Sem-4 Micro.pdf

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ECONOMTCS (HONSI - AT A GLANCE
Course Structure for B.A, (Hons.) Economics - Semester Model Approved by
Committee of Courses and Studies for Honours, Post-graduate and Research
Studies in Economics
Course Structure for B.A. (Hons.l Economics
Semester -I Semester -II
Economics Core Course 1 : Introductory
Microeconomics
Economics Core Course 3 : Introductory
Macroeconomics
Economics Core Course 2 : Mathematical
Methods for Economics-I
Economics Core Course 4: Mathematical
Methods for Economics-Il
Ability Enhancement Compulsory Course
TAECC)_I
Ability Enhancement Compulsory Course
(AECC)_II
Generic Elective (GE) Course - I Generic Elective (GE) Course - II
Semester - III Semester - IV
Economics Core Course 5: Intermediate
Microeconomics -I
Microeconomics-II
Economics Core Course 6'. Intermediate
Macroeconomics -i
Macroeconomics -iI
Economics Core Course 7: Statistical
Methods for Economics
Economics Core Course 10: Introductory
Econometrics
Skill Enhancement Course (SEC) - I Skiil Enhancement Course (SEC) - il
Generic Elective (GE) Course - III Generic Elective (GE) Course * IV
Semester - V Semester - VI
Economics Core Course 1 1:Indian Economy-l Economics Core Course 13:Indian Economy-Il
Economics Core Course 72: Development
Economics - I
Economics Core Course 14 : Development
Economics-II
Discipline Specific Elective (DSE) Course - I
(From List of Group-I)
Discipline Specific Elective (DSE) Course - III
(From List of Group-II)
Discipline Specific Elective (DSE) Course - II
(From List of Group-I)
Discipline Specific Elective (DSE) Course - IV
(From List of Group-II)
Group - I (Discipline Specilic Elective
IDSEI Coursesl
Group - II (Discipline Specific Elective
IDSEI Coursesl
(i) Economics of Health and Education (viii) Political Economy - II
(ii) Applied Econometrics (ix) Comparative Economic Development
t1850-1950)
(iii) Economic History of India (1857-1947) (x) FinancialEconomics
(iv) Topics in Microeconomics - I (xi) Topics in Microeconomics - II
ff) Political Economv- I (xii) Environmental Economics
(Vi) Money and Financial Markets (xiii) International Economics
(Vii) Public Economics (xiv) Dissertation / Project
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Subject
Course
Date of Meeting
Venue
Chair
Attended by:
UNIVERSITY OF DELHI
DELHI SCHOOL Or ECONOnarcs
DEPARTMENT OF ECONOMiCJ
Minutes of Meeting
: B.A. (Hons) Economics _ 4tr Sem. (CBCS)
: Jntermediate Microeconomics _ II
: l1th January,2Ol7
: Department of Economics, Delhi School of Economics,
University of Delhi
: Dr. Anirban Kar
1. Arjita Chand.na,
2. Surajit Deb,
3. Sandhya Varshney,
4. Neelam,
5. Manavi,
6. Naveen Thomas,
7. parul Gupta,
8. Valbha Shakya,
9. priyanka Singh,
10. Himani Shekhar,
1 1. Sandeep Kumar,
12. Neetu Khullar
13. Shirin Akhter,
14. Ravinder Jha,
15. Ram Gati Singh,
16. Swaran Lata Meena,
17. Pragra Nayyar,
18. Sanjeev Grewal,
19. J.R. Meena,
20. Rajiv Jha,
?) Meenakshi Sharma,
22. Sakshi God Bansal,
1. Syllabus and Readings
Course Description
This course is a seouel to Intermediate Microeconomics I. It covers general equilibrium
and welfare, imperiect markets
""J t"pi"" under irrror*uiio, l"o.romic.. To discuss
imperfect market
""d ;;;;;",r", *.'llio .r".a to introiuce students to strategic
interactions and game tr,.o,.v. ir,.-;;;#;" ;ri;.;;';;;ff;* conceptual clarity to
SPM College
Aryabhatta College
Dyal Singh CoIege
Satyawati College (E)
IP College
Jesus & Mary College
LSR College
Daulat Ram College
Daulat Ram College
Kalindi College
Kalindi College
Dyal Singh College
Zakir Hussain College
Miranda House College
sLC (E)
CVS
SGTB Khalsa College
St. Stephens College
SBSC
SRCC
SVC
JDMC
.
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..

2.
J.
the student coupled with the use of mathematical tools and analytical reasoning.
Abstract proofs can be complemented by numerical examples.
Textbooks
tfl R varian [v]: Intermediate Microeconomics: A Modern Approach, gth
edition, w.w. Norton and company/Affiriated East-west press (India), 2010.
The workbook by varian and Bergstiom courd be used for problems.
c' snyder and w. Nicholson [S-N]: Fundamentals of Microeconomics, cengage
Learning (India), 2010, Indian edition.
M. J. Osborne [O]: An introduction to Game Theory, Indian Edition
Course Outline
1. General Equilibrium, Efficiency and Welfare
Equilibrium and
-
efficiency under pure exchange and production; overarl
efficiency and welfare economics Readings:
(i) [V]: Chapters 31 and 33
(ii) [S-N]: chapter 13, p418 - p427. Numericars need not be done.
2, Strategic form game with perfect information;
(0 [O]: Chapter 2 (except 2.i0), p13 _ p5O
Mixed strates/ and extensive form ga-e" with perfect information
(ii) [S-N]: Chapter 8 (p231-p253, except concepts already covered above);
Market Structure and Game Theory
Monopoiy; pricing with..market power; price discrimination; peak_load pricing;
two-part tariff; monopolistic competition and oligopoly;
(i) [S-N]: chapter 14 (p464 - p485); chapter tst,.a92 - p507 and p511 - p519)
4. Market Failure
Externalities; pubric goods and markets with asymmetric information
Readings:
3.
(i) [V]: Chapter 34,
([V], p711-p715).
Assessment
Semester examination:
36 and 37, except 'Vickrey-Clarke-Groves Mechanism,
The question paper will have two sections. Section A will contain 4 questions from
topic 1 and4. students will be required to answer 2 questions out of 4. Section B will
contain 4 questions from topic 2 and. 3. Students wilr be required to answer 2
questions out of 4.
Internal assessment:
Th_ere will be two tests/assignments (at least one has to be a test) worth 10 and 15
marKs.
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2 Corrections and Clarifications
3[1f:::?1".11.:,T.f""T*# Non-smokers Diagram Figure:34.1, page:646,
A's money is measured horizontally from the lower left-hand corner of the box, and B.s
money is measured horizontarly from the upper right-hand corner. But the tora,
amount of smoke is measured vertically from the rower reft-hand corner.
Clarification 2: Bertral!_p^r1ce competition paragraph:6, page:494, Chapter:15,
Nicholson and Snyder,2OlO Indian daition
Case (ir) cannot be a Nash equilibrium, either. Let us look at two sub-cases separatell.
(ii- a) c<p1= pzand(ii- b)i.pr.pz'
"
f:r3#;,.*:#:i"::11 ,:,:,1,":."" incentive to deviate. rn this subcase Firm 2
::.;.."*{,,1:'i;L1tT:, j"T3io'}.'a1-",gifi i:,.ir:;;H:i"ff;#"';
ffi11?T:?:"I::i#,1-:?'-:{"i':1l"iT::rrri":fi;t!":#3T'j#ffi3":il
that market price and tot"r -ur*.tffiil';l+;i-iil:r*.t: t #ffl;#ffiil"*1:
that Firm 2 earns a profit (0, _ r)4p
7
by charging p2 and can earn
(p, - €-c) D(p, - e) by undercutting. change in profit due to price cut is,
[(
p,- e- c) o(p,- .)] -l f r, - ar@)1
L 2)
Because D(pr-.), O(pr) (downward sloping demand curve)
we want to show that Firm 2 can suitably choose the rever of price cut, that is, so that
the above difference is positive.
l(p,_ . _
drb,)l _
(G, _
4 y)=
^r,f,
u; _ .j
Since p2 > c, any choice of strictly positive smaller than 4 ! *orrtd be profitable
deviation for Firm 2. 2
(ii- b) 7f p < pzFirm2 earns zero profit. It can deviate to ptand earn positive profit.
;l1tf1;llr.Tf;r?r:rf"nr
constraint page: 501, chapter:15, Nicholson and snyder,
For the Bertrand model to generate the Bertrand paradox (the result that two firms
essentiaily behave as perfect competitors), firms -rr.t rr.rr.' unrimited capacities.
Starting from equal pri"e", if a firm to*"." its price the slightest amount then its
demand essentialiv doubles. fn. nrm-""n satisSz this increai.a de..r..rd because it
has no gapa-city constraints, giving n.*"
^ big incentive to undercut. If the
undercutting firm could not serve"all it .-a.-..4 .tlt. to*.. piic. be.a,rse of capacity
Kp,-.,)o(p,-.,-[[r, "y)],[b,_ e _,)o(p,)]_(r, _,4p)
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constraints, that wourd reave some residual demand for the higher_priced firm and
would decrease the incentiv. t;-;;;;;ut, The r",r*"g'ai""r"*.. a situation where
price competition does
"ot
f"uj to*_ligirrar co.t pricing.
consider the forowing simptified model, where two firms take part in a two stage
game. In the first-stage, fiims U"liJ
""p."ity Kr, r<,
"mrft.rreously. In the second
stage (Iirst stage choice.
