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Counterfactual and Welfare Analysis with an Approximate Model
Models are generally viewed as approximations. A common intuition in empirical work is that conclusions of a model are robust to “small” amounts of approximation error. Unfortunately, this intuition does not apply to many standard frameworks. For example, when performing a revealed preference analysis varian1982nonparametric, if a dataset is inconsistent with a model, then counterfactual predictions described by certain inequalities cross. Thus, if there is any violation of the model (no matter how “small”), then the model fails to generate coherent counterfactual or welfare statements.\footnote{Related concerns have been raised in the econometric literature on partial identification ponomareva2011misspecification,muller2016credibility.}
Alternatively, one can formally acknowledge approximation error throughout the analysis. Rather than treat approximation error as nonexistent, one can place restrictions on the magnitude of the approximation error. This paper does so for counterfactual and welfare analysis, with the following assumption on this magnitude.
This assumption is a natural extension of the standard approach to generate counterfactual predictions that assumes that both the observed data and counterfactual predictions are consistent with a model. We present a framework in which a model can be used even though it is not exactly consistent with observed data. In particular, this paper assumes that the approximation error of the model on observed and unobserved situations has the same magnitude. We call the counterfactuals that are consistent with Assumption (ref) adaptive counterfactuals because they adapt to approximation error present in the observed dataset. This assumption can be questioned, especially when the counterfactual setting is significantly different than observed data, yet provides a way to conduct counterfactual analysis taking approximation error seriously.
The conceptual framework of this paper is general and can be applied to different settings. In this paper, we formalize how to generate counterfactual predictions and measure welfare changes for the quasilinear utility model using the notion of approximation error from allen2020satisficing. In particular, we present bounds on counterfactual quantities at new prices, differences of utility over consumption bundles, and welfare differences involving a price change. The computation of all bounds is facilitated by linear programming and we demonstrate the methods in an illustrative empirical example using gasoline demand data from blundell2012measuring.
The quasilinear model is a suitable setting in which to study approximation error because it is implicitly viewed as an approximation.\footnote{A notable exception that studies the approximation error explicitly is willig1976consumer.} The most common criticism is that the model does not allow income effects. In addition, the model neglects dynamics, limited consideration, and peer effects, among many other omitted features. We present a framework in which one does not need to pick a single story why the baseline model is imperfect when generating counterfactual predictions on quantities. However, to make welfare comparisons we take a stand on the interpretation of the approximation error: the individual ranks bundles according to a quasilinear utility function but for reasons we do not model explicitly, the choices do not exactly maximize the function. Overall, the framework we propose permits many reasons why the baseline model is wrong, provided the approximation error is the same magnitude in the counterfactual setting.
Despite being viewed as an approximation, the quasilinear model is widely used. Examples include work in insurance choice einav2010estimating,bundorf2012pricing,tebaldi2018nonparametric and public health cohen2010free. In addition, the quasilinear structure is closely related to a large class of latent utility models (e.g. mcfadden1981econometric, ARidentification), and so the insights of this paper are directly relevant beyond a setting with just prices and quantities. In particular, many latent utility models used in applied work involve characteristics other than prices that shift the desirability of goods but not the budget constraint.\footnote{Our analysis is also relevant for specifications in which latent utilities depend on a nonlinear function of prices. For example, berry1995automobile specifies that the utility of alternative $j$ depends on several observables including a term $\ln(m - p_j)$ where $p_j$ is the price of good $j$ and $m$ is income. This is a quasilinear model in the variable $\tilde{p}_j = \ln(m - p_j)$.}
We now describe the framework in more detail. We begin by studing counterfactual bounds for the demand of goods at new prices. We construct the counterfactual bounds by looking for the maximal and minimal demand for each good in the presence of approximation error. We assume approximation error does not increase for the counterfactual predictions relative to the dataset we have seen. This leads to nontrivial restrictions on demand at new prices as long as the prices are not too low. For each counterfactual price, this procedure gives an interval for lower and upper demands for a given good.
For welfare analysis we view approximation error with a specific interpretation. In particular, we assume an individual cannot perfectly maximize their utility function because they are satisficers in the spirit of simon1947administrative. This means an individual chooses quantities that are “close” to the maximum utility possible, but not necessarily optimal. The interpretation of satisficing allows us to assign special welfare significance to the latent utility over bundles. This utility over consumption bundles is key for policy decisions involving the allocation of goods. In addition, we study welfare over price changes, which is key for tax policy and other policy thought to affect prices.
Two features limit the ability to measure welfare changes using data. First, the choices we see do not exactly maximize utility, which leads to a “measurement wedge” on the underlying utility function over bundles. Second, even if we knew the utility function over bundles exactly, we do not know which approximately-optimizing choices will be made at new prices. This leads to an additional “prediction wedge” when bounding utility differences obtained at different prices.\footnote{We thus complement the core analysis of bernheim2009beyond, which focuses on recovering ordinal information on preferences over consumption bundles and does not distinguish between these wedges.}
Taking into account these wedges, we present bounds on differences in utility over consumption bundles and robust consumer surplus bounds involving price changes. We present computational results for both bounds, as well as analytical results designed to interpret specifically how the data are used to measure welfare changes. These bounds generalize existing work in several directions: first, and most importantly, they are valid with approximation error; second, they apply to finite datasets rather than requiring demand functions; third, the bounds apply to the (approximate) indirect utility at a new price without needing to first bound the quantity at that price. In particular, we show that the bounds on (approximate) indirect utility is a generalization of the standard integral definition of consumer surplus and we establish a close connection between counterfactual quantities and welfare bounds.
In order to understand how the counterfactual/welfare bounds depend on data and prices, we present several shape restrictions. We are not aware of any work in the tradition of varian1982nonparametric that discusses shape restrictions of the counterfactual/welfare bounds (viewed as functions of data or counterfactual prices). Studying these shape restrictions is important to understand how data are used in an empirical analysis.
For counterfactual quantities, we find that when there is a single good and the counterfactual price changes, the upper and lower bounds on counterfactual demands are each weakly decreasing in price. Thus, the upper and lower bounds on demand are functions that satisfy the law of demand. For utility differences, we establish monotonicity and continuity properties of the bounds as the quantities being compared change. For a price change, the approximate indirect utility bounds satisfy convexity and monotonicity conditions as prices vary; these are also key shape restrictions for the indirect utility function for quasilinear utility.\footnote{More specifically, convexity holds for the money metric version of the utility function. In general one can only obtain quasiconvexity.} Finally, we establish several convexity properties describing how quantities data map to the bounds. To our knowledge these results are all new even under correct specification for quasilinear utility.\footnote{The closest work appears to be a computational approach to describing bounds for models related to quasilinear utility, without describing detailed shape restrictions chiong2017counterfactual,tebaldi2018nonparametric,ARidentification.}
Our analysis also establishes that the bounds satisfy a key continuity property: as the degree of approximation error limits to $0$, our analysis limits to the analysis under correct specification. In fact, we show a stronger property that the counterfactual and approximate indirect utility bounds are jointly continuous when viewed as a function of the quantities in the data and the degree of approximation error. This facilities plug-in estimation of the bounds in which we replace true quantities with estimated quantities. This is needed to cover our empirical application in which we apply the framework with data on gasoline purchases used previously in blundell2012measuring. The data is a single cross section, and we pre-process the data as in blundell2012measuring by kernel smoothing. We conduct a representative agent analysis with quantities (conditional means) estimated from the kernel smoothed data. Like blundell2012measuring, we find that for several natural choices of the bandwidth, demand is not downward sloping. Thus, it is inconsistent with the exact quasilinear model. Nonetheless, the minimal degree of approximation error need to describe data is small and welfare bounds are surprisingly narrow for all bandwidths we consider. In contrast, the informativeness of the counterfactual bounds depends on the bandwidth.
