Id .4 II I I .n w NIIIIIIIIIII {i ZavtéLSS‘BS LIBRARY Michigan State University I 1‘ his is to certify that the dissertation entitled THREE ESSAYS OF INTERNATIONAL TRADE IN DIFFERENTIATED PRODUCTS: INTRA—INDUSTRY TRADE AND TRADE POLICY presented by SAN GHO KIM has been accepted towards fulfillment of the requirements for Ph . D . degree in Economics IIMilli'llfllllllllllIiiil'Illllllll'llllllllfilllliil1| 3 1293 00620 7330 572W fiafiq Majoi" professor Steven Matus Date February 19, 1990 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE'DUE 553 0 6 1995 29o ‘ lube-A "_ ‘, “JAN 16 1097 /'~ MSU Is An Affirmative Action/Equal Opportunity Institution THREE ESSAYS OF INTERNATIONAL TRADE IN DIFFERENTIATED PRODUCTS: INTRA-INDUSTRY TRADE AND TRADE POLICY BY Sangho Kim A DISSERTATION Submitted to Michigan State university in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1990 Lac 5 460K ABSTRACT THREE ESSAYS OF INTERNATIONAL TRADE IN DIFFERENTIATED PRODUCTS:INTRA-INDUSTRY TRADE AND TRADE POLICY BY Sangho Kim (1) International Trade in Vertically Differentiated Products under Perfect Competition This paper presents a theory of international trade in a two-sector, one-factor economy in which one sector is vertically differentiated. The paper shows that trade arises from the cost differences in goods in this sector between countries. Furthermore, this trade is characterized as inter— industry trade when cost différences are uniform and intra— industry trade when cost differences are biased. In both cases, an economy with either of these types of trade is more efficient than an autarkic economy' because production is increased. (2) The Effects of International Trade Policy on Vertically Differentiated Products: A General Equilibrium Analysis This paper presents a general equilibrium model of two- country, two-factor and two-commodity in which one commodity is vertically differentiated. The policy analysis of the model shows that quantitative restrictions (quotas and VERs) are elusive as restrictions on imports due to quality upgrading. Social welfare comparison between tariffs and quantitative restrictions reveals that the former instruments dominate the latter. Quantitative restrictions are shown to have the same equilibrium independent of their specific forms (quotas or VERs). Minimum quality standards can be used either to restrict imports or improve terms of tradeu Quality standards also cause the factor'market.distortion.in which one factor is under-utilized. (3) Intra-Industry Trade in Horizontally Differentiated Products: A One-Sector Model with Lancaster's Ideal Variety Approach The paper presents a one-sector, Chamberlinian monopolistic competitive model of intra-industry trade based on Lancaster's ideal variety approach. 2hr specifying the utility function, two different cases of consumer demand are distinguished by the value of the parameter related to price elasticity: the "arbitrary" case and the "general" case. 'This paper is concerned with the general case, and shows that intra-industry trade occurs in order to take advantage of the internal diversity of preferences within each country. Copyright by SANGHO KIM 1990 Dedicated to my parents: Samsoo Kim and Youngnam Choe ACKNOWLEDGEMENTS I am deeply appreciative of my dissertation committee chairperson Steven Matusz. He took me in his arm as a student when my former adviser departed and showed patience in reading through my mistake laden initial proposals. Without his time consuming guidance, encouragement and help, this dissertation could never have been started and completed. Through numerous discussions in which I inevitably found that I had been wrong in the ends, I came to See his integrity and excellence as a teacher and person. He taught me that a dissertation should have originality, no mistakes in logic and be able to make contributions to the field. The other two members of the dissertation committee, Mordechai Kreinin and Carl Davidson, have been helpful. Actually, the second essay of this dissertation started out from the idea I got from Dr. Kreinin's paper about VERs on Auto industry, and his comments made the dissertation a better one. Despite all of their- guidance, needless to say, any remaining defects of the dissertation are mine. My parents, Samsoo Kim and Youngnam Choe, have always greatly supported me making my studies much easier. For their vi endless love and devotion, I dedicate this dissertation to them as a small token of my gratitude. I would like to thank all the members of my family, Sanghee, Kwangyong, Hangsook, Kyungsook, and Jeongsook. In particular, my elder brother, Sanghee, loved me enough to put off starting his own business for five years. All my sisters have always believed in me and given me their warmest encouragement. I love you all. Choona Lee, my wife, has always encouraged me throughout the many hard times. She is the one I can count on day in and day out. Geesoo, my daughter, has endured the transitions of living in two countries. It has helped her to have grandparents like Inkyu Lee, who has since departed, and Jeongsook Kim to take good care of her during her stay in Korea. I would like to thank them for supporting us. TABLE OF CONTENTS page LIST OF FIGURES.... .......... . ...... . .................. X CHAPTER 1. Introduction and Review of Literature ......... 1 1.1. The nature of Product Differentiation in International Trade ............ l 1.2. Intra—Industry Trade in Vertically Differentiated Products ................ 3 1.3. Trade Policy on Vertically Differentiated Products ................ 6 1.4. Intra-Industry Trade in Horizontally Differentiated Products .......... ...... 10 1.5. Purpose and Basic Features of the Study......... ..... . ............. 13 1.6. Overview of the Results ................ 15 2. International Trade in Vertically Differentiated Products under Perfect Competition ............. ........ .............. 17 2.1. Introduction ........................... 17 2.2. The Model .............................. 19 A. Production ......................... 19 B. Consumers.. .................... 24 (8-1). Utility Function.. ......... 24 (B-2). Indifference Curve ......... 25 (B-3). Budget Constraint .......... 27 (B-4). Demand for Qualities: Utility Maximization ....... 30 2.3. Autarkic Equilibrium .................. 34 2.4. Technological Differences and International Trade.. .................. 38 A. Uniform Cost Differences. .......... 38 B. Biased Cost Differences............ 45 C. Intra-Industry Trade ....... . ....... 51 2.5. Summary and Conclusions ......... . ...... 58 viii 3. The Effects of International Trade Policy on Vertically Differentiated Products: A general Equilibrium Analysis ...... .... ...... 61 3.1. Introduction..... ...................... 61 3.2 The Model .............................. 63 A. Production.. ....................... 64 B. Production Possibility Frontier. 73 3.3. Equilibrium and Comparative Statics.... 77 A. Autarkic Equilibrium ............... 77 B. The Comparative statics of the Equilibrium .................... 78 3.4. Policy Issues (1): Tariff, Quota and Voluntary Export Restraint ......... 88 A. Tariff.. ........ ...... ............. 88 B. Quota. ............................ 94 C. Voluntary Export Restraint ..... .... 109 3.5. Policy Issues (2). Minimum Quality Standard ............... 112 A. Production ......................... 112 B. The Production Possibility Frontier ........................... 115 C. The Offer Curve .................... 120 (C-l). The Home Country ........... 120 (C-2). The Foreign Country ........ 122 D. International Trade and the MQS.... 123 (D—l). The MQS on the Home Goods.. 125 (D-2). The MQS on Foreign Imports. 127 (D-3). The MQS on Both Domestic and Foreign Imports ........ 130 3.6. Conclusion ......... ...... ............... 130 4. Intra-Industry Trade in Horizontally Differentiated Products: A One-Sector Model with Lancaster's Ideal Variety Approach ....... 133 4.1. Introduction ..... .. ..... ....... ........ 133 4.2. The Model... ........................... 135 A. Demand Side ......................... 136 (A-l). Utility Function ........... 136 (A-2). Market Demand .............. 145 B. Supply Side ..... ....... ............ 149 4.3. International Trade........... ......... 155 4.4. Conclusions........ ....... .. ........... 158 ’PENDICES A: Quality Dimension in the Cost Function... 160 B: Other Restrictions on the Price Schedule. 166 C: The Production Function When x is Labor Intensive ............................ IBLIOGRAPHY .................................... . ...... 174 ix Figure 2.1 Cost Curves in the Differentiated Sector ......... 2.2 Utility Functions... ........... . ................. 2.3 Indifference Map of a Consumer... ................ 2.4 Indifference Curves of two Different lVlVlVIVlV \000\IO\UI lv H Mia o HIAPJonoo\imcn4>o3NIAPJH UNI-’0 wwwwwwwwwwwwwuvnv o. I. 7 I-' 4:. 3.15 3.16 3.17 3.18 3.19 3.20 3.21 Consumer types........... ..... ......... ....... .. Utility Maximization... ....... ............ ....... Autarkic Equilibrium with Single Consumer type... Uniform Cost Differences.................... ..... Biased Cost Differences................ .......... Quality Zones, and Marginal Consumer ............................ LIST OF FIGURES Technological Frontier Autarkic equilibrium with Multiple Consumer Types.. Free Trade equilibrium ................... ... ..... Gains From the Trade.. ...... ... .................. Zero Profit Curves... ........... .. ............ Four-Quadrant Diagram of the Model ............... Harrod-Johnson Diagram ........................... Production Possibility Frontier... ............... Trade Triangle............. ........ .. ............ Offer Curve... ........... ... ..................... Rybczynski Theorem.... ....... . ................... Effects of Capital Increase ..... ..... ............ International Equilibrium......... ............... Trade Indifference Curves..... . _ International Equilibrium and Autarkic Pr1ces.. Effects of a Tariff............................ Quantity Version of the Production possibility Frontier................... .......... Two Versions of the Production possibility Frontier...................... ....... Effects of Quantity Restriction on PPFs. .. ....... Trade Triangle under Quantity Restriction... ..... Shift of the Offer Curve under Quantity restriction.......................... Effects of the Effects of the Effective Zero Effects of the Quota.............. .......... Voluntary Export Restraint.... Profit Curve............ ...... MQS on Factor Price Frontier... X coo-0.000.000.0000 page 22 28 29 32 37 46 102 105 107 108 111 116 117 Production Possibility Frontier under MQS........ Shift of the Home Offer Curve under MQS.... ...... Shift of the Foreign Offer Curve under MQS ..... .. MQS on Home Goods........................... ..... MQS on Foreign Imports................. .......... MQS on both Domestic and Foreign Goods ..... ...... Compensation Function............................ Consumer Demand.................................. Consumer Distribution............................ Approximation of the Actual Demand............... Monopolistic Competitive Equilibrium............. Graphical Solution of the Model.................. Effects of International Trade.................. . Cost Function with Quality Factored into Fixed Cost only............ Cost Function with Quality Factored into Variable Cost only.......... ooocooooooonIo-o- Cost Function with Quality Factored into Both Fixed and Variable Cost ................ Corner Solution: Corner Solution: A Case of p"(q) = 0 ............. A Case of p"(q) < o............. xi 119 121 124 126 128 129 138 143 146 148 151 154 156 163 165 167 169 170 Chapter 1 Introduction and Review of Literature 1.1. The Nature of Product Differentiation in International Trade Goods traded internationally are grouped in the same itistical “class for reporting because they are close >stitutes in either production or consumption, or both. iduct differentiation of goods of the same class can be sidered as being of two types, quality and variety. In : real world these analytical classes tend to overlap, but ically quality differentiation is based on measurable formance characteristics of products while variety ferentiation is based on product appearance and marginal formance characteristics. The former type of differentiation, known as vertical erentiation, arises from variations in the quality of a odity and is an important determinant of the pattern of odity trade. The first theoretical paper on this subject presented by Armington (1969). He assumed that consumers otherwise identical goods produced in different countries different. Therefore, consumer uncertainty about fferences in quality among producers might lead consumers look at origin of products as a signal of average quality. This view of products as vertically differentiated by ntry of origin also prevails in international trade ctice. For example, there is a grading system for coffee ins based on source of origin, nature and quality of the aduct [ see Marshall (1983) ]. Location of cocoa beans iys a critical role in the determination of quality, both :ause of climatic and soil conditions and because of the mrent characteristics of beans within countries [ see 'tis et a1. (1987) ]. Similarly, grains are divided into .sses and subclasses according to shape, texture, color of a kernel, and their source of origin [ see CBT (1982)]. Besides agricultural products and raw materials, which graded and differentiated by quality, manufactured ducts can be vertically differentiated, too. Automobiles, example, which differ in size, weight, engine power, ability of finish, etc., are considered to be quality ferentiated. A theoretical explanation for this type of vertical erentiation was presented by Linder (1961). Linder argued a country tends to specialize in the production and rt of that quality of products which is demanded by the rity of its population, while it imports the qualities nded by both the richest and the poorest segments of its pulation. The other type of product differentiation, called rizontal differentiation, is based on variety and can result cm the geographic origin of goods in an international trade antext. Commodities become horizontally differentiated when nporters differ in their choice of the geographic origin of 1e good as a result of attributes related to the export of product, despite the possible absence of quality variations :om country to country. The