".. ou..*Juf, L tr,i"
"i;;.i ;;;5 simuitaneousry choose
prlces pt and pz. Firms cannot sell more in the
"..oii "ffi rrr", the capacity built in
the first stage' Let qr be the ,",p"t ."ii
"r
rir- il;;.t:;., e, < Ki.Suppose that
the marginal cost oi p.oar"tiorr-i;;;;; ind capacity ouramg cost is c per unit. Let us
assume that capacity buitding cost is sufficiently high, 3
<c< 1.
Market demand curve is D(p) = 1 -p. If.the n*".rro*e different-prices, saypi > pj,
then the firm which,rru.
"Eir"*.. i'ri"" (Firm7) f.";1h;;;-and D(1t1)and se, the
minimum of D(p1) and, K11u.."r.. ii J#""t produce *or. trr* its capacity). That is
q = min{D(p),Kr}. Firm z, which rr". .rro".r, ._rrigrr.. p.i"e,';;"" the residual demand
at p', which is @(p)-qi). rrrererore,-s.ir or piri, ; ii ,r. ii.ri*,r* of the residual
demand and it's capacit]i, tfrat i" q,l rnmitpfpl _ q,),K,|.
If the firms choose the same pri". o, = pr = p, then the demand is equarly shared (that
is each firm faces dr D( nl
r
''nn,"",,.
"",".#.11n:#:;;::"#ff;. #..H
uu"' tn^nD@),
Before we start our analysis,
"r,. ,fr"iii
bounded uv trr. irlropoty profit, which ,.'" -*'-um gross profit a firm can earn is
max pD(p)=,rr* [p1t - il]= +
a
Thus the maximum profit net of capacity cost is ()-ru,J. a,r,". cis greater than I
a / - 4
to earn non-negative profit, firms will choose a capacity smaller than 1.
3
We will analyze the game using backward induction. Consider th
game supposing the firms have arready uurtt
"ap".iti.. "i,
rl'tjt;;'li","L::"il:
shatl show that pt = pz = p* =(l_fi_fl ) is a
Nash equilibrium. Note that at this price, total demand
output selis are Q, = Ki, S, = K).
Is a deviatio n pj < p* profitable?
rs o(p)= xi+ri. Hence
In case of such deviation Firm 7 charges a smaller price than Firm l, because
pj < p* = pi' This increase,s Firm 7's demand. However it does not increase Firm 7,s sell
3ffffi;lf,,"i1'"X*r"Ii"g ;Jit'-',r"",,, q.. rhi";";,;ls 7t pront and such

Is a deviation pj > p* profitable?
In case of such deviation FirmT charges a higher price than Firm i, because
pj > p* = pr. Firm j still sells Ki* and Firm j faces the residual demand
(DIP)-K4 = -prK*). Gross nrofit of 7 is [p,(1-pyK,*)1. If this profit is a decreasi:.rE
function of p1, then we can claim that ih. d.;i"ti;n lprice increasey was unprofitabie=
To check, let us differentiate lple - p:- K*)l with respect to p;.
alp (t - p, - y.1
-f=(t-zp,_
xil
< (l - 2p*- K;*) because pj > p*
= [1- 2(1 - Kix- K1*) - Kl*] because p* = (t-"; -",)= K,.+ 2K,^_l
< 0 because K;. K; <:
J
Therefore pt=h= p*=t-ri-rr)r is a Nash equilibrium of the second stage price
competition game. At this equilibrium firms use their ful capacity, that is
!.r=Ki,Qr= K). Gross profit of Firm 1 r [t-f, -K;)K;) and that of Firm 2 is
(,-
"i
- K;)K;)
It can be shown that the above is the only Nash equitibrium of the second stage game.
A situation in which P = p2 < P* is not a Nash .qrrilib.irr..r. At this price, total"qu"antity
demanded exceeds total capacity, so Firm t could increase its profits by raising price
slightly and continuing to sell K,-. Similarly, pt = pz r p* is not a Nash equilibrium
because now total sales fall short of capacity. Here, at least one firm (say, Firm 1) is
selling less than its capacity. By cutting price slightly, Firm 1 can increase its prohts
(formal analysis is similar to the case pi r-p* = pl.
Nowwe are ready to analyze the first stagi of thi" g"*". Firm i's profit net of capacity
cost is, r, = [(1 - K6 - I**)K*l - cKix. Firms are choosing
""pr"iti.. slmuttanelusty.
This is exactly like the cournot game. we can obtain .qrltiurir- choice of capacitiJs
by solving the best response functions. Equilibrium choice of capacities are
Ki = K, = !. thr. the price at the second stage wil b. p*-[,-+) which is greater
than zeto. Therefore unlike Bertrand competition, 'price-competition, in this game does
not lead to marginal cost pricing
-- re

CONTENT
INTERMEDIATE MICRO II
Topic
Unit - I
Unit - il
Unit - ilI
Previous Year Question 2016
Previous Year Question 2015
Previous Year Question 2Ol4
Previous Year Question 2Ol3
Practice Set Questions
Practice Questions
Page No.
t-79
20-29
30-49
50-53
54-60
67-64
65-68
69-80
8t-82

INTERMEDIATE MACRO II
1.
2.
Topic
Reading List
Unit I
3. Unit II
4. Unit III
5. Unit IV
Previous Year Questions
Page No.
84-86
87 -95
96 - 706
to7 - 123
124 - 726
127 - t29
130 -.131
132 - 133
134 - 135
136 - 138
6. 2076
2015
207+
2013
7.
8.
9.
10. Practice Set Questions

UNIT.I
MARKET STRUCTURE AND GAME THEORY
MONOPOLY
CHAPTER 14
i. A monopolist can produce at constant average and marginal costs of
AC = MC = 5. The firm faces
"
*";k;;;;and curve given by e = 53 _ p.
(a)
3*:l:l:":T.l*l_y:l2binsprice_quantity combination for the
monopolist. Also calculate the
"i,rri"p"jr},s
profits.
2.
what output level would be produced by^ this industry under perfect
competition (where price = marginal
"o"ija
Sfl"#,il:r,H,.":T:3;,_i1.lly: obtained by consumers in case (b).
:,},,#ii,i*,:,:"_.-1,.,::!;;"":;ii,i";:ii:i,f f:;i""ffi IIL
ffffffi;:, :Yfl:y:-:lT-1 -t1, "i" " i"r .
-
wr, Jt"il' ; 5' ;'.1:.":? ;f, :
'deadweight loss,, from monopolization?
A monopolist faces a market demand. curve given by e = TO _ p.
(b)
(c)
(a)
ll
t#
: f:o="':' l,:ff 1::.1y.".', 1 :" " :i?: t averase an d marginar c o s t s
i.l*,;#:=-l:;pi_"lltyll+i;i;:ffi ,-;,1li"ilffS:#;:;::
ln order
li"T#*ff,1i:T*yhai is theprice
",
,r,i" output lever? what are
the monopolist,s profits?
(b) Assume instead that the monopolist has a cost structure where total
costs are described by
(c)
(d)
(q)=o.zsQ -sq+:oo
with the monopolist facing the same market demand and marginal
revenue, what price_qrantity combination will be chosen now to
maximize profits? wtial uU pi"niU.i-*
Assume now that a third cost structure exprains the monopolist,s
position, with total costs given by
(Q)=o.ot s3Gse+250
Again' calculate the monopolist's price-quantity combination that
maximizes profits. What will proni UJa
'"
Graph the market demand curve, the MR curve, and the three
marginat cost curves
1.9* n"1" i"t,-^ibt, and (c). Notice that the
monopolist's profit -**i"q ,fiUty'iJ'"orr"t..rned by (1) the market
demand curve (along with it.
"""L"ir,.i nrn curve) and (2) the cost
structure underlying production.
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3. A single firm monopolizes the entire market for widgets and can proc'*-= :-
constant average and marginal cost of
AC=MC=10
Originally, the firm faces a market demand curve given by
Q=60-P.
(a) Calculate the profit-maximizing price-quantity combination for -_::
firm. What are the firm's profits?
(b) Now assume that the market demand curve shifts outward (becomr;
steeper) and is given by
Q=45-O.sP.
(c) Instead of the assumptions of part (b), assume that the marke:
demand curve shifts outward(becoming flatter) and is given by
Q=100-2P
What is the firm's profit-maximizing price-quantity combination no*'?
What are the firm's profits?
(d) Graph the three different situations of part (a), (b) and (c). Using you;
results, explain why there is no real supply curve for a monopoly.
Suppose the market for Hula Hoops is monopolized by a single firm.
(a) Draw the initial equilibrium for such a market.
(b) Now suppose the demand for Hula Hoops shifts outward slightlv.
Show that, in general (contrary to the competitive case), it will not be
possible to predict the effect of this shift in demand on the marke:
price of Hula Hoops.
(c) Consider three possible ways in which the price elasticity of demanc
might change as the demand curve shifts: it might increase, it migh:
decrease, or it might stay the same. Consider also that marginal costs
for the monopolist might be rising, falling, or constant in the range
where MR = MC. Consequently, there are nine different combinations
of t5,pes of demand shifts and marginal cost slope configurations
Analyze each of these to determine for which it is possible to make a
definite prediction about the effect of the shift in demand on the pnce
of Hula Hoops.
Suppose a monopoly market has a demand function in which quantiq-
demanded depends not only on market price (P) but also on the amount o:
advertising the firm does (A, measured in dollars). The specific form of this
function is
Q = (20 - P) (1 + 0.1A - 0.01A'?)
The monopolistic firm's cost function is given by
C=10Q+15+A.
4.
5.
