Dynamic Programming

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.