W" 1 d C' H '5. 2335.33? {41:- :4. Cr... ‘ 1 r. '44:} 4. 1%! 6416445 1."! 4.43453“ .4: I ab 3;": zkflé T?“ a . J'.) “.544 41"! :gaws’mazfic'w “:34 "I $53374! \‘Yqv‘! ijmwi.“ uflSfllfiy‘fi ”3'“ ”m If“ {If 74,. ixfififi‘iimi .4. 45.4.1.4 ‘ ‘4” II.-.» .' 1'}1.§“.‘A" F'. 51;; 11:" ‘ $1494: Islam, 5,3,4. -+“.‘ ‘L-Z" Fri.“ Wu" “F, 1‘} ‘1 1L?” w§oafifi~ u‘ w P k- .. t 1 l “’51.. nA; ll L :Hv Saki} * In... ' " . «gar 4w: 7 ,4 ~44” -I .‘t,:;‘..‘} 3" . , 9 IKIYADZ , ’JImtrrii m .. ‘1' 41¢ Ink“? . VII. , +3.,“ 4% fig??? ? Var, 'JI‘ i 7-; 4"; "fif’fl‘m‘. "4 mi”, E , . in I § ’I’H ! 1.. HM '3, II‘WU. Y . I > r u , r wax-gm. 4'4 u I?“ "rv ml. . I ..-\7./ tgg“? :‘liprv. né‘fii“ ‘ t:‘ 4) N “ U .4 4 W 4 I ‘ 2 "1336;: w uk‘i ‘ L2...- . in" 6.5, -, 99's 34%;; fikgfif ‘7'. _. I....I . ‘33::313-‘3‘ I . ' ”4“, $2513” '- - Q35? : ,, , “i‘ 4‘53? ' 4L ibm‘wfi M“-.. 7%?“ 4. @4457“ 4 3 3; . My}, $352, . 44., 4%. ‘1’:- v 345.44, 4;; I214. d44.44 . . 444.24% 44.44.24 I w . AT: 7-. ‘ .u v u . jam”; 01 .u i", u‘n‘x‘KBJ ~* Mt «Namath - 12 Cd. my .9 ' L. :IH‘ rag} .. .J .3 3L ». ~‘...fi.... -4... ”u... ~ .... 373—3.: “11-... - {nu-l 7:. V my." .. 5311‘. .4,” mu”...- . ... us. 1’-" an. 1- ‘.-“IL‘_] ., ... 5"“.4} ”v .'.I In, y.:""}:"."1'.1 ,1 llllllllllllllllllllllllllllllllllllllllllllllllllllll 3 1293 007905 LIBRARY Michigan State University '1 l k l 4 -‘ —A4 V ____.....-—-_..7_ f; A This is to certify that the thesis entitled A PROBABILISTIC CHOICE MODEL FOR ANALYZING THE DEWAND FOR FOOD IN SENEGAL presented by Aliou Diagne has been accepted towards fulfillment of the requirements for M.S. Agricultural Economics degree in gamma Major professor Date June 15, 1990 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. r—r——_——_.——'—=—_——————_— DATE DUE DATE DUE DATE DUE —_l l MSU Is An Affirmative Action/Equal Opportunity Institution 7 chrnJ-pd ‘ A PROBABILISTIC CHOICE MODEL FOR ANALYZING THE DEMAND FOR FOOD IN SENEGAL BY Aliou Diagne A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of EASTER OF SCIENCE Department of Agricultural Economics 1990 3C4 é45— 688] ABSTRACT A PROBABILISTIC CHOICE MODEL FOR ANALYZING THE DEMAND FOR FOOD IN SENEGAL By Aliou Diagne This paper develops a structural model of household food consumption in Senegal. The model is based on the assumption that the household does not maximize the utility of the raw food staples but instead maximizes the utility of the dishes derived by means of some technological transformation of these raw food staples. The household maximization problem is solved to show that both the unconditional indirect utility and expenditure functions depend on the relative prices of the raw foods only through the costs of the dishes. Methods of estimation are discussed in detail, and the asymptotic distributions of the estimators are derived. The traditional model of food demand is shown to be a special case of this model, corresponding to the restriction of no dish choice effects. Means of testing for this restriction are provided. Finally, the elasticities of demand are derived and the policy implications of the model are discussed. To Hy Parents, Brothers and Sister. For their continuous, moral and material supports. iii indivi profes :hesi: variOi to pu and e Mef 9°85 PIOr 80 f0. \ SE:- ACKNOWLEDGMENTS I would like to acknowledge gratefully the help of many individuals who made possible the completion of my master program. The person I owe a great debt is Dr. Eric Crawford, my major professor. From the beginning of my program to the completion of this thesis he was always available to guide me through my course work and various phases of the writing of this thesis. His early encouragements to pursue the main idea of the thesis up to completion were motivating and extremely useful. I also thank the two other members of my thesis committee; Dr. Peter Schmidt who took his precious time to read early drafts of parts of the thesis and make useful corrections, and Dr. Eileen van Ravenswaay who made useful contributions to improve this work. The basic idea of this thesis was presented in the 99S departmental seminar directed by Dr. John Staatz. I would like to thank him for his comments in the seminar paper and for directing me to part of the relevant literature. I also thank Dr. James Oehmke for his useful comments on parts of the thesis. This work could not have been possible without the excellent and stimulating learning environment provided by the faculty members and staffs of the department of Agricultural Economics. I would like to thank them all. Special thanks go to Sherry Rich who typed most of the paper. Her help was essential for the completion of the thesis. The staff in the Ag. Econ computer services (Chris Wolf, Margaret Beaver, and Elizabeth Bartilson) were iv also he remaini (Scepha Baird, were ve to my 5 Boubaca Haaado: suppor Ante D study develc Studie Econou and tc Contii trust State burea 0133 lade in Se also helpful during the editorial stage. Needless to say, all the remaining errors in the thesis are of my sole responsibility. My thanks go also to my fellow graduate students in the department (Stephan Goetz, Don Hinman, Paul Wessen, Valentina Mazzucato, Katie Baird, Bill Guyton. Odinga Jere, Joseph Siegle, Jim Sterns, etc..) who were very supportive and made my studies at MSU enjoyable. Thank also to my Senegalese friends at MSU (Ousseynou N'doye and his wife Aida, Boubacar Barry and his wife Maguette, Desire Sarr and his wife Julienne, Mamadou Badiane and his wife Mama, and Habibou Gaye). Their moral support is gratefully acknowledged. My deep gratitude to two of my professors of mathematics at Cheikh Anta Diop University at Dakar. Pr. Sakhir Thiam who encouraged me to study economics and his assistant Dr. Mary Teuw Niane who helped me develop the scientific and mathematical maturity needed for graduate studies. My gratitude goes also to Dr. Charles Becker director of the . Economics Institute of Boulder Colorado, from where I learned English and took my first economics course, for his trust, enoouragements and continuous concerns about my academic progress. . A very special thank goes to the US. AID mission in Dakar. Their trust and continuous financial support during all my master program is gratefully acknowledged. The administrative support of the USDA\OITD bureau in washington (especially James Culley) and of Bids Keaton of the 0188 at MSU are also gratefully acknowledged. Their combined efforts made my stay in the United States very smooth. My deepest thanks go to my parents, brothers, sister and friends in Senegal. To my farther M'baye Diagne and my mother Mareme Samb for their love Gesture for to mV‘ FOLKS! tater. Tag?" unlimited expressed people of the envir their love and permanent support, to my older brothers Mamadou and Ousmane for their moral and financial support through all my education, to my younger brothers and sister (Souleymane, Osseynou, N'deye fatou, Matar, Tapha, Gora, and M'baye) and to all my friends for their unlimited moral support. For all of you, my deep feeling cannot be expressed in words. Finally, but certainly most importantly, I would like to thank the people of Senegal and of the United States who provided the means and the environment for the success of this learning experience. vi CHAPTER 1. INIROD'C C1 LTEIDETI 2.11718 $0 2.2 Sepa: TABLE OF CONTENTS CHAPTER 1. INTRODUCTION . CHAPTER 2. THE DETERMINANTS OF THE DEMAND FOR CEREALS IN SENEGAL . 2.1 The Socio-Economics of Food Consumption in Senegal . 2.2 Separability in the Food Consumption Choices . CHAPTER 3. FOOD CONSUMPTION AND THE HOUSEHOLD PRODUCTION MODEL . CHAPTER 4. THE MATHEMATICS OF THE PROBABILISTIC CHOICE MODEL 4.1. The Structure of the Choice System: Definitions and Assumptions 4.2. The Mathematical Structure of the Model . 4.2.1. Construction of a topology on the alternative space 4.2.2. Existence of a utility function on the alternative space. 4.2.3. The Randoh Utility Model . 4.3. Solution of the Household Maximization Problem . 4.3.1. The General Solution. 4.3.2. The structure of the feasible set of alternatives 4.4. Properties of the Utility, Expenditure, and Demand Functions. 4.4.1. The direct conditional utility functions . 4.4.2. The conditional expenditure functions vii 13 21 39 42 47 47 48 49 51 51 53 S9 59 64 4.4.3. Proper functi “.4. The 2‘. 4.4.5. The \ “.6. Dish 5.5. Parent ”.1. Par II.5.2. The 5.23111“ 5.1. The 5'2. Met 5.2.1. , 5.2.2. 5.2.3. 4.4.3. Properties of the unconditional indirect utility and expenditure functions 4.4.4. The Marshallian and Hicksian demands for the dishes . 4.4.5. The unconditional demands for the raw foods 4.4.6. Dish selection as a cause for zero quantity reported. 4.5. Parametrization of the Conditional Indirect Utility Functions 4.5.1. Parametrization of the meal budget. 4.5.2. The AIDS conditional indirect utility functions CHAPTER 5. ESTIMATION OF THE MODEL 5.1. The Model Under Normality 5.2. Methods of Estimation Under Normality . . 5.2.1. Assumptions and notations . 5.2.2. Maximum likelihood estimation . 5.2.3. Heckman's two-step estimator. CHAPTER 6. CONCLUSIONS AND POLICY IMPLICATIONS OF THE MODEL. APPENDICES Appendix A1 . Appendix A2 . Appendix A3 . Appendix A4 . viii 65 66 69 81 84 85 86 94 94 . 100 . 100 . 102 . 102 . 123 137 139 140 141 Appendix A5 Appendix A6 v'DSOTES WCES Appendix A5 . Appendix A6 . ENDNOTES REFERENCES ix 143 144 145 146 Sen Iajor stu bad perfo seventies consumpti A s inapprop; causes of P13?! an Performer up“: or effort; 1 “equate In ‘mculo 1) 2) 3) 30 goals ha: Cereals ‘ CHAPTER 1 INTRODUCTION Senegal's agricultural and food policies have been the focus of major studies in recent years. These studies have been motivated by the bad performance of Senegal's agricultural sector since the early seventies, and its increasing dependence on food imports to satisfy the consumption needs of its population. A series of severe droughts, a high population growth rate, and inappropriate agricultural policies have been identified as the major causes of Senegal's chronic food deficits. But the agricultural sector plays an important role in the Senegalese economy, and its overall performance along with the level of cereal imports has a tremendous impact on the balance of payments and on government revenues, most efforts to solve the crisis have been directed toward designing adequate agriculture and food policies. In short, the government is very concerned about having a agricultural policy that can: 1) Increase farm income. 2) Insure a high level of food self-sufficiency for the country. 3) Generate revenues for the government and contribute to reducing the balance of payments deficit. So far, the policy followed by the government to achieve these goals has been to change the relative prices between locally produced cereals and imported ones - especially between millet/sorghum and. imported ' boost the However. resources sufficien Han achieve I have focu uinly be urgent ne Constitut Man; the food . Potential Populatio' rele‘tamt 2 imported rice - and to provide general producer price incentives to boost the production of export crops - mainly peanuts and cotton. However, food crops and export crops tend to compete for land and scarce resources. Thus there is an apparent conflict between food self sufficiency and increase in export earnings. Many studies have been carried out to analyze the ways Senegal can achieve Increased food self-sufficiency. However, most of the studies have focused on the problems constraining the agricultural production, mainly because of lack of adequate data on food consumption and/or the urgent need to improve the living conditions of the rural people which constitute around 702 of Senegal's 6.5 million people. Many studies emphasizes the infeasibilty (and economic costs) of the food self-sufficiency goal. Indeed, given Senegal's present and potential resource endowments, along with the consumption habits of its population, this goal is not achievable unless a miracle happens (see, for example, Martin, 1988). Thus, the concept of food security is the relevant one for Senegal. One study that attempted to deal with food consumption is a world Bank policy study conducted in 1983 (Braverman et a1. 