Demand and Welfare Analysis in Discrete Choice Models under Social Interactions

Transcription

1 Demand and Welfare Analysis in Discrete Choice Models under Social Interactions Debopam Bhattacharya University of Cambridge Pascaline Dupas Stanford University April, Shin Kanaya University of Aarhus Abstract We develop tools for empirical welfare-analysis in discrete choice models with social interactions. We show that even under parametric preferences and a unique equilibrium, standard cross-sectional demand data contain inadequate information for exact welfare analysis when interactions are present. Nonetheless, under Brock-Durlauf-type single-index restrictions, the distribution of welfare-effects resulting from price-interventions can be bounded. We illustrate our results using experimental data on mosquito-net adoption in rural Kenya. We show that under interaction, means-tested subsidies may reduce average private welfare, and increase deadweightloss estimates, compared to settings with no interaction. We are grateful to Steven Durlauf, James Heckman and seminar participtants at the University of Chicago for helpful feedback. Bhattacharya acknowledges financial support from the ERC consolidator grant EDWEL. 1

2 Synopsis We develop tools for empirical welfare-analysis in discrete choice models with social interactions. We show that even under parametric preferences and a unique equilibrium, typical crosssectional datasets contain inadequate information for exact welfare analysis, when interactions are present. Identification failure arises from potential dependence of non-purchase utility on aggregate purchase-rate. However, in Brock-Durlauf-type single-index settings, the distribution of welfareeffects resulting from price-interventions can be bounded; the bounds width increases with (i) the strength of aggregate spillover and (ii) the impact of policy-induced price-change on average choice. We illustrate our results using experimental data on mosquito-net adoption in rural Kenya. We show that under interaction, means-tested subsidies may reduce average private welfare, and increase deadweight-loss estimates, compared to settings with no interaction. Ignoring interactions would also imply over (under)-estimation of demand at less (more) generous eligibility-rules. 2

3 1 Introduction This paper concerns the problem of program evaluation under social interactions, i.e., where an individual s utility from choosing an action is affected by her belief about the fraction of her peers choosing that action. In such situations, a policy intervention such as a means-tested price subsidy, can potentially affect the outcome and welfare of ineligible individuals. In this paper, we develop econometric tools for empirical demand and utility-based welfare analysis for binary choice with such interactions. Our main theoretical finding is that in presence of interactions, the distribution of welfare effects resulting from price changes is generically not identified from demand data, even when utilities and the distribution of unobserved heterogeneity are parametrically specified, equilibrium is unique, and there are no endogeneity concerns. This inadequacy stems from the fact that the impact of average peer-choice on individual utility must necessarily differ by the alternative chosen; these different effects are not separately identifiable from choice-probabilities (only their difference is), but they appear separately in welfare calculations. As an example, suppose in choosing whether or not to enrol in college, a student s utility depends on the fraction π of her peers choosing to enrol in college. Under a preference for conformity, the higher this fraction, the lower is the utility from choosing the non-enrolment option and higher is the utility from choosing the college option, ceteris paribus. In this setting, for the purpose of calculating welfare-effects, say of a tuition subsidy, one cannot simply set the utility from nonenrolment to zero (an innocouous normalization in standard discrete choice models), because the utility changes as π changes with the tuition subsidy. Moreover, this change must necessarily differ in sign from the effect of π on the utility from enrolment. This is in contrast to discrete choice without spillover (c.f. Bhattacharya, 2015, 2018), where utility from the outside option, i.e. non-enrolment, does not change due to the price change. Despite the lack of point-identification, we show that under a semiparametric, linear index specification along the lines of Brock and Durlauf, 2007, one can calculate bounds on welfare distributions, and consequently mean or quantiles of welfare, based solely on the (structural) choice probability functions. The width of the bounds increases with (i) the extent of net social spillover, i.e. how much the average peer-choice affects individual choice probabilities, and (ii) the difference in average choice before and after the price-change corresponding to any realized equilibrium. The index structure, which has been nearly universally used in the literature on social interactions, leads 3

4 to dimension reduction that plays an important role in identifying spillover effects. We therefore continue to use the index structure as it simplifies our expressions, and comes for free, because social spillovers cannot in general be identified without such structure anyway. Under (untestable) restrictions on the nature of spillover, these bounds can shrink to a singleton, implying pointidentification. One such, albeit arbitrary, restriction is symmetric peer-effects, i.e. when the effect of an increase in average peer-choice on individual utilities from the two choices are equal in magnitude and opposite in sign. In the absence of such symmetry restrictions, the same structural demand, i.e. choice probability function, can be consistent with both a positive and a negative net welfare effect of price interventions such as a means-tested subsidy. To gain some intuition about this, consider the college enrolment example above. In this case, a means-tested tuition subsidy can have a net negative welfare effect if the negative externality on non-eligibles resulting from higher enrolment of eligibles exceeds the welfare gain of eligibles due to price reduction. On the other hand, if negative externalities are negligible, and all of the spillover arises from positive externalities on enrolled children, then the net welfare effect would be positive. The structural demand function only identifies the effect of average enrolment on utility diff erences, but not its effect on utilities for each option separately, so the same aggregate peer effect could be consistent with both a positive externality for enrolment and zero externality for non-enrolment as well as a zero externality for enrolment and negative externality for non-enrolment. More broadly, this insight implies that choice probability functions (inclusive of a term capturing social interactions) has limited information about a crucial policy parameter, viz. welfare effects, unlike the case without externalities, where the structural demand function in fact contains all the information required to calculate welfare (c.f. Bhattacharya, 2015, 2018). We emphasize that our conclusion for the interactions case has nothing to do with multiplicity of equilibria; the same conclusion holds even when equilibrium (both before and after the intervention) is unique. We illustrate our theoretical results with an example of a targeted public subsidy scheme for anti-malarial bednets. In particular, we use micro-data from a pricing experiment in rural Kenya (Dupas, 2014) to estimate a structural model of demand for bednets, where spillover can arise from a preference for conformity and also, possibly, a perceived negative externality arising from neighbors use of a bednet. In this setting, we calculate predicted effects of hypothetical income-contingent subsidies on bednet demand and welfare. We perform these calculations by first accounting for social interaction, and then compare the results with what would be obtained if one had ignored 4