pattern of international trade in products .fferentiated by variety takes the form of countries porting styles most popular in their own population while .ey importing styles appealing to the minority. Empirical udies by Dreze (1960, 1961) supported these hypotheses using lgium trade data. 1.2. Intra-Industry Trade in Vertically Differentiated Products A careful observation of differentiated products in ernational trade reveals that vertically differentiated ds are at least as popular as horizontally differentiated ds. Grubel and Lloyd (1975, ch. 6), for example, showed t there is significant intra-industry trade in both tically and horizontally differentiated products. refore, intra-industry trade theory in horizontally 4 ferentiated goods can be a partial explanation of the total int of trade in differentiated products. The above recognition has largely been ignored in tomics literature which concentrates on horizontally Eerentiated products in intra-industry trade theory. The t of literature on vertically differentiated products in srnational trade results from the fact that there has not n any micro economic theory for vertically differentiated ducts. This contrasts with the extensive research on ra-industry trade theory in horizontally differentiated ducts following the development of monopolistic competition ory by Dixit and Stiglitz (1977) and Lancaster (1979). Theoretical attempts to explain patterns of trade in tically differentiated products date back to Linder (1961). envisioned trade in quality differentiated products on the imption that income is the dominant determinant of tastes. refore, the quality of products which are well developed lin, a country is the quality that is demanded by the Ilation of average income level of that country. From this lmption, Linder's hypothesis says that a country tends to dalize in the production and export of that quality of ,ucts which is demanded by the majority of its population, e it imports the qualities demanded by both the richest the poorest segments of its population. Linder supported his hypothesis with trade data from his 1e country, Sweden. However, detailed empirical support 5 the theory has not been found, and tests of propositions ight to reflect Linder's hypothesis have had only mixed ilts. Donnenfeld and Ethier (1984) combined the demand 1cture of Linder with the factor endowment model of trade explain inter-industry trade as well as intra-industry ie in vertically differentiated products. They showed that if trade in commodities does not lead factor price equalization, then a country will export the 3e of qualities which are relatively intensive in its ndant factor and import the range of qualities which are snsive in its relatively scarce factor. Donnenfeld (1986) extended Donnenfeld and Ethier's model .nclude imperfect information about quality and explain the :ern of trade. In a separate development, Grubel and Lloyd (1975) rested the life-cycle theory of Vernon (1966) as a possible .anation of intra-industry trade in vertically Ferentiated products. They showed that trade resulting 1 the life-cycle theory is intra-industry trade if goods differentiated by quality. In such trade, a country at Lgher technological state produces and exports higher ity goods and imports lower quality goods from a country lagging technological state. They used the pharmaceutical industry in which there is ’ge amount of trade in European- and U.S.—developed drugs 6 medicines to support their theory. This theory emphasizes the dynamic nature of hnological development as in the life-cycle theory but ls to provide the basic reason why' certain goods are tially developed by certain countries in the first place. 1.3.Trade Policy on Vertically Differentiated Products The current literature on international trade policy in tically differentiated products has been stimulated by the irical findings that quantitative trade restrictions lead a shift in the composition of trade toward higher valued, her quality products. This hypothesis of quality upgrading been confirmed in all cases of quantitative restrictions various industries. Patterson (1966) and Meier (1973) showed that quality ading exists in the textile industry. Their study aled that higher quality textile imports resulted from a restriction measured in yardage of textile imports. hee (1974) reported that the voluntary tonnage restriction teel exports to the U.S. lead to the increase of price per age of imported steel. Mintz (1973) has noticed quality ading in dairy products, sugar and meat resulting from the quotas on these imports. Recent examples reporting ity upgrading include Anderson (1985) for cheese products, d Roberts (1986) for footwear, Boorstein (1987) for steel 7 mports, and Feenstra (1988) for auto imports. These empirical findings have stimulated studies on trade olicy in vertically differentiated goods. The first heoretical models were presented by Rodriguez (1979) and .antoni and Van Cott (1979). These models assumed that uality can be varied continuously by recognizing' multi- imensional characteristics of goods (quality and quantity). iven utility generating multi characteristics of goods, the arket response to a quota will encompass the complete set of haracteristics, not just the characteristic which is formally imited by a quota. Thus, market participating profit ximizing individuals will exploit potential gains by ibstituting the product's unregulated characteristics for the agulated characteristic. Santoni and Van Cott used the shoe industry as an example rshow that when the unit of shoe imports is restricted, the .restricted characteristic of shoe quality (durability) is creased as a rent maximizing behavior of imports. Rodriguez presented a profit maximizing supplier who ooses the quality level to minimize cost per unit of rvices provided. Under this circumstance, he compared nsumer welfare between tariffs and quotas. In both models quality level is denoted by total amount services provided by goods and becomes an explicit variable ntrolled by economic agents. This approach has provided a rtial equilibrium analysis of competitive foreign producers e ' EF‘ 5. _;1 1.4.4. 8 which foreign producers regard price of services as given. These models have been extended to the case of a foreign nopolist instead of perfect competitive foreign producers Das and Donnenfeld (1987) and Krishna (1987). Das and nnenfeld showed that both quotas and minimum quality andards dominate tariffs as policy instruments. Krishna monstrated that the effects and desirability of various ade restrictions depend on the valuation of quality crements by the marginal consumer relative to the average luation of quality increments by all consumers. The initial models have been further extended by Mayer 982) who presented a simple general equilibrium model and owed the possibility of replacing tariffs with equivalent iimum quality standards. In. Mayer's model quality is :luded in the production function, and raising quality luces output at an increasing rate. In other models quality :ers into the cost function, and raising quality increases a unit production cost at an increasing rate. In another :ension of the initial models, Donnenfeld_and Mayer (1987) iwed that voluntary export restraints can be used as policy truments to increase social welfare under the existence of ormational externalities. In general there is no incentive individual firms to increase quality because their quality perceived as an average quality of industry. VERs Luntary Export Restraints), however, force firms to improve Lity, thus increasing social welfare. 9 One common assumption prevailing in the above literature is associated with Swan (1970). In the Swan model demand is essentially for services produced by goods, and higher quality goods provide more services. The profit maximizing quality choice under monopoly or competition can be shown to be that which minimizes cost per unit of services and to be independent of the level of services produced. All the above models ranked quotas, tariffs, and minimum quality standards in terms of consumer welfare. Contrary to traditional assumptions, quotas are shown to be preferred to tariffs because of the greater welfare induced by quality adjustment under quotas. However, the partial equilibrium or ad hoc nature of the literature prevents the studies from investigating various policy instruments fromaasocial welfare standpoint. 1.4. Intra-Industry Trade in Horizontally Differentiated Products The interest in intra-industry has arisen from the empirical studies by Balassa (1966), Kravis (1971) and Grubel and Lloyd (1975). These studies revealed a strikingly new :haracteristic of world trade which is that a trade among the )Cs features a large and growing volume of intra-industry Lrade, both absolutely and relative to inter-industry trade. 10 Intra-industry trade - the simultaneous presence of imports and exports of the products of a given industry, presents a substantial challenge to traditional trade theories. Two-way trade flows of similar products between countries with nearly identical factor endowments can not be explained with the standard H-O-S framework. Earlier theoretical attempts to explain intra-industry trade in differentiated. products were provided by Grubel (1970), Gray (1973) and Barker (1977). They had tried to model firm level product differentiation with monopolistic :ompetition. But only since 1980 have models appeared that successfully incorporate monopolistic competition with the yeneral equilibrium requirements of trade theory. In fact, a new wave of theoretical developments began Iith two studies of 1979 - Krugman (1979) and Lancaster (1979, :h. 10), which presented one sector model in which all .nternational trade is intra-industry trade. These studies irovided first formalized models explaining the effects of Iroduct differentiation, monopolistic competition, and conomies of scale on problems of international trade. mmediately, these simple models were extended to two sector odels in order to integrate the H-O-S approach to nternational trade with the theory of intra-industry trade see Lancaster (1980), Dixit & Norman (1980, ch. 9), Helpman 1981) and Krugman (1981) ]. Integrated models try to explain rade within an industry consisting of close substitute with ll .milar technologies, as well as trade of the products of the idustry for outputs of other industries. They relate the merminants of the two kinds of trade to the underlying aasons for trade, and show how intra-industry trade can be plained by product differentiation while conventional H—O- explanations apply to inter-industry trade. One difference that can be observed among the many udies of this topic is the specification of consumer eferences for differentiation. One approach following ence (1976) and Dixit & Stiglitz (1977) assumes that a presentative consumer likes to consume a large number of rieties [ for example, Dixit & Norman (1980, ch. 8), Krugman 981), Helpman & Razin (1980) and Lawrence & Spiller (1983) In these models, every variety is assumed to command the ne value from consumers and be produced by the same aduction function. Therefore, all varieties of a given aduct are equally priced at equilibrium. The alternative approach is derived from Lancaster's I79) characteristic approach to consumer's demand [ see icaster (1979, 1980) and Helpman (1981) ]. It is assumed It products are differentiated by the combination of some :ic characteristics. Every consumer has an ideal product, :. his most desired combination of characteristics. If a iety is represented by a point on a line or the cumference of a circle, the variety closest to ideal iety will be chosen by consumers if ideal variety is not pr wi re tr; HIOI di: inc Ric 0f a 1 12 available. In this approach, every available variety produced by firms of the same production function is equally priced and spaced in a symmetric equilibrium. Despite the fact that both approaches used a different specification of preferences, they reached the same broad conclusions regarding the nature of intra-industry trade and gains from specialization which are obtained by taking advantage of economies of scale. Another difference that distinguishes the various studies is whether the model deals with the trade in final products ( consumer's good ) or middle products (producer's good ). All of the papers mentioned so far confined attention to final products only. The theory can be modified in order to deal with trade in. middle products, without altering its main results [ see Ethier (1982) and Helpman (1983) ]. 1.5. Purpose and Basic Features of the Study In the following chapters, three different international rade models in differentiated products are presented. Each odel is associated with one of the three pieces of literature iscussed above. The first model presents a theory of intra- idustry trade in vertically differentiated products in a .cardian economy. The second model investigates the effects commercial policy on vertically differentiated products in H-O-S economy; The third model develops a monopolistic com dif dif. faC‘ bot] COUI dete teci the (197 as Prod comm diff Conn. lite: in t Prodl aSsun Cost to Sr 13 ampetition model of intra-industry trade in horizontally ifferentiated products. In the first essay, patterns of trade in vertically ifferentiated products are studied in a two-sector, one- actor, Ricardian economy in which there are differences in oth technological factors and consumer types between ountries. The model emphasizes technological factors in the stermination of patterns of trade. The emphasis on achnological factors in international trade originated from me life-cycle theory of Vernon (1966). Grubel and Lloyd L975) suggested the life-cycle theory in international trade 5 a possible explanation of vertically differentiated :oducts in intra-industry trade. In the second essay, a two-sector, two-factor,- two- >mmodity model in which one commodity is vertically ,fferentiated is presented. This general equilibrium model nnects partial equilibrium or ad hoc models of the terature to the standard H—O-S model. In the model vertically differentiated goods are measured total services, and total services are determined by a >duct of unit quality and physical quantity. Firms are :umed to choose an optimal quality to minimize their total t in providing services of differentiated goods according Swan (1970). The general equilibrium nature of the model enables us di CO tr 14 to compare the desirability of tariffs, quotas, voluntary export restraints and minimum quality standards from a social welfare standpoint instead of the consumer welfare standpoint of partial equilibrium models. In the third essay, a one-sector, Chamberlinian monopolistic competitive model of intra-industry trade based on Lancaster's ideal variety approach is developed. This model is an attempt to formalize a general idea suggested by Lancaster (1979). He suggested that gains from intra-industry trade could result from internal diversity of preferences within each country between identical countries. He specifies the utility function for differentiated products based on his characteristic approach, and it is called the ideal variety approach in the literature. This approach contrasts with the ove of variety approach of Dixit and Stiglitz (1977). In resenting the model in this essay, the different features of The two approaches are clarified. 