UCLAEconomics 11 - Fall 2015Professor MazzoccoMIDTERM 1, Version 1NAME: ID:TA:Part I: Multiple Choice Questions (3.5 points each):1. If all prices and income double, the Marshallian demands for goods X and Y will(a) both increase since the consumer has more income(b) both decrease since the two goods are now more expensive(c) remain unchanged because the prices and income increase by the same proportion(Answer)(d) the demand for X might increase or decrease depending on the preference of theconsumer2. If Angelina’s utility function is given by U(x, y) = min{2x, 4y} and she is consideringthe two consumption bundles A = (x = 100, y = 25) and B = (x = 50, y = 50), thenwe can say that the consumer(a) is indifferent between A and B (Answer)(b) prefers A relative to B(c) prefers B relative to A(d) can’t compare A and B because preferences are not complete3. Alice’s utility function is such that she is always indifferent between 1 unit of x and3 units of y. Suppose initially px= 2, py= 1, and I = 10. Alice chooses x and y tomaximize her utility. Then the government imposes a 100% sale tax on x (the priceof x doubles). How much MORE does the person have to earn in order to achieve thesame level of utility as before? (Find the increase in income required to achieve thesame utility level)1(a) 0(b) 5 (Answer)(c) 10(d) 184. Suppose the good x is graphed on the x-axis and the good y is graphed on the y-axis.The slope of the budget constraint is:(a)pxpy(b) −pxpy(Answer)(c) −pypx(d)pypx5. Suppose that the price of champagne is pc= 70 and that the price of cheap wine fromTrader Joe’s is pt= 10 and Alessio has income I = 210. Alessio’s MRS of cheap winefor champagne (that is,MUcMUt) is equal to 8 for any consumption bundle (for each pointof the indifference curves). If he spends his income optimally, then Alessio will(a) Buy only champagne (Answer)(b) Buy 8 units of champagne and 2 units of wine(c) Buy a 4:1 ratio of champagne to wine–we don’t know exactly how much he canafford(d) There is not enough information to answer the question6. Which of the following functions is not a homothetic function?(a) U(x, y) = x2+ y2(b) U(x, y) = [xρ+ yρ]1/ρ(c) U(x, y) = xαy1−α(d) U(x, y) = xy + x + y (Answer)7. Which of the following utility functions represent the same preferences as U (x, y) =√xy(a) U(x, y) = 10√xy2(b) U(x, y) = xy(c) U(x, y) = ln x + ln y(d) All of the above represent the same preferences. (Answer)8. If the utility function is U(x, y) = 6x + 3y, then the bundle (3,2) provides the sameutility as(a) (2,4) (Answer)(b) (5,0)(c) (1,5)(d) None of the above9. If an individual’s utility function is given by U(x, y) = y ln(x2), then MRS(a) Depends only on y(b) Depends only on x(c) Depends on x and y (Answer)(d) Is not determined10. If Marlon’s utility function for goods X and Y is given by U(X, Y ) = minX, 3Y , theindirect utility function will be given by(a) V (X, Y ) =3Ipx+ py(b) V (X, Y ) =I3px+ py(c) V (X, Y ) =Ipx+ 3py(d) V (X, Y ) =3I3px+ py(Answer)3Part II: Essay QuestionsQuestion 1 (36 Points)1. Arnold has the following utility function:U(x, y) = xy + x(a) For a given income I and prices pxand py, find Arnold’s Marshallian demands forx and y. (7 Points)(b) Find the indirect utility function. (5 Points)(c) Suppose Arnold has 20 dollars to spend on X and Y, and the prices of these goodsare px= 1 and py= 4. What is his optimal consumption bundle? What is hisutility at that consumption bundle? (5 Points)(d) The government wants to increase Arnold’s welfare. It decides to give Arnold acash transfer of T = 8 additional dollars. The prices stay at px= 1 and py= 4.What will be the new optimal consumption bundle and his new utility level?(5Points)(e) Suppose that instead of making the cash transfer, the government subsidizesArnold by providing a price subsidy τ for the consumption of the good X. Thegovernment chooses the subsidy so that the total amount paid in subsidies toArnold is exactly equal to T = 8. Find the value of the subsidy τ. What ishis new consumption bundle? By how much will his utility increase under thisarrangement? (7 Points)(f) Which program should the government use if the goal is to maximize Arnold’sutility: the transfer or the subsidy program? (5 Points)4Question 2 (29 Points)1. Pearl is a student who cares only about two goods: books (B) and food (F ). Her utilityfunction is given by u(B, F ) = B0.5+ F0.5and the prices of books and food are pBandpF.(a) Find Pearl’s Hicksian demand functions for B and F. (8 Points)(b) Find her expenditure function. (8 Points)(c) Suppose that pB= 1, pF= 4, and that Pearl want to achieve the utility levelU = 10. What is the optimal consumption bundle that allows Pearl to achievethat utility level? What is Pearl’s expenditure at the optimum? (5 Points)(d) Now suppose the price of books rises to p0B= 4. Pearl wants absolutely to stayat the same utility level U = 10. How much income does she needs to stay atU = 10 now that the price of books has increased to p0B= 4? (Hint: rememberthe economic definition of the expenditure function) (8 Points)5Question 1 Solution(a) Taking a strictly increasing transformation,u(x, y) = ln(x) + ln(y + 1)From the tangency condition px/py= ux/uy, we getpx/py= (y + 1)/x =⇒ x = (y + 1)py/pxPlugging this into budget constraint,(y + 1)py+ pyy = IThusy∗=I − py2pyx∗=I + py2px.(b) The indirect utility function isV =I + py2px(1 +I − py2py) =(I + py)24pxpy(c) x∗= 12, y∗= 2, V = 36(d) Now the income is I0= I + T . New consumptions arex∗=I + T + py2px= 16, y∗=I + T − py2py= 3And new utility isV ==(I + T + py)24pxpy= 64(e) The consumptions are x∗=I+py2(px−τ)and y∗=I−py2py. And the utility is V =(I+py)24(px−τ)py.The total amount of subsidies must satisfyτx∗= TThereforeτ =2(px− τ)I + pyT =⇒ τ =2pxTI + py+ 2I= 0.46And.x∗= 20, , y∗= 2, V = 60.(f) Comparing the two utilities, the cash transfer provides higher utility.Question 2 Solution(a) From the tangency condition pB/pF= uB/uF, we getB = (PFPB)2FUtility constraint is B0.5+ F0.5=¯U, sopFpBF0.5+ F0.5=¯U =⇒ F∗= (pB¯UpF+ pB)2ThenB∗= (pF¯UpB+ pF)2.(b) The expenditure function isE = pB(pF¯UpB+ pF)2+ pF(pB¯UpF+ pB)2=pFpB¯U2pF+ pB(c) F∗= 64, B∗= 4, E = 80.(d) After pBis increased to 4, the income needed is the expenditure function at the newpriceE =pFp0B¯U2pF+ p0B=