1983). This study tried to link the supply side of the agricultural sector to the demand of food, by using a multimarket model based on a farm household model. Despite the poor data, which affected the reliability of the estimated structural parameters, the model gave some insights into policy outcomes (income changes, production changes, and sizes of the deficits) under different scenarios of producer and consumer relative prices for the Ilajor crops and food staples. However, this model (in our opinion), is weakened Senegal. Th consmpt the pape Senegal utility dishes c' raw fooc focussec the deg: 1980b; ,; items a househo: the sod. differei 0f Subs 3 weakened by certain misspecifications of the nature of food demand in Senegal. This paper is an attempt to contribute to the understanding of the consumption pattern underlying food demand in Senegal. Specifically, the paper develops a structural model of household food consumption in Senegal based on the assumption that the household does not maximize the utility of the raw food staples but instead maximizes the utility of the dishes derived by means of some technological transformation of these raw food staples. Previous studies of food demand in Senegal have fecussed only on substitutability between cereals and particularly on the degree of substitutability between rice and millet (Ross, 1980a and 1980b; Josserand and Ross,l982). This model will incorporate other food items that are complements of cereals and that are important for the household when deciding which cereal to consume. But equally important, the model will incorporate information concerning how the nature of the different dishes consumed by the average Senegalese affects the degree of substitutability of the different cereals. Furthermore, within the household production model this food consumption model is shown to be a structural model whose reduced form corresponds to the traditional system of food demand equations but depends explicitly on the household's tastes, consumption technology and habits. A set of estimable elasticities including the traditional ones can be derived from both the structural model and its reduced form. These elasticities have policy implications that depart from the traditional food policy so far followed by the government, which is based pri e I A c1. *9) '01 eren: The Cha behavior consumpti with shor is analyz justifica chapters, assumptic Che economic economic ”amps: lPinyin; Che PTObabnj ludel. A utility 1 Se: (Se: conditior renneti one stat: Primarily Call dish 4 based primarily on the manipulation of the relative prices of the different cereals. The paper has six chapters including this introduction. Chapter 2 discusses the socio-and microeconomic consumption behavior of the average Senegalese household. The nature of the consumption technology which guides the household strategies for coping with shortages and/or increases in price of certain basic food staples, is analyzed in detail. This discussion serves as a background and justification for the food consumption model analyzed in subsequent chapters. It also discusses the consequences of the separability assumption in the household's food consumption choices. Chapter 3 reviews the household production model that will be our economic model for analyzing food demand.in Senegal. It reviews the main economic type results of this model which are relevant for our food consumption model. It also discusses the major limitations for applying it to our case. Chapter 4 presents and develops the mathematics of the prdbabilistic choice model (PCM) which will be used to estimate our model. A version of the fundamental axiom of the PCM is used to derive a utility function from an underlying preference ordering on the choice set (set of dishes). Duality theory is then used to derive the conditional and unconditional Marshallian demand functions. The restrictions implied by these demand functions are also investigated. One statistical consequence of viewing the household as choosing primarily among dishes rather than among raw foods is the presence of we . call dish selection bias which introduces some biases on the S coefficients estimates of the demand equations. It is also argued that the household's dish selection is the main reason why there are such a large number of "zero expenditures reported" usually found in food consumption surveys. Finally, the AIDS cost function (Deacon and Muellbauer, 1980) is used to present explicitly the model to be estimated. Chapter 5 is concerned with methods of estimation. First, the normal distribution is used to derive explicit expressions of the conditional moments that correct for the selection bias resulting from the household's dish selection. Then maximum likelihood estimation and Heckman's two-stage method are discussed. The asymptotic properties of the proposed estimators are also analyzed. Chapter 6 contains final remarks about the model, and ways of deriving elasticities of demand for the different raw foods used in policy analysis. Some measures of changes in household's tastes that can be used to evaluate implemented food policies are proposed.» The chapter also indicates possible ways of extending the model to capture taste variations both across time and households, and help design future food policies. CHAPTER 2 THE DETERMINANTS OF THE DEMAND FOR CEREALS IN SENEGAL 2.1 W The feasibility of a food policy which consists of forcing urban dwellers to change their food consumption habits by setting imported food prices very high, has two major limiting factors. The first one is political. The government was forced to decrease in May 1988 the prices of the basic food staples in order to ease the social and political tensions that followed the February 1988 general elections, during which food prices were the popular rallying point for the opposition. The second limiting factor comes from the possibility for people to smuggle part of their needed supply of food from Gambia where prices are much lower. Indeed, it was estimated that at least 85,000 tons of rice (about 252 of yearly rice imports) was smuggled into Senegal in 1987 when prices were at CPA. 160 (see, for example, N'deye, Ouedraogo, and Coat: 1989 or Lambert and Diouf, 1987)1. But, more importantly, this policy may not be effective in inducing urban consumers to switch from imported cereals to locally produced ones because of cultural practices and also because cereals are consumed along with other complements (fish, meat, oil, vegetables, I etc...) which are important for the household in deciding which cereals 1 N'deye, Ouedraogo, and Goetz (1989) estimated that the price differential between the smuggled rice and the official rice (of same quality) was up to 202 of the official price in some of the markets surveyed. to consume. constructet prices of consumer I The first Senegales ujor bas figure is in place degree 0 CorreSpo dishes h linear 1 “bait “15th tuba-1, fiddle histor fiftie prepot rural 7 to consume. The examination of the "food consumption matrix" constructed in Figure 2.1 is a first step toward understanding why prices of these complements are important to consider when evaluating consumer responses to changes in the relative prices between cereals. The first column of the figure shows the major dishes consumed by a Senegalese household. In the top row of the figure are presented the major basic ingredients used for the preparation of these dishes. The figure is read like a linear programming table with the difference that in place of the usual requirement coefficients we put signs to show the degree of substitutability of the food staple in the preparation of the corresponding dish. Had the transformation of the ingredients into dishes been linear for all dishes, the table would have been a true linear programming table. In general, for any given dish the degree of substitutability between two staples is measured by Allen's partial elasticity of substitution defined in the same way as in the substitutability between inputs in production theory. The dishes in the middle of the first column of Figure 2.1, couscous 1,2, and “lax“ are historically the major dishes consumed in Senegal up to the early fifties when imported rice from.the French colony Indochina, began to be preponderant in the urban diet. These dishes are still preponderant in rural Senegal (except maybe in Casamance). xduz edficu fits -cmc> twuqm mwam fiadfiuao .owe> fiOJOEm§ hOZDO‘ COOHUNU(OXNSI«E\UIO£3\UO-w2\00ux\ ? I6~Qwuw fi00k\ 9559.. no 95:25 £361. us posmcoo - mauuouaoaoadaoo 3.33 now avenue 0 9:93: on» ad non—3230 .35 use-.0333 canausuauage new «venue H 95:26 05 5 coeducoo we useaoadaoo inn—cocoa: you avenue +. 53.3.. on cognac". a. uses—c.3320 romance: you avenue G as «as ocean one 6 o 3:: .3300 .223 cease use: a + H H H «e no no: .350 Q + G H H + «:4. fidddm 600.30 Aumeauuoav o e . a . .53.. 0 + 0 + H H 0 is u esooesoo G + 6 0 we a maoomsoo @ as none: comes H H H H H H H 0 so:— “350 Aeoaem was ooumv 0 0 + H H o a. .550 «amaze Aswan ecu ooaav a o e 0 e e such. 238.. swam . «cause Ado vegan «eases as: 233 can 5»; 2:3 ,. wowsm cognac .omo> voxolm .350 53.5 use: :3...— usops cos; so; t wean—sum woo..— aHquulzaHHuuuwaaulaaau " aqulummuuu Bu: rural The growil its rela: which is other fac toward :1 of rural one of :1" study in d‘flt in c drought, While ch ”he! fa Indeed, interior fish ha; Slate t] found 11 9 But rural people are slowly adopting the urban style of consumption. The growing importance of rice in rural areas is usually explained by its relatively low price and easiness to prepare compared to millet which is time consuming and difficult to prepare. However, there two other factors at least equally important in influencing this shift toward rice in rural areas. The first one is the legitimate aspiration of rural people to diversify their diet. This fact probably explains one of the findings of the University of Michigan rural consumption study in Senegal in 1982. In this study, H. Josserand (1982), reported that in one village which was deficient in millet because of the drought, people were travelling far to other villages to buy millet while they could have easily bought rice in their same village. The other factor is the increasing availability of fish in rural markets. Indeed, in its preliminary survey of the marketing of fish in the interior regions of Senegal, C.R.0.D.T reported that the marketing of fish has been expanding at a steady rate (both in space and in time) since the early sixties; and before that time almost no fish could be found.in rural markets. Some of the traders interviewed still remember the arrival of the first lot of fish in their market (Kebe et a1. 1983). Since fish is the major complement of rice, one can easily understand why rice consumption is increasing in these areas. With respect to Senegalese agricultural and food policies, these factors point out the need to know to what extent increases in rural income (through agricultural price increases) will affect rice consumption. One of the striking facts in the table is that among the major dishes, only one-third are based on rice. None of the other dishes use any rice. Senegalese are consux confirmed (1978), a' the sampl Rig jenn'. 1 of prote: Hence, 1 the deci to use 1 Jenn' , 1 Wailab the O the gte- 10 any rice. One might ask then, why rice is so important in the Senegalese diet? Part of the answer is that these two rice-based dishes are consumed every other day at midday in urban areas. This fact was confirmed by two consumption surveys in Dakar, one done by C. Ross (1978), and the other done by Abt. Inc. (1984), In both surveys 991 of the samples declared eating exclusively rice at midday. Right now; the most common dish consumed in Senegal is 'cebbu jenn'. This dish is almost exclusively consumed at midday. As a source of protein, fish is an important complement of rice for this dish. Hence, its availability and its price are very important parameters in the decision of the average household to consume rice, and.how much rice to use in a given dish. In other words, for a given dish of 'cebbu jenn', the ratio between rice and fish depends to a large extent on the availability and the price of the latter. This ratio is determined as follows: when fresh fish is not available or its price is high, the household generally has two other alternatives. Either it can buy a small quantity and/or a low quality of fish and use more rice for the midday dish to make up the caloric deficiency, or it can buy dried and smoked fish and use much more rice and vegetables. Given the high price of meat, this strategy is usually the one adopted by the household to deal with the fish shortage, although some high income households may substitute meat for fish. Couscous is exclusively consumed in the evening; and meat along with fish - to a lesser extent - is an important complement of millet in the preparation of this dish. Thus, the decision of the household to prepare couscous depends primarily on the price of meat. For some poor househol sacked 1 Sc in the e Instead, cous. in anotl‘ relative for the Ir consider faced wj Very ric 11 households, or in some rural areas where meat is scarce, dried and/or smoked fish are substituted for meat. So, even if the Senegalese household prefers to consume couscous in the evening, the high price of meat will prevent it from doing so. Instead, it tends to substitute a poor quality rice based dish for cous- cous. Heat (or fish) is also a complement to green salad and potatoes in another vegetable-based dish consumed in the evening. This relatively meat-intensive or fish-intensive dish is usually out of reach for the poor. In any case, a rice-based dish in the evening is generally considered as an inferior alternative by the average household. But, faced with expensive substitute dishes, it tends to turn to cheap and very rice-intensive dishes for dinner. Another factor that increases Senegalese consumption of rice - which has a kind of income effect - comes from the cultural practice that gives more importance to the midday meal compared to the evening meal, so that the daily budget share of this meal is very high. Hence, with a perfect inelastic demand for this rice-based meal, an increase in the price of rice and/or fish will merely erode the budget share of the evening meal. Then, with not enough left for