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Unformatted text preview: ECONOMICS 201
SPRING 2010 EXAM 2 Instructions: Please mark your answers clearly. No class notes or other materials are allowed. 100 points total. Each question is worth
5 points with the exception of Q2, Q3, Q12 (15 points) and Q8, Q1 1 (10 points). 1. Characterize the returns to scale (decreasing, constant, or increasing) associated with the following
production functions. ay=fH%X5Xﬂ=XFX?+X§ b.y=fugxg=mmePsXJ 2. Dora the Explorer produces two outputs (output A and output B) using two inputs (input 1 and input 2).
The price of output A is PA , and the price of output B is PB. The price of input 1 is wl , and the price of input 2 is w2 . Dora produces output A using the production function y A = f A (X f , X1:1 ) , where X f1 is
the amount of input one and X f is the amount of input two allocated to output A. Analogously, she produces Output B using the production ﬁanction y B = f ,, (X ,3 ,X f ) . (15 points) a. What is Dora the Explorer’s PMP (proﬁt maximization problem)? b. What are the four ﬁrstorder conditions (with respect to X 1" , X 2" , X13 , and X f )? Mﬁ
MP,” ' Mat
MgA ’ is equal to MR TS of output B, c. Use the FOCs to show that MRTS of output A, a J. Robert Woodruff produces Coke (output y) using one input, sugar (X). Coke is valued at price P in the
market, and the price of sugar is w. His production function isy = f(X) = alnX , where a > 0. (g points) a. What is the PMP (proﬁt maximization problem)? b. What is the factor demand function, X (P, w) ‘? c. Taking a partial derivative, Show that the output of Coke increases when the market price of
Coke rises. 4. A ﬁrm makes a product with the production function y = 2X 1 + 3X 2. Input prices are wI = l and W2 2 2. What is the solution of the CMP (cost minimization problem) for y = 12? (Please circle one response.)
a. X1=6,X2=0
b. X1=0,X2=4
c. X1=4,X3=0
d. Xl=3 X =2 5. A ﬁrm makes a product with the production function y = min{2Xl ,3X2}. Input prices are WI :1 and w2 = 2 . What is the solution of the CMP (cost minimization problem) for y = 12 ? (Picase circle one
response.)
a. X 6
b. X 4
c. X126,
d. X 0 6. Which of the following expressions is the average proﬁt (i.e. proﬁt per unit output)? (Please circle one response.)
a. ACMC
b. MCAVC
c. AVC—AC
d. MCAC
e. AVCMC 7. If proﬁt if negative (7: < 0) and production is positive (y > 0), it must be that... (Piease circle one
response.) a. AC>MC>AVC
b. AC>AVC>MC
c. MC>AC>AVC
d. AVC>AC>MC 8. Alexa’s cost function is C(y) = y2 + 25 . (10 points) a. At what price does proﬁt equal zero?
b. At or below what price does Alexa choose not to produce? 9. Consider the CRS technology y 2 2X . The output price is P, and the input price is w. How big does the output price have to be (relative to the input price) for a competitive ﬁrm to make inﬁnite proﬁts in the
PMP‘? 10. Based on the Utility Possibility Set, clearly label the points corresponding to the Pareto Set (all PO
allocations). “a 11. Bert and Ernie are competitive consumers in a pure exchange economy. Their endowments of good I
and 2 are em = (10,10) and em“, = (10,15). The prices of goods 1 and 2 are 1‘91 and P2. Bert’s demand 3
mm” . Emie’s demand function for good 1 is X 15m” = mm” . (10 points) function for good 1 is X13” =
41:; 2P, a. What is the aggregate amount of good 1? b. What are m Ber, and m Em (as a function of prices and endowments)? c. Normalize Pi = 1. What are the equilibrium prices, i.e. what is P2 ? 12. Suppose that the social planner wants to implement a competitive equilibrium in which Bert consumes
exactly 5 units of good 2. To do so, the social planner transfers good one from Bert to Ernie. Bert’s demand function for good 2 is X f e” = ’2?” .Normalize P1 =1, as above. {15 points)
2 a. Denote the transfer from Bert to Ernie as T. What are m BM and m Em now? (Hint: take your
answer to 1 lb, subtract T from Bert’s income, and add Tto Ernie’s income.) b. The social planner wants Bert to consume exactly 5 units of good 2. Write the equation that sets
Bert’s demand for good 2 equal to 5. (Hint: it should only be a function of P2 and T.) c. Write the equation that determines the equilibrium prices, i.e. that determines P2 , since 1”] = 1.
