Assignment 4

1. Consider the following q problem in continuous time:

v[K0, 0] = max(It)t≥ 0 ∫ -r t AF[Kt] - It -  χ    t

subjectto  = It,  K0 isgiven.

Here r is the interest rate, and A > 0 is the total factor productivity. Note that in the problem just stated the capital stock cannot be adjusted instantaneously without incurring prohibitive adjust- ment costs. More precisely, if It is the amount of capital invested (or dis-invested), then the total cost of investment is given by It +   , where χ is a positive constant.

(a)  Does the model you have a stationary equilibrium? If your answer is affirmative, draw a phase diagram to explain the convergence to the stationary equilibrium.

(b) Show that the value of the Hamiltonian - evaluated at each instant t along the optimal trajec- tory - gives the permanent level of utility, i.e., the permanent level of real income, of the representa- tive agent from instant t on until the end of time.

2. A firm begins at time t = 0 with a capital stock of size K0 . The firm has been carrying out an R & D program to raise its productivity. The firm expects that at some time T > 0, the R & D program will be completed, and its fruit harvested.

Before the completion of the R & D program, the technology of the firm is represented by the following production function: Yt= AF[Kt], 0 ≤ t < T, where Yt is output, and Kt is the capital stock - both at time t, 0 ≤ t < T. Also, A > 0 is a parameter representing the total factor productivity before the R & D program is completed.

After the R & D program has been completed, the technology of the firm is represented by the production function Yt= (A + ΔA)F[Kt], t ≥ T, where ΔA > 0 is the rise in productivity, and Kt  is the capital stock at time t, t ≥ T.

The problem faced by the firm is to find an investment program (It)t≥ 0 to maximize the fhe present value of the stream of profits associated with an investment program. Formally, this problem can be stated as follows: