. the retailer can threat the Wholesaler by varying its purchase. The comparative static predictions for the case Where the single monopoly retailer serves the whole market are as below: (4. (4. (4. (4. (4. (4. (4. (4. (4. (4. (4. (4. (4. 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) The comparative static results show that. OPw'/aa an'/ab OPw'lac OPw'/OL on*/at awr*/aa Owr'/Ob awr'/ac awr'/6D owr'/0t owr*/0L ow,'/az OWK'/Ofx ) (1/4)(1/b) < -(a/4)(1/b3) 3/4 < 3/8 = -t/16 > ~L/16 > 118 -5t/32 -5L/32 (9/32)DL(a/b - (5/8)(1/b) 6P*/dc oP*/aL aP*/at tL/4 - C) > 6P*/Oa 0 > -(5a/8)(1/b2) = op'/ab -(9/64)(a/b)DL((a/b)3 - (tL/4 + c)3] < o -(9/32)bDL(a/b - tL/4 - C) (9/64)bL(a/b - tL/4 - c)2 > o -(9/128)bDLa(a/b - tL/4 - c) < o < 0 (9/64)bD(a/b - tL/4 - c)(a/b - 0 (3/2)awm'/oz -1 for all K = for all 2 P. m 3. 3tL/4 - C) b. the C. D. t. L equilibrium Wholesale price is positively affected DY the follow1ng Parameters: good. the steepness of the demand curve. marginal cost. Also. the consumer's preference for the the wholesaler's Parameters like transport cost. market Size. it is negatively affected by Furthermore, those Parameters such as the Population density and the fixed costs have no impact on the wholesale Price determination. function 0f the wholesale price. Since the equilibrium retail Price is an increaSing both the wholesale and the 119 retail price have the same sign patterns for their comparative static predictions. The magnitudes of the impacts are greater on the retail price When the impacts come from.the demand-side parameters. (Note that spatial factors are treated as demand-side parameters since spatial factors are incorporated in the demand function.) The magnitudes of the impacts are greater on the wholesale price When the impacts come from the cost-side Parameters. Generally speaking. the sign patterns of parameter changes on the profit levels are the same as those on the price levels. The exceptions are as below. The population density Which has no impact on the equilibrium prices has a positive impact on equilibriumLprofit levels. The fixed costs of each.firm have a dollar for dollar negative effect on its profits. The marginal cost of production. although it has positive impacts on prices. has negative impacts on profit levels. Also. the size of the market which.has negative effects on the prices. has positive effects on the profit levels. Case a. All the retailers are local monopolists When 3th/4[(a - wa)) > n z i. In this case R = (1/t)(a/b - P) and a/b > P > a/b - tL/a. This is the case where some of the consumers may not be served by the monopoly retailer(s) because the prevailing retail price is high. Furthermore. in this case. there may be a single monopolist if n = 1 < 3th/[4(a - wa)]. or 120 several local monopolists if 1 < n < 3th/[4(a - wa)]. Substituting R = (1/t)(a/b - P) into assumption (A8). we get (4.24) x : D(a - bP)3/bt. For any given PW. to maximize vr. every retailer will choose (4.25) P” : (1/3)(a/b) + an/B (4.26) R“ : (2/3)(a/b - Pw)/t (4.27) wr* (4/27)(bD/t)(a/b - Pw)3 - fr (4.25) am” (4/9)(an/t)(Pw - c)(a/b - Pw)a - fm (4.29) 9' = (16n/243)(bD/t)3(1=w - c)(a/b - Pw)5. If the Wholesale price is determined unilaterally by the monopoly wholesaler to maximize its profits, then P" : a/(3b) + ac/3. By the same logic. the wholesale price will be Pw = o to maximize "r if it is determined unilaterally by the retailers. The Wholesale price levels preferred by players of both sides are different. 80. we have a bargaining situation between the wholesaler and the retai l ers. The Hash bargaining equilibrium exists when Q” is maximized. The first order condition to maXimize Q” is OQ’VOPw : 0. which implies (4.30) (an/t)[(a/b - Pw)5 - 5(a/b - Pw)4(Pw - c)] = 0. So. (4.31) PW“ : (1/6)(a/b + 5c) 121 (4.32) P” : (1/9)(4a/b + 5C) (4.33) gr” (125/1458)(bD/d)(a/b - c)3 - fr (4.34) um” (25/466)(an/t)(a/b - c)3 - fm. As the number of retailers must be an integer. it is not likely that all the consumers are served in this case. Spatial factors. therefore. have no influence on the equilibrium prices. Equations (4.31) and (4. 32) reflect this fact. Also. as P” : (1/9)(4a/b + 5c) > a/b - tL/a. it is necessary that the marginal cost of production. c. satisfy c > a/b - (9/10)tL. Furthermore. the equilibrium Wholesale price in this case is higher than that in the Previous case. From equations (4.33) and (4.34). we found.that Wm” : (3n/5)ur' if fr = fm : o. It means that the monopoly Wholesaler benefits from the increase in the number of retailers. Also. it means that when all the retailers are local monopolists and When n > 1. the monopoly wholesaler gets higher profits than any individual retailer does. However, when n : 1. we return to the bilateral monopoly situation and have Wm” : (3/5)wr'. This implies that the monoposony retailer earns more profits in the bilateral monopoly situation since the retailer has strategic advantage in bargaining. This strategic advantage is enhanced When the retailer needs not to serve the entire market . 122 Comparing this bilateral monopoly case With the previous one. we find that the retailer’s profits in this case is relatively larger. (That is. up” = (5/3)wm' in this case While a,” : (3/2)wm* in the preVious case.) The monopoly retailer has a relatively stronger bargaining position in this case because it does not serve the entire market now. When the wholesaler tries to ask for a higher Wholesale price. the retailer may threat to reduce its demand for the wholesaler's product by decreasing the market area it serves. The comparative statics for this bargaining equilibrium Where 1 1 n < 3th/[4(a - wa)] are (4.35) 6Pw”/6a : (1/6)(1/b) < (4/9)(1/b) : 0P'/0a (4.36) OP"/db -(a/6)(1/b3) > -(4a/9)(1/b2) = aP*/ab (4.37) OP"/6c = 5/6 > 5/9 = 6P'/6c (4.3a) awr'/6a (125/486)(bD/t)(a/b - c)3(1/b) > 0 (4.39) awr'/ab = -(125/1458)(D/t)(a/b - c)3(c + aa/b) < 0 (4.40) owr'/oc = -(125/486)(DD/t)(a/b - c)2 < 0 (4.41) owr'/an = (125/1458)(b/t)(a/b - c)3 > 0 (4.42) dwr'/0t = -(125/1458)(bD/ta)(a/b - c)3 < 0 (4.43) arm'/an = (25/486)(bD/t)(a/b - c)3 > 0 (4.44) 0wm*/az = (3n/5)6flr'/OZ for all z = a. b. c. D. t The comparative static predictions in this case are qualitatively the same as those in the prev10us case. The exceptions are: i) no spatial factor has any impact on the eQuilibrium Wholesale or retail price. 11) the market size, 123 L. not only has no impact on the prices but also has no impact on profit levels. Those results are shown in the equations above and in Table I. We do not repeat them wordily here. (B). the case where the retailers are not local monopolists When the number of retailers. n. is greater than maxti. 3th/[4(a - wa)]l. the retailers are not local monopolists. It is because the number of firms is large relative to the market. With the Cournot-Hash.noncooperative behavioral assumption. there exists a unique. symmetric equilibrium7 in which every retailer's market area8 is determined by (4.45) R' : (1/2)(L/n). For any given P'. to maximize "r. every retailer will set its retail price. P. as (4.46) P” : (1/a){a/b + Pw + (3/a)(tL/n) - [(a/b - PW - (1/2)tL/n)a + 3(tL/n)311/33. then. (4.47) up” = (th/8)(L/n)3((a/b - PW) - (25/2)(tL/n) + 7((a/b - P. - (1/a)tL/n)a + 3(tL/n)311/3) - fr. (4.46) um” : (bDL/aupw - c)((a/b - PW) - (BtL/n) + [(a/b - Pw - (1/2)tL/n)a + 3(tL/n)3]1/3) — fm. So. (4.49) 0' = (tL/16)(bDL/n)3(Pw - c)t8(a/b - Pw)3 - (43/a)(tL/n)(a/b - Pw) + (191/4HtL/n)a + [6(a/b - PW) - (53/2)(tL/n)][(a/b - PW - (1/2)tL/n)a + 3(tL/n)211/3). 124 The Nash bargaining solution of PW is the one maximiZing 9‘. which comes from 60'/6Pw = 0. This implies (4.50) [8(a/b - P...)a - (43/2)(tL/n)(a/b - PW) + (191/4)(tL/n)a + 8((a/b - P.) - (53/2)(tL/n)][(a/b - P. - (1/2)tL/n)a + 3(tL/n)3)1/3) + (P' - c)[-16(a/b - P.) + (43/a)(tL/n) - 6[(a/b - pw - (1/2)tL/n)3 . 3(tL/n)311/a - [8(a/b - PW) - (53/2)(tL/n)][(a/b - Pw - (1/2)tL/n)a + 3(tL/n)3)-(1/3) [a/b - Pw - (1/2)(tL/n)]l = 0. Equation (4.50) shows PW is a function of all parameters except for population density. D. However. it is algebraically too complicated to solve PW in equation (4.50). Furthermore. the solution of PW will be lengthy and with complicated feature. Therefore. we made a computer simulation by setting a = b = t = D = L = 1 and fr = fm : 0 to check the equilibrium values of P“. P. wr and am. when the values of c and n are changed. We found that 1 > 0P"/6c > 6P'/6c > 0. and 6Pw*/6n > 6P'/6n > 0. This means an increase in either the marginal cost of production or the number of retailers Will increase the bargaining equilibrium prices. It is interesting that the number of retailers affects the equilibrium prices. In the prev10us case. the number of retailers does not affect the equilibrium prices as long as all the retailers are local monopolists. Furthermore. the Increase in the number Of retailers increases rather than 125 decreases the equilibrium prices. The reason is as below. Other things being unchanged. an increase in the number of retailers decreases the market area of each retailer. When the market area of each retailer decreases. the total cost 0f transportation paid by the most distant customer is reduced. So. the retailers are able to raise their retail prices. This. then. gives the wholesaler a chance to raise its Wholesale price. Also. we got the result that 6x'/0n < 0. OQ'/6n < 0. owr'/on < 0. and 6wm*/6n > 0. That is. an increase in the number of retailers decreases quantities demanded for this good because the prices are higher. Also, because the effect of the increase in the wholesale price outweighs the effect of the decrease in demand. the wholesaler's profits rises When.the number of retailers increases. Contrarily. an increase in the number of retailers decreases the retailer's profits. The reason is two-fold. First of all, the result that on*/6n > 6P”/0n > 0 implies that the rise in the retail price is less than the rise in the wholesale price. That is. every retailer's profits margin falls when the number of retailers increases. Secondly. an increase in the number of retailers causes both a smaller market area to every retailer and a higher retail price to every consumer. So. the quantity of good sold by each retailer is decreased. which is shown as 6X*/0n < 0. Therefore. each retailer's profits falls. 126 The result that 1 > on'/0c > 0P*/0c > 0 has the same implication as that in the Previous cases. That is. the egpilibrium Wholesale and retail prices ri_e as the marginal gpgt of progpction rpggg. Also. because the rise in the price is less than the rise in the cost. the wholesaler and“ every retailer has a smaller profit margin when the marginal cost of production increases. Furthermore. as 0X*/6c < 0 and 60'/oc < 0. the rise in the marginal cost of production causes the quantities demanded to fall. Therefore. profit levels decrease When the marginal cost of production increases. That is. owr*/0¢ < 0 and owm*/ac < 0. In the following sections. we will study the situation where the consumer demand is completely inelastic. The reason is two-fold. First of all. it reduces the complication in computation. Secondly. for some types of goods. the consumer demand is quite Price inelastic. 127 IV. Assumptions and Notation for the Completely7Inelastic Demand Case All the assumptions and notation in section II. except for (A1) and (A6). Will be used for models With.the completely inelastic demand. Here. we replace (A1) by (A1)' Besides good x. there is a homogeneous outside good y.9 The consumer gets a constant surplus of utility. s. from the consumption of one unit of y. Also. we replace (A6) by (A6)’ X = 1. if U(u. P) : v - tu - P 2 8. That is. the consumer buys one and only one unit of good x if U. (the surplus of utility in good x). exceeds 3. (the surplus of utility in good y). Here. V is the maximum of the utility a consumer could get from the consumption of good x. u and P are distance and the retail price respectively. R Furthermore. X : 2D] x(P. u)du : aDR. where R is the 0 max1mum distance of each side of the market area a retailer can reach. R S (1/a)(L/n). Also. R is a function of P as R = (1/8)(P' - P + tL/n)/t. (where P' is the mill price charged by a neighboring retailer). when the retailer is not a local monopolist. Or, R : mintL/a. (v - P)/tl When the retailer is a local monopolist. 128 And we add one more notation (A15) Denote v = V - s > 0 v is the effective reservation price for consumers to consume 800d X. V. The Analysis of the Completely inelastic Demand Case (A) cases of monopoly retailer(s) When n 1 maxii. tL/(v - Pw)l.1° the number of retailers is small relative to the market size. With the situation that n = 1 l tL/(v - PW). there is a single monopoly retailer serv1ng the entire market. For other situations where 1 S n < tL/(V - PW). there may be a single monopolist or several local monopolists serving part of the market.11 Therefore. R : minIL/a. (v - P)/t) for those cases. We will study each of these situations below. Case 1. The single monopolist serves the entire market when n = 1 l tL/(v - PW). In this case R : L/a and P S v - tL/a. This is the case where the prevailing retail price is so low that the single monopoly retailer serves the entire market. This retailer is a monopolist to the consumers and a monopsonist to the manufacturer. Since we have assumed one manufacturer in the industry of good x. this case becomes a situation of bilateral monopoly. When R : L/a, by assumption (A6)’ we know that X : DL. 129 So. for any given PW. (4. 