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Unformatted text preview: 2 In Problem 155. the demand function for (bangles is given by DUI) = 02+ 1)’2 If the price 4. in the some football conference as the university in Problem 15.9 is another univemity where oidmngles is so, then the price elasticity ni‘demand is the demand for football tickets at each game is 180,000 — 12.00011. If the capacity of the
a D p .3 P stadium at that university is 80,000 seats, what is the revenuemiiximizing price for this uni
o.) 77.20. E: .37).... : v2(P+l . — _2 versitytochargeper ticket? [W W P l? JUN;
(b) —3.60. D (loH) TR 5 p. F: POEM,”  my“ ,0 z i l ‘ +0
~5Al].
(c) p T :2’19‘ .. im— _ wing (at) $1.50 ol‘l’K _, 0' .. 2%!th : 0 M”)! FHA: ‘
(d) 0.90. Vii ._ +  $353 :17, _ i509.”
_i.so. ‘l i (c) 515 P“ WM” 7 5, Sell 3’1”” ﬂake[y {5
(d) 33.75 ’ W _ i a” 1M0 {3mm — PM”)? e 80de I
.ipblem.,iisl niifdlhEb' b 1' m . 07. =— r ..
8 pi. .‘i’ .qun‘f‘oﬁsimei’. 3.33033? 3035. visit s“..§°.ﬁi';.s'éys§§i sittt'éfe’s'gt'iﬁs (9) $22.50 g. HM“ l'z‘.W ( i l ‘ X100 _ lzP
22::iiinri;.ii.:i:.“;:h.‘i"gzrimitt:“i“mm1 MW“ so .. A. sum semis em Wk F
ii {0 W‘ > ' u (i M‘ 4. In Problem 5.3, Ambrose's utility is ”(21.22) = 42:” + 11. If the price of nuts (good 1) is
(a) 320. l ‘E ' 1 P1 $1, the price of berries (good 2) is $3, and his inwme is 60, how marry units of NUTS will
a» 2m. (owl) 32in): (“X3247 m) ”New“: BIA/M J ,{h
m. — —
@ Li (ga : 9(57'4’.) (s) s ~—+” : Fax«J ﬂ 1%”: __ “Z
(d) 30.22. 4% 0'“ 0%  M _. _——_.
(8) $4.40. 200 : lfo —5‘ r“, @ 3‘ Z i if},
5r : 5—0 (c) 72 Sipp< y—C Lida/‘76; in” IKS :11 “i
' (d) as F : ,3,
main (9)18:}_*—l:> mw ‘
5. in Problem 15.9, the demand for tickets is given by D01) = 200.000 710.000,», where p is the 07,),l _ 3 i Q, = 3K price of tickets. If the price of tickets is $17. then the price elasticity of demand for tlchets is 5. Ambmm8 “may mm”, is 445;!“ + a. If the price of nuts (good 1) is $1. the price of berries (good 2) is $3, and his income is 051. how many units of BERRlES will Ambrose choose? —1l.33. l? m m.“ r _
i” a 3 13 s strum]. Fro P W’i’m/ 443; : M I (a) 31 ‘
(b) 45.50. at? 0 301”“ @D s But/99} E? wimp, ﬁ/k. + P1 W1 : M
{c’ "17‘ ~17 52; 1° "36 +31%: 1;»
d —2.03. ... —
(J  3 (e, a 3 $1 2 l;
—5.67. 4/1, :
' ._ SA 6 7 a. Amhrm’e brother Christophe haa an income of 8265 and I utility function Mann) 
_‘ l 104/144," Thepﬂoeofgoodl(nuts)in$5andﬂwpﬂmofgood2(hurrisa)iu91. How
2. Ma. Quasimodo in Problem 14.3 has the utility function U (1mm) = 10052 — 2"}2+m, where 1 many unis. oi'nuu will Christopher demand? l/
is her consumption of earplugs and m is money left over to spend on other stuﬂ. If she has 59 A”: 3 u ///b. 1%) 7”” w. ‘— ; 5_
$10,000 to spend on earplugs and other stuff and if the price of earplugs rises from $50 to :3 45 a]; = air/11»; E i ' —: W $75, then her not consumer’s surplus all“ hat/U34; “Clio43?] ISh‘Pcr'l‘Ei: “grgfa (c 47 Slur/r :3 Bin/54+ Lin“ : Vizj: ___3’
