The Empirical Content of Binary Choice Models

TL;DR

This paper establishes nonparametric shape-restrictions for binary choice models based on economic rationality, enabling credible demand and welfare predictions without arbitrary assumptions.

Contribution

It introduces global, closed-form shape-restrictions for binary choice probabilities derived from utility maximization, applicable across various models and data sets.

Findings

Shape-restrictions are equivalent to Slutsky-like conditions under heterogeneity.

Restrictions are global, not dependent on observed budget-sets.

Provide simple bounds for demand and welfare predictions.

Abstract

An important goal of empirical demand analysis is choice and welfare prediction on counterfactual budget sets arising from potential policy-interventions. Such predictions are more credible when made without arbitrary functional-form/distributional assumptions, and instead based solely on economic rationality, i.e. that choice is consistent with utility maximization by a heterogeneous population. This paper investigates nonparametric economic rationality in the empirically important context of binary choice. We show that under general unobserved heterogeneity, economic rationality is equivalent to a pair of Slutsky-like shape-restrictions on choice-probability functions. The forms of these restrictions differ from Slutsky-inequalities for continuous goods. Unlike McFadden-Richter's stochastic revealed preference, our shape-restrictions (a) are global, i.e. their forms do not depend on which and how many budget-sets are observed, (b) are closed-form, hence easy to impose on parametric/semi/non-parametric models in practical applications, and (c) provide computationally simple, theory-consistent bounds on demand and welfare predictions on counterfactual budget-sets.

Equations

$$\frac{\partial }{\partial p}\bar{q}\left(p,y\right)\le 0\text{, and }\frac{\partial }{\partial p}\bar{q}\left(p,y\right)+\frac{\partial }{\partial y}\bar{q}\left(p,y\right)\le 0\text{.}$$

$$q\left(y,y-p\right)=\int 1\left\{W_0\left(y,\eta \right)\le W_1\left(y-p,\eta \right)\right\}dG\left(\eta \right)\text{,}$$

$$\frac{\partial }{\partial p}\bar{q}\left(p,y\right)$$

$$\frac{\partial }{\partial p}\bar{q}\left(p,y\right)+\frac{\partial }{\partial y}\bar{q}\left(p,y\right)$$

$$\frac{\partial }{\partial p}{q}^{\ast }\left(p,y\right)+q^{\ast }\left(p,y\right)\frac{\partial }{\partial y}{q}^{\ast }\left(p,y\right)\le 0\text{ for all }p,y\text{.}$$

$$\frac{\partial }{\partial p}\bar{q}\left(p,y\right)+\bar{q}\left(p,y\right)\frac{\partial }{\partial y}\bar{q}\left(p,y\right)\le 0\text{ for all }p,y\text{,}$$

$$q\left(a_0,a_1\right)$$

$$\frac{{\partial }^2}{\partial a_0\partial a_1}\ln \left[-\frac{\frac{\partial }{\partial a_1}q\left(a_0,a_1\right)}{\frac{\partial }{\partial a_0}q\left(a_0,a_1\right)}\right]$$

$$W_1\left(y-p,\eta \right)<W_0\left(y,\eta \right)<W_1\left(y-p^{\prime },\eta \right)\text{,}$$

$$W\left(x,y-px,\eta \right)<W\left(x^{\prime },y-p^{\prime }{x}^{\prime },\eta \right)\text{,}$$

$$\bar{q}\left(p,y\right)=F\left({\gamma }_0+{\gamma }_{1}p+{\gamma }_{2}y\right)=F\left({\gamma }_0+\left({\gamma }_1+{\gamma }_2\right)y-{\gamma }_1\left(y-p\right)\right)\text{,}$$

$$F\left({\gamma }_0+\left({\gamma }_1+{\gamma }_2\right)y-{\gamma }_1\left(y-p\right)\right)$$

$$W_1\left(y-p,V\right)$$

$$W_0\left(y,V\right)$$

$$\mathrm{Pr}\left(1|price=p,income=y\right)$$

$$U_1\left(y-p,\eta \right)=h\left(y-p,V,\theta \right)\text{; }{U}_0\left(y,\eta \right)=y\text{,}$$

$$\mathrm{Pr}\left(1|price=p,income=y\right)=\int \Phi \left({\gamma }_{1}p+{\gamma }_{2}y\right)dF\left({\gamma }_1,{\gamma }_2,\theta \right)$$

$$\frac{\partial }{\partial p}\bar{q}\left(p,y\right)+\frac{\partial }{\partial y}\bar{q}\left(p,y\right){|}_{p=1,y=1.2}$$

$$q^{-1}\left(u,y-p,x\right)\stackrel{def}{=}\sup \left\{z:q\left(z,y-p,x\right)\ge u\right\}\text{,}$$

$$q\left(y,y-p\right)=\mathrm{Pr}\left(1|Y=y,Y-P=y-p\right)\text{.}$$

$$\bar{L}\left(p^{\prime },y^{\prime }\right)$$

$$\bar{U}\left(p^{\prime },y^{\prime }\right)$$

$$\bar{q}\left(p,y\right)$$

$$\mathrm{Pr}\left[W_1\left(y^j-p^j,V\right)\ge y^j\right]$$

$$\text{and }\mathrm{Pr}\left[W_1\left(y^{\prime }-p^{\prime },V\right)\ge y^{\prime }\right]$$

$$\bar{L}\left(p,y+p-p_0\right)$$

$$\le$$

$$\equiv$$

$$L\left(a_0,a_1\right)$$

$$L\left(a_{0},a_{1}\right)=\left\{\begin{array}[]{l}q\left(k,a_{1}\right)\text{ if }l\leq a_{0}\leq k\\ L\left(k^{+},a_{1}\right)\text{ if }k<a_{0}\leq u\end{array}\right.$$

$$Q\left(a_{0},a_{1}\right)=\left\{\begin{array}[]{l}L\left(k,a_{1}\right)\text{, if }l\leq a_{0}\leq k\\ L\left(k,a_{1}\right)\times\left[1-\frac{a_{0}-k}{\delta}\right]+L\left(k^{+},a_{1}\right)\frac{a_{0}-k}{\delta}\text{ if }k<a_{0}\leq k+\delta\\ L\left(k^{+},a_{1}\right)\text{, if }k+\delta<a_{0}\leq u\end{array}\right.$$

$$\displaystyle\left[\begin{array}[]{ccc}0&1&1\\ 0&0&1\end{array}\right]\left[\begin{array}[]{c}\pi_{1}\\ \pi_{2}\\ \pi_{3}\end{array}\right]$$

Full text

1 Introduction

Many important economic decisions faced by individuals are binary in nature, including labour force participation, retirement, college enrolment, adoption of a new technology or health product, participation in a job-training program, etc. This paper concerns nonparametric analysis of binary choice under general unobserved heterogeneity and income effects. The paper has two goals. The first is to understand, theoretically, what nonparametric restrictions does utility maximization by heterogeneous consumers impose upon choice-probabilities, i.e. whether there are analogs of Slutsky restrictions for binary choice under general unobserved heterogeneity and income effects, and conversely, whether these restrictions are also sufficient for observed choice-probabilities to be rationalizable. This issue is important for logical coherency between theory and empirics and for prediction of demand and welfare in situations involving counterfactual, i.e. previously unobserved, budget sets. It is important in these exercises to allow for general unobserved heterogeneity because economic theory typically does not restrict its dimension or distribution, and does not specify how it enters utility functions. To date, closed-form Slutsky conditions for rationalizability of demand under general heterogeneity were available only for continuous choice. The present paper, to our knowledge, is the first to establish them for the leading case of discrete demand, viz. binary choice.

The second goal of the present paper is a practical one. It is motivated by the fact that in empirical applications of binary choice, requiring the estimation of elasticities, welfare calculations and demand predictions, researchers typically use parsimonious functional-forms for conditional choice probabilities. This is because fully nonparametric estimation is often hindered by curse of dimensionality, the sensitivity of estimates to the choice of tuning parameters and insufficient price variation, especially in consumer data from developed countries. The question therefore arises as to whether the economic theory of consumer behavior can inform the choice of such functional forms. Answering this question is our second objective.

Since McFadden 1973, discrete choice models of economic behavior have been studied extensively in the econometric literature, mostly under restrictive assumptions on utility functions and unobserved heterogeneity including, inter alia, quasi-linear preferences implying absence of income effects and/or parametrically specified heterogeneity distributions (c.f. Train 2009 for a textbook treatment). Matzkin (1992) investigated the nonparametric identification of binary choice models with additive heterogeneity, where both the distribution of unobserved heterogeneity and the functional form of utilities were left unspecified. More recently, Bhattacharya (2015, 2018) has shown that in discrete choice settings, welfare distributions resulting from price changes are nonparametrically point-identified from choice probabilities without any substantive restriction on preference heterogeneity, and even when preference distribution and heterogeneity dimension are not identified.

In the present paper, we consider a setting of binary choice by a population of budget-constrained consumers with general, unobserved heterogeneity, producing an individual-level cross-sectional dataset that records prices, individual income and the choice made by the individual.111As a referee has correctly commented, income plays a prominent role in this paper, unlike many existing empirical applications which ignore the role of income. In this setting, we develop a characterization of utility maximization which takes the form of simple, closed-form shape restrictions on choice probability functions in the population. These nonparametric shape-restrictions can be consistently tested in the usual asymptotic econometric sense and are extremely easy to impose on specifications of choice-probabilities – akin to testing or imposing monotonicity of regression functions. Most importantly, they lead to computationally simple bounds for theory-consistent demand and welfare predictions on counterfactual budgets sets – an important goal of empirical demand analysis. Interestingly, our shape-restrictions differ in form from the well-known Slutsky inequalities for continuous goods.

The above results are developed in a fully nonparametric context; nonetheless, they can help guide applied researchers intending to use simple parametric or semiparametric models. As a specific example, consider the popular probit/logit type model for binary choice of whether to buy a product or not. A standard specification is that the probability of buying depends (implicitly conditioning on other observed covariates) on its price $p$ and the decision-maker’s income $y$, e.g. $\bar{q}\left(p,y\right)=F\left(\gamma_{0}+\gamma_{1}p+\gamma_{2}y\right)$, where $F\left(\cdot\right)$ is a distribution function. We will show below that these choice-probabilities are consistent with utility maximization by a heterogenous population of consumers, if and only if $\gamma_{1}\leq 0$, and $\gamma_{1}+\gamma_{2}\leq 0$. While the first inequality simply means that demand falls with own price (holding income fixed), the second inequality is less obvious, and constitutes an important empirical characterization of utility maximization.

For the case of continuous goods, Lewbel 2001 explored the question of when average demand, generated from maximization of heterogeneous individual preferences, satisfies standard properties of non-stochastic demand functions. More recently, for the case of two continuous goods (i.e. a good of interest plus the numeraire) under general heterogeneity, Dette, Hoderlein and Neumayer 2016 have shown that constrained utility maximization implies quantiles of demand satisfy standard Slutsky negativity, and Hausman and Newey 2016 have shown that the two are in fact equivalent. The analog of the two goods setting in discrete choice is the case of binary alternatives. Accordingly, our main result (Theorem 1 below) may be viewed as the discrete choice counterpart of Hausman and Newey 2016, Theorem 1. Note however that quantiles are degenerate for binary outcomes, and indeed, the forms of our Slutsky-like shape restrictions are completely different from Dette et al and Hausman-Newey’s quantile-based conditions for continuous choice.

An alternative, algorithmic – as opposed to closed-form and analytic – approach to rationalizability of demand is the “revealed stochastic preference” (SRP, henceforth) method, which applies to very general choice settings where a heterogeneous population of consumers faces a finite number of budget sets, c.f. McFadden and Richter 1990, McFadden 2005. When budget sets are numerous or continuously distributed, as in household surveys with many income and/price values, SRP is well-known to be operationally prohibitive, c.f. Anderson et al 1992, Page 54-5 and Kitamura and Stoye 2016, Sec 3.3. Furthermore, the SRP conditions are difficult to impose on parametric specifications commonly used in practical applications, they change entirely in form upon addition of new budget sets, and are cumbersome to use for demand prediction on counterfactual budgets, especially in welfare calculations that typically require simultaneous prediction of demand on a continuous range of budget-sets. In contrast, our approach yields rationality conditions which (a) are global, in that they characterize choice probability functions, and their forms remain invariant to which and how many budget sets are observed in a dataset, and (b) are closed-form, analytic shape-restrictions, hence easy to impose, standard to test, and simple to use for the important practical problem of counterfactual predictions of demand and welfare. As such, these shape-restrictions establish the analogs of Slutsky conditions – the cornerstone of classical demand analysis – for binary choice under general unobserved heterogeneity and income effects.