51) "r 3 (P ‘ P')X ' fr 3 (P ' P')DL " fr WhiCh is a monotonically increasing function Of P. (as our/6P = DL > 0) When P s v - tL/a. Therefore. for the purpose of up maximization. the retailer Will charge P'I to its upper limit. That is. (4.52) P” : V - tL/2. With.P' : v - tL/a. the retailer’s profit level is (4.53) ”r” = (v - tL/a - Pw)DL - fr. The Wholesaler's profit level. With X = DL. becomes (4.64) "m” : (Pw - c)DL - fm Which is a monotonically increasing function of Pw. Also. (4.66) n“ = (DL)3(v - tL/a - Pw)(Pw - c) If P, is determined unilaterally by the monopoly wholesaler to maximize Wm”. then. as owm'/0Pw : DL > 0. the wholesaler Will charge PW as high as possible. If PW is determined unilaterally by the monopsony retailer to maximize up”. then PW” : 0. The wholesale price levels preferred by both players are different. 80. both players Will bargain over the Wholesale price in this bilateral monopoly case. The Nash bargaining equilibrium wholesale price is the PV! Which maximizes Q”. The first order conditions to 130 maximize Q“ is (4.66) (am/6Pw (DL)2[(v - tL/a - PW) - (Pw - c)] (DL)3(v - tL/a - 2Pw + c) = 0. The solution for equation (4.56) is (4.57) PW” : (1/2)(v - tL/a + c) (4.56) "r” : (1/a)(v - tL/a - c)DL - fr. and (4.59) "m” : (1/2)(v - tL/a - c)DL - fm Equations (4.56) and (4.59) show that. when fr = fm. both the wholesaler and the retailer have the same level of profits. This implies that the retailer's strategic advantage is insignificant when the demand is completely inelastic and the retailer serves the entire market. It is because the quantities demanded is the same regardless of the Wholesale price. Equation (4.5?) shows that the Wholesale price is a function of every parameter. except for the population density, D, and fixed costs. fr and fm. Also. to assure that the players earn non-negative profits. it is necessary that the wholesaler's marginal cost of production. C. is smaller than v - tL/a. The comparative static predictions for this case are (4.60) 6Pw”/6v = 1/2 < 1 : 6P*/6v (4.61) 0Pw*/6t : -L/4 > -L/2 = 6P*/6t (4.62) 0Pw”/0L : -t/4 > -t/2 = 6P*/6L (4.63) 6Pw*/oc : 1/2 > 0 : 0P*/6c (4.64) owr'/6v = 6wm*/6v = 1/2 > 0 (4.66) owr*/6t = 6wm*/6t : -L/4 < 0 131 (4.66) Owr*/OL = awm*/6L : (1/2)D(v - tL - c) > 0 (4.67) 6wr*/6c = 6wm*/6c : -DL/2 < 0 (4.66) owr*/an = 6wm*/60 = (1/2)(v - tL/Z - c)L > 0 Equations (4.60) to (4.66) show that. in general. the spatial factors and those parameters on the demand side have the positive effects on the equilibrium Wholesale and retail prices.12 Also. they have positive effects on the profit levels. Contrarily. the cost-side parameters have negative effects on the profits. Among them the fixed costs have no effect on the equilibrium prices while the marginal cost of production. c. has a positive impact on the wholesale price. Case 2. All the retailers are local monopolists when tL/(v - P") > n z 1. In this case R : (v - P)/t and v > P > V - (tL/2). This is the case Where some of the consumers may not be served by the monopoly retailer(s) because the prevailing retail price is high. Furthermore. in this case. there may be a single monopolist if n : 1 < tL/(v - PW) or several local monopolists if 1 < n < tL/(v - PW). Substituting R : (v - P)/t into assumption (A6)', we get (4.69) X = (2D/t)(V - P). For any given PW. to maximize up”. the retailer will choose 132 (4.70) P” = (1/2)(v + P”). So. (4.71) R“ : (1/2)(v - Pw)/t (4.72) x' : (D/t)(v - P"). Then. (4.73) up” : (1/2)(D/t)(v - Pw)3 - fr (4. 74) 17m. 2 (rib/t)?” " C) (V " P") ‘ fm (4.76) n'l = (1/2)(nD3/t3)(Pw - c)(v - Pw)3. If PW is determined unilaterally by the monopoly Wholesaler to maximize um”. then P,” = (v + c)/2. If PW is determined unilaterally by the retailers to maximize up“. then P.” : 0. The wholesale prices preferred by the players of both sides are different. 80. players on both sides will bargain over the Wholesale price. The flash bargaining equilibrium solution of PW is the one Which maximizes 0* in equation (4.75). It is (4.76) P,” : (1/4)(v + 3c). So. (4.77) P” : (1/8)(5V + 3C) (4.76) w,” (9/32)(D/t)(v - c)a - fr (4.79) um” (3/16)(nD/t)(v - c)2 - fm. Equations (4.76) and (4.79) show that Wm” : (2/3)nwr*, if fr : fm_: 0. That is. the monopoly wholesaler will benefit from the increase in the number of retailer. Also. the monopoly Wholesaler earns a bigger profit than any individual retailer does. when all the retailers are local ‘monopolists and When n > 1. However. when n = 1. the 133 monopoly Wholesaler earns less profits than the monopoly retailer in the bilateral monopoly situation. Again. this implies that the retailer has strategic advantage in bargaining. Comparing this bilateral monopoly case With the previous one. we find that the retailer's profits in this case is relatively larger. (That is ”r” : (3/2)wm' in this case While vr' : um” in the previous case.) This situation is the same as that When the consumer demand is linear. and the intuition behind.this phenomenon is same as before. Also. equations (4.76) and (4.77) show that the equilibrium Wholesale price and retail price are not affected by the spatial factors. This phenomenon is the same as when the demand function is linear. That is. as long as the retailers are local monopolists. the spatial factors have no impact on the equilibrium prices. no matter Which.of the two types of demand function is used. Furthermore. equation (4.77) implies that the marginal cost of production. c. must satisfy c > v - (4/3)tL. to make the equilibrium retail Price higher than v - tL/2. The comparative static predictions for the case where 1 s n < tL/(v - P") are as below: (4.60) OPw'/ov 1/4 < 5/6 6P*/6v 3/4 > 3/8 (4.61) 6Pw'/oc 6P'/6c (4.62) owr*/6v (9/16)(D/t)(V - C) > 0 134 (4.63) owr'/6c -(9/16)(D/t)(v - c) < 0 (9/32)(1/t)(v - c)2 > 0 (4.64) owr'/0D (4.66) 6wr*/0t -(9/32)(D/t3)(v - c)2 < 0 (4.66) awm*/6z (2n/3)(owr‘/OZ) for all 2 : v. c. D. t (4.67) owm'/6n = (3/16)(D/t)(v - c)2 > 0 The comparative static predictions shown in equations (4. 60) to (4. 67) are qualitatively the same as those when the demand function is linear. and thus need not be discussed again. (B). retailers are not local monopolists When n > maxti. tL/(v - Pw)). the retailers are not local monopolists because the number of retailers is large relative to the market size. With a Cournot-Hash behaVioral assumption. there exists a symmetric equilibrium. in which every retailer’s market area is determined by (4.88) R” : (1/2)(L/n). Substituting equation (4.66) into assumption (A6)'. we know that X : DL/n. Also for any given PW. every retailer maximizes its profit by choOSing the retail price. P. as14 (4.69) P“ = P, + tL/n. then (4.90) "r“ tDLa/na - fr (4.91) um“ (Pw - c)DL - fm (4.92) n” = (tD3L3/n3)(Pw - c). Equation (4.69) has an important implication. It shows 135 that in this Bertrand type price—setting game. the equilibrium retail price can be higher than the retailer's marginal cost. (that is. Pw). When.the retailers have the market power arising from the spatial factors. Also. as the retail Price is a mark-up 0f the wholesale price. the retailer's profits. wr'. are not a function of P". Furthermore. am” is a monotonically increasing function of P'. (because ovm'VOPw = DL > 0) according to equation (4.91). These special features show that. with.a completely inelastic demand function. When the retailers are not local monopolists. they will accept whatever price charged by the Wholesaler. So. the Wholesaler Will charge the Wholesale price as high as possible. The only limitation is that the Wholesale price may not cause the retail price over its upper limit. When the retail price is:g_ggrk-up of the Wholegglg ppice. there nggin fact. no bargaining over tpe wholeggig ppigg. The equilibrium wholesale price is the one which makes the retail price very close to its upper limit. That is. the equilibrium retail price will be (4.93) P” = v - (1/2)(tL/n) - 6 Where 6 is a very small positive number. Trivially. we omit 6. So. the equilibrium retail price is (4.94) PM : v - (1/2)(tL/n). and (4.96) Pw'* : v - (3/2)(tL/n). so (4.96) wm'* = DL[v - (3/2)(tL/n) - c] - fm. (4.97) "r”' = 6r” : tDLa/na — fr. 136 Equations (4.94) and (4.95) show that the equilibrium Wholesale and retail prices are at their upper limits. Also. every demand-side parameter (including the spatial factor). except for the population density. D. affects the equilibrium prices. But the cost-side parameters have no impact on the equilibrium prices. Furthermore. to assure that the Wholesaler's profits is non-negative. it is necessary that the marginal cost of production. c. satisfy C S (3/2)(tL/n). The comparative Static predictions for this case are (4.96) 6Pw**/6v 1 : 6P*'/6v (4.99) on"/0t -(3/2)(L/n)< -(1/2)(L/n) : 6P**/at (4.100) 0Pw'*/6L -(3/2)(t/n)< -(1/2)(t/n) = 0P**/6L (4.101) 0Pw“*/6n (3/2)(tL/n3) > (1/2)(tL/n3) = 6P**/0n (4.102) owr"/ov 0 < DL : 6wm*'/ov (4.103) our'*/6t DLa/na > 0 > -(3/2)(DL3/n) = 6wm**/6t (4.104) onr*'/6L 2tDL/na n D[v - 3(tL/n) - c] owm”*/6L (4.106) aflr"/0c 0 > 'DL 3 811m”*/0C (4.106) owr'”/0D tLa/na 4 L[v - (3/2)(tL/n) - c] owm"/0D fl! - - I! (4.107) our /0fr - -1 < 0 - arm /0fm (4.106) owr**/an -2(tDL3/n3) < 0 < (3/2)(tDL3/n3) owm*'/6n Some of the comparative static results are special. 137 Only the consumer’s evaluation on the good. v. has a positive effect on the equilibrium wholesale and retail prices. Spatial factors t and L. have negative effects on equilibrium prices because they have negative effects on the consumer's demand. Also. as in the linear demand case. an increase in the number of retailers causes the equilibrium Wholesale and retail prices to rise.15 when the retailers are not local monopolists. We have explained the reason of this phenomenon in Section III. VI. The Long-Run Egpilibrium at the Retailing Level In the preVious sections, we studied short-run equilibria by assuming that the number of firms is fixed. In this section. we assume that new firms may freely enter the market if the short—run equilibrium profit level is positive and if their expected profits after the entries are nonnegative. Also. the existing firms may freely leave the industry when the profit level in the short—run equilibrium is negative. Therefore. ”a market is defined to be in (long-run) equilibrium when no firm currently in the market wishes to exit the market and no potential entrant wiShes to enter."16 For simpliCity. we assume that the entry and exit happen at the retailing level only. (That is. there is always a monopoly Wholesaler in the wholesale level.) Also. we assume that the retailers are always evenly located on 138 the market after the entries and exits. With the assumptions stated above and the results from preVious sections. we are going to study. under What condition. a short-run equilibrium may become a long—run equilibrium. The analyses are as below. (I). Cases of Linear Demand Function We know that. in a short-run equilibrium. if the number of retailers is small relative to the size of market. (that is. n 1 max(1. 3th/[4(a - wa)]l). there is either a single monopoly retailer serving the entire market With n : 1 l 3th/[4(a - wa)]l. or all the retailers are local monopolists with 1 1 n < 3th/[4(a - Pw)]. Also. when the number of retailers is large relative to the market. (that is n > maxti. 3th/[4(a - wa)])). the retailers are not local monopolists in the short-run equilibrium. Case A. In the short-run equilibrium. the number 0f retailers is small relative to the size 0f market. That is. 1 s max(1. 3th/[4(a - wa)]). 1. The case where there is a single monopoly retailer serv1ng the entire market in the short-run equilibrium When n = 1 z 3th/[4(a - wa)]. A necessary condition for this equilibrium is C i (a/b) - (11/12)tL. This short-run equilibrium is characterized by 139 (4.7) PW” = (1/4)(a/b - tL/4 + 3c) (4.6) P” = (1/8)(5a/b - 5tL/4 + 3c) (4.9) "r“ = (9/64)bDL(a/b - tL/4 - c)3 - fr (4.10) em“ = (3/32)bDL(a/b - tL/4 - c)a - fm. If the existing single monopoly retailer earns zero profits in the short-run. that is fr : (9/64)bDL(a/b - tL/4 - c)3. then there is no incentive for any new firm to enter the market. This short-run equilibrium is also the long-run equilibrium. That is. in the long-run. there is a single monopolist in the retailing level. If the existing retailer earns positive profits, that is. fr < (9/64)bDL(a/b - tL/4 - c)3. then there is an incentive for new firms to enter the market. However. it is not guaranteed that new firms will surely enter the mabket. Instead. With complete information. potential entrants Will figure out if the profit level is positive after their entry. If the retailer’s fixed cost is pp; lower than a certain level.17 the profit level in the equilibrium With new entrants will be negative. Then. no new firm will enter the market. Therefore. it is possible that. in.the long- run. there is a single monopoly retailer serv1ng the Whole market and earning positive profits. Only when the profit level in the non-monopoly equilibrium is non-negative. it is possible that new firms enter the market. Then. in the long-run equilibrium. n > 1 2 3th/[4(a - wa)] and all the retailers are not local 140 monopolists. However. let us remark that the existing firm may try to deter18 the new comer(s) if its total expected discounted profits of being a single monopolist is higher than that of being a non-monopolist. This may be an interesting extension for the future study. 2. The case Where all the retailers are local monopolists in the short-run equilibrium when 1 S n < 3th/[4(a - wa)]. A necessary condition for this equilibrium is c > a/b - (9/10)tL. This short-run equilibrium is characterized by (4.31) Pw' = (1/5)(a/b + 50) (4.32) P” = (1/9)(4a/b + 50) (4.33) fir” (125)/1456)(bD/t)(a/b - c)3 - fr (4.34) am” (25)/466)(an/t)(a/b - c)3 - fm. By the same logic. we summarize the discussion for the long-run equilibrium in regard to this case as below: 1). If fr = (125/1456)(bD/t)(a/b - c)3. (that is wr* : 0 in the short-run equilibrium), there is no incentive for potential entrants to enter the market. This short-run equilibrium is also the long-run equilibrium. That is. in the long-run equilibrium. all the retailers are local mbnopolists with 1 1 n < 3th/[4(a - wa)] and some of the consumers may not be served. 