Identification of Preferences with Unobserved Budget Using Vertical Differentiation within Brands

A lot of discrete-choice analyses routinely skip over the affordability constraints that affect consumers’ consideration sets, thereby crediting taste heterogeneity for behavior that is, in truth, driven by limited opportunity. This paper develops a framework for identifying preferences and budget distribution in discrete choice models where unobserved budget constraints determine consumers’ consideration sets. Building on the vertical differentiation within brands—where products are ordered by quality and price—we exploit the single-crossing property to recover preference parameters and the distribution of unobserved budgets. Our key insight leverages the asymmetric consumer response to price and quality changes for products that are “Unaffordable First-Best” for marginal consumers. When a product’s quality attributes do not affect consumers’ choices but its price does, this asymmetry reveals information about consumer budget constraints. We show that preference parameters can be identified without knowing the budget distribution, and once these parameters are identified, the budget distribution can be nonparametrically recovered. We extend our analysis to settings with subsistence levels and show that both preference parameters and subsistence levels can be jointly identified using panel data. Our approach contributes to the literature on nonseparable models and offers practical insights for markets with large-ticket items where affordability constraints significantly influence consumer choices.

Recommended citation: Son, Woohun. (2025). "Identification of Preferences with Unobserved Budget Using Vertical Differentiation within Brands."http://SonWoohun.github.io/research/2025-08-05-unobs-budget
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