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Suppose there is no adyellising (A = 0). What output will the profit_
maximizing firm choose? whatla.t.ii.rce w,r this yierd? what w,l
be the monopoly,s profits?
--- ---*^""' v
Now let the firm fl"o choose its optimal level of advertising
expenditure' In this situation, *t
"t
oltp.rt level will be chosen? what
price will this vield?. what w,' ih" i"ri of advertising be? what are
the firm's profits in this case? .
suppose a monopory c3n produce any rever of output it wishes at a constant
marginal (and average) cost or $s pJ.-"nit. Assum; ;h;;;r"poly sells its
goods in two different markets
".p"r.i.J-b;"#;
jil""*. The demand
curve in the first market is given by'
Qr = 55 -Pr
And the demand curve in the second market is given by
Qz=70-2Pz
(a) If the monoporist can maintain the separation between the two
market, what lever of output strouta_ u. p.oar".Jl"
""ii,
market, and
y,lil,ilXT wilt prevait i'...r"r,-*"rket?'whal ;;" ;iJ profits in this
How would your answer change if it costs
transport goods between the
"two
mart<eta
monopolist,s new profit level in this situationa
How would your answer change if transportation costs were zero and
then the firm was forced to foff?w a si"g'i'._p.i". policy?
suppose the Iirm courd adopt a rinear two-part'tariff under which
marginal prices must__be
"q".f i" it
" two market" tri-lr*p_sum
entry fees might vary. what pticint-pori"y should the firm follow?
Suppose a perfectly competitive industry can produce widgets at a constant
marginal cost of $to pei unit. M";;;;'lired marginar r""i. ,i"" to g12 per
unit because $2 per unit *""t u"
-i"ia
to robb;s;. ,1" r.,*r, the widget
;roducers'favored
position. s"ppo". iil marto.t dlmana-ror widgets is given
Qn =1,000 -50P
(a)
(b)
6.
(b)
demanders only $5 to
What would be the
(c)
(d)
7.
(a)
(b)
(c)
calculate the perfectly competitive and monopoly outputs and prices.
;ffi:iT:"ht"
total loss of consumer surplus from monoporization of
ffi?trJ""r
results and explain how they differ from the usual
suppose the government wishes to combat the undesirabre arlocationar
effects of a monopoly through tfr" us. oi"a suUslav.
Why would a lump-sum subsidy not achieve the government,s goal?
8.
(a)
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-
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-
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:-=:
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-
5=-:=€
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:
:
:
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:
-
-
:
-
-
-
-
-

ee,.
(b)
(c)
use a graphical proof to. show how a per-unit-of-output subsidy might
achieve the government,s goal.
Suppose the government wants its subsidy to maximize the difference
between the totar value of the good to consumers and the good,s total
cost' show that in order to ,"f,i.u. this goar, tt. gov.rnm?n1 shouii
set
I
where t is the per-unit subsidy and p is the competitive price. Explain your
result intuitively.
s"ppg::^ a. monoporist produces alkarine batteries that may have various
useful lifetimes (x). suppose also that consumers,(inverse) demand depends
on batteries'lifetimes and quantity (e) purchas"d ;";;;i;!i" ,rr" function
P(Q,x)=e(x.e)
where g'< 0. That is, consumers care only about the product of quantity
times lifetime:
lhey are wilring to pay equany for many short-lived batteries
or few long-lived ones. Assum"
"f"o
that tattery costs are given by
c(Q, X) = C(X)Q
where c'(x)= 0. show that, in this case, the monopory wilt opt for the same
level of X as does a competitive industry even though levers of output and
prices may differ. Explain your result.
9.
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1.
CHAPTER - 15
IMPERFECT COMPETITION
Assume for simplicity that a monopolist has no costs of production and faces
a demand curve given by e = f SO I p.
(a) calcurate the, profit-maximizing price-quantity combination for this
monopolist. Also calculate the monopolist," pro,fit----
(b) suppose instead that there are two firms in the market facing the
demand and cost conditions just a."".iu"J--ro, their identical
products. Firms choose quantities
"ir"rt.r,*"lly"as in the cournot
moder' compute the outputs i" ,!. N""h .q;ili#irlm. Arso compute
market output, price and-Iirm profits.
(c) suppose the two firms choose prices simultaneously as in the
Bertrand moder.. compute the prices in the trlash-equrtibrium. Arso
compute firm output and profit as well
". _*t .i ortp,.,.
(d) Graph the demand curve and indicate where the market price-
quantity combinations from parts (a)_ (c) appear;;;;" curve.
Suppose that firms'marginal and average costs are constant and equar to c
and that inverse market demand is given by p =a _ be, where a, b > 0.
(a) calculate the profit-maximizing price-quantity combination for a
monopolist. Also calculate the minopolisis profii.
---
(b) calculate the Nash equ,ibrium quantities for cournot duoporists,
which choose quantities for their identicar proa""i. simurtaneously.
ll]},::"'o"te
market output, market p.i"" .rrJ-fi.*
"ra industry
(c) calculate the Nash equ,ibrium prices for Bertrand duopotists, which
choose price for thiir idendiicar proarr"t"
-
"i-rrit"rr"or"ly. Arso
compute firm and market output a" *i, as firm and inaustry profits.
(d) suppose now that there are n identicar firms in a cournot moder.
Compute the- Nash equilibrium quantities as functions of n. Also
compute market output, market price, and nr*
"o*irrdustry
profits.
(e) Show that the monopoly outcome^from part (a) can be reproduced in
part (d) by setting tr = 1, that the cournot a"il.rv-""icome from part
(b) can be reproducgd-in nart (d) by settingn = 2in part (d), and that
letting n approach infinity yi.ias the same market price, output and
industry profit as in part (c).
Let c i be the constant marginal and average cost for firm i (so that firms
may have different marginal costs). Suppose demand is given by p = 1 _ e.
(a) calcurate the Nash equ,ibrium quantities assuming there-are two
firms in a cournot market' Also compute market output, market
2.
3.
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(b)
price, firm profits, industry profits, consumer surplus, and :c-:
welfare.
Represent the Nash equilibrium on a best-response function diagra=
Show how a reduction in firm l's cost would change the equilibr:--::
Draw a representative isoprolit for firm 1.
suppose that firm 1 and 2 operate under conditions of constant average a;:
marginal cost but that firm 1's marginal cost is c, = 10 and fir 2,s is c. =E
Market demand is Q = 500 - 20P.
Suppose firms practice Bertrand competition, that is setting prices for
their identical products simultaneously. Compute the Nash
equilibrium prices. (To avoid technical problems in the question,
assume that if firms charge equal prices then the low-cost firm makes
all the sales.)
Compute firm output, firm profit, and market output.
Is total welfare maximized in the Nash equilibrium? If not, suggest an
outcome that would maximize total welfare, and compute the
deadweight loss in the Nash equilibrium compared to your outcome.
4.
(a)
(b)
(c)
5. consider the following Bertrand game involving two firms producing
differentiated products. Firms have no costs of production Firm l's demand
is
er=1-pr+bpz
Where b > 0. A s5rmmetric equation holds for firm 2's demarid.
(a)
(b)
(c)
Solve for the Nash equilibrium of the simultaneous price-choice game.
Compute the firms'outputs and profits.
Represent the equilibrium on a best-response function diagram. Show
how an increase in b would change the equilibrium. Draw a
representative isoprofit curve for firm 1.
6. Recall Example 6, which covers tacit collusion. Suppose (as in the example)
that a medical device is produced at constant average and marginal cost of
$t0 and that the demand for the device is given by
Q = s000 -100P
The market meets each period for an infinite number of periods. The
discount factor is 6.
Suppose that n firms engage in Bertrand competition each period.
suppose it takes two periods to discover a deviation because it takes
two periods to observe rivals' prices. Compute the discount factor
needed to sustain collusion in a subgame-perfect equilibrium using
grim strategies.
(a)
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(b)
L"J;T'ffir'ff"i'""r:T,lion. th-at, as in exam pte z,deviations are
rather i" J"t"riiii..;1;ii11o-: N,t*t' assume tr,.t ., i"
""ii#i',i u",
[il::l*",ilil,fl 1,i,*L'"r+Tj.Tj,:il:][*;+:: j:;,il:i:t:
7 fn:ff 6'='ir8:81'f;"l"lT:#,i,lJT: #:',1:
production costs, racing
(a) compute th!
""PFT9-relfecj equilibrium of the Stackerberg version
;:".
game in *ti""t, n.Ir-'r
"rr".i""'0,'-ilr";'"r, then firm f"hoo".s
(b) Now add an entry stage afterrr,T., choose q,. In this stage, firm2
decides whether o, ,rot to enter. If it enters tr,.r, it must sink cost K
after which it is all0wed to choose ,r.
"o-orr,".
the threshota ,ratue ii
K, above which firm 1 prefers to deter nr* it entry.
(c)
m::::$:[:."ffiffi]"Tl"Hl'l*- and entry-dete*ence outcome on
8' Reca' the Hotening.model 0f competition on a linear beach from Example 5
;iifffi}:simRricirr 'r'"
i""
"'"rm stands
"", i;;;; onrv at the two ends
ii il ffii;:ff':'Ji::',il-"T:ft0..': ":,"",;,;r',;;"j,""pment in the middre
involving product proliferation.
--
"ou
to analyze an entry-der..ri"g
""",.!)
(a)
(b)
(c)
consider the subeame in-which firm A
r". ryo ice cream stands, one
at each end of tfre beach,
-;;
;t;e^".arong-ii,#'r, the right
endpoint. What is the Nash
"qrifii'ri,.iiof this subgame?
If B must sink an entrv n^ef v -__- , , .
