Final Exam ECON 101 Spring 2012 Directions • There are 4 questions on this exam. All questions will be graded. • Write your answers in your blue book. • Write your name and perm number on the blue book. • Please label each subquestion clearly. • Please start a new page in your blue book for each of the questions. • The goal of this test is to get as few questions wrong as possible. If you misinterpret a question, you will most likely get it wrong. Therefore, if you have any questions about what a question asks, or how to interpret it, ask me or one of the TAs. • Good luck.
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Final Exam ECON 101 Spring 2012 Question 1 (15 Points) Consider the Solow model with constant returns to scale production function zF (K, eN ), employment rate e, depreciation rate d, and population growth rate n. Use graphical analysis when necessary to support your answers. (a) [3 pts] Derive per-person output y as a function of per-person capital k and the employment rate e. (b) [3 pts] Fix savings rate s. What is the relationship between K and K’ ? Use this to derive the relationship between k and k’. (c) [3 pts] Derive equations characterizing steady state capital per person k ∗ , output per person y ∗ , and consumption per person c∗ . (d) [3 pts] Suppose e rises. How will this affect k ∗ , y ∗ , and c∗ ? Can changes in e generate unbounded growth in y ∗ ? (e) [3 pts] Suppose s rises. How will this affect k ∗ , y ∗ , and c∗ ? Can changes in e generate unbounded growth in y ∗ ? Question 2 (30 Points) Consider a competitive endowment economy with two consumers, a government, and two consumption goods (c1 , c2 ). Consumer one pays an exogenous lump-sum tax T 1 to the government, which is used to finance an endogenous lump-sum transfer T 2 to consumer two. Consumer one’s endowment ω 1 = (1, 0), and consumer two’s endowment ω 2 = (0, 1). Let c12 denote consumption of good 2 by consumer 1. Consumers have utility functions u1 (c1 , c2 ) = 2 log(c1 ) + log(c2 ) u2 (c1 , c2 ) = log(c1 ) + 2 log(c2 ) where the superscript denotes the consumer. The price of c1 is 1, and the price of c2 is p. (a) [10 pts] Define a competitive equilibrium in THIS economy. (b) [5 pts] Solve for both consumer’s demand functions. (c) [2 pts] How does the equilibrium price level affect each consumer’s equilibrium utility? (d) [2 pts] Solve for aggregate demand for both goods. (e) [5 pts] Solve for the competitive equilibrium allocations and price p. (f) [6 pts] Consider an economy which is identical except that consumer two’s endowment is twice as large, meaning ω 2 = (0, 2). Solve for the competitive equilibrium allocation and price in this economy. Do consumers one and two benefit? Explain why or why not. 2
Question 3 (40 Points) Consider a static production economy with two consumers, one firm, and a government. Each consumer has one unit of time available. Both consumers have utility function u(c, l) = log(c) + α log(l). Consumption is produced by the firm according to production function Y = zN1γ N21−γ , where Ni denotes labor demanded from consumer i, which is hired by the firm at wage rate wi . The government uses a lump-sum tax T on consumer one to finance government purchases of consumption G. The price of c is normalized to 1, and the wages are denoted w1 and w2 . Let c1 denote consumption by consumer 1. The firm earns zero profit in equilibrium, and there is no need to mention firm profits. (a) [10 pts] Define a competitive equilibrium in THIS economy. (b) [5 pts] Solve for each consumer’s demand for c and l. (c) [5 pts] Solve for the firm’s optimal demand for N . (d) [3 pts] Suppose z rises and no prices change. How will this affect consumer and firm decisions? (e) [7 pts] Suppose z rises. How will this affect competitive equilibrium prices and allocations? (f) [3 pts] Suppose G rises and no prices change. How will this affect consumer and firm decisions? (g) [7 pts] Suppose G rises. How will this affect competitive equilibrium prices and allocations? Question 4 (15 Points) Consider the intertemporal choice model with one consumer, a government, and two time periods. The consumer has utility function u(c, c0 ) = log(c) + β log(c0 ), where c denotes consumption today and c0 denoted consumption tomorrow. The consumer has income in both periods, denoted (y, y 0 ), pays taxes in both periods, denoted (t, t0 ). The consumer purchases consumption in each period at price one, and can borrow and lend at rate r to transfer resources across time. The government purchases consumption in both periods, denoted (g, g 0 ). (a) [3 pts] Write down the consumer’s budget constraints in each period. (b) [3 pts] Derive the consumer’s lifetime budget constraint and graph it. Label the consumer’s endowment point E1 . (c) [3 pts] Suppose that the government raises taxes in the first period by 1, and reduces them in the second period by (1 + r). Derive and graph the consumer’s lifetime budget constraint and label the new endowment point E2 . (d) [3 pts] How will the consumer’s decisions change after the change in taxes? Why? (e) [3 pts] Suppose that the consumer enters a training program which reduces income in the first period by 1, and increases income in the second period by 2(1 + r). How will the consumer’s decisions change? Why? 3