On the Long-Run Steady State in a Simple Dynamic Model of Equilibrium with Heterogeneous Households

Abstract

This paper examines the long-run behavior of a one-good model of dynamic equilibrium with heterogeneous households.1 The main result is a characterization of the long-run steady state that verifies a conjecture of Ramsey [1928].2 If an agent's lifetime utility function over an infinite horizon is represented by a stationary, additive, discounted function with a constant pure rate of time preference, then the income distribution is shown in the long-run steady state to be determined by the lowest discount rate. The household with the lowest rate of discount owns all the capital and earns a wage income; all other households receive a wage income. If discount rates are equal between households, than the steady state distribution of income is indeterminate. Rader [1979] shows the emergence of a dominant consumer in the long run for a model without an explicit capital accumulation structure. The dominant consumer has the lowest pure rate of time preference in the economy. Advantages of our work over Rader's presentation are that the analysis is related to the theory of economic growth and the mathematics is elementary. One difference from the Rader result is that nondominant consumers have positive steady state consumption. Normally, those with high discount rates would want to contract to trade future labor for present consumption. These agents would then incur a debt equal to the discounted value of their wage income. Rader's result admits intertemporal trades of future labor for current consumption in a loan market extending into the indefinite future. Each consumer's budget constraint requires total assets to be