141 11). Even though fr < (125/1456)(bD/t)(a/b - c)3, (that is. fir” > 0 in the short-run equilibrium). this short-run equilibrium may also be the long-run equilibrium. That is. if fr is pp; too smalli9. then up can not be non-negative in the equilibrium With n > 3th/[4(a - wa)]. Therefore. it is possible that all tag local monopolists earn positive profits in the long-run egpilibrium but they do not serve the ent ire marke t . iii). When fr is so small that the profit level in the equilibrium with n > 3th/[4(a - wa)] is non-negative. new firms will enter the market. In the long run. all the retailers are not local monopolists and the number of firms is n > 3th/[4(a - th)). Case B. In the short-run equilibrium. the number of retailers is large relative to the size of market. That is. n > maxti. 3th/[4(a - wa)]3. For the case where retailers are not local monopolists in the short-run equilibrium. the same logic applies. If they earn non-negative profits. this short-run equilibrium is also the long-run equilibrium. Otherwise. some of the existing firms will exit from the market and make the long- run equilibrium characterized by either that all the retailers are local monopolists with some consumers not being served or that there is a single monopolist serving the entire market. The actual outcome depends on the value 142 of fr° (II). Cases of Completely Inelastic Demand Function We know that. in a short-run equilibrium. if the number of retailers is small relative to the size of market. (that is. n 1 maxti. tL/(v - Pw)). there is either a single monopoly retailer serv1ng the entire market Where n = 1 l 3th/4(a - wa)]. or all the retailers are local monopolists with 1 1 n < tL/(v - PW). Also. when the number of retailers is large relative to the market. (that is. n > maxti. tL/(v - Pw)l). all the retailers are not local monopolists in the short-run equilibrium. Case A. In the short-run equilibrium. the number of retailers is small relative to the size of market. That is. n S maxii. tL/(v - Pw)). 1. The case where there is a single monopoly retailer serVing the entire market in the short-run equilibrium when n = 1 z tL/(v - PW). A necessary condition for this equilibrium is C S V - tL/2. This short-run equilibrium is characterized by (4.52) P” : v - tL/2 (1/2)(V - tL/2 + C) (4.67) Pw' (4.56) up” (1/2)(v - tL/a - c)DL - fr 143 (4.59) w * : (1/2)(v - tL/2 - c)DL — f . m m By the same logic as in the linear demand cases. we summarize the discussion for the long-run equilibrium as below: i). If fr : (1/2)(v - tL/a - c)DL. (that is. hr” : 0 in the short-run). there is no incentive for potential entrants to enter the market. This short-run equilibrium is also the long-run equilibrium. That is. in the long-run equilibrium. there is a single monopoly retailer serv1ng the entire market. ii). If fr < (1/2)(v - tL/a — c)DL. (that is "r” > 0 in the short-run). but fr > tDLa/na. (that is “r” < 0 in the equilibrium With new entrants). then no new-comer enters this market. Therefore. it is possible that there is a single monopoly retailer serv1ng the whole market and earning positive profits in the long-run equilibrium. iii). When fr 1 tDLa/na < (1/2)(v - tL/2 - c). new firms will enter the market since fir“ z 0 after their entries. Therefore. the long-run equilibrium will be characterized by the Situation that all the retailers are not local monopolists With n > 1 Z tL/(v - PW)' 2. The case Where all the retailers are local monopolists in the short-run equilibrium when tL/(v - Pw) z n 2 1. A necessary condition for this equilibrium is 144 c > v - (4/3)tL. This short-run equilibrium is characterized by (4.76) PW” = (1/4)(v + 30) (4.77) P” : (1/8)(5V + 3C) (4.76) w,* (9/32)(D/t)(v - c)2 - fr (4.79) Wm” (3/16)(nD/t)(v - c)a - fm. By the same logic as in the linear demand cases. we summarize the discussion for the long-run equilibrium as below: i). If f, = (9/32)(D/t)(v - c)3. (that is. fir” : 0 in the short-run). there is no incentive for potential entrants to enter the market. This short-run equilibrium is also the long-run equilibrium. That is. in the long-run equilibrium. all the retailers are local monopolists earning zero profits and serVing part Of the market. 11). If fr < (9/32)(D/t)(v - c)3. (that is up“ > 0 in the short-run). but fr > tDLa/na. (that is up“ < 0 in the equilibrium with.new entrants). then no new comer enters this market. Therefore. it is possible that all the local monopolists earnapoaitiveaprofita in the long-run eguilibrium but they do not serve the entire market. iii). When fr 1 tDLa/n2 < (9/32)(D/t)(v - c)3. new firms Will enter the market since up” 2 0 after their entries. Therefore. the long-run equilibrium Will be 145 characterized by the Situation that all the retailers are not local monopolists with n > tL/(v - Pw). Case B. In the short-run equilibrium the number of retailers is large relative to the Size 0f market. That is. n > maxil. tL/(v - Pw)). A necessary condition for this equilibrium is C < V - (3/2)(tL/n). This short-run equilibrium is characterized by (4.94) P** : v - (1/2)(tL/n) (4.95) Pw”* v - (3/2)(tL/n) (4.96) am DL[v - (3/2)(tL/n) - c] - fm (4.97) tDLa/ne - fr. 3 5 a m The discussion on the long-run equilibrium in this case is qualitatively the same as that in previous cases. That is. when fr : tDLa/na this short-run equilibrium is also the long-run equilibrium. If fr > tDLa/na. some of the firms will exit from the market and make the long-run equilibrium characterized by either a single monopolist serv1ng the entire market or several local monopolists serving part of the market With 1 1 n < tL/(v — Pw). The actual outcome will depend on the value of fr- VII. Conclusions In this section. we summarize the most important 146 findings of this chapter as follows. i). In a bilateral monopoly bargaining game With spatial factors incorporated into the analysis. the monopsony buyer (that is. the retailer) earns more profits than the monopoly seller (that is. the wholesaler). The only exception occurs when the consumer demand is completely inelastic and the retailer serves the entire market. In that situation both players earn the same level of profits. (Here. we assume fixed costs are the same for both players.) This phenomena illustrates that when the demand for the wholesaler’s product is a derived demand. the retailer has strategic advantage in bargaining since it chooses quantity. X. This advantage becomes insignificant when the demand is completely inelastic and the retailer serves the entire market. Since X is going to be the same regardless of PW. 11). Similar to theories in non-spatial microeconomics. an increase in the gagginal cost of production causes the equilibrium prices to rise and the profit levels to fall.20 This statement holds for every case except for the case Where retailers are not local monopolists while the demand function is completely inelastic. In that case the marginal cost of production has no effect on either the equilibrium Prices or the retailer's profit level. 111). Spatial factors do not influence the equilibrium wholesale or retail price when all the retailers are local monopolists. In all the other cases. the spatial factors. 147 (except for the population density. D.). have influences on the equilibrium prices. iv). When the demand for the Wholesaler’s product is a derived demand and the retailer’s marginal cost (i.e. the wholesale price) is determined through the bargaining process. the equilibrium retail price in a Bertrand type price-setting game (that is. the equilibrium retail price in the case Where the retailers are not local monopolists) is higher than retailer's marginal cost. This result is different from that of the traditional price—setting game in which the marginal cost is exogeneously given and the equilibrium price is equal to firm's marginal COSt. In Table I. we show the sign patterns of comparative static predictions for all cases21 studied in this chapter. In the next chapter. we Will study the models in which the monopoly Wholesaler cooperates With all the retailers to maximize their sum of profits. However. we do not assume that the retailers cooperate with each other at the retailing level. The sum of all firms' profits in this cooperative game Will never be less than that in the bargaining game. How to diVide the pie to each player in this game becomes an important issue. If there is any player who earns less profits in this cooperative game. he will prefer going back to the bargaining game for higher profits. Therefore. we also construct a formula of side- 148 payment scheme to guarantee every Player in this game no worse off than in the bargaining game. L Table I: Cases Of small Number Of Retailers With Linear Demand comparative Static Results for Bargaining Cases Cases 1 max(1. 3th/[4(a - wa)]) n : 3th/[4(a - b? )1 3th/[4(a - b? )1 > n 1 Para- " w meters P w w P P w W W r 4m 1 r m a t” + +' + +*' D -II _ _ _l* _ _44' t _ _II _ o o _ _!l' D +* + o O + +*' L - +' + O O O 0 c +4 _Il _ +4 + _ -II’ f 0 - 0 0 0 - 0 r f 0 0 — 0 0 0 — m n O O 0 + 2a Cases of Small Number of Retailers With Completely Inelastic Demandi Cases 1 maxil. tL/(V - P“)) n = - P ) tL/(v - P ) > n z Para- V “ meters P w w P P w w W r m W r m v + + : + + +* + +" t - — : — o o _ -If’ D O + : + 0 O + +' L - + : + O 0 0 0 6 +4 _ = _ +4 + _ _II’ f 0 - 0 0 0 - 0 r f 0 0 - 0 0 0 - m n 0 0 0 + 149 150 2b. Cases of Large Number of Retailers With Completely Inelastic Demand Cases n > maxti. tL/(v - P )) Para- V meters P P w w aw r n1 v + : + O + t _I§ _ + __ D 0 0 + ? + L -"* - + ? ? c 0 0 0 - f 0 0 - 0 r f 0 0 0 - m n +* + - + Notes: Besides the directions of impacts on PW. P. fir. and Wm. we use some symbols indicating the relative impacts on PW and P (and on up and um) when they ear the same sign. The meanings of the symbols are as below: 1). u is associated With the one with greater positive impact. 11). In is associated with the one with greater absolate value of impact when the impacts are negative. 111). 4' or 44' indicates that if we compare Wm with nwr (the profit level for the entire retailing level) then the one with x' or 44' Will bear smaller (absolute) value of impact. (Otherwise. it bears a greater (absolute) value of impact.) iv). ... means no impact of n could be considered since n : 1 is fixed. V). When a ”z" mark stands between two columns of any row With same sign. it means they bear the same degree of impact. v1). When a ”?" mark stands between two columns of any row with the same sign. it means that we can not make any comparison unless some other conditions hold. 151 Footnotes of Chapter Four This concept is the same as that used by Salop [1979]. For example. the Robinson-Patman Act requires that all firms perceived in competition be treated equally. That is. the manufacturer can not offer a better price to one retailer other than another unless the differences can be justified in terms of cost variations. Assumption (A7) implies that the wholesaler charges a uniform price to all retailers. This condition is similar to that in Novshek [1960]. p.316. Hovshek's condition is simpler because he assumes the constant marginal cost being zero. In this thesis. the retailer's marginal cost is the wholesale price. PW. Which should be non-negative always. (In a trivial sense. it should be positive.) A detailed derivation of this condition (and the condition for other cases) is in Appendix A. . Because the number of retailers must be an integer, it is most likely that some of the consumers are not served by those local monopolists. We are indebted to Professor Paul Segerstrom for his initiation of this explanation. The proof of this equilibrium is similar to that in Novshek [1960]. pp. 322-325. In this thesis. there is a simplification that the locations of firms are assumed to be fixed, and there is a modification that the marginal COSt is positive rather than zero. 10. 11. 12. 13. 14. 15. 152 See Novshek [1980] p.316. Salop [1979] p.147 (with the same spirit but different notation). This concept and part of the analysis in this model are similar to those in Salop [1979]. However. we have made some modifications and extensions. The conditions shown in this section can be derived in the same manner as those in Appendix A. The only difference is the parameter a/b must be replaced by v during the process of calculation. See note 5. There are some exceptions. The population density. D. does not have any effect on the equilibrium prices. The market length. L. has negative effects on the equilibrium prices. See Salop [1979]. p.147. This is the same as equation (16) in Salop [1979]. p.147. but the notation is different. In nonspatial economics. an increase in the number of firms causes the equilibrium price to fall because of the increased competition. In this chapter where spatial factors are incorporated into the analysis. we have two different cases. In the case where all the retailers are local monopolists. an increase in the number of retailers does not affect the equilibrium wholesale or retail price as long as the retailers are still local monopolists. However. in the case where the retailers are not local monopolists, an increase in the number Of 16. 17. 18. 19. 20. 21. 153 firms causes the equilibrium prices to rise. See Greenhut. Norman and Hung [1967]. p.36. OWing to the computational complexity in Case B of Section IV. we do not know the exact value of "r which is non-negative when n > max(1. 3th/[4(a - wa)]l. Or. in other words. we do not know the exact value of P. Where "r = P - fr. Only when fr 1 F. it is possible that up i 0 in the equilibrium with n > maxt1. 