Unformatted text preview: ECONOMICS 201
FALL 2010
E XAM 2
Instructions: Please m ark your answers clearly. No class notes o r other materials are allowed. 100 points totaL E ach question is worth
5 points with the exception o fQ3, Q6, Q7, and Q12 (10 points) and Q5, Q8 (15 points). 1. Characterize the returns to scale (decreasing, constant, or increasing) associated with the following
production functions. 2. A firm's production function is y
w2 , respectively. = /(X1' X 2) X t2 , output price is P, and input prices are WI and a. What are the firstorder conditions (FOCs) o f the longrun Profit Maximization Problem (PMP)? b. What is the FOC o f the shortrun PMP, when X2 = 4? 3. Julia's production function is y = / (XI'X 2 , X3 ) = min{2X1'25X2 , 5X3 } . Prices are
W3 = 1. Suppose that the required level o f output is y WI 1 , w 2 = 5 , and 100. (1 0 points) a. What is her CMP (cost minimization problem)? b. What are the solutions to the CMP (Xl" , X; ,X;)? Hint: Your answer should be numbers. c. How much is she spending at the optimal input bundle? Hint: Your answer should be a number. 1 4. True or false: Assuming that the production function is n ot DRS, i f there are fixed costs, it is n ot
possible for the firm to m ake zero profits. Explain y our answer w ith o ne o r t wo sentences and/or a
graph. 5. Dora the Explorer p roduces t wo outputs, fun (output 1) a nd learning (output 2), using input X. The
output price o f fun is p. , a nd the o utput price o f learning is P2' The i nput p rice is w. D ora produces fun
using the production function YI = a In X I' where X I is the a mount o f t he i nput allocated to output 1.
She produces learning u sing t he production function Y2 = PIn X 2 , w here X 2 is the amount o f t he input
allocated to output 2. B oth a and Pare positive constants. (15 points)
a. What is D ora the E xplorer's P MP? b. What are the F Oes? c. W hat are b oth factor d emand functions? 6. Given the production function Y f ( XI' X 2) = min {2X I , X 2}, w hich c ombination o f i nput and o utput
prices yields infinite p rofit in the P MP? (Please circle one response.) (10 points)
a. WI = 1
w2 = 1
P =1
P =l
b. W I = 2
w2 = 1
P =l
W 2 = 0.5
c. WI = 0.5 d. WI =4 w2 =2 P =4 2 7. Alexa's cost function is C (y) = y2 + 5y + 16. (10 points)
a. Below what price does Alexa choose not to produce? b. Above what price does she obtain positive profits? 8. In class, we said that the marginal cost o f production was equal to the multiplier o f the e MP, i.e., BC( WI , W 2 ' Y) = p . L et's show this in three steps. ( 15 points) ay a. Take the derivative o f the cost function C(wp W 2 ,Y) with respect to
C(W] , w2,Y) = W]XI (w], w2,Y) + W2X 2(W], w 2,Y)· y. Hint: Remember that b. Plug the F Oes o f the e MP into what you obtained in part a. Hint: One o f the two F Oes is
B f(X p X 2)
WI = p
. ax] c. The constraint o f the e MP is f (X I (wp w2,Y),X2(W., w2,y)) = y . Differentiate this with respect
to y. Use the result and what you obtained in part b to show that a C(wp w2,Y) = p . ay 3 9. Ernie (consumer 1) a nd B urt ( consumer 2) are the competitive c onsumers i n the economy. Their
endowments o f g ood 1 a nd g ood 2 are eErnie = (2,5) a nd eBurt = (3,5) . D raw the E dgeworth Box. Label
the width a nd h eight o f the box, and label the e ndowment i n the box. 10. Based on the Utility P ossibility Set, clearly label t he points c orresponding t o t he Pareto Set (all P O
allocations). \f,. 11. Given the E dgeworth B ox, shade the feasible allocations t hat m ake b oth consun1ers at least as well o ff
relative to the allocation " x".
