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Unformatted text preview: ECONOMICS 201
FALL 2010 EXAM 1 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 Questions 4, 8, 9, 10, and 15 (10 points). 1. Dora likes hugs (good 1) and kisses (good 2). To be able to consume any positive amount of good 1, she
must pay $10. Once she pays this ﬁxed cost, the price (per unit) of good 1 is $1. The price (per unit) of
good 2 is $2. Her income is $100. Draw Dora’s budget set. Label the intercepts and slopes. good. 3‘ good 1 Let’s say that Lady Gaga is indifferent between bundles (5,10) and (9,2). Which of the following leads
us to conclude that her preferences are convex? ﬂint; The line connecting (5,10) and (9,2) is X 2 = —2X1 + 20; her preferences satisfy monotonicity. (Please circle one response.) a. She strictly prefers (5,10) to (8,3).
b. She strictly prefers (6,9) to (6,7).
c. She strictly prefers (7,7) to (5,10).
d. She strictly prefers (6,8) to (9,2). . Which of the following utility functions exhibit diminishing marginal utility with respect to good 1? (Please circle as many responses as necessary.)
1 a. u(X,,X,)=2XE+5X,
b. u(X],X2)=3X1+2X2
c. u(X],X2)=0.51nX1+0.3lnX2 Timon’s utility function is u(X1,X2). Suppose that income, m, and both prices, R and P2 , are
multiplied by 0 , where 6 > 0 . (10 points) a. What is the Lagrangian of the UMP (Utility Maximization Problem)? What is the ﬁrstorder
condition (F 0C) with respect to X1? b. How does the value of the multiplier, 21 , change when we multiply income and prices by 0 ? True or False: Take a utility function that represents a consumer’s preferences. If we add 10 units of
utility and divide the result by 2, we obtain another utility function that represents the same preferences.
Explain your answer with a mathematical expression, graph, and/or one or two sentences. . True or false: Suppose there are two goods. If neither is inferior, the goods must be complements.
Explain your answer with a mathematical expression, graph, and/or one or two sentences. . True or false: Suppose there are two goods. If they are complements and good 1 is not Giffen, good 1
must be normal. Explain your answer with a mathematical expression, graph, and/or one or two
sentences. ﬂint: Remember that only relative prices matter in the sense that if we increased income and
both prices by the same percentage, there would be no change in the consumption of good 1. . Rafael Nadal likes cereal (good 1) and oatmeal (good 2). Prices are P1 and P2 , and income is m. His
1 utility function is u(Xl ,X2) = Eln X1 + £1nXZ. (10 points) a. What is his UMP (utility maximization problem)? b. What is the Lagrangian? What are the ﬁrstorder conditions with respect to X1, X 2 , andl ? c. What is the demand function X10731 ,P2 , m) ? d. Take a derivative of the demand function to demonstrate that good 1 is ordinary. 9. Every day at school Debby eats tater tots (good 1), pizza (good 2), and soda (good 3). Her preferences
are u(X],X2,X3)=min{5X1,2X2,X3}. Prices are P] =1, P2 = 3, and P3 =2. Suppose that the required level of utility is E = 10. (10 points) a. What is her EMP (expenditure minimization problem)? b. In this speciﬁc case, what are (X I" , X ;,X 3' ) , the solutions to the EMP? Hint: Your answer
should be numbers. 0. In this speciﬁc case, how much is she spending at this optimal bundle? Hint: Your answer should
be a number. 10. A consumer likes good 1 and good 2. Prices are R and P2 , and income is m. The utility function is
u(X1,X2) = X12 + X22. Suppose that we also know that PI > P2. (10 points) a. What is the UMP (utility maximization problem)? b. What is the MRS? c. What is the optimal bundle, (X 1' ,X ) , the solutions to the UMP? ll. The following graph depicts an increase inP1 . Is good 1 ordinary or Giffen? Jon) :5 .3004 I 12. The following graph depicts a decrease inP1 . Point “A” is the initial bundle. Indifference curves are
linear; their slope is the same as that of the initial budget line. Draw and label bundles “B” and “C.” 300;! 3,
im‘h‘d .