-_
--
"qv6q,rc:-
nrm A,
"
* o
"
in .,#T,,Ti _k:,T.1.:"i:ffi ::" ;,:: LLTr,n",
'trt:f.'X11T"';j:1:l:t'"^l
stratesz credibre? or would A exit the
"orp",e^",-n;ilit{il?,"#::i}TiTffi :".r,*."t':[:
and both it and B have stands o, tt"-risht to.the case in *ti"r, A has
one stand on the teft;n{
""J-B"h;.,"".1,"r0 ", ,r*irgi,i,..,a 1"o e,"
entry has driven A out of the,il;;;;itne market).
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CHAPTER 8
STRATEGY AND GAME THEORY
1. Consider the followin
(a)
(c)
(d)
Find the pure-strategr Nash equilibria (if any).
g'".
"
J;l :}; #fi:L":Tl:fl,J':Le
quili brium in whi ch e ach praver
fff.po;*
players'expected payoffs in the equitibria found in parts (a)
Draw the extensive form for this game.
The mixed-strategr Nash equ,ibrium in^the Battre of the sexes in Tabre 3
3il,3;: ::3"n If :. H:j:.'."^* r+:!" .
r";
3* l"v;ir" :' ;;' *. neralize thi s
sotution, assume that the payoff *",r" ,".in':';"Ifi";,t;# r?,
Xif:."r?1.
Show how the mixed_strategr Nash equitibrium depends on the
Draw the extensive form.
The game of chicken is played by two macho teens who speed toward each
;,;,"H;;..Tf:.jffi;i1,,"*::ilj, veer orr is rranila the chicken,
l:'ffJ"ff fi "i"1Tl*m",::;;iF1"-"t?ffi:';-#ffi ::'[T"3,:;#Thl
2.
3.
(a)
e:
Player 1
A
D E F
0.o
7.6 5,9
B 5.8 7,6 1, 1
C 0,0 1, 1 4,4
Player 1 (Wife)
Ballet ioxir
q._q
1,K
Ballet K, 1
Boxing 0,0
Teen 2
Teen 1
Veer Don't veer
1.3
0,0
Veer ,c
Don't veer
0, 1
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r rIr r:rirlrrlFr:rlrlfl[flF
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(b) Find the pure_strategr Nash equilibrium or equilibria.
(c) Compute the mixed_strategv *"";-:^-:,,.:":
answer, draw the b..,_jlt^Y-
Nash equilibrium. As part of your
strategies. ---'-r'€spoflse function diagram r* ,rr"'#*.a
(d)
:H[ffiJl',.*ffi;:"i,'fl'S-sequentiauv, with teen A moving nrst and
".. t."I'-"e,j :H,,X"J:?[ Y_1|-y,tg away the steering wheer. what
"r,"r.9,
B,s contingent strategies?- #.il."ao*r, the normal and
(e),TJt-.:1";lT#f
'T ;"''"":;T*:,:,""H::fl;",
;,
(0 Identift the proper subgames in the extensive f
version of the g.*".'u"'. 03"*;;"ffi",# f:'ll;:.?;,"",,#
:iir-ffi f :fl:i|-",H,j':i:{*"*311r[-1i.''o,n".rv."r,"q.,iiiu.i,
4. Two neighboring homeowners, . = 1, 2, simultaneously choose how many
hour is o _ vvqqL'ur rawn. Ihe average benefit per
I
l}-t +!
'2
And the (opportunity) cost per hour for each is 4. Ho
benefit is increasing-in the rio".."r.igtbor j spend. ,rT,.:y_ler,i's
average
:li'ffff ffi H:Ji"'fi*;;;.&-"aep.na"'H";;;nl;."i:+;;;;lf:
(a) Compute the Nash equilibrium.
(b)
ffifl: Jl."rff"t
*sponse function and indicate the Nash equilibrium
(c) On the C.Tph, show how the equilibrium would chr
:lffi :.,jJ*iaeigr,uo.,s;;;;s"uen.nti;;;,ilif "r.,j;r:,i",.Jitri
5. The.Academy Award_winning movie A F
Nash dramitir."-w".rr,. scholarlv ..].:1Y,,if"l
Mind about the life of John
equilibrium concept dawns;;;;;t*;iltlb"rjo" .in a singr"
""";;;;;
ma r e grad u ate s tu d en t s. rh ey il;.:.'.!lt ii# J,JJ :.ll*, :lm f
,jt
$;,-fi ;*l#f;:Jl i[il: Xl"flH" [J'"
a.
"
o' tr
"',i ", .r,"" 0*.,
"
t t e s rh e
students, arong the rouowing--rlil:
:{l#:"Tn'#
jf T"t"fJf*XT
:l+!'trft:'"Y;',?:13#rfi;n:lia
#,,r"
'i',;'" ;J,.ttes rf ma,e i
ffil
jl1: ixfit; ll "". ; ; ;; * iil: ffi:t!il f;::il
-,;.tf
*,1 l":;
ott.."'whJ,";;HH:;;T:::"'$T#r,:,,Ctlftl{ilT#l,Ll}i
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:
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=
=
:
:
=
-
-
:
-
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=

(b) Find the pure_strategz Nash equilibrium or equilibria.
(c) Compute the mixed_strateg' *"";-^^-:,,:r:
answer, draw the b".t_j:ry
Nash equilibrium. As part of your
strategies. --'t-r'eSponse function dragram for the mixed
(d)
*xq{i,rAisffi: :"i#Tr",?.:#Hy;"yJ,il:-:l A moving rirst and
ffi *','":,nJi#,,il:"*::irF:$::il[.]::H1""[:-xffi 1,,*T
(e),Tji-.:1""f:#:T ;.'*'"":;X;X::?? ;_", "":.
(0 Identi$z the proper subgames in the extensive f
version or the same.
-u".
u3"r*#"ilil",# i:' :h:'?;,.",,i,1
;ii'-ru;ffi :i-',T:i:f)l;.:*tlrpl1t".otr,.,,v""'r,"q,,i#"
4. TWo neighboring homeowners, . = 1, 2, simultaneously choose how many
hours {, to spend maintaining a beautiful lawn. Tht
hour is o * vvquL,Lu rawll. lhe average benefit per
I
l0-r, +J
'2
And the (opportunityJ cost per hour for each is 4. Ho
benefit is increasing-in the f,o"r""rr"rsrrbor j spend"
""T::Ter,i's
average
:ffSff il;x".:di"T:il;;;";:r''a.-p",a.'H"r*,;ni;""X:+ru;U:
(a) Compute the Nash equilibrium.
(b) Graph the best-response function and indicate the Nash equ,ibrium
(c) On the C.lph, show- how the equilibrium would chr
of one of the n.igrruor,s ;;;;*" benefit function ilF."]f the intercept
smallernumber. -'!'q6L ucrreur runction fell from o to some
i:lif,"i['ffi,]x*$":il1'::#;r'i"o"f iyi,*, Mind about the rire orJohn
equiribriumconceptd,,*1";;;##*iil!T;:,;.,:?*#,r*"#t'
mare graduate students'
lhey ";il"';;."1 *."-"r,rr.'fr5ro and the rest
I$,'iE#:.?T:Jl ",'#'*:lT[* [J
re a e s i ra ur.' ti,, ir,l o *,,., t e s rh e
students, arong the rouowinr"ljil:
${l#:.it|,-
j|". Tt"fJfffi*
:l,j!'tri:":"3;"?:i,l#lii;t":yia #"1. ,i,;. ;T,"ttes rf ma,e i
ff*l
jirr'""riml"ff:ii*:L*ii"*:ilt!ilffi!il1;"t***";
others who approa"r,"a r,.ri".rr"., o#u,lr?.]..o
r3l
,ilro.|$"rffI, H#:
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-
-
-
c==:
-
:
:
==:
-::-
:
-===-:
:
:
:
-
:
-
:

(b)
(c)
earns a payoff of b > 0 from approaching a brunette , since there are lnc::
brunettes than males, so i is certain to get a date with a brunette. T:-:
desirability of the blond implies a > b.
Atgy-._ that this game does not have a symmetric pure_stratery Nash
equilibrium
Solve for the symmetric mixed-strategr equilibrium. That is letting p
be the probability that a male appro."h.s ihe blond, nna p-.
Show that the more males there are, the ress likely it is in the
equ-ilibrium frorn part (b) that the blond is approach.a [y rt t""st one
of them. Note: This paradoxical result *"" .rot.d by s. Anderson and
M' Engers in "Particlpation Games: Market Entry, coordination, and
the Beautiful Blond," Journal of Economic Behavior & organization
63 (2007): r2O - 371
compute a player's minmax varue if the rivar is restricted to pure
strategies. Is this minmax value different than if the rival is allowed to
use mixed strategies?
Suppose the stage game is played twice. Charactenze the subgame_
perfect equilibrium providing thL highest total payoffs
lraw a graph of the set of feasible per-period payoffs in the limit in a
finitely repeated ga.me according to [fr" firu theorem.
Return to the game with two neighbors in problem 5. continue to suppose
that player it's average benefit p.iho.r, of work o" r""a"".fiig t"
I
l0-l +-r
,2
continue to suppos.e that player 2's opportunity cost of an hour of
landscaping work is 4. Suppose that l's opportunity cost is either 3 or Swith
equal probability and that this cost is l's prlvate iniormation.
(a) Solve for the Bayesian-Nash equilibrium.
(b) Indicate the Bayesian-Nash equilibrium.
which tyge of player 1 would like to send a truthful signal to 2 if it
could? Which type would like to hid its private information?
6.
(b)
(c)
7.