3th/[4(a - wa)]l. Hilgrom and Roberts [1962] discussed the deterrence and entry with a non-spatial model. See note 17 except that the phrase "n > maxti. 3th/[4(a - wa)])" should be replaced as "n > maxti. tL/(v - PW)!" for this case. Some spatial economists. like Capozza and Attaran [1976] and Capozza and Van Order [1976]. found that under some situations cost increases (fixed. marginal. or transport) may result in lower equilibrium prices. Except for the case of large number of retailers With a linear demand function. Chapter Five: Wholesale Price Determination (III) - cooperative cases With side-payments - I. Introduction The only difference between the analytical frameworks of Chapters Four and Five is the assumption regarding the goal of firms. In this chapter. we assume that the goal of all firms (the monopoly wholesaler and the oligopolistic retailers) is to maximize the sum of their profits when they determination the wholesale price. All the other elements. such as the shape of the market. the types of the consumer demand function. the types of competition among the retailers. are the same in both chapters. The equilibrium Wholesale price levels. however. are quite different between the corresponding cases in these two chapters. as are the profit levels. The sum of profits in this chapter is higher than or equal to that in the previous chapter. However. the individual firm’s profit level in this cooperative game may not always be higher than if they bargain over the wholesale price. So. the side-payment scheme. which redistributes the total profits to guarantee everyone no worse-off. is important. The situation in which the wholesaler and the retailers cooperate is similar to the case of an integrated-firm where a manufacturing firm has many subsidiaries or sales units to distribute its goods. The models studied in this chapter. however. is different from the case Of an integrated-firm.‘ 154 155 First of all. we do not assume the wholesaler owns the retailing facilities. Besides. the retailers are noncooperative at the retailing level. The cooperation between the wholesaler and the retailers can be best described as below. Assume that there is a multi-national company which wants to introduce its product to a new market of a certain country. The company may ask some established firms of this country to be its retailers. The retailers. therefore. are not the subsidiaries or the sales units of this company. They may compete With each other at the retailing level. And. of course. they may choose to bargain With the Wholesaler over the wholesale price. However. the goal of maximizing the sum of profits places an upper bound of total profits under all other settings. The structure of this chapter is as below. Section II contains assumptions and notation for models with a linear demand function. Section III contains the analySis of those linear demand models diVided into three cases. Studies on the side-payment scheme are accompanied. Section IV contains assumptions and notation for models with a completely inelastic demand function. Section V contains the analysis of models with a completely inelastic demand function. In addition. studies on the side-payment scheme are contained in Sections III and V. Section VI contains the conclusions of this chapter. Furthermore. because the logic for the study Of long-run equilibrium in this chapter 156 is completely the same as that in Chapter Four. the discussion of long-run equilibrium Will not be in the text but in Appendix B of this thesis. II. Assumptions and Notation for the Linear Demand Case For readers’ convenience. we list all assumptions and notation as below. However. those who are familiar with the materials of Chapter Four may skip everything of this section except for assumption (A12). (A1) A single homogeneous good. (A2) A one dimensional market of length L shaped as the circumference of a circle. (A3) There are n symmetric retailers evenly located on the market. Neither entry nor exit occurs. The distance between any two neighboring retailers is L/n. Novshek [1960] shows. With appropriate conjecture concerning the response of competing firms. this can be an equilibrium pOSition if firms choose both the price and location as their decision variables. (A4) Consumers are symmetric and are uniformly distributed along the market With a density D. (A5) Each retailer charges a mill price to all consumers. (A6) Each consumer has the same demand function X a - b(P + tu) if a/b > P + tu; = 0. otherWise. where x is the quantity demanded by a consumer; P is the mill price charged by the retailer. 157 P l 0; t is the transport cost per mile regardless of direction; u is the distance between the customer and the retailer. u z 0; a. b are positive constants. (A7) One manufacturer produces 800d X. Owing to the legal (A8) restrictions. the wholesaler charges the same wholesale price. PW. to all retailers.a Furthermore. we assume no shipping costs for delivering goods from this monopoly wholesaler (or manufacturer) to the retailers. or equivalently. the costs are constant per unit and the same for each customer. This assumption is equivalent to assuming that the wholesaler is located at the center of the circular market. With constant shipping cost to each point on the circumference. The cost function for each retailer is or = wa + fr where Cr is the total costs of the retailer; P" is the wholesale price of good x. which is the retailer’s marginal cost. PW z 0; fr is the fixed costs to run a retailing store. X is the amount of good x sold by a retailer x 11 R R 2D] x(P. u)du = 2D] [a - b(P + tu)]du 0 0 2DR(a - bP - btR/2) 158 where R is the maximum distance of each side of the market a representative retailer can reach. R s L/an. Also. R is a function of P as R : (1/2t)(P’ - P + tL/n). where P’ is the mill price charged by a neighboring retailer. when the retailer is not a local monopolist. Or. R : mintL/a. (a/b — P)/t) when the retailer is a local monopolist. (A9) The profit function for each retailer is fir3PX‘Cr 2DR(P - Pw)(a - bP - btR/a) - fr 2 0 With PW given. the retailer chooses a retail price. P. to maXimize its profits. (A10) The cost function for the monopoly wholesaler is Cm = C0 + fm Where Cm is the total costs of the monopolist; c is the constant marginal cost of production. c l 0; fm is the fixed costs of production. fm l 0; Q is the quantity sold by the wholesaler. which is a derived demand from consumers’ demand for the retailers’ goods; 0 : nX. (A11) The profit function for the monopolist is 17m PwQ ‘ Cm 3 (PW " C)nX " fm 2nDR(Pw - c)(a - bP - btR/a) - fm 2 0 (A12) The goal of the monopoly wholesaler and all the 159 retailers is to choose a PW to maximize their sum of profits. Here. we denote w“ : nwr + Wm as their objective function. The wholesaler and the retailers cooperate in the wholesale price determination. However. the retailers are non-cooperative at the retailing level. According to Novshek [1960]. there are three possible types of competitive situations at the retailing level: a single monopolist. several local monopolists. and several retailers while they are not local monopolists. (A13) Complete information is assumed in this cooperative game. III. The Analysis Of the Linear Demand Case (A). the case of monopoly retailer(s) When n s maxli. 3th/[4(a - wa)]l.3 the number of retailers is small relative to the size of market. With n = 1 z 3th/[4(a - wa)]. there is a single monopoly retailer serving the entire market. For other situations where 1 S n < 3th/[4(a - wa)]. there might be a single monopoly retailer or several local monopolists serving part of the market. Therefore. R : miniL/e. (a/b - P)/tl for the cases mentioned above. We Will study all of them as below. Case 1. A single monopolist serves the entire market when n = 1 2 3th/[4(a - Pw)]. In this case R : L/2 and P S a/b - tL/2. 160 This is the case Where the prevailing retail price is so low that the single monopoly retailer serves the entire market. This retailer is a monopolist to the consumers and a monopsonist to the manufacturer. Since we have assumed one manufacturer in this industry. this case becomes a situation of bilateral monopoly. However. different from the bargaining case in Chapter Four. the bilateral monopolists cooperate in the wholesale price determination to maximize their sum of profits. This situation is similar to the case of an integrated firm with a manufacturing unit and a sales unit. Nevertheless. we do not assume that the manufacturing unit and the sales unit belong to the same owner. By substituting R : L/2 into assumption (A6). we get (5.1) X : bDL(a/b - tL/4 - P). For any given PW. to maximize fir in assumption (A9). the retailer will choose (5.2) P' = (1/2)(a/b - tL/4 + Pw). which.implies (5.3) "r” (1/4)bDL(a/b - tL/4 - P...)a - fr The sum of profits then Will be (5.5) w” : up” + um” (1/4)bDL(a/b - tL/4 - Pw)(a/b - tL/4 + P..., - 2c) - (fr + fm). Equation (5.5) is a function of PW. The first-order 161 condition to maximize the sum of profits. w”. is o1r*/0Pw : 0. That is (6.6) (1/4)bDL[(a/b - tL/4 - PW) - (a/b - tL/4 + P... - 2c)] : 0. The solution for equation (5.6) is (6.7) Pw' : c. So. (5.8) P” : (1/2)(a/b - tL/4 + c) (6.9) u,‘ = (1/4)bDL(a/b - tL/4 - c)a - fr (6.10) "m” = -fm (6.11) w“ = (1/4)bDL(a/b - tL/4 - c)a - (fr + fm). Equation (5.7) shows that the equilibrium wholesale price is egaal to the wholesaler’s marginal cost of production. With the wholesale price being equal to its marginal cost. the wholesaler runs a loss of its fixed costs. Unless there exist side-payments from the retailer to the wholesaler. this equilibrium wholesale price can only be justified for the case of the integrated firm. The equilibrium retail price shown in equation (5.6) implies that the wholesaler’s marginal cost of production. c. must satisfy 6 1 (a/b) - (3/4)tL to ensure that P” S (a/b) - tL/2. The sum of profitg. w”. in this cogpapative game is pagher than that in the bargaining game of Chapter Four. (That is. w” in equation (5.11) is greater than the sum of fir” in equation (4.9) and um” in equation (4.10).) With higher total profits. every player in this cooperative game 162 is possible to be better-off than in the bargaining game. if there exists a well-designed side—payment scheme. The side-payments. here. are the transfer of funds from the retailer to the Wholesaler. It is not just to make the Wholesaler earn a non-negative profits. Instead. it should make the Wholesaler earn a level of profits no less than that in the bargaining game. The negotiation for a side- payment scheme is. in fact. another bargaining game. With the assumption of complete information. both players know that they Will be back to the bargaining Situation. if they can not reach an agreement on diViding the cooperative total profits shown in equation (5.11). The threat-point payoffs to the players. hence. are the bargaining equilibrium profit levels shown as fir” and um” in equations (4.9) and (4.10). Let us rewrite the threat-point payoffs. With different notation. as below: (6.12) wrb (9/64)hDL(a/b - tL/4 - c)a - fr (5.13) am? (3/32)bDL(a/b - tL/4 - c)a - rm Where wrb is the retailer’s profits in the bargaining game: Wm? is the Wholesaler’s profits in the bargaining game. Also. let us denote a as the portion of total profits to the Wholesaler when the side-payment agreement has been reached and carried out. So. according to the Nash criterion. the equilibrium 163 With a side-payment scheme exists if there is an a” which maximizes (6.14) 08 : (an' - wmb)[(1 - a)w* — wrb]. The first-order condition to maximize equation (5.14) is dos/ad : 0. Which results in (5.15) a“ (1/2)t(w* - wrb + me)/w'1 (1/2)([(13/64)bDL(a/b - tL/4 - c)a - afml/[(1/4)bDL(a/b — tL/4 - c)a — (fr + fm)]}. Therefore. the side-payments from the retailer to the wholesaler. Sp. are (6.16) sp up“ - (1 - a*)w* : a'fl* _ "ml (13/126)hDL(a/h - tL/4 - c)3. The profit levels of both players. then. are (6.17) vr*' (19/126)hDL(a/b - tL/4 - c)a - fr (6.16) wm*’ (13/126)bDL(a/h - tL/4 - c)a - fm. As wa" > «Pb and wm" > WmP. every player is better- O'ff than if they bargain With each Other. The comparative static predictions for the case Where the single monopoly retailer serves the entire market are as below: (6.19) 6P*/0a (1/2)(1/h) > 0 (6.20) 0P'/6b -(1/2)(a/b2) < 0 (5.21) 6P*/0t -L/8 < 0 (6.22) 6P*/6L -t/8 < 0 164 (6.23) 6Pw*/6c : 1 > 1/2 : 6P*/6c (5.24) ow”/6a (1/2)DL(a/b - tL/4 - C) > 0 (5.25) ow*/ob -(1/4)DL(a/b - tL/4 - c)(a/b + tL/4 + C) < 0 (6.26) aw*/6t -(1/6)hDL3(a/b - tL/4 - c) < 0 (6.27) aw*/6D (1/4)bL(a/b - tL/4 - c)2 > 0 (5.26) 6w*/6L (1/4)bD(a/b - tL/4 - c)(a/b - 3tL/4 - C) > 0 (6.29) aw'/6c -(1/2)bDL(a/b - tL/4 - C) < 0 (5o 30) OW./°f1~ 3 "l 3 Oflr*/6fr (5.31) 0W”/0fm : -1 : OWm'/0fm (5.32) 6wr*’/az = (19/32)6w*/6z for z = a. b. c. D. t. L (5.33) 6wm*’/6z : (13/32)6w*/6z for z = a. b. c. D. t. L. Since the egpilibriam wholeaale price is equal to the wholesaler’s marginal cost of production. it is affected by this parameter only. Also. as the monopoly retailer serves all the consumers in the market. any changes in the consumer demand function cause the retailer to adjust its retail price. Therefore, every demand-side parameter. except for the population density. D. has an impact on the equiliprium retail price. The population density. D. is not in the indiVidual consumer’s demand function so it has no effect on the retail price. Equations (5.19) to (5.22) show that anything. which has a positive effect on the consumer demand. has a positive effect on the retail price and Vice versa. Also. equation (5.23) shows that an increase in the 165 marginal cost of production will cause both the equilibrium Wholesale and retail prices to rise. Furthermore. we find that the fixed costs have no effect on the equilibrium Prices. According to equations (5.25) to (5.32). thg sum 9f profitsI w” is affected by every parameter. Generally speaking. those parameters. Which.have negative effects on the individual’s demand function (such as b. t. L). have negative effects on the total profits. Also. every cost parameter (c. fr. fm) has a negative effect on the total profits. Furthermore. although the population density. D. is not in the individual consumer’s demand function. it has a poSitive effect on the total profits because it has a positive effect on the retailer’s total demand. Case 2. All the retailers are local monopolists when 3th/4[(a - wa)] > n z 1. In this case R = (1/t)(a/b - P) and a/b > P > a/b - tL/2. This is the case where some of the consumers may not be served by the monopoly retailer(S) because the prevailing retail price is high. Furthermore. in this case. there may be a Single monopolist if n : 1 < 3th/[4(a - wa)]. or several local monopolists if 1 < n < 3th/[4(a - wa)]. Substituting R : (1/t)(a/b - P) into assumption (A6). we get (6.34) x = D(a - hP)3/ht. 166 For any given PW. every retailer maximizes its profits by choosing P” as (5.35) P” (1/3)(a/b) + an/B. then (5.36) R” (2/3)(a/b - Pw)/t. Therefore. the profit levels will be (6.37) er” : (4/27)(bD/t)(a/b - Pw)3 - fr (5.36) um” = (4/9)(nhD/t)(Pw - c)(a/b - Pw)3 - fm (5.39) v” = nwr” + Wm” (4/27)(an/t)(a/b - Pw)3(a/b + 2p... - 3c) - (nfr + fm). Equation (5.39) is a function of Pw and the first-order condition of w‘ maximization is 6w'/6Pw = 0. which implies (5.40) (a/b - P..)