Final Exam Solution Sketches

Final Exam Solution Sketches - Economics 50 ' Stanford...

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Unformatted text preview: Economics 50 ' Stanford University Spring Quarter 2010 Name: (ii/x TA: wwaiii’ziI”Exafii"" June 8, 2010 , Part I (Questions 1 through 3; 48 points total) Write your name and your TA’s name (N adeem Karmali, Anqi Li, Sanaa Nadeem, Irina Weissbrot, or Xiaoling Zhou), 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 III 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. 1 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 I of 17 Economics 50- Stanford University A Spring Quarter 2010 Question 1: Draw me a Diagram [4 points each; 12 points total] For each part, sketch a curve (or curves) consistent With the given information. .Be as specific as 'you’ canyandjaheLyOnrnxeswfiwawe,anaemia,a a , , a , is, , ' a) A demand curve for X, for a consumer Whose preferences are described by the utility function U=2X+Y. ' ' PX. m b) An isoquant for a firm with production function F(L,K)=L“2K“2, Where the firm is a price— taking employer both of labor (at price w per unit) and of capital (at price r per unit). L. c) A (regular) demand curve for X and a Hicksian demand curve for X, where X is an inferior, but not a Giffen, good (be sure to label which curve is which!) ix Page 2 of 17 Economics 50 ‘ Stanford University Spring Quarter 2010 Question 2: Will we becompensated for this? [12 points] Crissy, whom you have met previously, has an income of $1 that she allocates to refined sugar (X) and-t0 freSh Vegetab1€S.(X).,.X!hiCh she can buy almarhet pricesExtantinliyirespectiyebeimmune“ preferences are desoribed by the utility function U(X,Y)=20[nX + Y. Draw a well—labeled Slutsky diagram illustrating the effects of an increase in the price of X. Assume that Crissy chooses aninterior solution both before and after the price change; label your diagram so that the “total effect” of the price change is from point A to point B, the “substitution effeCt” is from point A to point C, and the “income effect” is from point C to point B. Be sure to show Crissy’s “new” and “old” indifference curves, as well as all relevant budget lines. i ‘ o meow/Le £§<C€dr c‘ no {A M fi»flamz§~.m Page 3 of 17 Economics 50 Stanford University Spring Quarter 2010 Question 3: Saving, or Borrowing, Grace? [24 points] ‘ Consider a group of university students, all of Whom derive utility from their consumption now (C1) w-w«WWW—wwafidiheinonsump-tioninextyeartcg),rwhere-Lcrand €27 are-both»me-asured~intdollvarsawEaewhflfitheserm—ww ‘ students receives an income of11=$10,000 this year, and Will receive an income 0f12=$40,000 next ' year. Any unspent income this year can be saved (to be consumed next year), and will earn the interest rate 7:0; if students wish to consume more than their income this year, they can borrow from a loan shark at a 100% interest rate (that is, r=l). a) Each student faces the same budget constraint. Write down a mathematical expression for this constraint, and sketch it in a precisely-labeled diagram. [8_points] Now assume that all students have preferences that can be described by a utility function of the form U(C1, C2) = C1“2 + aCZI/Z, where ais some constant greater than or equal to zero — but that ais different for each student. all/(3C, m. [2 points] - b) Compute each student’s marginal rate of substitution, that is, Page 4 of 17 , Economics 50 Stanford University Spring Quarter 2010 , c) For what values of awill a student borrow in period 1? For what values of _a will a student save? For what values of awill a student neither borrow nor save? Show your work. [6 points] . ' l ((0 0 "CD ~ " r , WWWMMWJJ “1... than; aw; ,V,,,_v , . , ;_-.,- Ma, y. an: :7 W_____L.._‘i__.,___>a._. W _i(/.)O new than,....aw._.