2 The Result

Consider a population of heterogeneous individuals, each choosing whether or not to buy an indivisible good. Let $N$ represent the quantity of numeraire which an individual consumes in addition to the binary good. If the individual has income $Y=y$, and faces a price $P=p$ for the indivisible good, then the budget constraint is $N+pQ=y$ where $Q\in\left\{0,1\right\}$ represents the binary choice. Individuals derive satisfaction from both the indivisible good as well as the numeraire. Upon buying, an individual derives utility from the good but has a lower amount of numeraire $y-p$ left; upon not buying, she enjoys utility from her outside option and a higher quantity of numeraire $y$. There is unobserved heterogeneity across consumers which affect their choice, and so on each budget set defined by a price $p$ and consumer income $y$, there is a (structural) probability of buying, denoted by $\bar{q}\left(p,y\right)$; that is, if each member of the entire population were offered income $y$ and price $p$, then a fraction $\bar{q}\left(p,y\right)$ would buy the good. For now, we implicitly condition our analysis on observed covariates, and later show how to incorporate them into the results. We will show that these choice probabilities will be consistent with utility maximization by a heterogeneous population if and only if the following Slutsky-like conditions222Our main result does not need smoothness; we write the conditions with derivatives here to show the Slutsky-like form of the result. hold:

[TABLE]

For establishing this result, it will be convenient to rewrite the choice probabilities in an equivalent way as $q\left(y,y-p\right)=\bar{q}\left(p,y\right)$. Indeed, one can go back and forth between the two specifications because $\bar{q}\left(c,d\right)\equiv q\left(d,d-c\right)$ and $q\left(a,b\right)\equiv\bar{q}\left(a-b,a\right)$. The $q\left(y,y-p\right)$ formulation is motivated by the fact that given the budget set $\left(P,Y\right)=\left(p,y\right)$, an individual faces choice between the bundles $\left(0,y\right)$ and $\left(1,y-p\right)$; thus $q\left(\cdot,\cdot\right)$ is an equivalent representation of choice probabilities as functions of the income left over upon choosing options 0 and 1, respectively. For ease of exposition, we will state our results in terms of $q\left(\cdot,\cdot\right)$, and show that under smoothness they reduce to restriction (1) on $\bar{q}\left(\cdot,\cdot\right)$.

The following theorem establishes conditions that are necessary and sufficient for the conditional choice probability function to be generated from utility maximization by a heterogeneous population, where no a priori restriction is imposed on the dimension and functional form of the distribution of unobserved heterogeneity or on the functional form of utilities.

To formally state the theorem, we introduce some notation. Let $\bar{\Omega}$ denote the support of $\left(P,Y\right)$; let $\Omega_{1}=\left\{y-p:\left(p,y\right)\in\bar{\Omega}\right\}$ denote the support of $Y-P$, and for any $a_{1}\in\Omega_{1}$ let $\Omega_{0}\left(a_{1}\right)=\left\{y:\left(p,y\right)\in\bar{\Omega}\text{, }y-p=a_{1}\right\}$. Corresponding to the support $\bar{\Omega}$ of $\left(P,Y\right)$, denote the support of $\left(Y,Y-P\right)$ by $\Omega$, as short-hand for $\cup_{a_{1}\in\Omega_{1}}\cup_{a_{0}\in\Omega_{0}\left(a_{1}\right)}\left\{a_{0},a_{1}\right\}$.

Theorem 1

For binary choice under general heterogeneity, the following two statements are equivalent:

(I) The structural choice probability function $q\left(\cdot,\cdot\right):\Omega\rightarrow\left[0,1\right]$ satisfies that (A) (i) $q\left(\cdot,y-p\right)$ is non-increasing, and (ii) $q\left(y,\cdot\right)$ is non-decreasing; (B) $q\left(\cdot,y-p\right)$ is continuous; (C) corresponding to any fixed value $a_{1}\in\Omega_{1}$, there exist a small enough real number $y_{L}\left(a_{1}\right)\in\Omega_{0}\left(a_{1}\right)$, satisfying $\lim_{y\searrow y_{L}\left(a_{1}\right),y-p=a_{1}}q\left(y,y-p\right)=1$ and a large enough real number $y_{H}\left(a_{1}\right)\in\Omega_{0}\left(a_{1}\right)$, satisfying $\lim_{y\nearrow y_{H}\left(a_{1}\right),y-p=a_{1}}q\left(y,y-p\right)=0$.

(II) There exists a pair of utility functions $W_{0}\left(\cdot,\eta\right)$ and $W_{1}\left(\cdot,\eta\right)$, where the first argument denotes the amount of numeraire, and $\eta$ denotes unobserved heterogeneity, and a distribution $G\left(\cdot\right)$ of $\eta$ such that

[TABLE]

where (A’) for each fixed $\eta$, (i) $W_{0}\left(\cdot,\eta\right)$ is continuous and strictly increasing, and (ii) $W_{1}\left(\cdot,\eta\right)$ is non-decreasing; (B’) for any $p,y\in\bar{\Omega}$, it holds that $\int 1\left\{W_{1}\left(y-p,\eta\right)=W_{0}\left(y,\eta\right)\right\}dG\left(\eta\right)=0$; (C’) corresponding to any fixed $a_{1}\in\Omega_{1}$, there exist a small enough real number $y_{L}\left(a_{1}\right)\in\Omega_{0}\left(a_{1}\right)$ and a large enough real number $y_{H}\left(a_{1}\right)\in\Omega_{0}\left(a_{1}\right)$, satisfying $\lim_{y\searrow y_{L}\left(a_{1}\right),y-p=a_{1}}\Pr\left[W_{0}\left(y,\eta\right)\leq W_{1}\left(y-p,\eta\right)\right]=1$ and $\lim_{y\nearrow y_{H}\left(a_{1}\right),y-p=a_{1}}\Pr\left[W_{0}\left(y,\eta\right)\leq W_{1}\left(y-p,\eta\right)\right]=0$.

Proof. In Appendix

The key step in the proof is showing that (I) implies (II). This is done by constructing the utility functions $W_{0}\left(y,\eta\right)=y$ and $W_{1}\left(y-p,\eta\right)=q^{-1}\left(V,y-p\right)$ with $q^{-1}\left(\cdot,y-p\right)$ denoting a suitably defined inverse of the function $q\left(\cdot,y-p\right)$ with respect to its first argument, and the random variable $\eta=V\sim Uniform(0,1)$. Under conditions A, B, C of Theorem 1, this construction is then shown to imply that $\Pr\left[W_{1}\left(y-p,\eta\right)\geq W_{0}\left(y,\eta\right)\right]=q\left(y,y-p\right)$. The formal proof appears in the Appendix.

Interpretation of conditions: Intuitively, conditions (A/A’) mean that having more numeraire ceteris paribus is (weakly) better for every consumer, i.e. preferences are increasing in the amount of income left over after any choice. Condition (B/B’) – the “no-tie” assumption – is standard in discrete choice models, and intuitively means that there is a continuum of tastes. Condition (C) adds to condition (A); it says that holding fixed the income left over upon choosing option 1, if the income left over upon choosing option 0 is, hypothetically*,* made small enough, then everyone, i.e. all $\eta$, will choose option 1. In particular, $y\searrow y_{L}\left(a_{1}\right),y-p=a_{1}$ means that starting from a situation with $y-p=a_{1}$, we are lowering $p$ and $y$ by equal amounts, keeping $y-p$, i.e. the income left over upon choosing option 1, fixed at $a_{1}$ while $y$, the income left over upon choosing option 0, is lowered toward $y_{L}\left(a_{1}\right)$, i.e., $q\left(\underset{\searrow y_{L}\left(a_{1}\right)}{\underbrace{y}},\underset{\text{fixed at }a_{1}}{\underbrace{y-p}}\right)\nearrow 1$. A symmetric interpretation applies to $y_{H}\left(a_{1}\right)$. The following examples illustrate Condition C.

Example 1 (High and Low Price): Suppose $0,1$ denote respectively not buying and buying a binary good. Suppose preferences are such that at any income $y$, if price takes a high enough value $p^{H}$, e.g. close to the highest income in the population, no one would buy the good; conversely, when price takes a low enough value $p^{L}$, e.g. the good is free ($p^{L}=0$) or there is a high enough reward $r>0$ for choosing option 1 (i.e. $p^{L}=-r<0$) as in conditional cash transfer programs for school-attendance, everyone (i.e. all $\eta$) will choose option 1. Then starting from $y-p=a_{1}>0$, raising $p$ towards $p^{H}$ while simultaneously increasing $y$ by equal amount keeping $y-p$, the income left upon buying, fixed at $a_{1}$, we have that $q\left(y,y-p\right)\equiv q\left(a_{1}+p,a_{1}\right)\searrow q\left(a_{1}+p^{H},a_{1}\right)=0$; similarly, letting $p\searrow p^{L}$ and $y\searrow a_{1}+p^{L}$ while keeping fixed $y-p=a_{1}>0$, we have that $q\left(\underset{\searrow a_{1}+p^{L}}{\underbrace{y}},\underset{\text{fixed at }a_{1}}{\underbrace{y-p}}\right)\nearrow q\left(a_{1}+p^{L},a_{1}\right)=1$. Thus $y_{H}\left(a_{1}\right)=a_{1}+p^{H}$, and $y_{L}\left(a_{1}\right)=a_{1}+p^{L}$.

**Example 2 **(Labour supply): Suppose $0,1$ denote not working and working, respectively, $y$ is non-labour income (e.g. spousal earning or interest income from investment), and $p=-w$ is the negative of net wage received upon working, so that $q\left(y,y-p\right)=q\left(y,y+w\right)$. Here it is natural to assume that if non-labour income $y$ is zero, then an individual must work at any positive net wage $w$ for subsistence, so that $q\left(0,w\right)=1$, and thus $y_{L}\left(a_{1}\right)=0$ for any positive $a_{1}$. Similarly, if net wage is zero, then no one with positive non-labour income will work, i.e. $q\left(y,y\right)=0$, and thus $y_{H}\left(a_{1}\right)=a_{1}$.

Remark 1

Condition C/C’, which simplify the proof of the Theorem, can be dropped. In the appendix, we provide an alternative version of the theorem without conditions (C/C’), but with a slightly stronger continuity requirement (B/B’) and a significantly longer proof.

Remark 2

Note that assumptions (A)-(C) place no restriction on income effects, including its sign.

In statement (II) in Theorem 1, the functions $W_{j}\left(x,\eta\right)$ will correspond to the utility from choosing alternative $j\in\left\{0,1\right\}$ and being left with a quantity $x$ of the numeraire, and with $\eta$ denoting unobserved heterogeneity. This notation allows for the case where different vectors of unobservables enter the two utilities, i.e. where the utilities are given by $u_{0}\left(\cdot,\eta_{0}\right)$ and $u_{1}\left(\cdot,\eta_{1}\right)$, respectively, with $\eta_{0}\neq\eta_{1}$; simply set $\eta\equiv\left(\eta_{0},\eta_{1}\right)$, $W_{0}\left(\cdot,\eta\right)\equiv u_{0}\left(\cdot,\eta_{0}\right)$, $W_{1}\left(\cdot,\eta\right)\equiv u_{1}\left(\cdot,\eta_{1}\right)$. In the proof of the above theorem, when showing (II) implies (I), $\eta$ will be allowed to have *any arbitrary and unknown *dimension and distribution; in showing (I) implies (II) we will construct a scalar heterogeneity distribution that will rationalize the choice probabilities (see further discussion on this point under the heading ”Observational Equivalence” in the next section).

3 Further Discussion

A. Slutsky Form: To see the analogy between the shape restrictions in Theorem 1 and the traditional Slutsky inequality constraints with smooth demand, rewrite the choice probability on a budget set $\left(p,y\right)$ in the standard form as a function of price and income, viz. $\bar{q}\left(p,y\right)\equiv q\left(y,y-p\right)$ i.e., $q\left(a_{0},a_{1}\right)\equiv\bar{q}\left(a_{0}-a_{1},a_{0}\right)$. Then, under continuous differentiability, the shape restrictions (A) from Theorem 1 are equivalent to

[TABLE]

for all $p,y$.333I am grateful to a referee for suggesting this way of showing the equivalence. The forms of these inequalities are distinct from textbook Slutsky conditions for nonstochastic demand $q^{\ast}\left(p,y\right)$ for a continuous good, which are given by

[TABLE]

For a continuous good and under general unobserved heterogeneity, Dette, Hoderlein and Neumeyer 2016 (building on earlier work of Hoderlein 2011), and Hausman and Newey 2016 show that (4) also holds with $q^{\ast}\left(p,y\right)$ denoting any quantile of the demand distribution for fixed $\left(p,y\right)$. Thus, for binary choice with general heterogeneity, the forms of the Slutsky inequality (2) and (3) are different from the continuous choice counterpart (4).444Bhattacharya, 2015 (see also Lee and Bhattacharya, 2018) noted that (2) (resp, (3)) is necessary for the CDF of equivalent variation (resp., compensating variation) resulting from price-changes to be non-decreasing. In particular, the inequalities (2) and (3) are linear in $\bar{q}\left(\cdot,\cdot\right)$ (and $q\left(\cdot,\cdot\right)$), unlike (4), and hence easier to impose on nonparametric estimates of $q\left(\cdot,\cdot\right)$ using, say, shape-preserving sieves that guarantee that $\frac{\partial}{\partial a_{1}}\hat{q}\left(a_{0},a_{1}\right)\geq 0$, and $\frac{\partial}{\partial a_{0}}\hat{q}\left(a_{0},a_{1}\right)\leq 0$ for all $a_{0},a_{1}$.

Remark 3

It is tempting to think of (2) and (3) as (4) with the level $q^{\ast}\left(p,y\right)$ replaced by 0 and 1 corresponding to either of the two possible individual choices. However, this interpretation is incorrect, since $\bar{q}\left(p,y\right)$ is average demand, and takes values strictly inside $\left(0,1\right)$. In other words, $\bar{q}\left(p,y\right)$ is neither a quantile, nor individual demand at price $p$ and $y$, and generically (e.g. in a probit model) does not take the values of 0 and 1. Thus (2) and (3) cannot be rewritten as

[TABLE]

and, as such, are different from the continuous choice counterpart (4).

