Necessary Luxuries: A New Social Interactions Model, Applied to Keeping Up With the Joneses in India

Transcription

1 Necessary Luxuries: A New Social Interactions Model, Applied to Keeping Up With the Joneses in India Arthur Lewbel, Samuel Norris, Krishna Pendakur, and Xi Qu Boston College, Northwestern U., Simon Fraser U., and Shanghai Jiao Tong U. original July 2014, revised Jan Preliminary and Incomplete Abstract Models where utility depends on the difference between the consumed quantity vector and a vector of perceived minimum needs go back to Samuelson We model peer effects by letting the average consumption of one s peers affect the quantity vector that defines one s perceived needs. We show how dollar costs of these peer effects on welfare can be obtained in these models. In addition to the usual obstacles to identification in peer effects models, here only a small number of members of each peer group are observed, so true mean group consumption levels cannot be consistently estimated. We propose and apply a new identification and estimation strategy for spatial or social interactions data to deal with the resulting identification problems. We implement our model with Indian household-level consumption microdata. We find that peer effects are important for luxuries and not necessities. Our estimates imply that, in terms of utility, the cost of keeping up with the Joneses effects equaled about about 15% of total Indian GDP growth from 1994 to

2 1 Introduction Samuelson 1947 and Gorman 1976 propose models where an individual s perceived needs are represented by a quantity vector of goods. People get utility from the difference between the quantity vector they actually consume and this quantity vector of needs. We model peer effects in consumption choices e.g., keeping up with the Joneses by letting the average consumption of one s peers affect the quantity vector that defines one s perceived needs. The resulting model is an example of a social interactions or spatial regression framework, where the quantity one consumes of goods, particularly luxuries, depends not only on one s own budget and other characteristics, but also on the average consumption of luxuries and/or other goods consumed by one s peers. In our model, there are consumption externalities across people within groups. We provide a model of utility wherein the welfare effects of these consumption externalities are easy and clear in their characterisation, identified from consumer behaviour and estimable via GMM methods. We bring our model to the data with repeated cross-sections of Indian consumption micro-data. We find that perceived needs depend quite a lot on group-average consumption, and in particular, that perceived needs for luxuries depend positively on the luxury consumption of peers. Under the model, these externalities imply substantial costs on individuals, which may substantially reduce the gain from overall income increases. For the purpose of defining perceived needs, we define an individual s peer group as everyone in the population that shares a similar education level, works in the same industry, and lives in the same geographic region as the individual. However, our data are from sample surveys of expenditures by consumers. In this type of data one can only observe a small number of individuals in each each peer group. It is therefore unreasonable to assume either that the econometrician can observe the entire network asymptotically, or that the econometrician can asymptotically observe true mean consumption within each group of peers. Therefore, in our model, observed average consumption within a group is not assumed to consistently estimate true mean group consumption. Rather, the difference between observed group average consumption and the true group mean is treated as measurement error, which does not disappear as the sample size grows. This corresponds to asymptotics where the number of observed groups goes to infinity, but the number of individuals observed within each group is assumed to be fixed, and where some group members are never sampled. This necessitates our deriving a new method for identifying and estimating a social interactions or spatial regression type model. The novelty of the method for social interactions is that it is applicable when only a small number of individuals within each group is observed. Alternatively, our results can be interpreted as a new way to identify and estimate spatial 2

3 regression or network models where the usual assumptions regarding suffi cient interconnectedness of the network are violated. Our identification strategy and associated estimator also deals with additional complications, including Manski s 1993 reflection problem, the presence of group level fixed effects, nonlinearities resulting from utility maximization, and a multiple equation system where each equation depends on the vector of peer means from all of the equations. The steps of our analysis are as follows. We first consider a simple, generic model containing peer effects, and show how it can be identified and estimated with our type of data. This new methodology for identifying and estimating a peer effects model should be of general interest, since it is potentially applicable to other contexts where one only observes a small number of members of each peer group. We then discuss the existing theory of utility and demand function specification, and associated welfare analyses, that incorporates needs. This includes quantity shape invariance as in Pendakur 2005 and the absolute equivalence scale exactness of Blackorby and Donaldson We adapt this analysis to our context where needs are assumed to depend on the purchases of one s peers. We then specify demand functions for this analysis that can be identifed and estimated using the same techniques provided in our generic peer effects model. Given the above, we implement our estimator and associated welfare analyses using a few annual cross-sections of household-level expenditure data from India. Our data offer a good laboratory for this analysis because we observe relevant characteristics of each household for constructing peer groups, including education level, industry of employment, and detailed geographic area of residence, along with other household characteristics such as household size, age, religion and caste. We find that peer effects are large and important for luxuries but not necessities. A one rupee increase in peer group luxury spending increases one s own perceived needs for luxury spending by about half a rupee. So a large fraction of luxury purchases are perceived as necessary. The estimates imply that, in terms of utility, the costs of keeping up with the Joneses accounted for about one seventh of Indian GDP growth from 1994 to Relevant Literature - Peer ffects in Consumption, Income, and Demand Income gains in the upper parts of the income distribution may not increase well being much if utility depends on other people s consumption levels. See e.g., Frank 1999,

