Revealed Preferences over Risk and Uncertainty

Abstract

We develop a nonparametric method, called Generalized Restriction of Infinite Domains (GRID), for testing the consistency of budgetary choice data with models of choice under risk and under uncertainty. Our test can allow for risk-loving and elation-seeking attitudes, or it can require risk aversion. It can also be used to calculate, via Afriat’s efficiency index, the magnitude of violations from a particular model. We evaluate the performance of various models under risk (expected utility, disappointment aversion, rank-dependent utility, and stochastically monotone utility) using data collected from several recent portfolio choice experiments. (JEL C14, D11, D12, D81)

A lattice test for additive separability

“…While the lattice approach developed in this paper relates in concept to Polisson, Quah, and Renou (2017), it differs on two important accounts: (1) here we construct some arbitrary number subutility functions, one for every consumption good, each of which is required to be increasing and continuous, rather that a single Bernoulli utility function over state-contingent consumption; and (2) our method of proof involves constructing subutility functions which are piecewise linear, and this can be done explicitly, whereas Polisson, Quah, and Renou Returning to the example, we conclude that these data are, in fact, rationalizable by additive separability. Furthermore, they are also rationalizable by concave additive separability (see Varian (1983) and Diewert and Parkan (1985)), i.e., the consumer's choices are consistent with a smoothing across goods.…”

“…Notice, crucially, that the budget set B t (e) is not convex. 15 Since the lattice approach does not require this, it can be applied straightforwardly to check for rationalizations on nonconvex budget sets, and as in Polisson, Quah, and Renou (2017), this is one of its important features, particularly with regard to empirical work. 16…”

“…For each synthetic household, a set of 50 budgets has again been selected at random from among the sample of 4,027; however, to generate random choices from these 50 budgets which obey GARP, we do so sequentially. See Polisson, Quah, and Renou (2017) for the details of such a procedure. are rationalizable by utility maximization, and 15% by additive separability; naturally, the power measures are greater still at higher efficiency thresholds.…”

“…Furthermore, unlike previous revealed preference work on additive separability (see Varian (1983), Diewert and Parkan (1985), and Fleissig and Whitney (2007)), we do not require convexity of the preference or of the budget set a priori. Instead we extend the lattice approach (see Polisson, Quah, and Renou (2017)) to achieve domain reduction; crucially, the lattice approach does not appeal to the techniques of convex optimization, which allows for an easy extension to accommodate departures from full economic rationality, or errors, in the form of cost inefficiencies (see, e.g., Afriat (1972Afriat ( , 1973 and Varian (1990)).…”

“…The latter, crucially, can be checked. InPolisson, Quah, and Renou (2017), the lattice method allows for revealed preference tests of many different models of decision making under risk and under uncertainty, and without requiring concavity of the Bernoulli utility function a priori. In these settings, many utility representations take a separable form, and so the problem inPolisson, Quah, and Renou (2017) has a functional structure similar to additive separability as described in this paper.…”