inlleby937.50. “14,5 9W9", : i {ism pm $3: ‘3’— : 5 a I (1:) tall by2,937.50. #
i”) fa":"3525' “Pd’: 'DM—kj :3. Pas: lMﬂ?‘ m w 3r: 5 m. ”ﬁ 4"”
(‘9 ‘mmm‘iBii C9 41+ Rem; Miami): izw ‘ 7 = my. (a) increases by 1,875. C 5 aii 3:3 751.: (29(3) : '3 iZJ’ a 3.. Nut F.;.i , 1 .12 ~q'__
4C3 i250‘3i2fszq'37if ”BIA/4:}. w(ig‘iﬁv j—ji KL" I’sl‘ "M q I
= 0’47. 'l— ~24 4m=24 1. In P113an 5.], Charlie has a utility function Memes) = 2413, the price of apples is $1, and 3 “/3354 l You uc work HARDER! 07». V— 2
the price of bananas is $2. If Charlie’s income were $320, how many units of bananas would W = 6 he consume if he chase the bundle that maximized his utility subject to his budget constraint? 4/: = 35
1.. Ambrose‘s brother Anselm has In innorne of $159 and 3 utility I'mlotion (“ch”) = 48::"+n. The price of nut: (in) is 84 end the price of berries (:1) in ll. Haw many unit: of hurries will , .J = I I
.80 ErFlVieVi'l' K e‘ M Q’A’ﬁb Ammdm?ﬁ¢.+rz%1m l _ 15
(b) so Jim: 04% 2) ifggyjziéy: Am ‘DﬂgJ/éﬂ (5?“ $3541.41. m (c) 100 (c) a“ $2 = ‘5’
”L m _ F J. ’5 8. In Problem 16.6, the demand function for Schreckliche is 200— AR; —2PL and the demand func
(d) 16 i ‘H ' 5 Q8 :9 2 [325!) :' {gt} 3; 2 , (4L; :) éi/ ,_. J: 331 tion for LaMerdes is zoo—sprig. where P5 and P1, are respectively the price of Schrecklichs
(e) 240 {3 3 ' 2 and LaMerdes. If the world supply of Schrocklicha is 100 and the world supply of Lumenlea
is 140, than the equilibrium price of Schreckliohs is 2. Charlie’s utility function is momma) = “my. If Charlie’s income were $40, the price of $18. loo c200.— qu 2FL C) Li P3 +1.?! : ’00 apples wane 4, and the price of bananas were $6, how many apples would there be in the best
bundle that Charlie could aﬁord? , I (b) 325 [km _ :2 1 :5 f’ i7 L] (603? ) +2f£ : [00
Eirrnessi Eur/e Li: ”A ”if (a) $32. ’ 5 2 L d
(a) 10 I ( Li“ “it; P : so 3 aiOIZP +2? 2”
(b) 12 “am; 04:71 =33 40:20: 45;, =7474— q 25’ (d) m S " f: L z.
(c) 8 Hr J 2” 4' 1 P hi 312 3115:)t1b336.m n. h d t f bndl ‘Lw : [08
. n m m . it’s mo 'me e e I' r II 6
(d) 9 [I] M : P543 2 2. [$15) 32/ : 5 $13 1‘5) 6/3 J‘ 2" : 3 from herefovurile lfucndle fir Taming! by the sum 1‘: enigma Valﬁzs :fethe Billiesneer.
Her tavuﬁte bundle for ’lbmmy is (2,1), ills: is, 2 cookies and 7 glass: of milk. Tommy‘s i ‘1 : P
@ 5 mother’s indifference curve that passes through the point (a, m) = (d. 5) also passes through L
(a) thewint(5,3). (4,5) iiAS 2 ”My" P : éo“ 3 (“1)
eoins.,,.and,. .
a. In Problem1.5.2,. Clara’s utility function is irony) = (x +2)(Y+ 1). If Clara’s marginal rate Q film: (:21?) (a 7) ‘4 9) m l<. «5 CM ol 2 1e; 5 z 15
gifjgﬂzﬁizgr —2 and she is consuming 11 units of good X. how many units of good Y is 3'; 3w p°i:l;(4'::‘v:2'5)'m (2,9). 3 [“965 I _p Pu I ”C ‘i—A a H
8 one 0 l e a .
(J 2 MES: .42: ~MX— Hitill. —(‘{9Li) 5 WC Mr5.TWi‘i’ prefers
8. ~— : .
