Spr2008FinalExamSolutionSketches

Spr2008FinalExamSolutionSketches - Economics 50 Stanford...

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Unformatted text preview: Economics 50 Stanford University ' Spring Quarter 2008 Final Exam June 6, 2008 Part I (Questions 1 through 3; 50 points total) Write your name and your TA’s name atai Ater, Molly Goldstein, Pedro Miranda, Eduardo Perez, or Betsy Walls), and sign the statement on the cover of Parts I, II, and III of the exam. Pace yourself carefully, and provide clear, concise answers — lengthy explanations are not necessary! Write all of your answers in the space provided. If you need extra room, please use the back of each sheet. You can work on Parts I, II, and H1 in any order. If you must make any additional assumptions in order to answer a question, please state what those assumptions are. At least one member of the Econ 50 staff will be available outside the classroom at all times. We usually cannot answer questions, but please notify us if you feel you’ve found a mistake in the exam or if you observe a classmate engaging in suspicious behavior. At the end of the exam, turn in Part I, Part II, and Part III in the boxes provided. Your numerical answers should be as precise as possible. If you’re pressed for time, don’t worry about simplifying your answers perfectly. Make sure you show your work. You will have a total of 180 minutes to complete this exam. The exam is worth a total of 150 points, so you should allocate approximately one minute per point. Remember that no calculators, notes, books, headphones, or visible cell phones are allowed. Good luck! “The answers written on these pages are entirely my own. I attest that in taking this exam, I am fully complying with all provisions of Stanford’s Fundamental Standard and Honor Code.” Signature: Do not open this exam until it is time to begin. Page 1 of17 Economics 50 Stanford University Spring Quarter 2008 Question 1: Some stretching exercises, to begin [10 points] a) Compute the (own-)price elasticity of demand for X, where XD=I/(Px+3Py). Show your Vilork. [2 poi:1t's]a)(p PX __ T PX [P1 + X (19% + 3P4) I b) Compute the output elasticity of total costs, Where TC=4O+20W1/2r1/2Q. Show your work. [2 points] 6? 4/0 + QOa/"zr £0 ZOwI/Z rVZ ' .9 $5 0 1E: . w if: QM; ' P2 mms= aF/BL it“! K3“ I Kyla bF/DK K’Wq...,_, LIB/q ’BMRV$ K/LB'J /3[K),l/q i -t ‘ ° : ~’— 1- .1: K 34 : ’0 c/L ""113 [(9 Li (Z) l 3 o '4: fi’g )’ ’ B.U’"K~U" "— 5) J (6(1) 'k’W’" " "9 MU“ ' ‘1' Page 2 of 17 Economics 50 Stanford University Spring Quarter 2008 Question 2: Will the skulls decompose? [20 points] Indiana’s preferences over whips (X) and hats (Y) are summarized by the utility function U(X,y) = x + 4 In y. Indiana has an income of $1; he can purchase whips and hats at the market prices PX and Py, respectively. a) Derive expressions for Indiana’s utility-maximizing demands for whips and hats, XD(PX, Py, I) and YDGDX, Py, l), for the case where his decision is an “interior solution” (i.e. to simplify, you can ignore the possibility of a corner solution). Be precise, and show your work. [6 points] ' , ' . ‘ w/ [nfm‘a/ 50/079694 ‘ MES ~ Pj V W i 5 w, a y——- = 9/4 ‘44 ’ — All/4y ‘77— Mr 21 r9 6/" fiat/my 1W V41; PY=$2? [2 points] c) How many whips and hats will Indiana purchase if the price of a whip falls to Px=$3, while everything else remains as in part b? [2 points] $73,240) g '7 ‘ M’ “"7" y”(3,g,wj.— 5.3,; : [aha/3 Page 3 ofl7 Economics 50 Stanford University Spring Quarter 2008 d) Sketch a Slutsky diagram showing the effect of the decrease in the price of a whip as described in part c. Make sure you draw all necessary curves, including the budget line and indifference curves pertaining to Indiana’s choice before and after the change. Label the key points A, B, and C, such that the substitution effect is from A to B, the income effect is from i B to C, and the total effect is from A to C. Make sure your diagram is as legible as possible; in cases Where your work from parts b) and c) means that you have a precise numerical answer, your diagram should be consistent with that. [10 points] 30 fiLz VJ {ILL Pam/Is ; r g . , I, p. ‘ 3'" i , . 3/ sew/w. W k ' i fix Matty/ya? mm ami- 5? I g i S 5 COW/€674 3/??? i ,va 5fimfi€i i fir Ska 67$ ikizrsfzfiaérwuf gem/fl V 5;” v i W 1/?726y 51,1711 . r / ,4} a; at, 3 Kg?” [email protected]? 