5 these interactions. We find that allowing for interaction leads to a prediction of lower demand at low levels of eligibility and higher demand at higher levels of eligibility, relative to ignoring interactions. As for welfare, allowing for social interactions may lead to a welfare loss for ineligible households, in turn implying higher deadweight loss from the subsidy scheme, relative to estimates obtained ignoring social spillover; the latter would imply, by definition, that welfare effects for ineligibles are zero, as opposed to being negative. Previously, in Bhattacharya et al, 2013, we had pursued a reduced-form analysis to investigate the effect of price subsidies on demand under spillovers. In that work, we modeled individual choice as a function of own price and the average price faced by neighbors. That framework is not useful for welfare analysis, since the utility structure underlying such a reduced form model was left unspecified. More importantly, because of issues relating to multiple equilibria, a reduced form may not even exist in such settings. Instead, in the present paper, we lay out an underlying structural model, where individual choice is a result of maximizing utility that depends on own price and belief about neighbors choices. This formulation is well-founded in economic theory, is consistent with numerous models of peer-effects in the econometric literature, and also enables us to analyze welfare effects of price changes when there are social interactions. Literature Review and Contributions: Social interactions/spillover/peer-effects have been studied extensively in many branches of Economics. Seminal contributions to the econometrics of social spillover include Manski, 1993 on continuous outcomes, and Brock and Durlauf, 2001 on binary outcomes. More recently, there has been a surge of econometric research on the related theme of network models, c.f. de Paula, On the other hand, the econometric analysis of consumer welfare in discrete choice settings (without social spillover) started with McFadden, 1973, Domencich and McFadden, 1977, with later contributions by Daly and Zachary, 1978, Small and Rosen, 1982, Berry et al, 1995, McFadden and Train, 2002, and Bhattacharya, 2015, The present paper brings these two separate strands of the literature together by examining how social interactions influence welfare effects of interventions and its identifiability from standard demand data. \in the context of discrete choice with social interactions, Brock and Durlauf (2001, Sec 3.3) discussed the ranking of different equilibria resulting from policy changes in terms of average social as opposed to individual welfare. They use popular log-sum type formulae, such as Anderson et al 1982, to calculate the average indirect utility for a specific realized values of covariates and average peer choice. Such calculations are not useful for our purpose. This is because the income transfer that restores social average utility after a price change does not equal the average of individual 5

6 compensating variations that restore individual utilities to their pre-change level. It is the latter that is related to the concept of deadweight loss and thus the effi ciency cost of interventions, and consequently has received most attention in the recent literature on empirical welfare analysis, c.f. Hausman and Newey, 2016, Bhattacharya, 2015, McFadden, It is this notion of welfare that we are interested in. Our primary contribution is to develop empirical tools for individual welfare analysis under social interactions, given a structural demand function. For identification of the demand function itself in settings involving spillovers, we refer the reader to the large, existing literature on this topic, c.f. Brock and Durlauf (Handbook chapter). We briefly touch upon this issue in our empirical application, and utilize the experimental nature of our data to estimate structural demand. But it is the theory and empirics of welfare analysis that this paper primarily focuses on, and therein lies its main contribution. Furthermore, in order to focus on welfare-analysis, we use the simplest and most popular specification of peer-effects, namely that physical neighbors constitute an individual s peer group. This also seems reasonable in the context of our application, which concerns adoption of a health product in physically separated Kenyan villages. It would be interesting to extend our analysis to other network structures, e.g. those based on ethnicity, caste, social distance etc. We leave that to future work. The rest of the paper is organized as follows. Section 2 describes the set-up and notations. Section 3 lays out the results on welfare analysis. Section 4 describes the empirical context, Section 5 reports the empirical results and reports some robustness checks. Finally, Section 6 concludes. Some technical derivations are reported in the Appendix. 2 Set-up and Assumptions Consider a group of individuals or households indexed by i, residing in a large village. Suppose each individual faces a binary choice between alternatives 1 and 0, and their utilities from the two choices are given by U 1 (Y i P i, Π i, η i ) and U 0 (Y i, Π i, η i ), respectively, where Y i, P i and η i denote respectively the income, price and unobserved heterogeneity for the ith individual, and Π i is i s subjective expectation of what fraction in her peer-group would choose alternative 1. The dependence of utility on Π captures the presence of spillovers. As in Brock-Durlauf, we will assume that expectations can be consistently estimated by sample averages. Assumption 1: Assume that U 1 (, π, η), and U 0 (, π, η) are continuous and strictly increasing 6