The rest of this paper is organized as follows. After reviewing the related literature, Section (ref) describes the setup and conceptual framework of approximate counterfactuals for the quasilinear framework. Section (ref) studies counterfactuals. Section (ref) studies welfare. Section (ref) presents additional shape restrictions and discusses plug-in estimation. Section (ref) contains the application to gasoline demand. Section (ref) concludes.
This paper is part of a long literature that uses the revealed preference approach to do counterfactual and welfare analysis. The primary model used is the general model of utility maximization subject to a budget constraint, whose empirical content has been characterized in afriat1967construction, diewert1973afriat, and varian1982nonparametric. Recent econometric work considering counterfactual or welfare bounds includes blundell2003nonparametric, blundell2008best, blundell2012measuring, blundell2014bounding, hoderlein2015testing, kline2016bounding, blundell2017individual, cosaert2018nonparametric, aguiar2018meas, adams2019mutually, cherchye2019bounding, and kitamura2019nonparametric. Several proposals have been made to assess the fit of a model using revealed preference tools, including afriat1973system, houtman1985determining, varian1990goodness, and echenique2011money.\footnote{See ARmeasuring for additional references and discussion of units.} Other papers outside of the revealed preference literature that discuss fit of an approximate model include kydland1982time, vuong1989likelihood, and hansen1997assessing. The primary way in which we differ from existing work is that we use a measure of fit to adjust bounds on counterfactuals and welfare. In addition, relative to the general model with income effects, which has been the focus of the revealed preference literature, we conduct counterfactual analysis fixing prices at a new value without also fixing expenditure.\footnote{Work that uses revealed preference techniques to go beyond measuring the fit of the model includes varian1990goodness, halevy2018parametric, and gauthier2019, which study parameter recoverability. See also chetty2012bounds for a related approach.}
A growing econometric literature has studied sensitivity analysis and other ways in which a model can be used formally viewing it as an approximation. Examples include imbens2003sensitivity, conley2012plausibly, kline2013sensitivity, andrews2017measuring, manski2018right, masten2018identification,masten2019inference, armstrong2018sensitivity, bonhomme2018minimizing, christen2019, d2018rationalizing, fessler2019use, salanie2019fast, and andrews2019inference. See masten2018salvaging for additional references and discussion. We differ from this work by focusing on a notion of approximation derived from approximate optimization. We complement the robustness approach hansen2008robustness by focusing on the theoretical bounds for welfare/counterfactuals rather than focusing on an optimal decision.
This paper is naturally related to other frameworks for welfare analysis that go beyond the classic revealed preference tradition. Our approach differs substantially from an approach that posits individuals have consistent choices (that can be modeled as solutions to a “decision utility”), but whose consistent choices do not reveal the “true utility.” Subtleties with this approach have been discussed by bernheim2009beyond among others. A framework of “behavioral welfare analysis” is presented in bernheim2016good and bernheim2018behavioral, which provide further summaries of what has become a large literature. Broadly, we differ from welfare proposals in behavioral economics by addressing welfare questions given data (only) on prices and quantities, without observing decision frames or other variables thought to alter the choice process. In addition, when conducting welfare analysis concerning price changes, we introduce the “prediction wedge” because given prices, we do not know precisely what an individual would choose, even if utility were known a priori.
The goal of this paper is to provide a framework where a researcher begins with a baseline model that is taken seriously as an approximation. Since the model is an approximation, the researcher does not expect all data to be consistent with the model. Nonetheless, the researcher may want to use the model for counterfactual and welfare analysis. We present an adaptive framework framework for this by operationalizing Assumption (ref) for the quasilinear utility model. Recall that Assumption (ref) maintains that a researcher considers counterfactuals that are “no worse” than the observed data. To do this, we enlarge the baseline model to fit the data, which introduces a measurement wedge and prediction wedge. The measurement wedge concerns limits on what an analyst can learn due to approximation error for objects defined in the existing dataset (e.g. utility functions). The prediction wedge describes limits on what can be said in new settings where the model may not be perfect (e.g. counterfactual quantities). We formalize Assumption (ref) by making these wedges as small as possible while still fitting the observed data, using the notion of approximation error from allen2020satisficing. We describe the framework more below.
We formalize the baseline model of quasilinear utility. A consumption bundle $(x,y) \in \mathbb{R}_+^K \times \mathbb{R}$ is evaluated according to $u(x) + y$, where $u:\mathbb{R}_+^K\rightarrow\mathbb{R}$ is a utility function over bundles $x$. The numeraire good is given by $y$ and has a price of one. Given prices $p\in \mathbb{R}_{++}^K$ and income $I \in \mathbb{R}$ decisions in a quasilinear utility model follow
where consumption of the numeraire good is allowed to be negative for unobserved borrowing.\footnote{Allowing negative expenditure also avoids boundary issues for chosen consumption bundles.} We study the quasilinear utility model since it is regularly used in applications described in the Introduction, including adaptations to handle non-price characteristics .\footnote{ARidentification show many applications including the additive random utility model mcfadden1981econometric are quasilinear models with utility indices playing the role of prices.} In addition, it has a tractable notion of welfare in terms of units of the numeraire.
Now we present an enlargement of the baseline model that relaxes the assumption of exact maximization to a notion of approximate optimization. Because we focus on the empirical analysis, we define the enlargement in terms of a finite datasets of the form $\{ (x^t,p^t) \}_{t=1}^T$. There are $T$ observations, quantities are weakly positive $x^t \in \mathbb{R}_{+}^K$, and prices are strictly positive $p^t \in \mathbb{R}_{++}^K$. Importantly, quantities can be discrete or continuous, and $0$ quantities are permitted in this framework.
The value $\varepsilon$ is in the same units as the price of the numeraire good, e.g. dollars per time period. When $\varepsilon > 0$, the observed bundles are within $\varepsilon$ dollars of the maximum utility possible at a given price. One interpretation is that $\varepsilon$ captures “unstructured” deviations from the quasilinear utility model (cf. chetty2012bounds, hansen2018structured), without a single interpretation of the nature of the deviations. Instead, the magnitude of the deviations is controlled. This interpretation can be pursued when making counterfactual predictions. In contrast, to make welfare predictions an interpretation of the model is crucial. When discussing welfare, we follow allen2020satisficing and interpret the value $\varepsilon$ as a level of satisficing in the spirit of simon1947administrative. In this case, a higher value of $\varepsilon$ means there is a larger set of consumption bundles that are “good enough" to be chosen.
