Micro Econ Practice Test 2

Micro Econ Practice Test 2 - ECONOMICS 201 FALL 2010 E XAM...

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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 first-order conditions (FOCs) o f the long-run Profit Maximization Problem (PMP)? b. What is the FOC o f the short-run 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 first-order conditions (FOCs) o f the long-run 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 short-run 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. -r-l +),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, JX-z. -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 f-t -"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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This note was uploaded on 03/29/2012 for the course ECON 201 taught by Professor Ninkovic during the Spring '08 term at Emory.

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