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Unformatted text preview: University of Wisconsin
Economics 301: Intermediate Microeconomic Theory
Korinna K. Hansen Answers to Practice Problems (Part 1) 1a). False. The budget constraint would shift
inward in a parallel way. The slope is still the
same —p1/p2 but the opportunity set is a lot smaller. ' '64qu ), 7L
b). True. You know one intercept and the slope. 7' That’s enough to draw the budget line. 0). True. The relative prices are still the same (so the slope is still the same), but the denominator I y
in the intercepts increases by more (than the numerator so the intercepts are now smaller and the budget line 521/ .
shifts in. 3 PM I .
3 /P
d). True. This utility ﬁmction is a monotonic transformation of U = min (x, y), thtfulility ﬁmcti‘o'n for a lﬁ d' tshoe.
e an ngh 44“ 6). False. When preferences are quasilinear each indiﬁ‘erence
curve is a vertically shifted version of a single indiﬁ‘erence curve. Therefore the slope of the indiﬁ‘erence curves is constant
along a vertical line if utility ﬁmction is linear in good 2 (quantities
of which are measured along the vertical axis.) U 00112.): VCMH 12“ ’14 f). True. This is a Cobb Douglas utility ﬁmction of both exponents equal to 1. Therefore this is a
strictly convex utility ﬁmction with a diminishing MRS. g). False. Steve’s utility ﬁmction is not a monotonic transformation of Alice’s utility ﬁmction. It
cannot be written as a ﬁmction of Alice’s utility alone (plus a constant). It’s Alice’s utility ﬁmction plus
2x. Therefore these are two diﬂ‘erent ﬁmctions. h). True. This utility ﬁmction is a monotonic transformation of our basic utility ﬁmction for perfect
substitutes: U(X1, X2) = X1 + X2. i). False. The two utility ﬁmctions should rank all preferences in exactly the same order, assigning
higher numbers to more preferred bundles. j). False. This statement is not necessarily true. Suppose that Anne has convex preferences (a
diminishing marginal rate of substitution) and has A LOT of movie tickets. Even though she would
give up movie tickets to get another basketball ticket, she does not necessarily like basketball better. k). True. As income increases the consumption of some goods will have to increase if the consumer is
to spend all his income towards these goods. So at least one good will have to be a normal good. it; qu,>x,):\/cx4)+ia 1). True. With quasilinear preferences each indiﬂ‘erence
curve is a vertically shifted version of a single indiﬂ‘erence
curve. Increasing income does not change the consumption
of xi at all. All extra income goes to the consumption of
good 2. The income offer curve goes through bundles like:
(xr‘, xz‘) and (X1: x2‘+k) for any constant k. 1* i
I ) x2+h) m). True. The law of demand is not valid for a Giffen good. As the price of a Giifen good increases less will'be consumed ﬁom that good by Ivan. Since Ivan spends his entire income on the two goods, he will be buying more of the other good. V0 Ugo9 .
b 2). In 10 hours Felicity can read up to 300 pages of political
science or 50 pages of economics. She cannot go beyond the
“budget” time constraint with the intercepts 50 and 300 along
the economics and political science axis. Since she wants to
complete at least 30 pages of economics, she has to be to the
right of the vertical line at economics hours = 30. It takes 6
hours for Felicity to complete 30 pages of economics, and if
she does that she only has 4 hours left to devote to political 1ao
science, therefore she can read a maximum of 120 pages of
political science (4x30 =120). Ifshe chooses to read more
economics she will have to read less political science than 120.
Any combination in the triangle bounded by (30, 0), (50, 0) and
(30, 120) satisﬁes these constraints. 300 3). In this graph we have miles ﬂown, M, against all other goods,
G, in dollars. The slope of the budget line is —PM/PG. The price of go
miles ﬂown changes as she ﬂies more and there so the budget line is kinked at 25,000 and 50,000. Ifwe assume that PM = $1 per mile for fewer or equal to 25, 000 miles, PM = $.75 if mileage is between a;
25, 000 and 50,000 miles, and PM = $.50 for miles beyond that, then
we can graph the budget constraint. Also assume that PG = $1. The slope of the ﬁrst segment in the budget constraint is 1, the slope for
the second segment is .7 5 and the slope for the last segment is 0.5 crouch” 4). The indifference curves would look something like inverted Us.
