Stochastic Complementarity

Abstract

The Hicksian definition of complementarity and substitutability may not apply in contexts in which agents are not utility maximisers or where price or income variations, whether implicit or explicit, are not available. We look for tools to identify complementarity and substitutability satisfying the following criteria: they are behavioural (based only on observable choice data); model-free (valid whether the agent is rational or not); and they do not rely on price or income variation. We uncover a conflict between properties that any complementarity notion should intuitively possess. We discuss three different possible resolutions of the conflict. J.E.L. codes: D0.

A note on stochastic complementarity for the applied researcher

“…According to this criterion, two products are classified as complements if their purchases are positively correlated, see Brynjolfsson and Milgrom (2013). Manzini et al (2018) illustrate that the correlation criterion partitions the sampling space T into three subsets C, I, and S: the collections of possible values of (P xy , P x , P y , P 0 ) that exhibit complementarity, independence, and substitutability. Monotonicity Axiom.…”

“…1 Introduction Manzini et al (2018) propose various criteria to classify two products or services as complements. The proposed criteria satisfy three requirements:…”

“…(a) they do not rely on price/income or choice set variation, (b) they only rely on observed data on bundle-level consumption/purchase probabilities, and (c) they place no restriction on the data generating process. Manzini et al (2018) specifically attempt to go beyond the classical definition that two products are complements whenever their (compensated) cross-price elasticity is negative (e.g., Samuelson (1974)) and characterize complementarity also in cases of "items" for which there are no prices or choice set variation, such as a park and a museum or different business practices within a firm. 1 As a striking finding, Manzini et al (2018) show that there cannot be any criterion for complementarity satisfying (a), (b), and (c) that also fulfils all of the desirable axioms they propose.…”

“…Manzini et al (2018) specifically attempt to go beyond the classical definition that two products are complements whenever their (compensated) cross-price elasticity is negative (e.g., Samuelson (1974)) and characterize complementarity also in cases of "items" for which there are no prices or choice set variation, such as a park and a museum or different business practices within a firm. 1 As a striking finding, Manzini et al (2018) show that there cannot be any criterion for complementarity satisfying (a), (b), and (c) that also fulfils all of the desirable axioms they propose. In particular, the intuitively appealing correlation criterion (i.e., two products are complements if their purchases are positively correlated, see Brynjolfsson and Milgrom (2013)), in general violates the axiom of monotonicity: 2 products classified as complements can turn into substitutes following an increase in their joint purchases.…”

“…The purpose of this note is to show that by placing restrictions on the data generating process (point (c) above), as typically done by applied researchers, the correlation criterion can satisfy all the axioms suggested by Manzini et al (2018)-including monotonicity. In particular, we demonstrate that by restricting attention to mixed logit models along the lines of Gentzkow ( 2007)'s, one can prevent violations of the monotonicity axiom.…”