Remark 4

Our rationality conditions (A) take the form of simple monotonicity restrictions on the regression function $q\left(\cdot,\cdot\right)$. There are several papers in the Statistics literature on testing monotonicity of nonparametrically estimated regressions, e.g. Ghosal et al 2000, Hall and Heckman 2000, Chetverikov 2012, etc. which can therefore be used here.

B. Observational Equivalence: The construction in our proof of (II) $\Rightarrow$ (I) shows that a rationalizable binary choice model with general heterogeneity of unspecified dimension is observationally equivalent to one where a scalar heterogeneity enters the utility function of one of the alternatives in a monotonic way, and the utility of the other alternative is non-stochastic.555For quantile demand in the continuous case, a result of similar spirit is discussed in Hausman-Newey, 2016, Page 1228-9, following Theorem 1. In general, a result holding for the continuous case with two goods does not necessarily imply that it also holds for the binary case. For example, welfare related results are different for the binary and the two-good continuous case, c.f. Hausman-Newey 2016, and Bhattacharya 2015, and so are Slutsky negativity conditions, as discussed above. An intuitive explanation of this equivalence is that in the binary case, choice probabilities are determined solely by the marginal distribution of reservation price (given income) for alternative 1, and not the relative ranking of individual consumers in terms of their preferences within that distribution. So, as income varies, choice probabilities change only insofar as the marginal distribution of the reservation price changes, irrespective of how individual consumers’ relative positions change within that distribution.

It is worth pointing out here that a binary choice model with additive scalar heterogeneity – the so-called ARUM model – is restrictive, and not observationally equivalent to a binary choice model with general heterogeneity. To see this, suppose choice probabilities are generated via the ARUM model, viz.

[TABLE]

Assuming smoothness and strict monotonicity of $F_{\eta_{0}-\eta_{1}}\left[\cdot\right]$, $W_{1}\left(\cdot\right)$ and $W_{0}\left(\cdot\right)$, and thus of $q\left(\cdot,\cdot\right)$, it follows that

[TABLE]

for every $a_{0}$ and $a_{1}$. This equality is obviously not true for a general smooth and strictly monotone $q\left(\cdot,\cdot\right)$ satisfying conditions (A)-(C) of Theorem 1.

Remark 5

The construction of $q^{-1}\left(V,\cdot\right)$ in our proof of (II) $\Rightarrow$ (I) is unrelated to the almost sure representation of a continuous random variable $X$ as $F_{X}^{-1}\left(U\right)$ with $U=F_{X}\left(X\right)$, where $F_{X}$ and $F_{X}^{-1}$ denote the CDF and quantile function of $X$, and $U$ is distributed $U\left(0,1\right)$. Indeed, if we were to apply this so-called ”probability-integral transform” to $X=W_{1}\left(a_{1},\eta\right)$ for a fixed $a_{1}$, we will have $W_{1}\left(a_{1},\eta\right)\overset{a.s.}{=}F_{W_{1}\left(a_{1},\eta\right)}^{-1}\left(U\left(a_{1}\right)\right)$, where the scalar-valued uniform process $U\left(a_{1}\right)\equiv F_{W_{1}\left(a_{1},\eta\right)}\left(W_{1}\left(a_{1},\eta\right)\right)$ will vary with $a_{1}$, unlike $V$ in the proof of our theorem above, and therefore cannot represent unobserved heterogeneity in consumer preferences. In other words, our constructed $q^{-1}\left(V,a_{1}\right)$ will not equal the data generating process $W_{1}\left(a_{1},\eta\right)$ almost surely, but the probability that $q^{-1}\left(V,a_{1}\right)\geq a_{0}$ will equal the probability that $W_{1}\left(a_{1},\eta\right)\geq W_{0}\left(a_{0},\eta\right)$ for all $\left(a_{0},a_{1}\right)$.

C. Giffen Goods: Our rationalizability condition (2) says that own price effect on average demand is negative. This condition has no counterpart in the continuous case, appears to rule out Giffen behavior and may, therefore, appear restrictive. We now show that that is not the case: indeed, Giffen goods cannot arise in binary choice models if utilities are non-satiated in the numeraire. To see this, let the utility of options [math] and $1$ be given by $W_{0}\left(\cdot,\eta\right)$ and $W_{1}\left(\cdot,\eta\right)$ as in Theorem 1 above. Now note that if option 1 is Giffen for an $\eta$ type consumer with income $y$, then for some prices $p<p^{\prime}$ she buys at price $p^{\prime}$ but does not buy at $p$. Therefore,

[TABLE]

which is a contradiction, since $W_{1}\left(\cdot,\eta\right)$ is strictly increasing. In contrast, consider a continuous good with utilities $W\left(x,y-px,\eta\right)$, where $x$ denotes the quantity of the continuous good, and $W\left(\cdot,\cdot,\eta\right)$ is increasing in both arguments. Now it is possible that $x$ is bought at price $p$ and $x^{\prime}$ is bought at price $p^{\prime}$ with $p<p^{\prime}$ and $x<x^{\prime}$. That is, we can have

[TABLE]

if $x^{\prime}$ is preferred sufficiently over $x$. The intuitive reason for this difference between the discrete and the continuous case is that in the former, the only non-zero option is 1. Indeed, in the continuous case, it is also not possible that $W\left(x,y-px,\eta\right)<W\left(x,y-p^{\prime}x,\eta\right)$ for any common $x$ if $p<p^{\prime}$.

Also, note that although Giffen behavior cannot arise in binary choice, there is no restriction on the sign of the income effect. Indeed, (2) and (3) are compatible with both $\frac{\partial}{\partial y}\bar{q}\left(p,y\right)\geq 0$ and $\frac{\partial}{\partial y}\bar{q}\left(p,y\right)\leq 0$.

D. Parametric and Semiparametric Models: For a probit/logit specification of the buying decision, viz.

[TABLE]

where $F\left(\cdot\right)$ is a strictly increasing CDF, the shape restrictions of Theorem 1 amount to requiring $\gamma_{1}\leq 0$ and $\gamma_{1}+\gamma_{2}\leq 0$. While the first inequality is intuitive, and simply says that own price effect is negative, the second condition $\gamma_{1}+\gamma_{2}\leq 0$ is not a priori obvious, and shows the additional restriction implied by budget-constrained utility maximization. Now, applying Theorem 1, we obtain

[TABLE]

where $V\simeq U\left(0,1\right)$,666We implicitly assume that for fixed $y-p$, the function $q\left(y,y-p\right)$ varies with $y$ somewhere on $S\left(y-p\right)$, and thus $\gamma_{1}+\gamma_{2}\neq 0$. implying the rationalizing utility functions

[TABLE]

Remark 6

Note that since the restrictions $\gamma_{1}\leq 0$ and $\gamma_{1}+\gamma_{2}\leq 0$ are linear in parameters, it is computationally straightforward to maximize a globally concave likelihood, such as probit or logit, subject to these constraints.

The above discussion also applies to semiparametric binary choice models (c.f. Manski 1975, Han 1987, Klein and Spady 1993) where one need not specify the exact functional form of $F\left(\cdot\right)$. For example, the methods of Cavanagh and Sherman (1998) and Bhattacharya (2008), which only utilize the strict monotonicity of the CDF $F\left(\cdot\right)$, can be applied to estimate the binary choice model, subject to our sign restriction and standard scale-normalization, viz. $\gamma_{1}=-1$ and $\gamma_{1}+\gamma_{2}\leq 0$, i.e. using the specification that $\bar{q}\left(p,y\right)$ is a strictly increasing function of the linear index $-p+\gamma_{2}y$ with $\gamma_{2}\leq 1$.

E. Random Coefficients: An alternative parametric specification in this context is a random coefficient structure, popular in IO applications. It takes the form

[TABLE]

where $\gamma_{1}$ and $\gamma_{2}$ are now random variables with joint distribution $G\left(\cdot,\cdot,\theta\right)$, indexed by an unknown parameter vector $\theta$, and $F\left(\cdot\right)$ is a specified CDF (e.g. a probit or logit). Theorem 1 then implies that the distribution $G\left(\cdot,\cdot,\theta\right)$ must be such that the choice probability function $H\left(\cdot,\cdot,\cdot\right)$ satisfies $\frac{\partial}{\partial y}H\left(y,\cdot,\theta\right)\leq 0$ and $\frac{\partial}{\partial\left(y-p\right)}H\left(\cdot,y-p,\theta\right)\geq 0$. One way to guarantee this would be to specify the support of $\gamma_{1}$ and of $\gamma_{1}+\gamma_{2}$ to lie in $\left(-\infty,0\right)$. Using Theorem 1, a utility structure that would rationalize such a model is:

[TABLE]

where $V\simeq U\left(0,1\right)$, and $h\left(y-p,v,\theta\right)$ is $\sup\left\{x:H\left(x,y-p,\theta\right)\geq v\right\}$.777Note that an alternative preference distribution producing the same choice probabilities is given by $U_{1}\left(y-p,\eta\right)=-\gamma_{1}\left(y-p\right)$, $U_{0}\left(y,\eta\right)=\gamma_{0}-\left(\gamma_{1}+\gamma_{2}\right)y$, $\gamma_{0}\perp\left(\gamma_{1},\gamma_{2}\right)$, $\gamma_{0}\simeq F\left(\cdot\right)$, $\left(\gamma_{1},\gamma_{2}\right)\simeq G\left(\cdot,\cdot,\theta\right)$, $\gamma_{1}<0$, $\gamma_{1}+\gamma_{2}\leq 0$ w.p.1. This shows that the rationalizing preference distribution may not be unique.

It also follows from the above discussion that not every distribution of random coefficients $G\left(\cdot,\cdot,\theta\right)$ will lead to rationalizable choice-probability functions. In particular, the commonly used assumption that $\left(\gamma_{1},\gamma_{2}\right)$ is bivariate normal (so that the support of $\gamma_{1}$ and of $\gamma_{1}+\gamma_{2}$ do not lie in $\left(-\infty,0\right)$), can lead to choice probability functions $H\left(\cdot,\cdot,\theta\right)$ that would violate the shape restrictions of Theorem 1, and thus are not rationalizable.888As a numerical illustration, consider a random coefficient probit model

$\Pr\left(1|price=p,income=y\right)=\int\Phi\left(\gamma_{1}p+\gamma_{2}y\right)dF\left(\gamma_{1},\gamma_{2},\theta\right)$

where $\gamma_{1}\sim N\left(-1,0.1^{2}\right)$, $\gamma_{2}\sim N\left(3,0.2^{2}\right)$ and $\gamma_{1}\perp\gamma_{2}$, implying each of the probabilities of $\gamma_{1}\leq 0$ and $\gamma_{2}\geq 0$ exceeds 0.9999. Yet it can be verified numerically that e.g.

$\displaystyle\frac{\partial}{\partial p}\bar{q}\left(p,y\right)+\frac{\partial}{\partial y}\bar{q}\left(p,y\right)|_{p=1,y=1.2}$

$\displaystyle=$ $\displaystyle E\left[\left(\gamma_{1}+\gamma_{2}\right)\times\phi\left(\gamma_{1}+1.2\times\gamma_{2}\right)\right]\simeq 0.03>0\text{.}$

F. Observed Covariates: One can accommodate observed covariates in our theorem. For example, let $X$ denote a vector of observed covariates, and let $\bar{q}\left(p,y,x\right)\equiv q\left(y,y-p,x\right)$ denote the choice probability when $Y=y$, $Y-P=y-p$ and $X=x$. If for each fixed $x$, $q\left(\cdot,\cdot,x\right)$ satisfies the same properties as (I) A-C in the statement of Theorem 1, then letting

[TABLE]

we can rationalize the choice probabilities $\bar{q}\left(p,y,x\right)$ by setting $W_{1}\left(y-p,V,x\right)\equiv q^{-1}\left(V,y-p,x\right)$ and $W_{0}\left(y,V,x\right)\equiv y$, where $V\simeq U\left(0,1\right)$.

G. Endogeneity: Our results in Theorem 1 are stated in terms of structural choice probabilities $q\left(\cdot,\cdot\right)$. If budget sets are independent of unobserved heterogeneity (conditional on observed covariates), then these structural choice probabilities are equal to the observed conditional choice probabilities, i.e.,

[TABLE]

Early results on rationalizability of demand under heterogeneity, including McFadden and Richter 1990 and Lewbel 2001 worked under such independence. If the independence condition is violated (even conditional on observed covariates), then Theorem 1 continues to remain valid as stated, since it concerns the structural choice probability $q\left(\cdot,\cdot\right)$, but consistent estimation of $q\left(\cdot,\cdot\right)$ will be more involved. In applications, if endogeneity of budget sets is a potential concern, then it would be advisable to estimate structural choice-probabilities using methods for estimating average structural functions. A specific example is the method of control functions, c.f. Blundell and Powell 2003, 2004 and Imbens and Newey 2009, which require that $\eta\perp\left(P,Y\right)|V$, where $V$ is an estimable “control function” – typically a first stage residual from a regression of endogenous covariates on instruments. The structural choice probability function can then be recovered (under regularity conditions) as the integral of the conditional choice probability given $p,y$ and realizations $v$ of the control variable $V$ over the marginal distribution of $V$. Hoderlein 2011, Hoderlein and Stoye 2014, Hausman and Newey 2016, and Kitamura and Stoye 2018 have previously discussed using control functions to estimate demand nonparametrically.