6W; 7% M1 .1 "QM A“??? 5‘ campie‘mm ( EL - W)? 1 fl <h¥fr flaw {WW/Q aw“- MC} /M V , , i I ‘ g (r ¢ hwWZwmrl’k! flhwmwt; é? ' _ .. w 'r C K? r ;2 a 7% gm. #m 5 j Page 4 of 17 Economics 50 Stanford University Spring Quarter 2008 Question 3: Is this really equivalent? [20 points] Samantha’s preferences over whips (X) and vodka (Y) are summarized by the utility function U(x,y) = x + 2y. Samantha has a total weekly income of $1. The market price of vodka is $5 per bottle. The market price of each Whip is $4, but since Samantha has a free membership in a Whip Club, she receives a coupon booklet every week that allows her to receive up to ten whips at a reduced price of $2 each (i.e. she receives ten “$2 off” coupons per week, each good for one whip). a) Derive expressions for the number of whips (X*) and vodka (Y*) Samantha will purchase ' each week, as a function of her income, I. Show your work, and assume throughout that the prices of each product remain as stated above (i.e. you can plug in the precise values, rather than keeping the prices as “Variables”). [8 points] 6L 5 X + 23 *7 warped Subsfi'fi’v‘fi/ g2er xv . ‘ 1 fl // (:‘ay’k/ é>Tgl 0,7005% all X) "C «9(ij Wye“ ‘7 PX ,‘I/ SVbSb’K/kS X$H7l PX/ly :2/5 “my, V750] 5 01400;: all x Wm X40, my: all j X" (I) .- I/; #1120 y’YI) ; 0 74 I520 ago if b) Suppose I=$120. Show Samantha’s optimal choice in a simple (but clearly labeled) indifference curve — budget set diagram, showing the shape of each relevant curve. [4 . t E poms] / 75/ MAE/5» gr? 5066i / £05,“ 52%}me EL «We Page 5 of 17 +3 Economics 50 Stanford University Spring Quarter 2008 ‘ 0) Following up on your results from part a (that is, allowing I to vary, but plugging in for the specific prices), write down expressions for Samantha’s “Hicksian” demands for whips and vodka, XH(u) and YH(u). Show your work, and again assume that the market prices for each product remain fixed at the values indicated above. [4 points] 1: 22M. 19 “gm 10’ngva X1: and Yr 13095;: L010 X“, a it Mélo Y” 0 1; “$10 ‘ ’ M 5 :r mm [0 IF _b(?l0 a " ~QL H: Canal/Wang We. Wifiimiéd 0V" Magma/u “‘1 if “0 Wm stratum d) Now suppose again that Samantha’s income is I=$120, but that the Whip Club notifies her that it will begin charging a cash membership fee. What is the largest weekly fee Samantha will be willing to pay in order to remain a member of the club (and keep receiving the coupon booklets)? Show your work. [4 points] me m Ciao, Sfimanf’ha W01 Wm +0 canfl/mz (“4% j (beau/5e 5(35) and danqu 7597 b 5% Met/S7. m In (14 club, Samantha cm ‘9 0? Mile ([1418) M411 mu: m s I‘ iu’f -9 Clo—C = g; L/y] 5 e}, 717% Sam (4“, com a P [/Mrc) .-. 7‘3 be in da C/ub. +32 Page 6 of 17 + Q 143W mm Economics 50 Stanford University Spring Quarter 2008 i Name: - TA: Final Exam June 6, 2008 Part II (Questions 4 through 6; 50 points total) “The answers written on these pages are entirely my own. I attest that in taking this exam, I am fully complying with all provisions of Stanford’s Fundamental Standard and Honor Code.” Signature: Do not open this exam until it is time to begin. Page 7 of17 Economics 50 Stanford University Spring Quarter 2008 Question 4: A sketching exercise, now [4 points each; 12 points total] For parts a)-through c), sketch 3 curve with the appropriate shape for the given information. Be as specific as you can (labeling any relevant intercepts, kinks, etc.), and label your axes. a) A conditional demand curve for capital for a firm with production function F(L,K)=(L+K)m, where the firm is a price—taking employer both of labor (at price w per unit) and of capital (at price r per unit). . C <12” \< b) A conditional demand curve for labor for a firm with production function F(L,K)=4Min(L,K), where the firm is a price-taking employer both of labor (at price w per unit) and of capital (at price r per unit). U.) c) A Long-Run Industry Supply curve for a constant—cost competitive industry in which each firm’s production technology is summarized by the total cost function TC(q)=100+ qz. V Qofl Page 8 of 17 Economics 50 Stanford University Spring Quarter 2008 Question 5: A slightly mathematical di—lemma [12 points total] Consider a firm that is a price taker both in the output market (it sells each unit of its product for the market price p) and in the input markets (it purchases