7 for each fixed value of π and η, i.e., all else equal, utilities are non-satiated in the numeraire. Assumption 2: For each y and η, U 1 (y,, η) is continuous and strictly increasing and U 0 (y,, η) is continuous and weakly decreasing, i.e. conforming yields higher utility than not conforming for each individual. Define q 1 (p, y, π) to be the structural probability of an individual s choosing 1 when own price is p, own income is y and the expected choice is π, i.e. q 1 (p, y, π) = 1 {U 1 (y p, π, η) > U 0 (y, π, η)} df (η), (1) and q 0 (p, y, π) = 1 q 1 (p, y, π). Under rational expectations corresponding to a Bayes-Nash equilibrium, π must satisfy the fixed-point constraint: π = 1 {U 1 (y p, π, η) > U 0 (y, π, η)} df η,p,y (η, p, y) = q 1 (p, y, π) df P,Y (p, y), where F P,Y (, ) denotes the joint distribution of price and income and η is assumed independent of P, Y. Manski, 1993, Sec 3.2 considered coherency in this model. Brock and Durlauf, 2001 investigated parametric identification of such models, where q 1 (p, y, π) is specified through a logit probability function, and Brock and Durlauf, 2007 generalized this to semiparametric identification, where q 1 (p, y, π) is partially specified through an index structure. We will return to the issue of identifying q 1 (p, y, π) in the application. For now, assume that we have identified the q 1 (p, y, π) function, and consider the problem of welfare analysis. Policy Change: Start with a situation where the price of alternative 1 is p 0 and the value of π is π 0. Then suppose a price subsidy is introduced such that that individuals with income less than an income threshold τ become eligible to buy the product at price p 1 < p 0. This policy will alter the equilibrium adoption rate; assume that the new equilibrium adoption rate changes to π 1, given by the solution to the fixed point problem π 1 = [1 {y τ} q 1 (p 1, y, π 1 ) + 1 {y > τ} q 1 (p 0, y, π 1 )] df (y). (2) For fixed p 0, p 1, the RHS viewed as a function of π 1 is a map from [0, 1] to [0, 1]. If q 1 (p 1,, y) and q 1 (p 0,, y) are continuous, then by Bruower s fixed point theorem, there is at least one solution in π 1 to the above display, implying "coherence" of the model. However, there may be multiple 7

8 solutions. Below, we consider welfare analysis corresponding to any realized value of π 1, and show that even for a given π 1, welfare predictions will be non-unique. 3 Welfare Analysis We now investigate the calculation of welfare effects resulting from the above subsidy-scheme when there are spillover effects. By welfare effects we mean the compensating variation (CV), viz. what hypothetical income compensation would restore the post-change indirect utility for an individual to its pre-change level. For a subsidy-eligible individual, for any potential value of π 1 corresponding to the new equilibrium, the individual compensating variation is the solution S to the equation max {U 1 (y + S p 1, π 1 ; η), U 0 (y + S, π 1 ; η)} = max {U 1 (y p 0, π 0, η), U 0 (y, π 0, η)}, (3) whereas for a subsidy-ineligible individual, it is the solution S to max {U 1 (y + S p 0, π 1 ; η), U 0 (y + S, π 1 ; η)} = max {U 1 (y p 0, π 0, η), U 0 (y, π 0, η)}. (4) Note that we do not take into account peer-effects again in defining the CV because the income compensation underlying the definition of CV is hypothetical. So the impact of actual income compensation on neighboring households is irrelevant. Since the CV depends on the unobservable η, the same price change will produce a distribution of welfare effects across individuals; we are interested in calculating that distribution and its functionals such as mean welfare. Existence of S: Under the following condition, there exists an S that solves eqn. (3) and (4): Condition 1 For each η and any fixed (p 0, p 1, y), we have that (i) lim S U 1 (y + S p 1, 1; η) < U 1 (y p 0, 0, η), and (ii) lim S U 0 (y + S, 1; η) > U 0 (y, 0, η). Intuitively, this condition strengthens Assumption 1 by requiring that utilities can be increased and decreased suffi ciently by varying the quantity of numeraire. Existence follows via the intermediate value theorem. Under index structure, existence is explicitly shown below. Finally, uniqueness of the solution to (3) and (4) follows by strict monotonicity in numeraire. Since the maximum of two strictly increasing functions is strictly increasing, the LHS of (3) and (4) are strictly increasing in S, implying a unique solution. 8

9 Linear Index Structure: With few peer-groups (e.g. there are eleven villages in our application dataset), one cannot consistently estimate the impact of π on choice probabilities, holding other regressors constant. 1 given by and thus Accordingly, we assume a linear index structure, viz. that utilities are U 0 (y, π, η) = δ 0 + y + α 0 π + η 0, U 1 (y p, π, η) = δ 1 + (y p) + α 1 π + η 1, (5) U 1 (y p, π, η) U 0 (y, π, η) = (δ 1 δ 0 ) + ( ) y p + (α 1 α 0 ) π + (η 1 η 0 ) c 0 + c 1 p + c 2 y + απ + ε. (6) We assume throughout that > 0, > 0, i.e., non-satiation in numeraire, and that α 0 < 0 < α 1, i.e., compliance yields higher utility. 2 Given this utility specification, the structural choice probability for alternative 1 is given by q 1 (p, y, π) F (c 0 + c 1 p + c 2 y + απ), where F ( ) denotes the marginal distribution of ε. For welfare expressions outlined below, we do not need to know the functional form of F ( ). In the application, we will consider various ways to estimate the structural choice probabilities, including standard Logit and Klein and Spady s semiparametric MLE. The condition α 0 < 0 < α 1 makes the model different from standard demand specifications for differentiated products. In the standard case, for an individual consumer, coeffi cients on a specific characteristic remains identical across alternatives. For the so-called outside option, i.e. not buying any of the available alternatives, the utility is normalized to zero. For example, in modelling choice of automobiles, the coeffi cient on miles/gallon in an individual s utility function is assumed to be identical for each potential car, and utility is zero for the alternative of not 1 In particular, the fixed point constraint does not help because of dimensionality problems. In particular, in the fixed point condition: π = q 1 (p, y, π) f (p, y) dpdy, where f (p, y), the joint distribution of p, y is identified, the unknown q 1 (p, y, π). has higher dimension than the observable f (p, y). 2 In our empirical setting of anti-malarial bednet adoption, there are multiple potential sources of externality (i.e. α 1, α 0 0). The first is a pure compliance motive; the second is increased awareness of the benefits of a bednet when more villagers use it; the third is a free-riding motive, i.e. that increased bednet use by neighbours deters mosquitoes, reducing the incentive for adoption. While the first two sources are consistent with α 1 > 0 > α 0, the third would reverse the sign. Nonethless, the overall effect of the first two are empirically seen to dominate the third, leading to a significant positive effect of average adoption on own adoption. It is also unclear how large the deterrence effect might be, and whether potential users are sophisticated enough to consider such effects. 9