1.6. Overview of the results The first essay shows that trade arises from the cost lifferences in vertically differentiated goods between :ountries. urthermore, this trade is characterized as inter-industry rade when cost differences are uniform and intra-industry 15 trade when cost differences are biased. Uniform cost differences occur when there is a difference in labor productivity in the homogenous goods or a difference in the fixed cost required for the differentiated goods between countries. Biased cost differences result from changes in the parameter of the cost function representing the rate of change in cost in relation to quality. In both cases, an economy with either of these types of trade is more efficient than an autarkic economy because production is increased. The second essay shows that quantitative trade restrictions (quotas and VERs) are elusive as restrictions on imports due to quality upgrading. Tariffs dominate quantitative restrictions because the former increases social welfare more than the latter. Quantitative restrictions are shown to have the same economic result independent of their specific forms (quotas or VERs). Minimum quality standards can be used either to restrict imports or improve terms of trade, but ambiguous results of these quality standards require careful consideration in their imposition. Quality standards also cause factor market distortion in which one factor is under—utilized. This essay also investigates inter-relations of factor arket abundance and factor intensity in association to the ality of differentiated goods. The third essay ShOWS that intra-industry trade occurs 'n order to take advantage of the internal diversity of pref vari econ diff Lanc thei mixti In tl of e‘ chap1 16 preference within each country. Free trade provides more varieties than in a closed economy, and the welfare of the economy arises. The essay also shows that there are two different ways of specifying the utility function of Lancaster's ideal variety approach. In the "arbitrary" case, the consumer either specializes in one variety or consumes a mixture of varieties which offer the lowest effective price. In the "general" case, the consumer chooses a positive amount of every variety. Each essay is presented separately in the next three chapters. inter: goods goods. body ¢ differ differ CHAPTER 2 (ESSAY 1) International Trade in Vertically Differentiated Products Under Perfect Competition 2.1. Introduction A careful observation of differentiated products in international trade reveals that vertically differentiated goods are at least as popular as horizontally differentiated goods. However, contrary to abundance of a well developed body of literature on intra-industry trade in horizontally ifferentiated products, literature on that of ‘vertically ifferentiated products is scarce. This paper attempts to fill this vacuum by presenting model which can explain causes and results of intra-industry rade of vertically differentiated products. This paper emphasizes technological factors in the etermination of patterns of trade along with the demand tructure of the Hedonic price model. The emphasis on echnological factors in international trade originated from he life-cycle theory of Vernon (1966), and Grubel and Lloyd 17 (1975) as a product explicj paper 5 there a types I: Th differe between intra-i; more ClI trade I horizon1 The (1974), differer Cost fo In New Which re there ex the Same The which (1 h°ri20nt qualitie: 18 (1975) suggested the life-cycle theory in international trade as a possible explanation of vertically differentiated products in intra—industry trade. This paper presents an explicit model based on the idea of Grubel and Lloyd. The paper studies causes of trade and the resulting gains when there are differences in technological factors and consumer types between countries. This paper shows that intraetrade arises from the cost differences in goods in the vertically differentiated sector between countries. This paper also shows that the gains from intra-industry trade of vertically differentiated products more closely resemble the gains resulting from inter-industry trade rather than those based on intra-industry trade of ‘horizontally differentiated products. The present paper uses a utility function of Rosen (1974), and assumes there is a: competitive market in the differentiated sector with free entry with the usual U-shaped cost function. In every quality, there is perfect competition and free entry which reduces each firm's profits to zero. It is assumed that here exists a sufficiently large number of firms producing he same quality in every quality. The situation described in this paper is an economy in hich (1) no firm ever has any market power, and (2) no orizontal differentiation (varieties) exists within alities. 2.3, expla intei with disc1 brie! consi the c facto be us Produ quali to as q ind; 19 In the next section, the model is presented. In Section 2.3, autarkic equilibrium is derived, and its nature is explained. In Section 2.4, implications of the model on international trade are presented. This section is concerned with an open economy compared to the autarkic equilibrium discussed in Section 2.3. In the final section, summaries and brief conclusions will be stated. 2.2. The Model Consider an economy made up of two sectors, one consisting of vertically (quality) differentiated goods, and the other of composite (outside) goods. Labor is the only factor in the production of both goods. Outside goods will be used as a numeraire. In the market for differentiated products, there are many qualities of goods available. The quality level of these differentiated products is represented by a one-dimensional hedonic attribute q, which is referred to as "product quality". A larger value in the subscript of q indicates higher quality products. A. Production The production function for the outside goods (composite goods) is: (2.1) flhere I. employs of M. goods a Th assumed model 0 Further costsz. costs a Producii (2 .2) Where g Producec respectj quality (2'2) mu \ 1 is: ' The CI 1 .' 0t finch“ a??? 20 are M represents the outside goods, and 1h is the labor ployed in the outside sector; Each worker produces anunits ’ M. Thus, the wage rate simply equals am if the outside rods are produced, because M is the numeraire. The cost function in the differentiated goods sector is :sumed to be similar to the cost function of the one-factor rdel of Krugman (1979)1 modified to give a U-shaped AC curve. Lrthermore, quality is added. to both fixed and 'variable )StSZ. Because labor is the only factor of production, total )sts are always equal to wage costs. The labor used in foducing each quality is: .2) 1 0’ = C(Qi'v o,- the demand for the output of this industry is some integral \ For detailed discussion and calculation, see Appendix A. 22 MC(q2) AC(q2) A0011) f1”: Q Figure 2.1 Cost Curves in the Differentiated Sector multip] equili}: BI the pri (2-5) Fo consume necessa (2.7) Intuitii i“Crease Price 1; of qualj °h°0ses level 0 °°nsumez TheI‘t-‘Afor elists, ' decrEasi p "(‘l) < will be quality 23 ultiple of Qf, then each firm will produce at Q3, and the quilibrium price will be p'==(f. Thus, profit will be zero. By substituting IE for Q’ in AC derived from eq. (2.3), 1e prices of qualities are derived as: 2.6) pi = min. of AC = 2wh(q)/P = 2amh(q)./P For the unique solution for the quality demanded by insumers, A condition on the price schedule (2.6) is :cessary as follows: ~7) mm 2 0 p'(q) > 0 p"(q) > 0 tuitively, condition (2.7) implies that price should crease at an increasing rate as quality level rises. If ice increases at a constant rate with the rate of increase quality ( dp = dq; a case of p"(q) = O ), any consumer who Doses to buy a quality product will be indifferent to the vel of quality, because every quality yields the same nsumer surplus ( = utility - price ) for consumers. erefore, an infinite number of consumer quality choices ists (indeterminate solution). If price increases at a :reasing rate as quality level rises ( dp < dq; a case of (Q) < 0 ), any consumer who chooses to buy quality goods .1 be better off by upgrading quality, because higher [lity will provide him with more quality for the money spen yiel high next C356! (3'1) quali Produ funct be re (2.8) where qualh 0, 1 h l inde money °“tput Consul; 24 spent. Thus, the consumer's decision problem in this case yields a corner solution consisting only of zero or the highest quality level. This intuition will be clear in the next section of demand; see also Appendix B for two other cases which yield indeterminate and corner solutions. B . consumers (B-l). Utility Function Consumers are assumed to differ in their preferences for qualities, each buying either one unit of a differentiated product or none. A particularly useful form of the utility function, U(M,q,0,X), originated by Mussa & Rosen (1978), can 3e represented by: 32.8) U(M,q,X,€) = M + aqx X = 1 if buy quality, X = 0 if not. 'here M is the composite goods. X denotes the total units of uality goods bought by consumers, ( X takes binary values of I 1 because each consumer either buys zero or one unit ), and indexes consumer types. a is proportional to the amount of oney that each consumer is willing to pay for one unit of UtPUt of quality q of the differentiated products. Thus, onsumers valuations of quality vary in proportion to 9, 5° that dist: be a the d usefu it ig price stron diffe a cor depent (B-Z). maximj From t curves the gj equal (2.9) This ix 0f 9. 25 that the taste patterns of consumers are characterized by a distribution of parameter 9 among consumers. 9 is assumed to be a distribution on the interval of real numbers [0,R] with the density f(0). The utility function (2.8) has convenient properties useful for the study of quality differentiated goods. First, it ignores the income effects because it is defined only by price, quality and parameter a, space. Second, it assumes a strong separability between the composite goods and the differentiated products in question. Third, each consumer has a constant. marginal utility’ with regard to quality' which lepends on his preference 0. This is drawn in Figure 2.2. fB-Z). Indifference Curve The demand for quality is derived from the utility laximization subjected to the budget constraints of consumers. 'rom the utility function (2.8), we can draw the indifference urves on (M, q) space. By total differentiation of (2.8) for he given value of U, the indifference curves have slopes qual to -0. 2.9) if x = 1, dM/dq = -0 his indifference map is drawn in Figure 2.3 for a given value E 0. 26 Honth 93) U(qut 92) U(m1q! 01) Figure 2. 2 Utility Functions two C indif. in Fii consuu as cor of co: point utilit h indiff bundle qualit same f. the hi (IS-3). in this (2.10) where q 27 The property of the curve can be explained by considering :wo different individuals represented by 01 and 02. Two indifference curves representing consumers a. and 92 are drawn in Figure 2.4. At point A, both consumers have the same utility, so :onsumer 1's indifference curve guarantees the same utility as consumer 2's indifference curve guarantees. All utilities of consumers are from the consumption of composite goods at point A. Therefore, the difference in 9 does not affect the itility level of individuals. Notice that bundle B, lies below consumer 1's indifference curve, so consumer 2 gets greater utility from Dundle B than does consumer 1. In general, consumer 2 values zuality more than consumer 1; i.e. given both m and q are the same for both consumers, utility is higher for the person with :he higher 6. ‘B-3). Budget Constraint Consumer income is only from labor with the wage rate w n this one-factor economy. The budget constraint is: 2.10) M + p(q) = Tw here T and p are the total amount of labor time supplied by onsumers/workers and the price of the quality differentiated 28 Slope - —0 ./ Figure 2.3 Indifference Map of a Consumer mix or Isa-b 1 29 k, U(m,q,0o U(m9q, 92) Figure 2.4 Indifference Curves of Two Different Consumer Types the pre slo con fir 30 ood respectively, and p is dependent on g. From the total differentiation of (2.10), the slope of me budget constraint is: 2.11) am = —pl(q) 53 budget const. :esumably, p'(q) > 0.4 I-4). Demand for Qualities: Utility Maximization The utility maximization of consumers requires that the ope of the indifference curve equals that of the budget nstraint at optimum consumption bundle (M, q). This is the rst-order-condition of utility maximization. It is intuitively right to assume that higher qualities and to higher prices. But the proof of this is as follows. a competition with free entry requires that profit must be r all q, yielding a zero-profit condition in the long-run. I) = minimum of AC(q) (1) am the assumptions of the cost function, which was discussed ion .A, the cost function is an increa51ng function of , that is, (Q) > 0, C"(q) > 0 for all qualities q (2) and (2), p'(q) > o is implied. _ . :e that the quantity differences between qual1t1es does not :he proof because the min. AC of lower qualities lS always ian that of higher qualities. (2.12 To ge restr last (2.13 with consm 0f p(c derive for th and ti Shows ”h iet°t61 d ~pll ( q) We that t] 31 12) p'(q) = 0 generate the interior solution to (2.12), the following triction on the p(q) which is discussed intuitively in the t section is required. 