4 Empirical Implications

A practical implication of Theorem 1 is that it can be used to bound predicted choice probabilities on counterfactual, i.e. previously unobserved, budget-sets, e.g. those arising from a potential policy intervention. Such predictions are more reliable when made nonparametrically, i.e. without arbitrary functional-form/distributional assumptions on unobservables, and instead based solely on economic rationality. We now show how to obtain these nonparametric bounds using Theorem 1.

Counterfactual Demand Bounds: Let $\Omega$ denote the domain of definition of $\bar{q}\left(\cdot,\cdot\right)$. Let $A=\left\{\left(p^{j},y^{j}\right),j=1,...N\right\}\sqsubset\Omega$ denote the set of $\left(p,y\right)$ observed in the data, with corresponding choice probabilities $\left\{\bar{q}^{j},j=1,...N\right\}=\left\{q\left(y^{j},y^{j}-p^{j}\right)\text{, }\left(p^{j},y^{j}\right)\in A\right\}$, satisfying condition (A) of our Theorem. Suppose we are required to predict the probability $\bar{q}\left(p^{\prime},y^{\prime}\right)$ of buying at a counterfactual (i.e. previously unobserved) price $p^{\prime}$ and income $y^{\prime}$ with $\left(p^{\prime},y^{\prime}\right)\in\Omega$ $\backslash$ $A$. Then Theorem 1 implies the following bounds on this choice probability:

[TABLE]

The above calculation is extremely simple; for example, the lower bound $\bar{L}\left(p^{\prime},y^{\prime}\right)$ requires collecting those observed budget sets $\left(p,y\right)$ in the data that satisfy $y\geq y^{\prime},$ $y-p\leq y^{\prime}-p^{\prime}$ (a one-line command in STATA), evaluating choice probabilities on them, and sorting these values.

Note also that for all $\left(p,y\right)\in A$, we have that $\bar{L}\left(p,y\right)=\bar{q}\left(p,y\right)=\bar{U}\left(p,y\right)$.

Proposition 1

The bounds (9) and (12) are sharp.

Proof of Proposition. Define $W=\left\{\left(y,y-p\right):\left(p,y\right)\in A\right\}\cup\left(y^{\prime},y^{\prime}-p^{\prime}\right)$. Set $\bar{q}\left(p^{\prime},y^{\prime}\right)\equiv q\left(y^{\prime},y^{\prime}-p^{\prime}\right)=c$ for any $c$ belonging to the interval defined by the bounds in (9) and (12). Then the elements of the set $\left\{q\left(y,y-p\right):\left(p,y\right)\in A\cup\left(p^{\prime},y^{\prime}\right)\right\}$ satisfy the shape restrictions (A) of Theorem 1 on $W$. In particular, if $\left(p,y\right)\in A$ satisfies $y>y^{\prime},$ $y-p=y^{\prime}-p^{\prime}$, then

[TABLE]

on the other hand, if $\left(p,y\right)\in A$ satisfies $y=y^{\prime},$ $y-p>y^{\prime}-p^{\prime}$, then

[TABLE]

Next, note that conditions (B) and (C) of our theorem have no empirical content vis-a-vis the countably finite set of values $\left\{\bar{q}^{j}\text{, }j=1,...,N\right\}\cup\left\{c\right\}$, in that there are no set of values $\left\{\bar{q}^{j}\text{, }j=1,...,N\right\}\cup\left\{c\right\}$ which can imply a violation of conditions (B) and (C). Therefore, the choice probabilities $\left\{\bar{q}^{j}\text{, }j=1,...,N\right\}\cup\left\{c\right\}$ corresponding to $A\cup\left(p^{\prime},y^{\prime}\right)$ are compatible with a choice probability function $q\left(\cdot,\cdot\right)$ on a domain $G$ containing $W\cup\left(y^{\prime},y^{\prime}-p^{\prime}\right)$ and satisfying conditions (A)-(C) of Theorem 1 (for an explicit construction of such a function, see discussion on discrete support of $\left(P,Y\right)$ in the paragraph preceding Theorem 1 above). Therefore, applying Theorem 1, we conclude that there exist utility functions $W_{1}\left(a_{1},V\right)$ and $W_{0}\left(a_{0},V\right)=a_{0}$ with $V\simeq U\left(0,1\right)$ that satisfy the restrictions (A’)-(C’) of Theorem 1, and $\Pr\left[W_{1}\left(a_{1},V\right)\geq a_{0}\right]=q\left(a_{0},a_{1}\right)$ for all $\left(a_{0},a_{1}\right)\in G$; in particular,

[TABLE]

Welfare bounds: Given bounds on choice probabilities, one can obtain lower and upper bounds on economically interesting functionals thereof, such as average welfare. For example, the average compensating variation – i.e. utility preserving income compensation – corresponding to a price increase from $p_{0}$ to $p_{1}$ at income $y$ is given by $\int_{p_{0}}^{p_{1}}\bar{q}\left(p,y+p-p_{0}\right)dp$ (c.f. Bhattacharya 2015). This requires prediction of demand on a continuum of budget sets, viz. $\left\{\bar{q}\left(p,y+p-p_{0}\right):p\in\left[p_{0},p_{1}\right]\right\}$. Now, it follows from our discussion immediately above, and by Theorem 1, that pointwise sharp bounds on $\bar{q}\left(p,y+p-p_{0}\right)$ are given by

[TABLE]

This implies that average CV at $y$ is bounded below by $\int_{p_{0}}^{p_{1}}\bar{L}\left(p,y+p-p_{0}\right)dp$, and above by $\int_{p_{0}}^{p_{1}}\bar{M}\left(p,y+p-p_{0}\right)dp$.

As for sharpness, let $L\left(y,y-p\right)=\bar{L}\left(p,y\right)$ be defined analogous to $q\left(y,y-p\right)=\bar{q}\left(p,y\right)$ above. Then the lower bound on average CV becomes $\int_{p_{0}}^{p_{1}}L\left(y+p-p_{0},y-p_{0}\right)$. Now, by definition,

[TABLE]

is non-increasing in $a_{0}$ and non-decreasing in $a_{1}$, and $L\left(y,y-p\right)=q\left(y,y-p\right)$ when $\left(p,y\right)\in A$. Furthermore, for fixed value of $\left(y-p_{0}\right)$, as $p$ varies over the interval $\left[p_{0},p_{1}\right]$, the function $L\left(y+p-p_{0},y-p_{0}\right)$ can assume at most finitely many values (viz. $q\left(y^{m},y^{m}-p^{m}\right)$, $m=1,...,N$), and therefore, must necessarily be piecewise flat in $p$, with at most countably finite number of discontinuity points. Therefore, one can construct a function $Q\left(\cdot,\cdot\right)$ (see footnote below for an illustration) that (1) is continuous in the first argument, (2) equals $L\left(\cdot,\cdot\right)$ (and therefore $q\left(\cdot,\cdot\right)$) on $A$, (3) equals $L\left(\cdot,\cdot\right)$ everywhere else on the domain except in arbitrarily small (semi-closed) intervals around the points of discontinuity of $L\left(\cdot,\cdot\right)$, and (4) satisfies the same shape restrictions as $L\left(\cdot,\cdot\right)$ ; also, (5) $Q\left(\cdot,\cdot\right)$ can be trivially made to satisfy the limit conditions (C) of Theorem 1 by defining the limit points $y_{L}\left(\cdot\right)$, $y_{H}\left(\cdot\right)$ lower than the lowest and larger than the highest values respectively attained by $y$ in $A$ corresponding to any fixed value of $y-p$. Using (1), (4) and (5) and applying Theorem 1, we can rationalize $Q\left(\cdot,\cdot\right)$ – which equals $q\left(\cdot,\cdot\right)$ at all the observed data points, i.e. corresponding to $\left(p,y\right)\in A$ – via a pair of utility functions and a uniformly distributed unobserved heterogeneity, and at the same time, $\int_{p_{0}}^{p_{1}}Q\left(y+p-p_{0},y-p_{0}\right)dp$, is arbitrarily close to $\int_{p_{0}}^{p_{1}}L\left(y+p-p_{0},y-p_{0}\right)=\int_{p_{0}}^{p_{1}}\bar{L}\left(p,y+p-p_{0}\right)dp$, since they differ only on at most finitely many intervals of arbitrarily small length. Therefore, $\int_{p_{0}}^{p_{1}}\bar{L}\left(p,y+p-p_{0}\right)dp$ is a sharp lower bound for average CV $\int_{p_{0}}^{p_{1}}q\left(y+p-p_{0},y-p_{0}\right)\equiv\int_{p_{0}}^{p_{1}}\bar{q}\left(p,y+p-p_{0}\right)dp$.999As a simple illustration, consider a fixed $a_{1}=y-p_{0}\in\Omega_{1}$, and suppose the point $\left(k,a_{1}\right)\in A$, and $l<k<u$ for some real numbers $l,u$ belonging to the interval $\left[y,y+p_{1}-p_{0}\right]$ where the first argument of $L\left(y+p-p_{0},y-p_{0}\right)$ takes its values as $p$ varies over $\left[p_{0},p_{1}\right]$. Now suppose the lower bound function $L\left(\cdot.\cdot\right)$ satisfies

L\left(a_{0},a_{1}\right)=\left\{\begin{array}[]{l}q\left(k,a_{1}\right)\text{ if }l\leq a_{0}\leq k\\ L\left(k^{+},a_{1}\right)\text{ if }k<a_{0}\leq u\end{array}\right.

with $L\left(k^{+},a_{1}\right)<q\left(k,a_{1}\right)$. That is, $L\left(\cdot,\cdot\right)$ equals $q\left(\cdot,\cdot\right)$ at the point $\left(k,a_{1}\right)$ in $A$, is non-increasing in the first argument and is (right) discontinuous at $k$ with $L\left(k^{+},a_{1}\right)<L\left(k,a_{1}\right)$. Choose $\delta\in\left(0,u-k\right)$ and define the function $Q\left(\cdot,a_{1}\right)$ as

Q\left(a_{0},a_{1}\right)=\left\{\begin{array}[]{l}L\left(k,a_{1}\right)\text{, if }l\leq a_{0}\leq k\\ L\left(k,a_{1}\right)\times\left[1-\frac{a_{0}-k}{\delta}\right]+L\left(k^{+},a_{1}\right)\frac{a_{0}-k}{\delta}\text{ if }k<a_{0}\leq k+\delta\\ L\left(k^{+},a_{1}\right)\text{, if }k+\delta<a_{0}\leq u\end{array}\right.

Then (1) $Q\left(\cdot,a_{1}\right)$ is continuous in the first argument, since $Q\left(a_{0},a_{1}\right)\nearrow L\left(k,a_{1}\right)$ as $a_{0}\searrow k$, and $\searrow L\left(k^{+},a_{1}\right)$ as $a_{0}\nearrow\left(k+\delta\right)$, (2) at the point $\left(k,a_{1}\right)\in A$, $Q\left(k,a_{1}\right)=q\left(k,a_{1}\right)=L\left(k,a_{1}\right)$, (3) $Q\left(\cdot,a_{1}\right)$ equals $L\left(\cdot,a_{1}\right)$ except on the semi-open interval $(k,k+\delta]$ of length $\delta$, (4) $Q\left(\cdot,a_{1}\right)$ is non-increasing, and $Q\left(a_{0},\cdot\right)$. is non-decreasing since $L\left(\cdot,a_{1}\right)$ is non-increasing, and $L\left(a_{0},\cdot\right)$. is non-decreasing. Finally, $\int_{l}^{u}Q\left(a_{0},a_{1}\right)da_{0}-\int_{l}^{u}L\left(a_{0},a_{1}\right)da_{0}$ equals the area of the triangle with base $\delta$ and height $L\left(k,a_{1}\right)-L\left(k^{+},a_{1}\right)$ thus equalling $\frac{L\left(k,a_{1}\right)-L\left(k^{+},a_{1}\right)}{2}\delta$ which can be made arbitrarily close to 0 by choosing $\delta$ arbitrarily close to 0.

A symmetric line of argument implies that $\int_{p_{0}}^{p_{1}}\bar{M}\left(p,y+p-p_{0}\right)dp$ is the sharp upper bound.

5 Connection with Revealed Stochastic Preference

The welfare calculation above requires prediction of demand on a continuum of budget sets indexed by $p\in\left[p_{0},p_{1}\right]$, which is operationally difficult – if not practically impossible – to implement, using the finite-dimensional matrix equation based SRP approach. But in simple cases where there are a small, countably finite number of budget sets, and it is easy to verify the SRP conditions, a natural question is whether our shape restrictions (A) of Theorem 1 are compatible with the SRP based criterion for rationalizability; condition (B) and (C) of Theorem 1 are of course irrelevant in such cases. Below, we show that our shape restrictions (A) are in fact necessary for the SRP criterion to be satisfied.

Proposition 2

The shape restrictions (A) in Theorem 1 are necessary for McFadden Richter’s SRP conditions to hold.

