Micro Econ Practice Test

Micro Econ Practice - ECONOMICS 201 FALL 2011 EXAM 2 Instructions Please mark your answers clearly No class notes or other materials are allowed

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Unformatted text preview: ECONOMICS 201 FALL 2011 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 unless indicated otherwise. ‘ 1. Characterize the returns to Scale (decreasing, constant, or increasing) ass0ciated with the following production functions. ' a. y = f(X) = 5X b. y: f(X1,X2) =min{2X1,X2}+X1°'5 2. True or false: If profit is infinite and there are no fixed costs, the production function can be IRS or CRS, but if profit is infinite and there are fixed costs, then the production function must be IRS. Explain your answer with one or two sentences and/ or a graph. 3. True or false: If there are no fixed costs, it is optimal for a competitive firm to produce no output when output price is above average cost. Explain your answer with one or two sentences and/ or a graph. 4. Tiger Mom’s production function is y = f(X1,X2,X3) = min{Xl,5X2,10X3}. Prices are w1 =1, - w2 = 2 , and W3 = 3 . Suppose that the required level of output is} = 50. (10 points) a. How much is she spending at the optimal input bundle in the short-run when E =10 ? b. How much is she spending at the optimal input bundle in the long~run? 5. Chipotle produces burritos (output y) using only “yummy goodness” (input )0. Burritos are vaiued at price P in the market, and the input price is w. Chipotle’s production function is y = f (X ) m X a, where 0 <a<l. (10 points) a. What is Chipotle’s PMP? b. What is the FOC? c. What is the factor demand function? d. Taking a partial derivative, show that the production of burritos (output) decreases when the price of yummy goodness (input price) rises. ' 6. Let’s continue the above Chipotle example. (10 points) a. What is Chipotle’s CMP? Suppose that the required level of output is} . b. What is the conditional factor demand function? e. What is the cost function? Hint: Your answer should be an expression involving w, E, and or. 7. Dooley’s production function is y = f (X 1,X 2) = 0.5X1 ~+~ 2X2 . Output price is P, and input prices are w1 and w2 , respectively. I 10 points) a. What is Dooley’s PMP? b. What are the conditions that ensure profit is infinite? 8. Multiple choice. Given the graph, at which price would a competitive firm produce a positive amount of output but obtain negative profits? (Please circle one response.) M. L, a, OMS-Hon l' a. PriceA 6' P MC b. PriceB 4 c. PriceC d. PriceD 9. Alexa’s cost function is C (y) '= y2 + 2y + 4. (10 points) Ft 8 . a. Above what price does Alexa choose to produce? C D b. Below what price does she obtain negative profits? 10. Based on the Utility Possibility Set, clearly label the points corresponding to the Pareto Set (all P0 allocations). it; ' ' ' 1 ' ' ‘4 1 11. Bert and Ernie are competitive consumers in a pore exchange economy. Their endowments of good 1 and 2 are 636”. = (5,5) and 8};ij = (5,5) . The prices of goods 1 and 2 are P; and P2 . Bert’s demand . . m . . . . - 3m .8 function for good 1 1s X139” = Be” . Ernle’s demand function for good 1 is X 15”“ = m . 10 oints 4P1 4P] (_p > a. What is the aggregate amount of good 1? b. What are mBm and m Erma (as a function of prices and endowments)? c. Normalize P} :1. What are the equilibrium prices, i.e. what is P2 ? 