4 The possible mechanisms for this are varied: Veblen 1899 effects make consumers value consumption of visible status goods; reference-dependent utility functions make consumption valuable only inasmuch as it exceeds what we see around us; keeping up with the Joneses" makes our effective consumption smaller the more our peers consume; the consumption of our peers affects what we perceive as necessary ; etc. What these stories have in common is that they imply that each individual s consumption may have externalities on the utility functions of others around them. We bring this intuition to data using a model of consumption externalities in which the welfare cost of such externalities is easily expressed. We exploit the intuition that the consumption of a person s peers affects their own perception of their needs. In the context of utility and cost functions, needs are fixed costs, representing the minimum quantity vector one requires to start getting any utility. The idea that preferences have fixed costs that need to be met before expenditures start increasing utility is an old one, going back at least to Samuelson s 1947 note on the implications of linearity. Samuelson called the minimum quantity vector corresponding to needs the necessary set of goods, and defined supernumerary income as one s remaining income, after subtracting off the fixed cost of these needs. Utility is then obtained by spending supernumerary income. The classical Stone 1954 and Geary 1949 Linear xpenditure System incorporates this construction. More generally, Gorman 1976 showed that these kind of fixed costs which he calls overheads can be introduced into any utility function and will generally vary across consumers. This type of model has implications for the specification of demand functions that take the form of shape invariance in quantity demands see Pendakur 2005; Pollak and Wales 1992 refer to this as demographic translation. This class of specifications is analogous to the more well known shape invariance in budget shares popularized by, e.g., Pendakur 1999 and Blundell, Chen and Kristensen See Pendakur 1999 and Lewbel 2010 for details. Blackorby and Donaldson 1994 show how valid social welfare functions can be constructed based on differences between budgets and needs, using what they call "Absolute quivalence Scale xactness" or AS. Blackorby and Donaldson 1994 also show that needs themselves are not identified from observable consumer behaviour, but differences in needs across consumers with different values of exogenously varying conditioning variables are identifiable from behaviour, and that is all that we require for our welfare analyses. Identification and welfare calculations are more complicated in our model, because in our context needs depend on peer group expenditures which are endogenous rather than exogeneous. Nevertheless we find we can adapt their framework to our setting. Our model starts with Gorman s specification of overheads or needs as a quantity vector. One then derives utility from the vector of differences between the vector of goods purchased 4

5 and this minimum needs quantity vector. Our innovation is that we specify this vector of needs as a function of the consumption of one s peers. Thus, consumption externalities arise because one s perceived needs depend on the consumption of one s peers. If my peers consume more, my perceived needs and hence my fixed costs go up. This in turn makes my supernumerary income go down, resulting in both an observable change in expenditure patterns and a loss in utility. The model therefore has testable implications, and it implies welfare losses that are quantifiable in both an individual and a social sense. Specifically, we can simply add up the estimated increases in needs across people, which then provides the dollar social cost of keeping up with the Joneses. Note that these costs of keeping up with peers are not necessarily all wasted resources. For example, there may be a value to society if everyone has internet access, even if that makes internet access become a perceived need. We do not take a stand on what fraction of the costs of peer effects is not a waste from the standpoint of society, though the peer effects of luxuries are unlikely to be useful for the most part. Regardless, it is clearly useful to quantify these costs of peer effects, and separate those from the direct impacts of income increases on consumer s utility. Our model fits into the large literature on income-evaluation and income reference points, where one s valuation of income depends on one s reference group. Surveys of this literature include Kahneman 1992 and Clark, Frijters, and Shields A smaller literature focuses on the difference between consumption and income, allowing the valuation of one s consumption to depend one s reference group. These are are mostly intertemporal models that are intended to address macroeconomic puzzles. See, e.g., Gali 1994 or Maurer and Meier At the other extreme, some papers in psychology and marketing focus on how the valuation of particular individual goods or brands depend on one s peers. See, e.g., Rabin 1998 and Kalyanaram and Winer Chao and Schor 1998 find that these effects are particularly important for goods that are visible in their case, cosmetics, linking their findings to Veblen effects. Many analyses of peer effects of these types are essentially nonstructural, including the above Chao and Schor 1998 paper. Boneva 2013 regresses household quantity demand vectors on household budgets total expenditures and on the average budgets of reference groups, using PROGRSA/Opportunidades related variables to instrument group averages. Ravina 2008 and Clark and Senik 2010 regress self-reported utility on own budgets and reference group average budgets. Virtually all such studies find these group average variables to be significant, though Ravaillon and Lokshin 2010 say that it is not important for the poor. But the magnitude or statistical significance of coeffi cients on group variables do not measure welfare effects. This requires a model of utility with suffi cient structure to permit 5

