UCLAEconomics 11 - Fall 2016Professor MazzoccoMIDTERM 1 SOLUTIONSPart I: Multiple Choice Questions (3.5 points each):Version 1 Version 2 Version 31 B B A2 B C B3 D C A4 B A C5 C B B6 C A B7 A C D8 B B B9 A B C10 C D C1Part II: Essay QuestionsQuestion 1 (34 Points)1. Janet loves eating large salads (S) while drinking Odwalla juices (J). Drinking the juicesimproves the taste of salads. Therefore, her utility function has the following form:U(S, J) = JS + αS(Version 1: α = 2, Version 2: α = 3, Version 3: α = 4)(a) For a given income I and prices psand pj, find Janet’s Marshallian demands forS and J. (8 Points)Apply the tangency condition to getMRS =SJ + α=pjpsThusS =pjps(J + α)Replace in the budget constraint to getpjJ + pj(J + α) = ISoJ∗=I − αpj2pjS∗=I + αpj2psVersion 1 Version 2 Version 3J∗I−2pj2pjI−3pj2pjI−4pj2pjS∗I+2pj2psI+3pj2psI+4pj2ps(b) Find the indirect utility function. (8 Points)V (pj, ps, I) =I − αpj2pjI + αpj2ps+ αI + αpj2ps=(I + αpj)24pjps2Version 1 Version 2 Version 3V(I+2pj)24pjps(I+3pj)24pjps(I+4pj)24pjps(c) Suppose Janet has 50 dollars of income (version 1 and 2, for version 3 I = 40) tospend on salads and juices, and the prices of these goods are ps= 10 and pj= 5(version 1, version 2 ps= 4 and pj= 2, version 3 ps= 6 and pj= 2) . What is heroptimal consumption bundle? What is her utility at that consumption bundle?(5 Points)Version 1 Version 2 Version 3J∗4 11 8S∗3 7 4V 18 98 48Janet’s mother is really proud of her daughter’s healthy diet. She therefore wantsto find a way to improve her daughter’s happiness (utility). To do this she comesup with two possible plans:i. The first plan is to increase Janet’s income by 10% from its current value of50 (version 1 and 2, for version 3 I = 40).ii. The second plan is to subsidize Janet’s consumption of juices by paying 20%of the price of each juice she buys.(d) Which of the two plans should Janet’s mother choose if she wants to maximizeJanet’s utility? (To obtain the points, show the calculation that enables you tochoose the plan) (8 points)Indirect utility changes in the following way for each plani. New income is I0= I + 0.1I = 1.1I, so the new indirect utility isV (pj, ps, I0) =(0.9I + αpj)24pjpsReplace numbers to getVersion 1 Version 2 Version 3V 21.125 116.281 56.3333ii. New price of juice is p0j= pj− 0.2pj= 0.8pj,so the new indirect utility isV (p0j, ps, I) =(I + α0.8pj)24 · 0.8pjpsReplace numbers to getVersion 1 Version 2 Version 3V 21.025 117.306 56.067So plan 1 is the best for version 1 and 3, plan 2 for version 2(e) How much money does Janet’s mother have to spend to implement each plan? (5points)The cost of plan 1 is 10% of income, so 5 for version 1 and 2, 4 for version 3. Thecost of plan 2 is given by the subsidy multiplied by the number of juices bought.That isS = 0.2pj∗ J∗(0.8pj, ps, I) =I − α0.8pj8orVersion 1 Version 2 Version 3S 5.25 5.65 4.24Question 2 (31 Points)1. Janet got tired of eating salads and drinking juices and she decides that from now onshe will only eat steaks (s) and drink red wine (w). Her preferences for those two goodsare such that she always wants to eat one steak with α glasses of wine. Her utility isthereforeU(s, w) = min{αs, w}.(Version 1: α = 2, Version 2: α = 4, Version 3: α = 2) The prices of steak and wineare psand pw.(a) Find Janet’s Hicksian demand functions for s and w. (7 Points)Hicksian demand is always at the corner of the indifference curve, thereforeαs = w = UWhich returns the HicksiansVersion 1 Version 2 Version 3hsU2U4U2hwU U U(b) Find her expenditure function. (7 Points)Version 1 Version 2 Version 3E Ups2+ pwUps4+ pwUps2+ pw(c) Suppose that ps= 10, pw= 5 (version 1 and 2, version 3 ps= 8, pw= 4). Howmuch income does Janet needs to achieve the utility level U = 100 (version 1,version 2 U = 200, version 3 U = 400)? (4 Points)Evaluate the expenditure function to getVersion 1 Version 2 Version 3E 1000 1500 3200A new scientific article has provided evidence that excessive consumption of redmeat reduces life expectancy. As a consequence, the government has decided toimpose a sale tax of 50% on steaks.5(d) If Janet wants to stay at the same utility level , how much income does she needto spend after the introduction of the tax? (6 Points)The new tax changes the price of steak to 1.5pw.The new expenditure isVersion 1 Version 2 Version 3E 1250 1600 4000(e) Janet realizes that she cannot afford to spend that much and decides to spend thesame amount she was spending before the introduction of the tax. What will hernew level of utility be after the introduction of the tax? (7 Points)The utility level can be found by solving for the utility U that will give you thesame level of expenditure with the increased price. That is solvingVersion 1 Version 2 Version 3U152+ 5= 1000 U154+ 5= 1500 U122+ 4= 3200Which solves toVersion 1 Version 2 Version 3U 80 187.5