Consider the following stage game:
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8' In Blind rexan Poker, player 2 draw a card from a standard deck and places
it against her forehead'*iirro"t ir"r.i"s at it but so prayer 1 can see it. praver
1 moves first; deciding whetheri"
"1w
or fold. ii;#; 1 fords, rr" mrst pav
player 2 $50. If p^r"rgI r stavs, tlie action goes to prayer 2. prayerz can iori
or call. If player 2 folds,
"t.."".ipay player r SioOllrit i. a high card (9,
10, jack, queen, king, or."";, fffi. 1 pays player 2 $100.
(a) Draw the extensive form for the game.
(b) Solve for the hybrid equilibrium.
(c) Compute the players,expected payoffs.
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1.
PROBLEM SET
MONOpoLy BEHAVTOR {VARTAN WORr(ouTl
CHAPTER 25
Ferdinand Sludge has just written a disgusting new book, Orgr in il--:
Piggery. His pubrishel,
-cr1*
Mcswili estimates-that the Jemana for *us
book in the United States is q, __5o,oi0 _2,000p,, where p, is the price ::-
the u's' measured in U.S. donars. The demand for sludge,s opus in Englanc
is q, =10,000 -500p2, where p, i" l;";rice in England measured in u.S.
dollars. His publisher has.
"o"t
functi,on c(q)=$50,000 +2e, where e is
the total number of copies of Orgr that it produces.
(a)
::#::Til"Til'i,"*;? the same price in both count.i.:: hg* many
m aximiz e it"' p;;;;"_
"
*t*#""*i,r?H:1.,,r:H%.i
I*:jy,lt:=.:hl1T a different price in each country and wants to
maximizer.il,f:1"_:1""{""pii;;;iii."ji#iiilrrI,ilJ3l"ff
;
__-ae, --vrr 'rq'J uw1lrss sloulct rt sell tn the United States?
what price should it charge i* lil- United states?
.*r"*
-*_ I"^y,Tl"{ copies should it setl in England?
;;h-,,,rr +I.n?1,1::;l1o3ta it
"r'"!t in England? '"t'ffi;
much will its total profit be?
A monopoly faces an inverse demand curve, p(y) = 100 _ 2y, and has
constant marginal costs of 20.
(b)
2.
(a)
(b)
(c)
(d)
(e)
(0
What is its profit-maximizing level of output?
What is its proht-maximizing price?
What is the socially optimal price for this firm?
What is the socially optimal level of output for this firm?
3.
what is the deadweight ross due to the monoporr"o" o.nloll*r"
Iirm?
Suppose this monopolist courd operate as
I Rgrfectly discriminating
monopolist and setl
;1ch yit_gf output at the highest pri""'i, would
fetch. The deadweight loss in thi" ;;;;ould be
Banana computer company sens Banana computers in both the domestic
and foreign markets. Recause of differerrces in the power suppries, a Banana
purchased in one market cannot u"
"".a
in the other mark"t. *r. demand
and marginal revenue curves associated with the two markets are as follows:
Pa=20,00e20Q
& =25,ooo *5oe
M& =20,00G40Q M Rr = 25,ooo _tooe
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Banana's production process exhibits constant returns to scare.and it takes
$t,000,000 to produce 100 computers.---
Banana's long-run average cost function is AC(e) = .------..- and
its long-run marginal cosi functio" i"-fr,fCfOf
If Banana is maximizing its profits, it will sell
in the domestic market at
---. --::r_ computers in the for.igr,,rr"rk.t
"t
each. What are Banana,s total prontii
At the profit-maximizing price and om"
elasticity of demand in tie domestic mlrket? _ .- Wfr"t l"
the price etasticity of demand i" th;ro..igr, *.rtEi]_ r"
demand more or less elastic i" th;;;;iet where the higher price is
charged?
(d) Given that costs haven't changed, how many Banana computers
should ,*T3 sell?_
_
What p.i"" *itt it charge?
How will Banana,s profits cfranle now'that it can no
longer'practice price discrimination?
A monopolist has a cost function given by 4v)=t' and faces a demand
curve given by P(y) = t2O - y.
What is its profit-mT^Tl1r:c tevet of output? What price
(b) If you put a lump sum tax of $100 on this monopolist, what would its
output be?
(c) If you wanted to choose a price ceiling for this monopolist so as to
maximize consumer prus produ".r .r.itr", *t",'p1"" ce,ing should
you choose?
How much output will the monopolist produce at this price ceiling?
Srrppose
!_!-at Vou put- a specific tax on the monopolist of $20 per unit
output' what wourd its prolit-maximizing d;-""f li,",pr, be?
The Grand rheater is a movie house in a medium-sized colrege town. This
theater shows unusuar nms anJ ir""t"
"".ty-arriving
movie goers to rive
orgarl music and Bugs
lunny cartoons. If th; trr.atlii"-Jp.rr, the owners
have to pay a fixed nightty amounl-ti'gSoo for films ,.fri.", and so on,
regardless of how ma,,y people come to the movie.--ir.-.i-pricit5r, assume
that if the theater is closed, iis costs-ar J ,rro. The nightry demand for Grand
Theater movies by students if e =220-40R, *rr.* e i" trre number of
movie tickets demanded. by students at price pr. The nightly demand for
nonstudent moviegoers is Q =146-26*.
(a)
(b)
computers
dollars each and
dollars
(c)
4.
(a)
(d)
(e)
5.
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:--::
:
_:=:
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(a) If the Grand Theater charges a single price, p, , to everybodr_. rhe:- a:
prices between O.and $5.S0, the aggregate demand function for.rc,-._-:
tickets
,t"
qr(pr)= ou.'. trri" range of prices, the .nr_e=:
demand function is then p, (e , ) = _.
what is the profit-maximizing number of tickets for the Grand rhea:::
would this number of tickets U"
"iai- How many ticke:s
would be sold to students?
-
==- To nonstudents?
suppose that the cashier can accurately separate the students frorn
the nonstudents at the,door
!., f*;i;tudents show their school ID
cards' students cannot reseri their ti?kets and nonstudents do not
have access to student tD carJs. Th; the Grand can increase its
profits b.y charging students *a
"o""lidents differe-.rt'p^riJe*. wrrat
price will be charged to studentsi
tickets wlr be
";ldi
-- "'**"""wn.t
Ei"" #?,-#Til#Ij"iJ
nonstudent How rnany nonstudent tickets will be sold?
How much profit *iff
-
tfr" Grand Theater make?
6' The Mail street Journal is considering offering a new service which wilr send
news articres to readers by email. The-ir marka .;;;;;h indicates that there
are two types of potentiar users, impecuniou"-
"Jra"rr,"
and high_revel
executives. Let x be the number of articlis that a ,".i *qr."ts per year. The
executives have an inverse demand f"";;;Ai=il _ x and the students
have an inverse demand functionpu(")=go-*. (prices are measured in
cents.) The Journal has a zero marginal cost of sending articles via ema,.
Draw these demand functions in the $aph below and raber them.
(a) Suppose that the Journal can identi$z which users are students and
which are executives. It offels- each type or ,r"", a different all or
nothing dear. A student can either b"r;;"; i" ao articres per year
iJ ;:;i"?'.:: i'l,TH' j:,,'I: s'*-"
; ;il ; s tu d en t *ii t! *rirr" g
Suppose that the Journal can't tell which users are executives and
wnilf are undergraduates. rtrus
-lt
"L', be sure that executives
wouldn't buy the student package ti1h.y found it to be a better deal
for them. In this case, theior..r?l l"ol,,r offer two packages, but it
will have to let the users serf-select the one that is optimar for them.
suppose that it offers two packages: one that a-,ows up to go articres
peryear the other that arows rf to too articres peryear. what,s the
highest price that the underg;i;";;" will pay for the 80 article
subscription?--
(b)
(c)
(b)
(c) What
year?
is the total value to the executives of reading g0 articles per
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what in the maximum price that the Journar can charge for 100
articles per year if it wanis
"".""uu""'lo prefer this deJ-t-oiuying go
articles a year at the highest p;"; il"
"nd.rgraduai"" ,i"'*,rirrg to
pay for 80 articles?_
suppose that the Malr street Journar decides to incrude only 60
articles in the student package. wil i" the most ii
"rrra
lir"rge and.
still get student to buy iti" pi"t .g;"
'"
If the Mall street Journal offers a "student package,, os 60 articles at
this price, how much net consumer
",iirr";;dL";",ir"" get from
buyrng the student package2
----'-^ "*'t'
.
what is the most that the Mall street Journal could charge for a 100_
articte package and expect executives ,";;;lhi";i"?jlln"r than
the student p."k"g.?
.-.'"
If the number of executivesin the population equal the number of
students' would the
-Mall str."i-.lirinar make. higher profits by
ffi:i:$^
student package ,i 8t*;;: or a student package of 60
Bill Barriers, CEO of MightySoft softwa
stratesr:b";;Gil,.iiue"t_";xi;;'il";#;::::[rH,f -,ft
[.:rlXf$:t?
together and selling the pair of softwle products for one price.
From the viewpoint of
.
the cgmpany, bundling software and selring it at a
discounted price has tow effecis on s.r.s' (1) revenues go up due to to
additional sales of the.bundlel i"a'tzl revenues go down since there is less
of a demand for the individual corpJi.",. of the bundle.
The profitabilitv of P""llilq depends- on which of these two effects
dominates. suppose*that uigitviJrt'"'.tt" ,rr" wordprocessor for $200 and
the spreadsheet for_g2so. a. irait<eti.rj'rrr*"y. of 100 peopre who purchases
either of these packages in the 1;"d;? turned up the iolowing facts:
(1) 20 peopte bought both.