10 buying any car. This common coeffi cient (or its distribution in random coeffi cient models) can be identified from demand data because it appears in the choice probabilities as a coeffi cient of the observed differences in characteristics. In a social spillover setting, this common value assumption cannot be maintained for the coeffi cient on π. If higher π increases one s utility from buying, captured by a positive α 1, then it is natural that this will (weakly) decrease the utility from not buying, captured by a negative α 0. One cannot simply normalize the utility from not buying to be zero, because that utility depends on π. As we will see below, in welfare evaluations of a subsidy, α 1 and α 0 appear separately in the expressions for welfare-distributions, but cannot be separately identified from demand data, which can only identify α 1 α 0. This is why even when a distributional assumption (e.g. extreme value) on ε is maintained, one cannot simply use a log-sum type approach to calculate welfare effects, corresponding to any realized value of π in the postsubsidy equilibrium. Indeed, we will show below that in presence of externalities, welfare analysis cannot be done solely based on choice probabilities, unlike the case without externalities. The latter case was analyzed in Bhattacharya, 2015, 2017, where it was shown that welfare distributions can be expressed as closed-form functionals of structural choice probabilities even when no parametric assumption was imposed on utilities or demand functions. Note from Brock and Durlauf, 2007, that the structural choice probabilities F (c 0 + c 1 p + c 2 y + απ) identify c 0, c 1, c 2 and α, i.e. (δ 1 δ 0 ),, and (α 1 α 0 ), up to scale. Note that we are not assuming that the probability distribution of ε is known. Now consider a hypothetical policy of moving from a situation where everyone faces a price of p 0 to one where people with income less than an eligibility-threshold τ are given the option to buy at the subsidized price p 1 < p 0. This policy will alter the equilibrium take-up rate. Assume that the take up rate changes from π 0 to π 1. Note that π 0 and π 1 can be identified (there may be multiple solutions), since they satisfy fixed effect constraints (2). Given this, the welfare effect of the policy change can be calculated as described below. We first lay out the results in detail for the case where π 1 > π 0, which corresponds to our application. In the appendix we present results for the case where π 1 < π 0. For the rest of this section, assume that π 1 > π 0. 10

11 3.1 Welfare for Eligibles The compensating variation for eligibles is given by the solution S to max {δ 1 + (y + S p 1 ) + α 1 π 1 + η 1, δ 0 + (y + S) + α 0 π 1 + η 0 } = max {δ 1 + (y p 0 ) + α 1 π 0 + η 1, δ 0 + y + α 0 π 0 + η 0 } (7) Since LHS is strictly increasing in S, the condition S a is equivalent to max {δ 1 + (y + a p 1 ) + α 1 π 1 + η 1, δ 0 + (y + a) + α 0 π 1 + η 0 } max {δ 1 + (y p 0 ) + α 1 π 0 + η 1, δ 0 + y + α 0 π 0 + η 0 }. (8) We refer to eqn. (8) when referring to LHS and RHS. If a < p 1 p 0 α 1 (π 1 π 0 ) < 0, then each term on the LHS is smaller than the corresponding term on the RHS. If a α 0 (π 0 π 1 ) > 0, then each term on the LHS is larger than the corresponding term on the RHS. This gives us the support of S: 0, if a < p 1 p 0 α 1 β Pr (S a) = 1 (π 1 π 0 ), 1, if a α 0 (π 0 π 1 ). Remark 1 We note in passing that the above reasoning also helps establish existence of a solution to (7). We know from above that for S < p 1 p 0 α 1 (π 1 π 0 ), the LHS of (7) is strictly smaller than the RHS, and for S α 0 (π 0 π 1 ), the LHS of (7) is strictly larger than the RHS. By continuity, and the intermediate value theorem, it follows that there must be at least one S where (7) holds with equality. Back to calculating the CDF, now consider the intermediate case where a [p 1 p 0 α 1 (π 1 π 0 ) β, α0 (π 0 π 1 )). }{{ 1 β }} 0 {{} <0 >0 If p 1 p 0 α 1 (π 1 π 0 ) a < α 0 (π 0 π 1 ), then first term on LHS of (8) is larger than first term on RHS for all η 1, and 2nd term on LHS of (8) is smaller than the 2nd term on the RHS for all η 0, and thus (2) is equivalent to which is the same as δ 1 + (y + a p 1 ) + α 1 π 1 + η 1 δ 0 + y + α 0 π 0 + η 0, δ 1 + (y + a p 1 ) + α 1 π 0 + α 1 (π 1 π 0 ) + η 1 δ 0 + y + α 0 π 0 + η 0, (9) 11

12 which is not point-identified due to the presence of the term α 1 (π 1 π 0 ) on the LHS, since α 1 is not identified. Therefore, one can only bound this probability by varying α 1 over its possible range, viz. [0, α], where α = α 1 α 0 is identified from choice probabilities. For any specific choice of α 1, we have that the probability of (9) reduces to F (c 0 + α 1 (π 1 π 0 ) + c 1 (p 1 a) + c 2 y + απ 0 ) ( = q 1 p 1 a, y, π 0 + α ) 1 α (π 1 π 0 ). (10) The intercept c 0, the slopes c 1, c 2 and α are all identified from conditional choice probabilities; but α 1 is not identified, and therefore (10) is not point-identified from the structural choice probabilities. However, since α 1 [0, α], for each feasible value of α 1 [0, α], we can compute a feasible value of (10), giving us bounds on the welfare distribution. Note also that the thresholds of a at which the CDF expression changes are also not pointidentified for the same reason. However, since π 1 π 0 > 0 and > 0, > 0, the interval [ p 1 p 0 α 1 ] (π 0 π 1 ) (π 1 π 0 ) a < α 0 [ = p 1 p 0 α 1 (π 1 π 0 ) a < α α ] 1 (π 1 π 0 ) will translate to the left as α 1 varies from 0 to α. Putting all of this together, we get that: Theorem 1 If assumptions 1,2 and the linear index structure hold and π 1 > π 0, then given α 1 [0, α], the distribution of the compensating variation for eligibles is given by ( ) Pr S Elig a = 0, if a < p 1 p 0 α 1 (π 1 π 0 ), q 1 ( p1 a, y, π 0 + α 1 α (π 1 π 0 ) ), if p 1 p 0 α 1 (π 1 π 0 ) a < α α 1 (π 1 π 0 ), 1, if a α α 1 (π 1 π 0 ). (11) Discussion: The width of the bounds depends on the extent to which q 1 (,, π) is affected by π, i.e. the extent of social spillover, and also the difference in the realized values π 1 and π 0. From 12

