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Testing stochastic rationality and predicting stochastic demand: the case of two goods

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Abstract

This paper precisely delineates the empirical content of consumer rationality in the following setting: Data are from a repeated cross section; unobserved heterogeneity is completely unrestricted; however, there are only two goods. Simple closed-form expressions determine whether (population level) data are consistent with these assumptions. Bounds on counterfactual distributions of demand, and parameters thereof, follow. A striking finding is that any rationalizable collection of cross-sectional distributions can be rationalized by pretending that the ordering of individual consumers on the budget line is maintained across budgets. Hence, this seemingly strong assumption does not tighten the bounds.

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Notes

  1. The distribution \(G\) defines a random utility model; thus, \(\left\{ F_{t}\right\} _{t\in \mathcal {T}}\) is stochastically rationalizable iff it can arise as population distribution from some random utility model. In principle, the model could also be interpreted as describing single-agent stochastic choice, but our assumptions are geared to demand analysis with repeated cross sections.

  2. The name “rank invariance” is inspired by Chernozhukov and Hansen (2005), who impose the assumption in a treatment effects context. Hoderlein and Stoye (2014) call the same assumption “quantile constancy.”

  3. Andreoni and Miller (2002) observe repeated choices by the same consumers in a sequence of linear budget sets with two goods. Rank invariance is frequently violated; one violation is visually apparent from comparing panels c and d of their Figure 2. We are not aware of direct tests of rank invariance outside the laboratory. This would require consumer panel data, which are rare; see Christensen (2014) for references and a discussion. That said, the heterogeneity of income effects estimated by Hoderlein et al. (2010) and Christensen (2014) would preclude rank invariance in two-good versions of their models.

  4. See McFadden and Richter (1991), Matzkin (2006, for the continuous case), and HS. We provide an independent proof because it is very short and instructive. (iii)\(\Rightarrow \)(i) can also be proved by combining results in Bandyopadhyay et al. (2002, 2004).

  5. For clarity, we here abstract from any measurement error or any other “\(\varepsilon _{it}\)” term that would induce violations of rank invariance in observed data even if it were a true feature of the distribution of preferences.

  6. Recall that SARP excludes revealed preference cycles of arbitrary length. In this paper’s notation, call \(q(\mathbf {p}_{s},\mathbf {a})\) revealed preferred to \(q(\mathbf {p}_{t},\mathbf {a})\) if \(p_{s}^{1}\left( 1-p_{s}^{2}q( \mathbf {p}_{t},\mathbf {a})\right) /p_{s}^{1}+p_{s}^{2}q(\mathbf {p}_{t}, \mathbf {a})\le 1\), then there must not exist any finite length cycle of revealed preference in which at least one of the inequalities is strict.

  7. This remark applies only to nonparametric revealed preference analysis along the lines of this paper, e.g., nonparametric extrapolation from data in Andreoni and Miller (2002). If one uses panel data to estimate more structured models, the panel dimension might certainly be useful. Also, a given panel data set might reveal violations of WARP. If one is willing to attribute them to data issues but wants to pose a well-defined extrapolation problem, one will have to “iron them out” in some manner; else, there will not exist a rationalizable joint distribution of choices over all budgets. Our remark then applies to the ironed out data.

  8. See Hoderlein and Stoye (2014) for definitions of these concepts.

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Authors and Affiliations

  1. Department of Economics, Boston College, 140 Commonwealth Avenue, Chestnut Hill, MA, 02467, USA

    Stefan Hoderlein

  2. Department of Economics, Cornell University, Uris Hall, Ithaca, NY, 14850, USA

    Jörg Stoye

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  1. Stefan Hoderlein
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  2. Jörg Stoye
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Correspondence to Jörg Stoye.

Additional information

Stoye gratefully acknowledges support through NSF Grant SES-1260980. We thank Chuck Manski and two anonymous referees for helpful comments but assume responsibility for any and all errors.

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Hoderlein, S., Stoye, J. Testing stochastic rationality and predicting stochastic demand: the case of two goods. Econ Theory Bull 3, 313–328 (2015). https://doi.org/10.1007/s40505-014-0061-5

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