(2) 40 people bought only the wordprocessor. They would be wilring to
spend up to g120
-orl fo. the sfreadsheet.
(3) 40 peopre bguqht only the spreadsheet. They would be willing to
spend up to $100 mor. fo, the wordprocessor.
In answering the following questions you may assume the foll0wing:
(1) New purchasers
.of Mightysoft products will have the same
characteristics as this grorij.'
(2) There is a zero marginal cost to producing extra copies of either
software package.
(3) There is a zero marginal cost to creating a bundle.
(a) Let us assume th-at Mightysoft arso
.offel the products separatery as
well as bundred. tn orJerio alt.rmirr. how to price the bundre, B,l
(d)
(e)
(0
(g)
(h)
7.
Bliss Point Studies
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fff:iJ","Tf: *:l*llt: rouowing questions rn
Iess than ,_,rocessor purchasirs the price
In order to sell the t
have to be less ,h",
,rrdl. to the spreadsheet users, the price wouH
What would Miehtv
pricedrh"il;1?,:i$31H profits be on a sroup or 100 users ir j:
.Ylt1::,y,0 M-iqht soft,s profits
What wilt the price of a ride be?
what wilr coroner To
--
what is the pareto .#:::"Jj:j:,,,'il:""",
,r," o""aill;'dffii:_itofits be on a sroup of 100 users irit priced
If Mighty Soft offers
--- '
whatwourdpro,ts*H::IJ.:,"_r,ff
o:I;i"X,,set?
What would be the profits with the orrr*J
'-'sr --------.
Is it more profitable to bundle or not Or*.,
=.---
ifiii":rTi.h-l"tf,:ort
worries about the reriabirity or their market
I 9o o.*; ;i#d
til iTI J""{::J,A,, $:i*,;t""lii;
3::ffi;i.'o:lil" # Jloo -"7ii;' ;i,l T, ,r,J
-"p,.#lilJet
onry
what are pronts -,JJl::fi;:t
- there is no b"";il;;l::
order to s: - ;__-1
would har-e :: :":
(b)
(c)
(d)
(e)
(0
(g)
(h)
(i)
(j)
8.
0) l!r". .."rrsis so far has been <
would puichase at ]east orr. or'ol"erned
only with customers who
prices. is trr.r. ,n], ffiffiT ""::1.
programs at the originat set of
io"" ttri"...Jre
€my additional sou,rcg or aI_.ri;;.;" bundle? What
prontabiiity;%#;,il-iT '1'"'"tro". *.- i1,"" iirro" about the
_Colonel Tom Barke
world, ei",.'ffitcr is about to open
]! n.eyest amusement park, Elvis
lT'"r.'';1;;".rffi"5ni T,ffli.,i, :":i,i"; il..Iil;s : you can ride
dinner in the Heartburn"il;:,-#:1.:,"*-o the Jaithouse iock
";;^;;;
at trac t 1 d; p.'"fi""qTi Xr:':i o"lllt I I. * nsr," J;;; tvi s worr d w,r
Il"1t p is the price of
"
ria".'er.*"i"rierson will take x =
the same and negati
essentiary zero.
ve rides il;#.riltr-o "rsrts
etffiil:3.rttf #i:il
wed. The marginal cost of a ride is
(k) At what values of t would ,, O"..--.-..-.-
_D ur L would it be unprofitable to offer the bundle?
3 ,T:::j::Tl_:*"""n
inverse demand function for rides?
(b) If Colonel Ton
be taken ^^- r::1': the Price to maximize
-
be taken p". a.v i'fi#;'ffJ:,#r?-". profit, how many rides wlr
(c)
(d)
(e)
Bliss Point Studies
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(0
(g)
If Colonel- Tom charged the pareto efficie
ndes would be purchased? nt price of a ride, how many
decided to use,l^,*:;r::, tariff, he would
";
of
I:y .T"^"n consumers, surplus would
quantity? _ be generated at this price and
(h) If cotonet
admission
Tom
fee
9. The city of String Valley is squeezed between two moun
long' running IIom north to .o,rr,r.orlT::;." Y^' Tou.ntains and is 36 mles
town, the poor
of the .o"&'t":l1tt:"
h; ;ffi#il tx",*lb;t' ;31?"u
*d" wi;;;' il"
". .ih; ;;'h .#3'|;,}*.df j:A !1,ji!"
ir,.'
"i,r",,fii.f 'e
per mle' Becaus e
only three boUI",."ii:;:'' :-:'"":e.ol strict zonins reg,rtrti^-."er ûrc,uorrn
north edge
"r,:1Tg
a,evs.
"; ;i:'"I"t'i"t 'onii#*l?ii!ij!#?!:
or town, *6 o,f^*..l,,
one or,h;; ;",.J""fi j:I]"ft,:ff
1*:
ci ty, rimits ;; ;;
:'fl:T#ffi :::"ri1:i1l'[:,::xlla"i,T:""1,1:H']itl"j:dtililti;
H3Jx;,_"ff",#[:*lii;l,T*yl,ir
j:i;::""il1ti]it jffi ;1,g
(a) Consider
$t0 for ,":,::thê,bowlins alleysât either edge o
w,ring ,; ffit"lfl,ft3:: far wi,
" ",8;,":'i5,1;#U,Lr1:
would thir
(b)
i*#iffi;';"*;'*n#:'Tffi'#il#;
the edge of town *ri l*" i';i"."J '
customers that a bowling aley at
(c) write a a.-r,, ,^","-::::t
charges $p per night of bo*t;;;::l ''
(d) suppose ,rt"'t
for this bowring alrey,s inverse a.mana function._
cost of $o :1t-
the
,bowling alleys at the end of towr
the time ,;l!t ""tomer and sL
from the otlng.assY*.
,n"r-ri.lth'eir
pricel ;; ffil
have a marginal
have?
rr.i ur*rirs J"rlf:: -Pyi'-"c "ilv"
rc-.7mtze profits' (For
e ffi :::T-*#xtx*"i,##::?}l#flHHiru#;r
p.i".offi l;ilT1";H5,#.J,JLlffi ;:*1"".,J::ft;;';;",
(0 Ir the bowring allev in the
".;;"; ";;:,--
'::: P-er weel
-
per custom;;;; ;J1l;:""ilr1:*: arso has marginar costs org3
t "*-'f*,r"':Lff;1ffi1i'r;T}""-fi *k: w,r it
"nT*::
--_____. -- and Charg" . :-;^-^ 'vvurq set an
arge a price per ride of
r**l*]ryr.,#,f"':L::::"rT;:T:11,,1.n,""o".#".?'.wlrlltcharge?
(g) Suppose that the city relaxes its
3i#,ll *:g: :?1, " +.,
-;;
""
;T"i.:, i"*, [:T,ll,^,:.: " wh ere th e
:3_I : boyling .[.yr
**' ulrt contrnut"
l:,1::"" operating li".r"". ,o
about . r"""irii.',""'""":jn ::t:ilill?5l 1,"r". at ihe
".,a'orio*, *"
anywhere in town ,fr"r ,fr"i)
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rike at about the same cost. The bowring a_lley in the center of ton:: :.
committed to stav where it is. wouiJ;;;il^;i tie aleys at the edse ::
,I;I1.,TH?:: :T,:Tf :"
uy ro.,ting-nlx;i" i*,.,.*,",ing bowung i_ =.
rocation r*;uJ ;iin"".;;;*,Y"11,U$ .. ' p'oi,-*""ii.--l-.
10
i{fl,T;:i::;#:ai:*'::""iT;:l*.
in rhe r
" r"",l.,"*,.presenra'.i.
t.'. "iiv .'*r:
I : 1
; i;;: ;# :;'. :ffi j1T: ".'H i.rr"j:, :# :l jU,
#*.
citizen's political
'iews in tf," rdito*in* *",'ilJ.ir.n with the mos:
extreme reft-wing views is
""ij t.-L.- at point o
"rrJirrJ-"itizen with the mos:
extreme right-wing views is
"*Jt"'u. it poirl i.li"
"r,zen
has views tha:
are to the right of the
"r"*" "iit..aactin x oi trr"-"i.t"t poputation thai
citizen's views are-said to u" r"""i.i .t trr. p"iri
".';"*rrates for office are
forced to publicalry
",",. trr"r."lin, poritic"i p"Jro,lon tne zero_one left_
right scare' voters alwavs ;;;; ;; tir. ...rJii.i.;;":. stated position is
nearest to their own-views. 1ff tfr"r. i"
" ti. f..l;"#il:
a coin to decide which to
"ot!-fojr!
ro 4 Lrc ror nearest candidate, voters flip
(a)
}ff""ff;,j[ !TSi:t::
ror the congressionar seat. suppose that
,t
"." "rr-.frri1i.i,llt1'' "l?"J getting as
1lanv votes as possibre. Is
position give the o."lr,# ^Yll:n^.t:"h
candida"te ;;;;J;e best
ffi lffi l"-#].po"iiio";;1"J"fr'l#;,T:ftTf :::"r""""Jff"Hij
11.
iyiim:ffiil'Hfi',ff#:',i;;.,i":: lr_.,1." previous probrem, ret us
:"Hff l,*:',,L:'*iil;';;;ff ,'f,iy^Fl'f l'ff
"o:*j,:I?:Hi?;
ideologicar to"rtiolt .
thev receive' Thereiore .;;i";;;il.t. .rrooses his
contribution. 1'r. ,.'^'-t -lYth
t way as too *ux-Lt ;;;1
Let us denne
" '"--::]:"j::l::"
tr" fo"itio"
"i'h';;;a,ouot
or campaign
,"1ei;;-i.;;';*-iff iJil,:jT_i:ffi!_I",:: j,:T':3r.[: j..;i:::
political views lie t. tn....igrri-.i,ir.*ii*rr,rn8"i-"""iij.r.i"rra
a moderate
voter as one whose.ooritic"al ;;;;"ri."u.rr".., r#'frJr,,"ns of the two
candidates' Assume tr,"t .""r, ;;#;i:, voter contrit,i;;;. the candidate
whose position is closest to nis or frer^J1r, ,ri.r1" and thai moderate voter as
iHyX?"1.nX,'""*1' "i'*" ri' LJt*..',"i*,.
