Consider an economy with a representative consumer like the one described above, and a representative firm like The production function of the firm is given by zF (K, N) where: F (K, N) = KαN β and α + β < 1. The firm can hire workers in the labor market at a wage w and can rent capital market at a rental rate r (that is, the firm can rent K units of capital for which it would have to pay rK). For this problem assume that there is one unit of capital owned by a foreign investor. The investor’s capital supply is perfectly inelastic (that is she will supply one unit of capital at any rental rate).
a) The equilibrium rental rate must be such that the demand and supply of capital are equalized. Find the equilibrium rental rate. Hint: You can express the equilibrium rental as a function of the wage. For this is is useful to use the FOC of the firm with respect to capital.
b) The equilibrium wage must be such that the demand and supply of labor are equalized. Find the equilibrium wage. Hint: You can get an easier expression for the labor demand when you use the labor demand as a function of capital (obtained by combining the firm’s FOC) and set capital to its market clearing value. Then you can solve for the wage. Remember to replace by the equilibrium interest rate from the previous point.
c) With the equilibrium prices solve explicitly for the equilibrium labor. Hint: You can get a better expression for the equilibrium labor using the labor supply and replacing by the wage you obtained above.
d) In equilibrium, what happens to the wage, interest rate, labor and consumption when the government increases the labor taxes?
Hint: Find out first what happens to the wage, and then use your result for the equilibrium interest rate to find out what happens to that variable. The effect on labor can be obtained directly from your answer to (c) or from the FOC of the firm with respect to labor. Finally, get the effect on profits before finding out the effect that extra taxes have on consumption.