13) P(0) = 0 P'(q) > 0 P"(q) > 0 1 the above restriction,5 the unique choice of c; by sumers of 9 is illustrated in Figure 2.5. The concavity >(q) is guaranteed by the positiveness of the second-order Lvatives, p"(q) > O. This critical condition is required the unique tangent solution between the budget constraint the indifference curve. The solution in Figure 2.5 also is not only that the utility' maximization solution is me but also that consumers with high 6 maximize utility :hoosing higher q than consumers with low 0. Also note , if 9 is low enough, the consumer specializes in M, and he restriction of eq. (13) is, in fact, a second-order- of utility maximization. The max1m1zation of utility, q = (w - p(q)) + oq, results in a first-order-cond1t1on: q) + 9 = 0 differentiation of a first-order-condition, results in order-condition: q)<0 this is the same as eq.(l3). 32 High 92 Figure 2.5 Utility Maximization if I can rewr (2.1 wher and Qual. pr0p< must that (2.15 Where 00mpe > 0 w (2.16 This. 33 9 is high enough, the consumer specializes in q. The restriction (2.13) corresponding to a unique solution be expressed as a restriction of the cost function. By riting the cost function as: 14) C(Q, Q) = h(Q) [ V(Q) + F ] re, Q represents the quantity of the differentiated goods, V(Q) and F are variable and fixed costs respectively. lity is added to both variable and fixed costs portionately. A detailed discussion was offered in Section The price schedule with respect to the change in quality t equal the marginal change of cost as quantity changes, t is: 15) 2cm”, q)/ as: = h(q) we") p(q) = h(q) [awo‘v ac] re Q. represents the optimum level of production of the aetitive firm. Therefore, p(q) > 0, P'(Q) > 0: and P"(Q) will require the following restrictions: -5) h(q) > 0 h'(q) > 0 and h"(q) > 0 : will be satisfied for the specific functional form h(q) 1(Q) = qr, if r>l. This restriction will be used in the specif with t model, (2.17) By rew the utj (2.18) Thus, t 0.19). Th( l eq. (2 9.20) 34 ecification of the cost function in later sections. 2.3. Autarkic Equilibrium Now consider'an economy consisting of L workers/consumers :h the same type of preferences 9. For the solution of the lel, we will use a specific functional form for h(q): 17) h(q) = q’. r > 1 rewriting the price schedule (eq. (2.6) ), and F—O-C of : utility maximization ( eq. (2.12) ): 18) Price schedule: p(q) = Zafif/E F ~ 0 - C: 9 = p'(q) s, the quality produced at equilibrium can be solved as: 19). q = < e/ 2ra,./F)”""’> The equilibrium price of the quality produced in the tomy can be obtained by substituting equilibrium quality I. (2.19) ) into the price schedule: :0) p = 2am]? ( 9/ 2ram/F ) "(’4’ prod unit For of q COHS' sati; time, CORSt (2.22 BY 5 const If the g°0ds 35 The equilibrium quantity of the differentiated good oduced with L workers/consumers is equal to the total number consumers L. This is because each consumer demands one it of the group goods according to his utility function. r this model, the utility attainable from the consumption quality goods q ( =9q ) is always larger than that from isuming composite goods ( =p ), because 9q - p > 0 is :isfied at equilibrium for any 9 > 0, i.e.: 21) 9g - p = (1/r) MN) > (l/r) MM), for any 9 > 0 Given prices of quality goods, wage rate and total labor .e, the demand for the composite goods from the budget straints of the economy can be derived: 22) TwL = p(q)L + M substituting the equilibrium price into the budget straint, the income spent for the composite goods M is: 23) M = TWL - 2amqrfF'L = TamL - 2amqrfEL = amL( T - qu/F) :he composite goods consumed is positive, assume T 3 2qr/F. The total number of firms existing in the differentiated ls sector can be derived by dividing the total number of mar whi ECO. are good equi cont The Whic is c deri‘ Iq, 36 irket demand L by the optimum production of each firm Q* = IE LiCh corresponds to the minimum AC in a perfect competitive :onomy. The number of firms in the differentiated sector is: . 24) L/Q' = L/JF Equilibrium values of the quality goods and their prices e depicted graphically in Figure 2.6 for the differentiated ods sector. From the F-O-C of the market demand, and the nilibrium price of quality goods, the following equilibrium adition for quality and price is derived: .25) F-O-C: 2ram/Fq’" = 9 : LHS of eq. (2.25) is derived from p(q) = ZawThr, r>l, .ch in turn depends on the cost conditions. Therefore, it called the "supply factors". The RHS of eq. (2.25) is ived from the utility function, U = M + 9q = (Tw - p) + Thus, call this the "demand factors" from now on. For a given utility level U, the "demand factors" is ived by substituting M in the budget constraint, M + P = into the utility function, U = M + 9q. Therefore, the mand factors" in the graph is the same as in the ifference curve, and utility rises as one moves in a theast direction. This implies a tangency solution of the sumers' choice given the "supply factors", the price 37 Supply Factors Demand Factors p——-—_-__—_. .0 I .0 Figure 2.6 Autarkic Equilibrium with Single Consumer type eql prc W0! pri BY pro< red) leI of i Char uni: Ther 900d equi 38 equation. The closed economy equilibrium is represented by L aroduction of g by a tangent solution of (q, p) for each rorker/consumer. 2.4. Technological Differences and International Trade .. Uniform Cost Differences Suppose changes are introduced in the parameters of the rice schedule shifting the "supply factors" of the economy. y innovating the production processes of its quality roducts, through R&D investments for example, a country can educe its production costs, which may be expressed in lowered ixed cost F, or it can maintain a higher wage rate because f its higher productivity in the outside goods sector. These danges in the parameters am and F shift the price schedule niformly. This is depicted in Figure 2.7. In Figure 2.7, the shift-out of the price schedule from 1e original state (p, the home country) to the starred state f, the foreign country) corresponds to a lowering of either 1e value am or F. The shift-out occurs because each quality roduct can be supplied at a lower price with the new state. Lerefore, the consumer's tangency solution for each quality vod will force the consumer to choose higher quality at a new milibrium ( q < q' ). This can be shown by the partial dez oth The qu 39 lerivatives of eq. (2.19) with regard to am and F, keeping >ther exogenous variables ( 9, r ) constant: .2.26) flq/aam = { 1/(r._l)) { a/Zramfi )(Z-r)/(r-1) {- 9/2r/F'am2 ) < o 2.27) 9q/9F= { 1/(r-1)) ( 9/2ram/F )‘2‘”’“"“’ { [(‘F'mH/Z} wzram) < 0 rom the partial derivatives of the equilibrium price eq. 2.20), the effects of change of anand F on equilibrium price an be derived. The result is: 2.28) 9p/9am = 27"qu [ 1 - r/(r-l) ] < o 2.29) 9p/9F = (2//F)qr [ 1 - r/(r-l) 1 < 0 1e effects of changes in am and F on the equilibrium values 5 q and p, can be restated as: 1.30) 9q/9am < o 9q/9F < 0 9p/9am < 0, 9p/9F < 0 liS result shows that when there are uniform changes in price :hedule ( shift-out ) resulting from the lowered values of and F, the equilibrium quality consumed is raised, and its ice is raised in both cases. Thus, consumers can get higher Po. _ Po 40 Figure 2.7 Uniform Cost Differences fa di fa co ad' di. tEI rel goc prc twc of of pro in Cha: whi. the one eitt one. Spec 41 quality at higher prices in this new equilibrium. Now consider two countries, one with the original "supply factors" and the other' with new "supply factors" in ‘the differentiated products represented.by the shifted-out "supply factor" in Figure 2.7. In this one factor economy, the country with a shifted-out price schedule has a comparative advantage over the other in the production of the differentiated products. In addition, the country with lower :echnology in the production of quality goods will have a relative comparative advantage in the production of composite {oods which are assumed to require the same labor per unit of aroduction in both countries. Once trade opens between the :wo countries, each country will specializes in the products if its relative comparative advantage. This specialization if production after trade will increase total world wide (roduction to the benefit of both countries. The trade resulting from the technological innovations n the production of the differentiated products is haracterized as inter-industry trade between countries in hich one country specializes in differentiated products and he other in composite products. In fact, at free trade equilibrium there exists at least 3e country completely specializing in the production of ither composite or differentiated goods in this two-sector, ie-factor Ricardian economy. The exact determination of the >ecialization depends on the parameter values of the model. Th fr inc EVE Ass sam bot for. the COU} 42 The total world demand for qJ'( * = the foreign country ) with free trade is: (2.31) D = 2L The total labor required ( I. ) to produce the amount of quality goods demanded can be derived by dividing the total income spent on quality goods by income Tw (=Tam), because every worker/consumer earns the same income.6 [2.32) L = 2Lp*/Tam = 4LquF'/T fisuming the total labor force of the two countries is the :ame, if the labor required for quality goods demanded by >oth countries is matched by the exact labor force of the 'oreign country, both countries will completely specialize in he sector of their relative advantage. The home (foreign) ountry will produce only composite (differentiated) goods if: 2.33) 4]?" Lq"'/T = L (=L') ncomplete specialization in one country occurs if the total This is true in a closed economy of diversified production, : necessarily true in an Open economy. Generally, wage (w) qn(productivity in the outside sector) as discussed before. :ialization exists, wage rate used in this discussion refers : of the foreign country. la la bu go Fo: Spi is shc tot cou cal Con all In 43 labor demand for quality goods production is not equal to the labor force of one country. If the former is greater than the latter, the foreign country will specialize in quality goods, but the home country will diversify by producing both quality goods and composite goods.7 The condition for this is: (2.34) 2fF Lq"/T > L Foreign country diversification and home country specialization in composite goods also occurs if inequality is reversed in the above equation. For either of the above situations, free trade can be shown to be Pareto efficient than an autarkic economy, because :otal world wide production increases. Total production gain from the trade between the two :ountries can be shown in terms of composite goods. To :alculate production gain the production of quality is held :onstant at autarkic level ( L = qb, L = q. ) rather than .llowing it to change as expected under trade ( 2L = q. ). n this case, the production in the trade between the two ountries decreases the price of qr Even though this is possible, there remains a question how the home country can be competitive in the quality entiated products. A pricing scheme will be required for If less efficient home quality goods are still preferred by ers over the outside goods, then consumers will buy them. r, if foreign quality goods are preferred over those of the consumers are willing to pay a premium for foreign quality A pricing scheme must solve this questions. 1e (2 F11: Tht 0n the the C03~ 44 The price of q can be lowered from p0 to p" by the reallocation of the production between the two countries. [ See Figure 2.7 ] Now, the total production of composite goods after trade (MW, keeping the qualities produced constant at autarkic level, can be derived from eq. (2.22), and MH is greater than (M+M') (2.35) M” = (wTL - p“L) + (wTL - p'L) > (wTL - poL) + (wTL — p‘L) M + M' ( * = Foreign ) because p” < p Furthermore, the increase of production with free trade is: (2.36) dM=w-(M+M')=(po-p”)L Phus, the increase of production or gains from trade depend >n the price differentials of the two countries resulting from :he difference in "supply factors." a Biased Cost Differences Now consider the effects of changes of parameter "r" on he p(q) schedule. An increase in r raises the production ost of the differentiated products if q > 1, but lowers the cost funC' (2.3' Thus, which quali diffe: r kee consta derive (2.38) (2.39) (2.40) 45 cost if q < 1. This is clearly seen by looking at the cost function. (2.37) cm. q) = qr { oz + F } w If q < 1, q' falls as r increases If q = 1, qr unchanged as r increases If q > 1, q'- increases as r increases Thus, changes in "r" cause the twist in the p(q) schedule which is drawn in Figure 2.8. The effects of changes in r on the equilibrium values of quality and price can be derived by the partial differentiation of eq. (2.19) and eq. (2.20) with respect to r keeping’ all other' exogenous variables ( am, F3 and 9) :onstant. since r appears in the exponents of q and p, the lerivatives can be found by logarithmic differentiation. 2.38) ln q = [1/(r-1)] ln(9/2ram/F) = [l/(r-1)] [ln(9/2ram/F) - lnr] 2.39) (1/q)(dq/dr) = —t1/(r-1)"-1 [1n (o/zamff) -1nr1 + [1/(r-1)](-l/r) = -[1/(r-1)l [1nq + l/r] 2.40) Sign (dq/dr) = - Sign ( 1nq + l/r ) 2.41) dq/dr >< 0 iff q <,q' < 1 46 High r .... p——-——-—.