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 maximize the present value of her discounted utility, U intertemporal 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. d. What is the significance of the transversality condition (TVC) for equilibrium? That is, why is it necessary to hold for an equilibrium to exist? Interpret.

Answer 1

The transversality condition entails:

$\lim_{t\rightarrow \infty}\rho^t u'(c(t))\lambda_t = 0....(1) \: \\where \:\: \lambda_t \text{ is the lagrangian multiplier corresponding to the constraint}.\\$

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.