(b) 26 aid/oi ll (Xi—2) 13 um nimble 0!.IJHMM; [5 l2 [ 4. {Zf ; q Mr; Twhi’is ”Jiffy, "
(c) 13 “9 MW" (5 vhf/”J y
25 2(i3): *(l’H) All c/mw; whfka /£2/+ {WWI = 4/ 7 re) 5 26: ‘(Jrl 7 g a \r 1. In Problem 14.1. Sir Plus has a demand function for mead that is given by the equation
“u n d h. D(p) = 100 — 12. If the price of mead is $95, how much is Sir Plus’s net consumer‘s surplus? W 12.50 C5 : £_ Jugs,» heiﬁfh'i' , (b) 5 _. .L 5‘ a;
(a a) .. J m
Foils ‘ (d) 5.25 = i2.r Perfect Suhntinued; Utahzl) = an + .322 With perfect lubeiitutee, there ale three pnesibllitla: The onneunia will spam. her entire income on :1. the
consumer will spend her entire income on :1, or the camumei will he mdllheient between any point on hu
huiiget line. Sinee an] pimiden the same level of utility ad 5124122 WE will spend her eniiie tom
on 2. it up. < i924. Mathematically, the additional phase at information for this one u 22 = 0. Beginning
with the budget equation. pm +Piln = m nuhetltute the additional Information that 11 : El. and. rearrange to Iolve for 2‘. F111 +P22i 5 m
Pm +1120?) = "I
2i : mini oedemmmnmmeambempmdmemeammm m ET] 0171 < ﬁm
:liPhFivm) = u 5 2] g 2 up, = 5P2
in
0 am > in
Demand for 2. ie determined ia the me way.
in
7 > 3
P2 C'11! PI
zﬂlpliplvm): ugzlgﬂ apl=ﬁpi
P1
'1 am < in»: Quasiilnenr: Uilivla) = 1(2)) + z:
Hem, the quantity at In ehneen is determined by the tangency oi the indih'emme curve a: the budget line. Win!” w 9:3 I Il‘CﬂVM’ is (25)(1)i~(20)(3) : 255'. In Problem 4.2, Anibmee nae the utility luneilmi ”[21,53] = 42?} :3. u Ambrose we
initially consuming 25 units of nuta (mod I) and 21 unite of herriw {good 2). then what. in
the largest nunher of berries that he would he willing to give up in retren for an additional 11 uri ftlu '2‘
m w (2520 We (36/ 31) @532:‘[ [‘I‘i’Jii‘y lfV‘vl Mu$+ 1" 55”” 4} f; :, WWI). Lure—rm : “ll
@4 ”[3‘sz):qm+/lpl ci‘lt/Z‘L
('3 2 437’: l7 (I) Awhrﬁt Weill le‘C Mr 11“l7=Ll tumult; If Heroine (whose utility function is min{z,y], where it is her consumption of earrings and u is money left for other atoll) had an income of $24 and was paying a price of $5 for earring A partieuiaily Itraightlonvlrd example ll when ﬂay]  1n(r.). 4
am.) = i_.
021 Pi was
1 _ Fr
:7...
lilpiipiiml=ﬁ (b)
C
Substituting this into the hiniget equation and rearranging in naive int iii, 5d:
P121 Wm ="I (e)
nun/n) Hm = m
m: = "I —m
”L 5
. . = — u 1 
with P: V“) m when the price of earring went up to $8, then the equivalent variation of the price change (05'; MC ‘(lVl(I Olin/Q; 47L Fihﬂl amt: ”and [1.4, 58' 3'24: 343+?) angina! JanaP: $12. = $24, :g:+¢ §%l+E;—:ié
$4. _ zl “a m.¢i,ov% EU: 2i~t=$ In Problem 14.7, Lolita’s utility function is (may) = n — 29/2 + y where in is her consump
tion of cow feed and y is her consumption of hay. If the price of now feed is $0.20, the price of hay is $1, and her income is $2 and if Lolita chooses the combination of hay and CnhhDmiglu: “21.12) = :fzg
We derived the emaaem rule by maximizing utility subject m the budget maeimim. Demand hmctinns m
be derived by simple maximum. .i the men mie. n (b)
mm=l71=l (C)
needing: (d)
WWW“ (e)
lzlﬁiimvml: $5 Perfect [in'mpllsnle: No.12): mln{ﬂzi.ﬂl'2l molwo good we. Foranuiiieu'nal example, ﬁndthe utility Inuiniizingchnicewliere
U(x,zz)x:”rf”, 1213, pz 2,and iii3i]. It‘ll? ofresourcesaiea'pemon x]
ammofreemucamapenton x=,$lﬂwillbespeiitnngnnﬂ lanﬂSZUwillheapent manual
x‘]D.i3 x, 3.33
ii1 2012
1:1 10 Linear use; nun.)  m, 1 ﬁx: Inthiicaiie, ml isaperfmrubacimnefor myanilthereisoompleieileedonim
whalinitelietwaenﬂielwo. Hencetheimiieiaiiamntive, ml or Iﬁzz,willhechosen.