(Hint: it should only be a function of P2 and T.) d. Use the equations in b and c to solve for T (get a number). This is the transfer that implements
the social planner’s desired equilibrium. ECONOMICS 201
SPRING 2010 EXAM 2 Instructions: Please mark your answers clearly. No class notes or other materials are allowed. 100 points total. Each question is worth
5 points with the exception of Q2, Q3, Q12 (15 points) and Q3, Q11 (10 points). 1. Characterize the returns to scale (decreasing, constant, or increasing) associated with the following
production functions. 21' y:f(X!ngX3)=X10'5X§2+X303 05' 0.1 0.3
0.1 .5 0.0. OJ 0'3 6X 1: +9)! '3
rcexvexhexﬂc 6 x\ xi +9 K: < 1 1' 3 9 £(Xu11ﬁ3) DR; gig at; [A
e: ' e x 53x3 (MS: Er
b. y:f(X1,X2)=min{2X]°'5,3X2} 'FLQX'JGXL) Ming 3 L; 1 3 (qsgs_..
Lose 1‘ QJ? (xi, ‘1) 1 We {91%. , 933313 '3' («362111 0.: 
I} 3K1»a&‘ I 3K1< ax: m: 1‘ 3x1< axla‘
and 9"“? < 93:“ a 996 m! 9” 13‘s" 3' “X: z (“5' —_._ 2. Dora the Explorer produces two outputs (output A and output B) using two inputs (input 1 and input 2).
The price of output A is PA , and the price of output B is PB. The price of input 1 is wl , and the price of input 2 is w2 . Dora produces output A using the production function y A = f A (X 1A , X f ) , where X 1” is
the amount of input one and X 2" is the amount of input two allocated to output A. Analogously, she produces output B using the production function y3 = f 3 (X B,X 2.3 ) . 115 points) a. What is Dora the Explorer’s PMP (proﬁt maximization problem)?
A I
MAX WI *Xj) '* a
{xu 2:1J
’ﬁ'ﬂi
b. What are the four ﬁrstorder conditions (with respect to X 1” , X f ,X 13 , and X 23 )?
3:8. I B Jr. .a
[KT] 9* mew,~—O [X13 f’ta'——.3‘K:,*i,tai 0
a 31F 6 a;
3x: a i J ‘9); 1, W1
MR“ . MR3
c. Use the FOCS to show that MRTS of output A, A , 15 equal to MRTS of output B, B .
MP, MP2
“a \ eggi Jail w
A —— A
3“? 1". 9,. 1’. ___..'(" 27—7 3"? _ ____i_
T w} W1— 3
1 A“ .5 I DJ, I
"5;? “Mr, mRrs _ ..__ a. mars
{it La. W2
W1 W3. W 7. 3. Robert Woodruff produces Coke (output y) using one input, sugar (X). Coke is valued at price P in the
market, and the price of sugar is w. His production function is y = f (X) = a lnX , where a > 0. (1; points) a. What is the PMP (proﬁt maximization problem)? max P «lnX wX
{x3 b. What is the factor demand function, X (P, w) ? ’0‘ M
Ex] Livy: :0 7? >((P,w) : 
x W c. Taking a partial derivative, show that the output of Coke increases when the market price of
Coke rises. ' ii 93,4i‘=5_>0
5:0H"(w)' 3;“ m w r 4. A ﬁrm makes a product with the production function y = 2X 1 + 3X 2 . Input prices are w1 =1 and w2 = 2. What is the solution of the CMP (cost minimization problem) for y = 12 ? (Please circle one res onse. .