the evening meal, the Senegalese household tends to consume a low cost rice-intensive dish instead of couscous, green salad, or other vegetable based-dishes which are far preferred for dinner. This role of rice as preferred dish at midday and security dish in the evening, is so important for the urban household that the first food staple secured for a month of consumption is rice, bought in bags of 100 kilogram ituatio surveys daily be rice req Ar. Senegale curdled areas) a sugar at decimate been re; Tl behavlm Plrtly , dOublin; 1“sight the deg! inform no tax The Sam. conSump rice an. 12 kilograms. The strategy is to have enough rice to face all possible situations. This systematic behavior was confirmed by the consumption surveys cited earlier. In these surveys, only the very poor buy rice daily because their income does not allow them to buy the whole monthly rice requirement at once. Another traditional dish that is losing its place in the Senegalese diet is “lax“. This millet-intensive dish, prepared with curdled milk and sugar, used to be consumed (especially in the rural areas) at midday and for breakfast. Now, because of the high price of sugar and the scarcity of curdled milk since the drought, which decimated the livestock population in the mid seventies, this dish has been replaced by rice at midday, and by coffee and bread in the morning. This brief, but relatively detailed discussion of the consumption. behavior of the Senegalese household can help understand - at least partly jzwhy rice imports have doubled between 1978 and 1987, despite a doubling of its retail price. This discussion also suggests some insight on.why previous food demand studies in Senegal, which analyzed the degree of substitutability between rice and millet by using information on these two food staples only (thus ignoring the technical and taste constraints), embodied an incomplete food consumption model. The same criticisms apply to the policy makers' approach that views food consumption in Senegal simply as a problem of the relative price between rice and millet. cereals a complete long, es; quite in; the Univ: Tat share of in the t! these ta': scape of tables: QR food Almost - Cannot b. °f fresh substitu That is, Cowpensa dish. an low Cost of "ilk the .lax matrix ‘ mienthi‘ l3 2-2 WWW For modelling purposes, the starting point is to recognize that cereals are always consumed with other complementary food staples. The complete list of these complementary food staples may be relatively long, especially in urban areas. Even in some rural areas, the list is quite impressive, as reported by the 1982 rural consumption survey of the University of Michigan (Josserand and Ross, 1982). Table 2.2 and Table 2.3, reproduced from the cited study, give the share of income spent on food and the amount spent on major food items in the three villages surveyed, respectively. A full interpretation of these tables in relation to the food consumption matrix is beyond the scope of this paper, but one may notice some interesting facts about the tables: (1) the correlation between non-farm income and diversity of the food basket; (2) the relatively high rice consumption in Thienthie - almost three times higher than in the other villages . which certainly cannot be explained only by its low millet harvest. Indeed the absence of fresh fish and meat in the diet points to the likely presence of substitution and income effects of the types described in Section 2.1. That is, the unavailability of fresh fish (and vegetables) is compensated for by a high ratio of rice to smoked fish in the midday dish, and the absence of meat leads to the replacement of couscous by low cost rice-intensive dish in the evening. Note also that the absence of milk (curdled, powdered or fresh) would rule out the consumption of the ”lax" dish in Thienthie. Thus, in relation to our food consumption matrix, one can infer from Table 2.3 that the 27 households surveyed in Thienthie were consuming at that time almost exclusively the third, 14 .nm .oHSuCOHSH .Aumaav new .oconnom new .anom .0.0 was .m .ccouonnoa "oousom .onohsd unnaonoesom mo .oz .aaaa .ma sneeze - as an: nonsense any .Hsuou n.uooa scum enucoa n uo>o oomeuo>< Adv Madman ans om~.aea osm.oen ose.aoa aan.an~ mezezmze an aa~.ae «ma.naa an~.aaa na~.mee mzmmmmm use oee.aaa an~.ea~ Nom.noo nna.maa ma<~n. Rfii c E, and (T‘1(E), E) is a measurable space where E is the feasible set and T is the mapping defined in 32. There exists a probability measure PE defined in 8, such that (T'1(E), E, PE) is a probability space with X P3(a) - l. PE(a) measures the probability that the aEE household chooses alternative a. The household preference ordering as is described by: a a: b if and only if PE(a) z P5(b). -3! '4 4L .— jog-tents Assumption C: the usual ide orthant of Cl {technologies specifies tl‘u that associa' share a spec feasi le cho Ieasurabilit can make sen h0kl$ehold ch discussed in “flutes the th‘ househol dishes are e intuitive a: discrece am dish is an this we do I the outCOme hotnehold ) selectiOn O The °°nditi hokehOId m Iodified ve 46 Cements Assumption C1 identifies a dish with a Borel set of Rd+. This parallels the usual identification of the commodity space with the non-negative orthant of the Euclidean space. Indeed, given that a dish is a (technological) transformation of some vector of raw foods, Cl merely specifies the nature of this transformation which is a correspondence that associates each vector of Rd+ with a subset of Rd+ whose elements share a special relationship. Assumption C2 excludes Rd+ from the feasible choice set although it is an alternative by C1. The measurability assumption is needed so that the following assumption C3 can make sense. Assumption CB formalizes the randomness underlying the household choice process. The reasons for this randomness were discussed in 38. In some sense the postulated probability distribution measures the "chance“ of a given dish of being selected or preferred by the household. Hence, in some sense, at least for the observer the dishes are events of the underlying choice process. This has some intuitive appeal given that the choice of a dish is observationally discrete and that for the econometrician the household's selection of a dish is an action, the consequence or outcome of which is an event. By this we do not mean that the choice action itself is random but rather the outcome of the choice process (which dish is selected by the household ) is random for the observer. In summary, C3 says that the selection of a dish is an event with a probability of occurrence PE(a). The condition XPE(a) - l is just to formalize the fact that the aeE household must select one alternative from E. Assumption C4 is a' modified version of a probabilistic definition of utility usually found 4.1:... in the probab 1 Block and liars interpreted a: The pro the consumpti But also it . change from 4,2 ’1'»! ‘2 E: “-2-1 Cons Sues finite ch. into 10.1 1) d3, 11) d5 11:) ‘11-: by. wi- Prmar - fl V0.1: W inplie! alter“ Probab topole. all 47 in the probabilistic choice literature1 (see, for example, Luce (1965), Block and Marshak (1960), or Debreu 1958). PE(a) z PE(b) is interpreted as "a is preferred to b ' (Debreu, 1958, p. 440). The probability distribution PE depends indirectly through E on the consumption technology, the household characteristics and on time. But also it depends directly on time, since for an unchanging E, PE may change from one decision to the next. 4.2 Iha_nashsmasisal_§£rssturs_2f_ths_nsdeli 4.2.1 92natrusti2n_2f_a_t22212sz_2n_ths_alsernatizs_snasei Suppose that the fundamental axiom holds, and let ECA be the finite choice set given by the axiom. Define the function dE from AxA into [0.1] as follows: 1) dE(a,b) - IPE(a) - 23(b)| if aEE and has 11) dg(a,b) - 1 if aeE and her or was and has 111) dE(a,b) - 0 1r a¢E and be: Eggpggigign: dB is a pseudometric for A. Proof - see Appendix Al. figmngntg: We note that d: is not a metric since for a,b e E dE(a,b) - 0 implies PE(a) - PE(b), which does not imply a - b, because two alternatives in the feasible choice set may well have the same probability of being selected yet have different attributes. Given d3, (A, dB) is a pseudometric space. hence d3 defines a topology in A, namely the topology of which base is the collection of all open balls 3(a,r) - ( bEA; dE(a,b)0. Let (A, 4.2. that The Rex 48 FE) the corresponding topological space. Because dE is not a metric, (A, FE) cannot be a Hausdorff topological space. 4.2.2. W By definition a utility function on A is any real valued function that takes its values in R4,. Theorem Assume that the fundamental axiom holds, then there exists on (A,dE,ap) a continuous (ordinal) utility function that represents the household preference ordering 29. Before proving the theorem, we need the following lemma. lama: Under the axiom, (A, d3) is a perfectly separable pseudometric space. 21:22:: See Appendix A2. Wastes By C4. , the household's preference ordering as on A was defined as follows: (4.2.2) a b b if and only if PE(a) 2.- PE(b) for all a and b e A Then, a is a complete ordering on A (see Appendix A3 for the proof). Next, FE, the topology corresponding to d3 is a natural topology for a, that is, the sets Fl-{bEA/ ebb} and F2_(bEA/ baa} the One uti uti the C02 49 are closed for all aeA. This is the same as saying that the preference ordering a: is continuous for PE. See appendix A4 for the proof. 80, A is a completely ordered and perfectly separable topological space whose topology is natural for the ordering 2L Hence from a theorem of Debreu (1954, pp. 159-65), there exists on A, an ordinal continuous utility function U3 that preserves the preference ordering as, that is (4.2.3) a b b if and only if UE(a) 2 UE(b) for all a and b e A. In summary, there exists a continuous ordinal utility function in the choice set A such that the alternative yielding the maximum utility has the best chance to be chosen by the household. Since by definition a random variable on the measurable space (83+, A) is any measurable real valued function defined in (Rn.+ ' A), one point that comes immediately to mind is the measurability of the utility function UE° This would enable us to speak of ”E as a random utility function. However, this cannot make sense since the domain of "E is A instead of R“+. 4.2.3. WW- If in addition to the fundamental axiom one makes the standard assumption of utility maximization behavior, that is, the household always chooses the alternative yielding the maximum utility, then, from the observer's viewpoint, the alternative_yielding the maximum utility corresponds to the one with the highest probability of being chosen. pro: 58 1'1! COP. 63 var W in 30 ch at eh 50 For a formal development of the random utility model, one can proceed as follows: Let UE(A) be the range of UE and consider the a-field BE generated by the collection of all half lines ]-o, UE(a)] aeA, such that ]-¢, 05(a)[ C UE(A). Then (UE(A), BE) is a measurable space. Now, consider the a-field AB on A generated by Us that is Ag - 1 034(3); 3 e BE ). Then by construction UE is a measurable function from (A, Ag) onto (UE(A), 83). In that sense, we would say that “E is a random variable meaning that it is measurable with respect to the a-field defined above. Then, based on our postulated probability space, we can econstruct a probability measure on A5. More precisely, we would like to show that there exists a probability measure P defined on.AE such that P[Ug(a) z:UE(b)] - P[{ beA; UE(a) z UE(b) )1 - P3(a) for all aEA. (A, A3, P) would be then a probability space with P[ UE(a) z UECb) ] - PE(a) for all aEA. For our purpose, we will take the existence of this probability space as a conjecture. In Appendix A5, we give some indications on how one may proceed to show this existence. An alternative way to introduce randomness in the utility is to note that "E depends implicitly on the household vector of characteristic‘hc and on the attributes of the alternative's vector of attributes xa, because both E and PE depend on these. But all these characteristics and attributes cannot be completely known and observed. Only a finite number of them can be observed and measured (with some measurement errors). Then, one may want to partition the vectors of characteristics and attributes into two parts: one part observable and measurable, the other part not observable. That is the approach taken by Hanski (1977). Formally, he wrote hc and x8 as hc - (hco, he“); xa - (Xaol "1 O ’1 A. 11. « 4.3. 4.3. «or fol] 51 (x xau) with hco and xao observable but hcu and xau unobservable. aov This approach yields the random utility model where the source of randomness of the utility is attributed to the inability to observe all the household characteristics and goods attributes. Manski (1973) cited in Ben-Akiva et a1. (1985) identified the following sources of randomness of the utility function: 1 --?unobserved attributes 2 -- unobserved taste variations 3 -- measurement errors and imperfect information 4 —- instrumental (or proxy) variables For a detailed analysis of these sources of randomness, see Ben-Akiva et a1. (1985, pp. 55-57). 