Long-run survival in limited stock market participation models with power utilities

We extend the limited participation model in Basak and Cuoco (1998) to allow for traders with different time-preference coefficients but identical constant relative risk-aversion coefficients. Our main result gives parameter restrictions which ensure the existence of a Radner equilibrium. As an application, we give further parameter restrictions which ensure all traders survive in the long run.

💡 Research Summary

The paper revisits the classic limited‑participation Radner equilibrium model of Basak and Cuoco (1998) and introduces heterogeneity in agents’ time‑preference rates while keeping a common constant relative risk‑aversion (CRRA) coefficient γ∈(0,1). Two agents are considered: Agent 1 can trade both a risky stock and a risk‑free bond, whereas Agent 2 is restricted to the bond only. Both agents maximize power‑utility objectives of the form (c^{1‑γ})/(1‑γ) discounted at rates β₁ and β₂ respectively. The authors derive the agents’ optimal consumption‑investment policies, which are driven by a single state variable Y_t representing Agent 1’s consumption share.

A central technical contribution is the analysis of a singular, path‑dependent first‑order ordinary differential equation (ODE) that characterises the equilibrium price‑impact function h(y). The ODE (equations (2.9)–(2.10)) contains linear terms a₀(y), a₁(y), a non‑local term a₂(h,y) defined via an integral of h, and a new cubic term proportional to the difference β₂‑β₁. By imposing the parameter restrictions

 δ = 2(β₂‑β₁)σ_D² ∈ (‑γ, 0) and A = 2β₂ + σ_D² – (1‑γ)(2μ_D‑γσ_D²)/σ_D² ∈ (1+δ‑2δγ, ∞),

the authors prove (Lemma 2.3) that there exists a unique C¹ solution h on