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Waiting for Trade in Markets with Aggregate Uncertainty

Justus Preusser

TL;DR

The paper develops a dynamic model of learning under aggregate uncertainty about gains from trade, featuring a patient seller and privately informed, randomly arriving buyers with interdependent values. It shows that with commitment, the seller optimally waits for the most favorable buyer signal and exits at a type-dependent time, achieving the maximal expected surplus via a Lehmann-most informative experiment. Without commitment, equilibria exhibit distortions: exit times are delayed relative to the commitment benchmark and trade is more frequent than efficient, yielding a welfare gap between commitment and equilibrium. The analysis highlights how opacity and information structure shape equilibria, and suggests policy tools, like taxes, to mitigate inefficiencies by encouraging earlier exits. Overall, the work links dynamic information aggregation, Lehmann order, and intertemporal frictions to a nuanced view of efficiency in markets with aggregate uncertainty.

Abstract

This paper studies learning in markets with aggregate uncertainty about whether trade is efficient. A long-lived seller offers prices to buyers, who are short-lived and arrive according to a Poisson process. A hidden state determines whether the buyers' common value exceeds the seller's reservation value. All parties observe noisy, private signals about the state. With small intertemporal frictions and when the seller has commitment power, the seller waits for a buyer with the most favorable signal to arrive up to an exit time that depends on the seller's private information. This strategy profile maximizes both the seller's profit and the expected surplus. Without commitment, the commitment profit is unattainable. Instead, there is an equilibrium in which the seller also waits for a buyer with the most favorable signal, but, relative to the commitment case, the seller exits inefficiently late, and the trade probability is inefficiently high.

Waiting for Trade in Markets with Aggregate Uncertainty

TL;DR

The paper develops a dynamic model of learning under aggregate uncertainty about gains from trade, featuring a patient seller and privately informed, randomly arriving buyers with interdependent values. It shows that with commitment, the seller optimally waits for the most favorable buyer signal and exits at a type-dependent time, achieving the maximal expected surplus via a Lehmann-most informative experiment. Without commitment, equilibria exhibit distortions: exit times are delayed relative to the commitment benchmark and trade is more frequent than efficient, yielding a welfare gap between commitment and equilibrium. The analysis highlights how opacity and information structure shape equilibria, and suggests policy tools, like taxes, to mitigate inefficiencies by encouraging earlier exits. Overall, the work links dynamic information aggregation, Lehmann order, and intertemporal frictions to a nuanced view of efficiency in markets with aggregate uncertainty.

Abstract

This paper studies learning in markets with aggregate uncertainty about whether trade is efficient. A long-lived seller offers prices to buyers, who are short-lived and arrive according to a Poisson process. A hidden state determines whether the buyers' common value exceeds the seller's reservation value. All parties observe noisy, private signals about the state. With small intertemporal frictions and when the seller has commitment power, the seller waits for a buyer with the most favorable signal to arrive up to an exit time that depends on the seller's private information. This strategy profile maximizes both the seller's profit and the expected surplus. Without commitment, the commitment profit is unattainable. Instead, there is an equilibrium in which the seller also waits for a buyer with the most favorable signal, but, relative to the commitment case, the seller exits inefficiently late, and the trade probability is inefficiently high.

Paper Structure

This paper contains 31 sections, 18 theorems, 95 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model
  3. The seller's problem with commitment power
  4. Maximizing the expected surplus
  5. The commitment profit
  6. Optimal exit times
  7. Comparative statics
  8. Equilibria
  9. The commitment profit is unattainable in equilibrium
  10. An equilibrium with passive beliefs
  11. Comparison to the commitment solution
  12. Encouraging early exits
  13. Proofs: commitment
  14. A fundamental single-crossing lemma
  15. The posterior surplus and optimal exit times
  16. ...and 16 more sections

Key Result

Lemma 1

There exists $\bar{\delta} > 0$ such that for all breakdown rates $\delta\in [0, \bar{\delta})$ it holds $\bar{V} = \sup_{\sigma\in\Sigma} V(\sigma)$.

Figures (1)

  • Figure 1: The exit times $\bar{s}$ in \ref{['example:supermodf']} for various priors. Here, $[\underline{y}, \bar{y}] = [\underline{\omega}, \bar{\omega}] = [1/4, 3/4]$. The surplus is given by $v(\bar{\omega}) = - v(\underline{\omega}) = 1$. The likelihood ratio of the seller's type $y$ (which suffices to pin down the exit times) is given by $g(y\vert\underline{\omega}) / g(y\vert\bar{\omega}) = 9/10 - y$.

Theorems & Definitions (36)

  • Lemma 1
  • Remark 1: Learning interpretation
  • Remark 2: Discrete signals and breakdowns
  • Remark 3: Comparison to static environments
  • Theorem 1
  • Proposition 1
  • Example 1: Submodular arrivals
  • Proposition 2
  • Example 2: Supermodular arrivals
  • Proposition 3
  • ...and 26 more