Proof. Consider two price and income combinations $\left(p^{1},y\right)$ and $\left(p^{2},y\right)$. Suppose WLOG that $p^{1}\,<p^{2}$, i.e., $y-p^{1}>y-p^{2}$. Let $q\left(y,y-p^{1}\right)$, $q\left(y,y-p^{2}\right)$ denote choice probabilities of alternative 1 on the two budgets, respectively. Assume, if possible, that out shape restriction A(ii) is violated, so that $q\left(y,y-p^{1}\right)<q\left(y,y-p^{2}\right)$. We will show that this implies violation of McFadden-Richter’s SRP condition. Toward that end, consider three bundles $\left(0,y\right),\left(1,y-p^{1}\right)$ and $\left(1,y-p^{2}\right)$. Under nonsatiation in numeraire, there are 3 possible preference profiles in the population, given by (i) $\left(0,y\right)\succ\left(1,y-p^{1}\right)\succ\left(1,y-p^{2}\right)$, (ii) $\left(1,y-p^{1}\right)\succ\left(0,y\right)\succ\left(1,y-p^{2}\right)$ and (iii) $\left(1,y-p^{1}\right)\succ\left(1,y-p^{2}\right)\succ\left(0,y\right)$; assume the population proportions of these three profiles are $\left(\pi_{1},\pi_{2},\pi_{3}\right)$, respectively. Then McFadden-Richter’s SRP condition is that the matrix equation

[TABLE]

has a solution $\left(\pi_{1},\pi_{2},\pi_{3}\right)$ in the unit positive simplex. But if our hypothesis holds, i.e. $q\left(y,y-p^{1}\right)<q\left(y,y-p^{2}\right)$, then (29) implies $\pi_{2}+\pi_{3}<\pi_{3}$ i.e. $\pi_{2}<0$, a violation.

Next, consider the two price and income combinations $\left(p^{1},y^{1}\right)$ and $\left(p^{2},y^{2}\right)$ with $y^{1}<y^{2}$ and $y^{1}-p^{1}=y^{2}-p^{2}\equiv a_{1}$, say. Let $q\left(y^{1},a_{1}\right)$, $q\left(y^{2},a_{1}\right)$ denote choice probabilities of alternative 1 on the two budgets, respectively. Now suppose our shape restriction A(i) is violated, so that $q\left(y^{1},a_{1}\right)<q\left(y^{2},a_{1}\right)$. Consider the three bundles $\left(0,y^{1}\right)$, $\left(0,y^{2}\right)$ and $\left(1,a_{1}\right)$. Under nonsatiation, there are 3 possible preference profiles in the population, given by (i) $\left(0,y^{2}\right)\succ\left(0,y^{1}\right)\succ\left(1,a_{1}\right)$, (ii) $\left(0,y^{2}\right)\succ\left(1,a_{1}\right)\succ\left(0,y^{1}\right)$ and (iii) $\left(1,a_{1}\right)\succ\left(0,y^{2}\right)\succ\left(0,y^{1}\right)$; assume the population proportions of these three profiles are $\left(\pi_{1},\pi_{2},\pi_{3}\right)$, respectively. Then SRP requires a solution $\left(\pi_{1},\pi_{2},\pi_{3}\right)$ in the unit positive simplex to

[TABLE]

But $q\left(y^{1},a_{1}\right)<q\left(y^{2},a_{1}\right)$ implies that $\pi_{2}+\pi_{3}<\pi_{3}$ implying $\pi_{2}<0$, which is a violation of $\left(\pi_{1},\pi_{2},\pi_{3}\right)$ lying in the unit positive simplex.

With more budget sets, the corresponding higher dimensional matrix equations analogous to (29) and (38) quickly become operationally impractical and cumbersome, as is well-known in the literature (see introduction). In contrast, our shape-restrictions, by being global conditions on the $q\left(\cdot,\cdot\right)$ functions, remain invariant to which and how many budget sets are considered. Furthermore, we already know via Theorem 1 above, that these shape restrictions are also sufficient for rationalizability for any collection – finite or infinite – of budget sets.101010It does not seem possible to show directly, i.e. without using Theorem 1, that our shape restrictions are also sufficient for existence of admissible solutions to the analog of (29) and (38) corresponding to every arbitrary collection of budget sets. But given theorem 1, this exercise is probably of limited interest.

Appendix

1. Proof of Theorem 1

Proof. That (II) implies (I) is straightforward. In particular, letting $W_{0}^{-1}\left(\cdot,\eta\right)$ denote the inverse of $W_{0}\left(\cdot,\eta\right)$, we have that

[TABLE]

whence (B’) implies (B), (C’) implies (C), and (A’) implies (A).

We now show that (I) implies (II).

Note that (C) implies that for any $v\in\left[0,1\right]$ and $a_{1}\in\Omega_{1}$, the set $\left\{a_{0}\in\left[y_{L}\left(a_{1}\right),y_{H}\left(a_{1}\right)\right]:q\left(a_{0},a_{1}\right)\geq v\right\}$ is non-empty; for any fixed $a_{1}\in\Omega_{1}$ and for $v\in\left[0,1\right]$, define

[TABLE]

which takes values in $\left[y_{L}\left(a_{1}\right),y_{H}\left(a_{1}\right)\right]$.111111Here we are implicitly assuming that $\Omega_{0}\left(a_{1}\right)$ equals (or contains) $[y_{L}\left(a_{1}\right),y_{H}\left(a_{1}\right)]$. If however the support of price and income are discrete, then $\Omega_{0}\left(a_{1}\right)$ can be a strict subset of $[y_{L}\left(a_{1}\right),y_{H}\left(a_{1}\right)]$. Then $q\left(\cdot,\cdot\right)$ is not defined at the points ‘in between’ the points of support, and therefore, $q^{-1}\left(\cdot,a_{1}\right)$ in (39) is not well-defined. To cover this case, one can extend $q\left(\cdot,\cdot\right)$ to a continuous function $q^{c}\left(\cdot,\cdot\right)$ defined on a rectangle $\Omega^{c}$ containing $\Omega$ such that (i) $q^{c}\left(\cdot,\cdot\right)$ equals $q\left(\cdot,\cdot\right)$ on $\Omega$, (ii) $q^{c}\left(\cdot,\cdot\right)$ satisfies the same shape restrictions on $\Omega^{c}$ that are satisfied by $q\left(\cdot,\cdot\right)$ on $\Omega$, and (iii) $q^{c}\left(\cdot,\cdot\right)$ satisfies the limit conditions C of Theorem 1. In the online appendix, we provide an explicit construction of such a function. The proof of Theorem 1 then holds with $\Omega$, $\Omega_{0}\left(\cdot\right)$ and $q\left(\cdot,\cdot\right)$ equalling their corresponding extensions in the case where $\left(P,Y\right)$ have discrete support. Also, by condition (A), $q^{-1}\left(v,\cdot\right)$ must be non-decreasing.

Now, consider a random variable $V\simeq Uniform\left(0,1\right)$. Define $W_{0}\left(a_{0},V\right)\overset{defn}{=}a_{0}$ and $W_{1}\left(a_{1},V\right)\overset{defn}{=}q^{-1}\left(V,a_{1}\right)$. We will now show that $W_{0}\left(\cdot,V\right)$ and $W_{1}\left(\cdot,V\right)$ will rationalize the choice-probabilities $q\left(\cdot,\cdot\right)$, and satisfy properties (A’)-(C’) of our theorem.

To do so, first note that for any fixed $a_{1}\in\Omega_{1}$, the function $1-q\left(\cdot,a_{1}\right)$ is a continuous CDF by conditions A(i), B and C of the theorem, and $q^{-1}\left(v,a_{1}\right)$ is, by definition, the corresponding $\left(1-v\right)$th quantile. Standard properties of quantiles, c.f. Pfeiffer 1990, Sec 11a, Page 266-7, then imply the following three results (for completeness, we state and prove these results formally as a Claim below this proof):

Result (i): for any $a_{1}\in\Omega_{1}$ and $v\in\left[0,1\right]$, we must have that $q\left(q^{-1}\left(v,a_{1}\right),a_{1}\right)=v$ (Pfeiffer 1990, page 267, property 6);

Result (ii): for any $a_{1}\in\Omega_{1}$, $a_{0}\in\left[y_{L}\left(a_{1}\right),y_{H}\left(a_{1}\right)\right]$ and $v\in\left[0,1\right]$, we have $q\left(a_{0},a_{1}\right)\geq v\Leftrightarrow$ $a_{0}\leq q^{-1}\left(v,a_{1}\right)$ (Pfeiffer 1990 page 266 property 1);

Result (iii): for any $a_{1}\in\Omega_{1}$, the function $q^{-1}\left(\cdot,a_{1}\right)$ is one-to-one on $\left[0,1\right]$ (Consequence of Result (i)).

Now, for $V\simeq Uniform\left(0,1\right)$, it follows from Result (ii) that

[TABLE]

Therefore, the utility functions $W_{0}\left(a_{0},V\right)\equiv a_{0}$ and $W_{1}\left(a_{1},V\right)\equiv q^{-1}\left(V,a_{1}\right)$ with heterogeneity $V\simeq Uniform\left(0,1\right)$ rationalize the choice probabilities $q\left(\cdot,\cdot\right)$, and satisfy all the properties specified in panel (II) of Theorem 1. In particular, $W_{1}\left(a_{1},\eta\right)$ is non-decreasing in $a_{1}$ (see right after eqn. (39)), so (A’ii) holds; $W_{0}\left(a_{0},\eta\right)=a_{0}$ trivially satisfies (A’i). Next, for $v,v^{\prime}\in\left[0,1\right]$ with $v\neq v^{\prime}$, we cannot have that $q^{-1}\left(v,a_{1}\right)=q^{-1}\left(v^{\prime},a_{1}\right)$ by Result (iii); therefore,

[TABLE]

which implies property (B’). Finally,

[TABLE]

By an analogous argument, $\lim_{y\nearrow y_{H}\left(a_{1}\right),y-p=a_{1}}\Pr\left[q^{-1}\left(V,y-p\right)\geq y\right]=0$, thus satisfying (C’).

2. Proof of Results (i), (ii) and (iii) in Theorem 1

Claim: Suppose $q\left(\cdot,\cdot\right):\Omega\rightarrow\left[0,1\right]$ satisfies conditions (A), (B), (C) of Theorem 1, and $q^{-1}\left(\cdot,\cdot\right)$ is as defined in (39). Then (i) for any $a_{1}\in\Omega_{1}$ and $v\in\left[0,1\right]$, we must have that $q\left(q^{-1}\left(v,a_{1}\right),a_{1}\right)=v$; (ii) for any $v\in[0,1]$, and any $\left(a_{0},a_{1}\right)\in\Omega$, we have that $q\left(a_{0},a_{1}\right)\geq v\Longleftrightarrow a_{0}\leq q^{-1}\left(v,a_{1}\right)$; (iii) for any $a_{1}\in\Omega_{1}$, the function $q^{-1}\left(\cdot,a_{1}\right)$ is one-to-one on $\left[0,1\right]$.

Proof. **Claim (i): **Pick $a_{1}\in\Omega_{1}$. For $v=0$, we cannot have that $q\left(q^{-1}\left(v,a_{1}\right),a_{1}\right)<v$, since $q\left(\cdot,\cdot\right)$ takes values in $[0,1]$. So let $v\in\mathcal{(}0,1]$, and suppose if possible that $q\left(q^{-1}\left(v,a_{1}\right),a_{1}\right)<v$. Note that $q^{-1}\left(v,a_{1}\right)>y_{L}\left(a_{1}\right)$ because if $q^{-1}\left(v,a_{1}\right)=y_{L}\left(a_{1}\right)$, then $q\left(q^{-1}\left(v,a_{1}\right)\right)=q\left(y_{L}\left(a_{1}\right),a_{1}\right)=1\geq v$. Therefore, $q\left(q^{-1}\left(v,a_{1}\right),a_{1}\right)<v$ implies by the continuity condition (B) that there must exist $\varepsilon>0$ such that $q\left(x,a_{1}\right)<v$ for all $x\in\left[q^{-1}\left(v,a_{1}\right)-\varepsilon,q^{-1}\left(v,a_{1}\right)\right]$. But by condition (A) and the definition of $q^{-1}\left(\cdot,a_{1}\right)$ as the supremum in (39), we must have that $q\left(x,a_{1}\right)\geq v$ for all $x<q^{-1}\left(v,a_{1}\right)$, and in particular for $x\in\left[q^{-1}\left(v,a_{1}\right)-\varepsilon,q^{-1}\left(v,a_{1}\right)\right]$, which contradicts $q\left(x,a_{1}\right)<v$.

Next, for $v=1$, we cannot have that $q\left(q^{-1}\left(v,a_{1}\right),a_{1}\right)>v$, since $q\left(\cdot,\cdot\right)$ takes values in $[0,1]$. So let $v\in[0,1)$ and suppose $q\left(q^{-1}\left(v,a_{1}\right),a_{1}\right)>v$. Condition (B) and (C) imply via the intermediate value theorem that $\ni x\in\Omega_{0}\left(a_{1}\right)$, such that $q\left(x,a_{1}\right)=v$. But by hypothesis, $q\left(q^{-1}\left(v,a_{1}\right),a_{1}\right)>v=q\left(x,a_{1}\right)$, so (A) implies that $x>q^{-1}\left(v,a_{1}\right)$, which, together with $q\left(x,a_{1}\right)=v$, contradicts $q^{-1}\left(v,a_{1}\right)$ being the supremum in (39). Therefore, $q\left(q^{-1}\left(v,a_{1}\right),a_{1}\right)=v$ for all $v\in\left[0,1\right]$, and Claim (i) is proved.

Claim (ii): To prove claim (ii), note that for any $v\in[0,1]$, and any $\left(a_{0},a_{1}\right)\in\Omega$,

[TABLE]

Also, by definition of $q^{-1}\left(\cdot,a_{1}\right)$ as the supremum in (39), we have by (A) that

[TABLE]

Therefore, from (42) and (43), we have that $q\left(a_{0},a_{1}\right)\geq v\Longleftrightarrow a_{0}\leq q^{-1}\left(v,a_{1}\right)$, which proves claim (ii).