6 social welfare analysis. 3 Relevant Literature - Identification Our model where each individual s outcome depends on the mean of the outcomes of one s peer group is a form of social interactions model. It can also be seen as a spatial model, where all individuals within a group are equidistant from each other. A well known obstacle to identification of this kind of model is the reflection problem described by Manski 1993, See also Brock and Durlauf 2001, and Blume, Brock, Durlauf, and Ioannides Our model will have specific behaviorally derived structure in particular some nonlinearity that overcomes reflection problems. In some peer effects model, network information is available and can help identification. For example, Bramoullé, Djebbari, and Fortin 2009 show identification of peer effects in social networks that are suffi ciently interconnected, and where for each member of group g, the peer effect is linear in a mean taken over all other group members. These types of models exploit variation in group sizes to aid identification, and requires that the number of observed members of each group increases with sample size See, e.g., Devezies et. al, Using a somewhat related approach, Graham 2008 estimates peer effects in a linear model by comparing the covariance of test scores in large versus small classrooms. In our context, variation in group size does not provide any identifying power, both because we only see a small number of members of each group, and because we do not know actual group sizes. The interactions of peer group members may be modeled as a game. Suppose there is private information that cannot observed by econometricians. We assume that group members have utility functions that depend on peers only through the true mean of the peer group s outcomes. If group members also all observe each other s private information and make decisions simultaneously corresponding to a complete information game, then each individual s actual behavior will only depend on others through the group mean. Complete games are generally plausible only when the size of each group is small, and are typically estimated assuming the econometrician s data includes all members of each observed group. An example is Lee However, in our case the true group sizes are large, but we only observe a small number of members of each group. An alternative model of group behaviour is a Bayes equilibrium derived from a game of incomplete information, in which each individual has private information and makes decisions based on rational expectations regarding others. This type of incomplete game of group interactions can result in the reflection problem again, where endogenous effects, exogeneous effects, and the correlated effects cannot in general be separately identified. In either type of game there is also the potential problem of no 6

7 equilibrium or multiple equilibria existing, resulting in the problems of incompleteness or incoherence and the associated diffi culties they introduce for identification as discussed by Tamer We do not take a stand on whether the true game in our case is one of complete or incomplete information. We just assume that players are basing their behavior on the true group means. This assumption is most easily rationalized by assuming that consumers either have complete information, or can observe a suffi ciently large number of members in each group that their errors in calculating group means are negligible. A more diffi cult problem would be allowing for the possibility that each group member also only observes group means with error. We do not attempt to tackle that issue in this paper. In that case we would need to model how individuals estimate group means, how they incorporate uncertainty regarding group means into their purchasing decisions, and how all of that could be identified in the presence of all of the other obstacles to identification that we face. These obstacles include only observing a small number of members of each group, the reflection problem, group level fixed effects, nonlinearities resulting from utility maximization, and a multiple equation system where each equation depends on the vector of peer means from all of the equations. Identification depends on what we assume is observable from data. Standard models of within group interactions with large groups assume that there are no interactions between groups, and that both the number of groups G and the number of observed members n g within each group goes to infinity. However, for reasonable definitions of peer groups, standard consumer expenditure surveys only sample a relatively small number of individuals within each group even in our relatively large Indian data set, n g is less than one or two dozen for many groups. So while it is reasonable to assume that G goes to infinity, we take a completely new approach to identifying and estimating peer effects, by assuming that n g is small and fixed. This means that observed within group sample averages are mismeasured estimates of true within group means, and that these measurement errors do not disappear asymptotically as the sample size grows with G. Moreover, these measurement errors are by construction correlated with individual specific covariates, further exacerbating the diffi culties listed earlier in obtaining identification and constructing consistent estimators of model parameters. Measurement error more broadly has long been recognized as potentially important in social interactions models e.g., Moffi tt 2001 and Angrist, 2014, though this work focuses on standard issues of mismeasurement in regressors, recognizing that, unlike in ordinary models, outcomes are also regressors and hence measurement error in outcomes matters. This is quite different from our situation, which recognizes that only observing a limited number of individuals in each group results in measurement errors in group means. This 7

8 can also be interpreted as a missing data problem where what is missing is the outcomes of most group members. Others have looked at different missing data problems in peer models. For example Sojourner 2009 considers peer effects in Project STAR classrooms, where the missing data consists of pre-intervention information on student achievement. In his model, the diffi culties of missing data are addressed in part by assuming a linear model where student are randomly assigned to their peer groups, defined as classrooms. As is standard in models with measurement errors, we will assume we have valid instruments that are correlated with true group means. However, even with instruments, the obvious two stage least squares or GMM estimator that assumes model errors are uncorrelated with instruments after replacing true group means with their sample analogs will not be consistent in our context. This is because: a in a linear model such instruments will not overcome the reflection problem; and b in a nonlinear model we will have interaction terms between the measurement errors and the true regressors. An analogous problem arises in the polynomial model with measurement errors considered by Hausman, Newey, Ichimura, and Powell We show that overcoming these issues requires some novel transformations that ultimately lead to a valid GMM estimator. Finally, even given complete identification of model parameters, the Blackorby and Donaldson 1994 result discussed earlier still applies, namely, that only relative needs across consumers are identified, not the absolute level of needs. However, this will suffi ce for all of our welfare analyses. 4 Generic Model Identification Before introducing our general model of peer effects in consumer demand, in this section we consider a simple generic model where individual outcomes depend on group means. We use this model to illustrate the diffi culties associated with identification in our general context, and to show how we overcome these diffi culties. This generic model should also be useful in other spatial and/or peer effects models. Let y i denote an outcome and x i denote a regressor for an individual i. Let i g denote that the individual i belongs to group g. For each group g assume we observe n g = i g 1 individuals where n g is a small fixed number, which does not go to infinity. Let y g = y i i g, ŷ g = i g y i/n g, and ε yg = ŷ g y g, so y g is the true group mean outcome and ŷ g is the observed group average outcome in our data, and ε yg is the estimation error in the sample group average. Define x g = x i i g, x 2 g = x 2 i i g, and similarly define x g, x 2 g, ε xg and ε x 2 g analogously to ŷ g, and ε yg. Consider the following single equation model the multiple equation analog is discussed 8