When using the model for counterfactual or welfare analysis, a measurement wedge and a prediction wedge arise. These concepts will become more clear when we turn to specific analysis below, but we first provide an overview. When observed data is not exactly consistent with a quasilinear utility model, the econometrician knows there is no utility function that rationalizes the entire dataset, so at some observation the quantity is not optimal. Thus, if a researcher still wants to use the quasilinear model even when data is inconsistent with the baseline quasilinear model, then there is a measurement wedge when trying to recover information about candidate utility functions and indirect utility. Second, even after the econometrician has a set of candidate utility functions that match the original dataset, the econometrician cannot know that counterfactual choices will exactly maximize a candidate utility function. Thus, there is a prediction wedge when forecasting even after recovering information on the utility function.
We now discuss how to formalize Assumption (ref) for the approximate quasilinear utility model in relation to the measurement wedge and prediction wedge. Let $\varepsilon_M$ denote the measurement wedge and $\varepsilon_P$ denote the prediction wedge. In principle, the two wedges may not be the same, but Assumption (ref) allows us to treat these as equal.\footnote{See Appendix (ref) for additional discussion on this case and a more formal treatment of the measurement and prediction wedges.} To further formalize Assumption (ref), we introduce $\varepsilon^*$ as the smallest value of $\varepsilon$ such that the dataset is $\varepsilon^*$-rationalized by quasilinear utility. We also refer to $\varepsilon^*$ as the level of approximation error or approximation error of the quasilinear utility model for the observed dataset.
We note that $\varepsilon^*$ is a function of the dataset to a number, so for a dataset $D=\{(x^t,p^t)\}_{t=1}^T$ we can write $\varepsilon^*(D)$. When we discuss only a single dataset, we typically drop dependence on $D$. The value $\varepsilon^*$ will be used in our framework to place restrictions on the magnitude of the measurement and prediction wedges. Setting $\varepsilon_M \geq \varepsilon^*$ formalizes that the measurement wedge is large enough to explain the data we have seen. Similarly, setting $\varepsilon_P \geq \varepsilon^*$ formalizes that the model is no better at predicting in new settings than the data we have seen. We make these bounds as tight as possible, and formalize Assumption (ref) for this setting as follows.
This is a direct generalization of the standard approach to counterfactual and welfare analysis, which sets $\varepsilon_M = \varepsilon_P = 0$. The conceptual framework of the standard approach only applies to models that perfectly fit the data, which translates to $\varepsilon^* = 0$ here. We later develop a framework for counterfactual and welfare analysis when the measurement and prediction wedges are equal, $\varepsilon_M = \varepsilon_P$. For notational convenience, we will let $\varepsilon$ denote the common value. Assumption (ref) is the special case where $\varepsilon = \varepsilon^*$.
For the quasilinear utility model with approximation error that does not exceed $\varepsilon$, we consider sharp counterfactual bounds. More formally, we consider when the measurement and prediction wedges are both equal to a single value $\varepsilon$. We impose $\varepsilon = \varepsilon^*$ to implement Assumption (ref) and describe some properties on the counterfactual bounds. In Section (ref), we provide graphical intuition for the bounds. In Section (ref), we describe how to compute bounds on quantities fixing a price. In Section (ref) we describe additional restrictions that can be imposed to tighten the bounds, such as a priori bounds on expenditure at a new price.
To that end, for notational convenience, let $D =\{(x^t, p^t ) \}_{t = 1}^T$ denote the observed dataset. Suppose we have a candidate quantity-price tuple $(\tilde{x},\tilde{p})$. We can add this to the original dataset to form an augmented dataset $D \cup (\tilde{x}, \tilde{p})$. We consider candidates such that the approximation error of the augmented dataset is bounded by $\varepsilon$. In particular, the set of consistent demands and prices for the level of approximation error $\varepsilon$ is given by \[ C(D, \varepsilon ) = \left\{ (\tilde{x},\tilde{p}) \in \mathbb{R}_{+}^K \times \mathbb{R}_{++}^K \mid \varepsilon^* (D \cup (\tilde{x},\tilde{p}) ) \leq \varepsilon \right\}. \] To check whether a candidate tuple $(\tilde{x},\tilde{p})$ is in $C(D, \varepsilon )$, one can calculate $\varepsilon^*$ for the augmented dataset using Proposition (ref). If this measure of approximation error for the augmented dataset is below $\varepsilon$, then the candidate tuple is in the set $C(D, \varepsilon )$.
Our framework imposes Assumption (ref) to generate counterfactual predictions assuming the level of approximation error does not get worse. This amounts to setting $\varepsilon$ equal to the approximation error of the observed dataset. In particular, we focus on the adaptive counterfactual set
We collect some facts about $AC(\cdot)$ and $C(\cdot)$.
Thus, when a candidate observation in $AC(D)$ is added to $D$, the measure of approximation stays the same. This follows from the construction of $AC(D)$. This equality does not hold for all measures of model approximation error. For example, if we had chosen to take $\varepsilon^*$ divided by the number of observations $T$ as the measure of approximation error, then Fact (ref) would not hold in general since the measure of approximation error for the augmented dataset would divide by $T + 1$.
Higher values of $\varepsilon$ correspond to less informative counterfactual predictions. This follows from the fact that if a dataset is $\varepsilon$-rationalized by quasilinear utility, then it is also $\varepsilon'$-rationalized for $\varepsilon < \varepsilon'$.
This states that the observed data places a lower bound on the minimal amount of approximation error needed to conduct counterfactual anlaysis. If $\varepsilon \geq \varepsilon^*$, nonemptiness of $C(D,\varepsilon)$ is guaranteed by considering $\tilde{p}$ sufficiently high along each dimension and $\tilde{x} = 0$. Alternatively, for the dataset $D$, when $\varepsilon < \varepsilon^*$ even observations within the dataset cannot be $\varepsilon$-quasilinear rationalized.
This formalizes that setting $\varepsilon = \varepsilon^*$ for counterfactual values obtains the sharpest restrictions under Assumption (ref) subject to the constraint that counterfactuals are nonempty. This follows from the previous facts. To perform a sensitivity analysis, one could examine any $\varepsilon > \varepsilon^*$ and use $C(D,\varepsilon)$ as the counterfactual set. Our framework allows this yet focuses on $\varepsilon = \varepsilon^*$.
To gain intuition on the “shape" of the counterfactual sets $C(D,\varepsilon)$ and $AC(D)$, we present a graphical description of the restrictions on counterfactuals. For exposition we focus on some of the restrictions rather than all of them. First we describe a restriction that must hold for a dataset to be $\varepsilon$-rationalized. At price $p^r$ we must have \[ u(x^r) - p^r \cdot x^r \geq u(x^s) - p^r \cdot x^s - \varepsilon \] for some unknown function $u$. This states that $x^s$ cannot be much better than $x^r$ at price $p^r$. Flipping the role of observations $r$ and $s$ and basic algebra yields
This is a multivariate approximate law of demand. The usual multivariate law of demand obtains when $\varepsilon = 0$. For a given value $\varepsilon\ge 0$, this also places restrictions on counterfactual demand $\tilde{x}$ at prices $\tilde{p}$ since for any $r \in \{1, \ldots, T \}$, a potential counterfactual tuple must satisfy
When we apply Assumption (ref), we evaluate counterfacturals at $\varepsilon = \varepsilon^*$. This inequality places a restriction on candidate quantity-price tuples when compared with any observation in the dataset. In the one dimensional case ($K = 1$), this states that if the price increases from $p^r$ to $\tilde{p}$, then demand cannot increase by too much. The bound on the increase in quantities is inversely related to the magnitude of the price increase. That is, when $\tilde{p}-p^r>0$ we have $\tilde{x} \le x^r + \frac{2 \varepsilon}{\tilde{p}-p^r} $.
