Question

. Consider the following one-sector, closed, representative household economy. The production technology is given by the Cobb-Douglas production function where Y(t) is the output, K(t) is the capital stock, Lit) is the labor input, all at time t, 0 < a < and A(t) is the technology level at time t. Technological progress is at positive rate g. Let δ denote the depreciation rate for capital. This production function displays constant returns to scale in both K and L, hence each factor is paid its marginal product The instantaneous utility of the representative household is represented by the felicity function in the following form: u(c) = In(c) where c>0 is consumption per effective labor. The representative household wishes to masimize the present value ofr dsunted intertemporal intertemporal utility, U wice Assume that there is no population growth over time. Here, ρ > 0 represents the time preference rate. The representative household supplies labor inelastically in the labor market. For simplicity, assume that labor at time 0 is normalized to 1. b. Suppose that labor is paid w(t) per unit at time t, and interest rate on capital is r(t) at time t. Assume that the initial level of capital per effective labor owned by the household is ko. Carefully define a competitive equilibrium for this economy.

Answer 1

Answer:

The transversality condition entails:

lim ρυι,(e(t))X 0 (1) where λ, is the lagrangian multiplier corresponding to the constraint. t-+oo

Equation (1) says that for the optimality, either the marginal value of additional unit of consumption or the shadow price of the consumption, as t tends to infinity, must be zero. If this is not so, then the agent can always increase the consumption without violating the feasibility constraint.