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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)
andt0 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) ww«WWW—wwaﬁdiheinonsumptioninextyeartcg),rwhereLcrand €27 areboth»measured~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 preciselylabeled 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
_  ww  ~’—funetionuU(~X;Y~,N:)~=,~X¥NrWGusrcant/purchaseifoodiatethemarketprice oprfrperwounceandwater—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’ shortrun 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’ short11m maximized
(indirect) utility, U * ( pX, py,w,L). [2 points] .._‘ Jk  “swam APT.“ ‘ MLTW a . . . ,, WWW.Wwwmwth.,.,,,V..i.a.ws..r wwwmmwmwmfaw“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 longrun 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, @ CobbDouglas 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)=400X2Y2 Mil wwAolome/ 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 ‘ straightline, upwardsloping 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 proﬁtmaximizing 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 shortrun 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: Costminimizing, 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 proﬁtmaximizing 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 wageelasticity 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 proﬁtmaximizing ﬁrm with monopoly power in the market for its output. Its .
ra.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=6OP. 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 mam3:289:25)
‘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 proﬁtmaximizing 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] [PaMC] a} (21:30
wapob OwlLama [MLLMCB ﬂat FM (imam Zaal‘gz ZSI’MCﬂ’a'j WWW “ ZS 41,41 30 k Mt: go; Mg 2 6051
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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