Random Utility with Aggregation

Abstract

We study random utility (RU) rationality with aggregation when the underlying alternatives in each aggregate vary across consumers and are unobserved, as is typical for an outside option. RUM over the underlying alternatives is the natural assumption on the data generating process, while an aggregated random utility model (ARUM) is the standard empirical tool. We characterize RU rationality in three frameworks and show its testable implications are substantially weaker than those of an ARUM. We provide two independent conditions for their equivalence: non-overlapping preferences within aggregates and menu-independent aggregation. Simulations show that violating either condition produces meaningful estimation bias when imposing an ARUM.

Figures (20)

• Figure 1: RU polytope (blue), ARU polytope ${\mathcal{P}}_{\mathscr{A}}$ (red), and RU(3). RU(3) contains all vertices (i.e., $\rho^{\succ_i}_{\mathscr{E}_i}$) and all line segments connecting two vertices.
• Figure 2: Maximum positive bias (red), maximum negative bias (purple), minimum absolute bias (blue), and bias under menu-independent $\lambda$ (green), plotted across values of $\lambda_{\{x,y,a_0\}}$. The horizontal axis reports $\lambda_{\{x,y,a_0\}}(\{w\})$ and the vertical axis reports $\lambda_{\{x,y,a_0\}}(\{z\})$. The maximum and minimum biases are obtained by optimizing over all admissible values of $\lambda_{\{x,a_0\}}$ and $\lambda_{\{y,a_0\}}$.
• Figure 3: Random utility polytope (blue) and aggregated random utility polytope ${\mathcal{P}}_{\mathscr{A}}$ (red). The orthogonal projection illustrates the distance from $\rho$ to ${\mathcal{P}}_{\mathscr{A}}$
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Theorems & Definitions (61)

• Definition 2.1
• Example 2.1
• Remark 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.2
• Remark 2.3
• Definition 2.5
• Remark 2.4