13 (11), mean welfare loss is given by 0 ( q 1 p 1 a, y, π 0 + α ) 1 p 1 p 0 α 1 (π π 0 ) α (π 1 π 0 ) da } 1 {{} Welfare Gain (Smallest when α 1 =0) α α 1 + (π π 0 ) 0 ( [1 q 1 p 1 a, y, π 0 + α )] 1 0 α (π 1 π 0 ) da }{{} Welfare Loss (0 when α 0 =0) If the good is normal, so that the income effect > 0 (c.f. Eqn. (6)), then the above expression is increasing in α 1 ; therefore, the welfare gain is the largest when α 1 = α and the smallest when α 1 = 0. Corollary 1 In the special case where α 1 = α 0 in (5) (Brock and Durlauf, 2001 consider this case), so that α = α 1 α 0 = 2α 0, we get that α 1 α = α 0 2α 0 ( ) Pr S Elig a = 0, if a < p 1 p 0 2α (π 1 π 0 ), = 1 2, yielding point-identification q 1 ( p1 a, y, 1 2 (π 1 + π 0 ) ), if p 1 p 0 2α (π 1 π 0 ) a < α 2 (π 1 π 0 ), 1, if a α 2 (π 1 π 0 ). (12) 3.2 Welfare for Non-eligibles Recall that welfare for non-eligibles is defined as the solution S to the equation max {U 1 (y + S p 0, π 1 ; η), U 0 (y + S, π 1 ; η)} = max {U 1 (y p 0, π 0, η), U 0 (y, π 0, η)}. Then S a is equivalent to max {δ 1 + (y + a p 0 ) + α 1 π 1 + η 1, δ 0 + (y + a) + α 0 π 1 + η 0 } max {δ 1 + (y p 0 ) + α 1 π 0 + η 1, δ 0 + y + α 0 π 0 + η 0 }. (13) If a < α 1 (π 0 π 1 ) < 0, then each term on the LHS is smaller than the corresponding term on the RHS for each realization of the ηs. So the probability is 0. Similarly, for a α 0 (π 0 π 1 ) > 0, each term on the LHS is larger, and thus the probability is 1. In the intermediate range, a [ α 1 (π 0 π 1 ), α 0 (π 0 π 1 )), we have that the first term on the RHS exceeds the first term on the 13

14 LHS for each η 1, and the second term on the LHS is smaller than the second term on the RHS for each η 0. Therefore, (13) is equivalent to δ 1 + (y + a p 0 ) + α 1 π 1 + η 1 δ 0 + y + α 0 π 0 + η 0. The probability of this event is not point-identified since we do not know the values of α 1, α 0 separately. But for each choice of α 1 [0, α], we can compute the probability of this event as Putting all of this together, we have that F (c 0 + α 1 (π 1 π 0 ) + c 1 (p 0 a) + c 2 y + απ 0 ) ( = F p 0 a, y, π 0 + α ) 1 α (π 1 π 0 ). Theorem 2 If assumptions 1,2 and the linear index structure hold and π 1 > π 0, then for each α 1 [0, α], ( ) Pr S NonElig a = 0, if a < α 1 (π 0 π 1 ), q 1 ( p0 a, y, π 0 + α 1 α (π 1 π 0 ) ), if α 1 (π 0 π 1 ) a < α 0 (π 0 π 1 ), 1, if a α 0 (π 0 π 1 ). For non-eligibles, all of the welfare effects come from spillovers, since they experience no price change. In particular, for non-eligibles who buy, there is a welfare gain from positive spillover due to a higher π. For non-eligibles who do not buy, there is, however, a welfare loss due to increased π. This is why the CV distribution has a support that includes both positive and negative values. From (14), mean compensating variation is given by 0 ( q α 1 p 0 a, y, π 0 + α ) 1 1 (π π 1 ) α (π 1 π 0 ) da } 1 {{} W elfare Gain α α 1 (π π 0 ) 0 ( + {1 q 1 p 0 a, y, π 0 + α )} 1 0 α (π 1 π 0 ) da. }{{} W elfare Loss The first term captures the welfare gain of those ineligibles who would still buy, the gain resulting from higher π; this term would be zero if α 1 = 0. The second term captures the welfare loss of those ineligibles who do not buy, the loss also resulting from higher π; this loss would be zero if α 0 = 0. Of course, both would be zero if α = 0 = α 1 = α 0, reflecting the fact that welfare effect on ineligibles would be zero if there is no externality. (14) 14

