Question

5. A consumer who lives for two periods has a standard Cobb-Douglas utility func- tion: u(C1,C2) = ccm where ct = consumption in period t and a + b = 1. Her income in period one is 11 = 500 and 12 = 400, and she can lend or borrow at interest rate r = 0.2. (a) Find the optimal consumption demand. (b) What values of a, if any, makes the consumer a borrower? Interpret this result. (c) Suppose now that a = but that r is no longer 0.2. What values of r, if any, makes the consumer a borrower? Interpret this result. 6. Consider an individual whose utility function over money is u(w) = 1 + 2w 2. (a) Is the individual risk-averse, risk-neutral, or risk-loving? Does it depend on w? (b) Suppose the individual has initial wealth YW and faces the possible loss of YW The probability that the loss will occur is 1. Suppose insurance is available at price p, where p is not necessarily the fair price. Find the optimal amount of insurance the individual should buy. You may assume that the solution is interior. (c) Is there a price at which the individual will not want to buy any insurance? If so, find it. If no, explain.

Answer 1

ltr 1tr исс , Budget constant at C2 - I + I. » cit c = 500+ You - G+ c = 500 + 400 1+0.2 12 1.2G+C = 600 24 2 с. cl a) Forning kergeangian Fr. L = ca B + 1[600- 1.24 - 4 at , & G B 1.2A=0 ar to a r - dha Beat – 1=0 e te Bodet - 0:0, acier Baby a. [h= & G using this relation in B. c. 1:26, + & G = 600 1.224 + BC = 4600 1C = x 600 By pulling ina - B4 1.20+ B. lator we - bboo get G . 1.287B

- X 600 &+B=1 a Boi-d. a foo 1.24+ (14) - 0.28+1. (1-21600 (-2) 600 1.20+(1-x) = 0. 24+1 O (b) LA11212222PP???????? Borrower & a > &600 (1-4) 600 0.20+1 0.20 + 1 7 X600 (1) 600 22 > 1 2 > 12/ for all &> Y , Consumer is borromes. This imply that ginen er=0.2 when &> Y2 Marginal vtelity from spending a penny on G would be greater than the mu from a. Thus consumer will want more of Ge would become borrones. Green (6) Bc. cit - 5004 900 1+ (ltr) 500 + you BC: (1+r) Gta = 900 + 5000 veg, a) 737 2 = 174317 + 18900 45000 - (to) Ges] 22 a 4G (1+r)x=0 a da 4 G7 m 22 - 3 647 - 1-o a la 3 g 41 26 - 317 des 74317 + (1+rxon 711+r) 6,317 47 417 76

417 317 IC 317 OP , yo 3 412 © a na = 301&ssa a = 301 to Bing in B.C: 1 (HiG + 3 (1+2) = 900+ soor 7C1+nG = 4(900+ svor) CTE 4 (aoot soos) TH) 900+sour) 22222222222PPLITTP???PPPP Bomover i 3 (aootoor) 4900 + 5008) a Tita 2) 4 > 32 47 311+r)= 3+37 Ato) 18cY3 Fir & LY consunner becomes bonower. At a rate re the marginal utility from spending on G exceed mu poma. But @ 87 Y3 , C, becomes expensine and therefore consumer starts saving.