Brownian equilibria under Knightian uncertainty
The paper develops the modeling of dynamic stochastic general equilibrium with respect to generalization of agents' utilities representation. The target is being obtained under simplification of general equilibrium to exchange equilibrium, and with preference representation via a stochastic differential formulation of intertemporal recursive utility. The economy consists of \(m \in \mathbb N\) agents with utilities described by the following backward SDEs: \[ dU_t^i=-f^i(t,c_t,U_t^i,\sigma_t^i)dt + {\sigma_t^i}^TdB_t, \; U_T^i=0,\quad i\in \{1,\dots,m\}. \] Here, \(c_t\) is the time-realization of a stochastic consumption process, \(\sigma_t^i\) is the volatility of \(U_t^i\), \(f^i\) is an aggregator function, and \((B_t)\) is an \(n\)-dimensional Brownian motion. The initial value \(U_0^i = E[\int_0^Tf^i(t,c_t,U_t^i,\sigma_t^i)dt]\) is a utility functional of the \(i\)th agent named `generalized stochastic differential utility' (GSDU). Each agent has an endowment \(e^i\). Exchange is defined by utilities \(\{U^i\}\), endowments \(\{e^i\}\) and a price system \(\Pi\) (a linear functional on the commodity space). The equilibrium is defined in these terms as usual in the Arrow-Debreu setting. The main result of the paper is the existence theorem of an equilibrium \((\bar{c}_1,\dots,\bar{c}_m; \Pi)\). The setting generalizes the models of \textit{Z. Chen} and \textit{L. Epstein} [Econometrica 70, No. 4, 1403--1443 (2002; Zbl 1121.91359)] as well as \textit{T. F. Bewley} [Decis. Econ. Finance 25, No. 2, 79--110 (2002; Zbl 1041.91023)] with Knightian uncertainty.
scientific article; zbMATH DE number 6419178
scientific article