Dynamic Pareto Optima in Multi-Period Pure-Exchange Economies

Abstract

We study a problem of optimal allocation in a discrete-time multi-period pure-exchange economy, where agents have preferences over stochastic endowment processes that are represented by strongly time-consistent dynamic risk measures. We introduce the notion of dynamic Pareto-optimal allocation processes and show that such processes can be constructed recursively starting with the allocation at the terminal time. We further derive a comonotone improvement theorem for allocation processes, and we provide a recursive approach to constructing comonotone dynamic Pareto optima when the agents' preferences are coherent and satisfy a property that we call equidistribution-preserving. In the special case where each agent's dynamic risk measure is of the distortion type, we provide a closed-form characterization of comonotone dynamic Pareto optima. We illustrate our results in a two-period setting.

Figures (5)

• Figure 1: Distortion functions $k_2^{*(1)}$ (blue) and $k_2^{*(1)}$ (red) at time 2.
• Figure 2: Assessments of tail risks $k_2^{*(1)}({\mathbb{P}}(S_2>x))$ (blue) and $k_2^{*(2)}({\mathbb{P}}(S_2>x))$ (red) at time 2.
• Figure 3: PO at time 2 retention functions $g_2^{*(1)}(S_2)$ (blue) and $g_2^{*(2)}(S_2)$ (red).
• Figure 4: Assessments of tail risks $k_1^{*(1)}({\mathbb{P}}(S_1+{\overline{R}}_1>x))$ (blue) and $k_1^{*(2)}({\mathbb{P}}(S_1+{\overline{R}}_1>x))$ (red) at time 1.
• Figure 5: The PO at time 1 retention functions $g_1^{*(1)}(S_1+{\overline{R}}_1)$ (blue) and $g_1^{*(2)}(S_1+{\overline{R}}_1)$ (red) are plotted on the left. On the right, we plot the total risk retained by the agent minus the risk-to-go of the agent.

Theorems & Definitions (61)

• Definition 2.1: Conditional Risk Measure
• Definition 2.2: Properties of Conditional Risk Measures
• Definition 2.3: Dynamic Risk Measure
• Definition 2.4: Time Consistency
• Theorem 2.5: Recursive Relation
• Definition 3.1: Allocation - Dynamic
• Definition 3.3: Myopic Individual Rationality
• Definition 3.4: Myopic Pareto Optimality
• Proposition 3.5
• Definition 3.6: Dynamic Individual Rationality