2. Labor market equilibrium Consider an economy in which the labor market is competitive. There is a large number of firms, each with technology given by: Y = Aka [1-a where a e (0,1) Where Y denotes production, k the amount of capital, and I the amount of labor used in hours per week. There is a large number of identical consumers, with the following preferences: u (c , n ) = vc + 1 B gu Where n denotes leisure and c consumption. There are T hours in the week available to split between labor and leisure. Consumer's own K amount of capital stock. Assume the consumer's non labor income is R, and the price of consumption is normalized to 1. Use w to denote wages, and r the price of capital. a. Set up the optimization problem for the firm. Using the first order conditions, derive the labor and capital demand. Hint: the demand for capital is a function of labor, price of capital, and parameters A and a. The demand for labor is a function of capital, wages, and parameters A and a. b. Set up the optimization problem for the consumer. Using first order conditions, derive the expressions for labor supply, leisure, and consumption. Hint: leisure should be expressed as a function of wages and the parameters y, B. Consumption should be expressed as a function of wages, price of capital, capital stock K, and the parameters y, B, T, R. c. Find the reservation wage of the worker. d. Define the equilibrium of this economy. e. Solve for the equilibrium wage and labor. Assume a = 1 - B. f. Show in a graph how does the equilibrium wage and equilibrium labor change when T increases. Hint: graph labor supply and labor demand