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Unformatted text preview: Eco11, Fall 2008 Simon Board Economics 11: Solutions to Homework 2 October 28, 2008 Due date: Tuesday 28th October. Instructions: You are required to write up your solution separately and independently, al- though you are encouraged to discuss and work in groups. Please write your name, student ID number, and the name of your TA on the front page of the assignment that you hand in. Also, please put boxes around your final answer to each part. 1. Budget Sets with Price Discounts There are two goods: x 1 and x 2 . The seller of x 2 charges p 2 = 2. The seller of x 1 offers price discounts. If the agent buys x 1 < 10 she pays p 1 = 2 for every unit. If the agent buys x 1 10 she pays p 1 = 1 for every unit (including the first 10). (a) Suppose m = 30. Draw the agents budget set. (b) Assume the agent has monotone preferences. Do preferences exist such that the agent chooses (i) x 1 = 3, (ii) x 1 = 7 and (iii) x 1 = 12? Explain your answers. Solution (a) The budget line is given by x 2 = 15- x 1 for x 1 < 10 = 15- 1 2 x 1 for x 1 10 As a result, there is a discontinuity at x 1 = 10. (b) (i) Yes. (ii) No. Because of the discontinuity, the agent can obtain more of both goods. (iii) Yes. 1 Eco11, Fall 2008 Simon Board 2. Intertemporal Choice with Differential Interest Rates An agent allocates consumption across two periods. Let the consumption in period t be x t , and the income in period t be m t . The agents utility is u ( x 1 ,x 2 ) = ln( x 1 ) + 3 4 ln( x 2 ) The agent is poor in period 1 but wealthy in period 2. In particular, she has income m 1 = 3 in period 1 and m 2 = 4 in period 2. (a) Suppose the agent can borrow and save at interest rate r = 1 / 3, so $1 in period 1 is worth $(1+1/3) in period 2. Sketch the agents budget constraint. Solve for her optimal consumption. (b) Suppose the agent can still save at r = 1 / 3, but can only borrow at r = 1 / 2. Sketch the agents budget constraint. Solve for her optimal consumption. (c) Suppose the agent can still save at r = 1 / 3, but can only borrow at r = 1. Sketch the agents budget constraint. Solve for her optimal consumption. [Hint: Beware of kinks.] Solution (a) Writing everything in terms of period 1 money, the agents budget constraint is m 1 + 3 4 m 2 = x 1 + 3 4 x 2 The tangency condition states that MRS = p 1 /p 2 . That is, 4 3 x 2 x 1 = 4 3 which implies x 1 = x 2 . The LHS of the budget equation is 6. Using this, we obtain x * 1 = x * 2 = 24 / 7. Since7....
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This note was uploaded on 04/25/2009 for the course ECON 11 taught by Professor Cunningham during the Fall '08 term at UCLA.
- Fall '08