**Claim (iii): **To prove claim (iii), note that for $v,v^{\prime}\in\left[0,1\right]$ with $v\neq v^{\prime}$, we cannot have that $q^{-1}\left(v,a_{1}\right)=q^{-1}\left(v^{\prime},a_{1}\right)$; otherwise,

[TABLE]

contradicting $v\neq v^{\prime}$.

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Online Appendix

Abstract: This online appendix contains: (i) the construction of the continuous extension of the choice probability function to a domain containing $\Omega$, as mentioned in Footnote 11 in the proof of Theorem 1, and (ii) a version of Theorem 1 (called Theorem 2) with proof that does not require the limit conditions C/C’ of Theorem 1, but involves a slight strengthening of the continuity conditions B/B’.

1. Construction of Continuous Extension of Choice Probability Function

In the proof of Theorem 1, the definition of $q^{-1}\left(\cdot,a_{1}\right)$ in (39) implicitly assumes that $\Omega_{0}\left(a_{1}\right)$ equals (or contains) $[y_{L}\left(a_{1}\right),y_{H}\left(a_{1}\right)]$. If however the support of price and income are discrete, then $\Omega_{0}\left(a_{1}\right)$ can be a strict subset of $[y_{L}\left(a_{1}\right),y_{H}\left(a_{1}\right)]$. Then $q\left(\cdot,\cdot\right)$ is not defined at the points ‘in between’ the points of support, and therefore, $q^{-1}\left(\cdot,a_{1}\right)$ in (39) is not well-defined. To cover this case, one can extend $q\left(\cdot,\cdot\right)$ to a continuous function $q^{c}\left(\cdot,\cdot\right)$ defined on a rectangle $\Omega^{c}$ containing $\Omega$ such that (i) $q^{c}\left(\cdot,\cdot\right)$ equals $q\left(\cdot,\cdot\right)$ on $\Omega$, (ii) $q^{c}\left(\cdot,\cdot\right)$ satisfies the same shape restrictions on $\Omega^{c}$ that are satisfied by $q\left(\cdot,\cdot\right)$ on $\Omega$, and (iii) $q^{c}\left(\cdot,\cdot\right)$ satisfies the limit conditions C of Theorem 1. The proof of Theorem 1 then holds with $\Omega$, $\Omega_{0}\left(\cdot\right)$ and $q\left(\cdot,\cdot\right)$ equalling their corresponding extensions in the case where $\left(P,Y\right)$ have discrete support. Here we provide an explicit construction that achieves this extension.121212Alternatively, one can construct $q^{c}\left(\cdot,\cdot\right)$ as a smooth, tensor-product polynomial spline with coefficients chosen to satisfy the shape restrictions and a high enough degree to guarantee that $q^{c}\left(\cdot,\cdot\right)$ passes through the interpolating points $\left\{y^{j},y^{j}-p^{j},q\left(y^{j},y^{j}-p^{j}\right):\left(y^{j},y^{j}-p^{j}\right)\in\Omega\right\}$, along the lines of Costantini and Fontanella 1990.

Suppose the support of $\left(P,Y\right)$ is the discrete set $\bar{\Omega}=\left\{p_{1},...,p_{M}\right\}\times\left\{y_{1},...,y_{N}\right\}$, with $p_{1}<p_{2}<...<p_{M}$ and $y_{1}<y_{2}<...<y_{N}$. Suppose the choice probability $q\left(y,y-p\right)$, which is defined for $\left(p,y\right)\in\bar{\Omega}$, satisfies the shape constraints (A) of Theorem 1, i.e. $q\left(\cdot,\cdot\right)$ is non-increasing in the first and non-decreasing in the second argument. We want to construct an extension of $q\left(\cdot,\cdot\right)$, denoted by $q^{c}\left(y,y-p\right)$, which is (i) defined for all $\left(y,y-p\right)$ with $p_{1}\leq p\leq p_{M}$ and $y_{1}\leq y\leq y_{N}$, (ii) equals $q\left(y,y-p\right)$ for $\left(p,y\right)\in\bar{\Omega}$, and (iii) satisfies all three conditions A, B, C of Theorem 1. The construction proceeds in three steps.

Step 1: First, we extend $q\left(\cdot,\cdot\right)$ to the rectangular grid

[TABLE]

To do this, define $\tilde{q}\left(\cdot,\cdot\right):T\rightarrow\left[0,1\right]$ as:

[TABLE]

where $\lambda\in\left[0,1\right]$ is arbitrary, and for any $\left(y,y-p\right)\in T,$

[TABLE]

Note that $\tilde{q}\left(\cdot,\cdot\right)$, which is well defined on all of $T$, satisfies the shape constraints (A) of Theorem 1. This is because the set $\left\{\left(p^{\prime},y^{\prime}\right)\in\bar{\Omega}:\text{ }y^{\prime}\geq y,\text{ }y^{\prime}-p^{\prime}\leq y-p\right\}$ is decreasing in $y$ for fixed $y-p$, and increasing in $y-p$ for fixed $y$, so $\bar{L}\left(\cdot,\cdot\right)$ is decreasing in the first and increasing in the second argument; an analogous argument works for $\bar{U}\left(\cdot,\cdot\right)$. Furthermore, if $\left(p,y\right)\in\bar{\Omega}$, then

[TABLE]

whence the shape restrictions on $q\left(\cdot,\cdot\right)$ imply that $\bar{L}\left(y,y-p\right)=q\left(y,y-p\right)=\bar{U}\left(y,y-p\right)$, and hence $\tilde{q}\left(y,y-p\right)=q\left(y,y-p\right)$. Note, however, that $\tilde{q}\left(\cdot,\cdot\right)$ does not satisfy the continuity condition (B) and the limit conditions (C) of Theorem 1.

Step 2: The second step is to extend $\tilde{q}\left(\cdot,\cdot\right)$ to a continuous function $q^{c}\left(\cdot,\cdot\right)$ on the entire rectangle $\left[y_{1},y_{N}\right]\times\left[y_{1}-p_{M},\text{ }y_{N}-p_{1}\right]$, satisfying the shape constraints (A) of theorem 1, while also satisfying the interpolation conditions $q^{c}\left(y,y-p\right)=q\left(y,y-p\right)$ for $\left(p,y\right)\in\bar{\Omega}$. This is done using bilinear shape-preserving interpolation as follows.

Recall $y_{1}<y_{2}<...<y_{N}$, and define $w_{1}<w_{2}<...<w_{J}$ with $J\leq MN$ to be the ordered values of the set $\left\{y_{1}-p_{1},...,y_{1}-p_{M},...,y_{N}-p_{1},...,y_{N}-p_{M}\right\}$. We can have $J<MN$ if for some $\left(j,k\right)\neq\left(l,m\right)$, it holds that $y_{j}-p_{k}=y_{l}-p_{m}$. For each $i=1,...N-1$, $j=1,...,J-1$, and for $\left(y,y-p\right)\in[y_{i},y_{i+1}]\times[w_{j},w_{j+1}]$, let

[TABLE]

where $\tilde{q}\left(\cdot,\cdot\right)$ is defined in (44).

Step 3: The last step in the construction is to extend $q^{c}\left(\cdot,\cdot\right)$ beyond $\left[y_{1},y_{N}\right]\times\left[y_{1}-p_{M},\text{ }y_{N}-p_{1}\right]$ to ensure that the limit conditions (C) of Theorem 1 are satisfied. To do this, choose any pair of real numbers $y_{L},y_{H}$ s.t. $y_{L}<y_{1}$ and $y_{H}>y_{N}$. Let

[TABLE]

For any $w\in\left[y_{1}-p_{M},\text{ }y_{N}-p_{1}\right]$, define

[TABLE]

That is for $y\in\left[y_{L},y_{1}\right]$, $q^{c}\left(y,w\right)$ is the negatively sloped straight line joining $q^{c}\left(y_{1},w\right)$ to $1\equiv q^{c}\left(y_{L},w\right)$, and for $y\in\left[y_{N}-p_{1},y_{H}\right]$, $q^{c}\left(y,w\right)$ is the negatively sloped straight line joining $q^{c}\left(y_{N}-p_{1},w\right)$ to $0\equiv q^{c}\left(y_{H},w\right)$.

Proof that $q^{c}\left(\cdot,\cdot\right):D\rightarrow\left[0,1\right]$ equals $q\left(y,y-p\right)$ for $\left(p,y\right)\in\bar{\Omega}$ and satisfies conditions (A), (B) ,(C) of Theorem 1: To see the first assertion, observe that at the grid points $y=y_{i}$, $y-p=w_{j}$, we get from (47) that $\alpha_{i}\left(y\right)=0=\beta_{j}\left(w\right)$, so that $q^{c}\left(y,w\right)=\tilde{q}\left(y_{i},w_{j}\right)$. We have already seen that for $\left(p,y\right)\in\bar{\Omega}$, $q\left(y,y-p\right)=\tilde{q}\left(y,y-p\right)$. Now, since $\left(p,y\right)\in\bar{\Omega}$ implies $\left(y,y-p\right)\in T$, putting these two conclusions together, we get that for $\left(p,y\right)\in\bar{\Omega}$, it holds that $q^{c}\left(y,y-p\right)=\tilde{q}\left(y,y-p\right)=q\left(y,y-p\right)$.

As for the continuity condition (B) of Theorem 1, observe that holding fixed $w$, as $y\in[y_{i},y_{i+1})\nearrow y_{i+1}-$, we have that $\alpha_{i}\left(y\right)\nearrow 1$ whence from (47), it follows that

[TABLE]

On the other hand, for the same $w$ and for $y\in[y_{i+1},y_{i+2})$, we have that $\alpha_{i}\left(y\right)=\frac{y-y_{i+1}}{y_{i+2}-y_{i+1}}$ which at $y=y_{i+1}\in[y_{i+1},y_{i+2})$ equals [math], whence from (47) with $i$ replaced by $i+1$ and $i+1$ replaced by $i+2$, we get

[TABLE]

which equals (49). Therefore, for fixed $w$, $\tilde{q}\left(y,w\right)$ is simply a piecewise linear function of $y$ joined at the end-points $y_{2},...,y_{N-1}$, and therefore continuous in $y$ for $y\in\left[y_{1},y_{N}\right]$. For $y\in\left[y_{L},y_{H}\right]\backslash\left[y_{1},y_{N}\right]$, continuity is obvious from (48) and the fact that $\lim_{y\nearrow y_{1}-}q^{c}\left(y,w\right)=q^{c}\left(y_{1},w\right)=\lim_{y\searrow y_{1}+}q^{c}\left(y,w\right)$ and $\lim_{y\nearrow\left(y_{N}-p_{1}\right)-}q^{c}\left(y,w\right)=q^{c}\left(y_{N}-p_{1},w\right)=\lim_{y\searrow\left(y_{N}-p_{1}\right)+}q^{c}\left(y,w\right)$. An analogous argument shows that $q^{c}\left(y,w\right)$ is also continuous in $w$ for fixed $y$ (this property is not needed to prove Theorem 1 but is used in Theorem 2, the alternative version of Theorem 1 without the limiting condition, which appears below).

The limiting conditions (C) of Theorem 1 are satisfied, since (48) implies that $q^{c}\left(y_{L},w\right)=1$ and $q^{c}\left(y_{H},w\right)=0$ for each $w\in\left[y_{1}-p_{M},\text{ }y_{N}-p_{1}\right]$.

Finally, to see that the shape restrictions (A) of Theorem 1 hold on $\left[y_{1},y_{N}\right]\times\left[y_{1}-p_{M},\text{ }y_{N}-p_{1}\right]$, note from (47) that the coefficient of $y$ in $q^{c}\left(y,w\right)$ equals

[TABLE]

Similarly, the coefficient of $w$ in $q^{c}\left(y,w\right)$ equals

[TABLE]

From (48) it follows that the shape restrictions also hold on $\left[y_{L},y_{1}\right]\times\left[y_{1}-p_{M},\text{ }y_{N}-p_{1}\right]$ and on $\left[y_{N},y_{H}\right]\times\left[y_{1}-p_{M},\text{ }y_{N}-p_{1}\right]$, and thus condition (A) of Theorem 1 holds on all of $\left[y_{L},y_{H}\right]\times\left[y_{1}-p_{M},\text{ }y_{N}-p_{1}\right]$.

Thus $q^{c}\left(\cdot,\cdot\right)$ satisfies all three conditions of Theorem 1.

2. Main Result without condition (C/C’)

The following is a version of Theorem 1 that does not require the technical conditions C and C’ of Theorem 1, but involves a slight strengthening of the technical condition B. The proof of this version is considerably longer than that of Theorem 1. The proof works by constructing an extension $Q\left(\cdot,\cdot\right)$ of $q\left(\cdot,\cdot\right)$ which satisfies properties (A)-(C) of Theorem 1 although $q\left(\cdot,\cdot\right)$ itself does not satisfy property (C).131313The case where $\left(P,Y\right)$ have a discrete support is handled in exactly the same way as in Theorem 1 with two small modifications: (a) Step 3 in the construction immediately above is not required, and (b) continuity of $q^{c}\left(\cdot,\cdot\right)$ in the second argument is guaranteed by the construction in Step 2.

