Question: Consider the following Overlapping Generations economy: People live for two periods. The lifetime utility of an individual born at time t is given by Ut = ln(c1,t -x) + ln(c2,t+1) x > 0, where c1,t is consumption in the first period of life of a person born at time t, and c2,t+1 is consumption of the same person in the second period of life. Now assume that x is an exogenous constant.
Other than having x in the utility function in the ?rst period of life, the model is com pletely standard. People work in the first period of life, inelastically supplying one unit of labor. They do not work in the second period of life. The savings of the current elderly gen eration make up the capital stock used in production. Population size is constant. Factors are paid their marginal products. The rate of depreciation is zero.
The production function, in per worker terms, is given by y = ka.
a. Draw a diagram showing how capital evolves from period to period (i.e. a diagram with kt on the horizontal axis and kt+1 on the vertical axis.) Indicate any possible steady states and discuss their stability. Show how the number of steady states and their stability is a?ected by the size of x.
b. Now suppose that the value of x is endogenous. In particular, assume that x at time t (which applies only to the young people at time t) is based on the consumption of old people at time t : xt = c2,t. Discuss the conditions under which a stable steady state will exist in this economy.