.positions*oi'ir,.'r*. candidates.
position r"
"1""""i,.
extremist voter contrlo"t." io ?"'-"..raraate whose
camplign"";ffi
1ii:iil:.;H;:.,-ii!:r,L1,nt"i***,t*:;i:
contributes to his o.
1....
t""".iiJ'""i1,f.i,.
=
p.opo.tio.,ut to rhe distance
between the two candidates. ;;.;";;.*" ."".rrr. ,-i; there is some
rd!ii,:,x:l,[::[iT,f :,iljii:ir:tf i:;
j:it#and,he,Eil:
the left-wing candidr
receivedoril".,*,,ill,i#1,"H,3:;r*;,"J.*j,l5f
Ti,i[f H;ffi fi fl lI
(a) The right-wing candidate is located at y, the contributi
position for the reft-win!
""Jil.i. ," ,?iJlTiTLl"s
If the teft-win!
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(b)
(c)
:,Hilffi :#,ffi * i: ? =tn.
contribution-maximizing position ror the
other candidate, occurs where x =
rs' glven tne posl
/+andy=3/4
Solve for the unique pair of ideorogical positions for the two
candidates such that eacr, tJ." iie" posrtion 'that
maximizes his
campaign contributions given ,fr" p""iti"" of the other._
suppose that in addition to collecting contributions from extremists
on their side, candidates can ,r"o
"o,"-"i
campaign contributions from
moderates whose views are croser to their position than to that of
iHt#3L ff :,'X"r1;*t':*ir' it "'",oa
"ate
s,,r.-J' e*i.e mi
"
t
",
proportion i" ii"'ain ...,". u.t,"#'tnffi'"fft J3;r"JffittJt:il:
from their less-preferred candidJ;.;il; that in ,iri-
"r?"
trre unique
positions in which tt. r"rt
""i ffi-*irrg candidates are each
fiffl*f#:,j:f :T*si.::l*u,i"#., *,-*"*,i#-o"li iJn or the
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9811343411
,,"_ .,+ I ti I t:liili]:llililiril lilillliilil:lililiriliililjEliilillilifli$Efi
I ffi I I l ll ll ll lllll.llllilllllfflll I
irlll
ilfillnllllliliffi
t9
!EE_r:
E-
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UNIT - II
GENERAL EQUILIBRIUIII, EFFICIENCY AND WELT.ARE
(Varian/Workoutf Chapter O I
EXCHANGE
1. Frank,s utility function is
;"i: ;i;*k'J. ffi:;x#ht'i;r r'
"'ii"o ", *f,Y?,,;
of good i. tet r" nri?Jo"ro.1r,"l1"wment is 20 units
9f
good_1 and s unrts
set p, =r and nnd Frank,s rro or"l1i"'iT'H:Jffi:*ufirl:'fr:,
,:';
function o' 0, ., y"t"q ,n" ,.n"rn"es learned in chapter 6, we find that
Frank's demand function a. *""01 i5 m/2p , , *rr"."-rJis his income. Since
Frank's initial endowment i.
"o
,rri," or ggod 1 and io ,nit. of good 2, its
;::f'r':l[''
rhererore p'""t[-i"*"n-d r, s;;2 i" np,t2p: = 5. since
consumer *h", "tt perfect complementl
,
for Maggie, she wflr choose to
impries,n", *"i*,li= i;",Ili,ix"".;#*?,ffi #T,g m:i :,*f
!lilTT'il#r:9 units or gooa r"*,a,1 uiiS
", *,"i 2, herincome is
demand ,tr" ,r,'t"
at price p" Maggie's.d^emand i"ir"#io, )/(1+pr). Frank,s
ll:.'",ii"",,,i*ffi ?::i:;f ,,:::.,#:*fJL,J,,,J,#]#,*
unrt endowment, which .aa. ,"'i'slnits. rrr"."r"r. itrir,a equals suppry
s*
(20+5p;
)=, .
(l+p, )
Solving this equation, one finds that the equilibrium price is p, = 2. At the
;:H.'ll'iT.X,#;,?llt-r*' 0."*"d 5 units
"i;;;; and Massie w,l
Y:fl" Zapp and phitip SwaIow consu:
ini tial en d owmen t or ob u o or.. ;#. i ; ff*[j :r]1"::;i:i,#;X ]1T,,ffi
fi iTIo"":,;1.100 i.""t"
;; ; ;;Iu, J"
.", *i,
".
-
ii.v'nl'#' .,o other as s et s
a,,d a uriii.*Ji,i
with anyone other trran eacholi";;;. Morris, a book
u(b,.*) i['*'*,"i"n*"t are perfect substitutes. il
"*,, function is
.,rr,0",
"iu",iilJl,ire.
b is the number of books_he.;;-r";;;"" and w is the
and
"onv.-*l;:'i;i :'t'"ff ;:1-'i*"),llnitt p;;;;#"'*. *o,. subue
Edgeworth to"i.ior", Morris,s
"or.,r*r,1r1y,3":rro",
yq, w) = uw. I" H;
and ph,ip's is measured r.;;,i.
"pp:,'i
r,:il J:ff ::H:x jTa.,r," r.*"i r"iir
(a)
3,i"fi'#:?,"olimuT, where both peopre consume some of each
"L
r!i.I^i
""f"".iffir"" are equal.
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(b)
No matter what he
equal to "'
::fi,T#Ti,l1;.::.f"ff1c,i.d rate orsubstitution is
fffi "- r*lererore
.u",, ;;;; 1-::
th" ;;;i; ?o'' *
"
l, n*
equation .- amounti of
--roPtimal allocation where both
,"1"1'"T!;;;*r-:i:,i"*1.,*"i",f; 3'1,,*fl?0i0";""ff i"j*11i
lij ":*P"titive equilibrium, it w1r have
lifl:,f:ffi j*ii."1::i**.,".1nr,,;##iiry:::,:"#::ffi :
pnce or rrri^o
that ir *;;J. il'"fl:'["::?*5^i: - - -.-.-.--
3.
(d) At the cor,Defirirra ^_,,,,
-.'" qlrq
bottles of wine.
vvur
income 's
npetitive equilibrium
consuming*,,--; jg:tri{i}}'#,J."":1;:"iabove,Morris's
consume i
income. o,".Jln. "r-;-;": ;11^"1, "r
[" *"Jff:";J,?,,Y:T::#
bette-n.rr'.ln"l" pti"t",*l'.r":o:e than' less than) ;-; '
(e) suppose ,n
tn. o",iJr"G, it]'r'"tt" afford
' o""ar"- tr,"t rr" ri#i
#ji,XlH,","Tr:+:,!J";#ifi "*1,;?1?",il1f ,t,::_,ru:Mo*is
types had ,i:"lt and the ."-" ir"ti";."ir:;j'Tlis tvpes has the
::lh*Ti{ifl l*r;l[:#tr*::'i"",?*{xhff*"".'tfl:#
Fil,1,J'!lo"lll: or"r';'ri'" u"n=----.-
Ir each or
booki?_.*-oo,o.,*J,iJtJr:#,[1fi:ill["#tl#r;,T:,ri#
Consider a small
";.-=--
#"dlffi ,ti,$TI*l[#;$;],,:,,i$,#,f,,, ;nr,','i:"?
Birger r"
"
"'*"**::"""^_-1,:jrid's utitit/r""",i"rf""-'.o*-""' has no
,15;
l*iruI:,"i il,# "fr
l'
"." f
ili,= il:
"*ffi
:,''i lft
^
",,;;
Birger.)
- "vlrrq' ond H, and cu are amounts of herring and cheese for
price of ,"i.r"'
*.1" , we rlok€ b""u"^^11:.,""*"..ir"1trr*lir.
ts competitive equilibrium must be
(c) At the equilibr.
yrr a1 i s
-tii';Hi,:1I;fi
n f#",?#S, 11, ll. ras t p art o :
At these price
and . .-S,
"*1r^:ll "!"""".#;:,fl""J*"$::or
the question,
ffi,"._d-rrr
"rbottres
"r
r-". ir il#'i"}ffi ;^ffi,*H
consume--
*" wr tn€a::?: rhat PhitiR a".,i{t.9"".;;"",'he wi,
Let cheese be the nr
of-herring. ;r;;'J'-eraire (with
wi, d emaled ;; ;il::, ;T-?J::",""
r"'
Price 1) and let p de
ttr" u-orr.,i
"in""il""te
the price
rg that Birger
(a)
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Where.the price of cheese is 1 and p is
the value
"f a.tria
" initial endowmentthe
price of herring, whar :s
herring wilt Astrid demand ., p.i"" pT"'rl - ' How much
,I;"fi ffi "1*
.'tfl11::*';',Ti,";'. 1',,r Nigh ts oil exchan ge prati tu d e s
bromides, his utility is given by
)onsumes rr platitules ;d ;"
U,(B,,I )=8, +2.f
When professor Nightsoil consumes
utility is given by
u* (B*, T* )= B* + +./r;
Dean Interface's initiar endowment i"
,
r? oratitudes and g bromides.