Suppose the support of price $P$ and income $Y$ in the population is $[p_{l},p_{u}]\times\left[y_{l},y_{u}\right]$. Correspondingly, the support of $Y-P$ is $\Omega_{1}\overset{defn}{=}\left[y_{l}-p_{u},y_{u}-p_{l}\right]$. Pick any $a_{1}\in\Omega_{1}$. Corresponding to $Y-P=a_{1}$, the support of $Y=a_{1}+P$ is therefore

[TABLE]

Note that by definition, $L\left(\cdot\right)$ and $U\left(\cdot\right)$ are non-decreasing and continuous. Let $\Omega=\cup_{a_{1}\in\Omega_{1}}\cup_{a_{0}\in\Omega_{0}\left(a_{1}\right)}\left\{a_{0},a_{1}\right\}$.

Theorem 2

For binary choice under general heterogeneity, the following two statements are equivalent:

(I) The choice probabilities $q\left(y,y-p\right)$, defined above, satisfy that (A) $q\left(\cdot,y-p\right)$ is non-increasing, and $q\left(y,\cdot\right)$ is non-decreasing; (B) $q\left(\cdot,\cdot\right)$ is continuous.

(II) There exists a pair of utility functions $W_{0}\left(\cdot,\eta\right)$ and $W_{1}\left(\cdot,\eta\right)$, where the first argument denotes the amount of numeraire, and $\eta$ denotes unobserved heterogeneity, and a distribution $G\left(\cdot\right)$ of $\eta$ such that for any $\left(y-p\right)\in\Omega_{1}$ and correspondingly $y\in\Omega_{0}\left(y-p\right)$,

[TABLE]

where (A’) for each fixed $\eta$, $W_{0}\left(\cdot,\eta\right)$ and $W_{1}\left(\cdot,\eta\right)$ are non-decreasing; (B’) for each fixed $\eta$, $W_{0}\left(\cdot,\eta\right)$ and $W_{1}\left(\cdot,\eta\right)$ are continuous, and for any $\left(a_{0},a_{1}\right)\in\Omega$, it holds that $\int 1\left\{W_{0}\left(a_{0},\eta\right)\leq W_{1}\left(a_{1},\eta\right)\right\}dG\left(\eta\right)$ is continuous in $\left(a_{0},a_{1}\right)$.

Discussion of assumptions: Relative to Theorem 1, conditions (C/C’) are omitted, and condition (B/B’) is strengthened to continuity in both arguments. Note that under monotonicity in any one argument, the joint continuity of $q\left(\cdot,\cdot\right)$ is equivalent to coordinate wise continuity c.f. Kruse and Deely 1969.

To prove Theorem 2, we will utilize several lemmas.

Lemma 1** (Apostol, 1974, Ex 4.19)**

Suppose $r\left(\cdot\right):\left[c,b\right]\rightarrow\mathbb{R}$, is continuous on $\left[c,b\right]$. For $x\in\left[c,b\right]$, define $g\left(x\right)=\sup\left\{r\left(z\right):x\leq z\leq b\right\}$, and $h\left(x\right)=\sup\left\{r\left(z\right):c\leq z\leq x\right\}$. Then $g\left(\cdot\right)$ and $h\left(\cdot\right)$ are continuous on $\left[c,b\right]$.

Proof of Lemma 2. Fix any $x\in\left[c,a_{1}\right]$.

First, suppose $g\left(x\right)>r\left(x\right)$. Choose $\varepsilon=g\left(x\right)-r\left(x\right)>0$. Now by continuity of $r\left(\cdot\right)$, there must exist $\delta>0$ s.t. for any $z\in[x-\delta,x+\delta]$, we have that $r\left(z\right)<r\left(x\right)+\varepsilon=r\left(x\right)+g\left(x\right)-r\left(x\right)=g\left(x\right)$. Therefore, $\sup\left\{r\left(z\right):x-\delta\leq z\leq x+\delta\right\}<g\left(x\right)$. Therefore, $g\left(x-\delta\right)=g\left(x\right)=g\left(x+\delta\right)$, implying continuity of $g\left(\cdot\right)$ at $x$.

Next, suppose the sup is at $x$, i.e. $g\left(x\right)=r\left(x\right)$. By continuity, for any $\varepsilon>0$, there exists $\delta>0$, s.t. for all $u\in\left[x-\delta,x+\delta\right]$, we have that $r\left(x\right)+\varepsilon\geq r\left(u\right)\geq r\left(x\right)-\varepsilon$. For $u\in\left[x,x+\delta\right]$, $g\left(u\right)=\sup\left\{r\left(z\right):u\leq z\leq a_{1}\right\}\geq r\left(u\right)\geq r\left(x\right)-\varepsilon=g\left(x\right)-\varepsilon$, since $g\left(x\right)=r\left(x\right)$, by assumption. But $g\left(u\right)\leq g\left(x\right)$ by definition. Therefore, for all $u\in\left[x,x+\delta\right]$, we have that $g\left(x\right)\geq g\left(u\right)>g\left(x\right)-\varepsilon$. Next, for all $u\in\left[x-\delta,x\right]$, $r\left(u\right)\leq r\left(x\right)+\varepsilon=g\left(x\right)+\varepsilon$ implying

[TABLE]

Thus for all $u\in\left[x-\delta,x+\delta\right]$, we have that $g\left(x\right)+\varepsilon\geq g\left(u\right)>g\left(x\right)-\varepsilon$. Therefore, $g\left(\cdot\right)$ is continuous at $x$.

An exactly similar proof works for $h\left(x\right)=\sup\left\{r\left(z\right):c\leq z\leq x\right\}$.

Lemma 2** (Taylor, 1955, Chap 15.7, Theorem VII)**

Suppose the function $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is continuous, and the function $g\left(\cdot\right):\mathbb{R}\rightarrow\mathbb{R}$ is continuous w.r.t. the $L1$-norm. Then the function $h:\mathbb{R}\rightarrow\mathbb{R}$ defined as $h\left(x\right)=f\left(g\left(x\right),x\right)$ is continuous on $\mathbb{R}$.

Proof of Lemma 3. Pick any $x_{0}\in\mathbb{R}$, and $\varepsilon>0$. Continuity of $f\left(\cdot,\cdot\right)$ implies that there exists $\delta>0$ s.t. $\left|f\left(g\left(x\right),x\right)-f\left(g\left(x_{0}\right),x_{0}\right)\right|\leq\varepsilon$, whenever $\left\|\left(g\left(x\right),x\right)-\left(g\left(x_{0}\right),x_{0}\right)\right\|\leq\delta$. Now, continuity of $g\left(\cdot\right)$ implies that given the above $\delta>0$, there exists $\delta_{1}>0$ s.t. $\left|g\left(x\right)-g\left(x_{0}\right)\right|\leq\delta/2$ whenever $\left|x-x_{0}\right|\leq\delta_{1}$. Choose $\delta^{\ast}=\min\left\{\delta/2,\delta_{1}\right\}$. Then whenever $\left|x-x_{0}\right|\leq\delta^{\ast}$, we have that $\left|g\left(x\right)-g\left(x_{0}\right)\right|\leq\delta/2$ and $\left|x-x_{0}\right|\leq\delta/2$, and thus $\left\|\left(g\left(x\right),x\right)-\left(g\left(x_{0}\right),x_{0}\right)\right\|=\left|g\left(x\right)-g\left(x_{0}\right)\right|+\left|x-x_{0}\right|\leq\delta$, and therefore,

[TABLE]

Construction: The following construction will be used to prove the theorem. Pick $a_{1}\in\Omega_{1}$. Recall the definitions $L\left(a_{1}\right)\equiv\max\left\{p_{l}+a_{1},y_{l}\right\},$ and $U\left(a_{1}\right)\equiv\min\left\{p_{u}+a_{1},y_{u}\right\}$. Let $a_{0L}$, $a_{0H}$ be any pair of real numbers satisfying $a_{0L}<y_{l}$ and $a_{0H}>y_{u}$. For any $a_{0}<L\left(a_{1}\right)$ and $a_{0}>U\left(a_{1}\right)$, respectively, define

[TABLE]

Note that as $a_{0}$ decreases with $a_{1}$ fixed, or $a_{1}$ increases with $a_{0}$ fixed, the set $\left[a_{0},L\left(a_{1}\right)\right]$ expands, and therefore the sup over it weakly increases; thus $H\left(\cdot,a_{1}\right)$ is non-increasing and $H\left(a_{0},\cdot\right)$ is non-decreasing. Similarly, $h\left(\cdot,a_{1}\right)$ is non-increasing and $h\left(a_{0},\cdot\right)$ is non-decreasing. Now, define the function $Q\left(\cdot,\cdot\right):\left[a_{0L},a_{0H}\right]\mathbb{\rightarrow}\left[0,1\right]$ as follows. For any $a_{1}\in\Omega_{1}$,

[TABLE]

Claim 1

Suppose $q\left(\cdot,\cdot\right)$ satisfies (A) and (B) of Theorem 2. Then the function $Q\left(\cdot,\cdot\right)$ defined in (50) satisfies the following properties:

(1) $Q\left(\cdot,a_{1}\right)$ is non-increasing, and $Q\left(a_{0},\cdot\right)$ is non-decreasing for all $\left(a_{0},a_{1}\right)\in\left[a_{0L},a_{0H}\right]\times\Omega_{1}$

(2) $Q\left(\cdot,\cdot\right)$ is continuous in each argument, holding the other argument fixed.

(3) For any $a_{1}\in\Omega_{1}$, there exist real numbers $a_{0L}$ and $a_{0H}$ such that $\lim_{a_{0}\searrow a_{0L}}Q\left(a_{0},a_{1}\right)=1$ and $\lim_{a_{0}\nearrow a_{0H}}Q\left(a_{0},a_{1}\right)=0$.

Proof. Property (3) is obvious because $Q\left(a_{0L},a_{1}\right)=1$ and $Q\left(a_{0H},a_{1}\right)=0$, by construction. To show (1) and (2), fix $a_{1}\in\Omega_{1}$. Since $q\left(\cdot,\cdot\right)$ satisfies (A) and (B) on $a_{0}\in\left[L\left(a_{1}\right),U\left(a_{1}\right)\right]$, we only need to establish the properties over the range $a_{0}<L\left(a_{1}\right)$ and $a_{0}>U\left(a_{1}\right)$.

Property (1): First, we show that the shape restrictions hold for $Q\left(\cdot,\cdot\right)$. We have already noted that $H\left(\cdot,a_{1}\right)$ and $h\left(\cdot,a_{1}\right)$ are both non-increasing; further since $H\left(y_{l},a_{1}\right)\leq 1$ and $h\left(y_{u},a_{1}\right)\geq 0$, we have that $H\left(y_{l},a_{1}\right)+\left(1-H\left(y_{l},a_{1}\right)\right)\frac{y_{l}-a_{0}}{y_{l}-a_{0L}}$ is non-increasing in $a_{0}$ for $a_{0L}\leq a_{0}<y_{l}$, and $\frac{a_{0H}-a_{0}}{a_{0H}-y_{u}}h\left(y_{u},a_{1}\right)$ is non-increasing in $a_{0}$ for $y_{u}<a_{0}\leq a_{0H}$. Thus $Q\left(a_{0},a_{1}\right)$ is non-increasing in $a_{0}$ for all $a_{0}<L\left(a_{1}\right)$ and $a_{0}>U\left(a_{1}\right)$.

Next, pick $a_{0}\in\left[a_{0L},a_{0H}\right]$, and consider monotonicity of $Q\left(a_{0},\cdot\right)$. Let $a_{1}^{1},a_{1}^{2}\in\Omega_{1}$ with $a_{1}^{1}<a_{1}^{2}$, implying $L\left(a_{1}^{1}\right)\leq L\left(a_{1}^{2}\right)$ and $U\left(a_{1}^{1}\right)\leq U\left(a_{1}^{2}\right)$. Now there are 10 cases to consider, labelled (a)-(j) below, depending on the ordering of $L\left(a_{1}^{2}\right)$ and $U\left(a_{1}^{1}\right)$, and where $a_{0}$ lies. Case (a) $a_{0L}\leq a_{0}<y_{l}$, then

[TABLE]

Case (b) $y_{l}\leq a_{0}\leq L\left(a_{1}^{1}\right)$, i.e. $\left[a_{0},L\left(a_{1}^{1}\right)\right]\sqsubseteq\left[a_{0},L\left(a_{1}^{2}\right)\right]$, and so $H\left(a_{0},a_{1}^{1}\right)\leq H\left(a_{0},a_{1}^{2}\right)$, and therefore, $Q\left(a_{0},a_{1}^{1}\right)=H\left(a_{0},a_{1}^{1}\right)\leq H\left(a_{0},a_{1}^{2}\right)=Q\left(a_{0},a_{1}^{2}\right)$. Case (c): $y_{u}<a_{0}\leq a_{0H}$, and Case (d) $U\left(a_{1}^{2}\right)<a_{0}\leq y_{u}$, the proofs are exactly analogous to respectively (a) and (b) above.

So we are left with the following cases, where Cases (e)-(g) correspond to $U\left(a_{1}^{1}\right)<L\left(a_{1}^{2}\right)$, and (h)-(j) to $U\left(a_{1}^{1}\right)\geq L\left(a_{1}^{2}\right)$.