(Pw - c) = 0. With a/b > P 2 PW”. the solution for equation (5.41) is (5.41) PW” = c. then (5.42) P“ : (1/3)(a/b) + (2/3)c (6.43) wr' (4/27)(bD/t)(a/b - c)3 I H) r, (5.44) 17m” 3 ’fm < 0 (5.45) w” : nwr* + um” (4/27)(an/t)(a/b - c)3 - (nfr + fm). Equation (5.42) shows that the equilibrium wholesale price iapequal to the gagginal cost of proguction. This equilibrium wholesale price is the same as that in the monopoly retailer case. Furthermore. from equation (5.45). we know that a side-payment scheme is necessary. OtherWise. the wholesaler will not agree with PW” : c since this 167 wholesale price causes the monopoly wholesaler to run a loss Of its fixed costs. When the wholesaler cooperates With all the retailers. the sum of profits. w”. is higher than if the Wholesaler bargains With the retailers. (That is. v” in equation (5.45) is greater than the sum of nwr* in equation (4.33) and Wm” in equation (4.34)). However. the equilibrium of this cooperative game can not be sustained if there is no such side-payment scheme that makes the wholesaler no worse- off than if it bargains with the retailers over the wholesale price. Beside the side-payment scheme. the other necessary condition to make this equilibrium sustainable is the marginal cost of production must satisfy c > (a/b) - (3/4)tL. It is because the equilibrium retail price P” shown in equation (5.43) must be greater than (a/b) - (tL/2) for this case. The negotiation for a side-payment scheme. in fact. is another bargaining game. With the same logic as before. we know that the threat-point payoffs to each retailer and the wholesaler are the profit levels shown in equations (4.33) and (4.34) separately. Let us rewrite the threat-point payoffs. With different notation. as below: (5.46) wrb (125/1458)(bD/t)(a/b - c)3 — fr (6.47) wmb (25/466)(an/t)(a/b - c)3 — fm where wrb is the retailer’s profit level in the 168 bargaining game and um? is the wholesaler’s profit level in the bargaining game. Again. let us denote d as the portion Of total profits to the Wholesaler When the side-payment scheme has been agreed and carried out. So. according to the Nash criterion. the equilibrium with a side-payment scheme exists if there is an a” which maximizes (5.46) 03 = n(dw* - me)[(1 - a)w'/n - wrb). The first-order condition of 03 maximization is ans/ad : 0. Which results in (5.49) a” (1/2)[(w' - nwrb + um?)/w*] (1/8)l[(83/729)(an/t)(a/b - c)3 - 2fmJ/[(4/27)(an/t)(a/b - c)3 - (n1, + fm)]!. Therefore. the side-payments from.the retailers to the (5.60) sp nwr* - (1 - d*)w* II 5 (117 ’1'!“ (63/1456)(an/t)(a/b - c)3. The profit levels of each retailer and the wholesaler. then. are (5.61) ur*' (133/1456)(bD/t)(a/b - c)3 - fr (5.63) wm*' (63/1456)(an/t)(a/b - c)3 - fm. As wr" > if” and Wm”, > me. every player is better- 169 Off than if they bargain over the wholesale Price. The comparative static predictions are as below: (5.53) 6P'/0a (1/3)(1/b) > 0 (6.54) 6P*/6h -(1/3)(a/b3) < 0 (5.55) 6Pw'/6c 1 < 2/3 = 6P*/6c (5.56) 6v'/6a (4/9)(D/t)(a/b - c)2 > 0 (6.67) aw'/Oh -(4/27)(D/t)(a/b — c)3(c + 2a/b) < 0 (5.56) 6w*/6t -(4/27)(hD/t3)(a/h - c)3 < 0 (5.59) 0w”/0D (4/27)(h/t)(a/b - c)3 > 0 (5.60) aw'/0c -(4/9)(bD/t)(a/b - c)2 < 0 (5.61) ow'/0fr -n < 0 (6.62) 6w'/6n (4/27)(hD/t)(a/h - c)3 > 0 (5.63) aw*/6fm -1 < 0 (5.64) 6w,*’/6z : (133/216)(6w*/62)/n. for z = a. b. c. D. t (6.65) awm*’/6z (83/216)(6W'/OZ). for Z : a. b. C. D. t. As the equilibrium wholesale price is equal to the wholesaler’s marginal cost of production. it is affected by this parameter only. Also. when all the retailers are local monopolists. it is not necessary that all the consumers are served. so the spatial factors have no influence on the equilibrium retail price. From equations (5. 53) and (5.54). we know that the equilibrium retail price will be higher When there is a lump-sum increase in the consumer demand or when the slope of consumer demand curve is steeper. Also. from equation (5.55). we know that the equilibrium wholesale 170 and retail price Will rise when the marginal cost of production increases. The smm of profits. n”. is a function of every parameter. except for the market size. L. in this model. Generally speaking. those parameters which have positive effects on the indiv1dual consumer’s demand function have positive effects on the total profits and vice versa. Also. every cost parameter (c. fr.-fm)1has a negative effect on the total profits. Furthermore. as long as the number of retailers is kept small enough. (so that every retailer is a local monopolist). changes in the number of retailers have no effect on the equilibrium prices but have a positive effect on the total profits.' (This is expressed in equation (5.62).) (B). The case where the retailers are not local monopolists When the number of retailers. n. is greater than maxti. 3th/[4(a - wa)]l. the retailers are not local monopolists. It is because the number of retailers is large relative to the market. With the Cournot-Nash noncooperative behavioral assumption. there exists a unique. symmetric equilibrium4 in which every retailer’s market area5 is determined by (5.66) R” : (1/2)(L/n). For any given PW. every retailer maximizes its profits by choosing the retail price as (5.67) P” : (1/2)(a/b + Pw + (3/2)(tL/n) - [(a/h - Pw - 171 (1/2)tL/n)2 + 3(tL/n)3]1/31. then (5.66) "r” : (th/6)(L/n)a((a/b - Pw) - (25/2)(tL/n) + 7[(a/b - Pw - (1/2)tL/n)a + 3(tL/n)311/31 - fr (5.69) um” : (bDL/2)(Pw - c)l(a/b - PW) - (2tL/n) + [(a/b - Pw - (1/2)tL/n)a + 3(tL/n)211/3) - fm. The objective function. V”. is (5.70) w“ nwr* + um” (hot/6)(L3/n)1(a/b - Pw) - (25/2)(tL/n) + 7[(a/b - Pw - (1/2)tL/n)a + 3(tL/n)3]1/3) + (hDL/2)(Pw — c)((a/b - Pw) - (2tL/n) + [(a/b - Pw - (1/2)tL/n)a + 3(tL/n)311/31 - (fr + fm). The first-order condition Of maXimiZing 11" in equation (5.70) is (6.71) 61r'/6Pw = 0. WhiCh implies that (6.72) (tL/4n)[-1 — 7[(a/b - Pw - tL/2n)2 + 3(tL/n)3)('1/3) (a/b - PW - tL/2n)l + ((a/b - Pw) - 2(tL/n) + [(a/b - Pw - tL/2n)2 + 3(tL/n)3)(1/3)3 + (Pw - c)(-1 - [(a/b - Pw - tL/2n)3 + 3(tL/n)3](-1/3)(a/b - Pw - tL/2n)3 = 0. Equation (5.72) shows that Pw is a function of all parameters except for the population denSity. D. However. it is algebraically too complicated to solve PW. Furthermore. the solution of Pw will be lengthy and with complicated feature. which can not give us any information in economic sense. Therefore. we made a computer simulation 172 by setting a = b = t = D = L = 1 and fr : fm = 0 to check the equilibrium values of PW. P. wr. um. and w” when the values of c and n are changed. We have some interesting results as below: i). 1 > 6Pw/6c > 0. 1 > 6P/6c > 0. and OPW/ac > oP/oc with the same initial value of c. Also. oX/ac < 0. OQ/oc < 0. our/ac < 0. cum/ac < 0. and ow*/00 < 0. An increase in.the marginal cost of production will cause the equilibrium Wholesale and retail prices to rise. However. the rise in the wholesale price is less than the rise in the marginal cost of production. 80. the wholesaler’s profit margin falls. By the same logic. the retailer’s profits fall because the rise in the retail price is less than the rise in the wholesale price. Furthermore. the rise in the price causes the quantities demanded to fall. Therefore. the profit levels (each retailer’s profits. the monopoly wholesaler’s profits. and the sum of all firms’ profits) fall When the marginal cost of production increases. 11). on/On > 0. 6P/6n > 0 and OPW/On > 0P/6n With the same initial value of n. Also. OX/On < 0. OO/On > 0, Our/6n < 0. owm/on > 0. and ow'/6n > 0. An increase in the number of retailers makes the market area of each retailer smaller. So. the total costs of transportation paid by the farthest consumer is reduced. 173 This gives the retailer and.the wholesaler an opportunity to raise their prices. Therefore. an increase in the number of retailers increases the equilibrium wholesale and retail prices. It is different from the previous cases in this chapter. where the number of retailers has no effect on the equilibrium prices. Also. the retailer’s profit level is adversely affected by the increase of the number of retailers because the quantities demanded for each retailer’s goods fall and the rise of the retail price is smaller than the rise of the wholesale price. Contrarily. an increase in the number of retailers increases the wholesaler’s profits since it increases both the quantities demanded for the wholesaler’s goods and the equilibrium wholesale price. The smm of profits. w'. increases When the number of retailers increases implying that the increase in the wholesaler’s profits outweighs the decreases in the retailers’ profits. Note that the above Statement holds because we have assumed iii). wrb > up”. am? < um” and ab < v”. where wrb. wmb, and uh are the retailer’s profits. wholesaler’s profits and the sum of profits separately. when the wholesaler bargains with the retailers over the Wholesale price. The results above show that When the wholesaler cooperates With the retailers in the wholesale Price 174 determination. the sum of profits is higher than if the wholesaler bargains With the retailers over the wholesale price. Also. the Wholesaler earns a positive profit because the equilibrium wholesale price is not equal to but higher than.the Wholesaler’s marginal cost. The reason why the wholesale price. PW. is not chosen to be the wholesaler’s marginal cost. c. in this case is that PW is used to force the retailers not to compete With one another such that the total profits can be maximized. High.enough.P' forces high P and hence the retailers become almost local monopolists. Furthermore. the Wholesaler’s profit level is higher in this cooperative game than in the bargaining game. while the retailer’s profit level is lower than in the bargaining game. This phenomenon is opposite to that in the preVious cases. In the cases where the retailers are local monopolists. the retailers are better-off in the cooperative game than in the bargaining game. Without side—payments from the wholesaler to the retailers. the retailers Will not cooperate with the wholesaler. The side-payment scheme which we need is a plan to assure every retailer a profit level no less than that in the bargaining equilibrium. Similar to the preVious cases. the negotiation for a side-payment scheme is another bargaining game. The threat- point payoffs to each retailer and the wholesaler are wrb and Wm? separately. 80. according to the Nash criterion. 175 the equilibrium with the side-payments exists if there exists an a which maximize (5.73) as = n(dw* - nmb)[(1 - a)w*/n - wrb) where a is the portion of total profits to the wholesaler. The first-order condition of 03 maximization is oQS/od : 0. which results in (5.74) a” : (1/2)[(w* - nwrb + me)/w*]. So. the side-payments from the wholesaler to each retailer are (5.75) Sp (wm* - dw*)/n (1/2)[(wm* - wmb) + n(wrb - wr*)]/n. From the simulation. we know that aa/ac < 0 and ad/On > 0. That is. the portion to the wholesaler after the profits redistribution falls when the marginal cost of production increases. but it rises when the number of retailers increases. In the follow1ng sections. we Will study the situations where the consumer demand function is completely inelastic. The reason is two-fold. First of all. it reduces the complication in computation. Secondly. for some kinds 0f good. the consumer demand is quite price inelastic. 