Existence of an equilibrium with limited stock market participation and power utilities

Paper: arXiv 2402.07185

Authors: Paolo Guasoni, Kasper Larsen, Giovanni Leoni

Abstract

For constants $γ\in (0,1)$ and $A\in (1,\infty)$, we prove existence and uniqueness of a solution to the singular and path-dependent Riccati-type ODE \begin{align*} \begin{cases} h’(y) = \frac{1+γ}{y}\big( γ- h(y)\big)+h(y)\frac{γ+ \big((A-γ)e^{\int_y^1 \frac{1-h(q)}{1-q}dq}-A\big)h(y)}{1-y},\quad y\in(0,1), h(0) = γ, \quad h(1) = 1. \end{cases} \end{align*} As an application, we use the ODE solution to prove existence of a Radner equilibrium with homogenous power-utility investors in the limited participation model from Basak and Cuoco (1998).

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical proof for the existence and uniqueness of a solution to a complex, path-dependent Riccati-type ODE, establishing existence for a classic Radner equilibrium model. However, it is purely theoretical with no empirical data or backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Prove Radner Equilibrium in Limited Participation Model with Power Utility"] --> B["Inputs: Constants γ ∈ (0,1) & A ∈ (1,∞)"]
    B --> C["Formulate Singular Path-Dependent Riccati-Type ODE"]
    C --> D["Mathematical Analysis: Prove Existence & Uniqueness of ODE Solution h(y)"]
    D --> E["Apply ODE Solution to Basak & Cuoco (1998) Financial Economy"]
    E --> F["Outcome: Rigorous Proof of Radner Equilibrium Existence & Asset Pricing Characteristics"]