Consideranunierica] examplewhereﬁhirleymwammconmewmmf
beer. Hung167i, +3.32. LeI {ll3,192 2,aui1 iii 15. lfSliiiley apenilalier
miirebudgeionlﬁmcamshecanaﬁnﬂlnanl. Alternaiivelyﬁliccanaﬁ'crdliﬁ, manna. Since 5 Means inieraaci a higher indifference median 7.5 3 ounce
mashechocaenheSlnouncecana Thiaiipictuiedbelow: cow feed that she likes best from among those combinations she can aﬂ'oid, her utility will be 2.32. aw: :% e. ~ 063] K “Pd, _. “£13 __ ~,2 1.80. "Elk 3” 45/5 ‘ J _ ﬂy ‘ l . . — z __ M , '2 ﬂ ¢‘ '2‘”%”” uzaiiﬂiihﬂ 132: 3: h'l— @2449 :Z'<0.Z)(ﬂi5) 2
: "ng 1‘ 2‘32 In Problem 4.l, Charlie’s utility function is MAB) = AB where A and B are the numbered
apples and bananas. respectively, that he moinee If Charlie is consuming IO apples and '3",le : i . . . . . 3U banana, than if we put apples on the howinontll axis and henna» an the vertical axis,
ELI/9:, p; The the demand fnncimn can he solved hy locating the inn]: Eroni the miiiiferenue curve on the budget the 3101,, of hi! indiﬁeﬂm curve at his curiunt oomption is
iﬁﬂzlﬂﬂzl = 1 equation. The kink is described by on; : ﬁn.
1 p1 Pili+Pizr:m (a) '11' 35 d 6 _. —3u/nift haAE/JA
Renaming and simplifying. : (ﬂ 4. El  “E— 2:
no.) 7 ﬂ ""+”1‘°/ﬂ"‘ '” A a u 0 [g afiB/J B
311 7 P2 hIi + min/ﬁllzi = m (c) in.
The axaet form ante demand function depend. upon {(21). z.(p.,p.,m) = L (d) ﬁll _ 3
Pl ﬁne/ﬂ) (9) —1,'n : _...’'—
As an alternative. consider setting the slope of the indiiierauce curve equal to the alope of the budget 1 pm H1212 = in ID A f A
Manipulate the budget equation to solve for the slope of the budget line drawn with $1 on the hon'zo PAW/Cl)“ + p112 : m 5 if? is 2%! _ .3
. . . l 
axis and $2 on the vertical axis. MW“) Hell: = "1
P1111 +P2$2 :7" = L
I 71" _ 1 Zilpimii'nl MW“) ﬂ)!
[’2 2 111 1 Consider maximizing n CobbDouglas utility function
in 1 . . . . .
$2 =E * fTEi um E2) = fr” 3. Bernice in Problem 14.5 has the utility function 110:, y) = min{:i:,y}, whens e is the number of
2 ' ‘ 2 pairs of earrings she buys per week and y is the number of dollars per week she has left to
The slope of the indjﬂernice curve is subject in 5 budget wmumnt spend on other things. (We allow the possibility that she buys fractional numbers of pairs of
ilﬂ :—BU/B$1 m = p111 +1;an gaining; liar vfieek) If all; origimlly had 8%“ income of $6 pesgw'eﬁk and was paying option 3;
it all a“ , . . per pair 0 earnngs, t an I 9. price 0 earrings rose to . e compensating variation
1 / U’mg the Lagrangian mwmd' ,, 5 that price chang (measured in dollars per week) would be closest to .l kink
ﬁxing/311 t. = 1le 7 Mm ’Flml 711222) $3 it“ :3
=— o . _.
aﬁlg/awg 37: = Hafiz: 7 Apr () g/E ’3 w éb ilﬁ +l' ’5
, new: i when... m u G = M
WWI n ” MM“ keTE
71 i=m7p$ 7mm if $12
=_ﬂ BA 1 J ’ U [4i new one; .4 £454;
in." Solving for A. (e) $11. Q At > ll .