“V‘s” I.» but
lint (55“. ‘NA5 306’?) l 13. The expected utility ﬁinction is U (61 , CZ) = (1 — 7r)  01° '75 + 77  €375. Calculate the coefﬁcient of absolute risk aversion to determine whether preferences are risk averse, riskloving, or riskneutral. 14. Multiple choice. Consider a lottery or gamble that entails winning $10 with probability 0.5 and winning
$2 with probability 0.5. Which of the following is evidence that a consumer is risk loving? (Please circle
one response.) a. %u(10)+ $142) > 14(3)
1 1 b. a u(10) + E u(2) > u(8) c. %u(10) + 541(2) < u(8)
1 1 d. 5 u(lO) + 5 u(2) < u(3) 15. In the insurance example, suppose the consumer’s expected utility function is
U (c1 , c2) = (1 — 7r)  c1 + 7r  c2 , and the price of insurance is greater than the probability of state 2, i.e. 7 > 71' . As before, c1 = w — 7a and c2 = w — ya — D + a , where w is income; D is the loss; and a is
the number of units of insurance. (10 points) a. What is the UMP (utility maximization problem)? b. What is the ﬁrst—order condition (FOC) with respect to a , the number of units of insurance? c. What is the optimal a ? ECONOMICS 201
FALL 2010 EXAM 1 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 Questions 4, 8, 9, 10, and 15 (10 points). 1. Dora likes hugs (good 1) and kisses (good 2). To be able to consume any positive amount of good 1, she
must pay $10. Once she pays this ﬁxed cost, the price (per unit) of good 1 is $1. The price (per unit) of
good 2 is $2. Her income is $100. Draw Dora’s budget set. Label the intercepts and slopes. go 3001 1 Let’s say that Lady Gaga is indifferent between bundles (5,10) and (9,2). Which of the following leads
us to conclude that her preferences are convex? m: The line connecting (5,10) and (9,2) is
X 2 2 —2X] + 20; her preferences satisfy monotonicity. (Please circle one response.) a. She strictly prefers (5,10) to (8,3).
b. She strictly prefers (6,9) to (6,7).
c. She strictly prefers (7,7) to (5,10). She strictly prefers (6,8) to (9,2). . Which of the following utility functions exhibit diminishing marginal utility with respect to good 1? (Please circle as many responses as necessary.)
3
l L "’
@ u(X13X2)=2X12+5X2 2"" b. “(X],X2)=3X1 o '7.
@ u(X1,X2)=0.51nX, +031an  1.2x. <0 Timon’s utility function is u(X1,X2) . Suppose that income, m, and both prices, P1 and P2 , are
multiplied by 6 , where 6 > 0 . (10 points) a. What is the Lagrangian of the UMP (Utility Maximization Problem)? What is the ﬁrstorder
condition (F 0C) with respect to X1?
1 = ‘4 (X.,X1) X (9'31. 4' 991x13. em) on, _.
Cx‘j 13:12 —)t6?. _0 b. How does the value of the multiplier, A , change when we multiply income and prices by 6 ?
J H < X), X1.) , new all
A v 3x: . >\ : A“
99. 9 5. True or False: Take a utility function that represents a consumer’s preferences. If we add 10 units of
utility and divide the result by 2, we obtain another utility function that represents the same preferences.
Explain your answer with a mathematical expression, graph, and/or one or two sentences. True. v; uHO m v=.i.\4+5’ a which Is an Inc“845$: 'Fvnth'on. 6. True or false: Suppose there are two goods. If neither is inferior, the goods must be complements.
Explain your answe with a mathematical expression, graph, and/or one or two sentences. False 4+ c, ‘He god: «a Canqhmewrt
M c', +1.4 July we {nmil'uhp 7. True or false: Suppose there are two goods. If they are complements and good 1 is not Giffen, good 1
must be normal. Explain your answer with a mathematical expression, graph, and/or one or two
sentences. m: Remember that only relative prices matter in the sense that if we increased income and
both prices by the same percentage, there would be no change in the consumption of good 1. True. Sumac ﬂ, 0.. «a M ﬁre ‘1 the Same reamPeat
AX. 4, due 4'. hi rise 1'» f. («0+ ét‘ében)
AX. + be .9; +k¢ N‘SC i» 91 ((Om'leMﬁn'h) , .
Thus, AK. 1» due +. 4M m‘se .L. m (Noun), 5;“, Me +0?“ char M "v ’5
8. Rafael Nadal likes cereal (good 1) and oatmeal (good 2). Prices are P1 and P2 , and income is m. His '1'!” utility function is 14(X1,X2)=%lnX1 + gln X2. (10 points) a. What is his UMP (utility maximization problem)? MAX iin.¥‘$’INX\.
x""‘ 3.4. ﬁx. +9061." M
b. What is the Lagrangian? What are the ﬁrstorder conditions with respect to X], X 2 , and A ?