~.._mm..,,, _ 3mm .4 s > W i 67 (x We a Q low > a C‘ZIDIOED) C2; 40, day) d) Write down a precise expression for a. student’s optimal choice of C1, as a function of a. Show your work. [6 points] Exfol<l :' MS":- dbl—(“)3 Cg; dC‘Y’Z‘ : Z A}; 4.0LZCV‘GZ e) What level of Cl will be chosen by a student for whom a is very close to 'zero? What level of C1 will be chosen by a student for whom a is almost infinitely large? [2 points] R (#40 37 Cl-%_32>ooa> dam?) q a O “'9. w‘ Page 5 of 17 Economics 50 _ Stanford University Spring Quarter 2010 Name: I TA: 1‘ M WWWWWWWWWWWWWWWWWWWWWWWWW "WW """"""""""" ’ I J n n e 8 , 2 0 1 0 _ ' Part II (Questions 4 and 5; 48 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 6 of 17 Economics 50 Stanford University Spring Quarter 2010 Problem 4: Laboring, in the .short- and long—run [28 points]- Gus’ daily preferences over food (X), water (Y), and leisure time (N) are summarized by the utility _ -------- w-w ------ ~’—-funetionuU(~X;-Y~,N:)~=,~X¥NrWGusrcant/purchaseifoodiatethemarket-price opr-frperwounceandwater—w«we»» ~«~~~~~~~~~~- at the market price of pY per ounce. Gus has a total of 24 hours per day that he can allocate to leisure (N) or to labor (L); he receives a wage of $w for each hour of labor he works, and he has no income other than what he earns frOm labor. a) Suppose Gus currently required by a binding labor contract to work i hours per day, so that he enjoys N=_24— L hours of leisure. Write down an expression for his daily budget constraint, in terms of L , and Show what Gus’ optimal choice of X and Y will look like in a budget line — indifference curve diagram. [6 points] PXX“? Y WW ' ‘K b) Deriire expressions for Gus’ short-run demands for food and for water, X * ( p X, py,w,Z) and Y * ( pX, py,w,L) . Show your work. [8 points] 3 "CW'WS MM“ "- exK : PYYW $9 i" ' : (‘3: ZQY Y: 'Page 7 of 17 Economics 50 _ Stanford University Spring Quarter 2010 c) Using your results fiom part b),_write down an expression for Gus’ short-11m maximized (indirect) utility, U * ( pX, py,w,L). [2 points] .._‘ J-k - “swam APT.“ ‘ MLTW a . . . ,, WWW.Wwwmwth.,.,,,V..i.a.-ws..r www-mmwmwmfaw“WWW.,7, Zfié ZQY ' -— 'L — c; a?“ L (1) Now suppose Gus is able to renegotiate his labor contract, so that he can freely choose his daily level of leisure (and labor). Compute the numb er of hours Gus will choose to work each day, i L“ (for a “short cut,” you should be able to use your answers to part c), and then write down > expressions for his long-run demands for food and for water, XLR pX,py,w) and YLR (szpYaW)' POintsl Page 8 ofl7 I Economics 50 Stanford University 1 Spring Quarter 2010 e) Sketch a diagram showing how Gus’ maximum utility (on the Y axis) depends upon the number of hours, L, he works each day (on the X axis). [4 points] U Question 5: Multiple choice [2 points each; 20 points total] For each of the following, indicate the best answer by clearly circling its number (Le. draw a circle around the number 1, 2, 3, 4, or 5). - a) Suppose a consumer’s income consumption curve is a straight line, described perfectly by the equation Y=X. Which of the following preferences (over X and Y) could be consistent with such a curve? I ®( 41); X, Y 3 . . MLLK; My ALA, we“ “ACLMQ. CL. 1. XandYare perfect substitutes._ (Mb/1 ‘3'»; (’1 ’ (’Y 1; ma- \(::K, @ Cobb-Douglas preferences over X and Y. V55 V! 3. Preferences described by the utility function U(X,Y) = 40X1’?+ Y. M o - 4- B°thla“d?