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Answers_Practice1 - University of Wisconsin Economics 301...

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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 Ugo-9 . 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 = $1. The slope of the first segment in the budget constraint is -1, the slope for the second segment is -.7 5 and the slope for the last segment is -0.5 crouch” 4). The indifference curves would look something like inverted Us. The area under these curves does not have to be convex. The better of the two curves drawn is the higher one. R 5). a) b). R is the number of rock concerts and H the number of hockey games for Mark and Nick. At any combination of R and H, Nick is willing to give up less of R to get some of H than Mark is. Therefore Nick has a lower MRS of R for H than Mark has. Nick’s indifference curves are less steep than Mark’s at any point in the graph. 6) We omu Look 05‘ 4k!— Slope of m 'IMcfl-ch/mu. worm which {5 MMRS, For 337$: ”($12) :Slnu +57sz Fer Glenn 1 14(11)17)=Z,+7L,7Lf £13291 } dCLfiLL): “2+1:- guCX-hlz) 5 P.— , QUC 2,) U I MQS=___&Z.=L__: K—I in. MRS: mig— :- L'l’lq. MRS,W ”gm-‘3!- U -i’EL—z-fl ——”‘ .. 31317999) g5, 2., 9ch 20 1+1! Qucxh) 5‘12 7‘2 . 2' 2" 31L 912. A3 7% “mum“? “demo“ A "x, a}: 9g tutu-cm: Lia-e, a9 X4 Mum/pm 0M5? —SP/J 4M H123 IS CJWB. Md iszCItGSE/J sl‘x/U) 102W Wig (0)10 0U wwski-Aé, Y‘OJ‘io CMMRS} 'IS COWS, GM MR9) LMW- ”:11 iucR-‘tltreuu CLAW Jewiurswv‘a. W m lVLCQl'fiWA are ”usagov-L. CDMVL-L , Th iuchWu. UAW/'1') (MW are W!" ‘ m “‘4’": w‘ Comm» / 7. a. Q .6 m U (Mpg?) "'74 >62 / L071: (101-10423 :p—l—lz) lg}; imam; __4L__ 7L).TouaeiM<iemamcl fwdfiowa 3d up w {Mg/OWWM oon solve, 44/4. FOCQ +0 891' 1,1 )12. w: uCuwQ=xf7L§f 80 £7- 71¢7CZQ+ a (M47191 “”703 ‘74?) $065: 9% a", _ ala—t g —”‘ __,,_—- : 0-14 f z :7): ___1__I'_2._ 9Y4 . 7"; 3’0; 0 pi _ GLKL'PJ— ”EXT ~ xgxfxg-l 31/0 :o =>9 1579:7526?) — 61’ K _ 2 , /_ 9‘79 1 Pa 37752: 'fii'xl . 97% @ 03p; «wk, 0.. __ 4AA. 7&2: a I; E Mg 1% é,» : 2 fig?» 2__ ol £w MmL._ belvvm cho 10w Clio??? G i d..— ._ '3 ’01— =o " "’ 11 PJ— : L11... :7 2::‘EL— 5% “7'; We: =0 = J— ’1 £1"?- ”0-1 ”‘1 1 P61“?- @cbwwaw 6295 @7010?va gqx, Pg .. __ 4/14, 337%”me F74 ’0 =7 14:4” ‘4’ :7 7L[*"""1 pi PL- MAOWOQ $01 (3130an- ¥C> a: y 6, ____.ff‘2’r<—I’0V)L. ~ 2.) —/~,L)0:m =6oo)lfl=/o)/og:¢20 4%7Mm“ %,_———— K’inz 600 a “7% M 3E to 2360:“? 12.: £- Qfl- :13;- 600 :— 50: (O 12: 1)}. >2; fi)‘% 8,7/3 L 6,91 9.5 = z E WHW‘ 761:1;_,i :€E-L zso—izs‘q ‘ 0 7L2: /°_;_ _ i2 : ’/ PL" 2° 1 7’4) Show: xvi MW (pill—1‘— W. SFO‘K Ma ', fi— .21: /,_,i P1(a.+6 PL 2-2..— wfimwl‘s G‘M' MA. "I‘ w @«f 30(le % W "' mfg/Pl: ifl’ol- WWI/L» Van‘s Wm 8). a). C=10,B=3andU=127. Beam“ Foe”; Ive. W“ . .. . QUCcflg)_—_ a ?tué I’Wi'ol’td. Ld‘tag 'fiuuuhm. 37‘ QO’QC =0 =>QC=ZZO=>C=W u=&o.10-lol-+ l8.3—3.3L ’67UCC 6) six-9c 0—1004—5 - = 2? #:15948” => 65:18—57 8:3 ' 0 l 99 i b).C=4,B=1andU=79. Mow we. Low M QMNWQQ omnmm/c: as: 5 => 11mm): M 5-8) {swim-15:- 552: L00 —a08-'L35+Bz-'OBJ+'3‘351 =7 U («ammo 406—25 ‘Bz-HOB +13 5 — 3 at: - L1 sags +76. TMQM.