3.3 Consider a closed economy with a three-input Cobb–Douglas function: Y = AK^{alpha}(LH)^{eta} where H is human capital and K & L are the same variables as we saw in lectures. alpha > 0 and eta > 0 are constants. The exogenous parameter A > 0 represents technology. The economy has two types of workers: skilled labor and unskilled labor. The real wage rate for an unskilled worker is the same as in a standard Cobb-Douglas model. In addition to that, a skilled worker earns an extra marginal product of human capital. Assume that factor prices adjust to equilibrate supply and demand; and the population is composed of the aforementioned two types of workers.
a. Derive the condition on exogenous parameters for the economy to earn a zero economic profit.
b. Derive the condition on exogenous parameters for the economy to have diminishing returns on all three inputs.
c. Some economists argue that economic growth is likely to widen income inequality. Derive the wage gap between skilled and unskilled workers as a function of Y and H and discuss the impact of an increase in Y on the wage gap, holding all else constant.
d. The government in this economy aims to increase H by expanding educational programs. If the condition you derived in Part a is true, discuss the impact of increased H on the wage gap, holding all else constant.
e. Assuming that A = 100, alpha = 0.25, and eta = 0.375. The units of capital, human capital, and labor in the economy are 625, 256, and 256, respectively. All factors are fully utilized in the production. Calculate total output, real rental rate of capital, and real wage rate for skilled and unskilled workers respectively.