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

For constants $\gamma \in (0,1)$ and $A\in (1,\infty )$, we prove existence and uniqueness of a solution to the singular and path-dependent Riccati-type ODE$\{\begin{array}{c}{h}^{\prime }\left(y\right)=\frac{1+\gamma }{y}(\gamma -h(y\left)\right)+h\left(y\right)\frac{\gamma +\left(\right(A-\gamma )\exp \{{\int }_y^1\frac{1-h\left(q\right)}{1-q}dq\}-A)h\left(y\right)}{1-y},y\in (0,1),\\ h\left(0\right)=\gamma ,h\left(1\right)=1.\end{array}$ As an application, we use the ODE solution to prove existence of a Radner equilibrium with homogeneous power-utility investors in the limited participation model from Basak and Cuoco (1998).