Hicksian complementarity and perturbed utility models
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Hicksian complementarity and perturbed utility models Roy Allen1 · John Rehbeck2 Received: 26 July 2019 / Accepted: 7 November 2019 © Society for the Advancement of Economic Theory 2019
Abstract This paper studies aggregate complementarity without price or income variation. We show that for a class of utility functions, variation in non-price observables allows one to recover a measure of complementarity similar to Hicksian complementarity. In addition, the entire Slutsky matrix can be recovered up to scale without price variation. We then examine aggregate complementarity in latent utility models used in discrete choice, bundles, and matching. We show that classical linear instrumental variables can recover Hicksian complementarity for the special case of quadratic utility. Keywords Hicksian complementarity · Demand · Instrumental variables JEL Classification D01 · D11 · C10
1 Introduction The classical definition of complementarity labels two goods complements if the crossprice derivative of compensated demand is negative (Hicks and Allen 1934). The compensated nature of the Hicksian definition makes it difficult to apply, especially when income does not vary. Moreover, there are environments where prices do not vary or there are goods that are unpriced, such as the study of online and print news of Gentzkow (2007). Outside of standard demand settings, one may be interested in measuring complementarity when prices and income do not even hypothetically vary.
Some results in this paper have appeared in the deprecated working paper “Complementarity in Perturbed Utility Models”. We thank Christopher P. Chambers, Tim Conley, Ivana Komunjer, Andres Santos, Jeff Shrader, and Yixiao Sun for helpful comments.
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John Rehbeck [email protected] Roy Allen [email protected]
1
Department of Economics, University of Western Ontario, London, Canada
2
Department of Economics, The Ohio State University, Columbus, USA
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R. Allen, J. Rehbeck
In this paper, we study the measurement of complementarity using variation in nonprice observables that alter the desirability of goods. We propose a derivative-ratio measure of complementarity. We show that for perturbed utility models (McFadden and Fosgerau 2012; Allen and Rehbeck 2019b), a derivative ratio (involving variation in non-price observables) is closely related to Hicksian complementarity. An important feature of perturbed utility models for this analysis is that non-price observables for each good only shift the marginal utility for that good. Perturbed utility models arise upon aggregating several latent utility models, including the discrete choice additive random utility model (McFadden 1981), the bundles model of Gentzkow (2007), and a model of one-to-one matching (Fox et al. 2018). Since the perturbed utility model can be interpreted as an individual or aggregate model, the measure of complementarity can be used for individual or aggregate data. To measure complementarity, we exploit changes in non-price observables. For example, food items like yogurt contain info
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