4.3. W- 4.3.1. WW Let E-{a1, ..., an} be the finite set of feasible alternatives, for i-l,..;,m let d1 be the dichotomous random variable defined as follows: { 1 if at is selected by the household (4.3.1) d1 - 0 if not then we have : (4.3.2) P(d1-l} - PE{a1) - Pt UE(a1) - max UE(aj) j-1,...,m I This follows immediately from the conjecture defining P and the assumption of utility maximization, with utility maximized subject to the constraint defined by the set E of feasible alternatives. We note Be at indi 52 that the di i-l,...,m are observable. Here i is an index for the alternative a1, and since there is a one to one correspondence between at and i, we can just keep the index i standing for at. For instance, the finite set of feasible dishes of the household is designated as dish 1, dish 2, . . ., dish m. Now, since E depends on the alternatives' vectors of attributes x , and costs C(i) i-1,...,m., and the household vector of a1 characteristics he, a solution at of the maximization problem max{ UE(‘j) j- l,...,m ) subject to aJ e E, will be function of x31, the vector of attributes of the chosen alternative, hcv and a vector of costs of all the alternatives, noted v. That is we should have a demand function a1(xai,v,hc) for a chosen alternative a1 such that (4.3.3) UE[a1(xa1,v,hc)]- maxl UE(aj) j - l,...,m. }. Hence, when (xai,v,hc) varies over Xqu+xC, we have the conditional indirect utility function V1 (conditional on dish i being chosen) defined as:’ mn+x °°°°°° >R V1: (xai-Vvhc) ----- ~ v1(xa1vv'hc) - Uglai(xa1vv-hc)] -1;;:mUE(aj) with (4.3.4) Pld1-1}-P{ V1(xai,v,hc) - maxl UE(aj) j-l,...,m) ). a .M .' l.“"11"¥ and given that random. The ram into a observ (x , h ). au CU (£35) V where as usus unobservable Let Vk functions, t 4. ( 3.6) v1 Hence (5.3.7) Pl '1'] (MCFadden . . 53 and given that U is random in the sense defined above, V1 will also be E random. The randomness of V1 can also be derived by decomposing (xai-hc) into a observable component (xa , hco) and an unobservable component 10 (xau' hcu)' Then following Hanski (1977) we can write: (4.3.5) vi(xa1'v’hc)-vi(xaio’v’hco) + ei(x ) ,h aiu cu where as usual, ¢1(x ,h ) is a random disturbance summarizing the aiu cu unobservable components of the conditional indirect utility function. Let Vk(xa1,v,hc) k-l,...,m be the m conditional indirect utility functions, then we have: (4.3.6)- V1(xa1,v,hc) - max (03(aj), j-l,...m.} if and only if Vi(x‘1,v;hc) z Vk(xa1,v,hc) for all k vi, k-l,...,m. Hence, (4.3.7) Pidi-ll-P{ V1(xai,v;hc) z Vk(xa1,v,hc) for kfli and k-l,...,m} This is the familiar formulation of the random utility model (McFadden,‘l981). 43.2- W: Let x11 be the amount of raw food j used in the preparation of dish 1 for i-l,...,m and j-l,...,m. Hence xij-O if no raw food j is used in dish i. We can see that a priori, for a given dish i, x1] depends in general on observable household characteristics (ex: household composition, household income etc.). as well as on unobservable 'w-‘ a" .‘s; 'l . ‘1 Ii“ -" f . 1 household c etc...)« I content of raw foods u the intrins i). For a on the unot the amount conditiona'. variable X. 54 household characteristics (ex: household "tastes", technical knowledge, etc...). It depends also on the characteristics of raw food j (per unit content of calorie, protein, vitamins, etc.). the relative prices of the raw foods used in the dish, and some other unobservable attributes (ex: the intrinsic contributing flavor of the raw food j with respect to dish i). For a given raw food j, x13 depends generally among other things on the unobservable consumption technology. Hence, for a given dish 1, the amount of raw food j used in dish 1, which is the household's conditional quantity demanded of raw food j, is a real valued random variable X11 with observable and unobservable components that isS: (4.3.8) X11 - 111(P1, C) + 111] where c is a vector of observable household characteristics that includes: household size, number of adults and women in the household, household income etc... n11 is a random disturbance term. Pi-(P1,..., 231) is the vector price of the J1 raw foods used in dish 1, hi is an unknown function of P1 and c. Given i, one might expect hi to be a nondecreasing function of the household size and number of adults for most of the raw foods. It is also expected to be a non increasing function of Pj for j-1,...,J1 (ceteris paribus of course). Hence, an appropriate specification of the functions hi i-l,...,m would enable one to capture the economy of scale 5. In general we will consider all the raw foods characteristics as unobservable to simplify the analysis although it is possible to observe some of them (ex: per unit content of calorie, protein, vitamins, etc.). If one is interested in some nutritional aspects of food consumption, he may consider measuring these nutritional contents of the individual raw foods and incorporate them into the model. The following results can be easily changed to accommodate these nutritional aspects. eabodied i functional a solution In g dishes are Give observable consumptio depends on assured b: “:0! comp: hausehold ; h0useho 1d 1 55 embodied in the household consumption technology. The precise functional form of the hi's will be clear later, when we derive them as a solution of the household utility maximization problem. In general, the only observable intrinsic attributes of the dishes are the amount of time and energy needed for their preparations. Given the food prices and the consumption technology, the observable constraints facing the household when making its daily consumption choices are: (l) the household meal budget d(c) which depends on its vector of characteristics c, and (2) the time constraint6 measured by the maximum total time available for dish preparation. The major components of the vector of characteristics c in d(c) are: the household income y and its expenditure on non food items e". But, the household size, its number of adults and other observable and unobservable characteristics are included in c as well. Hence, if we assume some degree of separability between food consumption and non food consumption, than we can write: (4.3.9) d'- d(y,eN,c) where c now stands for the components of the household characteristics other than income and expenditure on non food items. One should expect d(y,eN,c) to be a nondecreasing function of income, household size and number of adults but a nonincreasing function of expenditure on nonfood items. However, since food is a necessity, beyond some range of income, d(y,eN,c) is probably insensitive to 5. Since energy and time enter the model in a similar way, to simplify the analysis we will concentrate only on the time variable. Alternatively, one can think of our time variable as a bi-dimensional vector with coordinates time and energy. changes in i that changes household sf the househo' Hence the unit to (4.3.10) 1 But if we ' with the“ the 56 changes in income or expenditure on nonfood items. It is also likely that changes in d(y,eN,c) is less than proportional to changes in household size, and composition because of economy of scale inherent in the household consumption technology. Hence, if we denote by 71 the time preparation of dish 1 and by w the unit cost of time, the set of feasible alternatives can be rewritten as: 31 (4.3.10) E "{ 81 E A; jflxij,Pj + wri S d(y -eN, C) 1-1,... ,lll.) But if we let ri-xio, Po-w, and noting that d1 6 {0,1} for i-l,..., m with m 2 d1 - 1 (since only one dish is chosen), i-l then the budget constraint can be rewritten as: (4.3.11) 2 2 d x p s d 1-1 j-O 1 13 3 where we have put d-d(y,eN,c) to simplify the notation. Introducing the variables C(i,j) - xiij the share cost of food j in dish 1 for j-O,...,J1 with C(i,o) - xioP°.- "'i being the share cost of time for dish i i-l,..., m . For i-l,..., m let the total cost of dish 1 be: (4.3.12) C(i) where .11 is the constraint can In .1 (“3.13) 2 2 1-1 j- Hence This is clear} the c“) repla 57 J1 (4.3.12) C(i) - 2 c<1.j) 3'0 where J1 is the number of raw foods used in dish i then the budget constraint can be rewritten as: m Ji m J1 m (4.3.13) 2 E di-x1 oPJ- 2 di°( 2 C(i,j)) - 2 dioc(i) sd 1-1 j-O 3 1-1 j-O 1-1 Hence II I (4.3.14) E -{a1 6 A; 2 died) 5d with die(0,1) and 2 di-l} ' i-l i-l This is clearly a linear budget constraint, with the cost of the dishes, the c(i) replacing the usual vector of prices. With this specification of E, the household maximization problem can be reformulated as: (l315) 3 Subje with (11 E {0 Since finite set 0 Let v- Define q-v/d Then, the un (k3.16) v(q) . .e the ‘ s metions inc In 0 ding t 58 (4.3.15) max{ UE(ai) ; i-l,...,m } Subject to m 2 d.c(i)sd . 1 1-1 m with di 5 (0,1); 131 di-l ; and PE(a1)-P[d1-l}. Since by assumption, the household must select one dish from the finite set of dishes E, this problem has a unique solution. Let v-(c(l),...,c(m)) be the m-dimensional vector of dish costs. Define q-v/d-(q1,...,qm) as the vector of real costs of the dishes. Then, the unconditional indirect utility function is defined as (4.3.16) E diq1 sl; die{0,l); f d -l; V(q) - max {HI-5‘1”, 1 lsism l i l PEIa1)-P(di-l} } I max V1(Pi/d) lsism where the V i-l,...,m. are the m conditional indirect utility it functions, and Pi-(Po""’Pi) the price vector of the raw foods including the time cost of preparation of dish i. 3,4 fie Utili: One way to the fact that, h technological t2 X:'(X ,...,X , , . 1 J1) dish 1. Hence the ditec1 Then. following h“15310141 choos the budget cons 59 4.4 n e d u c ons 4.4.1. Waist: One way to define the conditional direct utility would be to use the fact that, by assumption, each dish a1 is the result of some technological transformation T such that ai-T(x1), where xi-(x1,...,xJi), the vector of J1 raw foods used in the preparation of dish i. Hence the direct utility of dish i can be written as (4.4.1) UE(ai) " UE(T(X1)) I 010(1) Then, following McFadden (1981), for a selected alternative at, the household chooses xi to maximize the conditional utility 01 subject to the budget constraint J1 J1 c(i) - E c(i,j) - 2 xi P 5d 1.0 1.0 J .1 The result would yield the conditional indirect utility Ji (4.4.2) V1(Pi,d)-max(Ui(xi); c(i)-2 xiijsd; xiij; P120 } xi j-O . w "- .‘ -°.A .4!" —‘- . ~,... I! ' n. For 1.11' ' ' ' ' . functions derlv fmction given ((6.3) V(q) where q-(ql, . .. real costs of t real prices of Hence, we 0111You the tea ”Onditional j 111 the dishes. discuss“ in C} household conso 5°“. but rat 1'his der- fimetions v is ifUE and the apprOpriate r£ leCClOns Sat f‘ q ‘, v0.ditions D“; i) °°ntinu 01; (ii) UOndQCre 60 For i-1,...,m. we would get the m conditional indirect utility functions derived earlier with the unconditional indirect utility function given by: (4.4.3) V(q) - max V1(P1,d) - max v:(P‘{,1) 1515:: 1515:: where q-(q1,...,qm) - (c(1)/d,...,c(m)/d) is the m-dimensional vector of real costs of the dishes and PI- Pi/d-(Po/d, . . . ,Pi/d) is the vector of real prices of the raw foods and the real cost of time for i-1,...,m. Hence, we see that while the conditional indirect utilities depend only on the real prices of the raw food and on real cost of time, the unconditional indirect utility depend explicitly on the real costs of all the dishes. This gives a formal proof of the important fact already discussed in Chapter 2 and 3 that the variables relevant for the household consumption decisions are not the relative prices of the raw foods, but rather the relative costs of the dishes. This derivation of the conditional direct and indirect utility functions, is probably the most natural and familiar way. FUrthermore, if "E and the technological transformation function T satisfy appropriate regularity conditions so that the direct conditional utility functions satisfy the usual following conditions: W for 1-1.....m. 010:1) is (i) continuous (ii) nondecreasing 61 (iii) subject to local nonsatiation (iv) Quasi-concave then the conditional indirect utility functions will satisfy the following conditions: (see, Diewert; 1977 and 1981) and1§12n§_lnz for i-l,...,m. Vi(qi) is (1) Continuous (ii) nonincreasing (iii) subject to local nonsatiation (iv) Quasi-convex for PI>>O Furthermore, we have by duality J1 U (X.)-min {H (P ,d): C(i) - 2 x P 5d ; P >>O} i i Pi i i j-O ij j j 1-1.0.0.". Conversely, if the conditional indirect utility functions satisfy conditions ID, then the direct conditional utility functions will satisfy conditions DU and we have J i V1(Pi.d) " max {010:1 )2 C(1)‘ 2 "ij'Pde; X1120} xi ,J-O i-l,...,m. So the 01 i-l,...,m satisfying DU is dually equivalent to the v1 i-1,...,m. satisfying ID. (Diewert, 1977). 62 Since the conditions DU and ID are dually equivalent, one could alternatively derive the conditional direct utility functions from the conditional indirect utility functions which are already derived in (4.3.16) from the unconditional indirect utility function by V(q)- max( V1(Pi,d) i-l,...,m 1 then postulating that the V1 i-l,...,m. satisfy the conditions ID, we can define the conditional direct utility functions by J1 (4.4.4) U (x) - min{V (P ,d): C(i) - 2: x P sd, P »o i 1 P1 1 i jD ijj j i-l,...,m. and the 01's will satisfy the conditions DU. The advantage of this derivation, would be to avoid the use of the technological transformation functions T and to make any assumption about their regularity conditions. If in addition to conditions ID, we assume that the m conditional indirect utility functions are differentiable with respect to P1 and d, then.we have by Roy's identity, the conditional Marshallian demands given by: 63 avi(Pi,d)/apj (4.4.5) Xij(P1,d) -- j-0,...,Ji ; avi(Pi,d)/ad Since d - d(y,eN,C) The conditional Marshallian demands are of the form xij(Pi’d) - h1(P1,d) +n1j - hi(P1,c) + "ij j-O,...,Ji. and i-l,...,m. where "ij is a disturbance term. This is exactly the functional dependence predicted for the conditional Marshallian demands in (4.6.1) 4.4.2. Ihs_s2ndisi2nal_exnendisure_fnnssisnsa Given the conditional direct utility functions, we can define the corresponding expenditure functions by 31 (4.4.6) e.