0 ' I.. :te, O. 12. B ert and Ernie are c ompetitive consumers i n a pure e xchange economy. T heir e ndowments o f g ood 1
a nd 2 are eBert = (20,10) a nd eErnie = (10,10). T he prices o f g oods 1 a nd 2 are p. a nd P 2' B ert's d emand
function for good 1 is x :ert = mBert • E rnie's demand function for g ood 1 is X :'rnie 2P' a. What is t he aggregate a mount o f good I ? b. What are m Bert a nd m Ernie (as a function o f prices a nd e ndowments)? c. Normalize p. = 1. W hat are t he e quilibrium prices, i.e. w hat is P2 ? d. H ow m uch g ood 1 is B ert c onsuming in the equilibrium allocation? 4 2P' . (10 points) ECONOMICS 201
FALL 2010
E XAM 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 o fQ3, Q6, Q7, and Q12 (10 points) and Q5, Q8 (15 points). 1. Characterize the returns to scale (decreasing, constant, or increasing) associated with the following
production functions. y = ! (X p X 2) = O.9XI + 0 .2X2
(eXI/elf.,.) ::
( e)( I ) I o. ( ex ....) :: 9 [0.<1 x I 4 0.(1.)( ...] :: (} [ 0.1l() t o·a
b.  l(z.] e ( RS xg.5 , output price is P, and input prices are 2. A firm's production function is y = ! (XI , X 2) =
W2 , WI and respectively.
a. What are the firstorder conditions (FOCs) o f the longrun Profit Maximization Problem (PMP)?
p .,. W I)('  o· a p [ ) (\1 [X't1 o. 5 " 0.1
I o.S f )( \  w \ =.  VI ><"2. o .a ... ) c'1.  •. , X "" 0 ::: 0 b. What is the FOC o f the shortrun PMP, when X 2 ,.1.  [>( , ) o.a ., W,", LL  ( 1.) P = 4? TW'I. W\ ::: 0 3. Julia's production function i sy = ! (Xl'X2,X3 ) = min{2X p 2 5X2,5X3 } . Prices are
W3 WI = 1, w2 = 5 , and = 1. Suppose that the required level o f output is y = 100. ( l 0 points) a. What is her CMP (cost minimization problem)? w,)( I 1Y\1o'\ ... " "1.l("\,. ' .f. J. w1 X.) 5 Xl " )(' ... >::: 1 00 b. What are the solutions to the CMP ( X;, X ;, X ;)? Hint: Your answer should be numbers. ax I : :. A S)( 1.. x,'* = sO, :. S 1 '" Xl. 1 00 . : .Lf, "" , ""'0 P\ c. How much is she spending at the optimal input bundle? Hint: Your answer should be a number. 5 0· t f .f Y· 1s + 0. I1 Pi 0 4. True o r false: A ssuming t hat the production function is n ot D RS, i f there are fixed costs, i t is n ot
possible for the firm to m ake zero profits. Explain y our a nswer w ith one o r t wo sentences and/or a
graph.
J:.«''S,
if:=: 1Jf). :;'f e llS 15ell)(!I'4JI; IT'::.o o r 11= IJD • : t,s
T"ve.
wef"
tus.r.:.? S"'ppolt n::: f tl(lf) . w>(  F. rl +),C!
 r.
ti"..., pt'.d.H.tJ, i.{ p+()c) .. .,.,X :: fJ J
1\::. ... f .
p,..,,l"lfs, ,'1' p .$(X) ..... w'X,.o I t£,ell'l if=:. M :J.
5. D ora the Explorer p roduces t wo outputs, fun (output 1) a nd l earning ( output 2), using input X. T he
output price o f fun is , a nd the o utput price o f learning is P2 . The i nput p rice i s w. D ora produces fun
using the p roduction function YI = a In X l' where X l is the a mount o f the i nput allocated to output 1.
She produces learning u sing t he p roduction function Y 2 = /3 In X 2 , w here X 2 is the a mount o f the i nput
allocated to o utput 2. B oth a a nd p are positive constants. (15 points)
a. What is D ora the E xplorer's P MP?
.., 0 (1,,)(
"'" f
Ql,. X
1""\ \ "a. r "'L. )  \v ( X.+)(1. )(,,'<.... b. What are t he F Oes?
p\ t ( VI  [ ) (\1 )(, 0  LX'l.J f'1
VJ 0 :;0
x"\. c. What are b oth factor d emand functions? 6. Given the p roduction f unction Y = !(X1 ,X2 ) = min{2X1,X2 } , w hich c ombination o f input a nd o utput
prices yields infinite p rofit i n t he P MP? (Please circle one response.) (10 points)
a. WI =1 b. Wj = WI = 0.5 o d. WI 2 =4 w2
w2
w2
w2 =1
= 1 = 0.5
=2 P = 1 1 \ = P P=1
P =1
P =4 { ·
'S,...,{e
II ::. f ;;t)(' I IT ::
1l'==.2 v   I , ) ( ' I. ) WI  l (,  £.oJ ' 1 ' l.. • v
 "1. ."  &J VI I X,  I "
"L"\X )
I I W1.. ) (a p X\  WI  ') 0 . 7. A lexa's cost function is C (y) = y2 + 5y + 16. (10 points)
a. B elow w hat price d oes A lexa choose n ot to p roduce? p ; /V\ <. =. Av t:.
