This is a Macroeconomics question. Please answer it as precisely as possible.

Instruction: Answer all questions and provide full explanations. You may work in groups, but please hand in individually. Imagine an economy that lasts only one period and consists of the representative consumer and the representative rm. The consumer's preference is given by U (C, I), where C is consumption and l is leisure. The consumer is endowed with one unit of time, which can be allocated between work and leisure. Also, the consumer owns K units of capital, which can be rented to the rm. Finally, the consumer is the owner of the rm. The rm has a technology for producing consumption goods according to Y: 2F (K, N), where Yis output, K is the capital input, N is the labor input, and z is a parameter representing total factor productivity (TFP). There are no frictions. That is, all the markets are perfectly competitive. Denote real wages by w, the rental price by 7'. Also, suppose that U (C, l) and F(K , N ) satisfy all the assumptions that we have learned in class. 1. The Consumer's Problem a) Set up the consumer's budget constraint. b) Set up the consumer's optimization problem. 0) Find the consumer's rst order conditions and interpret the results. 2. The Firm's Problem a) Solve the rm's prot maximization problem. b) Find the rm's prot and discuss. 3. A Competitive Equilibrium a) Dene a competitive equilibrium b) Now, consider U(C,l) = 0:17,? +1! and F(K,N) = KaN1_\" where \"y > 0 and a 6 (0,1). Find the closed form solutions for C, N, and w. c) Suppose an increase in 2. What would happen to aggregate consumption, 0, employment N, and real wages 10? Provide your answer that involves business cycle facts covered in class