176 IV. Assumptions and Notation for the Completely_Inelaatic Demand Case Those who are familiar with the materials of Chapter Four may skip this section. since it is the same as section IV of Chapter Four. However. for reader’s convenience. we repeat them as below. All the assumptions and notation in section II. except for (A1) and (A6). Will be used here. We replace (A1) by (A1)’. Beside good x. there is a homogeneous outside good y.6 The consumer gets a constant surplus of utility. S. from the consumption of one unit of y. Also. we replace (A6) by (A6)’. x = 1. if U(u. P) : V - tu - P 2 S. That is. the consumer buys one and only one unit of good x if U. (the surplus of utility from good x). exceeds 3 (the surplus of utility from good y). Here. V is the maximum of the utility a consumer could get from the consumption of good x. u and P are distance and the retail price respectively. Furthermore. R X : 2D]0 x(P. u)du : 2DR. where R is the maximum distance of each side of the market area a retailer can reach. R 1 (1/2)(L/n). Also. R is a function of P as R : (1/2)(P’ - P + tL/n)/t. (where P’ is the mill price charged by a neighboring retailer). when the 177 retailer is not a local monopolist. Or. R : miniL/a. (v - P)/t} when the retailer is a local monopolist. And here we add one more notation. (A15). Denote v = V - s > 0 v is the effective reservation price for consumers to consume good K. V. The analysis of the Completely Inelastic Demand Case (A) cases of monopoly retailer(s) When n s maxil. tL/(v - Pw)l.7 the number of retailers is small relative to the market size. With the situation that n = 1 2 tL/(v - Pw). there is a single monopoly retailer serv1ng the entire market. For other Situations where 1 s n < tL/(v - PW). there may be a single monopolist or several local monopolists serv1ng part of the market. Therefore. R : miniL/a. (v - P)/t} for those cases. We will study all the situations as below. Case 1. The single monopolist serves the entire market when n 1 2 tL/(v - PW). In this case R : L/2 and P s v - tL/2. This is the case where the prevailing retail price is so low that the Single monopoly retailer serves the entire market. This retailer is a monopolist t0 the consumers and a monopsonist to the manufacturer. Since we have assumed one manufacturer in the industry 0f good X. this case 178 becomes the situation of pglateral monopoly. However. the bilateral monopolists cooperate in the wholesale Price determination to maximize their sum Of profits. As R - L/2. by assumption (A6)’ we know that X : DL. 80. for any given Pw. (6.76) «r = (P - Pw)X - fr : (P - Pw)DL - fr WhiCh is a monotonically increasing function Of P. (as Our/6P = DL > 0). when P 1 v - tL/2. Therefore. for the purpose of up max1mization. the retailer Will charge P“ to its upper limit no matter how much PW is. That is. (5.77) P” : V - tL/2. With P” : v - tL/2. the retailer’s profit level is (5.76) up” = (v - tL/2 - Pw)DL - fr. The wholesaler’s profit level. with X = DL. becomes (5. 79) 17m” 3 (PW ' C)DL " fm. WhiCh is a monotonically increasing function Of PW’ So. by assumption (A12). we know (5.60) w” : "r” + Wm”. : (v - tL/2 - c)DL - (fr + fm) Since w” in equation (5.60) is not a function of P". at the first glance the equilibrium wholesale priceaaeemaato be uncertain. However. as we assume that the ownerships Of the Wholesale and retailing units are independent. we may find 179 the equilibrium Wholesale Price through the following steps. As the cooperation of both players is to improve their profits from the non—cooperative bargaining situation. the profit levels after the distribution of w” should be no less than those in the bargaining game. OtherWise. the cooperation collapses and both players returns to the bargaining situation. 80. we may solve the profit levels for both players first and then find the equilibrium wholesale price by backward deduction. Let us denote a as the portion of total profits to the wholesaler and wrb and wmb as the profit levels for the retailer and the wholesaler in the bargaining game. According to equations (4.59) and (4.60). we know that wrb and me are (5 61) wrb = (1/2)(v - tL/2 - c)DL - fr (6.62) «m? : (1/2)(v - tL/a - c)DL - fm. Therefore. by the Nash criterion. the equilibrium with the redistribution of the total profits exists if there is an a which maximizes (5.63) 05 : (dfl' - wmb)[(1 - a)w* - wrb]. The first-order condition of OS maximization is ans/ad : 0. which results in (5 64) a” (1/8)[("” ' ”pb * Vmb)/W*] [(1/2)(v - tL/2 - c)DL - fmJ/[(v — tL/2 - c)DL - 160 (fr + fm)]. That is. after the redistribution of the total profits. each player gets (5.65) wm" : d*w* (1/2)(v - tL/2 - c)DL - fm um? and (5.66) "r”, (1 - d)v* (1/2)(v - tL/a - c)DL - fr wrb. Equations (5.65) and (5.66) show that players get the same payoffs as what they may have in the bargaining game. This is because the sum of profits. w”. here is equal to that in the wholesale-price bargaining game. (That is. v” : "rb + vmb.) Furthermore. with the "r”, and wm" given. we know that the wholesale price. Pw. is (5.87) Pw' = (1/2)(V - tL/2 + C). Therefore. when the consumer demand function is completely inelastic. the bilateral monopoly equilibrium in this cooperative game is the same as that in the bargaining Case 2. All the retailers are local monopolists when tL/(v - PW) > n 2 1. In this case R : (v - P)/t and v > P > v - (tL/Z). 181 This is the case Where some of the consumers may not be served by the monopoly retailer(s) because the prevailing retail price is high. Furthermore. in this case. there may be a single monopolist if n = 1 < tL/(v - PW) or several local monopolists if 1 < n i tL/(v - PW). Substituting R = (v - P)/t into assumption (A6)’. we get (5.88) X : (ED/t)(v - P). For any given PW. to maximize up”. the retailer will choose (5.69) P” : (1/2)(v + PW). 80. (5.90) R” = (1/2)(v - Pw)/t (5.91) x“ = (D/t)(V - PW) then (6.92) "r” = (1/2)(D/t)(v - P...)2 - fr (5.93) 17m” 3 (nD/t) (PW ‘ C)(V - Pw) ‘ fm. The sum of profits. n”. will be (5.94) w“ nwr* + Wm” (1/2)(nD/t)(v - Pw)[v + Pw - 2c] - (nfr + fm). To maximize n“. which is a function of Pw. we solve the first-order condition ow’VOPw : 0 and get the solution as (6.96) PW” = c. then (5.96) P” = (1/2)(v + c) (5.97) up” = (1/2)(D/t)(v - c)a - fr 182 (5. 96) "m” 3 ‘fm (5.99) w” : (1/2)(nD/t)(v - c)a - (nfr + fm). Equation (5.95) shows. once again. the equilibrium Wholesale price is equal to the marginal cost of production. This is the same as when the demand.is linear. Also. equation (5.96) implies that the marginal cost of production. c. must be as high as c > v - tL to assure P” > v - tL/2. Furthermore. equation (5.96) shows that side- payments from the retailers to the wholesaler are necessary. OtherWise. the wholesaler will not agree on such a wholesale price as to run a loss. By the same logic as before. the negotiation for a side-payment scheme creates a new bargaining game. The threat-point payoffs to each retailer and the wholesaler are those profit levels shown in equations (4.64) and (4.65) separately. Let us rewrite those threat-point payoffs. With different notation. as below: (5.100) wrb (9/32)(D/t)(v — c)a - fr (5.101) wmb (3/16)(D/t)(v — c)a - fm where wrb is the retailer’s profit level in the bargaining equilibrium and wmb is the wholesaler’s profit level in the bargaining equilibrium. Also. let us denote a as the portion of total profits 183 to the wholesaler when the side-payment scheme has been agreed and carried out. So. according to Nash criterion. the equilibrium with a side-payment scheme exists if there is an d which maximizes (5.102) as : n(dw* - wmb)[(1 - d)n'/n - vrb]. The first-order condition of 05 maximization is ans/ea : 0. which results in (5.103) a” (1/2)[(w* - nwrb + me)/n*] (1/2)l[(13/32)(nD/t)(v - c)a - Bfml/[(1/2)(nD/t)(v - c)a - (nfr + fm)]l. Therefore. the side-payments from the retailers to the wholesaler. Sp. are (5.104) Sp nflp' " (l " (1)17” I I an - «m o (13/64)(nD/t)(v - c)2 The profit levels Of each retailer and the wholesaler after the redistribution Of profits. then. are (5.106) w,*’ (19/64)(D/t)(v - c)a - fr (5.106) vm*’ (13/64)(nD/t)(v - c)a - fm. As WI." > 17,.” and 17m" > "m . we know that every player is better-off than if they bargain over the wholesale price. The comparative static predictions are as below: (5.107) 6P'/8V : 1/2 > 0 (5. 106) 6Pw*/6c : 1 > 1/2 = 6P*/6c 184 (5.109) ow*/6v : (D/t)(v - c) > 0 (5.110) 6w*/6t : -(1/2)(D/t3)(v - c)8 < 0 (6.111) 6w*/ao (1/2)(1/t)(v - c)2 > 0 (5.112) 6w*/6c : -(D/t)(v - c) < 0 (6.113) 6w'/6fr = -n < 0 (5.114) aw*/6n = (1/2)(D/t)(v - c)2 > 0 (5.115) 6w”/ofm : -1 < 0 (6.116) awr"/6z (19/64)(6w*/6Z)/n. for 2 ll 5 .Y D. c. or (6.117) 6wm*’/oz (13/64)(6w*/6Z). for z = v. t. D. c. or Equations (5.95). (5.96). (5.107) and (5.106) show that When the retailers are local monopolists. the spatial factors such.as t. D. L. do not affect equilibrium.prices. The egpilibrapm wholesale price is solely determined by the marginal cost of production. The egpilibriam retail pragg is a function of the consumer’s evaluation on the good. and the marginal cost of production. An increase in the consumer’s evaluation on the good will cause the retail price to rise but does not affect the wholesale price. An increase in the marginal cost of production will cause both prices to rise with the wholesale price rising more. Also. the fixed costs do not affect the equilibrium Prices. They have negative impacts on the total profits and individual firm’s profit level as shown in equations 185 (5.115) to (5.117). The ppm or profita. w”. is a function of every parameter in this model. except for the market Size. L. Generally speaking. those parameters. which have positive effects on the consumer’s surplus of utility. have positive effects on the total profits and vice versa. Also. every cost parameter (c. fr. fm) has a negative effect on the total profits. Furthermore. as long as the number of retailers is small enough (so that every retailer is a local monopolist). an increase in the number of retailers do not affect the equilibrium prices but has a positive effect on the total profits. (B). Retailers Are Not Local Monopolists When n > maxti. tL/(v - Pw)). the retailers are not local monopolists because the number of retailers is large relative to the market size. With Cournot-Nash behaVioral assumption. there exists a unique. symmetric eqeilibrium in which every retailer’s market area is determined by (5.118) R” : (1/2)(L/n). Substituting equation (5.118) into assumption (A6)’. we get X : DL/n. Also. for any given PW. every retailer maximizes its profits by choosing the retail price. P. as8 (5.119) P” : Pw + tL/n. then (6.120) wr* tDLa/na - fr (5. 121) 17m“ 3 (PW “ C)DL - fm. 186 Same as that in Chapter Four. we find that the equilibrium retail price is a mark-up of the Wholesale price. That is. in the Bertrand type price-setting game at the retailing level. the equilibrium price can be higher than the retailer’s marginal cost when spatial factors are incorporated into the analysis. Also. from equations (5.120) and (5.121). we find that the retailer’s profits are not a function of Pw while the wholesaler’s profits are an increaSing function of Pw. So. the objective function. which is the sum of all firms’profits. is (5.122) n” um” + nwr* (Pw - c)DL - fm + tDLa/n - nfr. Equation (5.122) is an increasing function of PW. too. as 67r*/6Pw : DL > 0. Therefore. to maximize w” in equation (5.122), PW” will be set as high as possible but not too high to cause P” higher than v - (1/2)(tL/n). because it is necessary that P + tR 1 v. So. at equilibrium, the retail price Will be (5.123) P” = v - (1/2)(tL/n) - e Where 6 is a very small positive number. TriVially. we omit 6. then (5.124) P” : v - (1/2)(tL/n). and (5.125) PW” : v - (3/2)(tL/n). So. the wholesaler’s profit level is 187 (5.126) "m. : [v - (3/2)(tL/n) - c)DL - fm. As before. the retailer’s profit level is (6.127) up” = tDLa/na - fr. Therefore. the sum of profits. w“. is (5.126) n” : nwr* + um” [v - (1/2)(tL/n) — c)DL - (nfr + fm). Equation (5.126) shows that the wholesaler’s marginal cost of production. c. must be as low as c S v - (3/2)(tL/n) to make sure that the wholesaler's profits are non-negative. Also. as the wholesaler’s profits and the retailer’s profits in this cooperative game are the same as those when they bargain over the wholesale price.9 there is no need for any side-payment scheme. In other words. when the consumer demand.function is completely inelastic. and.When the retailers are not local monopolists. fine cooperation in the wholesale price determination between the Wholesaler and the retailers does not make anyone better—off. The equilibrium outcome in this cooperative game is the same as that in the bargaining game. The comparative static predictions and their implications then are the same for both games. Therefore. there is no need to repeat the discussion of the comparative statics here. VI. Conclusions In this section. we outline two major findings of this chapter as below. 188 1). When the number of retailers is small relative to the market. (so that every retailer is a local monopolist or there is a single monopolist serv1ng the entire market). the equilibrium Wholesale price Will be equal to the Wholesaler’s marginal cost. ii). When the number of retailers is large relative to the market. (so that the retailers are not local monopolists). the equilibrium wholesale price is no longer the wholesaler’s marginal cost but is chosen to reduce the effect of competition at the retailing level. In Table II. we show the sign patterns of the comparative static results for all cases10 studied in this chapter. In next chapter. we summarize and compare all the major findings of this research. Also. at the end of that chapter. we list some possible extensions for future Studies. Table II: Comparative Static Results for Cooperative Cases 1. Cases Of Small Number Of Retailers with.Linear Demand Cases n S maxll. 3th/[4(a - DP )1] 412 n : 1 2 3th/[4(a - bP )] 3th/[4(a - DP )] > n 2 1 Para- !___ 1 meters P P w ’ w ’ w' P P w ’ w ’ w” ___l 1‘ m__ ___l r a O + + + 0 + + + + b O - - - - O - - - - t O - - - - 0 O - - - D 0 0 + + + O 0 + + + L O - + + + O 0 O O 0 c +* + - - - +' + - - - f 0 0 - 0 - O O - O - r O O O - — O O O - - m n O o O O O O 0 O O O O O O O o o o o + 2a. Cases Of Small Number Of Retailers With completely Inelastic Demandi Cases n 1 maxll. tL/(v - P!)} n = 1 2 tL/(V - P ) tL/(v - P ) > n 2 1 Para- I 2! meters P P w ’ w ’ w” P P w ’ w ’ w” ...! r 2IL_____..