(P ,u) - min { c(i)- 2 x P : U (x )2 u } 1 1 xi j-O ij j i i i-1,...,m. One of the major advantages of working with the expenditure function is that it satisfies many regularity conditions yet requires only that the direct utility function be continuous and that the level of utility u belong to the range of the utility function. Indeed if the direct utility functions U1(x1) are continuous and if u belongs to the 64 range of U1 i-l,...,m. then it can be shown (Diewert; 1977, 1981) that the corresponding expenditure functions have the following properties. M; For i-l,..., m °i(Pi' u) is: (i) a nonnegative function that is, e1(P1,u)20 for any Pi>>0 and u 6 range of U1. (ii) linearly homogeneous in Pi that is, e1(kP1,u)- ke1(Pi,u) for any P1>>O, u 6 range of Ui' and k>0. (iii) non decreasing in Pi that is, e1(P1i,u)z e(P?, u) for any Pi>P?. (iv) concave in Pi that is, e1(P1,u) is a concave function of P1 for any u 5 range of U1. (v) continuous in P1 that is, e1(Pi,u) is a continuous function of P1 for any u 6 range of Ui' (vi) nonedecreasing in u that is, e1(P1,u) is a non-decreasing function of u for any Pi: (vii) continuous from below in u that is, for any u and any non- decreasing sequence {Un. nEN ) in the range of U1 such that lim unfiu then lim e1(Pi,un) - e1(P1, u) for any P1>>O. (viii) twice differentiable with respect to P1 except possibly at a set of Lebesgue measure zero (see Deaton and Muellbauer, 1984 p.40 and, Fuss and HcPadden, 1978 for a proof). If in addition the conditional direct utility functions satisfy conditions DU, then they can be recovered by J 1 (4.4.7) U (x )-max {e (P , u): P .x - 2 x P V P 20 i i u i i i i j-O ij j i i-l,...,m. As usual, we can apply Shepherd's lemma to find the conditional Hicksian demand functions. 6S hij (P1, u) -a_°_1_(_l’1fil j-0,...,Ji. and i-l,...,m. 81’] Given our assumptions, the conditional Marshallian and Hicksian demands satisfy all the usual properties and restrictions which are: homogeneity of degree zero, adding up, symmetry and negativity of the Slustky matrix. They also satisfy a conditional Slustky equation. 4.4.3. W W The unconditional indirect utility function was derived in (4.6.9) as V(q) - max V1 (P1,d) lsism where q-v/d-(c(l)/d,...,C(m)/d) For convenience, we may write it in the form (4.4.8) V(v,d) - max V1(Pi,d) lsism If all the conditional indirect utility functions satisfy the conditions ID, then the unconditional indirect utility V(v,d) will satisfy the usual regularity conditions of an indirect utility function with the 66 vector of costs of the dishes replacing the vector of prices. If in addition the V1(Pi,d) are differentiable, then V(v,d) will be differentiable, and the Marshallian demands for the dishes are given by the usual Roy's identity (McFadden, 1981; pp. 207-208). As usual, we can define the unconditional expenditure function as: m (4.4.9) e(v,u) - min { Z d1c(i): UE(a1) z u; d16(0,1); Isism i-l m i-l,...,m; 2 d1 -1; PE(ai) - Pldi-l) i-l I min {e1(Pi,u); i-l,...,m.) Because U8 is continuous, e(v,u) has the usual regularity conditions of an expenditure function that is, it satisfies the properties EX. with the vector of costs of the dishes replacing the vector of prices. In particular, it is twice differentiable, and using Shephard's lemma we can derive the Hicksian demands for the dishes by taking the partial derivative of e(v,u) with respect to each dish cost. Before investigating the properties of the demands for the dishes, we point out again the fact that while the conditional expenditures depend on the vector of prices and the time cost of dish preparation, the unconditional expenditure depends only on the vector cost of the dishes. Hence, it is affected by changes in prices only through these costs of dishes. In other words, using the statistician's language, the vector cost of the dishes is a sufficient statistic for changes in relative prices. 67 4.4.4. 3 o e ' Under our assumption of differentiability, the household's demand for the dishes are given by Roy's identity (McFadden, 1981 p. 208): aV(V.d)/3C(1) (4.4.10) d1 - 01(v.d) - - aV(v,d)/ad { 1 if V1 2 Vk for kfli and k-l,...,m. - 0 otherwise i-l,...,m. where v-(c(1),...,c(m)) and c(i)-cost of dish 1 and V1 - V1(P1,d) 1-1,...,II. Similarly, and by duality, the Hicksian demands for the dishes are given by ae(v,u) 1 if e1 5 ek for kfli; k-l,...,m. (4.16.11) (11 - “107,10 - —— I 6c(i) 0 otherwise where ei-e1(P1,u) i-l,...,m. And by duality, if the alternative a1 maximizes utility "E at dish costs v* and daily expenditure d*, we have the identity: (4.4.12) Hi(v,u*) - Di(v, e(v,u*)) 68 where u*-UE(ai) Because 05 is continuous and the demand functions are derived from utility maximization, the Marshallian and Hicksian demands for the dishes satisfy all the usual properties and restrictions of demand theory which are: m m (i) Adding up: 2 c(i)Di(v,d) - E c(i)Hi(v,u) - d i-l i-l (ii) Homogeneity of degree zero in the vector of dish costs and total expenditure: D1(0v,0d)-Di(v,d), H1(0v,u)-Hi(v,u) for any positive scalar 0>0. (iii) Symmetry of the substitution or Slutsky matrix: 1 32e(v,u) the matrix J is symetric ac(i)ac(j) lsism lsdsm 8H1(v,u) aHj(v,u) i-l,...,m. or equivalently, -——————— - -————-——' ac<3) 66(1) j-1,..., m. (iv) Negativity of the substitution matrix: 82e(v,u) --—-—- ] is negative semi definite accnam lsism lsjsm 69 8Hi(v,u) which implies that ——-————— s 0 for all i. 6C(i) However, the Slutsky equations cannot be derived, because it would require differentiability of the Marshallian demands for the dishes, which is unlikely given the nature of our choice set. So far, these testable restrictions on the demands for the dishes and the conditional demands for the raw foods are the only ones implied by the model. 4.4.5 .Ihe_unc2ndisi2nel_dsnands.f2r_she_ras;fssds In this part, we will focus only on the Marshallian demands. The unconditional Hicksian demands can be derived similarly. If we introduce randomness in the conditional indirect utility functions by decomposing them into observable and unobservable components, then for i-1,...,m. we can write: (4.4.13). V1(P1,d,£1) — V1(P1,d) + £1 i-l,...,m. Where (1 is a disturbance term for the nonmeasurable components of the conditional indirect utility. Hence the conditional demands for the raw foods given by Roy's identity are: 8V1(Pi,d,e1)/8Pj 8V1(P1,d)/aPJ (4.4.14) xij - - - - + "13 3V1(Pi,d,e1)/ad avi(91,d)/ad for j-l,....,Ji and i-l,...,m. 70 Where n11 is an error term standing for measurement error and unobservable variables. It is clear from the derivation that the ”1] j-1,...,Ji are somehow connected to e1. Indeed, there are many reasons why the "ij and e1 are not independent. Dubin and McFadden (1984), Hausman (1985), and Duncan (1980) have given numerous reasons with models similar to ours. Without going into details, one reason in our food consumption model would be the fact that some unobservable household characteristics and/or unobservable alternatives' attributes may be correlated to unobservable attributes of some raw foods. For instances, if some dishes have increasing returns to scale with respect to household size, then, for a large household, the quantity demanded for some raw food is likely to be correlated with its dish choice. Thus in general we will assume that n11 and e1 are jointly distributed. More generally, let 01 - (n10....,n1J1)' i-l,...,m., q -vec(n1,...,nm), and e - (£1, ..., ‘m) i-l,...,m. with (n,e) jointly distributed with density f(n,e) and distribution function P(q,e). With this notation, the conditional mean demand for raw food j j-1,...,J1 is given by: (4.4.15) E[Xijld1-1] - ~3Vi(P1,d)/3PJ +flij | Vi+ eisz+ek Vk)‘ i 8V1(P1,d)/ad 71 or -avi(Pi,d)/apj (4.4.16) £[xij Id1 -1] - aVi(Pi,d)/ad + 5("ij l‘k 5 v1 - Vk + £1 V k w i) where we have put Vi-V1(Pi,d) i-l,...,m. to simplify the notation. Let A1 be the event (di-l), that is A1 - {V1 + 61 2 Wk + Gk, k i i and k-1,...,fl.} then we have by definition of the conditional mean: 1 (4.4.17) 3(011 IAi) - X Jaijf(fl,£)dfld¢ j‘l,...,J1 ‘ P(A ) 1 A1 m where f' is a multiple integral of order m + 2 J1 and A1 1']. m J1 m dfl - n H dnlj ; de - n dei i-l j-l i-l If we integrate out n and :1 in (4.4.17), we can express the conditional mean as a function of the V1 - Vk; kfll and k-1,...,m. that is: E(flij IAi) - 13(Vi-V1,...,Vi-V1_1, Vi-V1+1,...,Vi- m) 72 In the case where (n,e) is bivariate, A}(Vi—Vi,...,Vi-Vm) is known as the hazard rate. To simplify the notation we will write vi'vk as Vki for k-1,...,m. and kni. In the discussion of estimation, in chapter 5, we will see how the expression A}(.) can be simplified for some form of the joint distribution function F of (q,e). For instance, if (q,e) is distributed multivariate normal, a simplified expression of A}(.) can be derived (Tallis, (1961); Johnson and Kotz, (1972); Duncan, (1982); and Amemiya, (1985)). When (n,e) has the multivariate generalized extreme value (CEV) distribution, then A}(.) can be integrated (McFadden, 1978). The conditional mean demands for the raw foods are then: 8V1(P1,d)Ian (4.4.18) 3(X1j lei-1) - - aVi(P1,d)ad + 1j(v},...,v1'1,vf+l,....vg) j-0,...,J1 Hence, the estimable equations of the conditional demands for the raw foods are: 73 (4 4 19) x avi(Pym/6191+ 1 v1 vi'l v":+1 [P * . . ij - ‘ 31 ,..., , 1 ,...,V ) {-qu 8V1(Pi,d)/ad j-0,...,Ji. where ":1 - xij'Elxidei'll is an error term (conditional on i) with EInijl-O and it can be shown that its variance is (see Appendix A6 for details): (4.4.20) Var (”Ij) - ewijzl :1i -1) - [1}(v},...,vijl v3}... v: )1 Again, a simplified expression for Var(n§j) can be obtained with specific distributional form like the multivariate normal or the G.E.V distribution. In practice, we do observe the X11 the conditional quantities - demanded fer the raw foods. Hence, if we specify a functional form for the conditional indirect utility V1(P1,d) and distribution function F for (q,e), then equation (4.4.19) can be estimated. But, because A}(.) is nonlinear in the parameters, the regression will be nonlinear regardless of the functional form of V1(P1,d). In the next chapter, we will explore the different methods of estimation. In general, we do not observe the unconditional quantities demanded for the raw foods because the raw foods used by the household are conditional to the dish selected. However, we can estimate these unconditional demands by their unconditional means given by: 74 m z t[xij Idi-1]oP{di-1} j-1,...,J (4.4.21) a[x 1 - . 5 1-1 where J is the total number of raw foods used in all dishes. Using the expression of Elxij ld1‘1] in (4.11.7) and letting P{di) - ”i, we have: (4.4.22) E[Xj] - 2 1-1 6V1(Pi,d)/ad + 13(v},...,v1'1,v{+1,...,vT }.,1 j-1,...,J These unconditional mean demands for the raw foods can be consistently estimated, provided the parameters in the equations of the conditional demands and the choice probabilities «1 are consistently estimated. To summarize, the estimable equations for our food consumption model are: (4.4.23) «1 -P[d1-l] -F(q-O,ei-w,V1+e1,...,Vi'1+ei,V1+1+e1,...,VT+ei) i-l,....,m. 75 v. (e o ) ij J 1,000, ’1 ’0... i)+nij aVi(P1,d)/ad j-l,.H,Ji. (4.4.25) d-d(y,eN,c) + 2* (where 6* is an error term). From these 3 equations we can derive the unconditional mean demands m [6V1(Pi,d)/6Pj] on (4.4.26) a[xJ] - 2 1-1 aVi(P1,d)/ad + ; 13(v},...,v{'1,v1+1,...,V$)-x1 1-1 j-1,...,J These four equations completely describe the food consumption model and they contain all the information needed to evaluate and predict accurately the household's response to changes in exogenous variables. Indeed, equation (4.4.23) gives the dish choice probabilities, (4.4.24) gives the conditional demands of the raw foods, (4.4.25) determines the meal budget as a function of household income, expenditure on nonfood items, and other household characteristics, and (4.4.26) gives the unconditional demand for any food staple consumed by the household. This latter equation, from which the usual elasticities of demand are derived, is of major interest for policy analysis. It may be worth emphasizing that only the first three equations are relevant for the estimation procedure and that all the variables appearing in these equations are observable. we should also note the 76 particular way that household characteristics (including income) enter the model: they influence food consumption decisions only through the meal budget. Finally, if the conditional demands for the raw foods are independent of the dish choices (that is 011 and e1 are independent), then the second terms in equations (4.4.24) and (4.4.26) become zero. Hence this condition can be tested after estimating equation (4.4.24). The next