9 later. For each individual i in group g, let y i = y g a + x i b 2 d + yg a + x i b + v g + u i 1 where v g is a group level fixed effect and u i is an idiosyncratic error. The goal here is identification and estimation of the effects of y g and x i on y i, which means identifying the coeffi cients a, b, and d. We could have written the seemingly more general model y i = y g a + x i b + c 2 d + yg a + x i b + c k + v g + u i where c and k are additional constants to be estimated. However, it can be shown see the appendix for details, that by suitably redefining the fixed effect v g and the constants a, b, and d, that this equation is equivalent either to equation 1 or to y i = y g a + x i b 2 +vg +u i. Since this latter equation is strictly easier to identify and estimate, and is irrelevant for our empirical application, we will rule it out and therefore without loss of generality replace the more general model with equation 1. Next observe that, regardless of what we assume about within group or between group sample sizes, if this model were linear i.e., d = 0, then we would not be able to identify the effect of y g on y i, i.e., we would not be able to identify the peer effect. This is because, if d = 0, then there is no way to separate y g from the group level fixed effect v g. All values of a would be observationally equivalent, by suitable redefinitions of v g. This is a manifestation of the reflection problem, which we overcome by a combination of nonlinearity and functional form restrictions. We assume that the number of groups G goes to infinity, but we do NOT assume that n g goes to infinity, so ŷ g is not a consistent estimator of y g. We instead treat ε yg = ŷ g y g as measurement error in ŷ g, which is not asymptotically negligible. This makes sense for data like ours where only a small number of individuals are observed within each peer group. This may also a sensible assumption in many standard applications where true peer groups are small. For example, in a model where peer groups are classrooms, failure to observe even one or two children in a class of, say, 20 students may mean that the observed class average significantly mismeasures the true class average. Formally, our first identification theorem makes assumptions A1 to A3 below. Assumption A1: ach individual i in group g satisfies equation 1. Unobserved errors u i are mean zero and independent of x j for all individuals j and groups g, and v g are unobserved group level fixed effects. The number of observed groups G. For each 9

10 observed group g, a random sample of n g observations of y i, x i is observed. ach sample size n g is fixed and does not go to infinity. The true number of individuals comprising each group is unknown and could be finite. Assumption A2: The coeffi cients a, b, d are unknown constants satisfying d 0, b 0, and [1 a2x g bd + 1] 2 4a 2 d[x 2 gb 2 d + bx g + v g ] 0. Assumption A1 just defines the model. In Assumption A2, as discussed above d 0 is needed to avoid the reflection problem. Having b 0 is necessary since otherwise we would have nothing exogenous in the model. Finally, note that the inequality in Assumption A2 takes the form of a simple lower or upper bound depending on the sign of d on each fixed effect v g. This inequality ensures that an equilibrium exists for each group, thereby avoiding Tamer s 2003 potential incoherence problem. To see this, plug equation 1 for y i into y g = y i i g. This yields a quadratic in y g, which, if a 0, has the solution y g = 1 a2x gbd + 1 ± [1 a2x g bd + 1] 2 4a 2 d[x 2 gb 2 d + bx g + v g ] 2a 2 d 2 if the inequality in Assumption A2 is satisfied while if a does equal zero, then the model will be trivially identified because in that case there aren t any peer effects. We do not take a stand on which root of equation 2 is chosen by consumers, we just make the following assumption. Assumption A3: Individuals within each group agree on an equilibrium selection rule. For each i g, define demeaned variables ỹ i = y i ŷ g, = x i x g, x 2 i = x2 i x 2 g, and ũ i = u i û g. Note that by Assumption A1, and x 2 i are independent of ũ i. For identification, we need to remove the fixed effect from equation 1, which we do by subtracting off observable group averages. This yields ỹ i = 2abd y g + b 2 d x 2 i + b + ũ i = 2abd ŷ g + b 2 d x 2 i + b + ũ i 2abd ε yg 3 where the second equality is obtained by replacing y g on the right hand side with ŷ g ε yg. Note that in addition to removing the fixed effects v g, subtracting off the group averages also removed the linear term ay g, and the squared term da 2 y 2 g. The second equality in equation 3 shows that ỹ i is linear in observable functions of data, plus a composite error term ũ i 2abd ε yg that contains both ε yg and ũ i. It can also be shown see the Appendix that ε yg = 2abdy g ε xg + b 2 dε x 2 g + bε xg + û g 10

