W12 MIC Unit 1 Exam Solutions

W12 MIC Unit 1 Exam Solutions - Economics 310-1. Professors...

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Economics 310-1. Professors Hornsten and Braeutigam Page 1 First Midterm Solutions, Winter 2012 SUGGESTED SOLUTIONS FOR THE FIRST MIDTERM (See final page for info on scores) 1) a) An indifference curve illustrates the set of (X, Y) combinations that have a constant level of utility. The 32-util iso-utility curve passes through (X=4, Y=0), (2, 8), and (1, 24), because 4*8 = 2*16 = 1*32 = 32. The 64-util iso- utility curve passes through (8, 0), (4, 8), (2, 24), and (1, 56), because 8*8 = 4*16 = 2*32 = 1*64 = 64. It would also pass through (1/2, 120) because ½ * 128 = 64. Notice that these indifference curves intersect the X-axis (where Y=0), but never touch the Y-axis. [Scoring: 5 = 2 pts per IC and 1 for labels on axes & curves; -3 if IC cross; -2 if IC wd hit Y-axis; -1 per math error] b) If you pay $1 membership fee, leaving $5 of income, you qualify for Club Discount price of P X = ½ . max X , Y U [ X , Y ] = X ( Y + 8) subject to XP X + YP X " I # 1 2 X + Y = 5 (because you've paid the $1 fee) MRS X , Y = MU X MU y = Y + 8 X (which falls as X rises along an iso- U curve, so we cd hv an interior optimum) MU X MU y = P X P y # Y + 8 X = 1 2 1 # 1 2 X = Y + 8 (Tangency) Combining : 5 = 1 2 X + Y = ( Y + 8) + Y = 2 Y + 8 # Y = $ 3 2 (which is metaphysically troublesome) # Y * = 0 (a corner solution) # X * = 10 # U [10,0] = 10(0 + 8) = 80. Bang for the Buck verifies we want to max out on X : MU X P X > MU Y P y % Y + 8 1 2 > X 1 % 8 1 2 > 10 1 . [Scoring: 10 = 4 graph (proper intercepts, kink & slope) + 1 budget + 3 tangency/bang for buck + 2 answer; -1 for skipping vertical part of BC] U=32 utils (black) (8, 0) (4, 0) Good X Good Y (1, 56) (2, 24) (4, 8) (1, 24) (2, 8) U=64 utils (blue) BC DISCOUNT CLUB Good X 2 4 Good Y 6 4 2 10 IC:80 utils 8 6 5 0
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Economics 310-1. Professors Hornsten and Braeutigam Page 2 First Midterm Solutions, Winter 2012 c) If you Buy 4 Units at the regular price, then you Get 4 Units Free. The new budget constraint slopes downward, runs horizontally, and then slopes downward, so there are three zones we should consider when solving this problem. As we found in (b), MRS X , Y = Y + 8 X is diminishing in X, so for Y = 2, utility rises with X; thus, we needn' t to check the horizontal segment, because U [8,2] > U [ X ,2] for these X. TOP PART : max X , Y U [ X , Y ] = X ( Y + 8) subject to XP X + YP Y " I # X + Y = 6 MU X MU Y = P X P Y # Y + 8 X = 1 1 # X = Y + 8 (Tangency) Combining : ( Y + 8) + Y = 6 # 2 Y = $ 2 # Y < 0, so we won' t have an interior solution, and the best we can do on this segment is to buy as much X as possible at the "corner" at (4,2) . .. but once we get to (4,2), we can get 4 free units of X and slide over to (8,2). Effectively, P X = 0 for a while. Thus, our U - maximizing bundle can't have
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W12 MIC Unit 1 Exam Solutions - Economics 310-1. Professors...

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