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A Sparsity-Based Model of Bounded Rationality
[report]

Xavier Gabaix

2011
unpublished

I thank David Laibson for a great many enlightening conversations about behavioral economics over the years. For very helpful comments, I thank the editor and the referees, ABSTRACT This paper defines and analyzes a "sparse max" operator, which is a less than fully attentive and rational version of the traditional max operator. The agent builds (as economists do) a simplified model of the world which is sparse, considering only the variables of first-order importance. His stylized model and his
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... resulting choices both derive from constrained optimization. Still, the sparse max remains tractable to compute. Moreover, the induced outcomes reflect basic psychological forces governing limited attention. The sparse max yields a behavioral version of two basic chapters of the microeconomics textbook: consumer demand and competitive equilibrium. We obtain a behavioral version of Marshallian and Hicksian demand, the Slutsky matrix, the Edgeworth box, Roy's identity etc. The Slutsky matrix is no longer symmetric: non-salient prices are associated with anomalously small demand elasticities. Because the consumer exhibits nominal illusion, in the Edgeworth box, the offer curve is a twodimensional surface rather than a one-dimensional curve. As a result, different aggregate price levels correspond to materially distinct competitive equilibria, in a similar spirit to a Phillips curve. This framework provides a way to assess which parts of basic microeconomics are robust, and which are not, to the assumption of perfect maximization. An online appendix is available at: http://www.nber.org/data-appendix/w16911 This paper proposes a tractable model of some dimensions of bounded rationality (BR). It develops a "sparse max" operator, which is a behavioral version of the traditional "max" operator, and applies to general problems of maximization under constraint. 1 In the sparse max, the agent pays less or no attention to some features of the problem, in a way that is psychologically founded. I use the sparse max to propose a behavioral version of two basic chapters of the economic textbooks: consumer theory (problem 1) and basic equilibrium theory. The principles behind the sparse max are the following. First, the agent builds a simplified model of the world, somewhat like economists do, and thinks about the world through this simplified model. Second, this representation is "sparse," i.e., uses few parameters that are non-zero or differ from the usual state of affairs. These choices are controlled by an optimization of his representation of the world that depends on the problem at hand. I draw from fairly recent literature on statistics and image processing to use a notion of "sparsity" that still entails well-behaved, convex maximization problems (Tibshirani (1996) , Candès and Tao (2006) ). The idea is to think of "sparsity" (having lots of zeroes in a vector) instead of "simplicity" (which is an amorphous notion), and measure the lack of "sparsity" by the sum of absolute values. This paper follows this lead to use sparsity notions in economic modelling, and to the best of my knowledge is the first to do so. 2 "Sparsity" is also a psychologically realistic feature of life. For any decision, in principle, thousands of considerations are relevant to the agent: his income, but also GDP growth in his country, the interest rate, recent progress in the construction of plastics, interest rates in Hungary, the state of the Amazonian forest, etc. Since it would be too burdensome to take all of these variables into account, he is going to discard most of them. The traditional modelling for this is to postulate a fixed cost for each variable. However, that often leads to discontinuous reactions and intractable problems (fixed costs, with their non-convexity, are notoriously ill-behaved). In contrast, the notion of sparsity I use leads to continuous reactions and problems that are easy to solve. The model rests on very robust psychological notions. It incorporates limited attention, of course. To supply the missing elements due to limited attention, people rely on defaults -which are typically the expected values of variables. At the same time, attention is allocated purposefully, towards features that seem important. When taking into account some information, agents anchor on the default and do a limited adjustment towards the truth, as in Tversky and Kahneman's (1974) "anchoring and adjustment". 3 1 The meaning of "sparse" is that of a sparse vector or matrix. For instance, a vector ∈ R 100000 with only a few non-zero elements is sparse. In this paper, the vector of things the agent considers is (endogenously) sparse. 2 Econometricians have already successfully used sparsity (e.g. Belloni and Chernozhukov 2011). 3 In models with noisy perception, an agent optimally responds by shading his noisy signal, so that he optimally underreacts (conditionnally on the true signal). Hence, he behaves on average as he misperceives the truth -indeed, perceives only a fraction of it. The sparsity model displays this "partial adjustment" behavior even though it is deterministic (see Proposition 15). The sparse agent is in part a deterministic "representative agent" idealization of such an agent with noisy perception. 4 Dufwenberg et al. (2011) analyze competitive equilibrium with other-regarding, but rational, preferences. 5 Recall that the "offer curve" of an agent is the set of consumption bundles he chooses as prices change (those price changes also affecting the value of his endowment). 6 This notion is very different from the idea of a "thick indifference curve", in which the consumer is indifferent between dominated bundles. A sparse consumer has only a thin indifference curve. 7 The paper discusses the empirical relevance and underlying conditions for the deviations expressed here.

doi:10.3386/w16911
fatcat:ocwc7jv5n5d3bpbabkarbhgrfi