X)1___6,X2:0 MRTS : i; i : W‘HC “6310.
b. X1=0,X2=4 OM: We in?” I. c. X1=4,X2=0
d. X123,X2=2 Smilax f9 x116, X110‘ 5. A ﬁrm makes a product with the production function y = min{2X1 ,3X2 Input prices are W1 2 1 and w2 = 2. What is the solution of the CMP (cost minimization problem) for y = 12 ? (Please circle one
response.) a. X126,X2=0 \1 : RX. :BXL
b. X1=4,X2=6 @X1=6,X2=4
d. X1=0,X2=4 6. Which of the following expressions is the average proﬁt (i.e. proﬁt per unit output)? (Please circle one
response.) a. ACMC Tl' = (9* M3 3 b. MCAVC c. AVCAC (News: 1T = [97463 . 6’ MCAC ’ 1 WC : MC — A c
g AVOMC Since '9 MC, «mag 71' 2 7. If proﬁt if negative (71' < 0) and production is positive (y > 0), it must be that... (Please circle one response.) AC 3; M _
AC>MC>AVC 5‘ 17w" AC C
. AC>AVC>MC O ML 14 Y ,0 M( 7 Ava.
c. MC>AC>AVC 1" ’
d. AVC>AC>MC 1" 8. Alexa’s cost function is C(y) = y2 + 25 . 110 points) a. At what price does proﬁt equal zero? Meg] : Adj) P I
"A .» :MC 3 0,
a3 :j+5_'§. g 32%? 7? ﬂ~5
5
b. At or below what price does Alexa choose not to produce?
Mg : AVC awn93:0 P30 9. Consider the CRS technology y = 2X . The output price is P, and the input price is w. How big does the
output price have to be (relative to the input price) for a competitive ﬁrm to make infinite proﬁts in the PMP‘? 1T_ Paxﬂwx _: (PawJX : (aP—w)x ’ 11 9?!!— 3‘ proﬁt :3 id}. {+9. 4“ 10. Based on the Utility Possibility Set, clearly label the points corresponding to the Pareto Set (all P0
allocations). 11. Bert and Ernie are competitive consumers in a pure exchange economy. Their endowments of good 1
and 2 are em,” = (10,10) and eEm = (10,15) . The prices of goods 1 and 2 are P] and P2 . Bert’s demand 3m . . . , 
He" . Ern1e’s demand function for good 1 is X 15”“ : mm” . (10 points} function for good 1 is X?“ =
43 23 a. What is the aggregate amount of good 1? lotlo :30
b. What are m3,” and mm, (as a function of prices and endowments)?
“am 3 ﬁle 1 9,30 ‘5 10 + 109,, (erL P’s!) 1“ Ef'uiE. : l’tlo +10ng '5 lo * lg P‘— c. Normalize P; = 1. What are the equilibrium prices, i.e. what is .1"2 ? , E
xiiiX5330 :33” +£1.30 ﬂ3m6+3m "Sgo
‘1' A 3 309,430+ 09 2‘20
goermt34aéweuv;):80‘$ 0* 3 1 09430 “.._,L i".
M 12. Suppose that the social planner wants to implement a competitive equilibrium in which Bert consumes
exactly 5 units of good 2. To do so, the social planner transfers good one from Bert to Ernie. Bert’s demand function for good 2 is X fa” = ’23:" . Normalize 1’, =1, as above. 115 points)
2 a. Denote the transfer from Bert to Ernie as T. What are m am and m Em now? (Hint: take your
answer to 11b, subtract T from Bert’s income, and add Tto Ernie’s income.) mm, :. touchyr minJ0 1' lo 4' 91.. + T
b. The social planner wants Bert to consume exactly 5 units of good 2. Write the equation that sets
Bert’s demand for good 2 equal to 5. (Hint: it should only be a function of P2 and T.) xszs. g Maw a a nmzaopl 10+}OP1—T '2'. 2091 w. c. Write the equation that determines the equilibrium prices, i.e. that determines P2 , since I”1 =1.
(Hint: it should only be a function of P2 and T.) xhxf : no a 3.3+ 2.5: 20 ’37 d. Use the equations in b and c to solve for T (get a number). This is the transﬁr that implements
the social planner’s desired equilibrium. Solve Efﬂﬁ'lnS lWUI) sinl '1':ioroP,_ T:to—l0'i :—— 130
.7 cor «rows? 130 4
70‘ ~— m T: 3—9 "
Pl_you Pizi «v If: 9
u 7 __._ ...
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This note was uploaded on 03/29/2012 for the course ECON 201 taught by Professor Ninkovic during the Spring '08 term at Emory.
 Spring '08
 NINKOVIC
 Economics

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