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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 shortrun 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, OMSHon 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 05 0‘5 'waoa’“) :M;*£1(9xa),9X13+ “Why”: 9 minf'lKi,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'Ht infiniie prob»? 6V6" lid “31‘ e
are F.'xe Ll (OJf5
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 ﬁOi’ 4’6 Prague 31/ U ) fr _
75° y>o 0? Pia4‘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 liphind] 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, 510,,0.103,5o xlzso,
x1" ‘0' 4"" X3310" 704’“ was 4»: $l56t zitrm #340 : $100. _______
b. How much is she spending at the optimal input bundle in the longrun?
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
erI ’ 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 ““
(om1w” U :_ (fastD : « 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 : wE" 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‘lXﬂ “.34: " wax"), aux, b. What are the conditions that ensure proﬁt is inﬁnite?
' f4 1% 7 31. (use. Mia iqur 1)] "YT:09
wt I
.33.; _I, (3; (use only he” 1), 1T7.” “I aP'Wa 7 0*
V Wu. 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)? Fri71 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 xIC‘ 4m
X33 __ X15
f—uI 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 — LAY2&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" ___ _ _—
fitLA} .5 Xm X1A ('Y.)(x xiii) (X1 X1}! . O E Lanitrainer 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(1XZA) 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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 Spring '08
 NINKOVIC
 Economics, Bert, FOCs

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