section will specify an explicit functional form for the V1(P1,d). First, however, we want to summarize three major implications of the model: the first one is of an economic nature, namely, conditional on the distribution function F or A}(.) having the desired properties, the usual restrictions implied by demand theory (adding up, homogeneity, symmetry and nonnegativity of the substitution matrix) apply only to the conditional demands for the raw foods and (partially) to the unconditional demands for the dishes. These restrictions can be tested after estimation or imposed before estimation. Beyond that, apparently the model implies no restrictions for the unconditional demands for the raw foods. This is an open question that can be investigated using the equations of the unconditional demands for the raw foods. The second implication is of a statistical nature, namely that if the conditional demands for the raw foods are dependent on the dish choices, then estimations of food demand systems that do not incorporate information on the dish choices generally yield biased and inconsistent eatimates. The last implication is also of statistical nature, that is if household demands for food are conditional on the dishes it prepares, 77 then the unconditional quantities demanded for the raw foods cannot in general be observed. However, they can be estimated using equation (4.4.26). One important feature which turns out to simplify greatly the analysis of the model, is the fact that the vector price of all the raw foods P, and the meal budget d, is the same for all dishes. Only the dish preparation time 71 is different from one dish to another. Hence the observable components of the conditional indirect utilities can be decomposed as: (4.4.27) V1(P1,d) - V(P,d,0) + 1wri i-1,...,m where w‘is the opportunity cost of time and (0,1) is the vector of parameters common to all dishes. This parametrization with a common vector of observable attributes and the same vector of parameters for all alternatives, is standard in discrete choice models, and is made possible by putting all the alternative specific attributes in the error term e1. It then follows: (4.4.28) vlf - V1(P1,d) - “((21.4) - w-(ri-rk) - 1:11‘ for kvi and k~1,...,m. Where r¥ - V(Ti-Tk) 78 and the conditional demands are given by: aV(P,d,0)/aPJ . (4.4.29) Xij - + X}(‘1r},...,‘Yri-1,1r}:+l,...,1rgl) + "lj 3V(P,d,0)/3d if food j is used in dish 1 and xij - 0 if food j is not used in dish 1 And since m -8V1(P1,d)/8PJ m -aV(P,d,0)/8PJ 2 ox - 2 -x i-l aVi(Pi,d)/ad i-l 8V(P,d,0)/8d aV(P,d,0)/8Pj ° av - at»1 I £1 < vril - Rfilfi I 61 < 1:11 96 and (5.1.2) Cov(q:) - R cov(€i I 51 < 1ri) R' + Q This result is merely a multivariate generalization of the well-known facts about the conditional distribution of a random variable jointly normally distributed with another random variable. They can be obtained easily by noting that $1 and 01-351 are independent. The conditional moments of £1|£1<1r1 were derived by Tallis (1961) in terms of the correlation matrix of £1. Amemiya (1974) rewrote F Tmllis's results in terms of the covariance matrix of {1. We will h rewrite Amemiya's results in matrix notation in a way suitable to our model. Using Amemiya's notation, let £1} be the marginal density of the kth variable of £1, fiat) the joint conditional density of the remaining 1“-2 variables given that the kth variable of £1 is equal to 1rk1, ft! the joint marginal density of the kth and 1th variables of £1, and 5(b) the joint conditional density of the remaining m-3 variables Sivan that the kth and 1th variables of 61 are equal to 191 and 1r11 respectively . Also define: 1,1 1 “'1 7‘: 1 .. - A Where rik means r1 without its kth element 1 1 "1 7‘: 1 F‘7‘m"sflk 1 I“, fad) (A)dA where r1“ means r1 without its kth and 1th element. 97 let ai be the m-l dimensional vector whose kth element is i i i k i ak - ak(1r1) - fk(7r1) F(k)(7r1k) and Bithe (m-l) x (ml) dimensional matrix whose kth 1th element is: i i i k l i To further simplify the notation, let the truncated variable 61 I £1<1r1-V and F1 " I"1"”1) With these notations, it can be easily verified that equations (2.7) and (2.8) in Amemiya (1974, pp. 1002-1003) can be respectively rewritten in matr 1x form as : . 1rka: - («Di-{)1 (5.1-3) m' -01+%,—01[D( 1 1 kk) +1511“1L i ”kl: (5.1-4) law-Lag!” F1 1 "here 01 - 542,111 - Cov (61) was defined earlier. (at is the kth column of Q1 and “111k its kth diagonal element. D(-) is a diagonal matrix for which the term in the parentheses is the 11th diagonal element. bki is the 16* column of 31. From (5.1.3) and (5.1.4) we can get the covariance matrix of W by: (5-1- 5) Cov (v) - EW' - Ew-Ew' 1rkalt-w1bi -o +a[l—D( 1 kk)+-1—-Bi-—1—aia1']0 1 1 F1 1 F1 F2 1 “1:1: 1 98 Now with these notations we have from (5.1.1) and (5.1.2) at A1(7r1) - REV and Cov(qi) - RCov(W)R' + Q but (5 1 6) a - so, g'm'l- Em ewm'l - 2 A'o‘l ' ' i i i i i i :11: i i and -1 5.1.7 -2 -RA£ -2 -2 A") A2 -2 -ROR' ( ) Q 911 1 mi 111 "i‘ i i i "'1 "i 1 hence using (5.1.4) and (5.1.6) we have i l , i (5.1-8) A (1r)--—-2 Aa (aJ xlvector) i F1 ”it i i and using (5.1.5) and (5.1.7) we have (5.1 - 9) Cov(n*) - 2* i i ”12;. .. '1 _ 2" + 2" eliilii’T-IX 1 1 k k) + £981 - :—-2-a1a1 ]A12; 6 = i i 1 ”11k 1 1 i (a J1 x .11 matrix) xix th 1 j 7:1), the j element of A (vri), can be obtained from (5.1.8) and is given by: (5-1-10) A1(1r)-L[2 1(z -2 )1 j 1 F k-lak n e n e 110:1 11 1 13 k The expression of X13011) is very interesting and open to interpretations with respect to how the different dishes in the huusehold's choice set affect the conditional demands of the raw foodsz. We will see later that only the term 99 61k - 2 e - 2 e can be identified. "11 1 "13 k Note that the restriction that leads to the traditional model is: , 1 (5.1.11) 201‘ Aia (1ri) 0 which leads to 2 A'-0 or a1(1r ) E kernel of 2 A' n e i i nit i i 5.2. 11W 5.2.1. W In this section, in addition to the normality assumption, we assume a random sample of N households having the same set of dish choices. The case of different choice sets for different households can 1’0 handled by a slight modification of the procedure followed here (by allowing the umber of dishes m to depend on the household), by the ”“1 treatment of sample selection bias, or by stratifying the P°Pu1ation according to the choice set and perform separate regressions 5"" each section of the population- Before rewriting the model in estimation format, we introduce the f°llowing binary variables and notations. din - 1 if the nth heusehold selects dish 1 - 0 otherwise and P(d1n - 1} - F1n(1or1n) - E2*'1(1 92') Na. Nn—l in in N-n Nn—l J1 in in Ji- in at. finite nonsingular for every 1. 5.2.2. W With these assumptions and notations, the likelihood of observing a“ nth household selecting a particular dish along with the necessary qua“Cities of raw foods is given by: (5.2 -5) din l 140.12) - n ‘Smlsm - $1,521“. rm. 0. 1)] thrmn i-l '3“: the conditional density Sin of :11; is given by: 102 1r. 1 f* * Jo En * ‘ F1 (7:. ) ("1n) (éinl "1n‘ "in)d£i n in 6 in-«o where f is the joint density of (em, "in) and f* is the marginal density of "in: Using the expression of the conditional distribution of a random variable normally distributed, we get: (5.:2.7) 51n(s1n'§1n)‘i"TlE"7‘J [s1n'(IJ “zin)’1' Z 1 in 7 in i 1 "1 xo [ r . -(s -(1 oz' )0 )'z A'a ’1 o 1 m-l 7 in’ in 31 in 1 at: 1 1 ' 1 wherc ”(LA) is the density of the k-dimensional random variable, normally distributed with zero mean, covariance matrix A41, and m"Ill-laced at the point x. ¢k(x,u,A) is the cumulative distribution of d“ k-dimensional normal random variable with mean p, covariance matrix Kl and evaluated at the, point x. Hence, the likelihood function of a and-On sample of N households is given by: N m (5-2.8) L012)" ' {d S -(I 02'” Z ] ' ' .91 1E1 31‘ 1“ J1 in 11 "1 O ' '0_1 0 din x"malpyrin’ °(sin-(IJ1°zin)’i) 2:016 A1 1 ' 1“ The log likelihood is then : N m n-l i-l 1 q 103 N m -l + 2 2d Log @(11' ,-S -(I 82' )9 )'2. A'O 0) n_11_1in in in J1 in 1 file i i i '20.}: )+2(‘7.9.3 .3) 1 171 2 9716 61'. The way the likelihood decomposes is very interesting and has some implications with respect to the solutions of the likelihood equation of 2,71 . Indeed, since the second term does not involve 22,71, we can readily solve for the MLE of 2,” in terms of the one for 01. We can then use the concentrated log-likelihood to solve numerically for the other parameters. Note also that if the dish choices are not relevant, that is if m-l, din'l for all n and 2016-0’ then the second term of the log- likelihood is zero, and the first term reduces to the log-likelihood of the traditional model. This provides us with a means of testing directly the restriction implied by the traditional model by use of the likelihood ratio test or the Hausman test. 2 has i1(m +121J1)(m + 121.11 + 1) free parameters. But it is .rarely the “39 that all the parameters are identified. Indeed, it is known from multinomial probit estimation that in general only part of 2‘ can be identified. (See, for example, Hausman and Wise, 1978). One usually 1‘“ to normalize some elements of 2‘ and/or use some type of ”tithe trization like the one used by Hausman and Wise (1978). It will b. seen also later that without prior restrictions on the elements of 2 not all the elements of 2'11‘ are identified: only the elements of 2033A '1 can be identified in general. 104 Given our assumptions, the above log-likelihood satisfies the usual regularity conditions so that the MLE of the identifiable parameters are consistent and asymptotically efficient with the asymptotic covariance matrix given by the Inverse of the information matrix. The computation of the NLE's is done iteratively and involves the evaluation of a multivariate normal probability (which is an integral of p dimension m-l) using numerical methods. Until recently, this was I computationally unfeasible for mz3. The next section will discuss some of the recent methods of approximating the integral. The method of H iteration normally used, is the NewtoneRaphson method and its variants. E! (See, for example, Amemiya, 1985, pp. 137-141 and page 274). The difficulties involved with the computation of the NLE have led to a search for other computationally more feasible methods of estimation. The method usually used is the Heckman's two-stage method. This method consists of estimating first 1 and 2‘ by the probit HLE using equation 5.2.2 alone, and then performing an ordinary least squares on equation 5.2.3 after replacing the unknown parameters 1 and 8‘ by their probit NLE. However, the computational advantage of this method over the NLE which is more efficient is not obvious in our model since the probit HLB is obtained by iteration which also needs the evaluation of the m—l-dimensional multivariate normal integral. The only simplification would come from the reduction in the number of parameters that have to be simultaneously estimated. 105 5.2.3 KW Heckman's method is attractive not only because of its possible computational advantage but also because it reduces our food consumption model to a simple problem of correcting what is referred in the literature as selection bias, when estimating the conditional demand equations. This selection bias problem arises whenever the dependent variable (the quantity of raw food) is conditionally observed according E to some selection process (the dish choices). Without correcting for q the selectivity bias, the least squares estimates of the parameters of the demand equations are biased and inconsistent. Heckman’s method is a ‘Efi“‘“”“ relatively simple way to get consistent estimates. Furthermore, with Heckman's method the assumption of joint normality of e and n can be removed, since the derivation of the conditional moments involved in equation 5.2.3 depends only on the linearity of the conditional expectation of a given c, the normality of e, and the independence between‘c and the regression residual of n on e. (Lee, 1982; Johnson and Kotz, 1972, p. 70). The consequence of this is that the correction of the selectivity bias is insensitive to the distribution of n* (Lee, 1982). The two stages in Heckman's method are as follows: 5:33.11. vafie In this stage, we maximize the probit likelihood function N m * 2 (1, 2‘) - “E1 151 dialog °m-l(7rin’ 0, A12eA1) 106 to f19d 7 and.ik the probit MLE of 1 and 2‘. (normalization of some elements of 26 would be needed.) ; and T, are consistent. See, for example, Amemiya (1985, pp. 286-292) for its asymptotic distribution and other theoretical properties. All the remarks made in the previous section with respect to the computation of the MLE apply equally here. These computational difficulties has limited the use of the multinomial probit MLE in the ’7 past, despite its theoretical advantage over the multinomial logit . (Hausman and Wise, 1978). However, recent progress has been made in developing computationally attractive methods of approximating the value i of the multiple integral involved. Hausman and Wise (1978) reported that a series expansion method is feasible up to five alternatives in the choice set. An approximation method originally proposed by Clark in 1961 was refined by Daganzo and Sheffi (1977). They argued that their algorithm was computationally efficient. Finally, Lerman and Manski (l981),cused Monte Carlo methods to approximate the multivariate normal integral. According to them, the method was feasible up to ten alternatives. More importantly, they reported having a computer program that lets you choose between the Monte Carlo method and Clark's method. This is an interesting feature because, although Clark's method dominated Monte Carlo in their experiments, in some other instances the former performed poorly. (Amemiya, 1985, p. 309). In the process of getting 9 and.ik’ Finigrin) and a1(;r1n) can be computed at the same time. Hence, from this first stage, we should be able to compute 107 A .. e(vr) hi(7ri ) - _£__,£2__ . 