+ 5 ::: .f' 5'
::. 0 a" t1 W t .,. f f)' (. e It I e)( " Lei O\jJ 5 J 1, 0 (J Se..s ,., f }} fro dv L e. J..o P :. 6' b. Above w hat p rice d oes s he obtain positive profits? ... S :::. "" "It 4 Ib T .r 5 .r
(J == ,.."J,e.., fr)', e q hove 1"'3 J 1"'.(,I'.ij . I f l e>'4/I( '3
8. In class, we said t hat the marginal cost o f production was equal to t he m ultiplier o f the CMP, Le.,
8 C( t W y ) =f.1. L et's s how this in three steps. (15 ]Joints)
2, a. Take t he derivative o f the cost function C ( WI' w2 ' y) w ith r espect t o C(wI , w 2 , y) y. Hint: Remember that wJX1(WI' W 2 , y) + W 2 X 2 (W 1 , w 2 , y) . b. P lug the F OCs o f the C MP into w hat y ou obtained i n p art a. Hint: O ne o f the two F OCs is
a f(X p X 2 )
WI = p
.
a XI d ((""" ""oz., g') _ J.f ,}X, JXz. r JA dj .) g c. The constraint o f the C MP i sf(XJ(w p w2 , y),X2 (w p w2 , y)) = y. Differentiate this w ith respect
to y. U se the r esult a nd w hat y ou obtained in p art b to s how t hat a C(wp w 2 ,y) I I !d3. .!£ ),5 ::: I .
'S
J c (WI,
_ W 'l.1 g) rh vS [elf d3
f't dX'l t 3 I ay • . JX"] _
;rg  r" = p. 9. Ernie ( consumer 1) a nd B urt ( consumer 2) are the competitive c onsumers i n the economy. Their
endowments o f g ood 1 a nd g ood 2 are eErnie = (2,5) a nd eBurt = (3,5) . D raw t he E dgeworth Box. Label
the width a nd h eight o f t he box, a nd label the e ndowment i n t he b ox. • e L====_.. e,:s O£Nlit 10. Based o n the l Jtility P ossibility Set, clearly label the points c orresponding to t he Pareto S et (all P O
allocations). Lt, 11. Given the E dgeworth B ox, s hade the feasible allocations t hat m ake b oth c onsumers a t least as well o ff
relative to the a llocation " x".
Q 0,
12. B ert and Ernie are c ompetitive c onsumers in a p ure e xchange e conomy. T heir e ndowments o f g ood 1
and 2 are e Bert = (20,10) a nd eErnie = (10,10). T he p rices o f g oods 1 a nd 2 a re
a nd P2 . B ert's d emand
function for g ood 1 is X IBert = m Bert E rnie's d emand function for g ood 1 is X IEmie • = mEmie a. W hat is t he a ggregate a mount o f good I ? e" :: + 10 30 b. W hat are m Bert a nd m Ernie (as a function o f prices a nd e ndowments)?
:=
+ 10 p...
ep
p1 8 t I'.J. t , ,,'" \. l it £'''1'( P, e t"l. + "'L f e"J,."I... = 10 4 '0 P.,. = 1. W hat are the equilibrium prices, i.e. w hat is P2 ?
X
e... 10'). of' 10.,. 10 l"),. ::: c. N ormalize X,,'" 11..::: I P 3 0 .r 30 "'2.0 f"1. f,.. =" 0 "0 0 = "3 0
' 20 d. H ow n luch g ood 1 is B ert c onsuming in the e quilibrium a llocation?
e..,..,. M O ft
"1.0
10 P"'l.
, ,0 of' JO$
3S XI :::: _ : :
""Z. t> \  "'I :z. "2. f \ 4 _ z.... 1 7.S • ( l0 points) ...
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