15 Corollary 2 In the special case where α 1 = α 0 = α 2, we have point-identification ( ) Pr S NonElig a ( = q 1 p0 a, y, 1 2 (π 1 + π 0 ) ), if 0, if a < α 2 (π 0 π 1 ), α 2 (π 0 π 1 ) a < α 2 (π 1 π 0 ), 1, if a α 2 (π 1 π 0 ). (15) Remark 2 Note that given the index structure, we do not require a distributional assumption on the ηs to calculate the bounds in (11) and (14). Any existing semiparametric estimation method for index models can be used for calculations, e.g. Klein and Spady, 1994, which requires bandwidth choice and Bhattacharya, 2008, which does not. 3.3 Group-Specific Unobservables Our data for the application come from eleven different villages with approximately 180 households per village. It is plausible that utilities from using and from not using a bednet are affected by village-specific unobservable characteristics, such as the chance of contracting malaria when not using a bednet. Such effects were termed contextual by Manski, Brock and Durlauf, 2007 discussed some diffi culties with estimating social spillover effects in presence of group-specific unobservables. To capture this situation, modify the linear utility structure above to U 0 (y, π, η) = δ 0 + y + α 0 π + ξ 0 + η 0, U 1 (y p, π, η) = δ 1 + (y p) + α 1 π + ξ 1 + η 1, where ξ 0 and ξ 1 denote unobservable village specific characteristics. Therefore, U 1 (y p, π, η) U 0 (y, π, η) = (δ 1 δ 0 ) + ( ) y p + (α 1 α 0 ) π + ξ 1 ξ 0 + (η 1 η 0 ) c 0 + c 1 p + c 2 y + απ + ξ + ε. Since ξ is village specific and we have many observations per village, we can use a dummy γ v for each village, and estimate the regression (similar to well-known SUR for linear equations) of take-up on price, income and other characteristics that vary across households h within village v, together with village dummies, i.e. Pr (q vh = 1 p vh, y vh, v) = F (γ v + c 1 p vh + c 2 y vh ). 15

16 The identified coeffi cients γ v of the village dummies therefore satisfy γ v = απ v + c 0 + ξ v. We will only need to identify the sum ξ v c 0 + ξ v for welfare calculations below. However, in the equations γ v = απ v + ξ v there are as many ξ v as there are γ v, so we have a dimension problem. In our empirical exercise, we attempt to get around this problem by a homogeneity assumption. Homogeneity Assumption: If two villages are very similar in terms of observables, then it may be reasonable to assume that they have similar values of ξ, which leads to a dimension reduction, and enables point-identification simply by solving the linear system γ v = απ v + ξ v as there are as many ξ v s than the number of γ v less 1 (for α). Indeed, in our application, there are two villages out of eleven in our dataset that are very similar in terms of observables, and hence are amenable to this approach. 3 Welfare Calculation with Group-Effects: Once we have a plausible way to estimate the structural choice probabilities, we can proceed with welfare calculation in presence of social spillover and unobserved group-effects, as follows. Consider an initial situation where everyone faces the unsubsidized price p 0, so that the predicted take-up rate π 0v in village v solves π 0v = F ( c 1 p 0 + c 2 y + απ 0v + ξ ) v dg (y v), (16) where G (y v) is the distribution of income in village v, and c 1, c 2, α and ξ v are estimated as above. Now consider a policy induced price regime p 0 for ineligibles (wealth larger than a) and p 1 for eligibles (wealth less than a). Then the resulting usage π 1v in village v is obtained via solving the fixed point π 1v in the equation π 1v = 1 {y τ} F ( c 1 p 1 + c 2 y + απ 1v + ξ ) v +1 {y > τ} F ( c 1 p 0 + c 2 y + απ 1v + ξ ) v dg (y v), (17) Finally, average welfare effect of this policy change in village v can be calculated using W v = [1 {y τ} E v (y) + 1 {y > τ} N v (y)] dg (y v), (18) where E v (y) and N v (y) are average welfare at income y in village v, calculated from from eqn. (11) for eligibles and eqn. (14) for ineligibles, using π 0v and π 1v as the predicted take-up probability in village v (analogous to π 0 and π 1 in (11) and (14)), calculated using (16) and (17), α 1 [0, α] as above. 3 Alternative approaches to address the endogeneity of π v could be the use of control function, Lewbel s special regressor, and regression projection type approaches, which we do not explore in the interest of brevity, since our main focus is on welfare calculation, given a consistent estimate of (structural) choice probabilities. 16

17 4 Empirical Context and Data Malaria is a life-threatening parasitic disease transmitted from human to human through mosquitoes. In 2016, an estimated 216 million cases of malaria occurred worldwide, with 90% of the cases in sub-saharan Africa (WHO 2017). The main tool for malaria control in sub-sahran Africa is the use of insecticide treated bednets. Regular use of a bednet reduces overall child mortality by around 18 percent and reduces morbidity for the entire population (Lengeler, 2004). However, at $6 or more a piece, bednets are unaffordable for many households, and to palliate the very low coverage levels observed in the mid-2000s, public subsidy schemes were introduced in numerous countries in the last 10 years. Our empirical exercise is designed to evaluate such subsidy schemes not just in respect of their effectiveness in promoting bednet adoption, but also their impact on individual welfare and deadweight loss, in line with the classical economic theory of utility-based welfare analysis. In this setting, spillover can arise from (i) preference for conformity, (b) a perceived negative externality arising from neighbors use of a bednet, i.e. if I not use a bednet but a neighbor does, then mosquitoes are more likely to bite me, and (c) a positive health externality in that if one s neighbors are protected by bednet use, then one has less chance of contracting malaria, thereby encouraging free-riding. We ignore the third mechanism because we believe that in practical terms, it is unlikely that (i) such effects are perceptively strong, and (ii) village residents have suffi cient understanding of the malaria transmission mechanism that they would act on this notion. Experimental design: We exploit data from a 2007 randomized bednet subsidy experiment conducted in eleven villages of Western Kenya, where malaria is transmitted year-round. In each village, a list of 150 to 200 households was compiled from school registers, and households on the list were randomly assigned to a subsidy level. After the random assignment had been performed in offi ce, trained enumerators visited each sampled household to administer a baseline survey. At the end of the interview, the household was given a voucher for an bednet at the randomly assigned subsidy level. The subsidy level varied from 40% to 100% in two villages, and from 40% to 90% in the remaining 9 villages; there were 22 corresponding final prices faced by households, ranging from 0 to 300 Ksh (US $5.50). Vouchers could be redeemed within three months at participating local retailers. Data: We use data on bednet usage as observed from coupon redemption and verified obtained through a follow-up survey. We also use data on baseline household characteristics measured during 17