1:.in("rin) 11:29.21 MW om. 29mm and 2131,, In the second stage, we collect all the observations on the conditional demand equations for the raw foods corresponding to each dish. Then we can rewrite the system of conditional demand (5.2.3) by replacing the unknown values a (1 r ) . 8 (1r 7 F (7 r ) i in in 0 in Fin(1rin) obtained from stage 1. We will then have : (5.2.11) 3 - (1 I 1 v A ‘- 1n 0 Zia), + 2” A h (7r1n) + "in e i i J 1 i n-l,...,N i - l,...,m i where: N1 is the number of households having dish 1 in-the sample. And - * ‘ (5.2.12) "in - at“ + 201‘ Ai[hi(7rin) - h1(1r1n)] is the new'disturbance term. We can estimate the system (5.2.11) by ordinary least squares equation by equation to get consistent estimates of 01 and 2 ‘93: 016 That is, for each dish 1, we perform J1 0.L.S. using the J1 conditional demands with Ni observations for each variable. In total, we will perform 108 121 Ji 0.L.S to cover all the dishes. From (5.2.11), the jth conditional demand corresponding to dish i is: (5 2 13) Si - Z' 0 + 2 SJ hl(;r ) + 3 n - l N ° ° jn in j l-l it i in ijn ""' i lui where: ll 2:5.- 811 - 2 e - 2 e is the jth, lth element of 2 (A; , and "13 1 "11 2 ”1 1 “ 1 z " th “ ii h1(1rin) - ;——zw;——; x a1(1rin) is the 1 element of hi(7rin) in 7 in It is clear from (5.2.13) that we cannot estimate all the 2" e ij 1 for i,£-l,...,m only the differences 811 can be estimated. That is to say that only the elements of the matrix 2 A' i-l,...,m are identified. 1 at: i Note also that for each equation we have J1 + m + 1 parameters to estimate. Thus for 0.L.S. to be feasible for each i we should have N12J1+m+l. Furthermore, for the consistency to apply, Ni should be large. Provided this later condition is met, the second step of Heckman's procedure will yield consistent estimates of the 121 Ji(Ji + m + l) conditional demand parameters for the raw food corresponding to all dishes. If one were interested in only getting consistent estimates, he may stop here. However, inferences based on the standard errors given 109 by the regression output would be wrong. The reason for this is that (1) the disturbances of the regression in (5.2.13) are heteroskedastic - this can be seen from (5.2.12) - and (2) in (5,2,13) % was estimated before performing least squares so that the covariance matrix of the 0.L.S. estimator should take account of this fact. Although the exact covariance matrix of least squares is intractable, we can find its asymptotic distribution in exactly the same manner as in Amemiya (1985, pp. 369-70) or Heckman (1979). For this purpose, we write (5.2.13) in matrix notation as: j J T-fiLA—Iflb'ng—r—wu. _ 1 A (5.2.14) 81 - [21,H1] + "ij where S1 and 311 are the N dimensional vector the nth element of j 1 which are respectively Sin dimensional matrix of the original regressors of the jth conditional and zijn 21 is the Ni x (Ji + 2) demand, the nth row of which is 21“ H1 is the N1 x (m-l) matrix of the additional regressors - correcting for the selectivity bias - , the nth row of which is ‘, " , * 1 ch , h1 — h1(1r1n) . Finally, AJ is the j row of 201‘ A1 , a m-l vector of parameters, the 1th element of which is 511 with 1 v i. To simplify further the notation we will put X1 — (21, Hi) a Ni x (J1+m+l) matrix, 1 , i' , and £1 - (OJ, Aj ) a (J1+m+l)xl vector. 110 Similarly, the expression of gij can be found from (5.2.12) and is given by: (5.2.15) i + (H - H A ~ * "11 " "11 1 1) j where ”:3 is the N1 dimensional vector the nth element of which is ":1“ H1 is the N1x(m-1) matrix the nth row of which is hi - h1(7r1n)'. If we combine the J1 regression equations similar to (5.2.14), then for each dish 1, we have a system of seemingly unrelated regression (S.U.R.) with heteroskedastic disturbances and correlation across equations depending on the observations. The set of J1 regression equations can be written in S.U.R. format as one combined equation (5.2.16) 51 --(1J e 119,191 + "1 i where 4 S - (81' Si.)' ‘ 3 - (3' 5' )' i i ’ ° ° ' ’ J1 ' 1 il’ ' ' ° ’ iJ1 fl - (fli , . . . , Bi )' and I is the identity matrix of dimension J 1 1 J1 Ji 1 Similarly, we can combine the J equations defining 31 as: i - * . ‘ 1 (5.2.17) 01 - at + [IJ 9 (H1 - 81)]A i * *' where "i - (n11, . . . , "1J1) and i i' i' , , A - (Ai , . . . , AJi) vec (2015A1) It can be shown that a: and (IJ 0(H1 - Hi)]Ai are uncorrelated i u: .-. 'n mar-"u uu It‘d-w 111 (see Amemiya, 1985, p. 370). Hence we have (5.2.18) cov(;i) - cov(n:) + cov[(IJ 8 (H - Hi)Ai] 1 i From the expression of 2:“ - cov(n:n) in (15.1.9) we get: 5 2 19 * 2 1 * ( . . ) cov(q1) - q o N + 21 i i * wherex1 is the NiJi x NiJi square block matrix of J1 x Ji th square submatrices of size N the j , kch submatrix of which is i D (Ai'A* 1) a diagonal matrix of size N for which jk j inAk the term in the parentheses is its nth element; a with A1“ given by. i i' i * 1 7‘1n‘kn “k bkn 1 1 1 1 1' in F i F n n n in w in kk in where D(-) is as usual, the diagonal matrix for which the term in the parentheses is the kth diagonal element. From (5.2.14) or (5.2.16), the 0.L.S. estimates of 3: j-l,...,J1 are: (5.2.20) 5: - (x'x )‘lxi5§ - p; + (xixif1 + ‘1‘“1 - H1)A1] [X j 1 1 {"11 we can now derive the asymptotic distribution of 3; along the same lines as in Amemiya (1985, pp. 369-70). Because of our assumptions and the consistency of the probit MLE, we have: (5.2.21) plim N i 1 - lim fi—Xixi and Nine 1 Niea i 112 9 N(0, lim x'zixi) (5.2.22) N'HX N.“ i J 1 ' 'k 1 1"13 1 * . where X1 - (21,H1) and Zj - cov(qij) is the diagonal matrix of Size N1.. the nth element of which is the jth diagonal element of 2:“. From the expression of 2:“ in (15.1.9) or from (5.2.19) we get i i' * i i' * 1 5.2.23 2 - a I +10 A A A -I> + A A A E3 ‘ ’ J 11 N, ‘1 inJ’ (“11 J inJ) - . where ajj is the jth diagonal element of 20 . i A By a Taylor expansion of hi(1rin), the nth column of Hi , around 1 ‘E’“‘"‘”" we have: * 31‘ (1 r ) A i in l 37 (7 - 7) + 0(—N1) (5.2.24) h1(1r1n) - hi(1rin) - where 1* lies between 1 and 1 and lim 0(%—) - 0 N110 1 0 A Because of our assumptions and the consistency of 1, we have: * ah (1r ) ah (1r ) i in i in N-m i Hence, N;&Xi(H£ - H1)A; has the same limit distribution as: 31'! (7t ) A l , i in A1 h 11! if x1 31 a} N (1-1). N110 1 By taking the partial derivative of hi with respect to 1 and using the notation in section 5.1 we get: 113 6h (1r ) 1 in 1 1V, , - 1 k1 1 , (5'2'26) 81 ' F2 an VFinrin Fin°(7r1 akn ) + f VF(k) ik in . i where Fin - Fin(7rin)’ VF is the gradient vector of F, f is the (m-l)xl vector the kth element of which is f:(1r:), and D(~) is, as usual the diagonal matrix of size m-l for which the term in parentheses , is the kth diagonal element. a Notin that V? - a1 and f1VF1'r' -Bir' - 0(bh 1k) 8 in n (k) rik n in kri we have after some algebra: an (71- > P i in 1 i i l i l i i' (5.2.27) -—-5-;—‘ "' - [$.— Dhaka + bkk) ' r3“ + Tan an kin in in F in where bik is the kth diagonal element of B: 83 (1r ) Thus the nth row of -—£5;—£2— is: 3h (1: )' i in l i i l i l i i' 81 - - rin[§_—D(78kn + bkk) ' §_-Bn + -2_anan I in in_ F in Hence, we have: 83 (1t ) i in 1 i i 1 i 1 i i' (5°2°28) "'5?"" ' ' R1“?’°(7‘kn + bkk) ‘ E"Bn + '7' an‘n 1 ' in in F in ' ’ RiAin where R1 is the Ni x (m-l) matrix of regressors in the probit equation, the nth row of which is tin; and to simplify the notation we have put 1 i i l i l i i' Ain - F D(wakn + bkk) - §- Bn + 2 ana in in F 114 It follows from above that NiHXi(Hi - Himi has same limit distribution J as i -H xiRiAi nAjN (1 - 1) which converges to 1 N {0, lim [— X'R A A1] x F x lim [— —A1 A1 ”R N1 N1 i i in j "1 N1 j i X11} where I' is the asymptotic variance of Nf"(1-1) from the probit MLE which is given in Anemiya (1985, pp. 288-89). With our notation, we can ,3 write P as (5.2.29) 1' ' -1 N 1 F - lim N [ 2 F an tintinan 1] - lim N 1[ N2 %(r1nai)2] 1 ii .1. 1...: . 1 .1 N npl i N 101n-1 i' ' -l i -1 - lim N1[an R10 (Fin)Rian] N110 where D’1(F1n) is the inverse of D(F1n) a diagonal matrix of size N1 with Fin as the nth element. Finally, using the fact that qu and A A X’(H - H 1 A1 are uncorrelated, we conclude that fli. the o.L.s. °f 5§' 1 1’1 j-l,...,J1 are asymptotically normal with means fl; and asymptotic covariance matrices given by3: i -1 ' Ai' 1 (5.2.30) Acov(flj) :i:.(X& ) X1[D(ajj+ Aj A* mAj) i i' ' i -1Ai'A -1 + R 11A nAjlan Ri D 1(Fi n>Rianl Aj Ain Rulx (X? i) A computable consistent estimate of Acov(fi;) can be obtained by dropping the sign fiigb and replacing the unknown parameters by their consistent i 114 A It: follows from above that 5:59.011 - Hi)A1 has same limit distribution J as , i J! A . xiRiAinAjN (1 - 1) which converges to N {0 lim [1'- X'R A A1] x 1‘ x lim [LAVA R'X ]} ' N N i i inj N j in i i {to 1 Ni-«m 1 where I' is the asymptotic variance of thh-y) from the probit MLE which is given in Amemiya (1985, pp. 288-89). With our notation, we can write I‘ as (5.2-29) N N l i' ' i -l l i 2 -1 I‘ - lim Nil i F an tintinan] - lim Nil ’1: F (finan) ] Ni“ n-l in N140 n-l in i' ' -l i -1 - lim Nitan Rib (Fin)Rian] Ni“ "hare D'1(F1n) is the inverse of D(F1n) a diagonal matrix of size N1 with Fin as the nth element. Finally, using the fact that at.” and ii(Hi - NBA: are uncorrelated, we conclude that 8;, the O.L.S. of fli, 1'1, . . . ’Ji are asymptotically normal with means ,8; and asymptotic covariance matrices given by3: (5.2.30) Acov(fi;') - r11:1“.(x;x1)-1x;[1>(ajj + Aji'A:nA11) + RiAinA;[ai'R;D-1(Fin)kia:]-lAi'AinR;]X1(X;X1)-1 A cmhputable consistent estimate of Acoflfii’) can be obtained by dropping ‘1'“ Sign him“, and replacing the unknown parameters by their consistent 1 Juli, “'5'. fiufi‘ng. -.‘1 115 A . i. . estimates. Consistent estimates of 1, wk!’ are respectively 7 the probit MLE of 1 and ”1:: the km, 2‘“ element of ai probit ML; of Oi consistent estimates of A; j-l, .. . ,J iare the 0.L.S. estimates A; Then it remains to find a consistent estimate of a“. It can be shown that 1 (s 2 31) 3 - L :1 ‘2 - 31 [L :1? 121 '* ° ' jj N1 n-l "ijn 11 N n-l in j is a consistent estimate of a . Where 3 are the 0.L.S. -, JJ ijn Ev" residuals from the regression equation (5.2.10 and A?“ is obtained from Kin after replacing 1 by ’1. However, the computation of this °3t1nate of Acov(:8‘3) is very cmbersome as it requires the numerical °Valuation of many multivariate normal integral. A simpler consistent .3t1mat’e of Acovdfi) can be obtained by using the method of White (1980). Under this method, Acov(§§) is estimated by: A.‘ A A 1 (xix - “v -2 “ - 1) lximnijnmiaixi) A where D(;2 ) is the diagonal matrix of size N., the nth element ijn , l A of which isi§§ the square of the nth 0.L.S. residual. White (1980) 1‘33 proved the consistency of this estimator under general form of h°t°roskedasticity. Because of the heteroskedasticity in each equation and the °°rrelation across equations, we can get asymptotically more efficient ll6 estimators than 0.L.S. by using weighted least squares (WLS) equation by equation or SUR using the regression equation in (5.2.16). SUR is in general more efficient than WLS because it uses the correlation across equations. Note that without the heteroskedastic variances and correlation, OLS, WLS, and SUR would yield the same estimator since for a given dish 1, the regressors are the same in all J1 equations. The Wis estimate of 811 can be computed using a consistent estimate of the asymptotic covariance matrix of 7:13 (Amemiya, 1985, p. 371) . The asymptotic covariance matrix of 7)” can be found from (5.2 .15) and is given by the matrix within the outer square bracket in (5.2-30). A consistent estimate of this matrix can be obtained by following the procedure after (5.2.30). The WLS estimate of £11 will be Sivon then by: s- 51 - A,A-1 A -1 A'A-1 1 ( 2.33) 51m “1'13 x1) x1!” sJ j-1,...,J i Wherg $11 is the consistent estimate of ACovCfiij) the asymptotic covariance matrix of ‘51], obtained following the procedure outlined above. It can be shown in a way similar to the 0.L.S. case, that 213m is consistent and asymptotically normal with asymptotic covariance matrix given, by: (5.2.34) Acov(fl;‘st) - (XiIACOVGi-jndxifl A 1A -1 which A' ' can be estimated by (X1 '11 X1) 117 However, as in the OLS case, to avoid the computation of multivariate normal integrals involved in ‘3” we can use the method of White (1980) and estimate the WLS of flij and its asymptotic covariance matrix respectively by: A A -l ~2 -l " -l -2 i " -l ~2 -1 [X113 ("ijn”(il X10 (nijnfij and [XiD (maxi) where 0:16.211“) is the inverse of orfijn) . . ’3 To compute the SUR estimate of 81 in (5.2.16), we need a consistent estimate of ACov(n1), the asymptotic covariance matrix of '11 from (5.2.18), (5.2.19) and after (5.2.25) we have: (5.2-35). Acov(n1)-£ OI +2? + "1 N1 1 i ' ' -l i -l 1' N11: (1.11 O RiAin)A (aiRiD (Fin)Rian) A (I JiOAinRi) 1 ACOVOH) can be consistently estimated by replacing the unknown Parameters 1 and 01 by respectively ; and 61 their probit estimate, and AB j-l, . . . . ,J1 by A1) the 0.L.S. estimates obtained using least squares equation by equation. It can be shown that a consistent estimate of 2,, is given by i A A A N A ' (5.2.35) 2 -21+ A: [11?" 2:1 “:11”: "1 1 n-l where ii is the JixJi matrix of 0.L.S. residuals, the jth, kth element 0f which is 1 N1 A A A* ' ‘— 2 " “’ N1 “-1 "ijnnikn and A1 is the 01.5 estimate of 2" eAi i 118 a J1x(m-l) matrix, the jth row of which is A; The SUR estimate of 5i is then given by A, A A' A (5. 2.37) 31"”1 “1"1 ”(1 9x1” 1.,(1 exiuil 1 1 J1Si where 31 is the consistent estimate of Acov(n1) obtained by following the procedure outlined after (5.2.35). Again, it can be shown that 31 is consistent and asymptotically normal with asymptotic covariance matrix given by (5. 2.38) AcovCéi) - [(1J 1311;)(Accw(qi))‘1(1J exin’l 1 1 A,A which can be estimated by [(IJOX1)i;1J(I 01(1)]1 Ji Ji As in the previous cases, we can avoid the computation of the M]. tivariate normal integral in ‘1. by using White's method in the SUR Context to estimate pi and its asymptotic covariance matrix respectively by A,A [(IJ ex )011 (IJ ex )1 1.1a “1111)951 and J1 J1 J1 I A, A H J1°x1m11 (1331“11”) Where Di is the NiJixNiJisquare block matrix of JixJi diagonal submatrix of size “1' the jeh. kth diagonal submatrix of which is A A 0.11531 jnaikn) where the term in the parentheses stands for the nth 119 element of the matrix. Finally we can test the significance of the selectivity bias or dish choice effects after the second stage of Heckman's method. The null hypothesis of no selectivity bias would be then: . 1 Ho . HiAj 0 A sufficient condition for H0 to hold is: A3 - O. This leads to the following sequential test: (1) test W by using a standard F test of the significance of a subset of the P‘rameter vector. Note that under H1 the asymptotic covariance matrix of 0.L.S. is aJJ(X'X)'1, an estimate of which is given by the regression Output in most regression packages. (2) If H1 is not rejected then we can stop and conclude that the dish choice effects are not significant. If H1 is rejected then we test ”0 3 HiAji-O by using the fact that (5.2. ‘ .. - “19. l. 1 Li' 1 40) N1 (111 111M] N(O, N113 ("1111111511)1‘(N1Aj Anni» 3° that under Ho we have (5.2-4 A1.A, A1, , .1 A A1 A A1A1,A , + A A1 9 l) A H (an R1!) (Finmian )[RiAinAjAj AinRi] HiAj x 11 P1 120 where the sign "+" stands for generalized inverse, pi is the rank of the matrix within the square bracket of (5.2.41). The at“, Ein and Kin are obtained by replacing 1 with ; in air-1, Fin' and Ain respectively. Note that under H1 the limit distribution of the expression in (5.2.41) is degenerate. Before concluding the chapter, we should point out the possibility to simultaneously estimate 01 , A5 and 1 by using non linear least: squares applied to 5.2.13 without the probit MLE :1. See, Amemiya (1985, p. 372) for details. However, the potential gain (if any) in computational time and simplicity over the MLE, is not probably worth the loss of efficiency compared to the MILE which is always efficient. We finally conclude the chapter by summarizing the estimation procedure re<-‘.<>u|lnended for this model. Given the computational cost involved in completely estimating the “0461. we recommend that one should first do the Heckman's two-stage and 8": the 01.8 estimates which are consistent. Then, since the additional 81111:! inefficiency of the WLS, SUR and FILE estimators are mostly rOlavant only when selectivity bias is significant, one should test for this significance by following the testing procedure outlined above before proceeding further. If the selectivity bias is not statistically si-g'laificant, we can content ourselves with the 01.8 estimates. Otherwise, we can use these consistent estimates to get the more efficient WLS or SUR estimates. Or alternatively, with minor changes in a“ program that computes the probit MLE :ywe can use these consistent 01-3 estimates along with 1‘1 as starting values in the Newton-Raphson iteration that computes simultaneously the MU: of all the. parameters. In this case, it can be shown (see Amemiya, 1985, pp. 138) that the 121 estimator from the second round of the iteration is asymptotically efficient. nu..— CHAPTER 6 CONCLUSIONS AND POLICY IHPLICATIONS OF THE MODEL. As we were unsatisfied with the present models used to analyze the demand for food in Senegal, our purpose in this paper was to present an alternative model that represents more realistically the micro-behavior of the Senegalese household. Our experience in the Senegalese context made us feel that the consumption technology was equally if not more important than the relative prices of the raw foods in determining the demand for the cereals. Although this model was specifically designed "11:11 reference to the Senegalese context, it can be thought of as a ...s’ Senoral model of food demand having the traditional model as a special Case. Indeed, almost every society has its own consumption technology cllint: guides its selection of which raw foods to consume7. We have seen in the estimation procedure that correcting for this selection bias is uOctessary for the estimates of the demand parameters to be unbiased and cousistent. A major concern during this study was to have a realistic but workable model for forecasting food demand and for performing policy analysis. For this purpose, the relevant equations are the 7 This model may not be applicable in a society with more complex fOOd consumption habits (like in the United States or Europe) where for °a¢h meal, the household can cook more than one dish of varying sizes. In Such a complex setting, the household can be seen as making not a discrete but a continuous choice among all the raw foods used in all the dishes in the feasible choice set. Even in this case, the fact that the hmisehold has a specific set of dishes that guides its choice of raw :zods. if ignored, will introduce selection bias of the level 2 type s<3I-Issed in section 4.4.6. 122 123 unconditional demands for the raw foods and the dish choice probabilities. The unconditional quantities demanded are not observable, but can be estimated. These unconditional demand equations given in (4.5.12) can be used for forecasting the demands for the individual raw foods. Since the dish choice probabilities depend only on time and do not depend on prices and expenditures, the easiest way of computing the elasticities of demand for each food is to compute the different conditional elasticities from the conditional demand equations corresponding to the dishes where the food is used. Then we can get the unconditional elasticities by a weighted average of these conditional elasticities where the weights are the dish choice probabilities. More precisely, let ‘jki be the elasticity of demand of raw food j with respect to the price of raw food k when they are both used in dish 1 (with 'jki'o if one of them is not used in dish i) then the unconditional elasticity of demand of food j with respect to price k is: m m (6 O - - — 1’ ‘jk 151 ‘jk1P‘m1 1’ 1fl‘jk1p1hr1) whifih can be consistently estimated by m - § 3 1.. (es. “ 2) 'jk 1N1‘jk1 11‘“ dish frequencies Ni/N are known to be strongly consistent estimates Of the dish choice probabilities P(d1-l). The expenditure elasticities c“ be computed similarly. Then to compute the income elasticities and th‘ other household characteristic elasticities, we use the estimated ‘q‘lation corresponding to (5.2.1), which gives expenditure as a function of household characteristics, and then apply the chain rule. For 1t“fiance, the elasticity of income will be given by the product of the 124 expenditure elasticity and the elasticity of expenditure with respect to income; that is °jy - ejd'edy- Finally, we note that the elasticities with respect to dish preparation time will involve the density of the probability distribution. One immediate policy implication of this model is the statistical consequences of the selectivity bias. As already noted, the dish selection bias can have serious consequences on the reliability of the estimated parameters. In some empirical examples (Newey et a1. , (1990)), the presence of selection bias if not corrected, yielded wrong signs for some of the parameters. This should be of major concern when one bases policy recommendations on the magnitudes and signs of the J estimated price and income elasticities. The policy implications of the model go beyond this statistical consequence. The fact that the dish choice probabilities do not depend on prices of the raw foods not only simplifies the computation of the elasticities but also has important policy implications. As long as the ordering of the different dishes by their relative costs is not reversed, then for a given level of utility and meal budget, changes in the relative prices of the raw foods cannot alter the dish choice probabilities which describe the structural consumption behavior of the household. More precisely, we have from the equation giving the dish choice probabilities (ex: in (4.5.15)). aPldi-lllan - 0 for all prices Pj. To support this result, we can cite a preliminary finding of the CEEMAT/CIRAD ongoing survey in Dakar (Kelly and Reardon, 1989) . It is reported that in their sample, only the households with monthly income 125 .above CFA 100,000 consume couscous. Our interpretation of this finding :is that the poorest households cannot afford this millet-based dish because of the high cost of the complements going into it. Furthermore, these dish choice probabilities are important in determining the size of the unconditional elasticities. One consequence 143 that policies aimed at changing these choice probabilities may be more effective in changing the consumption behavior of the household titanlthose that change the relative price of the raw foods. There are at least two forms these policies could take: '1. The dish probabilities can depend on an exogenous shift parameter 0 in the control of the government so that 3F1(7r1, 0)/ao measure the effects that changes in 0 have on the household dish choices. 0 can include all the factors that can be used to influence the structural behavior of the household (ex: generation of technology that reduces the processing costs of local cereals used in some dishes). Increases in the number of dishes in the household choice set should decrease the dish choice probabilities. In particular, if the dish introduced is accepted and is maize-based rather than rice-based, this will decrease rice consumption. Nbreover, as noted at the beginning of chapter 4, the dish choice probabilities are time dependent, that is they are stochastic (Markov) processes. Hence their moments (variances, covariances and higher moments) can be used to evaluate the effectiveness of implemented food policies, by indicating the degree of change in household food preferences (dish choice probabilities) over time. usages- 126 For example, we have seen that because of the consumption technology constraint, any policy aimed at changing gubstangiany the demand for a particular raw food must necessarily change the relative size of the dish choice probabilities. That is for some dishes, we should have: < Ptlull-l) z Pc2{m1-1} _ where the subscripts t1, and t2 stand for the time when the choices are E] made. Hence these dish choice probabilities (how often a household consumes a particular dish) are good indicators of how the household has responded to a particular policy. To be more precise, let ”it - rum?” - Fit(7ri)- ”J Then As“: I- ‘it - “it-l measures the change in the probability (between t and t-l) that dish i is consumed by the household. It can be interpreted as a partial measure (with respect to dish 1) of the change in the i'household's taste.“ Thus: if A11: >0 we would say that the household taste has changed in favor of dish i if Ami: <0 we would say that the household taste has changed in defavor of dish i if As“: - 0 we would say that the household taste did not change with respect to dish 1. If dish 1 is a newly introduced dish, then As“: can be used to measure the rate of adoption or diffusion through time of the dish. Since the sum of the dish choice probabilities is always 1, a change in the household's taste in favor of one dish necessarily implies a change away from at least one other dish. Hence a good measure of 127 overall change in taste should take into account these correlations. TThis motivates our definition of the overall change in household's taste at time t by the covariance of Art where «tn-(alt, . . . -'mt) is the vector of dish choice probabilities at time t. cov(Ast) is a square matrix of size m, with its diagonal elements measuring the variances of the dish choice probabilities and its off diagonal terms measuring the degree of substitution between dishes. The household taste at time To (a state variable) can be measured by: To 1‘(To) II 2 cov(A1rt) c-«n and the change in the household's taste (structural behavior) between To and T1 (say, after one or two years) is measured by: r r - ng