We illustrate these bounds in two example datasets displayed in Figure (ref). Each dataset has four observations respresented as black dots. The gray area denotes the set of quantity-price tuples $(\tilde{x},\tilde{p})$ that are consistent with the existing dataset with minimal level of approximation error $\varepsilon = \varepsilon^*$. In panel (a), the observed dataset is exactly consistent with quasilinear utility and the counterfactual set has $\varepsilon = 0$. In this case, the constructed bounds have the property that when price increases quantity cannot increase. This leads to the “rectangular” bounds in panel (a). Note that for low values of prices, quantity has a lower bound but not an upper bound. Similarly, when prices are higher than any observed data the lower bound on counterfactual demand is zero.
In panel (b), the dataset is not consistent with quasilinear utility, because there is an instance in which price goes up and quantity goes up. Here we graphically obtain the counterfactual restrictions using the approximate law of demand constructed from Equation (ref), setting $\varepsilon = \varepsilon^*$.\footnote{There are additional restrictions beyond Equation (ref); here we provide a graphical illustration but the general framework uses additional inequalities.} This approach leads to “hyperbolic” bounds, in contrast with the rectangular bounds in panel (a). The fact that $\varepsilon = \varepsilon^*$ is the minimal approximation error needed to rationalize the data is demonstrated on the graph by two points touching dashed hyperbolas.
The sets $C(\cdot)$ and $AC(\cdot)$ completely describe counterfactuals. An analyst may not be interested in the entire set of counterfactual quantity-price tuples, but rather certain features of it. For example, an analyst may only be interested in quantities at a fixed counterfactual price $\tilde{p} \in \mathbb{R}^K_{++}$ allowing approximation error $\varepsilon$. This set may be written \[ X(\tilde{p},D,\varepsilon) = \left\{\tilde{x} \in \mathbb{R}_{+}^K \mid (\tilde{x},\tilde{p}) \in C(D,\varepsilon) \right\}. \]
Our first question is when this set is nonempty, i.e. when can we conduct counterfactual analysis.
This shows that by allowing enough approximation error, we can find counterfactual quantities for any price. This is stronger than Fact (ref) because it gives nonemptiness of the counterfactual quantity set for any price. Existing work has studied when observed datasets can be rationalized by quasilinear utility (brown for $\varepsilon = 0$ or with certain random shocks) or an enlargement (allen2020satisficing for $\varepsilon \geq 0$). This strengthens those results by showing when we can conduct counterfactual analsyis of quantities at a new price. The question is nontrivial because the utility functions constructed in brown and allen2020satisficing to explain data have no approximate maximizer for low prices because the indirect utility is infinite.\footnote{See the proof of Proposition (ref). See also aguiar2020rationalization for recent work concerning emptiness of counterfactual sets when using the weak axiom of revealed preference.}
We now discuss additional properties of $X(\tilde{p},D,\varepsilon)$.
When $M = 1$, the inequalities in Equation (ref) yield the approximate law of demand described in Equation (ref). In this case we compare an observation in the dataset with a conjectured counterfactual tuple $(\tilde{x},\tilde{p})$, which leads to two instances of $\varepsilon$ in Equation (ref), just like the approximate law of demand. Proposition (ref) shows there are other restrictions imposed on counterfactuals beyond the law of demand by considering more than one observation at a time ($M \geq 2$). These restrictions arise by adding up additional sequences of inequalities similar to constructing the approximate law of demand. By summing up appropriate sequences, the unknown utility function is removed so restrictions on counterfactual quantities is given only using observable data. We later use similar information to generate bounds on certain welfare objects.
Suppose now that we are only interested in bounding the quantity of the $k$-th good at a price $\tilde{p}$, allowing up to $\varepsilon$ approximation error. These bounds are extrema of $X(\tilde{p},D,\varepsilon)$ along the $k$-the dimension. That is, they they are the extreme points of the set
The following proposition discusses the bounds for the $k$-th good. In particular, when the bounds exist they can be computed by a linear program and the bounds satisfy monotonicity properties with respect to the approximation error $\varepsilon$.
The details on the linear program to compute bounds are found in Proposition (ref) of Appendix (ref). Recall $X_k(\tilde{p},D,\varepsilon)$ is convex from Proposition (ref). Thus, any quantity between $\underline{x}_k(\tilde{p},\varepsilon)$ and $\overline{x}_k(\tilde{p},\varepsilon)$ is a candidate counterfactual quantity for good $k$.
Next we elaborate on when these bounds are finite. We show the lower bound is always finite but the upper bound is finite only when prices are sufficiently high. To formalize this define the upper comprehensive convex hull of a finite set $\{ z^{\ell} \}_{\ell = 1}^L$ as
The inequality in the definition here is componentwise. In addition, let $\operatorname*{int} A$ denote the interior of a set $A$.
Finally, we show that for one good $(K = 1)$, the bounds on demand are downward sloping in own-price.
When $\varepsilon = 0$, the dataset satisfies the exact law of demand. When $\varepsilon^*$ (and $K = 1$), there is some pair of observations $r,s \in \{1,\ldots,T\}$ that violates the law of demand so \[ (x^r - x^s) (p^r - p^s) > 0. \] Proposition (ref) shows that while such violations can occur in the data, the bounds themselves satisfy the exact law of demand.
Additional assumptions can tighten the bounds on quantities in Proposition (ref). For example, one can assume that expenditure is the same at the counterfactual value as the last period of data, so $\tilde{p} \cdot \tilde{x} = p^T \cdot x^T$. Alternatively, one could place bounds on the expenditures so that $\underline{m} \le \tilde{p} \cdot \tilde{x} \le \overline{m}$. One may also impose a priori bounds on the quantities of other goods. These bounds can considerably shrink the set of counterfactual bounds, especially when there are multiple goods. In addition, computation with these additional restrictions is not challenging because these are inequality constraints that can be appended to the original linear program. When adding these additional constraints, however, it is possible that the counterfactual set can be empty.
We emphasize that in general, such expenditure bounds are not needed to deliver nontrivial counterfactual bounds. It is helpful to contrast our approach with the general model of utility maximization subject to a budget constraint, with preferences that need not be quasilinear. In the general model, even under correct specification the sharp bounds on quantities of each good at a given price are the trivial bounds $[0,\infty)$ unless the analyst places a priori bounds on expenditure at the new price.\footnote{The bounds $[0,\infty)$ are for bounding one good at a time (similar to $\underline{x}_k$ and $\overline{x}_k$ above). There are nontrivial restrictions on the entire demand tuple.} This is because the general model does not rule out expenditure of $0$ or arbitrarily high values at counterfactuals when we only fix prices.\footnote{The results in deb2018revealed can be used to show nontrivial bounds are possible in the general model when income is always the same value (inside and outside the dataset) and there is an unobserved good whose price is fixed.}
To study welfare, we must take a stand on the interpretation of approximation error. For this section, we follow allen2020satisficing and treat the approximation error as arising from satisficing in the spirit of simon1947administrative. In particular, an individual has a utility function that describes the ranking over goods, but satisfices by choosing bundles that are “good enough.”
