Solving Nonlinear Expectations Models by Approximating the Stochastic Equilibrium System: Application to Global Solution Methods

Project Start:08/2016
Researchers:Michael P. Evers, Markus Kontny
Category: Macro Finance
Funded by:LOEWE

Key elements in modern dynamic economic modeling are uncertainty and forward-looking behavior: The decisions about contemporaneous behavior of economic agents, be it consumers, entrepreneurs, or policy makers, are based upon expectations about uncertain future behavior of the agents in response to random exogenous innovations. While understanding the interaction of uncertainty and forward-looking behavior in nonlinear settings is of considerable interest, it is confronted with the non-trivial challenge to actually solve the models: the solution to contemporaneous endogenous variables of the model depends on the expectations about uncertain future evolution of endogenous and exogenous variables.


To overcome the demanding computational challenges, it is proposed to obtain the solution not directly from the original system of stochastic equilibrium conditions, but from an approximated equilibrium system (AES) instead: A Taylor series which expands the deterministic equilibrium system (when exogenous disturbances are absent) in future exogenous disturbances up to some order k accurately approximates the true equilibrium system with respect to the equilibrium implications of the first k moments of the exogenous disturbances.


The merits of the AES-approach are twofold: First, the new equilibrium system is non-stochastic and it preserves the nonlinearities in the endogenous variables, but it fully captures the equilibrium effects of uncertainty by the first k moments of the exogenous disturbances.


Second, the implicit solution to the new equilibrium system is parameterized in the first k moments, too, which allows to derive the risk decomposition of the solution by means of a series expansion in the exogenous moments. This representation of the solution is most useful as it explicitly decomposes the sources and implications of uncertainty on the equilibrium process.


The pseudo-code for the proposed solution procedure reads as follows: i) Compute the solution to the deterministic model; ii) Compute a Taylor series expansion about the deterministic system with the corresponding deterministic solution in future exogenous disturbances up to some order k (compute the AES); iii) Compute the solution to the approximated equilibrium system.


The goal of the project is to promote the conceptual foundation of the AES-approach and to provide a computational toolbox that implements the procedure based on global solution methods for steps i) and iii).