FINAL EXAMINATION
June 26, 2009
This is a clod-book and clod-notes exam. You have 3 hours to complete the exam. There are 4 questions on the exam for a total of 120 points.杨柳花
1. (40 points) Consider an economy where the reprentative houhold has a time-parable utility function of the form
where denotes hours worked. In addition, there is a reprentative firm which us a technology given by
Suppo that the growth rate of population and the rate of capital depreciates are 0, and the endowment of a houhold is .
(a) (10 points) Sep up the reprentative houhold’s utility maximization problem, and derive the first order conditions.
(b) (10 points) Set up the reprentative firm’s profit maximization problem, and derive the first order conditions.
(c) (10 points) Define a competitive equilibrium for the economy described above and write down all the necessary and sufficient conditions for the equilibrium.早泄怎么锻炼
(d)(10 points)Find the steady state for the economy described above. Write down as functions of the model parameters.
2. 火影忍者q版摩羯女双鱼男(40 points) Consider a neoclassical growth model where the reprentative houhold has a time-parable utility function of the form
where denotes the natural logarithm, and denotes hours worked.. Let be deposits that houhold holds in the end of period at the bank. The houholds’ budget constraints are
where are interest rate on deposits and rental price of labor (real wage), respectively. In addition, there is a reprentative firm which us a technology given by
Suppo that the growth rate of population and the rate of capital depreciates are 0.
(a) (10 points) Sep up the social planner’s problem and find a t of first-order conditions that an optimal solution to the planning problem must satisfy.
(b) (10 points) There are three tax rates in each period: the tax rate on labor income, the tax rate on capital income, and the tax rate on consumption. There are no transfers. Write down the houholds’ budget constraints with the taxes, the budget constraint for the government, and the resource constraint.
(c) (10 points) Sep up the reprentative houhold’s utility maximization problem taking tax rates as given, and derive the first order conditions.体型最大的猫
(d) (10 points) Discuss the impacts of tax rates on houholds’ optimal behavior, and consider the optimal choice of rates of taxation.
3. (20 points) A consumer eks to maximize her lifetime utility of consumption and leisure, which is given by , where is consumption in period and is leisure in period . The felicity function is strictly increasing and strictly concave in both of its arguments.
The consumer is endowed with one unit of time in each period, which she allocates between leisure and work. The consumer’s labor income in period equals , where is the wage per unit of human capita, is the consumer’s level of human capital in period 陈宏志, and is the amount of time that the consumer spends working in period . Human capital accumulates over time according to:
where is strictly increasing in both arguments, strictly concave in its first argument, and satisfies: . In other words, human capital depreciates at rate , but the consumer can invest in human capital by working. The consumer has human capital equal to in period 0. To keep things simple, suppo that the consumer does not participate in ast markets and instead simply consumes her entire labor income in every period.
(a)(5 points) Formulate the consumer’s dynamic optimization problem as a dynamic programming problem. That is, display the consumer’s Bellman equation and identify clea
rly the control (or choice) variable(s) and the state variables.
(b)(5 points) Derive the first order condition and the envelop condition.
(c)(10 points) Find an equation that determines (implicitly天堂鸟花语) the steady-state level of human capital. (hint: let )
4. (20 points) Let the consumer’s utility function be , where denotes hours worked. Houholds can hold tow asts: money (m) or government bounds (b), the latter earn the return. Suppo that a cash-in-advance constraint holds. is price index.
(a) (5 points) Write down a cash-in-advance constraint.
(b) (5 points)Write down a houhold’s budget constraint.学生期末自我评价
(c) (10 points) Solve houhold’s maximization problem as a dynamic programming problem, and find a t of first-order conditions and envelope conditions that an optimal solution to the problem must satisfy.