———-— Figure 2.8 Biased Cost Differences Simil (2.42 (2.43 (2.44 (2.45; From 4 into t (2.46) This 1 caUSes qualit: 47 for q1 = e'”r (from 1nq + 1/r = 0) Similar derivatives can be attained for the equilibrium price. (2.42) 1n p = 1n(2aW/F) + rlnq (2~43) (l/P)(dP/dr) = 1nq + r (l/Q)(dQ/dr) = 1nq + (r/q) I -[q/(r-1)l [ 1nq + l/r] by substituting (1/q)(dq/dr) from eq. (2.39) = -1/(r-1) [ lnq + 1 ] (2.44) Sign dp/dr = - Sign [ 1nq + 1 ] (2.45) dp/dr >< 0 iff q <> q2 < q1 < 1 for q2 = e‘1 ( from 1nq + 1 = 0 ) From eq. (2.41) and (2.45), we can divide the p(q) schedule into three zones according to the signs of dq/dr and dp/dr: (2.46) Zone I ( for q 5 q2 ): dq/dr > 0. dP/dr Z 0 Zone II ( for q2 < q S q1 )i dq/dr Z 0, dp/dr < 0 Zone III ( for q < q1 ); dq/dr < 0, dp/dr < 0 This is drawn in Figure 2.9. In zone I, the increase in r causes both quantity and its price to rise. Therefore, the quality consumed rises, and the price paid for this higher Cd Th at of Le llt in be am wil max Wit C01“, whi lab the excl tha. 48 quality also rises. for lower quality with low r. In zone II, the increase in r causes the quality consumed to rise but its price falls. Therefore, consumers pay less for higher quality than lower quality. In zone III, the increase in r causes both the quality consumed and its price to fall. Therefore, consumers buy lower quality and pay a lower price. Even though consumers' choice of quality and price is affected differently depending on the zone, they are better off as they move to the outer frontier of the price schedule. Let's call this outer envelop of the price schedule the "technological frontier." It is represented by a thick line in Figure 2.9. In Figure 2.9, "marginal consumers" who can be satisfied by the qualities produced in both countries, q0 and q00 are represented. These consumers are equally well off with high quality-high price (qm) or low quality-low price (go). People with 9's which are greater than those of marginal consumers buy higher quality goods from the country with low r, and people with lower 9's than those of marginal consumers buy lower quality from the country with high r. Therefore, the pattern of trade depends on the labor type which is assumed to be the same between countries. If the labor type 9 is lower than that of marginal consumers, then the country with high r will export differentiated goods in exchange for outside goods imported from the country with low r. On the other hand, if the labor type 9 is greater than that of marginal consumers, the country with low r will export 49 High 1: Marginal Consumer 'DN 9 . .fl .. .0 (100 Figure 2.9 Quality Zones, Technological Frontier and Marginal Consumer di fr th tw th in w 90' till in C01 eng the pat val can Wor LWC W103, Wham 50 differentiated goods in exchange for imports of outside goods from the country with high r. If labor type 9 is equal to that of marginal consumers, there will be no trade between the two countries. Thus, if consumers buy higher quality goods, the country with low r has a relative comparative advantage in the production of differentiated goods and the other country has a relative comparative advantage in the outside goods. Similarly, if lower quality is consumed by consumers, the country with high r has a relative comparative advantage in differentiated goods with, the other country having a comparative advantage in outside goods. For each case, once trade opens, the two countries will engage in inter-industry trade in which one country exports the goods of its relative comparative advantage. The exact pattern of specialization is dependent upon the parameter values by the same reasoning as in the last section, and it can be shown that trade is better than no trade because total world wide production is increased.8 C. Intra-Industry trade Now suppose the economy of the home country consists of L workers/consumers with three different types of preferences For the gains from the trade, see the proof of the next ion. The same method of proof can be used with slight fication. 9i de (2.4‘ where aSSlll (2.46 CODSU of t} goods compo (2.49 BY sul the i (2.50; 51 91' denoted by numeric subscripts on labor L: (2.47) L=L0+L1+L2 where Lirepresents workers/consumers with preferences 9“ and assume that: (2.48) o = 90 < 91< 92 The equilibrium in the closed economy will produce L1and L2 units of quality g1 and q2 respectively because each consumer of type 1 and 2 will demand one unit of quality goods of type of 1 and 2 respectively. Given prices of quality goods and the total income Tw, we can derive the demand for composite goods from the budget constraint of the economy. (2.49) TwLo + TwL1+ TwL2 = p(q1)L1+ p(q2)L2 + M By substituting pi= ZwJFqJ into the budget constraint, we get the income spent for the demand of the composite goods M: (2.50) M = wTLo + (wTL1 - p1L1) + (wTL2 - p21?) = wTLo + wL1 (T - Z/qu) + sz (T - zfiqzr) = 2) amLi (T - 2/qu), q0 = o i 3) amLi (T-2ffqi’), q0 = 0 llt wi fo Th in in in di1 COL adv a 1 Con qua the To ( con. 52 This autarkic equilibrium is depicted graphically in Figure 2.10. Suppose the foreign country has a higher "r" than the home country with the same labor types. The price schedules for the two countries are drawn in Figure 2.11. In Figure 2.11, q: and q2 are tangency solutions of the "technological frontier" of the two countries of consumers with labor types of 91 and 92 respectively. Consumers with 91 ( 92 ) will be better off consuming q: ( q2 ) from the foreign ( home ) country after trade, assuming 91 < 9"1 < 9r The exact changes in (q, p) with trade will depend on the initial position of q (zone I, II, and III) as we discussed in the last section. The trade resulting from biased cost differences (differences in r) between countries is intra-industry trade in which each country specializes in one part of the differentiated products and then trades with the other country. The home country has a relative comparative advantage in high quality goods and the foreign country has a relative comparative advantage in low quality goods. Both countries will gain from the trade because they can consume quality products at lower prices after trade. In fact, the total production gain from the trade between the two countries can be shown in terms of composite goods. To calculate production gain the production of quality is held Constant at autarkic level ( I” = Q“ I? = Qy 14 = Q1, and L2 P:- P1 53 Figure 2.10 Autarkic Equilibrium with Multiple Consumer Types 54 Marginal Consumer 91 Figure 2.11 Free Trade Equilibrium In 90 di Th1 55 = Q; ) rather than allowing it to change as expected under trade ( Q,‘ = 2L1, and Q2 = 21.2 ). In this case, the "technological frontier" in the trade between the two countries decreases the prices of q1and q;. This is drawn in Figure 2.12. The price of q1 (q;) can be lowered from p1 (p;) to pfi (pf) by the reallocation of the production between the two countries. Now, the total production of composite goods after trade (M”), Ikeeping the qualities produced constant at autarkic level, can be derived from eq. (2.22), and MH is greater than (M+ M'). (2.51) m" = 2W‘I’Lo + L1(ZWT - pf - p1") + 12(ZW’I‘ - p2 - p2") > 2qusz0 + L1(2WT — p1 - pf) + L2(2wT - p2 - pi) = M + M' because p1 > pf, and p; > pgv In addition, the production expansion in terms of composite goods (dM) depends on the differentials of (p1 and pf) and (p; and p;) which in turn depend on the technological differences between the two countries. (2.52) dM = m" - (M + M' ) = L1(p1- 101”) + Lam; - p2“) hus, We have shown that free trade is better than no trade. P2' 56 III-Il'"|l 'II'II| l'n'llll ll]ll|l“’ll|lllll-ll P2 . . _ . . _ _ . _ _ . _ . _ _ _ _ _ w P2 P1 P Figure 2.12 Gains from the Trade Thi ver gai has pro! Rica dif: wit} sec1 that «We: func It n or k and the firs Sine the dEte 57 This also shows that the gains from intra-industry trade of vertically differentiated products more closely resemble the gains resulting from inter-industry trade rather than those based on intra-industry trade of horizontally differentiated products. 2.5. Summary and Conclusions This paper presents a model for a two-sector one-factor Ricardian economy in which one sector is vertically differentiated. Perfect competition along with free entry with a U-shaped cost function is assumed in the differentiated sector. The discussion of the cost function with quality, shows that the quantity level of differentiated goods at the minimum average cost depends on how quality is factored into the cost function. There are three ways quality may be factored in. It may be multiplied with the variable cost, the fixed cost, or both. The minimum AC is the same in the first two ways, and only depends on the fixed cost and the quality. However, he minimum AC of the third way is different from that of the first two ways and only depends on the quality. Therefore, ince the cost function is not affected by quantity in any of he three ways, the utility maximization can be used to etermine optimum quality. The optimum quantity for each firm pr di eh Th1 in in inc C01! rel dif the 58 is determined irrespective of the quality goods produced in the market. Therefore, the results of this paper, which proceeds on the assumption that quality enters both variable and fixed cost, would remain the same if the cost function were defined using the first two methods. The closed economy equilibrium of the model shows the quality produced and its price given labor type. The equilibrium quantity of outside goods demanded is derived from the budget constraint of the economy. In the equilibrium, "supply factors" representing cost conditions and "demand factors" representing the consumers' problem played major roles. Changes in parameters am and F lead to shifts in the price schedule of the differentiated products. Unbiased cost differences resulting from decreases in am and F raise the equilibrium level of quality and cause its price to fall. Thus, two countries which have a difference in productivity in the outside goods sector (am) or a difference fixed costs in the differentiated goods sector (F) engage in inter- industry trade in which one country with high values of anand F has a relative comparative advantage in the production of composite goods, and the other country with low values has a relative comparative advantage in the production of differentiated goods. The change in value of parameter r leads to "twist" in the price schedule. This biased cost difference changes the 59 equilibrium quality and its price depending on the initial value of quality. Thus, countries with different values of "r" engage in intra-industry trade when the two countries have more than one type of labor. In this trade, the country with higher r has a relative comparative advantage in the production of low quality goods, and the country with low r has a relative comparative advantage in high quality goods. Furthermore, it is shown that if either of the above types of trade happens, total production efficiency is increased. This paper also shows that the gains from intra- Iindustry trade of vertically differentiated products more closely resemble the gains resulting from inter-industry trade rather than those based on intra-industry trade of horizontally differentiated products. The extension of the paper can be pursued by assuming increasing costs in the outside goods. For example, suppose capital is another factor of production that is specific to the production of outside goods. Extending this model to a two-factor economy may give us the basis for a trade model based on factor endowments and distribution of preference types. Another interesting question may be inquiring what if preference for quality depends on income which may be related to factor endowments in some way. As a direct application, this model could be used to examine commercial policy. Tariffs, and minimum quality stand analy 60 standards could be analyzed conveniently with same type of analysis used in this model. Ver1 emp: higk Part on ( fai] emph fail term CHAPTER 3 (ESSAY 2) The Effects of International Trade Policy on Vertically Differentiated Products: A General Equilibrium Analysis 3.1. Introduction The current literature on international trade policy in vertically differentiated goods has been stimulated by the empirical finding that quantitative trade restrictions.lead to a shift in the composition of trade toward higher valued, higher quality products. The resulting theoretical models are limited by their partial equilibrium and ad hoc nature because they concentrate on explaining the reason behind the hypothesis. Thus, they fail to analyze it thoroughly in a standard H-O-S economy. The partial equilibrium models of the literature only emphasize consumer welfare effects of policy instruments thus failing to consider social welfare effects resulting from terms of trade effects. Another defect of the literature of ad hoc and partial 61 eqt pol fac twc dif in det Fir the ace of tee] to Sim] Spec the abut of < quot Star Cone Qth equi 62 equilibrium medels is that they can not investigate trade policy within a whole economy, interaction between goods and factors and that between countries. The present paper presents a two-country, two-factor, two-commodity' model in. which one commodity is vertically differentiated. Vertically differentiated goods are measured in the total services they generate, and total services are determined by a product of unit quality and physical quantity. Firms are assumed to choose an optimal quality to minimize their total cost in providing services of differentiated goods according to Swan (1970). The economic situation of the model is similar to that of the standard H-O-S economy. In the production, Leontief technology is used as a specific example of constant returns to scale technology. Leontief technology is chosen for simplicity, but the basic results are not dependent on this specific functional form. The general equilibrium nature of the model enables us to investigate inter-relations of factor abundance and factor intensity in association to the quality of differentiated goods. The model also reveals the. desirability of tariffs, quotas, voluntary export restraints and minimum quality standards from a social welfare standpoint instead of the consumer welfare standpoint of partial equilibrium models. uotas and voluntary export restraints are shown to be quivalent in their final result and inferior to tariffs. Mi. se. of tel thc st; the rum di: tr: tr: pre 001' the quotas TherefOI West 5-9 trade DC 63 Minimum quality standards can be used to restrict total import services or improve social welfare but with underemployment of factor endowment. In the next section, the basic model of Leontief technology is set up. The production possibility frontier of the economy is derived based on the technology. In Section 3.3, autarkic equilibrium and comparative statics of the equilibrium are presented. This section proves the standard theorems of the H-O-S model in the context of the model. In Section 3.4 and 3.5, commercial policies are discussed. Section 3.4 is devoted to tariffs and quantitative trade policies, and Section 3.5 is devoted to qualitative trade policies. In the final section, brief summaries and conclusions are presented. 3.2. the Model9 Consider an economy made up of two sectors, one consisting of vertically (quality) differentiated goods, and the other of homogenous goods. Leontief (fixed coefficient) .9. The model of Chapter 1 can be used for analysis of minimum ity standards and tariffs but is not suitable for the study of as and VERs because of fixed optimum quality assumption. efore, the new model will be developed for a comprehensive stigation of both quantitative (quotas, VERs) and qualitative e policy (minimum quality standards) within the same context. th! whe uni Leo var tot. tlnj.