Optimal Control

In this paper, we first consider the role of rational expectations, the Lucas critique and the policy ineffectiveness debate in economic applications of optimal control theory. The problem of time-inconsistency in optimal control of macro- economic models with rational expectations will then be analyzed. The impact of reputation and the stochastic environment on the problem of inconsistency in dynamic choice together with the question of how can the developments in optimal control of macroeconomic models with forward-looking expectations contribute to the practice of econometric model building are the other topics which are discussed in this paper. We have adopted a historical approach in this paper, and the scope of our analysis is confined to the basic contributions made in the 20th century.

Optimality conditions for consumption behavior with liquidity constraints are obtained using the functional recurrence equation in Bellman’s dynamic programming and the generalized Hamiltonian function in Pontryagin’s maximum principle. The rejection of Hall’s random walk hypothesis is then established for liquidity constrained consumers. An explicit mathematical relation is formulated which demonstrates the effects of liquidity constraints on consumption, which implies that under certain conditions the liquidity constraint may shift the optimal consumption profile forward even when the rate of time preference exceeds the interest rate. Our analysis is further developed to time-varying interest rates. Using the Kuhn-Tucker conditions, we have shown the interactions between the time-varying interest rate, the utility discount rate and the severity of liquidity constraints. It is shown, using the coefficient of absolute risk aversion, that how the time-varying interest rate may affect optimal consumption through intertemporal elasticity of substitution. Simultaneous effects of the pure preference parameters, interest rates variations and the liquidity constraints on optimal consumption path are mathematically formulated. Limitations in optimal control applications in modeling optimal consumption with liquidity constraints in a stochastic environment are briefly examined.