11 where ε xg = x g x g and ε x 2 g = x 2 g x 2 g. A natural way to identify and estimate equation 3 might be to regress ỹ i on ŷ g, x2 i, and, perhaps using other demeaned nonlinear functions of x i as instruments for the ŷ g term. More generally, ε yg, ε xg, and ε x 2 g are equivalent to measurement errors in regressors, and a standard way to cope with measurement errors in regressions is to employ instruments that are correlated with the true regressors and uncorrelated with the measurement errors. An estimator like this could work if the number of individuals observed in each group grew with the sample size, making ε yg in shrink to zero. But we cannot estimate equation 3 using instrumental variables here, because of the ε yg component in the error term. It can be shown see the Appendix that any available instrument that is correlated with y g will also be correlated with ε yg, and so cannot be used as an instrument. This also precludes using other common tricks to gain identification, e.g., replacing ŷ g with a sample average that excludes observation i will not overcome this problem. To deal with the measurement errors we will in fact assume instruments are available as discussed below, but further transformations of the model are required to allow any instruments to work. We first need to remove the interaction between and ε yg which we do by dividing both sides of equation 3 by to get where the composite error U i is Next construct group level aggregates ỹ i x 2 i = 2abdŷ g + b 2 d + b + U i 4 U i = ũi 2abd 2abdy g ε xg + ε x 2 gb 2 d + ε xg b + û g Ŷ g = 1 n g i g ỹ i, Xg = 1 n g i g x 2 i and x Ûg = 1 U i. i n g i g Assumption A4: We observe a vector of one or more group level instruments r g such that x i x g x g, x 2g, v g, r g = 0, x 2 i x 2 g r g = 0, u i is independent of r g and r g ŷ g, X g, 1 r g ŷ g, X g, 1 is nonsingular. Most of Assumption A4 corresponds to standard instrument validity assumptions. The first two equalities in Assumption A4 are somewhat stronger than this, but a suffi cient condition for them to hold would be to let ε ix = x i x g, and assume that ε ix i g, r g, v g, x g = 0 and ε 2 ix i g, r g = ε 2 ix i g. ssentially, this corresponds to 11

12 saying that any individual i in group g has a value of x i that is essentially a randomly drawn deviation around their group mean level x g. The first two equalities in A4 are in turn only used to show that ε xg x g, x 2g, v g, r g = 0, and so could be relaxed. In particular, we really only require mean independence of the average value x i x g over the observed individuals in the group. Given Assumption A4, one can directly verify that U i r g = 0, that is, ỹ i 2abdŷ g b 2 d x 2 i b r g = 0. 5 This expression can also be immediately aggregated up to the group level, so Ûg r g = 0, which then implies Ŷg 2abdŷ g b 2 d X g b r g = 0. 6 These moments then identify the coeffi cients 2abd, b 2 d, and b, which we can immediately solve for the three unknowns a, b, d. The above derivations outline the proof of the following identification theorem details are provided in the Appendix. Theorem 1. Given Assumptions A1, A2, A3, and A4, the coeffi cients a, b, d are identified. Given the identification in Theorem 1, corresponding consistent estimators are easy to construct. Generally, given a vector of instruments r g, we may apply GMM to moments defined by equation 6. However, a natural question to ask is, where would we obtain suitable instruments r g? In our empirical application, the instruments will be group means of functions of x j and x 2 j constructed using individuals j that are observed in the same groups as individuals i, but in a different time period of our survey. Since the data take the form of repeated cross sections rather than panels, different individuals are observed in each time period. This produces the necessary uncorrelatedness instrument validity conditions in Assumption A4. The relevance of these instruments the nonsingularity condition in Assumption A4 will hold as long as group level moments of functions of x in one time period are correlated with the same group level moments in other periods. In our application, x i is the total expenditures of a consumer i, so assumption A4 will hold if the distribution of income within groups is suffi ciently similar across time periods, while the specific individuals within each group who are sampled change over time. Let X g and x g be the same as X g and x g, but constructed using observations x j from a different time period than the data used to construct Ŷg, X g, and x g. Define θ = θ 1, θ 2, θ 3 = 2abd, b 2 d, b. Then letting r g = x g, Xg, 1, GMM estimation simplies to the ordinary 12

13 instrumental variables estimator θ [ G = g=1 r g ŷ g, X ] 1 g, 1 G g=1 ŷgr g, from which and one can immediately recover consistent estimates b = θ 3, d = θ 2 / b 2, and â = θ 1 / 2 b d. Note that additional instruments can also be constructed, which could help effi ciency of estimation. In particular, the nonlinearity of y g in equation 2 shows that a reduced form expression for ỹ i and hence for ŷ g would include additional nonlinear functions of x g. This suggests that nonlinear functions of x g, such as x 2 g or x 1/2 g, would therefore be valid and potentially useful additional instruments. 5 Generic Model xtensions One extension of the generic model is to include more covariates. The simplest way to accomplish this, which will suffi ce for our demand application, is to include a vector of additional covariates z i that are discretely distributed. We can then extend the above analysis to estimate the model y i = y g a + x i b + z iγ 2 d + yg a + x i b + z iγ + v g + u i 7 The construction of ŷ g and X g for estimation involved demeaning y i and x i, and then averaging across the entire sample functions of these demeaned variables. The main difference in the identification and estimation of equation 7 is that we will need to demean within each cell of values of that z i take on, before averaging across all of the data. This extension is therefore only practical if the number of different values z i can take on is relatively small. One nice feature of this extension is that it allows u i to be correlated with z i. Describing this extension requires a little more notation. Analogous to the previous notation i g and ŷ g = i g y i/n g, let i gz denote observations i that are both in group g and have z i = z, let n gz = i gz 1 and let ŷ gz = i gz y i/n gz. So n gz is the number of individuals i in group g that have z i = z, and ŷ gz is the sample average of y i for these individuals. Let ỹ i = y i ŷ gz for i gz. So just as ỹ i was the difference between y i and that individual s group average ŷ g, we have ỹ i being the difference between y i and that individual s group and cell average ŷ gz. In short ỹ i demeans based on the average of everyone in the same group as individual i, while ỹ i demeans based on the average of everyone who is both in the same group and has the same value of z as individual i. Define, x2 i and ũ i analogously. Finally, define Ỹ g = 1 n g i g ỹ i and Xg = 1 n g i g x 2 i. 13