The area under these curves does not have to be convex. The
better of the two curves drawn is the higher one. R 5). a) b). R is the number of rock concerts and H the number of hockey
games for Mark and Nick. At any combination of R and H, Nick is
willing to give up less of R to get some of H than Mark is. Therefore
Nick has a lower MRS of R for H than Mark has. Nick’s indifference
curves are less steep than Mark’s at any point in the graph. 6) We omu Look 05‘ 4k!— Slope of m 'IMcﬂch/mu. worm which {5 MMRS, For 337$: ”($12) :Slnu +57sz Fer Glenn 1 14(11)17)=Z,+7L,7Lf £13291 } dCLﬁLL): “2+1:
guCXhlz) 5 P.— , QUC 2,) U I
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9).a). Max would consume 10 units of X1 and 10 units of X2. That’s what we ﬁnd when we set the
MRS= p1/p2. 2x4 :20 AwaQ \A‘li‘eﬂé 3
’é’ucnﬂtz) 22m U=Q!olO—l—i #“Qh:i jleﬂwl k => Hahn 'auoech '" 622,, 912. Also Buclgrzlr con/1mm b). We can solve for the new equilibrium quantity for X1. Now WL won/d". Also aha“? l .21. c» was 1:; “F 112.710.5111 MRS'JA, ‘3’ Also «flan5213 (Na—Hm “"3 9:19}
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at: . Also, check pages 8788inVan'an. Withaquantitytax 7%: Opﬁwlaﬂ cAm‘ce we have to set MRS = (p1 + t)/p2. Instead, without the
quantity tax we can set MRS = p1/p2. That means that the
optimal choice without the quantity tax will have to lie on
a higher indifference curve. Wlmauauﬂy {a1 Op'l‘p'w‘aﬂ ClA U‘lLL
LUNA. Mme. la 1. 10). The price of X1 must be at least $10. Her marginal rate of substitution at the bundle (6,0) is 10. If
the price of X1, is $10 or greater, she will choose to consume no X1. UChncz) =><4>cz+47q +12%? I1” 1.926 1% 11:0 QUCXD'ILJ
9‘10‘1111) Nil—fl 371 11). a). Her indifference curve is a broken line consisting of the outer envelope of the two lines 3X1 + 2X2=116 and 2x1+5x2= 116. Thepoint (12, '40) is onthe line 3x1+2x2 =116. Seopa ?
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2.. P; a PLM+psz=1M :7 1s 12+ —L+O * i8+4o 25:8
12). Ifwe plot X1 on the horizontal axis and X2 on the vertical axis,
the slope of David’s indifference curves is —MUX1/MUX2 = 2.
Therefore if the price of X1 is less than twice as much as that of B,
David buys only X1, the optimal bundle is at the X1 axis , at m/ P1,
where m is the income and P1 is the price of X1. If the price'of X1 is
is more than twice that of X2, David buys only X2. Ifthe price of X1
is exactly twice as much as that of X2 , he is indifferent between buying
any bundle along the budget line. 13). a) and b). U(X1, X2) = X1 + 8/3X2. Therefore X1= m/ p1,1fp1<3/8 p2; and X1= 0 1f p1>3/8 p2 V14 3)
aum,w * /P; ‘4 437/th PL .
x1 = P
Ma§—§L=.£:3 M 5E) IF 3/5—1/9 5/
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212. 3 0 IF 3/; é %o >101 c). Changes in p2 don’t affect demand until they reverse the inequality.
14). X1, X2, and X are the quantities of peanut butter, jelly and sandwiches respectively. Ifm=30, 001' p1=0.05 and p2 = 0.10 the budget constraint becomes: 0.05 X1 + 0.10 X2 = 30. The utility ﬁinction is: U(X1, X2)= min (X1 , 2X2) IfX* is the number of sandwiches David buys: 4) IF mfg]; ”3:00: ﬁzz0.1054 x4 XZ)2Z,WLCEMQ/ui'1+u‘90{ PB, Wﬁdauclmoh
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16). This statement is false. The substitution effect 1s always negative The substitution effect of an
increase in the price always leads to a decrease 1n the quantity purchased and vice versa Here the price
of good 1 increases and both the substitution and income eﬁ‘ects lead to a decrease in the quantity
demanded of good 1. The income effect is reinforcing the substitution effect. This is a normal good. \ PM =7 11"» P1 \ 311111513 bwmeosih’m ...
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 Spring '08
 Hansen
 Microeconomics

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