Question 3 Consider the problem of a firm who produces output in the present and the future period. This firm is endowed with K units of capital in the current period which is exogenously given. The firm can, however, choose future capital stock K' through investment I. In particular, we have the following capital accumulation equation:

K' = I.

Note that this capital accumulation equation implies that capital is fully depreciated in each period (d = 1).

The firm's profit is = Y wN I in the current period, where Y = zF(K, N) is output, w is wage, and N is employed labor. Similarly, the profit in the future period is given by ' = Y'w'N' , where Y' = z'F(K' , N' ) is future output, w 0 is future wage, N' is future labor demand. Note that there is no value of capital left over after production in the future period due to full depreciation (d = 1). The firm maximizes the present value of profits given by

V = + ('/ 1 + r).

Answer the following questions.

1. Write down the firm's profit maximization problem.

2. Derive labor demands for both the current period and the future period as functions of K and K'.

3. Derive the optimal rule of investment