14 A derivation analogous to that of equation 6, shown in the Appendix then gives ] [ Ỹ g 2abdŷ g b 2 d Xg 2bdẑ gγ b r g = 0 8 for suitable instruments r g. Similar to equation 6, equation 8 can be used to estimate the parameters a, b, d, and γ using GMM. In particular, we can let r g = x Xg, g, ẑ g, 1 where Xg, x g, and ẑ g are the same as Xg, x g, and ẑ g, but constructed using observations x j, z j from a different time period than the data used to construct the Ỹ g, ŷ g, Xg, and ẑ g variables. Our actual demand application has a vector of J outcomes and a corresponding system of J equations. xtending the generic model to a multiple equation system introduces potential cross equation peer effects, resulting in more parameters to identify and estimate. Letting y ji denote the j th outcome for individual i gives the model y ji = y ga j + x i b j + z iγ j 2 d j + y ga j + x i b j + z iγ j + vjg + u ji 9 where y g = y 1g,..., y Jg is a J-dimensional outcome vector and a j = a 1j,..., a Jj is the associated J-dimensional vector of peer effects for jth outcome. Analogous derivations to those above give moment conditions Ỹ jg 2b j d j ŷ ga j b 2jd j Xg 2b j ẑ gγ j d j b j r g = 0. for j = 1,..., J. For estimation, we need r g to be a vector of at least J + K + 1 instruments, where K is the dimension of z. Similar to before, Xg and ẑ g can serve as instruments for Xg and ẑ g. For ŷ g, one can see from the expression for ŷ jg that ŷ g contains terms y ga j 2, x g y g, ẑ g y g, x 2 g, ẑz g, xz g, and x g. As y g is itself implicitly a function of x g, x 2 g, zz g, z g, and xz g, we may 2, use x 2 g, ẑz g, xz g, x g, x g 2, x g x2 g, x2 g etc. to construct the instruments for ŷg. If J is small, even without the z covariates, we have enough instruments for the J-dimensional ŷ g from nonlinear forms of x. The vector of regressors z, including squares and cross products, provides additional instruments. 14

15 6 Utility and Welfare With Needs Before laying out the specifics of our specification of utility and demand function, we first summarize some existing theory regarding utility and welfare calculations in the presence of needs aka: fixed costs, overheads, etc. Let i index consumers and let q i = qi 1,.., qi J be a J vector of commodity quantities chosen by consumer or household i. Let p be the corresponding J-vector of prices of each commodity, and let x i be the total budget for commodities of consumer i. Commodities here are aggregates of goods or services that are assumed to be purchased and consumed in continuous quantities. ach consumer i is assumed to choose q i to maximize a direct utility function, subject to the budget constraint that p q i x i. Preferences and the utility function of a consumer i can be represented by an indirect utility function V i p, x i which gives the utility level attained by consumer i when facing the budget constraint defined by p, x i. The purchase choice q i of consumer i in all different price and budget regimes describes their demand functions q i = g i p, x i. The demand functions g i are related to V i by Roy s identity. If consumer s preferences are stable, then demand functions q i = g i p, x i may be observed from demand data, that is, by seeing what the consumer chooses to buy in every possible price and budget regime. For welfare comparisons, we need to be able to compare well being across consumers. Define the equivalent-income function X i p, x as the income budget needed by consumer i to get the same level of utility as that of some reference consumer i = 0 having a budget x. Blackorby and Donaldson 1994 define Absolute quivalence Scale xactness AS as a property of utility functions that holds if and only if, for all consumers i, X i p, x = x F i p for some function F i p. They show that AS holds if and only if F i p = F i p F 0 p where V i p, x i = V p, x i F i p Here V is some indirect utility function not the utility function V 0 of the reference household and the function F i p for each consumer i can be interpreted as the cost of satisfying the perceived needs of consumer i. This model implies that each consumer derives utility through a common utility function V 0 from the extent to which their available budget x i exceeds their perceived cost of needs F i p. One property of AS is that it involves both testable and untestable restrictions on utility. If follows from Roy s identity that quantity demand functions are given by: g i p, x i = g 0 p, x i F i p + F i p p. This is shape-invariance in quantity demands as in Pendakur Here, demand functions 15