We now discuss how satisficing relates to the measurement and prediction wedge. When trying to learn about utility from data, a measurement wedge arises since observed choices may not be optimal. When trying to predict welfare for a price change, the prediction wedge occurs since we only know the region of bundles that are “good enough." Assumption (ref) means that the measurement and prediction wedge are the same size as the smallest amount of satisficing needed to describe the data. We note that one can also apply the satisficing interpretation to counterfactual quantities, but it is not necessary. For this reason we did not distinguish between these wedges in Section (ref). Appendix (ref) provides additional discussion.
Since we are studying quasilinear utility there are two natural welfare objects. We look at differences in utility over consumption bundles and differences in (approximate) indirect utility over prices. An important asymmetry arises because learning about differences in utility only involves the measurement wedge because it does not involve choices in new situations. In contrast, differences in (approximate) indirect utility over prices involves both the measurement wedge and prediction wedge because one must consider choices in new settings. We elaborate more below.
Our first goal is to learn about the unknown utility function over consumption bundles using data. This is helpful when considering policies involving the direct distribution of goods.
In general, there is a collection utility functions that can $\varepsilon$-rationalize a dataset $\{ (x^t, p^t) \}_{t = 1}^T$. We study bounds on utility differences between consumption bundles. Specifically, given two consumption bundles $\tilde{x}^1, \tilde{x}^0 \in \mathbb{R}^K_+$ we consider the upper and lower bounds
Here, we consider all possible utility functions $u : \mathbb{R}^K_+ \rightarrow \mathbb{R}$ without additional restrictions such as monotonicity or concavity. A utility function $u$ is said to $\varepsilon$-rationalize the dataset $\{ (x^t, p^t ) \}_{t = 1}^T$ when for every $t\in \{1,\ldots,T\}$ the inequality \[ u(x^t) - p^t \cdot x^t \geq u(x) - p \cdot x - \varepsilon \] holds for every $x \in \mathbb{R}^K_+$.
To interpret these bounds, suppose for example that $\overline{u}(\tilde{x}^1, \tilde{x}^0, \varepsilon) < 0$. We conclude that the individual ranks $\tilde{x}^0$ above $\tilde{x}^1$, even when the individual's choices do not exactly maximize utility. Thus, there is no ambiguity in the ranking of these bundles according to the unknown utility function $u$. If $\overline{u}(\tilde{x}^1, \tilde{x}^0, \varepsilon) > 0$, then it is possible that the individual ranks $\tilde{x}^1$ above $\tilde{x}^0$. Lastly, if $\underline{u}(\tilde{x}^1, \tilde{x}^0, \varepsilon) > 0$, then we conclude the individual ranks $\tilde{x}^1$ above $\tilde{x}^0$. More broadly, these bounds provide cardinal information on utility differences, in units of the price of the numeraire.
To gain some intuition how bounds on differences of utility are informed by data, consider two bundles $x^r$ and $x^s$ in the dataset. Since $x^r$ is approximately optimal given prices $p^r$, we have the restriction \[ u(x^r) - p^r \cdot x^r \geq u(x^s) - p^r \cdot x^s - \varepsilon, \] which rearranges to
Differences in utility are thus bounded by changes in expenditure. Here, price is fixed and a change in quantity determines the magnitude of the expenditure change. The inequality in ((ref)) arises because the point in the data $x^r$ was approximately optimal at prices $p^r$. Thus, $\varepsilon$ here directly involves the observed data and is part of the measurement wedge. There is no prediction wedge because an analyst does not contemplate choices in new situations.
We first formalize computation of the bounds before providing additional interpretation. We show the bounds can be calculated as a linear program. An explicit description is relegated to Proposition (ref) in Appendix (ref).
Note that the set \[ \left\{u \mid u \text{ } \varepsilon-\text{rationalizes } \{ (x^t, p^t ) \}_{t = 1}^T \right\} \] is convex in the sense that if each $u^a, u^b$ $\varepsilon$-rationalize the dataset, then $\alpha u^a + (1 - \alpha)u^b$ does for $\alpha \in [0,1]$. This follows from inspecting inequalities such as \[ u(x^t) - p^t \cdot x^t \geq u(x) - p \cdot x - \varepsilon \] that define $\varepsilon$-rationalizability by a utility function $u$. This means that any value between $\underline{u}(\tilde{x}^1, \tilde{x}^0, \varepsilon)$ and $\overline{u}(\tilde{x}^1, \tilde{x}^0, \varepsilon)$ can be attained.
To gain further intuition how data bound utility differences, we provide an analytical characterization. This characterization builds on inequalities such as ((ref)) above, yet uses longer sequences (rather than just pairs) of observations to describe the tightest possible bounds. This parallels analysis of counterfactuals, where restrictions other than the law of demand arise by considering sequences of observations.
The sums inside the minimum are closely related to sums discussed in Proposition (ref) for counterfactuals. The sums differ because Proposition (ref) constructs sequences making a cycle (to remove the unknown utility numbers). In contrast, Proposition (ref) considers sequences that do not make a cycle because the goal is to examine differences of utility numbers.
Continuity and concavity fail at $\tilde{x}^1 = x^S$ (when $\varepsilon > 0$) because the difference in utility is zero when the quantity is the same. Since $\overline{u}(\tilde{x}^0,\tilde{x}^1,\varepsilon) = -\underline{u}(\tilde{x}^1,\tilde{x}^0,\varepsilon)$, analogous results hold for the lower bound $\underline{u}(x^F, \tilde{x}^1,\varepsilon)$ if the first argument $x^F$ is in the dataset. See Proposition (ref) for formal results.
An important feature for practical application is that the bounds on utilities are trivial unless an appropriate quantity is in the dataset. We formalize this as follows.
Recall that Proposition (ref) shows that $\overline{u}$ is strictly increasing and continuous over a region. Thus, the upper bound on utility differences has some shape restrictions like a “nice” utility function. Despite this, the bound is not concave/continuous in the first argument at $\tilde{x}^1 = x^S$ (when $\varepsilon > 0$). This means that imposing concavity/continuity can potentially tighten the bounds.\footnote{Continuity and concavity do not tighten the bounds when $\varepsilon = 0$ because the upper bound $\overline{u}$ is then continuous and concave for all values of $\tilde{x}^1$. See the proof of Proposition (ref) for more details.} Imposing these (or other) shape restrictions is important if one wishes to bound utility when neither quantity is in the dataset, since from Proposition (ref) we know the bounds are trivial without more structure.
We now turn to welfare analysis concerning price changes. Here both the measurement and prediction wedge play a role. Recall that a “measurement wedge” shows up for bounds on the utility over bundles as in Section (ref) since observations may not exactly maximize a quasilinear utility function. Here the prediction wedge also arises when $\varepsilon^* > 0$ because even when we know the utility function, we do not know which approximately-optimal choice would be made at a new price.
We first discuss the indirect utility, which is the standard welfare object for the exact quasilinear model. Later we introduce the approximate indirect utility to account for the fact that an individual does not exactly optimize. The indirect utility function associated with the utility function $u : \mathbb{R}^K_+ \rightarrow \mathbb{R}$ is given by \[ V_u(p) = \sup_{x \in \mathbb{R}^K_+} u(x) - x \cdot p. \] Since the researcher does not know the individual's utility a priori, we consider indirect utility associated with candidate utility functions.
We show how indirect utility interacts with the measurement wedge. If $x^t$ is within $\varepsilon$ of the maximum utility possible at price $p^t$, then we can write \[ V_u(p^t) \leq u(x^t) - p^t \cdot x^t + \varepsilon. \] The definition of the indirect utility yields for arbitrary $p \in \mathbb{R}^K_{++}$, the inequality \[ V_u(p) \geq u(x^t) - p \cdot x^t. \] Differencing these, we obtain
Here, $\varepsilon$ arises because the observed choices need not be exact maximizers and thus is part of the measurement wedge. With a restriction on the magnitude of $\varepsilon$, we can use observations of $x^t$ and $p^t$ to bound differences in indirect utility.
We now introduce the prediction wedge. This wedge arises because raw differences in indirect utility are not the natural welfare object in our setting for a price change because we focus on ex ante policy evaluation. Instead, we take into account that when $\varepsilon > 0$, an individual may choose bundles with different utility when facing the same prices. This is because we assume an individual satisfices.
The utility the individual attains for a given price and choice of consumption bundle is the approximate indirect utility. For observation $t \in \{1,\ldots,T\}$, the approximate indirect utility is \[ u(x^t) - p^t \cdot x^t. \] At price $p$, the approximate indirect utility is restricted to be somewhere in the interval \[ [V_u(p) - \varepsilon, V_u(p)]. \] In fact, (weakly) further restrictions take into account that the approximate indirect utility attained is bounded below by \[ \underline{V}_{u,A}(p,\varepsilon) = \inf_{x \in \mathbb{R}_+^K} u(x) - p \cdot x \qquad \text{s.t. } \quad u(x) - p \cdot x \geq V_u(p) - \varepsilon, \] while the upper bound is the indirect utility. The lower bound on approximate indirect utility is the lower bound $\underline{V}_{u,A}(p,\varepsilon)$, while the upper bound on approximate indirect utility is the upper bound $\overline{V}_{u,A}(p,\varepsilon) = V_u(p)$.
Now suppose we wish to bound the change in approximate indirect utility between prices $\tilde{p}^0$ and $\tilde{p}^1$. If the utility $u$ and level of satisficing $\varepsilon$ were known, then the welfare bounds would be \[ [\underline{V}_{u,A}(\tilde{p}^1,\varepsilon) - \overline{V}_{u,A}(\tilde{p}^0,\varepsilon), \overline{V}_{u,A}(\tilde{p}^1,\varepsilon) - \underline{V}_{u,A}(\tilde{p}^0,\varepsilon)]. \] Fixing $u$, this interval becomes wider when $\varepsilon$ increases. In general, $\varepsilon$ controls the prediction wedge, which arises even if we knew $u$ because we would not know what is chosen by the satisficer.
Since we do not know the utility function a priori, we consider bounds involving the smallest and largest changes in approximate indirect utility among all utility functions that $\varepsilon$-rationalize the dataset:
\footnotetext{Formally, we take the supremum over $u$ such that $\overline{V}_{u,A}(\tilde{p}^0,\varepsilon)$ is not $\infty$, and the infimum over $u$ such that $\overline{V}_{u,A}(\tilde{p}^1,\varepsilon)$ is not $\infty$.}
These bounds incorporate both the measurement and prediction wedges. The measurement wedge shows up when considering $u$ that $\varepsilon$-rationalize the data, while the prediction wedge arises when defining the approximate indirect utility. We use the same value $\varepsilon$ for both since we maintain Assumption (ref).
These bounds can inform a researcher about changes in welfare even in the presence of satisficing. If $\underline{V}(\tilde{p}^1,\tilde{p}^0,\varepsilon) > 0$, then we can conclude that given a price change from $\tilde{p}^0$ to $\tilde{p}^1$ the individual is better off at $\tilde{p}^1$. If $\overline{V}(\tilde{p}^1,\tilde{p}^0,\varepsilon) < 0$, then the price change from $\tilde{p}^0$ to $\tilde{p}^1$ makes the individual worse off. In contrast, ambiguity arises when $\underline{V}(\tilde{p}^1,\tilde{p}^0,\varepsilon)<0$ and $\overline{V}(\tilde{p}^1,\tilde{p}^0,\varepsilon) > 0$. In this case an individual may be better or worse given the price change, but the data alone are inconclusive.
We now state a computational result for the bounds. A specific description of the linear program is given in Proposition (ref) in Appendix (ref).
When $\varepsilon = \varepsilon^* = 0$, the approximate indirect utility equals the indirect utility, and these are the sharp bounds on consumer surplus with limited price variation. When we set $\varepsilon = \varepsilon^*$, these are the adaptive consumer surplus bounds. These bounds may be used for arbitrary prices $\tilde{p}^1, \tilde{p}^0$, not only at prices in $\{ ( p^t ) \}_{t = 1}^T$. In particular, these bounds provide welfare bounds at new prices without needing to first provide bounds on the quantities at the prices.
Recall ((ref)) established for $p^t$ in the dataset, \[ V_u(p^t) - V_u(p) \leq x^t \cdot (p - p^t) + \varepsilon. \] This states that differences in indirect utility are bounded by changes in expenditure. Here, the change in expenditure involves keeping the quantity fixed and changing prices. We present lower and upper bounds on $\overline{V}$ that build on this inequality. To state the result, first suppose $\tilde{p}^1 = p^S$ is in the dataset. Define \[ h(\tilde{p}^0) = \min_{\sigma \in \Sigma_S} \left\{ x^{\sigma(M)} \cdot (\tilde{p}^0 - p^{\sigma(M)}) + \sum_{m = 1}^{M-1} x^{\sigma(m)} \cdot (p^{\sigma(m+1)} - p^{\sigma(m)}) + M \varepsilon \right\}, \] where $\Sigma_S$ is the set of sequences that start with $\sigma(1)=S$, have no cycles, and have length at least $M\ge 1$.
This result is established by leveraging duality results we present in Appendix (ref). Analogous results exist for $\underline{V}$ because $\overline{V}(\tilde{p}^1,\tilde{p}^0,\varepsilon) = -\underline{V}(\tilde{p}^0,\tilde{p}^1,\varepsilon)$, and are omitted for brevity.
We reiterate that Proposition (ref) describes that $\overline{V}$ can be computed exactly as a linear program. The goal of Proposition (ref) is to make this process less of a “black box.” Note that when $\varepsilon = 0$, the lower and upper bounds coincide and we characterize $\overline{V}(\tilde{p}^1,\tilde{p}^0,0)$. We recognize $h$ as a function closely related to the construction of the Riemann integral, since it computes the sum of the area of certain rectangles. We may view $h$ as a “discrete” analogue of the consumer surplus formula, which states that differences in indirect utility are the area of a demand function. In fact, this integration intuition can be formalized in the special case of a single good $(K = 1)$, when $\varepsilon = 0$.
Proposition (ref) shows that in a certain case, there is a tight connection between bounds on quantities and welfare bounds. Further relationships between welfare and counterfactual quantities are left for future work.
To close this section, we present shape restrictions on $\overline{V}$ that hold for all $\varepsilon \geq 0$.