12. Suppose there are two competitive consumers in the economy, A and B. They have utility functions 14,, (XIA,X2A) = X845ng and uB(XIB,XZB) = XffiXgfi. There is a fixed amount ofeach good, X1 and Z. (15 points) a. In a competitive equilibrium, each consumer’s MRS equals the price ratio, which implies that A’s MRS equals B’s MRS. What is the expression MRS A = MRS B ? b. We wish to derive an expression for Pareto Optimality. To obtain this, let’s maximize A’s utility subject to B’s utility held constant at us . What are the FOCS with respect to X 1 A and X 2 A ? Hint: Prior to taking derivatives, plug in the following: X13 = R1 — X] A and X23 2 f2 — X 2 A . c. Every competitive equilibrium is Pareto Optimal. Show that the FOCs in part b yield the same expression as in part a. Hint: Remember that X13 = X1_"“ X 1 A and X 2 B = X 2 — X 2 A . ECONOMICS 201 FALL 201 l 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 unless indicated otherwise. 1. Characterize the returns to scale (decreasing, constant, or increasing) associated with the following production functions. ' a. ymf(X)=5X Hear) .1 5 (ex) = 9 (90. Hex) s e to) -—» c Rs "-——-_-.. eHx): err»). b- y=f(X17X2)=min{2XlflX2}+Xf)5 0-5 0‘5 'waoa’“) :M;*£1(9xa),9X13+ “Why”: 9 minf'l-Ki,x1)) *6 X. , award) a 9 mrnfufil) .L e xf‘5 . Hanan) (Magda 035 2. True or false: If profit is infinite and there are no fixed costs, the production function can be IRS or CRS, but if profit is infinite and there are fixed costs, then the production function must be IRS. Explain your answer with one or two sentences and/or a graph. Fa l se. C RS Is <15 nsi’srewt wi'H-t infiniie prob»? 6V6" lid “31‘ e are -F.'xe Ll (OJ-f5- SUrPose *F< Tl: 95(xjnwx~F< 9". ‘ "Hflfln e [P 5(a)“ wxj— F 13' fro L4, J's Hm PM?) ’Pfl'mal {we/H3 4P9 ’Iré'm'i'E. 3. True or false: If there are no fixed costs, it is optimal for a competitive firm to produce no output when output price is above average cost. Explain your answer with one or two sentences and/or a graph. FqISQ. 1'5 Of'h'mai fiO-i’ 4’6 Prague 31/ U ) fr _ 75° y>o 0? Pia-4‘3) =7 ((3)7 Pa :7 com» :> MM 11' pn‘ce 3'3 Iggy averaJe 605%) ,q. . 2:"! we wait AM; L3 9, . U . 1:5 lip-hind] to pro Jud 18rd cufffi‘ 4. Tiger Mom’s production function is y = f(X1,X2,X3) = min{X1 ,5X2,10X3 Prices are w1 =1, W2 2 2 , and w3 : 3 . Suppose that the required level of output is ; = 50. (10 points) a. How much is she spending at the optimal input bundle in the short—run when X; =10 ‘? in m SLoH’ m, 7:50: my. {50, 5-10,,0.103,5o xlzso, x1" ‘0' 4"" X3310" 704’“ was 4»: $l-56-t- zit-rm #340 : $100. _______ b. How much is she spending at the optimal input bundle in the long-run? ix“; ‘i’lgl land“ I‘dn’ 7:50 :3 Min é 5'0J 5'40) [0.53. 50 X): 50) :3) x - I J “09 X375. “fetal “fits are tusor 12.104 {3.5119‘L3'5. .———-.____ 5. Chipotie produces burritos (output y) using only “yummy goodness” (input X). Burritos are valued at price P in the market, and the input price is w. Chipotie’s production function is y = f (X ) = X ‘1 , Where 0<05<l.(10points) at a. WhatisChipotle’sPMP? Mod PX -* WX X er-I ’ O b. WhatistheFOC? [x] MW "44 - .L w "”‘ c. What is the factor demand function? X ( flu) -_-_ (1. Taking a partial derivative, Show that the production of burritos (output) decreases when the price of yummy goodness (input price) rises. * .25.. -I r.- -'v W 3‘- 3 or w ““ (om-1w” U :_- (fast-D : «-| J :— .——— (_.__. I < O 6. Let’s continue the above Chipotle example. (10 points] a. What is Chipotle’s CMP? Suppose that the required level of output is; . Min W‘x “ s.