‘ 64 technology is used in the production as a specific example of constant return to scale (CRS) technology. The economic situation of the model is similar to that of the standard H- O-S economy. A. Production The production function for the homogenous goods y, and the differentiated goods x are: (3.1-1) y = Min {lg/aw, H/aw } (3.1-2) x = Min { kx/axx, Lx/aLx } ll 3 P- :3 { kx/ (akx/Q) I Lx/aqu } with akx = an/q, and aLx = auq where ki and Li represent the capital and labor used in sector i respectively, and aij is the factor i required to produce one unit of goods j. The production function for goods y is usual Leontief technology, but that for goods x includes quality variables in the fixed coefficients which require an explanation. In sector x, each good is measured by the total services it generates because goods are differentiated by quality, and otal services are determined by quality level and physical nits of output as: wt CE CE 65 (3.2) x= qQ where x and q are services and the quality of a unit of goods x respectively, and Q represents physical units of output that can vary by quality. Thus, this equation captures the fact that higher quality grains (q), for example, yield more calories (x) than lower quality grains for given amount (Q). Q has the following production function: (3.3) Q = Min < Ig/akx, Lx/mzLx qz) } where akx and aqu2 are the capital and labor required to produce one physical unit of Q respectively. Thus, the physical units of output producible from given factors depend on quality q inversely. The way quality enters into a fixed coefficient of labor (auqfi specifies that higher quality requires more labor for a unit production of Q. This amounts to assuming that upgrading quality requires R&D investment of hiring more scientists and engineers. The production for x (3.1-2) is derived from the combination of (3.2) and (3.3). The cost functions for sector x and y can be derived from the production functions as: I! (3.4-1) cx (aLx q) w + (ah/q) r (3.4-2) Cy = aLy W + aky r w] 66 where Ci is the unit (or average) cost function for sector i, and w and r are the wages and rents respectively. Firms in sector x choose an optimal quality to minimize their total costs in providing services of x according to an idea originated by Swan (1970). From the cost function (3.4- l), the optimal quality can be derived by partial differentiation with regard to q, and it is a negative function of the wage-rental ratio. (3-5) q = J(r akx)/(w a“) Intuitively, increase in quality requires more labor such as scientists and engineers, and excess demand for labor raises wage-rental ratio. Therefore, optimal quality is inverslely related to wage-rental ratio. The cost function with an optimal quality can be derived from the substitution of (3.5) into (3.4-1). 1 (3.6) C = 2 x aankxwr The zero-profit curves in sector x and y can be written as: (3.7-1) 1 = w an + r aky (3.7-2) p = w 0:qu + r akx/q pl 011 be Fe an de pr go re fr- No: file 67 where goods y is used as a numeraire, and p is the relative price of x in terms of y. The slopes of the zero-profit curves which are equal to negative factor intensity ratios can be derived from the differentiation as: (3.8-1) dw/drhrx=0 = - aw/aw = _ ky (3.8-2) dw/clrlvry=o = - akx/qzaLx = - (w/r) = - R For the derivation of (3.8-2), the envelope theorem is used, and q. is substitued into q. The zero-profit curves are depicted in Figure 3.1. The figure shows that there exists a factor intensity reversal in the model because two zero- profit curves intersect twice at E1 and E2. At E1 the slope of ”X is steeper than that of my, and goods x are relatively capital intensive. At E2 goods x are relatively labor intensive. The price p can be solved explicitly as a function of w/r from (3.7) after the substitution of q, into q as: Note that (3—10) (w/r) = 0 ~ p = 0 and lim p = 0 (ti/P)“ also, from the differentiation of p 68 Figure 3.1 Zero Profit Curves Th th Op‘ an: dia pri int The Dos; Upg; 0th 69 (3.11) dp/d(w/r) = {f—‘aLXakx/[aLYW/r) + akyfn a..,(w/r)"’2 - a., o » akY/aLy > w/r Note that the critical point aW/aw (=ky) = w/r (=kx) is the point of factor intensity reversal. The following relationship between the output price and optimal quality can be derived from the combination of (3.5) and (3.12) as: (3.13) q = q(p) q' > 0 -’ R. > k, The relationships can be illustrated in the 4-quadrant diagram in Figure 3.2. Intuitively, increase in the output price of vertically differentiated products which is capital intensive will lower wage-rental ratio (the Stolper-Samuelson Theorem), and this lowered wage-rental ratio will make possible for firms to hire more labor which is required to Jpgrading quality (optimal quality relationship). The value of p at the factor intensity reversal is Dbtained by subtituting w/r = aky/aLy into (3.13): 70 w/r rI-IIII .l'l‘ --------*._- —-——~—-— _. I‘ll-I'l‘l 45° Figure 3.2 Four Quadrant Diagram of the Model Th ha 3D 71 m — (3 ' 14) p — V aankx; aLyaky Whether the useful equilibrium for the economy is E1 or E2 is decided by the relative factor abundancy of the economy. Once the factor endowment is given, the wage-rental ratio of the economy is determined, along with the equilibrium. This can be illustrated by the Harrod-Johnson diagram in Figure 3.3. If the endowment ratio of the economy (=k) is greater than ky, kx is greater than ky. The economy, in this case, will has an E1 type equilibrium. If k is smaller than ky, the economy will has an E2 type equilibrium. Therefore, both types of equilibrium are not possible at the same time. The following relationships represents the above discussion. (3.15) k>ky -' kx>ky -' E1 k E2 The model will proceed on the assumption that the economy is relatively capital abundant, thus goods x are relatively capital intensive. In this case, (3.13) can be written as: (3.16) q = q(p), q' > 0 This fits well with the fact that quality and the output price has a positive relationship. Appendix C shows the way of SPeCifying the production function when goods x are relatively 72 k."W/r kY - aky/aly w/r Figure 3.3 Harroerohnson Diagram de di 73 labor intensive with the same result as (3.16). Thus, the proceeding discussions will apply equally to both case irrespectively of the capital intensity of goods x. B. The Production Possibility Frontier The production possibility frontier (PPF) can be derived from the productions (1) for a given endowment of (K, L) of the economy. The sum of the factors used in both sectors must be equal to the endowment, and this creates the following restrictions: (3.17-1) aky y + (aH/q) x = k (KK) (3.17-2) aLY y + (aqu) x = L (LL) By re-arranging (3.17), we have: (3.18-1) y = k/aky - (av/an) X (KK) (3.18-2) y = L/aLY - (aqu/aw) x (LL) KK curve (18-1) is steeper than LL curve (3.18-2), since goods x are relatively capital intensive. These two curves are. depicted in Figure 3.4. The two intercepts Kq/au, L/auq change in opposite directions as the price changes in the same direction [see (3.16)]. For example, as the price increases from the K/a,Ky ~ 74 K/a” L/ay ° Kq/a... A L/akq x Figure 3 .4 Production Possibility Frontier or mo or F1 th in Fi fa LI. di Re Le 75 original price, which yields the intersection point.Eo,kq/czkx moves further from the origin 0, and L/aqu moves closer to the origin. The restrictions (3.18) change to the dashed line in Figure 3.4, yielding a new intersection point E1. IRepeating this operation on all prices and connecting the resulting intersection points such as E1, the PPF of the thick line in Figure 3.3 can be derived. At these intersection points, the factors are fully employed because two conditions of KK and LL are satisfied at the same time. Note that as the production of x increases, so does the quality. This relationship can be written as: (3.19) dq/dx > o The slope of the PPF can be derived from the total differentiation of (3.17), which is: (3.20-1) o akydy + 8|.de - (akxX/qmq (3.20-2) o = aLYdy + aLxdx + (aLxx/q)dq Re-arranging (3.20) into a matrix form, we have: (3 . 2 1) aky aky dy/dq akxx/q aw aLx dx/dq -aLxx/q Let: (3.2 Using (3.2: There (3.24 The t (3.25 There 76 (3.22) m = I aky akx aLY aLx my = qux/q an = 2akxaLxx/q -aLxx/ q aLx m" = aky akxx/q = -(akyaLx + akxaLy)x/q aw -a..X/q Using Cramer's rule, we have: (3.23) dy/dq = mY/m dx/dq = mx/m Therefore, the slope of the PPF can be derived from (3.23) as: (3.24) dy/dxlppf = my/mx = - ZakxaLx/(akyaLx + akxaLy) = - 2almaLx/(akyaqu + aLyakx/q) The total differentiation of (3.24) with regard to q gives: (3.25) (a/aq) (dy/dx) lppf 2 = 2akxaLx [akyaLx - akxaLy/q]/[akyaqu + aLyakx/q] Therefore, dc of an CU th Nc re tr be th ta be It he 77 (3.26) (d/dq) (dy/dx) Ippf < o if kx > ky The concavity of the PPF is derived from the combination of (3.26) and (3.19). 3.3. Equilibrium and Comparative Statics A. Autarkic Equilibrium The indifference curves of the economy are assumed to be downward sloped and convex to the origin. The homotheticity of the consumer's preference is sufficient for this. An autarkic equilibrium can be illustrated with indifference curves, the PPF and a price line as in Figure 3.5. The production/consumption point A in Figure 3.5 yields the highest level of utility in the economy assuming no trade. Not only is A, the "optimal" production point, it also represents the "autarkic" equilibrium. The marginal rate of transformation and the marginal rate of substitution at point A are equal to the price ratio p'. The existence of tangency between the price line and indifference curve is assumed from the well behaving indifference curves, but the existence of tangency between the price line and the PPF must be proven because of the specifiijroducti n function.used. n‘t e model. It is proven by showing that the price is equal to the negative of the slope of the PPF. The price (3.9) after the SU al or ex cc th va tr ge Or 91‘. an 78 substitution (w/r) from (3.5) is: (3.27) p = { 2./aankx / [a,_y(w/r)+akyl )./w/r eq. (3.9) { 2“ aankx / [aLy(akx/aLx) +aky:I }4akx/aLx substituting (w/r) from (3.5) = 2almalx / (aLyakx + akyaLx) = - dy/dxlppf eq. (3.24) Now suppose that the economy depicted in Figure 3.5 is allowed to engage in international trade. The excess demand or supply of products can be derived at each price. For example, at p1 and the corresponding production-cum- consumption combination (for example, E-cum-C) in Figure 3.5, there is an offer of exports (FE of y) for an equal market value of imports (PC of x), and this offer is represented by trade triangle of EFC. Placing all such triangles in Figure 3.6 (where triangle TBO represents the equal triangle of EFC) generates the offer curve OH. Note that as we moves along the OH further from the origin the price of x decreases, as does the quality of x. B. The Comparative Statics of the Equilibrium Now consider the effects of changing the relative factor endowment on the PPF and offer curve. These effects can be analysed by the Rybczynski theorem. In the following the P1 ;-——---—----¢— 79 h-——¢D--— /0 °r Figure 3.5 Trade Triangle RY mo At a< is Not cor. div let 80 Rybczynski theorem can be proven in this specific production model. Proposition 1: (The Rybczynski theorem) At constant prices, an increase in capital will increase by a greater amount the output of the differentiated goods which is intensive in capital and will reduce the output of the homogenous goods. Proof: From the total differentiation of (3.17), we have: (3.28-1) akydy + akxdx = dk (kk) dL (LL) (3.28-2) awdy + audx Note that akx and ag< are constant, since q is constant at constant prices. Re-arranging (3.28) in a matrix form after dividing it by dk and letting dL = 0, we have: (3.29) aky aLx dy/dk = 1 an aLx dx/dk 0 let, (3.30) m = aky akx = akyaLx - aLyakx a a Home Exports) Y 81 Figure 3.6 Offer Curve X (Home Imports) There: (3.31; Note * (3.32 Of on T0 ' T1 I The R: which the s. °ffer to 11: 82 my = l akx = aLX O aLx mx = aky 1 = -aLY a 0 Therefore, using Cramer’s rule we have: (3.31) dy/dk =IW/m = aLx / (akyaLx - aha”) dx/dk =Im/m = -aw / {akyaLx - aflaw) Note that: (3.32) dy/dk < 0 and dx/dk > 0 since kx > ky Q.E.D. Proposition 1 is depicted in Figure 3.7. The increase of one factor (capital) shifts the PPF outward from 15T1 to T0“Tf. Suppose the economy is at E before a factor increase. The Rybczynski theorem asserts that point E' on the new PPF, which has the same slope as E (= -p) on the old PPF, lies to the southeast of E, as illustrated. Now consider the effects of increased capital on the offer curve. The same diagram of the capital increase is used to illustrate the effects in Figure 3.8. As capital increases in pro: as « tra< of cap: Thi: to lie: pas: Thi: the tra: Offe inc: ace: a r Com eas: 83 in a relatively labor abundant country, the economy will produce relatively more capital intensive differentiated goods as explained by the Rybczynski theorem, and thus reduce the trade. The increase in capital will neutralize the difference of relative factor endowment between this country and the capital abundant country. Therefore, trade will be reduced. This is illustrated in Figure 3.8. ' An increase in capital shifts the PPF outward from T6“ to TO'T1', and production will change to the point E1 which lies southeast of the old production point B at given price p. Consumption will change to C1 which lies on the ray passing through C. Thus, trade will shrink from EFC to~EfifiC1. This reduction in trade is represented as an inward shift of the offer curve from OH to OH' in Figure 3.8-B. INote that the trade triangles EFC and E1F1C1 in Figure 3.8-A are equal to the offer triangles EKG and B'K'O in Figure 3.8-B respectively. By the same reasoning, we can analyse the case of increased capital in a capital abundant country. This will accentuate the relative factor endowment of the country. As a result, trade will expand, and the offer curve of the country will shift out. The following theorem on the pattern of the trade can be easily derived from proposition 1. Proposition 2: (The Heckscher Ohlin theorem) A relatively capital abundant country has a comparative 84 El Figure 3.7 Rybczynski Theorem es y . 1;“ To‘ Expansion Path P P 0 T1 T1' X Y P H H! 5) I I I I I I J L 0 K' K x Figure 3.8 Effects of Capital Increase adv: proc *0 I1 9 whe abu whe la}: of The Ex] OP] 86 advantage in relatively capital intensive differentiated products. Proof If a country is relatively capital abundant, proposition 1 tells us that: (3 . 