The shape restrictions in Proposition (ref) are those of an indirect utility function. We do not obtain global continuity here because the welfare bounds can be infinite. However, $\overline{V}(\tilde{p}^1,\tilde{p}^0,\varepsilon)$ is continuous in $\tilde{p}^1$ over the relative interior of $\operatorname{CCo}(\{p^t\}_{t = 1}^T)$ because it is convex and finite over this set (rockafellar2015convex, Theorem 10.1).
Recall that Proposition (ref) shows that bounds on utility differences are trivial when a quantity is not in the dataset. In contrast, Proposition (ref) shows that the bound on approximate indirect utility $\overline{V}(\tilde{p}^1,\tilde{p}^0,\varepsilon)$ is typically finite provided $\tilde{p}^1$ is not too low. In particular, neither $\tilde{p}^1$ nor $\tilde{p}^0$ need be in the dataset. The reason we obtain these contrasting results is that indirect utility functions must satisfy certain shape restrictions while we consider utility functions that need not satisfy shape restrictions such as concavity or monotonicity.
In classic revealed preference, a small amount of measurement error can lead to refutation of the model. In this case, there is no way to use the model for counterfactual or welfare analysis. Below we show continuity of the welfare/counterfactual bounds in both quantities and degree of approximation error. Thus, we provide a way to still conduct analysis when the model is not perfect, and do so in a way that is a continuous enlargement of the standard conceptual framework.\footnote{To be clear, results in this paper are also new under $\varepsilon = 0$ with a few exceptions that are noted.}
In more detail, here we study the joint mapping from quantities and approximation error to the bounds analyzed previously. One motivation for this is that in applications, an analyst may not observe a dataset of interest $\{ (x^t, p^t)\}_{t = 1}^T$ exactly, and may instead only have an estimate of the quantities. We show below that if we can consistently estimate quantities, then we can consistently estimate the bounds.
For concreteness, suppose an analyst is conducting a representative agent analysis, and quantities are mean quantities from a population at a each time period. We examine the mean demand vector at period $t$ so that $x^t = \mathbb{E}[ X^{i,t}]$, where $X^{i,t}$ is demand for individual $i$ at time $t$. Here $X^{i,t}$ is treated as a random variable that is identically distributed across individuals. An analyst estimates $\widehat{\mathbb{E}[X^{i,t}]}$ from a cross-sectional dataset in which individuals at each time period face the same prices. For example, the estimator could be the sample average of demands at time $t$ across many individuals.
We now turn to the formal results. While we allow estimation error associated with quantities, here we take each price $p^t$ as nonrandom and measured exactly. Recall that $\overline{x}_k(\tilde{p},\varepsilon)$ is the maximal quantity of good $k$ at counterfactual price $\tilde{p}$ assuming approximation error is no greater than $\varepsilon$. The definition of $\overline{x}_k$ is presented in Proposition (ref). We now treat $\overline{x}_k$ as a function of the dataset of quantities, and with minor abuse of notation we write $\overline{x}_k \left(d, \tilde{p}, \varepsilon \right)$, where $d = \left(d^1, \ldots, d^T \right) \in \mathbb{R}^{K \times T}_+$ denotes quantities across all goods at time periods. This allows us to study how the bound depends on quantities in the dataset while keeping prices, $\{p^t\}_{t=1}^T$, fixed. Similarly, $\underline{x}_k \left(d,\tilde{p},\varepsilon \right)$ denotes the lower bound. Finally, let $A \subseteq \mathbb{R}^{K \times T}_{+} \times \mathbb{R}_+$ denote combinations of quantities and approximation error such that the counterfactual/welfare objects are defined, i.e. \[ A = \{ (d,\varepsilon) \in \mathbb{R}_{+}^{K \times T} \times \mathbb{R}_{+} \mid \varepsilon \ge \varepsilon^* \left(\{ (d^t, p^t) \}_{t = 1}^{T} \right) \}. \]
We obtain a similar result for the bounds on approximate indirect utility $\overline{V}$ and $\underline{V}$ when we view them as a function of the dataset of quantities. To formalize this, with minor abuse of notation let $\overline{V} \left(d,\tilde{p}^1,\tilde{p}^0,\varepsilon \right)$ describe the upper bound as a mapping of the quantities $d \in \mathbb{R}^{K \times T}_+$ in a dataset. Similarly, $\underline{V} \left(d, \tilde{p}^1,\tilde{p}^0,\varepsilon \right)$ denotes the lower bound.
Recall Proposition (ref) shows that when $\tilde{p}^1 \in \operatorname{CCo}(\{p^t\}_{t=1}^T)$, $\overline{V}(d, \tilde{p}^1,\tilde{p}^0,\varepsilon)$ is finite for any $(d,\varepsilon) \in A$.
We need one more result. Here, we interpret the minimal approximation error $\varepsilon^*$ as a function of quantities for fixed prices $\{ p^t \}_{t = 1}^T$.
The previous continuity results imply the following consistency results.
This provides a theoretical foundation for plug-in estimation. We omit a formal description of the sampling scheme since the result applies to any collection of random variables $\left\{ \hat{d}^n \right\}$ that converges in probability to $d$. For example, if we have panel data and the quantities $(X^{i,t})_{t = 1}^T$ are independent and identically distributed across individuals, one can use sample averages so that $\hat{d}^n= \left( \frac{1}{n}\sum_{i=1}^n X^{i,t} \right)_{t=1}^T$ when estimating $d= \left( \mathbb{E}[X^{i,t}] \right)_{t=1}^T$.
Finally, we consider shape restrictions for the bounds on utility differences $\overline{u}$ and $\underline{u}$, viewed as functions of quantities. As before, fixing prices $\{ p^t \}_{t = 1}^T$, we study dependence on the quantities $d \in \mathbb{R}^{K \times T}_{+}$. With minor abuse of notation write $\overline{u}(d, \tilde{x}^0, \tilde{x}^1, \varepsilon)$ as a function of quantities and approximation error, and similarly for $\underline{u}$.
Recall that Proposition (ref) shows that when $\tilde{x}^0$ is in the dataset of quantities $d$, $\overline{u}(d,\tilde{x}^1,\tilde{x}^0,\varepsilon)$ is finite provided $\varepsilon \geq \varepsilon^*$. In contrast, Proposition (ref) shows that whenever $\tilde{x}^0$ is outside the dataset, $\overline{u}(d,\tilde{x}^1,\tilde{x}^0,\varepsilon) = \infty$. We conclude that when viewed as a mapping of quantities, $\overline{u}$ is no longer continuous. It is, however, continuous over a certain subset of $A$. To describe this, for a vector $\tilde{x} \in \mathbb{R}^K_{+}$ let $A(\tilde{x}) = \{ (d,\varepsilon) \in A \mid d^1 = \tilde{x} \}$. This restricts attention to quantities datasets that all contain a certain vector $\tilde{x}$ as the first component. We formalize continuity and concavity results as follows.
Continuity over all of $A$ does not hold and so one cannot directly apply the continuous mapping theorem to establish a consistency result like Corollary (ref). If some quantity vector $d^1$ is measured without error, however, then it is possible to consistently estimate the bounds on utility differences between $d^1$ and other bundles though we omit details for brevity.