‘ QW:_QB+3=D =7 E=i :7 C=5—B=4{~ K. (1:7? 9 IS 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-!o-lO—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 «flan-5213 (Na—Hm “"3 9:19} c). No. lUg Wu (779' Loewe. 09,13. Also 01 111565“ : _ Ullik. git/($11,131.,fy {at £4 M=DZO+I -S’:a?§ W41—1S-6 02‘ ‘1‘ fit ‘ ' 11.11.... gazgs- . at: . Also, check pages 87-88inVan'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 37-1 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 ? «.3 3xl+gmz¢i6 :7 If Mzo :7 121:9? x249 It w2=o =7)q:5c?.6 Z Qm—tgxz—ziié IF M=O =73tl=25~7—7L’H€/Z It X220 :7 KLzb’g [E‘ilgki slépe. ———‘-> 38 6 58 H b). The price of sugar must be $1 5 and her income must be $58 MRgzi llA‘l/l/‘a/‘l swam :FUY exp/of £12; :7 IF Pat-1:: 10121.3? 2.. P; a PLM+psz=1M :7 1s 12+ -—L+O * i8+4o 25:8 12). Ifwe plot X1 on the horizontal axis and X2 on the vertical axis, the slope of David’s indifference curves is —MUX1/MUX2 = -2. Therefore if the price of X1 is less than twice as much as that of B, David buys only X1, the optimal bundle is at the X1 axis , at m/ P1, where m is the income and P1 is the price of X1. If the price'of X1 is is more than twice that of X2, David buys only X2. Ifthe price of X1 is exactly twice as much as that of X2 , he is indifferent between buying any bundle along the budget line. 13). a) and b). U(X1, X2) = X1 + 8/3X2. Therefore X1= m/ p1,1fp1<3/8 p2; and X1= 0 1f p1>3/8 p2 V14 3) aum,w * /P; ‘4 437/th PL . x1 = P Ma§—§L=.£:3 M 5E) IF 3/5—1/9 5/ 5.? “591199) 3-. ‘8 ’9' PL 310; 212. 3 0 IF 3/; é %o >101 c). Changes in p2 don’t affect demand until they reverse the inequality. 14). X1, X2, and X are the quantities of peanut butter, jelly and sandwiches respectively. Ifm=30, 001' p1=0.05 and p2 = 0.10 the budget constraint becomes: 0.05 X1 + 0.10 X2 = 30. The utility fiinction is: U(X1, X2)= min (X1 , 2X2) IfX* is the number of sandwiches David buys: 4) IF mfg]; ”3:00: fizz-0.1054 x4 XZ)2Z,WLCEMQ/ui'1+u‘90{ PB, Wfidauclmoh Body) cowl/0100' Q05‘x4f0101a: 3 dc 7.1001242): A/LI/tfxr :2sz Ia’ 28’ 13 mastof Scwdwdw. Dwell My: ./ .2.0.o:+z.iv0.10:3 . 1 .1. 11— é; %) M 15:5?»3230 24 11:1:g 76+: 3 1: 3:1 + 3 : -_ §.0,0f+0./ .2, = 3 —.= : ._..___—— __ , )l: ‘1“ C) x 2_-0.or+.i$ 0.2: ‘3: 2 9w” ‘9 “Lg—Li ‘9‘ 2221 3% 1104)”): : a? €71 732. Onflaao fl- 2/ [92:2] 444:20 15) amt/941%?” F’Mfi’ /’ =2, Fz=1)/M=Zo "Den/Ugh DMQMQQ . MUM , PL 9%me { __ ..... MU. ' ”F3— - ’21e4 .. : P1. _ X it; 2- 79—Hu(n1)h)"02—’ 'P‘ ~7Z—l:_fii_[’;;;=°2PP__L WM? tum W" "2 2 (Pl 7' ’3‘"; 330%»:de 2 01064 I _. ' : ~:'37_fi:”27§f" ”5:11- depmd 0W1 1M . t; «,q p2~f__>: =P¢X4~W 2P1 7 A-=._..__. PT "i” 2 PL W M 7- _ _, ,,_,_ __,_j________i,0M-_L__- Q CQOW‘ 01 45% __— ‘57» \ 2 (W L, 1:5: 1 P52m=w s1; 2,:2— 9 (QM: Lo—z-Z. _ [g ~<2 Z w E——: M £2, PirllPZ—‘i/ M4220 =7 1;? 1 75,2043 @1101): XXV/"Z 1): 27:5: 15? (0 20 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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