Consider the Cobb-Douglas production function: Y = zK^a N^1-a where Y is output, z is total factor productivity, K is capital, N is labor employed, and is the share of capital in the production function.

1. Write down the problem of the firm using the Cobb-Douglas production function

2. Derive the optimality condition of the firm and provide an interpretation.

3. Show that the optimality condition of the rm can be written as a=1-wN/Y where w is the real wage.

Please, answer this as a firm situation

e.g. zF(K,N^d)

pie = Y - wN^d

pie = zF(K,N^d)-wN^d

The firms problem is then: max zF(K,N^d)-wN^d N^d.