We now illustrate the results in the paper with data on the demand for gasoline. Data are from the 2001 United States National Household Travel Survey, and have previously been used in blundell2012measuring. The data are from a single cross-section. For brevity we refer to blundell2012measuring for additional details, including construction of the particular sample.
The primary observables of interest are quantities and prices. Quantities are annual gasoline consumption, which is constructed from odometer readings and an estimate of fuel efficiency. Prices are the average tax-inclusive price per gallon, in the county where the individual lives.
First note it is possible to map the raw quantities and prices to a dataset $\{ (X^i, P^i) \}_{i = 1}^n$, and then apply our previous analysis. Here $i$ denotes the individual and $n$ denotes the sample size. We use this notation rather than $t$ and $T$ to emphasize we have a cross-section. We use upper case $X^i$ and $P^i$ to denote that these are random variables.
We do not use the raw dataset, and instead “pre-process it” to map to our framework. We do so because we have a cross-section of individuals. We wish to both to diminish the impact of sampling variability as well as incorporate heterogeneity along observable variables.\footnote{allen2020satisficing study how stochastic shocks and approximation error can be studied in a common framework. That paper provides several aggregation theorems, and discusses a representative agent in this setting.} As in, blundell2012measuring we pre-process by first considering a partially linear model given by \[ X = g(P,Y) + \beta'W + U, \] where $P$ is price, $Y$ is income, $W$ are observed covariates, and $U$ is unobservable heterogeneity. While blundell2012measuring interpret $g$ as a demand curve for a representative agent for the general model of utility maximization subject to a budget constraint, here we have a different interpretation. We interpret $g(\cdot, Y)$ as the demand curve for the representative agent with income level $Y$; $Y$ thus serves as a demographic characteristic that alters the shape of the demand curve. We close the model with the restriction \[ \mathbb{E}[ U \mid P = p, Y = y, W = w] = 0. \] This specification allows price sensitivity to depend on the level of income of an individual. For each level of income $\tilde{y}$, we consider a dataset (in the sense used previously in the paper) of the form \[ D(\tilde{y}) = \left\{ \hat{g}(P^i,\tilde{y}), P^i \right\}_{i = 1}^{\tilde{n}}, \] where $\hat{g}$ is an estimator of $g$ described below. Thus, $\hat{g}(P^i,\tilde{y})$ is akin to the structural quantity $x^t$ in the previous notation, and $P^i$ is akin to $p^t$. Like blundell2012measuring, we consider prices between the $5$-th and $95$-th quantile to mitigate endpoint issues, so $\tilde{n}$ enumerates these observations.
The estimator $\hat{g}$ is constructed similar to blundell2012measuring. We first estimate $\hat{\beta}$ by a double residual regression as in robinson1988root.\footnote{We use the biweight kernel with ad hoc bandwidth $.75$ after standardizing the data.} Then we set \[ \hat{g}(P^i, \tilde{y}) = \frac{\sum_{j = 1}^N \left(X^j - \hat{\beta}'W_j\right) K_{h_p}(P^j - P^i)K_{h_y}(Y^j -\tilde{y})}{\sum_{j = 1}^N K_{h_p}(P^j - P^i)K_{h_y}(Y^j -\tilde{y})}, \] where $K_h$ is a kernel with bandwith $h$. Following blundell2012measuring we use the biweight kernel. Throughout, the bandwidths $h_p$ and $h_y$ are chosen so that \[ \frac{h_p}{h_y} = \frac{\hat{\sigma}_P}{\hat{\sigma}_Y}, \] where $\hat{\sigma}^2_P = \frac{1}{n}\sum_{i = 1}^n (P^i - \overline{P})^2$, $\overline{P} = \frac{1}{n} \sum_{i = 1}^n P_i$, $\hat{\sigma}^2_Y = \frac{1}{n}\sum_{i = 1}^n (Y^i - \overline{Y})^2$, and $\overline{Y} = \frac{1}{n} \sum_{i = 1}^n Y_i$. Note that these are all constructed with all observations.
Figure (ref) presents analysis with two choices of bandwidths. These correspond to the ad hoc choices $.75$ and $1$ after standardizing. The top two panels display the kernel-smoothed “dataset” \[ D \left(\overline{Y} \right) = \left\{ \hat{g} \left(P^i,\overline{Y} \right), P^i \right\}_{i = 1}^{\tilde{n}} \] as well as counterfactual bounds, where $\hat{g} \left(P^i,\overline{Y} \right)$ is interpreted as a quantity for observation $i$ facing prices $P^i$. Recall $\tilde{n}$ denotes the middle $90\%$ of observations in terms of price, where we drop the lower and upper $5\%$ to mitigate endpoint issues. Income is evaluated at the sample mean $\overline{Y}$. The welfare bounds for approximate indirect utility are displayed in the middle panels. The bounds are evaluated relative to the mean price $\overline{P} = 1.334$. The bounds for differences in utility for certain quantities are displayed in the lower panels. The utility bounds are relative to the median quantity in the dataset. There are 101 comparisons, which for computational reasons are made between 101 of the points in the dataset of quantities $\left\{ \hat{g} \left(P^i,\overline{Y} \right) \right\}_{i = 1}^{\tilde{n}}$. Recall from Proposition (ref), comparisons in utility when one quantity is not in the dataset will have at least one trivial bound. This is why we restrict attention to comparisons in which both quantities are in the dataset. It is important to note that in practice, simply bounding quantities over a grid will lead to trivial bounds for many points in the grid.
As can be seen from the figures, the choice of bandwidth noticeably alters the informativeness of the counterfactual bounds (upper panels). In contrast, the bounds on approximate indirect utility (middle panels) and utility (lower panels) are relatively narrow for both bandwidths. Similar results obtain for alternative bandwidths and are available upon request.
In the lower panels, a contrast emerges between the lower and upper bounds on utility. Recall that these bounds are for $u(\tilde{x}^1) - u(\tilde{x}^0)$, where $\tilde{x}^0$ is the median quantity, i.e. in the dataset. Because the second argument is in the dataset, Proposition (ref) applies to the upper bounds and establishes monotonicity in the argument $\tilde{x}^1$. The graphs are consistent with this, since the upper bounds are monotone in quantities. In contrast, the lower bounds are not monotone. In order to make lower bounds on utilities monotone, it would be necessary to have the first argument be in the dataset and fixed ($\tilde{x}^1$). See Proposition (ref)(vi) in Appendix (ref).
This paper provides a conceptual framework for counterfactual and welfare analysis for approximate models. Our main conceptual assumption is that model approximation error has the same magnitude in new settings as the data we have seen. We formalize this for the quasilinear utility model. This assumption is portable to other settings, and generalizes the standard approach that requires correct specification in both the data we have seen and at hypothetical values.
Engaging with the possibility that a model may not perfectly match data is especially important using the nonparametric revealed preference approach. Indeed, a natural intuition is that if approximation error is “small,” then it is second order and we can ignore it for certain questions. Unfortunately, this intuition is false in the standard approach used in the revealed preference literature, since small violations of the model mean it cannot be used for counterfactual or welfare analysis. This paper presents an adaptive approach allowing the analyst to use the model formally viewing it as an approximation. Moreover, our counterfactual/welfare bounds are continuous in the degree of approximation error, and so they continuously transition to the standard framework when approximation error is negligible.
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