16 are identical across any two consumers i and j, except that they are translated in x by F i p F j p and translated in quantities by the vector F i p F j p. Figure 1 shows a p p quantity demand equation for 2 consumers where this shape-invariance property is satisfied. Here, the quantity demand equations over expenditure of both all consumers have the same shape, but may be shifted vertically in the quantity demand or horizontally in the budget x. This restriction must hold for all quantity demand equations. Identification of the differences in needs is easy to see. Given exogenous variation in a variable that shifts F i, we can recover the response of F i to that variation by estimating the horizontal shift in quantity demand curves for consumer i. The heavy lifting is in finding moment conditions that give us that exogenous variation. Shape invariance in quantity demands is testable. Testing involves comparing the demand functions g i p, x of each consumer i with the demand functions g 0 p, x of a reference consumer, and seeing if a function F i p exists for each consumer i that makes the above equation hold. However, AS is not fully testable. In particular, since quantity demands depend only on the ordinal properties of utility functions, we cannot test the cardinal restrictions imposed by AS. That is, the exact same observable restrictions on demand that is, shape-invariance will hold if, for each consumer i, V i p, x i = H i [V 0 p, x i F i p] for some strictly monotonically increasing function H i. The untestable restriction of AS is that H i equals the identity function. Another property of AS documented by Blackorby and Donaldson 1994 is that the cost of needs functions F i p cannot themselves be identified from demand data except possibly by arbitrary functional form restrictions. All that can be identified is the differences in needs functions across consumers, that is, F i p = F i p F 0 p is identified but not F i p itself. Consequenty, we normalize F 0 p = 0. Identification of F i p is in the following sense. Suppose utility functions satisfy AS. Then, preferences satisfy shape-invariance, and we can recover F i p from demand behaviour. However, by applications of monotonic transformations H i, we could generate an infinite number of other equivalent-income functions X i, and all of these would be consistent with observed behaviour. Blackorby and Donaldson show that only one of these satisfies the additive property of the AS equivalent-income function. The usefulness of AS for our purpose is that, since X i p, x = x F i p, we can use AS to directly measure, in dollar terms, the welfare impacts of changes in costs, which in our application will come from peer effects. It follows from the equivalent income function that, under AS, we can define anonymous social welfare functions over X i p, x instead over over utilities V i. One such valid social welfare function is the simple sum i x i F i p 16

17 it is not inequality-averse. Notice that this welfare function is easy to understand, and can be evaluated just from identification of the relative costs of needs functions F i p. If needs change, social welfare changes by the sum of those needs changes. In our work we will assume that F i p = p f i. This is the most natural form of AS model, where f i is a quantity vector. ach element f ji of f i equals the quantity of commodity j that consumer i perceives as needs, that is, f ji is the minimum amount household i feels must be consumed, before they can start to generate any utility. This implies the model V i p, x i = V p, x i p f i which by Roys identity has associated demand functions g i p, x i = gp, x i p f i + f i 10 where g is the vector of demand functions that correspond to the utility function V. For linear g, Samuelson 1947 calls the vector of needs f i the necessary set, and calls x i p f i supernumerary income. The Stone 1954 and Geary 1949 linear expenditure system is a special case of this model. Gorman 1976 and Pollak and Wales 1981 assume that f i is a function of observable household characteristics preference shifters and call f i overheads, interpreting V p, x i p f i as a consumer s production function where the ultimate product being produced is utility. They consider linear and quadratic specifications for g. Pendakur 2005 showed the semiparametric characterisation of this model, and showed how identification works with unspecified form for g. 7 Our Utility and Demand Model - Needs With Peer ffects We modify the existing literature on needs, as summarized in the previous section, by specifying utility functions in which the needs vector f i of a consumer i depend on quantities purchased by that consumer s peers. Let i g denote that consumer i belongs to group g. Let q g = q i i g, so q g is the mean level of quantities consumed by consumers in group g. ach peer group g may contain many consumers, but in our data we only observe n g consumers in each group, where n g = i g 1 is some small fixed number. Thus, q g is unobserved, but we can still use the group average q g = 1 n g i g q i to construct estimators. We specify f i as a function of peer demands, specifically, we let f i = Aq g + Cz i for 17

18 some J by J matrix A and some J by K matrix C, where g is the peer group of consumer i and z i is a vector of observed characteristics of consumer i. The larger the elements of A are, the greater are the peer effects. If A is a diagonal matrix, then the perceived needs for any commodity depend only on the group average purchases of that commodity. We more generally allow for nonzero off diagonal elements as well. If keeping up with the Joneses induces negative externalities across individuals, so that increases in group average consumption cause increases in needs, we should expect elements of A particularly diagonal elements to be nonnegative. However, they cannot be arbitrarily large. If the diagonal elements are too large, e.g. larger than 1 then stable equilibria may not exist analogous to the Assumption A2 inequality being violated in the generic model.. A long empirical literature on commodity demands finds that observed demand functions are close to polynomial, and have a property known as rank equal to three. See, e.g. Lewbel 1991 and Banks, Blundell, and Lewbel 1997, and references therein. Gorman 1981 showed that any polynomial demand system will have a maximum rank of three, and Lewbel 1989 shows that the simplest tractible class of indirect utility functions that yields rank three polynomials in x is V p, x = x F p 1 λ B p / 1 λ D p for some constant λ and some differentiable functions F, B and D. Combining the shape invariant AS model of the previous section with this specification of utility gives the model V i p, x = x i p Aq g p Cz i 1 λ B p / 1 λ D p Preserving homogeneity i.e., the absence of money illusion, which is a necessary condition for rationality of preferences, requires B p 1/λ 1 to be homogeneous of degree one in p and D p to be homogeneous of degree zero in p. Applying Roys identity to this indirect utility function then yields the demand functions q i = x i p Aq g + Cz i λ D p B p + x i p Aq g + Cz i B p λ 1 B p + Aq g + Cz i. The most commonly assumed rank three models are quadratic see the above references, and Pollak and Wales 1980, which corresponds to λ = 2. Convenient specifications of the price functions are ln B p = b ln p with b 1 = λ 1 and D p = d ln p with d 1 = 0, which yields the model q i = x i p Aq g + Cz i 2 e b ln p d/p + x i p Aq g + Cz i b/p + Aq g + Cz i, 11 where d/p and b/p are vectors with entries d j /p j and b j /p j for j = 1,..., J. The goal will be estimation of the parameters A, C, d, and b, and our welfare analyses will be based on 18