___J[, r flL_____. v + + + + + O + + + + t - - - - - O 0 - - - D O O + + + O O + + + L - - + + + 0 O 0 0 O c + 0 - - - +* + - - - f O 0 - O - O O - O - r 0 O O - - O O - - m. n O O O O O O 0 O O O o o o + * 189 190 2b. Case of Large Number of Retailers with Completely Inelastic Demand Case n > maxti. tL/(v - P )1 Para- Kt meters P P a w w” w. r m V + 2 + O + + t _II _ + _ _ D 0 0 + + + L -”* + + ? + c 0 0 O - - f O 0 — O O r f O O O - - m n +3 + — + ') Notes: Besides the directions of impacts on PW. P. fir. and am. We use some symbols indicating the relative impacts on PW and P (and on up and um) when they bear the same Sign. The meanings of the symbols are as below: 1). u is associated with the one With greater positive impact. ii). an is associated With the one With greater absolute value of impact when the impacts are negative. iii). ... means no impact of n could be considered since n : 1 is fixed. iv). When a "z" mark stands between two columns of any row With same Sign. it means they bear the same degree 0f impact. 191 Footnotes of Chapter Five We do not study the integrated-firm case because the importance of wholesale price determination in this case is insignificant. An integrated-firm chooses an optimal ratail price to maximize its profits. Wholesale price (or. the price of the intermediate good from the manufacturing units to the sales units) is of little importance Since it has no influence on the PrOfit level, When both the manufacturing units and the sales units belong to the same ownership. If the sales units and the manufacturing units belong to different shareholders. then the major concern of this integrated-firm. beside the retail price level. is the scheme to divide the total profits. Wholesale price level still has little importance. See note 2 of Chapter Four for the example of the legal restrictions. This condition is the same as that in Chapter Four. A detailed derivation is presented in the Appendix A of this thesis. The proof of this equilibrium is in Novshek [1960], pp.322-325. There is a minor modification in this dissertation that the marginal cost is pOSitive rather than zero. See Novshek [1960], p.316 and Salop [1979]. p.147 (with the same spirit but different notation). This concept and part of the analyses in this model are 10. 192 similar to those in Salop [1979]. However. we made some modifications and extensions. All the conditions shown in this section can be derived by the same way as that derived in Appendix A. The only exception is the parameter a/b must by replaced by v during the process of calculation. This is the same as equation (16) in Salop [1979]. p.147, but the notation is different. Equations (5.124) to (5.127) show that the price and profit levels are the same as those in equations (4.94) to (4.97). Except for the case of large number of retailers with a linear demand function. Chapter Six: Conclusions and Extensions Since firms’ bargaining power arises from two different sources: fewness in number and ability to differentiate their product. studies on Wholesale price determination (or on the Price determination Of any intermediate good) can use both.the theory Of bargaining and spatial economics as analytical tools. This thesis is one Of the first to integrate both tools for such an analysis. By using pure bargaining theory to investigate Wholesale price determination. we study six different cases from.the bilateral monopoly case (that is. the two-person bargaining game) to cases of two-by-two multi-unit bargaining games. By integrating bargaining theory and spatial economics to investigate Wholesale price determination. which enlarges the scopes of both fields. we are rewarded with some interesting results. Throughout this thesis. we assume that the retailers are identical With one another. and so are the wholesalers. Also. we assume that the demand for the Wholesaler’s product is a derived demand. The most important findings are summarized as follows. i). In the pure bargaining models. when there is only one Wholesaler at the wholesale level. the equilibrium wholesale price is invariant to the number Of retailers. However. this conclusion does not hOld When we incorporate spatial 193 194 factors into the analysis. ii). When the consumer demand function is linear. the retailer earns more profits than the wholesaler does in the bilateral monopoly situation. This shows that the retailer has a strategic advantage in the bargaining process because the demand for the wholesaler’s product is a derived demand. Furthermore. this conclusion holds in both the spatial and non-spatial model 5. iii). In traditional microeconomic theory. with marginal cost exogeneously given. the equilibrium price in a Bertrand type price-setting game is equal to firm’s marginal cost. In our spatial bargaining model. where the retailer’s marginal cost (that is. the wholesale price) is endogeneously determined through the bargaining With the wholesaler. the equilibrium retail price is not equal to but higher than the retailer’s marginal cost in the Bertrand type price-setting game. This is because spatial factors prov1de the retailers With market power at the retailing level. iv). In a two-by-two pure bargaining game. the bargaining outcome is very sensitive to the bargaining process. With a bargaining process in which the retailers take turns bargaining With the Wholesalers. the equilibrium wholesale price level is the same as that determined solely by the wholesalers to maximize their profits. That is. this equilibrium outcome is most dis-advantageous to the 195 retailers. Conversely. with a bargaining process in which the Wholesalers take turns bargaining With the retailers. the equilibrium wholesale price is equal to the wholesaler’s marginal cost of production. This outcome is the same as that where there is a monopsony retailer versus duopolistic Wholesalers and is most dis-advantageous to the wholesalers. V). When all the consumers on the market are served by a single monopolist or by multiple retailers which are not local monopolists. spatial factors affect the equilibrium wholesale price. Those factors which have pOSitive effect on the individual consumer’s demand have positive effect on the equilibrium wholesale price. and vice versa. When there are some consumers which may not be served by the local monopoly retailers. spatial factors have no effect on the equilibrium wholesale price. v1). The equilibrium wholesale price level in the cooperative game depends on the type of competitive situation among the retailers. The equilibrium wholesale price is equal to the wholesaler’s marginal cost if the number of retailers is small relative to the market size (that is. a single monopoly retailer serv1ng the entire market or several local monopoly retailers serving part of the market). The equilibrium wholesale price is higher than the wholesaler’s marginal cost for cases where the retailers compete With one another since higher wholesale prices reduce this competition. 196 Although we have many interesting findings stated above. the coverage of this thesis is by no means complete. There are many extensions worthy of further study. We outline them below. i). Bargaining models under incomplete information can provide a better description of the real world. Many economists. like Fudenberg and Tirole [1963]. Cramton [1964]. Rubinstein [1965]. and DaVidson and Cheung [1967]. have many striking developments in this approach. 11). Different pricing policies adopted by spatial firms (in this thesis. the retailers) may bring about different bargaining outcome. In this thesis we assume that the retailer charges a mill price to all the consumers and the wholesaler charges a uniform delivered price to all the retailers. It will be interesting to investigate the bargaining equilibria With different pricing policies. iii). Different bargaining solutions other than the Nash solution. (for example. solutions provided by Kalai and Smorodinsky [1975]). may be worth trying in the future. iv). In every spatial model of this thesis. we assume the retailers’ strategic variable is the price only. A further study. in which retailers choose both price and location as strategic variables. Will give spatial factors more important roles in the analysis. 197 v). Throughout this thesis. we use the linear demand function and completely inelastic demand function to represent consumers’ purchasing behavior. A generalized consumer demand function is a natural extension. Or. a thorough investigation on consumer’s purchasing decision over space and time. like what has been done by Bacon [1964]. may be a new dimension for future study. vi). In this thesis. we assume that the wholesale price is the only object bargained by the wholesalers and the retailers. In the future studies we may try the situation where the wholesalers and the retailers bargain over both the quantity and the wholesale price. The bargaining equilibrium outcome then Will be quite different. vii). The entry deterrence played by the active firms and potential entrants at either the wholesale or the retailing level may be another interesting extension for studies in the future . APPENDIX A conditions for Different Types Of Competitive Situations at the Retailing Level From assumption (A6) of Chapter Four. we know x n R R 2D I x(P. u)du : 2D I [a - b(P + tu)]du 0 0 2DR(a - bP - btR/2). --- (1) Also. from assumption (A9) Of Chapter Four and the equation above. we know 2DR(P - P')(a - bP - btR/2). --- (2) We now prove the following conditions. I). All the retailers are local monopolists if and only if n 1 max11. 3th/[4(a - wa)]l. 2220; Two situations are implied in the statement above. Those are: (1) n i 1. if 1 2 3th/[4(a - wa)]. As n is an integer. it is the case that n = 1 2 3th/[4(a - bP')]. (2) n 1 3th/[4(a - wa)]. if 1 < 3th/[4(a - wa)]. That is. 1 s n S 3th/[4(a - bP')). However. as the number of retailers. n. must be an integer. it is only by chance that n : 3th/[4(a - wa)]. Therefore. it is triv1ally the case 1 i n < 198 199 3th/[4(a - th)). Also. when all the retailers are local monopolists. the market area of each retailer is not affected by the price change of its neighboring firms. That is R : minlL/a. (a/b - P)/t). When R : L/2 1 (a/b - P)/t. Which implies P S (a/b) - (tL/2). there is a single monopolist serving the entire market. This corresponds to Situation 1. We Will Prove it latter. When R : (a/b - P)/t < L/2 which implies P > (a/b) - (tL/2). there may be a single monopolist or multiple monopolists in the market. As the number of retailers must be an integer. it is most likely that some consumers are not served. This corresponds to Situation 2. We Will Prove it latter. Furthermore. for any Pw given. by plugging the value of R into equation (2) and solv1ng the first-order condition of max wr(P). we get the equilibrium retail Price as P* : (1/2)(a/b - tL/4 + PW). if R = L/2. --- (3) Or. P” : (1/3)(a/b + 2Pw). if R : (a/b - P)/t. --- (4) We prove each situation as below. Case 1. There is a single monopolist serv1ng the entire market if and only if n : 1 2 3th/[4(a - wa)]. There are two steps to prove the statement above. (1) (11) 200 If there is a single monopolist serv1ng the entire market then n : 1 2 3th/[4(a - wa)]. When there is a Single monopolist serving the entire market. n : 1. R : L/2. and P S (a/b) - tL/2. Also. when R : L/2. from equation (3). we know that P : (1/2)(a/b - tL/4 + P'). Since P S (a/b) — (tL/2) is implied in this case. (1/2)(a/b - tL/4 + PW) E (a/b) (tL/2). That is. (v .0 I I I (1/2)(a/b) - (3/8)tL - (1/2)Pw Or. (a/b) - Pw - (3/4)tL 2 0. --- Equation (7) is equivalent to a - wa 2 (3/4)th. --- DiViding both sides of equation (6) wa). it is clear that 1 2 3th/[4(a - bP')]. If n : 1 2 3th/[4(a - wa)]. then there is a single monopolist serv1ng the whole market. n = 1 means there is a single monopolist in the market. So. what we need is to prove that R : L/2. 201 When R : L/2. every consumer is served by this single monopolist. Now. 1 2 3th/[4(a - wa)] implies PV, 1 (a/b) - (3/4)tL. --- (10) Suppose R i L/2 but R : (a/b - P)/t < L/2. then by equation (4). we know P’I : (1/3)(a/b + 2P"). By substituting equation (10) into P* above, we get P” S (1/3)(a/b) + (2/3)[(a/b) - (3/4)tL]~ --- (11) That is P' S (a/b) - tL/2. --- (12) This contradicts With P > (a/b) - (tL/2) for R : (a/b - P)/t < L/2. 80. R = L/2 and all consumers are served by this single monopolist. Q.E.D. Case 2. All the retailers are local monopolists and do not serve the entire market if and only if 1 S n < 3th/[4(a - th)]. As we have mentioned before. there may be a single monopolist or multiple local monopolists on the market When some consumers are not served. 1. There is a Single monopolist which does not 202 serve the entire market if and only if 1 2 n ( 3th/[4(a - wa)). There are tWO steps to prove the statement above. (1) (11) If there is a single monopolist which does not serve the entire market, then 1 3 n < 3th/[4(a - wa)]. For the single monopolist. it is sure n = 1. Now, we want to prove that 1 < 3th/[4(a - wa)]. As it was mentioned before. when the retailer does not serve the entire market. R = (a/b - P)/t < (L/a) and P > (a/b) - (tL/B). Also, by equation (4) we Know that when R : (a/b - P)/t, P” : (1/3)(a/b + an). So, it is implied that (1/3)(a/b + an) > (a/b) - (tL/a). --- (13) That is, (2/3)(a/b) - tL/a - (2/3)Pw < o. --- (14) Or, (2/3)(a - wa) < th/a. --- (15) Dividing both sides by (2/3)(a - wa), it is clear that 1 < 3th/[4(a — bP')]. -—- (16) Q.E.D. If 1 : n < 3th/[4(a - wa)], then there is 203 a single monopolist WhiCh does not serve the entire market. A Single monopolist which does not serve the entire market implies that R : (a/b - P)/t < L/a and P > (a/b) - (tL/E). Also. n = 1 < 3th/[4(a - wa)] implies that Pw > (a/b) - (3/4)tL. --- (17) Suppose R ¢ (a/b - P)/t but R : (L/a) < (a/b - P)/t. then by equation (3), the equilibrium retail price is PI : (1/2)(a/b - tL/4 + PW). Substituting equation (17) into the P” above. we get P” > (1/2)(a/b) - (1/8)tL + (1/2)[a/b - (3/4)tL]. --- (18) That is. p* > (a/b) - (tL/Z). --- (19) This contradicts with P s (a/b) - (tL/a) for R : L/a < (a/b - P)/t. So. R : (a/b - P)/t and some consumers are not serve DY this single monopolist. 