1. Consider the following economy. There is a representative household with inter-temporal utility

U = _t=0∫^∞(c^1-θ - 1/(1- θ)) e^-pt.dt

For notational simplicity, assume that there is no population growth, i.e. n = 0 and L(t) is constant. In the above function, c is consumption per capita, θ > 0 is the inverse of the elasticity of intertemporal substitution, and Ρ > 0 represents the time-preference rate.

The technology for producing output is given by

Y(t) = AK(t)^αH(t)^1-α

where Y(t) denotes the aggregate output, K(t) the physical capital input, and H(t) the human capital input, all at time t and 0 < a < 1. Here, think of H as the number of workers L multiplied by the human capital of a typical worker, h. The production function above exhibits all neoclassical properties, including the constant returns to scale in K and H. Assume that output can be used on a one-for-one basis for consumption, for investment in physical capital, or for investment in human capital. Since we assume that worker population, L, is constant, changes in H reflect only the net investment in human capital.

The stocks of physical and human capital depreciate at the rates tCH and bH. and they accumulate according to K = I_K - δ_KK and H = I_H - δ_HH, respectively. I_K is the gross investment in phsical capital, while I_H denotes gross investment in human capital. Throughout the model, it is given that

δ_K = δ_H = δ

a. Let R_K and R_H be the rental prices of physical capital and human capital, respectively. Initial stocks of physical and human capital are given by Ks and Ho. Define a competitive equilibrium for this economy.

b. Determine that in equilibrium there exists a unique, constant K/H such that a

K/H = α/1 - α

c. Using your results from part (b), show and argue that the equilibrium implications (i.e. in both transition and in balanced growth path) from this model are the same as those obtained from the simple AK model. Interpret your results, and provide economic intuition.

2. Consider the following production function for firm i in a model similar (but not the same!) to the learning-by-doing with knowledge spillovers model studied in class:

Y_i = AK_i^αL_i^1-α K^λ
where
0 < α < 1
0 < λ < 1
and K is the aggregate stock of capital.

a. Show that if λ < 1 - α and L is constant, the model has transitional dynamics similar to those of the Ramsey model.

b. What is the steady-state growth rate of Y, K and C in the case in (a)?

c. Show that if A = 1 - α and L is constant, the steady state and the transitional dynamics are like those of the AK model.