_ e, we: =. nd (3 GM? = 17. +4 lady
‘ ‘ — in ~
ﬁm‘ neiz‘“ A View a H C MI (42
Settin thetwoaln a uni —“= I"M
g P on on; p, I” CU: l2~62£
—ﬁ—$l = _ 11—2 Eliminating A, 6 I5
51137112 _ BI)" 271 5M" 0" d4? Airgun n.1uio
w: =ﬁpiwi p1 — p2 3mm...” 4 n9 u/Myﬂ _ Ll w 3% _ .1 w
Meal: anxgri CurV0 My}; —‘ W : /’r 5 _ 3
P212 :Eplml p Lalo = ”Lewis B ¢Aq ¢ﬁ
1 I 2 1 2 . . , . . . .
The ﬁnal steps m substituting this imam Domitian in“) the budget em“ a _ .5 " 1.233.121“.3727;:$313iii‘i'iftt’imaliiiﬁfizii‘. Em“; ii”t3‘.’t§iil‘.‘.i§f.‘i.t§
_ M _
m =;Ur1=i +1122? p.21 132.111 w never Slip; ,J; 614/701 LIVle : “PA I 3 _ ‘3
one = 5pm (D Me = axe. "l/k T}? W ‘
= 15”" + EP‘E‘ iting into the budget eminent, in) X5 = xi. —3 < J
m 7 a + 3p 1 (c) XA =4th 4734
— 1 1 mzpixilpzzi (d) X =4X
0: 5 an; nix =M 3 WA: LiA’q;
n+3?” =P1Ei :pimi+;piZi (9) I” E 
gunman: maﬁuﬁlxz} g(x,)+xl
And by substitution, in = —p 111
3 m7 1 Lump,“ Inmisiasauiiliryielimarmx: butnonliiieariiix]. w‘hileitispnssiiilefolaonmnnier
(2+1? —pz 2 ”WE ionhooneahuni‘llecnnmiuingallxlnndno 1:,folcertainﬁinclinnnlfnrmaofg(‘)such
And l’l’ substitution m the natural log below, it may lie Masaiy in consume at last me x] at any amt.
ii. In Problem 5.6, Elmer’s utility function is U(ai,y) f min(z,y"}, If the price of e is $25, the I _ Therefore a consumer duel not have complete ﬁeaimn to sum m“ x] and :r. .
prion ofy is 20, and Elmer dwoses to noneume 3 units at Y, what must Elmer r income he? “+57" 4‘21? In the '5 w: have m ihus far in A _ 80 be: m _ a m or a
(a) 570 a The ﬁnal two equations are known as the mic/neat, or budget share rule. For GobiDough! preferences, Millil lognnilinL For m ofllime was, the marginal mntrihuijon of x] exceeds that
(b) 135 Elie urgeniuu ofwtal income spent on z, is u/(u+ﬁ). The proportion uftutal income spt an m, is «x: forum ﬁmfewdnllara 5mm Alan Mlple, conﬁler “(Xile‘MTi)‘li
(a 385 3 /(o )iijohbnnuglan use: u(xx,)x'x’ Whﬂe p] 3. p1 2,anil ill24. Toaolvemathenintimilyﬁenheslapeoflhe
(a, 7‘. “‘ " ’ I inﬂiﬁelencemequaltotheslopeofﬂiehudgetline.
"5 . . ' FuthchbhDuuglaauaamthereisﬁeedommmnmx]ﬂirx1,bu1inamolnlimilod I3Ihl3x,—_p
(e) The” " “M "“5“ ”mum" “’ “I" W 55153 llian nie previnus example. NINE that uniiry Will be zero iteiiiier x] or x1 is equal But312 Pa
El 1153 ioziero. mesolvmgihispmhleminonmpletegeneraljwusingnmmutual 1i"x‘__3
may (OTISLUVI(S ﬂ‘j' Kink ‘4: meconsunierspenm a %Dflmﬂmeﬂlfx]anﬂ '8 %forxi.11ii's,m ] 2
F" V' a e ,s a e ,3 ‘ 2
' mnjunoiionwitiiiliemilityﬁinclimi: enough informiionto Inlve mathematicallyior ‘1'} Ifsagzisspenion r..ilieremaininghudget,524—222,:anliespemon r}. enm
Xzll Klemﬂﬂeeuirixalnﬁnlmﬂr.) Fumehnkodmmﬁmmmiheteiamﬁoedmiorubammbmnme
oomiiiipiinngnodi. Efﬁcient consumption munmuanlie lciiik. 'l'heloinllii described malhemaiiccllyby the condition axl  ﬁ‘i This inforlliiniiin1 plusihe equation oiihe
budget line, is ruineem ioﬁnil ineihemiiiaii Inhlnoii. Suppose
“(milmilling). 1:21 3. p1 2,anrl in 35. 21111. smmg lliis into
1112 budget emotion gives 3):. +731  35 3x. 1 Kb.)  33 7x.  35 Jr.  5 xi  2(5) x1 10 ...
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 Summer '12
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 Supply And Demand, Charlie

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