2. There are multiple local monopolists which do not serve the entire market if and only if 1 < n < 3th/[4(a - wa)]. 204 There are two steps to prove the statement above. (1) If there are multiple local monopolists which do not serve the entire market. then 1 < n < 3th/[4(a - wa)]. When there are multiple local monopolists. it is sure that 1 < n. So. what we want to prove is that n < 3th/[4(a - wa)]. When there are some consumers not served. the market area for each retailer is R = (a/b - P)/t < L/a which implies that P > (a/b) - (tL/Z). Furthermore. the total area served by those retailers is anR = 2n(a/b - P)/t < L. --- (20) As we have known that P” : (1/3)(a/b + an) when R : (a/b - P)/t. we substitute this P” into equation (20) and get 2n[a/b - (1/3)(a/b + ZPW)]/t < L. --- (21) That is. an[(2/3)(a/b - Pw)] < tL. --- (22) Or. n[(4/3)(a - wa)1 < th. --- (23) Div1ding both sides of equation (23) by (4/3)(a - wa). we get n < 3th/[4(a - wa)]. --- (24) So. 1 < n < 3th/[4(a - bP')] holds. (11) 205 G.E.D. If 1 < n < 3th/[4(a - wa)]. then there are multiple local monopolist which do not serve the entire market. 1 < n means that there are multiple retailers. Now. what we want to prove is when n < 3th/[4(a - wa)] some consumers are not served. When n < 3th/[4(a - wa)]. it is implied that PW > (a/b) - (3/4)tL. --- (25) Substituting equation (25) into P” : (1/3)(a/b + an). we get P” > (1/3)(a/b) + (3/8)[(a/b) - (3/4)(tL/n)]. --- (26) That is. P” > (a/b) - (1/2)(tL/n). -—- (27) suppose all the consumers are served. then ZnR : L. --- (28) That is 2n(a/b - P)/t : L. —-- (29) Or. P = (a/b) - (1/2)(tL/n). --- (30) This contradicts with equation (27). So. there are some consumers not served. 206 II. The retailers are not local monopolists if and only if n > maxtl. 3th/[4(a - wa)]3. 2.299.: Novshek [1980] had made a formal proof for this case except that he assumed the marginal cost being zero and assumed firms choosing both location and price as strategic variables. As we have assumed that firms are evenly located on the market. (this is one of his major conclusions). it is guaranteed by his proof that the statement above is true. So. we omit the tedious algebraic computation process. Instead. we explain it verbally as below. By assuming that all the symmetric firms are evenly located on the market. there are only three possible types of competitive situations among them: i) a single monopolist serv1ng the entire market. ii) one or multiple local monopolists which do not serve the entire market. (as the number of firms must be an integer). iii) multiple non-monopolists serving the entire market. The sufficient and necessary condition for the first two situations is n s max11. 3th/[4(a - wa)]}. Therefore. the sufficient and necessary condition for the last situation is n > maxii. 3th/[4(a - wa)]}- Those who are interested in a formal mathematical proof may refer to Novshek [1980]. pp. 322-325. APPENDIX B The Long-Run Equilibrium at the Retailing Level in the Cooperative Game In sections 11 to V of Chapter Five. we studied short- run equilibria by assuming that the number of firms is fixed. In this section. we assume that new firms may freely enter the market if the short-run equilibrium profit level is positive and if their expected profits after the entries are nonnegative. Also. the existing firms may freely leave the industry when the profit level in the short-run equilibrium is negative. Therefore. "a market is defined to be in (long-run) equilibrium when no firm currently in the market wishes to exit the market and no potential entrant wishes to enter. '1 For simplicity. we assume that the entry and exit happen at the retailing level only. (That is. there is always a monopoly wholesaler in the wholesale level.) Also. we assume that the retailers are always evenly located on the market after the entries and exits. Furthermore. we assume that when retailers are not local monopolists they compete with one another under Cournot-Hash behaVioral assumption. With the assumptions stated above and the results from previous sections of Chapter Five. we are gaing to study. under what condition. a short-run equilibrium may become a long-run equilibrium in this cooperative game. The analyses are as below. (I). Cases Of Linear Demand Function 207 208 We know that. in a short-run equilibrium. if the number of retailers is small relative to the size of market. (that is. n 1 maxt1. 3th/[4(a - wa)]l). there is either a single monopoly retailer serving the entire market with n : 1 z 3th/[4(a - wa)]l. or all the retailers are local monopolists with 1 1 n < 3th/[4(a - Pw)]. Also. when the number of retailers is large relative to the market. (that is n > maxt1. 3th/[4(a - wa)]l). the retailers are not local monopolists in the short-run equilibrium. Case A. In the short-run equilibrium. the number of retailers is small relative to the size of market. That is. n S maxil. 3th/[4(a - wa)]). 1. The case where there is a single monopoly retailer serving the entire market in the short-run equilibrium when n = 1 2 3th/[4(a - wa)]. A necessary condition for this equilibrium is c S (a/b) - (3/4)tL. This short-run equilibrium 15 characterized by (5.7) PW” : c (5.8) P” : (1/a)(a/b - tL/4 + c) (5.11) w“ : (1/4)bDL(a/b - tL/4 - c)a - (fr + fm) (5.17) wr*' (19/128)bDL(a/b - tL/4 - c)a - fr. (5.18) "m”, : (13/128)bDL(a/b - tL/4 - c)3 - fm. If the eX1St1n8 single monopoly retailer earns zero 209 profits in the short-run. that is fr : (19/128)bDL(a/b - tL/4 - c)3. then there is no incentive for any new firm to enter the market. This short-run equilibrium is also the long-run equilibrium. That is. in the long-run. there is a single monopolist in the retailing level. If the existing retailer earns positive profits. that is. fr < (19/128)bDL(a/b - tL/4 - c)3. then there is an incentive for new firms to enter the market. However. it is not guaranteed that new firms will surely enter the mabket. Instead. with complete information. potential entrants will figure out if the profit level is positive after their entry. If the retailer’s fixed cost is not lower than a certain level.2 the profit level in the equilibrium with new entrants will be negative. Then. no new firm W111 enter the market. Therefore. it is possible that. in the long—run. there is a single monopoly retailer serving the whole market and earning positive profits. Only when the profit level in the non—monopoly equilibrium is non-negative. it is p0851ble that new firms enter the market. Then. in the long-run equilibrium. n > 1 l 3th/[4(a - wa)] and all the retailers are not local monopolists. However. let us remark that the existing firm may try to deter the new comer(s) if its total expected discounted profits of being a single monopolist is higher than that of being a non-monopolist. This may be an interesting extension for the future study. 210 2. The case where all the retailers are local monopolists in the short-run equilibrium when 1 S n < 3th/[4(a - wa)1. A necessary condition for this equilibrium is c > a/b - (3/4)tL. This short-run equilibrium is characterized by (5. 4'1) PWfl = C (5.42) P” (1/3)(a/b + 2c) (5.45) w” : (4/27)(an/t)(a/b - c)3 - (nfr + fm) (5.51) "r”, (133/1458)(bD/t)(a/b - c)3 - fr ‘ s I (5.52) um (83/1458)(an/t)(a/b - c)3 - fm. By the same logic. we summarize the discussion for the long-run equilibrium in regard to this case as below: 1). If fr : (133/1455)(bD/t)(a/b - c)3. (that is. up” : 0 in the short-run equilibrium), there is no incentive for potential entrants to enter the market. This short-run equilibrium is also the long-run equilibrium. That is. in the long-run equilibrium. all the retailers are local monopolists with 1 S n < 3th/[4(a - wa)] and some of the consumers may not be served. 11). Even though fr < (133/1458)(bD/t)(a/b - c)3, (that is. fir” > O in the short—run equilibrium). this short—run equilibrium may also be the long-run equilibrium. That is. if fr is not too small. then "r can not be non-negative in 211 the equilibrium with n > 3th/[4(a - wa)]. Therefore. it is possible that all the local monopolists earn positive profits in theglong-run equilibrium but they do not serve the ent ire marke t . iii). When fr is so small that the profit level in the equilibrium with n > 3th/[4(a - wa)] is non-negative. new firms will enter the market. In the long run. all the retailers are not local monopolists and the number of firms is n > 3th/[4(a - wa)]. Case B. In the short-run equilibrium. the number of retailers is large relative to the size of market. That is. n > max{1. 3th/[4(a - wa)]). For the case where retailers are not local monopolists in the short-run equilibrium. the same logic applies. If they earn non-negative profits. this short-run equilibrium is also the long-run equilibrium. Otherwise. some of the existing firms will exit from the market and make the long- run equilibrium characterized by either that all the retailers are local monopolists With some consumers not being served or that there is a single monopolist serv1ng the entire market. The actual outcome depends on the value of fr. (II). Cases of Completely Inelastic Demand Function 212 We know that. in a short-run equilibrium. if the number of retailers is small relative to the size of market. (that is. n s max11. tL/(v - Pw)l. there is either a single monopoly retailer serving the entire market where n : 1 z 3th/4(a . wa)], or all the retailers are local monopolists with 1 1 n < tL/(v - PW). Also. when the number of retailers is large relative to the market. (that is. n > maxli. tL/(v - Pw)l). all the retailers are not local monopolists in the Short-run equilibrium. Case A. In the short-run equilibrium. the number of retailers is small relative to the size of market. That is. n S maxi1. tL/(v - Pw)). 1. The case where there is a Single monopoly retailer serv1ng the entire market in the short-run equilibrium when n = 1 z tL/(v - PW). A necessary condition for this equilibrium is C S V - tL/2. This short-run equilibrium is characterized by (5.77) P” : V - tL/2 (5.87) PW” : (1/2)(v - tL/2 + c) (5.80) w” : (v - tL/2 - c)DL - (fr + fm) (5.86) "r”, : (1/a)(v - tL/2 - c)DL - fr (5.85) "m”, - (1/a)(v - tL/a - c)DL - fm. BY the same logic as in the linear demand cases. we 213 summarize the discussion for the long-run equilibrium as below: i). If fr : (1/2)(v — tL/2 - c)DL. (that is. fir” : O in the Shor run). there is no incentive for potential entrants to enter the market. This short-run equilibrium is also the long-run equilibrium. That is. in the long-run equilibrium. there is a single monopoly retailer serving the entire market. 11). If fr < (1/a)(v - tL/2 - c)DL. (that is "r” > o in the short-run). but fr > tDLa/na. (that is fir” < O in the equilibrium with new entrants). then no new-comer enters this market. Therefore. it is possible that there is a single monopoly retailer serving the whole market and earning positive profits in the long-run equilibrium. 111). When fr 1 tDLa/na < (1/2)(v - tL/a - c). new firms will enter the market since up” 2 0 after their entries. Therefore. the long-run equilibrium Will be characterized by the situation that all the retailers are not local monopolists with n > 1 z tL/(v - Pw). 2. The case where all the retailers are local monopolists [in the Short-run equilibrium when tL/(v — PW) z n 2 1. A necessary condition for this equilibrium is c > v - tL. This short-run equilibrium is characterized by 214 (5. 95) Pw* : C (5.96) P” (1/2)(V + C) (5.99) w” (1/2)(nD/t)(v - c)a - (nfr + fm) (5.105) nr*’ (19/64)(D/t)(v - c)2 - fr (5.106) wm*' (13/64)(nD/t)(v - c)a - fm. By the same logic as in the linear demand cases. we summarize the discussion for the long-run equilibrium as below: i). If fr : (19/64)(D/t)(v - c)3. (that is. w?” z o in the short-run). there is no incentive for potential entrants to enter the market. This short-run equilibrium is also the long-run equilibrium. That is. in the long-run equilibrium. all the retailers are local monopolists earning zero profits and serVing part of the market. 11). If fr < (19/54)(D/t)(v - c)3. (that is up“ > o in the short-run). but fr > tDLa/na. (that is fir” < O in the equilibrium with new entrants). then no new comer enters this market. Therefore. it is possible that all the local monopolists earn positive profits in the long-run guilibrium but they do not serve the entire market. 111). When fr 5 tDLa/na < (19/64)(D/t)(v - c)3. new firms will enter the market since fir” 1 0 after their entries. Therefore. the long-run equilibrium will be characterized by the Situation that all the retailers are not local monopolists with n > tL/(v - PW). 215 Case B. In the short-run equilibrium the number of retailers is large relative to the size of market. That is. n > maxl1. tL/(v - Pw)). A necessary condition for this equilibrium is c < v - (3/2)(tL/n). This short-run equilibrium is characterized by (5.124) P” = v - (1/2)(tL/n) (5.125) PW” : v - (3/2)(tL/n) (5.128) W” = [V - (1/2)(tL/n) - C]DL - (nfr + fm) (5.126) "m”! DL[v - (3/2)(tL/n) - c] - fm (5.127) wr** tDLa/na - fr. The discussion on the long-run equilibrium in this case is qualitatively the same as that in previous cases. That is. when fr : tDLa/na this short—run equilibrium is also the long-run equilibrium. 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