labor at a price of w per unit, and capital at a price of r per Unit), with a (maximized) profit function given by: H =p2 / (Swl/Zrl/Z) a) Write down expressions for this firm’s profit—maximizing demand for labor and for capital. Show your work. [4 points] £21,”: “P2 09w [(0 u33/2 P‘IZ .0117. art 0; r‘ /g/Z b) Write down an expression for this firm’s output supply function. Show your work. [2 points] c) Use your answers from parts a) and b) to write down an expression for the firm’s conditional demands for labor and for capital. [These should simplify nicely, but if they don’t, just leave your expressions as they are] [4 points] d) Based upon your results from above, write down an expression for this firm’s production function, F(L,K). [2 points] 2 L .. C? I r/ZL __KV a/ :———— 092' w G??- Li Llf: C’Q ( 2 g; ”””””””””””””””””””””””””””””””””””””” "" Page 9 of 17 Economics 50 Stanford University Spring Quarter 2008 Question 6: Better, with technology [26 points] Mutt’s House of Paternity uses machine—hours (M), labor (L), and capital (K) to produce customized paternity tests (Q), according to the production function Q= [M + min(L,K) ]1/2. Mutt’s can purchase machines, labor, and capital at the market prices 771, w, and r, respectively. a) Does Mutt’s production function exhibit r constant returns to scale? [2 points] {- mun (XL/1K51VZ < 2 CM-l" mm (’LJKBBVL b) Write down expressions for Mutt’s conditional demand functions for all three inputs, that is, M (m,w,r,Q), L*(m,w,r,Q), and K*(m,w,r,Q). Be precise. [6 points] L=I< i'l: gall M) kSC’D/L:O F4“ W) >w+r\ :9 (Q:vm‘q((.,lcy/Z :>L=K) L501, }::cq>z - 5” . . 6 ll:- W\ 7 WW at "'P “‘7 WM av} "Me ‘ H W amw'“% is?) W:- W-H‘ stfi' “’1 m“ M 6 Nut!“ (4153‘ V 6)”, Air g o m0 m< new a: ’ C91 PP W7 ll" N 0:315“? ,'S‘— m : who 0) Suppose m=$16, w=$6, and r=$6. Write down an ex ression for Mutt’s total cost function Tng22, and sketch his total cost ction in a labeled diagram (showing its general shape). [6 points], 1 G 7 l1 TC(Q) = n—a'l _‘ Z l (— (595 " L26? Page100f17 Economics 50 ' Stanford University Spring Quarter 2008 d) Now suppose Mutt’s operates in a competitive industry, and can sell each paternity test at the market price, P. Following from your answer to part b (still assuming that m=$16, w=$6, and r=$6), write down an expression for Mutt’s individual supply function, q*(P). Show your work. [2 points] e) Suppose a technological improvement causes Mutt’s production function to change to Q: [2M + min(L,K) ]“2. Assuming all input prices remain the same (m=$16, w=$6, and r=$6), write down Mutt’s new total cost function, TC(Q), and Mutt’s new supply function, q*(P), after the technical change. Show your work, and explain your reasoning. [6 points] W\<'Z.Ccu1‘t‘> @m*:_@;__ K‘Y‘30’ L*:O ) T C (6252(6622 PzVMC f“ v: P Pr-ICpCQ ~—-> 55 T; f f) Sketch Mutt’s “old” and “new” supply curves (before and after the technical change) in a labeled diagram. Make sure you indicate which curve is which. [4 points] ? Sold 3 New C31 Page 11 of 17 Economics 50 Stanford University Spring Quarter 2008 Name: TA: Final Exam June 6, 2008 Part III (Questions 7 through 9; 50‘ points total) “The answers written on these pages are entirely my own. I attest that in taking this exam, I am fully complying with all provisions of Stanford’s Fundamental Standard and Honor Code.” Signature: Do not open this exam until it is time to begin. Page 12 of 17 Economics 50 Stanford University Spring Quarter 2008 Question 7: Choices can be revealing [10 points] Your research assistant presents you with the following data on the consumption choices of Irina, who is known to devote her entire budget to just two products: beets (X) and hair dye (Y). Unfortunately, owing to a “smudge” in the table, you are not sure what the price of beets was in June 2008. Price of a Price of a tube beet of hair dye $10 a) What price of (or range of prices for) a beet in June 2008 would make Irina’s behavior consistent with maximizing a Cobb—Douglas utility function? [2 points] SWMC prove/ifiefi Di’ EMAth 0“ each jocci‘, “Mr: 10° , —— :: QfCC< 0" kcci’ “((A3 \—a Fug 1 age-x 5 €— 9 XC’I b) What price of (or range of prices for) a beet in June 2008 would make Irina’s behavior consistent with maximizing a Leontief utility function? [2 points] None. («€ka he MKSVMP/ uflvk Catfish»? (room/LCM c) What price of (or range of prices for) a beet in June 2008 would make lrina’s behavior consistent with maximizing a linear utility function? [2 points] \detoctiwhmbr ga\q\\'\\ox (price firi‘io Mods [a ‘o& MI, Show: CA eack period d) What price of (or range of prices for) a beet in June 2008 would make Irina’s behavior inconsistent with utility maximization (i.e. a “revealed preference” argument provides a contradiction)? Show your work. [4 points] I, xi 4 Fer {kcopsx.s‘\(.k:7 we [\‘QQJ HM"- ‘N KUAQ. I XLi/au'wU-é W?