For Case (e) $L\left(a_{1}^{1}\right)\leq a_{0}\leq U\left(a_{1}^{1}\right)<L\left(a_{1}^{2}\right)$, since $L\left(a_{1}^{1}\right)<a_{0}<L\left(a_{1}^{2}\right)$, by continuity of $L\left(\cdot\right)$ and the intermediate value theorem, there exists $c\in\left[a_{1}^{1},a_{1}^{2}\right]$ s.t. $a_{0}=L\left(c\right)$. Therefore,

[TABLE]

where $\overset{(1)}{\leq}$ holds because $a_{1}^{1}\leq c$ and condition (A) of Theorem 1, and $\overset{(2)}{\leq}$ holds by definition of $\sup$. Next, suppose Case (f) $L\left(a_{1}^{1}\right)\leq U\left(a_{1}^{1}\right)\leq a_{0}<L\left(a_{1}^{2}\right)\leq U\left(a_{1}^{2}\right)$, then by continuity of $L\left(\cdot\right)$ and the intermediate value theorem, there exists $c\in\left[a_{1}^{1},a_{1}^{2}\right]$ s.t. $a_{0}=L\left(c\right)$; and by continuity of $U\left(\cdot\right)$ and the intermediate value theorem, there exists $d\in\left[a_{1}^{1},a_{1}^{2}\right]$ s.t. $a_{0}=U\left(d\right)$, with $d\leq c$. Then

[TABLE]

Next, for Case (g) $L\left(a_{1}^{1}\right)\leq U\left(a_{1}^{1}\right)<L\left(a_{1}^{2}\right)\leq a_{0}\leq U\left(a_{1}^{2}\right)$, using continuity of $U\left(\cdot\right)$ and the intermediate value theorem, we have $a_{0}=U\left(c\right)$ for some $c\in\left[a_{1}^{1},a_{1}^{2}\right]$ so that

[TABLE]

Next, consider Case (h) $L\left(a_{1}^{1}\right)\leq a_{0}\leq L\left(a_{1}^{2}\right)\leq U\left(a_{1}^{1}\right)$. Since $L\left(a_{1}^{1}\right)\leq a_{0}\leq L\left(a_{1}^{2}\right)$, by continuity and the intermediate value theorem, we have that $a_{0}=L\left(c\right)$ for some $c\in\left[a_{1}^{1},a_{1}^{2}\right]$, whence we have

[TABLE]

Next, if Case (i) $L\left(a_{1}^{1}\right)\leq L\left(a_{1}^{2}\right)\leq a_{0}\leq U\left(a_{1}^{1}\right)$, we have that $Q\left(a_{0},a_{1}^{1}\right)=q\left(a_{0},a_{1}^{1}\right)\leq q\left(a_{0},a_{1}^{2}\right)=Q\left(a_{0},a_{1}^{2}\right)$.

Finally, for the Case (j) $L\left(a_{1}^{1}\right)\leq L\left(a_{1}^{2}\right)\leq U\left(a_{1}^{1}\right)\leq a_{0}\leq U\left(a_{1}^{2}\right)$,  the same argument as in (g) applies.

This establishes the requisite shape restrictions, i.e. Property (1).

Property (2): First, consider continuity of $Q\left(\cdot,a_{1}\right)$. Note that $H\left(y_{l},a_{1}\right)+\left(1-H\left(y_{l},a_{1}\right)\right)\frac{y_{l}-a_{0}}{y_{l}-a_{0L}}$ is obviously continuous at $a_{0}$ for $a_{0L}\leq a_{0}<y_{l}$; next, at $a_{0}=y_{l}$, $Q\left(a_{0},a_{1}\right)=H\left(y_{l},a_{1}\right)+\left(1-H\left(y_{l},a_{1}\right)\right)\frac{y_{l}-y_{l}}{y_{l}-a_{0L}}=H\left(y_{l},a_{1}\right)$, while at $a_{0}=L\left(a_{1}\right)>y_{l}$,

[TABLE]

and thus $Q\left(\cdot,a_{1}\right)$ is continuous at $a_{0}=y_{l}$ and at $a_{0}=L\left(a_{1}\right)$. Finally, if $a_{0}\in(y_{l},L\left(a_{1}\right))$, then we can have $L\left(x\right)\in\left[a_{0},L\left(a_{1}\right)\right]$ only if $L\left(x\right)>y_{l}$ in which case $L\left(x\right)=x+p_{l}$ and thus $q\left(L\left(x\right),x\right)=q\left(x+p_{l},x\right)$ implying

[TABLE]

By Lemma 3, $q\left(x+p_{l},x\right)$ is continuous in $x$, and therefore, by Lemma 2, $Q\left(a_{0},a_{1}\right)$ is continuous in $a_{0}$ for fixed $a_{1}$. Thus we have that $Q\left(\cdot,a_{1}\right)$ is continuous on all of $\left[a_{0L},U\left(a_{1}\right)\right]$. An exactly analogous argument works for $a_{0}>U\left(a_{1}\right)$.

Finally, consider continuity in $a_{1}$ for fixed $a_{0}$. If (a) $a_{1}\leq y_{l}-p_{l}$, then $L\left(a_{1}\right)=y_{l}$, and therefore,

[TABLE]

which does not depend on $a_{1}$ and therefore trivially continuous in $a_{1}$. So consider (b) $a_{1}>y_{l}-p_{l}$, so that $L\left(a_{1}\right)=a_{1}+p_{l}$. Therefore, at $a_{0}=y_{l}$, $H\left(a_{0},a_{1}\right)=H\left(y_{l},a_{1}\right)$ equals

[TABLE]

The last equality $\overset{(2)}{=}$ follows because $\underset{=\max\left\{p_{l}+x,y_{l}\right\}}{\underbrace{L\left(x\right)}}\in\left[y_{l},a_{1}+p_{l}\right]$ if and only if $x\in\left[y_{l}-p_{u},a_{1}\right]$. Now, since $L\left(\cdot\right)$ is continuous, and so is $q\left(\cdot,\cdot\right)$, the function $x\mapsto q\left(L\left(x\right),x\right)$ is continuous in $x$ (see Lemma 3 above), and therefore, it follows from Lemma 2 that $\sup\left\{q\left(L\left(x\right),x\right):x\in\left[y_{l}-p_{u},a_{1}\right]\right\}$ is continuous in $a_{1}$. In particular, as $a_{1}\searrow\left(y_{l}-p_{l}\right)_{+}$, $L\left(a_{1}\right)$ approaches $y_{l}$ and so (5) tends to (52).

Finally,** **for any $a_{0}>y_{l}$, (recall $a_{1}>y_{l}-p_{l}$, so that $L\left(a_{1}\right)=a_{1}+p_{l}$), we have that

[TABLE]

which is continuous in $a_{1}$ by Lemma 2 and 3. Exactly analogous arguments hold for (a’) $a_{1}\geq y_{u}-p_{u}$ and (b’) $a_{1}<y_{u}-p_{u}$ respectively. Thus, we have that $Q\left(a_{0},\cdot\right)$ is continuous at each $a_{0}$.

Lemma 3

Suppose the function $Q\left(\cdot,\cdot\right):\left[a_{0L},a_{0H}\right]\times\Omega_{1}\sqsubseteq\mathbb{R}^{2}\rightarrow\left[0,1\right]$ satisfies on its domain that (1) $Q\left(\cdot,a_{1}\right)$ is non-increasing, and $Q\left(a_{0},\cdot\right)$ is non-decreasing; (2) $Q\left(\cdot,a_{1}\right)$ is continuous, and (3) for any $a_{1}\in\Omega_{1}$, $\lim_{a_{0}\searrow a_{0L}}Q\left(a_{0},a_{1}\right)=1$ and $\lim_{a_{0}\nearrow a_{0H}}Q\left(a_{0},a_{1}\right)=0$. For any fixed $a_{1}\in\Omega_{1}$, define for each $u\in\left[0,1\right]$,

[TABLE]

Then we must have that $Q\left(Q^{-1}\left(v,a_{1}\right),a_{1}\right)=v$, for any $v\in[0,1]$.

Proof of Lemma 4. Since $Q\left(\cdot,\cdot\right)$ satisfies the same properties as $q\left(\cdot,\cdot\right)$ of Theorem 1 (A)-(C), the proof of this lemma is identical to the proof of Lemma 1 used to prove Theorem 1.

Proof of Theorem 2. That (II) implies (I) is straightforward, since

[TABLE]

whence (B’) implies (B), and (A’) implies (A).

We now show that (I) implies (II). To do so, recall the definition of $Q^{-1}\left(v,a_{1}\right)$ in (54). Now, consider a random variable $V\simeq Uniform\left(0,1\right)$. Define $W_{0}\left(a_{0},V\right)\overset{defn}{=}a_{0}$ and $W_{1}\left(a_{1},V\right)\overset{defn}{=}Q^{-1}\left(V,a_{1}\right)$. We will now show that for $y-p\in\Omega_{1}$ and correspondingly, $y\in\left[L\left(y-p\right),U\left(y-p\right)\right]$, the functions $W_{0}\left(y,V\right)$ and $W_{1}\left(y-p,V\right)$ will rationalize the choice-probabilities $q\left(y,y-p\right)$.

To prove this, note that for any $v\in[0,1]$, and $\left(a_{0},a_{1}\right)\in\Omega$,

[TABLE]

Also, by definition of $Q^{-1}\left(\cdot,a_{1}\right)$ as the supremum in (54), we have that

[TABLE]

Therefore, by (55) and (56), we have that $Q\left(a_{0},a_{1}\right)\geq v\Longleftrightarrow a_{0}\leq Q^{-1}\left(v,a_{1}\right)$. Thus, for $V\simeq U\left(0,1\right)$, it follows that

[TABLE]

Recall that for $y-p\in\Omega_{1}$ and correspondingly $y\in\left[L\left(y-p\right),U\left(y-p\right)\right]$, we have that $Q\left(y,y-p\right)=q\left(y,y-p\right)$ by definition. Therefore, it follows from (57) that the utility functions $W_{0}\left(y,V\right)\equiv y$ and $W_{1}\left(y-p,V\right)\equiv Q^{-1}\left(V,y-p\right)$ with heterogeneity $V\simeq Uniform\left(0,1\right)$ rationalize the choice probability function $q\left(\cdot,\cdot\right)$ on its domain.

Next, note that $Q^{-1}\left(v,a_{1}^{\prime}\right)\leq Q^{-1}\left(v,a_{1}\right)$ whenever $a_{1}^{\prime}<a_{1}$. To see this, suppose $a_{1}>a_{1}^{\prime}$ and yet $Q^{-1}\left(v,a_{1}\right)<Q^{-1}\left(v,a_{1}^{\prime}\right)$. Choose $c$ s.t. $Q^{-1}\left(v,a_{1}\right)<c<Q^{-1}\left(v,a_{1}^{\prime}\right)$. Then by conclusion (i) of the previous lemma and by definition (54) of $Q^{-1}\left(v,\cdot\right)$, we must have $Q\left(c,a_{1}\right)<v\leq Q\left(c,a_{1}^{\prime}\right)$. But since $a_{1}>a_{1}^{\prime}$, this contradicts conclusion (1) of the Claim 1.

Next, it follows from (A) and (B) that $Q^{-1}\left(v,\cdot\right)$ is continuous. To see this, fix $v\in\left[0,1\right]$, and suppose to the contrary that $Q^{-1}\left(v,\cdot\right)$ is discontinuous at $a_{1}$; suppose there exists $\epsilon>0$ such that for any $\delta>0$, $Q^{-1}\left(v,a_{1}\right)>Q^{-1}\left(v,a_{1}^{\prime}\right)+\varepsilon$ for all $a_{1}^{\prime}$ satisfying $a_{1}^{\prime}<a_{1}<a_{1}^{\prime}+\delta$. For any such $a_{1}^{\prime}$ satisfying $Q^{-1}\left(v,a_{1}\right)>Q^{-1}\left(v,a_{1}^{\prime}\right)+\varepsilon$, it follows from the definition of $Q^{-1}\left(\cdot,a_{1}^{\prime}\right)$ that there exists $\varepsilon^{\prime}=\varepsilon^{\prime}\left(\varepsilon\right)>0$ s.t.

[TABLE]

Inequality $(1)$ follows because $Q\left(Q^{-1}\left(v,a_{1}^{\prime}\right),a_{1}^{\prime}\right)\leq Q\left(Q^{-1}\left(v,a_{1}\right),a_{1}^{\prime}\right)$ since $Q^{-1}\left(v,a_{1}\right)>$ $Q^{-1}\left(v,a_{1}^{\prime}\right)$, and if $Q\left(Q^{-1}\left(v,a_{1}^{\prime}\right),a_{1}^{\prime}\right)=Q\left(Q^{-1}\left(v,a_{1}\right),a_{1}^{\prime}\right)$ with $Q^{-1}\left(v,a_{1}\right)>$ $Q^{-1}\left(v,a_{1}^{\prime}\right)+\varepsilon$, then that contradicts the definition of $Q^{-1}\left(v,a_{1}^{\prime}\right)$ as the sup. Therefore, it follows from (58) that

[TABLE]

which contradicts that $Q\left(\cdot,\cdot\right)$ is continuous in its second argument for fixed value of its first argument (see property (2) in Claim 1 above), since $a_{1}^{\prime}$ can be made arbitrarily close to $a_{1}$ by choosing $\delta$ small enough.

Finally, $W_{0}\left(y,\eta\right)=y$ is obviously continuous and strictly increasing in $y$, thus (A’) holds. Finally, (B) ensures that (B’) is satisfied.

Reference

1. Apostol, Tom M. (1974): Mathematical Analysis, Addison-Wesley.

2. Kruse, R.L. and Deely, J.J. 1969. Joint continuity of monotonic functions. The American Mathematical Monthly, 76(1), pp.74-76.

3. Taylor, Angus E. (1955): Advanced Calculus, Ginn and Company.