a) Assume that the labor market is competitive (i.e., firms are price-takers) and frictionless. The representative firm's production function is given by F(L) = log(L) where L is the labor employed at the firm. Assume that wage is given by w and that the firm maximizes profits. There is a total mass of firms equal to one in this economy. Show that the labor demand curve in this economy is given by D(w) = 1/w

b) Let us take a reduced-form approach, by simply assuming that labor supply is given by S(w) = w. What is the equilibrium wage and employment in this labor market?

c) Now assume that labor supply is SN(w) = (1 p)w where p is the proportion of workers that are working. Show that the equilibrium wage in the private sector will be: w** = (1/(1 p) )^0.5