# Working Papers

**Mixed Logit and Pure Characteristics Models **joint with Jay Lu [working paper pdf]

Mixed logit or random coefficients logit models are used extensively in empirical work while pure characteristic models feature in much of theoretical work. We provide a theoretical analysis of the relationship between the two classes of models. First, we show an approximation theorem that precisely characterizes the extent and limits of mixed logit approximations of pure characteristic models. Second, we present two conditions that highlight behavioral differences between the two classes of models. The first is a substitutability condition that is satisfied by many pure characteristic models (including the Hotelling model of horizontal differentiation) but is violated by almost all mixed logit models. The second is a continuity condition that is satisfied by all pure characteristic models but is violated by all mixed logit models. Both conditions pertain to choice patterns when product characteristics change or new products are introduced and illustrate the limitations of using mixed logit models for counterfactual analysis.

**Repeated Choice: A Theory of Stochastic Intertemporal Preferences** joint with Jay Lu [working paper pdf] [submitted version pdf] [30 min presentation video]

We provide a repeated-choice foundation for stochastic choice. We obtain necessary and sufficient conditions under which an agent's observed stochastic choice can be represented as a limit frequency of optimal choices over time. In our model, the agent repeatedly chooses today's consumption and tomorrow's continuation menu, aware that future preferences will evolve according to a subjective ergodic utility process. Using our model, we demonstrate how not taking into account the intertemporal structure of the problem may lead an analyst to biased estimates of risk preferences. Estimation of preferences can be performed by the analyst without explicitly modeling continuation problems (i.e. stochastic choice is independent of continuation menus) if and only if the utility process takes on the standard additive and separable form. Applications include dynamic discrete choice models when agents have non-trivial intertemporal preferences, such as Epstein-Zin preferences. We provide a numerical example which shows the significance of biases caused by ignoring the agent's Epstein-Zin preferences.

**Decision Making under Uncertainty: An Experimental Study in Market Settings** Current Version: Dec 06 , 2019 joint with Federico Echenique and Taisuke Imai. [pdf]

We design and implement a novel experimental test of subjective expected utility theory and its generalizations. Our experiments are implemented in the laboratory with a student population and pushed out through a large-scale panel to a general sample of the U.S. population. We find that a majority of subjects' choices are consistent with the maximization of some utility function, but not with subjective expected utility theory. The theory is tested by gauging how subjects respond to price changes. A majority of subjects respond to price changes in the direction predicted by the theory, but not to a degree that makes them fully consistent with subjective expected utility. Surprisingly, maxmin expected utility adds no explanatory power to subjective expected utility. Our findings remain the same regardless of whether we look at laboratory data or the panel survey, even though the two subject populations are very different. The degree of violations of subjective expected utility theory is not affected by age nor cognitive ability,

but it is correlated with financial literacy.

**Axiomatizations of the Mixed Logit Model** First Draft:
July 29, 2017, Current Version: June 17, 2018. [pdf]

A mixed logit function, also known as a random-coefficient
logit function, is an integral of logit functions. Necessary and sufficient
conditions are provided under which a random choice function can be represented
as a mixed logit function. The axioms are based on the social surplus function
proposed by McFadden (1978, 1981).

**Approximate Expected Utility Rationalization** Current
Version: June 17, 2018 joint with Federico Echenique and Taisuke Imai. [pdf][online
appendix]

We propose a new measure of deviations from expected
utility, given data on economic choices under risk and uncertainty. In a
revealed preference setup, and given a positive number e, we provide a
characterization of the datasets whose deviation (in beliefs, utility, or
perceived prices) is within e of expected utility theory. The number e can then
be used as a distance to the theory. We apply our methodology to three recent
large-scale experiments. Many subjects in those experiments are consistent with
utility maximization, but not expected utility maximization. The correlation of
our measure with demographics is also interesting, and provides new and
intuitive findings on expected utility.

**A Relationship between Risk and Time** First Draft:
February 10, 2011, Current Version: April 23, 2015. [pdf]

This paper investigates a general relationship between risk
and time preferences. I consider a decision maker who chooses between
consumption of a particular prize in one period and a different prize in
another period. The individual believes that today's good is certain, and that,
as the promised date for a future good becomes increasingly distant, the
probability of his consuming the good decreases. Under these assumptions, this
paper shows that the individuals exhibits the common ratio effect, the
certainty effect, and the expected utility if and only if he discounts
hyperbolically, quasi-hyperbolically and exponentially, respectively.