18 the baseline survey. The three main baseline characteristics we consider are wealth (the combined value of all durable and animal assets owned by the household); the number of children under 10 years old; and the education level of the female head of the household. 5 Empirical Specification and Results We work with the linear index structure (5), where y is taken to be the household wealth, p is the experimentally set price faced by the household, π is the average adoption in the village (excluding the household in question). The health externality from bednet use is implicitly taken into account through the dependence of utilities from adoption and from non-adoption on the average adoption rate π (c.f. eqn. (5)). As additional controls affecting the take-up of bednet, we choose presence of children under the age of ten and years of education of the oldest female member of the household. A village-specific variable that could affect adoption is the extent of malaria exposure risk in the village. We measure this in our data from the response to the question: "Did anyone in your household have malaria in the past month?". Summary statistics for all relevant variables are reported in Table 1, and their village averages are shown in table 2, for each of the eleven villages in the data. Our first of results correspond to taking F ( ) to be the logit CDF, and including average takeup in village as a regressor. 4 The marginal effects at mean from the logit estimation are presented in Table 3. It is evident that demand is highly price elastic, and that average bednet adoption in the village has a significant positive association with private adoption, conditional on price and other household characteristics, i.e. α > 0 in terms of our notation above. The effect of children is negative, likely reflecting that households with children had already invested in other anti-malarial steps, e.g. had bought a less effective traditional bednet prior to the experiment. We also computed analogous estimates where we ignore the externality, i.e., we drop average take-up in village from the list of regressors. The corresponding marginal effects for the retained regressors are not very different in magnitude from those obtained when including the average village take-up, and so we do not report those here. Instead, we use the two sets of coeffi cients to calculate and contrast the predicted bednet adoption rate corresponding to different eligibility thresholds. These predicted 4 We do not impose the fixed effect constraint while estimating the logit parameters, i.e. for each village v, π v = Λ ( + p + β 2y + β 3z + απ ) d ˆF P,Y,Z (p, y v), While this would have improved effi ciency, the additional computational burden would be quite onerous. 18

19 effects are quite different depending on whether or not we allow for spillover, and so we investigate these further. In particular, we consider a hypothetical subsidy rule, where those with wealth less than τ are eligible to get the bednet for 50KSh (90% subsidy), whereas those with wealth larger than τ get it for the price of 250 KSh (50% subsidy). Based on our logit coeffi cients, we plot the predicted aggregate take-up of bednets corresponding to different income thresholds τ. In Figure 1, for each threshold τ, we plot the fraction of households eligible for subsidy on the horizontal axis, and the predicted fraction choosing the bednet on the vertical axis, based on coeffi cients obtained by including (solid) and excluding (small dash) the spillover effect. The 45 degree line (large dash) showing the fraction eligible for the subsidy is also plotted in the same figure for comparison. It is evident from Figure 1 that ignoring externalities leads to over-estimation of adoption at lower thresholds and underestimation at higher thresholds of eligibility. To get some intuition behind this finding, consider a much simpler set-up where an outcome Y is related to a scalar covariate X via the classical linear regression model Y = + X + ε where ε is zero-mean, independent of X and > 0. OLS estimation of this model yields estimators ˆ, ˆ with probability limits (and also expected values) = Cov [X, Y ] /Var [X] and = E [Y ] E [X], respectively. Corresponding to a value x of X, the predicted outcome has a probability limit of y := + x = E [Y ] + {x E [X]}. Now consider what happens if one ignores the covariate X. Then the prediction is simply the sample mean of Y which has the probability limit of y miss := E [Y ]. Therefore, y < y miss if x < E [X]. Thus, although the ignored covariate X has a positive effect on the outcome (since > 0), ignoring it in prediction leads to an overestimation of the outcome if the point x where the prediction is made is smaller than the population average of the ignored covariate. On the other hand, if x > E [X], then there will be under-estimation. Having obtained these (uncompensated) effects, we now turn to calculating the average demand and the mean compensating variation for a specific subsidy rule. We consider an initial situation where everyone faces a price of 250 KSh for the bednet, and a final situation where an bednet is offered for 50 KSh to households with wealth less than τ =8000 KSh (about the 27th percentile of the wealth distribution), and for the price of 250 KSh to those with wealth above that. The demand results are reported in Table 4, and the welfare results in Table 5. We perform these calculations village-by-village, and then aggregate across villages. To calculate these numbers, we first predict the bednet adoption when everyone is facing a price of 250, and then when eligibles face a price of 50 and the rest stay at 250 KSh, giving us the equilibrium values of π 0 and π 1, respectively, in our 19

20 notation above. In all such calculations with our data, we always detected a single solution to the fixed point π (i.e. a unique equilibrium) as can be seen from Figure 1, where we plot the squared difference between the RHS and the LHS of eqn. (2), i.e. [ π 1 ] 2 [1 {y τ} ˆq 1 (p 1, y, π 1 ) + 1 {y > τ} ˆq 1 (p 0, y, π 1 )] d ˆF (y) on the vertical axis, and π 1 on the horizontal axis, separately for each of the eleven villages. The globally convex nature of each objective function is evident from Figure 1. The minima are relatively close to each other around 0.15, except village 7 and 10, where it is larger. A similar set of globally [ convex graphs is obtained for π 0, which minimizes π 0 2. ˆq 1 (p 1, y, π 0 ) d ˆF (y)] These predicted values of π 0 and π 1 are used as inputs into the prediction of demand as per eqn. (1) and welfare as per Theorems 1 and 2. The first row of Table 4 shows the pre-subsidy predicted demand (using a logit F ) by subsidyeligibility. In the second row, we calculate the predicted effect of the subsidy on demand, and break that up by the own price effect (row 2) and the spillover effect (row 3). The own effect is obtained by changing the price in accordance with the subsidy but keeping the average village demand equal to the pre-subsidy value; the spillover effect is the difference between the overall effect and the own effect. It is clear that spillover effects on both eligibles and non-eligibles are large in magnitude. In particular, the spillover effect raises demand for ineligibles by nearly 33% of its pre-subsidy level. In Table 5, we report welfare calculations. First, in the row titled "Logit", we report the average CV of the subsidy rule for eligibles, corresponding to assuming no externality. In this case, we simply use the results of Bhattacharya, 2015 to calculate the (point-identified) average CV for eligibles as the price changes from 250 KSh to 50 KSh. This yields the value of welfare gain to be 56.3 KSh. As there is no externality, the welfare change of ineligibles is zero by definition, and therefore the net welfare gain, denoted by net CV is simply the fraction eligible (0.27) times the average CV for eligibles. This is reported in the 2nd column of Table 5. We next turn to the case with externalities. Using the predicted adoption rates π 0 and π 1, we compute the lower and upper bounds of the overall average CV using (11) for eligibles, and using (14) for ineligibles. These are reported in columns 3-6 of Table 5. The most conspicuous conclusion from these numbers is that ineligibles can suffer a welfare loss on average due to the subsidy. This is because the subsidy facilitates usage for solely the eligibles, raising the equilibrium usage π in the village, but the ineligibles keep facing the high price, and thus a lower utility U 0 (y, π, η) from not buying because π is now higher (in the index specification, α 0 0). However, the few ineligibles 20