+. "f :2 X" b. What is the conditional factor demand function? a. Solue {arm—.7. ". - ." Y x XU") 7.) "' j c. What is the cost function? Hint: Your answer should be an expression involving w, J, and or. J. ch,‘j) : w‘x : w-E" 7. Dooley’s production function is y = f(X1,X2) = 0.5X1 + 2X 2 . Output price is P, and input prices are w1 and w2 , reSpectively. (10 points) a. Whatis Dooley’sPMP? NMX P ( 05x1 "iax‘l-Xfl “.34: " wax"), aux, b. What are the conditions that ensure profit is infinite? ' f4 1% 7 31. (use. Mia iqu-r 1)] "YT-:09 wt I .33.; _I, (-3; (use only he” 1), 1T7.” “I aP'Wa 7 0* V W-u. 1:; W1 2 8. Multiple choice. Given the graph, at which price would a competitive firm produce a positive amount of output but obtain negative profits? (Please circle one response.) a. PriceA GM A 4:... flugsfiog 8 ML .......L._—v-——"""" b. PriceB _ @ Price C E}? d. Price D 9. Alexa’s cost function is C(y) = y2 + 2y + 4. (10 points] a. Above What price does Alexa choose to produce? MC : AVC a 25 + D. t: 3 4’ 3‘- L...’ 3 3 a 5 b. Below what price does she obtain negative profits? 1 C. 3M ( fh 3‘ +1 P s 6 _ \j 4* 3x .. ‘3 i’ 3 -——-—— 3 :* :5— 2; 3":Vé ‘51; 10. Based on the Utility Possibility Set, clearly label the points corresponding to the Pareto Set (all P0 allocations). ' ‘4 a i : .. 55% .. “if 11. Bert and Ernie are competitive consumers in a pure exchange economy. Their endowments of good 1 and 2 are em. = (5,5) and emf, = (5,5) . The prices of goods 1 and 2 are R and P2 . Bert’s demand . . If . . . 4 t ' 3 ( - function for good 1 IS X13” m Tile)” . Erme’s demand function for good 1 1s X19” = :9?“ l I . t 10 points) a. What is the aggregate amount of good 1? b. What are m 3m and m -, mm s 5 +59, Meme 5 5 + 5P?- c. Normalize P1 =1. What are the equilibrium prices, i.e. whatis P2? KH 4' XVI. LEI Sin; 4, 3(S+S?1) =10 :> 5+591 + ‘54 I591: 9‘0 1+ WET—g aoPlzao (as a function of prices and endowments)? Fri-71 and Pf-‘Ji 12. Suppose there are two competitive consumers in the economy, A and B. They have utility functions uA(X1A,X2A) = XflfXZDj and uB(X]B,XZB) m Xlofingfi. There is a fixed amount ofeach good, X1 and f2. (15 points) a. In a competitive equilibrium, each consumer’s MRS equals the price ratio, which implies that A’s MRS equals B’s MRS. What is the expression MRS A = MRSB ’2 -.s ,9 —.3 A: 3 0.; XIA x1.‘ 1 ' _: 01a X'B x15 : X15 .5 - M "u *1“: M 0.: x12 x-IC‘ 4m X33 __ X15 f—u-I w Xlg HLXIS b. We Wish to derive an expression for Pareto Optimality. To obtain this, let’s maximize A’s utility subject to B’s utility held constant at us . Consider the maximization problem: max m3: SJ. (f1 — LAY-2&2 — X211)”-8 = 53 What are the .FOCs with respect to X1 A and X 2 A ? Hint: First, write the Lagrangian With . 1- A ‘ a r 5 :5 a .- ‘1- .- A a __ '.5 ' 5 a' “' '-' - ! " Cm} .5 xm x“ M (-a) (erm) (*1: Yu) J" “.1. .5 .ns «p 4"- ___ _ _— fit-LA} .5 Xm X1A ('Y.)(x| xiii) (X1 X1}! .- O E Lani-trainer 0. Every competitive equilibrium is Pareto Optimal. Show that the FOCs in part b yield the same expression as in part a. Hint: Remember that X13 = X1 — XM and X23 = $2 — X”. __ . .r -, ,S’X,:x1: x _ -SXJA X1.“ . -;_ - .... I 1(il"xm)“C§( )‘yjfl’ _, , gC23,XM)'7'(S-(1-XZA) 1. -.s .5 .. «9 _ ,7 .sx.. x1, I -' ,wa) (MM) —-—-—-:-‘ h' .a ,1 "'-‘ _ '1 ,sxl‘: M: ’ 10w“) (x, X“) (F «x Xapr X15 XiA : 1 u) or '1 '1‘ I Km Li (“"x'”) X” m ...
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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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Micro Econ Practice - ECONOMICS 201 FALL 2011 EXAM 2 Instructions Please mark your answers clearly No class notes or other materials are allowed

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