33) sx/sx > 3x73; where Si and Si. are the supply of goods i by the capital abundant and the labor abundant country respectively. Assuming that two countries have the same tastes: (3.34) Dx/DY = Dx/Dy where Di and D; are the demand for good i by the capital and labor abundant country respectively. The world consumption of each good equals the world supply. Thus: (3.35) sx/sy > Dx/DY = (3x + sx')/(Sy + Sf) * * *8! = Dx/Dy > Sx/ y The first inequality says that the capital abundant country exports x and imports y, and the last inequality says the Opposite about the labor abundant country. Q.E.D. goc frc C01 Hec V6: ma: wh ab Pr 9)! la 87 The next proposition relating quality of differentiated goods with autarkic equilibrium prices can be derived easily from the patterns of trade. Proposition 3: (Quality in Autarky Economy) In autarkic equilibrium, the capital abundant country produces lower quality differentiated goods than the labor abundant country. 3% Proposition 2, which states the physical version of the Heckscher-Ohlin theorem, can be transformed into the price version of the Heckscher-Ohlin theorem assuming no factor market distortions as: (3.36) pA < p; where pA and p: are the autarkic prices of capital and labor abundant country respectively. This with (3.16) proves proposition 3. Q.E.D. Once trade opens, the capital abundant country will export differentiated products which will be imported by the labor abundant country. At free trade the equilibrium price of of eq he is 9Q tr At ec. 88 of the differentiated goods is determined by the intersection of the offer curves of the two countries. The international equilibrium is depicted in Figure 3.9. In Figure 3.9 OH and OF represent the offer curves of the home and the foreign country respectively. The home country is assumed to be relatively labor abundant. Op* is the equilibrium price at free trade» The law of one price at free trade gives the following proposition. Proposition 4: (Quality Equalization at Free Trade) At free trade the quality of the differentiated goods becomes equal between countries, and determined at world trade prices. 3.4. Policy Issues (1): Tariff, Quota and Voluntary Export Restraint A. Tariff In policy analysis trade indifference curves (TIC) , which were originated.by Meade (1952), are used to study the welfare effects of various trade polices. Trade indifference curves represent the locus of imports and exports which brings equal welfare to the economy. TICs are depicted in Figure 10 for the country which imports goods x for exports of goods y. TICs have the following properties which give rise to the concave shape: (1) An increase of imports (dx) is required to Cl g1 89 compensate for an increase of exports (dy), and (2) the more the country imports x on any TIC the greater is the increment of imports x which is required in order to compensate for a given increment of exports of y. As a country moves toward the southeast direction of the trade indifference map following the arrow in Figure 3.10, the country has more imports for less exports, and the welfare of the country increases. Every point of the offer curve is a tangency between the price line and TIC, since the offer curve is derived to maximize welfare. For the country which imports goods y in exchange for goods x, the TIC is concave to the axis Y which is the mirror shape of TICs of Figure 3.10. The welfare of the country increases as the country moves to the northwest direction. Suppose there are two countries, say home and foreign. The home country is assumed to be relatively labor intensive and imports differentiated products x. The international equilibrium is represented by the offer curve of the two :ountries in Figure 3.11. The offer curve of the home country DH intersects the foreign offer curve OF at point A generating the international equilibrium price p'. At autarkic equilibrium, the two countries do not engage in trade and remain at point 0 where the autarkic prices are 9; and p: for the home and the foreign country respectively. .. J (Home 90 {Home Exports) Y t a 0 X (Home Imports) Figure 3.9 International Equilibrium Thi: thai dif: quai a d: it Hec. end goo opt 0011 OH' bet tra Off 0K Pay Par onl 91 (3.37) pAF < p* < pAH This is a graphical exposition of proposition 3 which states that the capital abundant foreign country has a lower quality differentiated goods than the labor abundant home country. At international trade, goods x will have the same quality for both countries corresponding to pa This requires a decrease of quality for the home country and a increase of it for the foreign country. The pattern of trade between the countries is typical Heckscher-Ohlin type and determined by the relative factor endowments. The services, not quality, of the differentiated goods matters for consumers, and quality is determined by an optimal behavior of firms given price in this model. Now consider the effects of tariffs imposed by the home country. A tariff will shift the home offer curve from OH to OH' as in Figure 3.12. A shift in of the offer curve is due to the difference between the offer of consumers and that actually presented to trade after tariff. For example, at point B on the free trade offer curve, it indicates that in order to obtain the quantity OK of imports, consumers of the home country are willing to pay BK of exports at a price pt. But if there is a tariff, part of the total payment BK must be paid as a tariff, and only a portion will be left for the foreign country. Point A' in Figure 3.12 is drawn so that BK'/A'K equals the tariff (Home Ex 92 e Exports) 7 0 x (Home Imports) Figure 3.10 Trade Indifference Curves (Home E 93 5 Exports) X (Home Imports) Figure 3.11 International Equilibrium and Autarkic Prices rat inc pri The bet qua qua. 00113 of t fore PM brou rest in diff dime and ths 94 te t. At the new equilibrium A', the tariff-ridden price pt :reases from the free trade price p', and the international ice becomes p'. .38) pt = p'(1+t) > p' > p' arefore, tariffs will destroy the equality of quality :ween the two countries. The home country will produce the ality corresponding to pt, and consume a part of the lower ality corresponding to p' which is produced by the foreign intry. Considering the TICs for these two countries, the welfare the home country can only be improved at the expense of the :eign country. Quota A quota limits the physical units of goods that can be >ught into the country. For homogenous goods a quota :tricts total imports quantity since the goods are measured one-dimensional physical quantity. For vertically Fferentiated goods, the goods are measured in two- lensional total services which is a product of unit quality . physical quantity. In this case, quota which restricts 'Sical quantity leaves quality to change freely. Thus, the (Home EX] Y t 95 ExPorts) HI fi x (Home Imports) Figure 3.12 Effects of a Tariff B) in in Co: Po. do. 96 effects of quantitative restrictive policy are different between the two types of goods. If a quota is imposed by the home country, exports of the foreign country are subject to limits on the specific physical units but can be increased by providing higher qualities. Consider the PPF in terms of Q and y to understand how it is related with the PPF of x and y. The production function of Q is given in (3.3). From the production functions (3.1-1) and (3.3), the following factor endowment restrictions are derived: (3.39-1) aky y + akx Q = K (K) (3.39-2) aLy y + (aqu2) q = L (LL) By re-arranging (3.39), we have: (3.40-1) y K/aky - a)... Q/aky (KK) (3.40-2) - L/aLy - aquZ/aLYQ (LL) I< I KK is steeper than LL, since goods x are relatively capital intensive. These two curves are depicted in Figure 3.13. The PPF of Q and y can be derived by connecting the intersection points between KK and Id. as quality changes corresponding to the change of the price. The intersection points of the two restrictions lie on the KK line below the dotted horizontal LL line for zero price because only the LL our by -ak’ and Arr hav BY The Ser 97 rve is a function of quality. The resulting PPF is depicted the thick line in Figure 3.13. The slope of the PPF is simply the slope of the KK line X/aky. This can also be shown by the differentiation of K 1 LL. .41-1) aky dy + an dQ = o .41-2) aLY dy + aqu2 dQ + 2aquQ dq = 0 ranging (3.41) in a matrix form after dividing it by dq, we 16: . 42) aky akx dy/dq . = 0 aLy aquz dQ/dq -ZaquQ Cramer's rule, we get: 2 .43) dy/dq = 2ozquQorkx /(akyaqu ' aLyakx) dQ/dq = -2akyaquQ /(akyozqu2 - aLYakx) arefore, we get: -44) dy/dolppf = (dy/dq)/(do/dq) = «.../a, The transformation of the quantity version PPF into the TVice version PPF can be done by multiplying Q by q, since Fig int eqr deg V8] oh: 281 98 e = qQ. Notice that the PPFs of both the service version in ?igure 3.4 and the quantity version have the same vertical intercepts of KK and LL. The horizontal intercepts are also equal if q = 1. In this case x = Q. This comparison can be iepicted in the same graph in Figure 3.14. The quantity version PPF-is transformed into the service Jersion in Figure 3.14. The following relationship is observed because quality q is an index number greater than zero: (3.45) O 0, for v > 0 (d) h"(v) > 0 Property (a) follows directly from the definition of the compensation ratio. Property (b) is required for the consumer's tangency solution at the most preferred specification, implying that this is indeed an optimal specification. Property (c) means that every variety other than the ideal one requires positive compensation. Finally, property (d) assumes the convexity of the compensation function. A typical compensation function is drawn in Figure 4.1. Using the compensation function, the utility function can be defined as: (4.2) u(c1, ...,ci,...,cn) = 23 HICi/hWiH where ci is the consumption of the available variety, vi is the distance between variety i and the ideal variety, and h(vi) is Figure 4.1 Compensation Function 139 the compensation function which converts ci into the equivalent quantity of the ideal variety. Therefore, all of the available n varieties enter the utility function additively, and are measured in units of ideal variety. This utility function is assumed to have the following more specific function form for further analysis: (4.3) u(c1,...,ci,...,cn) = 33 [oi/h(vi)]b o < b IA H where b is a parameter related to the price elasticity. Consumer demand for variety can be derived from (4.3) by utility maximization subject to the budget constraint of: (4.4) 2% pkn = I where I is total income of a consumer; Depending on the value of the parameter b, the consumer's problem can be separated into two different cases. Case 1: b = 1 This case corresponds to the example presented by Helpman & Krugman (1985, ch. 6). The consumer's choice depends both on prices of available varieties and the distance of the available varieties from.his ideal varietyu A consumer either specializes in one variety or consumes a mixture of varieties which offer the lowest effective price, the price which 140 satisfies the first-order-condition. Forming the Lagrangian, we have: (4'5) L = 2i [Ci/h(ViIJ + I“ [ I ' Z Pici ] where p. is a Lagrangian multiplier. First order conditions are: (4.6) aL/aci = 1/h(vi) - )1 pi S 0, strict equality if ci > O Re-arranging terms: (4.7) 1/p. s p‘. h(vi), strict equality if ci > 0 Suppose we numbered goods in such a way that the following ordering was true: (4.8) p1h(v1) 5 p2 h(vz) 5....5 pn h(vn) u is adjusted so that: (4.9) 1/p. = p1h(v1) 5 p2 h(vz) S...S pn h(vn) If p1 h(v1) < p2 h(vz) , the consumer specializes in variety 1. If p1 h(v1) = p2 h(vz) < p3 h(v3) , etc, the consumer divides his 141 income between goods 1 and 2 but consumes none of the other varieties. Case 2: 0 < b < 1 This is a: more general case and the concern of this paper. This case eliminates the "arbitrary" problem in the choice decision of a consumer. A consumer chooses a positive amount of every variety. The Lagrangian is: (4.10) L = z. [Ci/h(vi)]b + )3 [ I - 2 pici ] I First order conditions are: (4.11) Mei/mmnb‘h/MVIH - up. 5 0: a strict equality if ci > 0 Re-writing the above equation, we have: b 64 . . . (4.12) b[1/h(vQ] ci S upi, strict equality 1f ci > 0 » 1/p S {pi/b}[h(vi)]b cfl'b, strict equality if ci > 0 Notice that if we set b = 1, this is the exact same conditions as we had earlier in equation (4.7). In this case of b = 1, the right hand side of the above relationship is independent of c? 142 However, if 0 < b < 1, the right hand side of the above relationship increases in ci. For given values of p, pi, vi and b, there exists a solution entailing positive c‘- for all i (i.e. consumers diversify their consumption.) [see Figure 4.2] To solve for u and get a complete demand specification, arbitrarily choose one of the varieties to be numeraire, e.g. variety 1. In this case., p1 = 1 and we have: (4.13) (p1/b) [h(vmb c.” = 1/ u = (p./b)[hp-""’"’ ( . ) ""‘b’ (+) <-> (4.20) ave/av. = II/I.II(.3"“"”(b/<1-b)>I.>‘2""”"'b’ h(VIIh' (Viv-[h(vi) )2 + [./-[. ]2]pib/(b°1) (ID/(3:10)){-}‘2'°"””'b’h(’\q)h'(V,-)(o)”‘1't”/{h(V.-)}‘2 (+3 (+3 (4.21) aci/aI = (.)b""b’(-)W'b’/[.] The partial derivatives with respect to pi and vi have two 145 parts which have opposite signs. Thus, ace/9P3 and dci/avi seem to have indeterminate signs. For a large number of varieties (n is large), [.12 dominates the second part making it close to zero, and the first part dominates total effects. Notice that n is assumed to be a large number. Therefore, we have the following properties of consumer demand: (4.22) aci/api < 0 aci/avi < 0 aci/JI > 0 (A-2). Market Demand Market demand can be derived from the individual demand (4.18), i.e. market demand is a total sum of (4.18) over all consumers. For an actual calculation, we need both a distribution of consumers and varieties along the circumference on which the varieties can be represented by points. It is assumed that preferences for ideal products are uniformly distributed over the unit length circumference of the circle and the population density on the circumference is equal to L. Notice that L is both the density and the size of the population. From the unit length circumference, the demand for citur a consumer whose ideal variety is i is represented by p01nt c in Figure 4.3. The minimum demand is from a consumer at point A with vi = 1/2, and the average demand is from a B (vi - 1/4) A (vi - 1/2) Figure 4. 