“ S{ CLW‘SQA muju x4 (3 oe‘xli—oriagic Kflfy > t7 ~Hoo Page 13 of 17 X 2? a, Economics 50 Stanford University Spring Quarter 2008 Question 8: We’ve got some power, here [20 points] Carrie’s Cupcakes (CC) produces designer cupcakes (Q) according to the production function Q= F(L,K) = L+K, where L is labor and K is capital. Carrie’s can obtain capital at the constant price of r=$16 per unit, but is a monopsony purchaser of labor: the supply of labor is summarized by the equation Ls=w. CC faces no costs other than those associated with labor and capital. a) Compute CC’s marginal rate of technical substitution. Show your work. [2 points] 3; NUS: girl b) Write down expressions for CC’s marginal expenditures on labor (ME) and on capital (MEK). [4 points] __ ’L \a; L whale web: usw c) Now formally solve CC’s cost—minimization problem mathematically to derive CC’s conditional in ut demands for labor and ca ital, and for its total cost function. Show your work. Assume the price of capital remains at r=$16 throughout your analysis. [10 points] Lei X (g Page 14 0f17 Economics 50 I Stanford University Spring Quarter 2008 (1) Sketch CC’s oumut expansion path in a well—labeled diagram. Be as precise as you can. [4 points] k Page 15 of17 Economics 50 Stanford University Spring Quarter 2008 Question 9: A long and winding run [20 points] Crystal skulls are produced in a perfectly competitive industry in which each firm has total cost function TC=4r + wq2/4, where r is the price of a barrel of oil and w is the price of a barrel of water. a) Suppose initially that the crystal skull industry is in Ion -run com etitive e uilibrium, with r=$25 and w=$1. What is the market price of a crystal skull? How many skulls does each firm produce? Show your work. [4 points] MCCKC {N LL QC‘\M-=> (\zlo 'leo b) Suppose the price of oil suddenly increases to r=$100. If, owing to competitive pressures, the market price initially remains the same as you calculated in part a), then what will be each firm’s profits? Show your work. [2 points] 0"“) C93)"; C‘NH‘SQ , Qn'bz (QMCM M Shot—v6: L TC; “00+ he? :§00 '\'('\: Zo‘lo" {—00 3 J50“ c) Describe what will happen to the structure of the crystal skull industry in the long run, assuming that oil prices remain at F$1002 what will be the long-run equilibrium price of a crystal skull? How many skulls will each firm produce? What will happen to the number of firms in the industry, as compared to the “cheaper-oil” equilibrium? Be as specific as you can. [6 points] Cl:°‘° L: NC 5) N 99M Geek Ir ‘ g M q ‘S Jpn/Lied (Xmoi irvhfl "Uaxki'((*7 \‘((h({,\ (i 0"” Qt‘ruj (5dr \‘CO'S'l’ Page 16 of 17 Economics 50 Stanford University Spring Quarter 2008 d) Now suppose there are four skull-producing firms, each with the total cost function noted above, and that no further firms are able to enter the industry. If the demand for crystal skulls is giVen by the function Q=640-16P and the four firms act as a cartel, then what will be the market price of a skull when r=$25 and w=$1? What will be the profits earned by each firm? Show your work. [4 points] Q r @1046? (L: V 9': {Q (C: , 6 (to q i {6 3 MC 1 C‘:"(0 / Q={GQ (KQCVO! :> fis'So‘Roe‘t‘ZY—h‘hflie ..' (3—00 l M$F3{WQ‘ (09* U the §0~M so (grt’ce \\S tat/U 30. K9520} {,M:‘10§ If? 71:. (Uo,goo ~\Qo:\\{oo 6) Repeat your analysis from part (1), now for the case where r=$100 and w=$1. Show your work. [4 points] Good luck with the rest of your exams, and have a great summer! Page 17 of17 ...
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This note was uploaded on 03/19/2011 for the course ECON 50 taught by Professor Abramitzky,r during the Spring '07 term at Stanford.

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Spr2008FinalExamSolutionSketches - Economics 50 Stanford...

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