I = i’hx, +é’lnxt‘ A ( P.x.+r.X1.’m)
Exa 15“.— M’, .:.O
E "13 ﬁxt'API '1 o
C X] 9.x, + v‘xtc m c. What is the demand function X I (P1,P2,m)?
Solve Kup. = 2X,P.. an ramm, zln _ x‘ (Fur11A) : 3P:
I d. Take a derivative of the demand function to demonstrate that good 1 is ordinary. 3X\(9.,Pu’”) = anl’r; <0 ‘ Good I f: ordinal), ﬂ 9?. 3f) 9. Every day at school Debby eats tater tots (good 1), pizza (good 2), and soda (good 3). Her preferences
are u(X1,X2,X3) = min{5Xl,2X2,X3}. Prices are P1 =1, P2 = 3, and P3 = 2 . Suppose that the required level of utility is 5 =10. (10 points) a. What is her EMP (expenditure minimization problem)?
min ﬁx. + ﬁx... + Pix)
‘9x‘lx3 Saf‘ min fgx,’ 1x" X33 : 17
In this speciﬁc case, what are (X 1" ,X2‘ ,X3') , the solutions to the EMP? Hint: Your answer lldb b. 3?
Siou enumers _’ xrzal x::S x3 zlo I In this speciﬁc case, how much is she spending at this optimal bundle? Hint: Your answer should
be a number. 4.1,“;3 +10% :t37. 10. A consumer likes good 1 and good 2. Prices are P] and P2 , and income is m. The utility function is
u(X1,X2) = X]2 + X22. Suppose that we also know that P1 > P2 (10 points) a. What is the UMP (utility maximization problem)? a
max X. + x:
xu"‘ Sal.
0.x“? 9,)(1: m b. What is the MRS? C. M311"; = 331'. z X' No+e 44.8 MRS m'ses win. X,,
“"1. X}.
What is the optimal bundle, (X 1" ,X; ) , the solutions to the UMP?
J4
xl :o
«k 31‘.
X1 ’ 1>1 11. The following graph depicts an increase in P] . Is good 1 ordinary or Giffen? \
\s
A OI‘JI‘nqr] 12. The following graph depicts a decrease in P1 . Point “A” is the initial bundle. Indifference curves are linear; their slope is the same as that of the initial budget line. Draw and label bundles “B” and “C.”
3003 31 0A ‘
a c 3°
13. The expected utility function is U (cl , cl) = (l — 71001075 + E  0375. Calculate the coefﬁcient of absolute risk aversion to determine whether preferences are risk averse, riskloving, or riskneutral.
h  .15
TA“): _. u (a) L‘1‘“): 0.7s’c in (km: .715 > o
u'lx) «"00 = ""5 ‘ a"; c ‘—
RI‘Sk averse v 14. Multiple choice. Consider a lottery or gamble that entails winning $10 with probability 0.5 and winning
$2 with probability 0.5. Which of the following is evidence that a consumer is risk loving? (Please circle
one response.) 1 1
a. —u(10)+—u(2) >u(3)
2 2 JvW(10)+JI“(3\)> 14(7)) “(q
(:6) lu(10)+lu(2)>u(8) " ‘
2 2
c. %u(10)+%u(2)<u(8)
1 1 d. Eu(10) + 5 21(2) < u(3) 15. In the insurance example, suppose the consumer’s expected utility function is
U (c1,cz) = (1— ﬂ)  c1 + 71"62 , and the price of insurance is greater than the probability of state 2, i.e.
y > 71'.As before, 01 = w — 7a and c2 = w — 7a — D + a, where w is income; D is the loss; and a is
the number of units of insurance. (10 points) a. What is the UMP (utility maximization problem)? Max (1w)(w«v.<) + 17 (wax—0+4)
o( b. .What is the ﬁrst—order condition (FOC) with respect to a the number of Lmits of insurance?
[a] (12100?) + 11 (7; +1)
 X .1 TTY  Tr" ~r Tl
1T ' Y < 0 c. What is the optimal a ? it “911406) since Hie E06 3" 095QJ’J'V‘I) a( :0.
The. N'SK nCVYrql “owner does 100? 1315va
in ~14.“ L45€~ ...
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 Spring '08
 NINKOVIC
 Economics, Microeconomics, Utility, Mathematical Expression, Utility maximization problem

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