19 estimates of the system of equations 11. To allow for unobserved heterogeneity in behavior, we append the error term v g +u i to the above set of demand functions, where v g is a J vector of group level fixed effects and u i is a J vector of individual specific, mean zero error terms that are assumed to be independent of x i, z i, and p. The group level fixed effect v g is not assumed to be independent of peer effects q g. These error terms and fixed effects can be interpreted either as departures from utility maximization by individuals, or as unobserved preference heterogeneity. Assuming that the price weighted sum p v g + u i is zero suffi ces to keep each individual on their budget constraint. Under this restriction, if desired one could replace Cz i with Cz i + v g + u i in the indirect utility function above, and treat error terms as unobserved preference heterogeneity. Individual consumers within a group are all assumed to face the same prices this is not necessary for identification, but it simplifies notation. However, prices may vary across groups, either because groups vary by time or place, or because the mix of goods that comprises broad commodities like luxuries or necessities may vary across groups. Let p g be the vector of prices faced by group g. With these substitutions into equation 11, the system of demand functions we have to identify and estimate are q i = x i p gaq g + Cz i 2 e b ln p g d/p g + x i p gaq g + Cz i b/p g +Aq g +Cz i +v g +u i. 12 Proof that the parameters A, C, d, and b are identified from the system of equations 12, analogous to the generic model, is shown as Theorem 2 in the Appendix. For ease of exposition, the specific assumptions needed to prove identification, analogous to Assumptions A1 to A4 in the generic model, are also deferred to the Appendix. Identification there is proven in two steps. First, suppose there was no price variation, so that p g = p for all groups, where p is a vector constants. Then equation 12 reduces to the system of ngel curves q i = x i α q g γ z i 2 m + xi α q g γ z i δ + Aqg + Cz i + v g + u i 13 where α = A p, γ = C p, δ = b/p, and m = e b ln p d/p with constraints of b 1 = 1 and d 1 = 0. This has a very similar structure to the generic multiple equation system of equations 9, and by a similar derivation we can show that α, γ, δ, and b are identified. However, unlike the generic multiple equation system, only α and γ rather than A and C are identified in equation 13, because only α and γ appear inside the nonlinear part of the model. ssentially, the required demeaning steps in the identification end up removing the additive terms Aq g and Cz i. 19

20 Once identification of the ngel curve system 13 is proven, this means that for a single given price vector p, what can be identified is p A, p C, d, and b. Now let P be a matrix consisting of J different values of p. Since we can identify the ngel curve system separately in each of these J different price regimes, we can therefore identify P A and P C. It follows that we can then identify A and C, as long as P is nonsingular. This means that we only need to observe a relatively small amount of price variation to identify the entire model. For example, if prices only vary by time, but all groups observed in the same time period face the same prices, then for identification we only need to have data from J different time periods. In terms of our welfare analyses F i p = α q g + γ z i, so from ngel curves 13 without price variation we can identify and estimate the relative cost of needs F i p for the given price regime p. However, if we want to know how the relative cost of needs changes when prices change, or if we want to know the full matrix of own and cross peer effects A, then we must estimate the demand system stimating the Demand Model Our identification strategy is constructive, providing moments that can be used to estimate the parameters using standard GMM. To construct variables for these moments, we use essentially the same notation as in the generic model, though we now add a t subscript to denote time. Our survey data takes the form of repeated cross sections. This means that we see individuals from the same peer group defined by region, occupation, etc. in a few different time periods, but different individuals within the group are sampled in each time period. So now, instead of denoting groups by g, we denote them by gt. Let G denote the set of peer groups we observe, and let T be the number of time periods for which we have data. So now, for any g G and any t {1,..., T }, the set of individuals i gt refers to everyone we observe in peer group g in time period t. Similarly, let the set of individuals i gtz consist of all individuals in group g in time t that have z i = z. The resulting notation is then essentially the same as for the generic model, but with the addition of a t subscript. We have n gt = i gt 1 being the number of individuals observed in group g in time t, and n gtz = i gtz 1 is the number of individuals observed in group g in time t who have z i = z. For any matrix of variables w i, let ŵ gt = i gt w i/n gt and w i = w i ŵ gt for i gt. Similarly let ŵ gtz = i gtz w i/n gtz and w i = w i ŵ gtz for i gtz. So every element of w i is demeaned by subtracting off the average of the corresponding element of w, where the average is taken over all of the observed individuals in the same group and time period as individual i. In contrast, w i is demeaned by subtracting off the average of all observed 20