3 Consumer Distribution 147 consumer at point B with vj = 1/4. We will approximate a market demand by multiplying the average demand (demand by a conSumer with,vq:= 1/4) by the total number of population L. The above approximation becomes an actual market demand if the second derivative «Jaci/avi2 becomes zero. This is depicted in Figure 4.4-As If aQn/avf is less (greater) than zero, an approximation exaggerates (decreases) the actual demand. These two situations are illustrated in Figure 4.4- B and 4.4-C. An approximated demand.xh can be written as: (4.23) x. = L Ci" = L ci(vi.. = 1/4) I = L [ Ith.)/h(v">>"""‘” I/[szthn/thj)WM” pib/(b-1)] ] (l/pi)1/(1-b) The denominator of (4.23) can be simplified if we denote: (4.24) {h(v1)/h(v")}b’(1'b’ pub/w") = (l/n)[ 2,- {h(v.)/h(v,)I”"""’p.b"”"’3 Then: L [ { [h(v1)/h(v")]b’“'b’ I }/{ n p"b/(b-1) (4.25) x. I [h(V1)/h(V")]bm'b’ I I <1/p.>"“'b’ b/(b-1) (l/pi)1/(1-b) (L I/n) (l/P") The market demand (4.25) is shown as a function of a share of 148 A . c«;\ (A) claim I CI cm can (3) 0cmucmhl/2 - Actual Demand 0AB1/2 - Approximated Demand Cmm I vi 0 1/4 1/2 1 cm: q A c B In C \- ( ) cm“ I vi 0 1/4 1/2 Figure 4.4 Approximation of the Actual Demand 149 variety (S) from total GDP (LI): (4.26) s = L I/n and its own price (pi) and an average price of all other varieties (p). The price elasticity of the demand can be easily calculated as: (4.27) ep = 1/(1-b) B. Supply Side All goods are assumed to be produced with the identical cost function. The labor used in producing each good is a linear function of output xi: (4.28) 14 = a + B xi where li is the labor used in the production of good i, x5 is the output of good i, and a is the fixed cost. This input requirement function specifies economies of scale with decreasing cost and constant marginal cost as output increases. Monopolistic competition of the Chamberlinian type is assumed.in theidifferentiated.goods market» ZEach firm chooses its price given cost conditions which are known to everyone. 150 The cost conditions of all the different types are assumed to be the same as (4.28). Thus, firm i's problem is to maximize its profit: (4.29) n} = ppn - (a + B x) w where w is wage rate. In its maximization solution, a firm earns a positive profit if its price lies above the AC curve. This situation is termed a short-run equilibrium of monopolistic competition. In the long-run, the entry of firms into the industry will drive profit to zero. Therefore, in the long-run, each firm must charge a price pi and produce at output xi with zero profit: (4.30) n = ppg - (a + B X” mr==0 This means that price must equal average cost for each firm in the long-run equilibrium. In addition, each firm must be at the maximal profit point on its demand curve: any inefficient firm will be driven out of business by the entry of other firms. Thus, the demand curve facing firm i must be tangent to its average cost curve, see Figure 4.5. From profit maximization of (4.29), firm i chooses its price given the market demand for its products (4.25). The profit maximization price depends on marginal cost (8) and on n 151 Demand -9 AC Figure 4.5 Monopolistic Competitive Equilibrium 152 the elasticity of demand (4.27): (4.31) pi (1- 1/ep) = 3 w or gn/w = B/b Since elasticity of demand and marginal cost are constant in this model, the profit maximization prices of firm i. are proportional to wage rates as in (4.31). From the zero-profit condition (4.30), the price of the firms in the market equals AC: (4.32) pi = (a/x + E) w or pi/w = B + a/x In addition to the two conditions of (4.31) and (4.32), we have a factor' market equilibrium condition with full employment. Full employment implies a sum of factor employments of n firms equals total labor L: (4.33) L = Eilfi = 2 [ a + B xi] Notice that there are three endogenous variables: pL/w, the price of each good relative to the wage; x, the output of each good; and the number of goods produced, n. To make the analysis simple, we assume a symmetry in every good produced which requires every variety having the same price and quantity of production. Thus, from now on, we can use 153 variables without subscript i: (4.34) p = pi, x = xi, c = c. and 1 1i, for all i We can re-write (4.33) with symmetry. (4.35) L n (a + B x) or n = L/(a + B x) The number of goods produced is determined by the total labor force divided by the labor requirements of each firm. This is shown in (4.35). By re-writing (4.23) in a shorthand notation, we have: (4.36) x = L c" Therefore, the consumption of an average consumer (c") can determine the output of each firm (x). Once p/w and cTare solved from (4.31) and (4.32), n can be determined from (4.35). The graphical solution of (4.31) and (4.32) is shown in Figure 4.6. The profit maximization condition is line PP and is horizontal because the elasticity of consumers is constant. The zero profit condition line 22 is negatively sloped because it decreases as average consumer demand increases. Notice that B/b is above 8, since 0 < b < 1. If b = 1, 154 P/w z t P B/b ' p I I I I I 8 --3- ----- - ~ --I- -‘- - -'- -3- - - - - -‘- I I I l . I I ' I 0 can C" Figure 4.6 Graphical Solution of the Model 155 then B/b equals 8, and there is no solution in this economy, since there is no intersection between 22 and PP schedules. This fact shows that the "arbitrary" case:of consumer decision is not compatible with the monopolistic competitive model of this paper. In the next chapter, we use the model to analyze the effects of trade. 4.3. International Trade Suppose there are two countries which are identical in every respect. In standard H-O-S models, there is no reason for trade because trade results from the difference of factor endowments between countries. In this model, there will be both trade and gains from it. Two countries with identical technology and tastes can be integrated as one country as trade opens. Thus, the effects of trade are identical to the effects of labor growth in the economy. Furthermore, the effects of trade can be analyzed as a change of the parameter, labor, of the model. The effects of labor growth are depicted in Figure 4.7. As labor grows, PP is constant because the profit maximization condition (4.31) does not depend on labor, but the 22 schedule shifts to the left because p/w is negatively related to labor in the zero profit condition (4.32). Therefore, the equilibrium of the model changes from A to B, which is the new intersection point of PP and Z'Z'. 156 NW fl/b p cl! Figure 4.7 Effects of International Trade 157 At B consumer's demand (c) falls and p/w remains unchanged. Thus, the output of each firm will not change because there is no change in equilibrium price. The output of each firm can be derived explicitly from (4.32) as: (4.37) x = a/(p/w - 8) There is an increase in total number of firms in the economy, which can be derived from (4.36) as: (4.38) n = L/(a + BLc) In (4.38), n increases as L increases and c decreases. Intuitively, this result implies that an increase in labor requires each consumer to spend less on each variety for the firm to stay at zero-profit because each firm's output remains constant. The consumer's budget now spreads out over the increased variety given constant income. .As a result of trade, the number of varieties will increase, and each variety will be produced in the same amount irrespective of trade. Notice that the firms' output is independent of the labor force because of the constant equilibrium price, which in turn is based on the constant elasticity of demand. Note that SP is assumed to be a parameter defined in (4.27), and this assumption is fundamental to a Lancaster type specification of the 158 preferences in the model. Therefore, free trade and resulting market integration increases the total number of varieties available in the economy, Krugman (1979) presented a similar model with Dixit- stiglitz type preferences in which both the output of firms and the total number of varieties in the market increase in free trade. His conclusion is based on the assumption that elasticity of demand decreases as consumer demand increases. Therefore, the elasticity with regard to demand is critical in determination.of the change of each firm's output" In this model of Lancaster type preference, the output of each variety does not change. Consumers of the economy will gain from trade because of the increased variety. Gains from trade can be seen from the utility function of consumers (4.3); it increases with the new increased number of varieties. From (4.3) an increase in welfare results from an increased number of varieties because variety is valued in itself: An increase in variety will increase utility. 4.4. Conclusions This paper presents a monopolistic competitive model with Lancaster type preferences, in which each consumer has an ideal variety and compensation function. It shows that intra- industry trade occurs in order to take advantage of the 159 preferences between symmetric countries which have the same technology and tastes. ZFree trade and the resulting extension of market will provide more varieties than a closed economy, and the welfare of the economy increases. Contrary to Krugman's (1979) model which is based on Dixit-Stighitz type preference with a variable elasticity assumption, this paper shows that individual firms' output is unchanged with trade. Firms have no incentive to increase their output if demand elasticity is fixed as in this model. Therefore, the extent of the utilization of scale economies by each firm depends on the elasticity of demand. However, the limitation of this model is its use of the approximated demand for the market demand. The market demand should be solved for more explicit analysis of the model. APPENDICES A: Quality Dimension in the Cost Function The structure of the cost function with. a quality dimension can be developed from the general form of the cost function used in economics. Without considering the quality dimension, the cost function can be expressed as the sum of variable costs plus fixed costs: (34.1) C(Q) = V(Q) + P where Q represents the total number of quality goods produced. For a U-shaped AC curve, V(Q) will take a quadratic form such as, V(Q) = Q2. Now there are three different ways in which the quality dimension (quality) can be entered into the cost function of (A.1). Each of the cost functions depends upon the different assumptions on how the change of quality level affects the costs of production. First, if the quality level produced only affects the fixed cost, the cost function looks like this: 160 161 (A-Z) C(Q, q) = V(Q) + h(Q) F Second, if the quality levels are assumed to affect only the variable cost, the cost function can be written as: (A-3) C(Q, q) = h(Q) V(Q) + F Third, if the quality level produced affects not only the fixed cost but also the variable cost, condition which are more true of reality, the cost function can be written as: (A-4) C(Q, Q) = h(Q) E V(Q) + F ] The above three cost functions have different curvatures depending on how the quantity corresponding to the minimum AC changes with respect to the quality levels. When the quality only affects fixed cost, the cost function has the following properties. From (A.2), AC and MC are as follows: (A-S) AC = C(Q, q)/Q = V(Q)/Q + h(q) F/Q MC = 'aC(Q, q)/aQ = V'(Q) The minimum point of AC can be determined by equating AC to MC. Using the example of V(Q) = Q2 for the U—shaped curve, the output level compatible to the point of minimum AC, denoted by Q', can be determined as: 162 (23.6) 0' = fqu) F Thus, as quality level q increases, Q' increases too. This is because a higher quantity of goods must be produced to absorb the higher' fixed cost -required for' higher quality goods. Therefore, as quality increases, the AC curve reaches the minimum point at a greater quantity. This is illustrated in Figure A.1. By substituting Qi and h(q) = qr into AC, we can derive the minimum of the AC as: (A.7) min. AC = 2/F q"2 When the quality level is assumed to be added only to variable costs, we can derive AC and MC from eq. (A.3). Using the specific functional form, V(Q) = Q2, we get the following minimum point of AC: (A-8) AC = h(Q) V(Q)/Q + F/Q MC = h(q) V'(Q) Q' = J F/HIqI Therefore, for higher quality goods, Q' has a lower value. This is because the higher variable costs corresponding to higher quality goods increase AC at an earlier stage compared 163 AC AVC(qo) - AVC(q1) AC(ql) I AC(Qo) I I I l I I l ' I l I I AFCIq.) l I I I g I AFC 0 p"(q) = 0 In this case, the budget constraint, which is a straight line (by p" = 0), and the indifference curves are drawn in Figure B.1. Three possible cases of the consumer's maximization are as follows: (1) 9 < p'(q): Consumers purchase q = 0 (equivalent to X = 0) (2) 0 > p'(q): Consumers purchase the highest quality available, might entail m = 0 if the highest quality available is high enough. (3) 0 = p'(q): There are an infinite number of solutions to the consumer problem. The other restriction is stated as follows: (B-Z) P(0) = 0 P'(CI) > 0 P"(CII < 0 The consumers' maximization problem is drawn in Figure B.2 with the budget constraint which is convex to the origin ( p" <0). Except in the indeterminate case (3) from ‘the first restriction, consumer'maximization yields corner solutions (1) and (2) above. 169 Budget Constraint Figure 3.1 Corner Solution: A Case of p"(q) - O 170 Budget Constraint \ Figure B.2 Corner Solution: A Case of p"(q) < 0 171 C: The Production Function When x is Labor Intensive The production functions for the homogenous goods y, and the differentiated goods x are: (C.l-1) y Min ( ky/aky, LI/aLy I (CA-2) X = Min { Jg/akx. Lx/aLx) = Min I kx/akxq, Lx/(aLx/q) I with akx = akx aLx = au/q The production function of (C.1) is derived from the combination of (2) and the following production function of Q: (C2) 0 = Min { Ig/akxqz. Lx/au I The cost function for sector x and y can be derived from (C. 1) as: (C.3-1) cx‘- (an/q) w + (auq) r (C.3-2) c a Y Lyw+akyr The optimal quality is derived from the partial differentiation of (C.3-1) with regard to q: (C.4) q' = ./ wakx/rau 172 The cost function with an optimal quality can be derived from the substitution of (C.4) into (C.3-1) as: t (C.5) ox = Zjauauwr The zero-profit curves in sector x and y are: (c.6-1) 1 = w a + r aky LY (C.6-2) p = wan/q + rakx where goods y are used as a numeraire. The slopes of the zero-profit curves are: (c.7-1) dw/drl1ry=0 = - aw/aw = -.ky (c.7-2) dw/drlnx=o = - akqu/aLx = - w/r = - kx Note that these are the same as (8) of the text. The price p can be solved as a function of w/r from (C.6) with the substitution of q' into q as: (C-8) 9 = { ZJaankx/(aLyW/r) + aky) ) Jw/r (C.8) also equals p of the text, (9). 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