Question I [40 marks] Assume the aggregate production is produced using the technology function Y = 2z √ N and the utility of the representative consumer is given by U(C, l) = ln(C). This utility function implies, in particular, that the consumer does not value leisure. Let’s assume that G = T = 0. The consumer has a total number of h hours to split between leisure and work time.
1. Write down the consumer’s budget constraint.
2. Solve the consumer’s problem and solve for the optimal C ? and Ns
3. Solve the firm’s problem and derive the optimal labour demand Nd .
4. Compute the equilibrium wage, consumption and employment in this economy.
5. How do these variables react to an increase in total productivity factor z?
Hint: Since the consumer doesn’t value leisure, he will optimally choose l ? = 0. The MRS can not be calculated here and there is no need to use the MRS condition to solve for the optimal C ? and l ? . When solving Question 4), use the definition of an equilibrium, that is, Ns = Nd .
Question II [60 marks] Suppose households preferences are described by the utility function U(C, l) = 2β √ C + δl, where C stands for consumption of market goods l stands for leisure, δ and β are positive constant parameters. The total number of hours available to the representative consumer is 1, and the market real wage is w. Output is produced using the production function Y = z √ Nd , where z > 0 is the total factor productivity. In answering the first three questions assume, for simplicity that there is no government and π = 0. 1
1. What are the optimal values C ? and l ? .
2. Derive the labour supply and labour demand curves and graph them. Discuss if the income
effect can ever dominate the substitution effect or not.
3. Compute the competitive equilibrium wage rate and employment for this economy.
4. Was it reasonable to assume a zero profit when solving the first three questions?
5. Now suppose that there is a government in this economy and its purchases are equal to G, where it finances them with imposing lump-sum taxes equal to T. Does the labour supply depend on G in the presence of the government. Why or why not? If G increases, how do output employment and employment respond? Explain clearly.
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