Afriat's Theorem Research Articles

A Compact, Logical Approach to Large-Market Analysis

Many economic-theoretic models incorporate finiteness assumptions that, while introduced for simplicity, play a real role in the analysis. Such assumptions introduce a conceptual problem, as results that rely on finiteness are often implicitly nonrobust; for example, they may depend upon edge effects or artificial boundary conditions. Here, we present a unified method that enables us to remove finiteness assumptions, such as those on market sizes, time horizons, and datasets. We then apply our approach to a variety of matching, exchange economy, and revealed preference settings. The key to our approach is Logical Compactness, a core result from Propositional Logic. Building on Logical Compactness, in a matching setting, we reprove large-market existence results implied by Fleiner's analysis, and (newly) prove both the strategy-proofness of the manoptimal stable mechanism in infinite markets and an infinite-market version of Nguyen and Vohra's existence result for near-feasible stable matchings with couples. In a trading-network setting, we prove that the Hatfield et al. result on existence of Walrasian equilibria extends to in_nite markets. In a dynamic matching setting, we prove that Pereyra's existence result for dynamic two-sided matching markets extends to a doubly infinite time horizon. Finally, beyond existence and characterization of solutions, in a revealed-preference setting we reprove Reny's infinite-data version of Afriat's theorem and (newly) prove an infinite-data version of McFadden and Richter's characterization of rationalizable stochastic datasets.

Samuelson's Approach to Revealed Preference Theory: Some Recent Advances

Since Paul Samuelson introduced the theory of revealed preference, it has become one of the most important concepts in economics. This chapter surveys some recent contributions in the revealed preference literature. We depart from Afriat's theorem, which provides the conditions for a data set to be consistent with the utility maximization hypothesis. We provide and motivate a new condition, which we call the Varian inequalities. The advantage of the Varian inequalities is that they can be formulated as a set of mixed integer linear inequalities, which are linear in the quantity and price data. We show how the Varian inequalities can be used to derive revealed preference tests for weak separability, and show how it can be used to formulate tests of the collective household model. Finally, we discuss measurement errors in the observed data and measures of goodness-of-fit, power and predictive success.

Revealed Preference, Afriat's Theorem, and Falsifiability: A Review Essay on Revealed Preference Theory by Christopher P. Chambers and Federico Echenique

This paper uses the 2016 book by Chambers and Echenique as the focal point for a general discussion of the revealed preference literature: starting with Samuelson (1938) but with a particular emphasis on its relationship to general equilibrium theory and recent empirical applications. The main purpose is exposition, not criticism, but Chambers and Echenique's commitment to Karl Popper's logical falsificationism is criticized in the final section of the paper.

A Comprehensive Approach to Revealed Preference Theory

We develop a version of Afriat's theorem that is applicable in a variety of choice environments beyond the setting of classical consumer theory. This allows us to devise tests for rationalizability in environments where the set of alternatives is not the positive orthant of a Euclidean space and where the rationalizing utility function is required to satisfy properties appropriate to that environment. We show that our results are applicable, amongst others, to choice data on lotteries, contingent consumption, and intertemporal consumption. (JEL D11, D81)

Nonparametric Demand Forecasting and Detection of Energy Aware Consumers

To increase the reliability of the power grid and reduce the risk of power supply failure, demand-side management (DSM) is of central importance. In this paper, a nonparametric test is applied to detect if the demand behavior of consumers is consistent with time-of-day electricity tariff initiatives. The test is based on Afriat's theorem in economics and has the unique feature that it provides necessary and sufficient conditions to detect if the price-demand behavior is consistent with utility maximization (i.e., the test detects demand-responsive consumers) without prior knowledge of the consumer's utility function. For consumers that are responsive to time-of-day pricing initiatives, a nonparametric learning algorithm is used to forecast power demands for unobserved electricity tariffs. The nonparametric learning algorithm can be used in anticipatory control structures in a DSM framework to achieve power usage objectives. Real-world data from Ontario's power system and numerical examples illustrate the accuracy of the nonparametric test and nonparametric learning algorithm for forecasting consumer demand.

Remarks on the Afriat's Theorem and the Monge-Kantorovich Problem

The famous Afriat's theorem from the theory of revealed preferences establishes necessary and sufficient conditions for existence of utility function for a given set of choices and prices. The result on existence of a homogeneous utility function can be considered as a particular fact of the Monge-Kantorovich mass transportation theory. In this paper we explain this viewpoint and discuss some related questions.

Afriat's Theorem for Indivisible Goods

We identify a natural counterpart of the standard GARP for demand data in which goods are all indivisible. We show that the new axiom (DARP, for discrete axiom of revealed preference) is necessary and sufficient for the rationalization of the data by a well-behaved utility function. Our results complement the main finding of Polisson and Quah, Am. Econ. J.: Micro. 5(1) p.28-34 (2013), who rather minimally modify the original consumer problem with indivisible goods so that the standard GARP still applies.

The housing problem and revealed preference theory: duality and an application

This paper exhibits a duality between the theory of Revealed Preference of Afriat and the housing allocation problem of Shapley and Scarf. In particular, it is shown that Afriat's theorem can be interpreted as a second welfare theorem in the housing problem. Using this duality, the revealed preference problem is connected to an optimal assignment problem, and a geometrical characterization of the rationalizability of experiment data is given. This allows in turn to give new indices of rationalizability of the data, and to define weaker notions of rationalizability, in the spirit of Afriat's efficiency index.

Revealed Preference in a Discrete Consumption Space

We show that an agent maximizing some utility function on a discrete (as opposed to continuous) consumption space will obey the generalized axiom of revealed preference (GARP), so long as the agent obeys cost efficiency. Cost efficiency will hold if there is some good, outside the set of goods being studied by the modeler, that can be consumed by the agent in continuous quantities. An application of Afriat's Theorem then guarantees that there is a strictly increasing utility function on the discrete consumption space that rationalizes price and demand observations. (JEL D11)

Afriat's theorem for general budget sets

Afriat proved the equivalence of a variant of the strong axiom of revealed preference and the existence of a solution to a set of linear inequalities. From this solution he constructed a utility function rationalizing the choices of a competitive consumer. We extend Afriat's theorem to a class of nonlinear, nonconvex budget sets. We thereby obtain testable implications of rational behavior for a wide class of economic environments, and a constructive method to derive individual preferences from observed choices. We also show that by increasing in a regular way the number of observed choices from our class of budget sets one can fully identify the underlying preference relation.

Afriat's Theorem for General Budget Sets

Afriat (1967) showed the equivalence of the strong axiom of revealed preference and the existence of a solution to a set of linear inequalities. From this solution he constructed a utility function rationalizing the choices of a competitive consumer. We extend Afriat's theorem to a class of nonlinear budget sets. We obtain testable implications of rational behavior for a wide class of economic environments, and a constructive method to derive individual preferences from observed choices. In an application to market games, we identify a set of observable restrictions characterizing Nash equilibrium outcomes.