21 who buy, despite the high price, get some welfare increase from a rise in the average adoption rate, that explains the small but positive upper bound corresponding to the case α 0 = 0. As for eligibles, the lower and upper bounds on average welfare gain do not contain the estimate that ignores externalities, suggesting over-estimation of welfare gains in the latter case. This is also consistent with Figure 1, where we see that at 27% eligibility and lower, demand is overestimated when externalities are ignored. The overall welfare gain, reported in the column with heading net CV, includes the negative welfare effects on ineligibles, further lowering the average effect, relative to ignoring externalities and incorrectly concluding no welfare change for ineligibles. Deadweight Loss: To compute the average private deadweight loss, we subtract the net welfare from the predicted subsidy expenditure. The latter equals the amount of subsidy (200 KSh) times the average demand at the subsidized price 50 KSh of the eligibles. Thus the expression for DWL is given by D = {y τ} q externality 1 (50, y, z, π 1 ) 1 {y τ} µ E (y, z, π 1, π 0 ) 1 {y > τ} µ NE (y, z, π 1, π 0 ) df (y, z), where y denotes wealth, z denotes other covariates, q externality 1 (50, y, z, π 1 ) denotes predicted demand at price 50 including the effect of externality, and µ E and µ NE refer to average welfare gain for eligibles and non-eligibles, respectively. Ignoring externalities leads to the point-identified deadweight loss [ ] D = {y τ} q No externality 1 (50, y, z) 1 {y τ} µ No externality (y, z) df (y, z). 5.1 Robustness Checks Group-specific Effects and Homogeneity: It is evident from table 2 that villages 6 and 10 are highly similar in terms of the average values of key regressors, except that the (randomly assigned) average price in village 6 is much higher than in village 10, which explains the much lower average adoption in village 6. Given this, we assume that villages 6 and 10 are likely to be similar in terms of their unobservables, and as such, we estimate a single ξ v for them. Specifically, we first estimate Pr (q vh = 1 p vh, y vh, x vh, v) = F ( γ v + c 1 p vh + c 2 y vh + c ) 3x vh, where x vh is a vector containing presence of children and female education, the γ v s are villagespecific intercepts (estimated using dummies for the villages), and p vh and y vh are price faced by 21

22 the household in the experiment and its wealth, respectively. In the second step, we solve the linear system γ v = απ v + c 0 + ξ v = απ v + ξ v, for α and ξ v, for v = 1,..., 11, where γ v is obtained in the previous step, and the π v s are the average adoption rates in individual villages in the experiment. In solving this system, we set ξ 6 = ξ 10, which incorporates the homogeneity assumption discussed above. We can do all of this in one step by adding nine dummies for villages 1-5, 7-9, 11 and one for villages 6 and 10, and then running a regression of individual use on the regressors p, y and x, the average use in each village, as well as the village dummies. In the second row in Table 5, we report the average welfare effects of the same hypothetical policy change as described above, using expression (18). The numbers do vary a bit from those obtained without accounting for village-level unobservables, but not by much. Semiparametric Estimates: In the third row of Table 5, we report welfare results from a semiparametric index estimation of the conditional choice-probabilities, i.e. retaining the index structure but dropping the logit assumption. This is achieved by using the sml routine (de Luca, 2008) in Stata which implements Klein and Spady s estimator for single index models, using (i) a default bandwidth of h n = n 1/6.5 to estimate the index, and then (ii) a local cubic polynomial for regressing the binary outcome on the estimated index to produce the predicted probabilities, using a bandwidth of h n = cn 1/5 where c was chosen via leave-one-out cross-validation. These estimates suggest similar average welfare but somewhat larger deadweight loss, primarily due to higher estimates of choice probabilities at the subsidized price. The overall conclusion from Table 5 is that accounting for spillovers leads to much lower estimates of net welfare gain from the subsidy program and higher deadweight loss. Most of this difference arises from welfare loss suffered by ineligibles that is missed upon assuming no spillover. In Table 6, we report the standard errors where we first ignore spatial correlations, and then include them. In Table 7, we show how the welfare effects change as we vary the generosity of the subsidy scheme; the wealth threshold for qualification is varied so that either 20%, 40% or 60% of the population is eligible. It is apparent from Table 7 that the upper bound on welfare loss for ineligibles increases as more people become eligible (since equilibrium take up is higher), and the deadweight loss larger still due to both a larger extent of subsidy induced distortion, as well as the higher welfare loss of ineligibles. The lower bound on the welfare gain for eligibles decreases as the share eligible increases, in fact it becomes negative when 60% are eligible. This is because those among the eligible who are too poor to buy the bednet even at the 50Ksh price are now experiencing a 22