Existence and uniqueness of the robust stationary equilibrium

Establish a rigorous proof of existence and uniqueness of the stationary Markovian competitive equilibrium with model uncertainty described in Proposition “Equilibrium with Model Uncertainty.” Specifically, prove that for given parameters (r, l, η, γ, α, θ) and demand D(p) = 1 − (1/(αη)) p, there exists a unique pair of functions p*(M) and u*(M) defined on [underline M, overline M] together with positive free boundaries underline M and overline M such that: (i) the market-clearing condition D(p(M)) = M [ (θ/(u(M)η)) ( p(M) + (u′(M)/u(M)) D(p(M)) η ) − (u′(M)/u(M)) D(p(M)) ] holds for all M in (underline M, overline M); (ii) the HJBI-derived condition 2r = (θ/u(M)) ( p(M) + (u′(M)/u(M)) D(p(M)) η )^2 + [ u″(M)/u(M) − 2 (u′(M)/u(M))^2 ] D(p(M))^2 η^2 holds for all M in (underline M, overline M); and (iii) the boundary conditions u(underline M) = 1 + γ, u(overline M) = 1, u′(underline M) = u′(overline M) = 0, together with monotonicity and boundedness 1 ≤ u(M) ≤ 1 + γ and u′(M) ≤ 0, are satisfied.

Background

In the benchmark model without model uncertainty, Henriet et al. (2016) proved existence and uniqueness of the stationary Markovian equilibrium. In the robust extension of this paper, the equilibrium conditions and boundary requirements change due to ambiguity-averse preferences and the associated HJBI system, making the analysis of existence and uniqueness more delicate.

The authors present numerical evidence that solutions satisfying the required properties exist for standard parameter choices and argue that uniqueness follows under fixed boundaries via local Lipschitz conditions. However, a general rigorous proof with free boundaries is not provided, and establishing uniqueness in the presence of free boundaries is noted to be challenging.