(°nly)-<—=~i+lxu‘cs mm was 4190 QCLQQIJ'e‘él/‘J 5. 1,2,and3. ‘ gang 4w; UvUL YzK is a 84g} of Milk/PX : W\c/‘9\( | b) iWhich of the following utility functions describes “strictly convex” preferences? 1. U(X,Y) = X+2Y .2. U(X,Y) =Min(3X,Y) . - 3. U(X,Y)=X2+2Y2 » V l U(X,Y)=400-X2-Y2 Mil- wwA-olome/ but comm . 5. None of the above. ' Page 9 01°17 Economics 50 Stanford University Spring Quarter 2010 c) The production function F(L,K)= L3/4K3/4 exhibits: 1. Diminishing marginal products and decreasing returns to scale. _ Diminishing marginal products and increasing returns to scale. V 3'. Increasing marginal products and decreasing returns to scale. 4. Increasing marginal products and increasing returns to scale. 5. None of the above. (1) What is the elasticity of substitution for the production function E(L,K)=4L+K? 1.4 ' '_ W13 . /.. y 2. 1 ' * .firw FL/L, . 3. 1/4 4. 0 @ infinity e) Which of the following supply functions exhibits a constant price elasticity of supply? @ QS = 2P/(w+r) .2. QS = 2+P 3. QS = Min(2,l?) 4. QS=ZlnP 5. None of the above. Page 10 of 17 Economics 50 Spring Quarter 2010 Stanford University f) If a firm experiences labor—saving (capital—enhancing) technical change, then which of the following will be true of its isoquant corresponding to Q=400? [Assume, as usual,- that labor is plotted on the X axis. ] ‘ 1. It willslihiftloiltward, and will become flatter. ' \ Q) It will shift inward, and will become flatter. 3. It will shift outward, and will becorne steeper. 4., It will shift inward, and will become steeper. . 5. None of the above. g) _ Which of the following is not necessarily true? 1. If X and Y are perfect complements, then a consumer will always purchase them in the same proportion, regardless of their prices. ' 2. aXH(Px, Py, u)/ 0Py = 0YH(PX, Py, u)/ an , 3. If a consumer has Cobb—Douglas preferences over X and Y, then both X and Y have ‘ straight-line, upward-sloping Engel curves. ‘5. None of the above. If P<MC, then a firm is earning negative economic profits. “MAM ' 0‘ ‘F‘J‘M W much P> M: ‘H M W! 44.0,. . . ml as = M . h) -Wh1ch of the followmg cost functions corresponds to the production function - F(L,K) = [L + K]2 ‘2 1. TC = szin(w,r) 2. To = (w+r)Q2 @ TC = Qmmin(w,r) 4. TC = (w+r)Q“2 5. None of_the above. Page 11 of 17 wt (’1? MC— Mia Economics 50 . _ Stanford University Spring Quarter 2010 i) Consider a profit-maximizing monopolist for which marginal cost equals $10 and the price elasticity of demand is -3. What price will this monopolist charge for its product? A __________ C. "a _ .1 _____ ~ \ 2. $13.33 I ’ T v “’5 ~ 2‘ 3. $6.67 4. 1 $5 5. None of the above. j) Which of the following would'be a vertical line? ‘ ,- 1. , The demand curve for X, where X and Yare perfect complements. K I The short—run conditional demand curve for labor, Where F (L,K) = LUZKU2 and W e capital is fixed at K = K. ow 3. The short-run profit—maximizing demand curve for labor, for a competitiVe firm I M where F (L,K ) = L“sz and capital is fixed at K :K . ( Thevincome consumption curve for X, for a consumer with preferences summarized B ~- by U(X,Y)=Y—X. .5. All of the above. Page 12 of17 Economics 50 Stanford University Spring Quarter 2010 IName: 1M Q Jr: 1.4 Q3 TA: ““““ Exam ‘ June 8, 2010 Part III (Questions 6 and 7; 54 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 13 of17 Economics 50 Stanford University, Spring Quarter 2010 Problem 6: Cost-minimizing, or profit—maximizing? [24 points] After careful examination bf data, you estimate the following total cost function for a firm, Where Q is ,WWWMW the {luaEEELOf QLliPlltRYOduCEd: W islhfiprice'ofgach unit of labor,(L),.,and maths priceof each unit“ ________ ____________ an”, of capital (K): _ ~ _3 2/31/3 4/3 TC—TW I” Q a). Write down expressions for this firm’s marginal cost and aVerage cost functions, and compute the output elasticity of total costs for this‘firm. [6 points] '6. 