UCLAEconomics 11 - Fall 2014Professor MazzoccoMIDTERM 1, Version 1NAME: ID:TA:Part I: Multiple Choice Questions (3.5 points each):1. John likes 2 sugar packets (S) in his coffee (C). Which utility function would bestrepresent his preferences?(a) U (S, C) = min12S, C(b) U (S, C) = min {S, 2C}(c) U (S, C) = min {2S, 4C}(d) All of the aboveSolution : d2. Robert’s preferences are described by the following utility function U (x, y) = 2x + y.If px> 2py, then the optimal consumption for good x is given by(a) x = I/px(b) x = I/ (2px)(c) x = 0(d) x = I/pySolution : c13. Suppose we have the following equations for the MRS of a utility function U (x, y).Which of the following corresponds to a homothetic utility function?(a) MRS (x, y) = xy(b) MRS (x, y) =x2y(c) MRS (x, y) = 2(x + y)(d) MRS (x, y) =x2+ y2xySolution : d4. Suppose a consumer’s utility function U (x, y) is given by max{2x, y}. Suppose youknow that when the price of x is 1, the consumer’s demand for x is 4. What is theconsumer’s income?(a) 2(b) 4(c) 8(d) None of the aboveSolution : b5. Pizza and pepsi are complements. Suppose that at current consumption level of 4pizzas and 3 pepsis, an individual’s marginal utility of consuming an extra pizza is 12whereas the marginal utility of consuming an extra pepsi is 3. Then the MRS of pepsifor pizza (that is, the number of pizzas the individual is willing to give up to get onemore pepsi) is...(a) 3(b) 4(c) 1/3(d) 1/4Solution : d26. Andy is indifferent between consuming M units of x or N units of y. Which of thefollowing utility functions of x and y can represent Andy’s preference?(a) U (x, y) =√Mx + Ny(b) U (x, y) = ln (Nx + My)(c) All of the above(d) None of the aboveSolution : bIf Andy views M units of x to be the same thing as N units of y, then they areperfect substitutes. The usual form for perfect substitutes is U (x, y) = Nx +My. And since (b) is a monotonic transformation of Nx + My (b) representsthe same preferences. Note that this is the correct form of the utility functionsince U (M, 0) = U (0, N) = ln (NM). One can also check this by calculatingMRSx,y= N/M.7. If the preferences of an individual are represented by the utility function U (x, y) = xαyβand the consumption bundle {x1, y1} lies on a higher indifference curve than {x2, y2},it must be that(a) x1> x2and y1= y2.(b) x1= x2and y1> y2.(c) x1> x2and y1> y2.(d) Any of these can happen.Solution : d38. If an individual’s utility function for good x and y is given by U (x, y) = min{4x, y},the indirect utility function will be given by:(a) V (px, py, I) =I4px+ py(b) V (px, py, I) =4Ipx+ py(c) V (px, py, I) =4Ipx+ 4py(d) V (px, py, I) =Ipx+ 4pySolution : c9. If an individuals utility function is given by U (x, y) = x0.5y0.5and I = 20, px= 2, py=1, his or her preferred consumption bundle will be:(a) x = 10, y = 20(b) x = 5, y = 10(c) x = 4, y = 8(d) x = 20, y = 5Solution : b10. Which of the following demand functions is not homogeneous of degree zero in px, py,and I?(a) x = I/ (px+ 3py)(b) x = I0.5p−0.3xp−0.2y(c) x = Ipy/p2x(d) x = I/ (pxpy)Solution : d4Part II: Essay QuestionsQuestion 1 (35 Points)1. Jane has the following utility function U (x, y) = xay1−a, with 0 < a < 1. Prices are pxand pyand income is I.(a) Set up the budget constraint for Jane and use the Lagrangian multiplier methodto solve for the optimal consumption of x and y. (7 points)Solution :The budget constraint isI = pxx + pyyTherefore the Lagrangian for the utility maximization problem is given byL(x, y, λ) = xay1−a+ λ (I − pxx − pyy)Taking first order conditions (FOC’s)∂L(x,y,λ)/∂x = axa−1y1−a− λpx= 0 (1)∂L(x,y,λ)/∂y = (1 − a)xay−a− λpy= 0 (2)∂L(x,y,λ)/∂λ = I − pxx − pyy = 0 (3)(1), (2), and (3) can be re-written asaxa−1y1−a= λpx(4)(1 − a)xay−a= λpy(5)I = pxx + pyy (6)Dividing (4) by (5)a1 − ayx=pxpyRearranging to isolate yy =1 − aapxpyx (7)(6) and (7) now describe a system of two equations and two variables. Sub-5stituting (7) into (6) we can find the optimal consumption of x.I = pxx∗+ py1 − aapxpyx∗⇒ I = (px+1 − aapx)x∗⇒ I =1apxx∗⇒ x∗= gx(px, py, I) =aIpxSubstituting x∗into (7) we can find y∗y∗=1 − aapxpyx∗⇒ y∗=1 − aapxpyaIpx⇒ y∗= gy(px, py, I) =(1 − a)Ipy(b) Find the indirect utility function for a = 1/3. (7 points)Solution :V (px, py, I) = U (gx(px, py, I) , gy(px, py, I))=aIpxa(1 − a)Ipy1−a⇒ V (px, py, I) =apxa(1 − a)py1−aISo if a = 1/3 thenV (px, py, I) =13px1/323py2/3I6For the rest of the question use px= 10, py= 5, I = 300, and a = 1/3.(c) What is the optimal choice of x and y given those prices, income, and a? (5 points)Solution :x∗= gx(10, 5, 300) =(1/3) ∗ 30010= 10y∗= gy(10, 5, 300) =(2/3) ∗ 3005= 40V (10, 5, 300) =13 ∗ 101/323 ∗ 52/3300 ≈ 25.2The government wishes to raise taxes by using one of the following two taxes: (i)a tax on Jane’s income of T = 100 dollars (with the tax, the new income is I −T );or a sale tax τ on each unit purchased of the good y (the new price of y is thereforepy+ τ ).(d) Using the prices, income, and a given above, find the sale tax τ that makes the taxrevenues collected from Jane using the sale tax equal to the tax revenues collectedfrom Jane using the income tax. (8 points)Solution :The following is the general form the of the equation that will allow us to findthe sales tax, τ. (Remember the sales tax changes the price of y).τgy(px, py+ τ, I) = T⇒ τ(1 − a)Ipy+ τ= T⇒ (1 − a)Iτ = T (py+ τ )⇒ τ[(1 − a)I − T ] = pyT⇒ τ =pyT[(1 − a)I − T ]With the given numbers we getτ =5 ∗ 100[(2/3)300 − 100]=5 ∗ 100[200 − 100]=5[2 − 1]⇒ τ = 57(e) Should the government use the income tax or the sale tax if the objective is tomaximize Jane’s utility? Show why you answered in a particular way. (8 points)Solution :To answer this question we need to compare Jane’s utility under both policies.Thankfully we have already computed her indirect utility function.V (px, py, I) =13px1/323py2/3IIncome TaxV (px, py, I −T ) = V (10, 5, 200) =13 ∗ 101/323 ∗ 52/3∗ 200 ≈ 16.8Sales TaxV (px, py+ τ, I) = V (10, 10, 300) =13 ∗ 101/323 ∗ 102/3∗300 ≈ 15.9Therefore, in this case Jane would be better off with the income tax. Ifthe student didn’t calculate the numbers, full credit is still given if the for-mulas were presented and the decision rule was