3 Zfevs‘lg' _ Tc. L _ fir Q . b) Use Shephard’s Lemma to write down expressions for this firm’s conditional input demand functions for labor and for capital (you can uSe a different method to derive these functions if you Wish, but that wouldprobably be much more difficult!) Also compute this firm’s wage elasticity of conditional demand for labor. Show your work. [6 points] m a er- _._Lr7 Page 14 ofl7 ' Economics 50 ' I Stanford University Spring Quarter 2010 r c) Assuming that this is a profit-maximizing firm in a perfectly competitive industry, write down an expression for its supply function, q*(w,r,P) , and compute the firm’s price elasticity of supply. Show your work. [4 points] P \ m.” ~ : 6543(48 r: r a.“ q“ _ ’ d) Use your previous results to write down an expression for this firm’s production function. Show yoUr work. [4 points] i' ‘ ' Z V V; Rm £931. H” a, 2L. to]; D 4K ‘7 (as; 2 21: I V "‘ v - V A . . (A) '5 Q3 CV3 ,q/Z ' 2/ ZL. ., 3 5")..J. 5i an” e) Finally, use your previous results to write down an expression for the finn’s profit- maximizing demand for labor. Compute the wage-elasticity of this firm’s profit— rnaxirnizing demand for labor. Show your work. [4 points] Ya 4/ v 4 . ~ a". if?) ' . _L€9~ “"‘“""”f§r:‘2’r/3'("FWWQ3 G‘d’g (A) (, Page 15 of 17 if . m- , ,,,,,,,,,,,,,,, t, .- W m“ ............................ I Economics 50 . Stanford University Spring Quarter 2010 Question 7: How much power? [30 points] Micro Magic is a profit-maximizing firm with monopoly power in the market for its output. Its . r-a.m...warm:sofrpIQductionaarcadomesticlab.or_.(L1). andioreignlabortL2),.-accordingioatherproducti011M ............... W; ............ -. ' function Q=F(L1, L2): L1 + L2. Micro Magic can hire up to 25 units of domestic labor at the price w1=$20 per unit; any additional units of domestic labor will cost $40 each. The firm can hire as many units of foreign labor‘as it wants at the constant price w2=$30 each. The market demand for Micro Magic’s product is described by the function QD=6O-P. Micro Magic incurs no costs other than those associated with domestic and foreign labor. a) Derive expressions for Micro Magic’s conditional demands for domestic labor, L1*(Q), and for foreign labor, L; (Q). Show your work. Write down an expression for the firm’s (minimized) total cost function, TC*(Q). [10 points] \%r Qzagl usemia L\ ~>2' La=éQ_‘)TC:f206L» _ ‘ carols: mu :_= a5) L12Q»a§'3'izz mam-3:289:25) ‘2; 30'Q_ 2,5‘3 b) Sketch the firm’s Marginal Costs, Average Costs, and its demand curve in a diagram. Be as precise as you can, and label your axes. [8 points] Page 16 of 17 Economics 50 Stanford University Spring Quarter 2010 0) Calculate Micro Magic’s profit-maximizing choice of output, Q*, and its profit;maximizing demand for both types of labor, L1,“ and 142*. Show your work. [8 points] Meal/KC: Go;7.¢{:29 ——.—52..cq_: 40 we ZQ (1) Compute the deadweight loss attributable to Micro Magic’s monopoly power. Show your work. ' [4 points] [Pa-MC] a} (21:30 wapob Owl-Lama [MLLMCB flat FM (imam Zaal‘gz ZSI’MC-fl’a'j WWW- “ ZS 41,41 30 k Mt: go; Mg 2 60-51 Val—z“ 32:52) + gag) ‘+ g <53) (wage ea mew flagm M L») Page 17 of 17 éoqpef‘ § MK: 60- ZQ éfikf. (attic) 3. ...
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Final Exam Solution Sketches - Economics 50 ' Stanford...

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