Professor Nightso,'s initial ;;J;;;, ,r"" o or",r,roes and a iiomrdes.
(a) If Dean Interface consumes rr platitudes and B, bromides, his
marginal rate of substitution ,ili #
--"**'- -,,1j,
^
:fiil[:t*,Pratitudes ; ;. ;""'"* r[';:T:;,.:*":1
On the contract curve, Dean Interface,s mnroi-ar _^+^ _r
equals Professor Nigh;;il'," "'#;ff"r" marginal rate of substitution
condition. _
-----;;,"'
^:1:'_:^ i". equation that states this
","r,' i",;";;*r" ili: :?"ffi1,,1:.,.:o":,dry .i*pi" because
consumptionorpradtud.";;;;;,;;il,J::,::"*f,f,iirily_,,ot.l,J
I:x.:i':#::Hl::.::: :n"t t/f,n=- at al, points on the
and .
s glves us one equation in the two unknowns r,
But we arso know that arong the contract curve it must be that
equal the total endowment of platitudes.
Solving these two equations in two unknowns, we find that
ever5rwhere on the c<
to __________._ and
rntract curve, T, and T,
"r"
constant and equal
(b)
4.
T11 platitudes and BN bromide, his
(b)
(c)
(d)
(e)
(0
(s)
F the Edgeworth box, label the initial
D^ean Inteila"" rr..1ti"t srA/ hFh^ir^, ,_ -l**-ent with the letter E.
"',"ffi
::TT""B;;,*:T,f, .fl"i':::?}jrfi,Er"ti:#i:B:,]*fl*
i::1":1T#""Sm*11,,"#,'#t!i";::Ti#ffi ';"',:i:f n:
lr^:y_. :r,pareto d,t;;1;#J"inTT" use blue ink to show the
horizontal, aiagonai) _ lina i- -'l^'T:n curvg
,rs a (verticall
horrzontar, diagon aLjf
- --'*' v "" ""' rr'r?"rr"i#TiH:ffi ;:":
;;;o,,."1"*, u,"*
what the prices have to hc or ^^*_^.,-,
what the prices h.r-
-=*"'urru,r pnces and quantities. You know
e to be at competitive equilibrir* U""".r"e you
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know what the
Pareto optimum
marginal rates of substitution have to be at every
5. A little exchange economy has just two consumers, named Ken and Barbie,
and two commodities, quiche ,rra *i.r". Ken s initii .ra"*-"rt is 4 units of
quiche and 2 units of wine. Barbie sirritiat .rrao*-.rrt-i" i ,rrrrt of quiche
and 6 units of wine. Ken ,"a e".bi.- i"* td";;i;;iiity rrr"tions. we
write Ken's utitity function * ,(a;,;;)= e.*" and Barbi,s utility function
": ,(a,,w,)=e"w", where q. I"i** *" the amounts of quiche and
wine for Ken and e
"
md w, ,r. amounts of quiche and wine for Barbie.
(a) At arry-Pareto optimal a,ocation where both consume some of each
good, Ken's
lnarginal rate of
"rt"Utrtio, between quiche and wine
i:H::?t1i:3:!1:::I1* ?" equation that states this condition in
terms of the consumptio"" or
"""i-goo"J;ffi ;;.? j;
on your graph, show the locus of points that are pareto efficienr
equilibrium, pq/Fw =
(d) If competitive
(b)
(c) In this example, at any pareto efficient arlocation, where both persons
consume both goods, the slope of Ken,s indifference curve will be
:::]:::*", T",:,
o" pareto .fficient,'r". k,o* that at compe,tive
6. Linus Straight,s utility function is U(a, b) = a + 2b, where a is his
consumption of apples and b is his.consumption*of bananas. Lucy Kink,s
utitity tunction i: yl.:. b) = min{a, Zt}. L""V i"ii*rf, has rZ apples and no
bananas' Linus- initially'rr." iz'u*anas^and no apples. In the Edgeworth
box below, cooqs f91 LLcy .r-L.."r."{ from the upper right corner of the
box and goods for Linus *" -L.""red from trr. rt*ii reft corner. Laber the
initial endowment point on tt. grapt with the lett"i E- o."* two of Lucy,s
indifference curves in red ink aiJ two of Linus,s indifference curve in brue
fir1"y,li,_lrack
ink to draw
"-rir," tr,.ougr, ;; the pareto optimal
(a) In this economy, in competitive equ,ibrium, the ratio of the price of
apples to the price of banlnas must be
_equilibrium, Ken,s
,consumption bundle must be
How about Barbie,s
"orr""*ptio., bundle?
Let as be Linus,s consumption of apples and let bs be his
consumption of bananas. At competitive equilibrium] ,rrr.r.,"
consumption will have to satis$r the budget constraint, as +=_---
br=-. This gives us one equation in two unknowns. To find a
second equation, consider Lucy's consumption. In competitive
equilibrium, total consumptio" oi
"ppt"" "qr.l" the total supply of
apples and total
"on".rmplion of banlnas equal the total suppty of
(b)
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bananas. Therefore Lucy will consume l2_a" apples and __=.--
-b' brnanas' At a competitive equilibrium, Lucy will be consuming at
one of her kind points. The kinks occur at bundles where Lucy
consume
"rPl.,a^
for every banana that she consumes.
Therefore we know that 'z - 3,
=
12-b"
7.
(c) You can solve the two equations that you found above to find the
quantities of apples .t a bananas consumed in competitive
equilibrium by Linus and Lucy. iirr. *iff
"o.r"r*J .=.-..---.- units
of apptes and -.------
""it" or b;;r;;".T;* *n consume
units of apples and 3 units of U*.rr"l*",
consider a pure exch-ange economy with two consumers and two goods. At
some given pareto efficient alrocation iii. t rro*., ilr.i-trirr'"onsumers are
consuming both good-s and that lor"r-", A has a marginal rate of
substitution between the two goods
-ii--2,
wrrat is co.r:,r;;. B,s marginat
rate of substitution between th;r;;;;oods? _
8'
:,1"1':*r
roves apples and hates bananas. Her utility function is
u(a.b)= u-;0" where a is the number of apples she consumes and b is the
number of bananas she consumes. w,bur likes both appres and bananas.
His utility function is u(a,b)=a +znli. charrotte has an initial endowment of
ffi#"*::."nd
8 bananas. wilbur has an initiar endowment of 16 apples and.
If charl0tte hates bananas and w,bur rikes them, how many bananas
can charrotte be consuming at *- p"r.lo
--;;il;*rlocation?
(a)
(b)
S:,*il":^il"rr^"."^f::itive
equitibrium. ailocation must be pareto
:X,'f;fl #*ii'"::ii":::."::::";E;;:T#*ruT#':T:I
XLll,I; i"we
kn ow that at ; ;_o. ;u*;ff ,ffi ,ffi:.#fffl H: ii#
;:ffi I, T, ;;;;, _ * ", *41?:,li;",#,il#,, ft Ll_.?,Xl#*" "l,l!
number * o,Tr"??:;:::,-f#' y,ii,,rl"i"."bffH:I,?,ir,Ti
11,hi" marginat ,rlitty ,f
"ppf". *iff t
r__--"*11'l::iii::#::*:il1t?-;i,;,i,,?Ti*i,_.".,_*;
l:J,^,I, :.1r1,o., .:
ol
" "-. ;;iy'tu";'jd;:: ::
competitive equilibrium, i"r',l"tcrr"rr-o"o"o"
rS -.-=.--.--.--. In
will consume
rrrv vrrarruLl
1- bananas and
:" apples and
bananas and
Charlotte will consume
apples.
9. Mutt and Jeff have t .gro" of m,k and g cups of juice to divide between
themsetves. Each has th":.r+.";,iri,i'ii"",i"r, gir.ri ty rtir,:t = max{m, i},
;,ffi:J##""r.H.Hr":::,LFjt"",n. amount of juice that each has.
rhat is,
"""h ;i;;;:;;#;#':d#,'ff.T:;X, ;t iil:illffiil,l*;
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10.
liquid that he has and is indifferent to the liquid of which he has the smaller
amount.
This problem combines equilibrium analysis with some of the things you
learned in the chapter on intertemporal choice. It concerns the economics of
saving and the life cycle on an imaginary planet where life is short and
simple. In advanced courses in macioeconomics, you wourd study more-
complicated versions of this model that build in mtre earthly realism. For
the present, this simple moder gives you a good idea of how the analysis
must go.
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J 1.
(Varian/Workoutf Chapter 83
WELFARE
A social planner has decided that she wants to ailocate income between 2
people so as to maximize ff.aE where y, is the amount of income that
person i gets. supqgs: that the planner has a fixed amount such that
Y, + Y, = w , where w is some fixed amount. This planner would have
ordinar5z convex indifference curves between y, and y, and a ,,budget
constraint" where the "price" of income for each person rs r. rnerefore the
planner would set her marginal rate of substitution between income for the
two people equal to the rerative price which is 1. when yo,
"or.r.
this, you
will find that she sets y, = y, = w2 . Suppose instead that it is *more
expensive" for the pranner to give money to person 1 than to person 2.
(Perhaps person 1 is forgetful and loses money, or perhaps person 1 is
frequently robbed.) For example, suppose that the planner,s budget is
2Y, + y, = ry Then the planner maximizes fi*afl subject to
2Y , + Y. = w . Setting her MRS equar to the price ratio, we Iind that E =, .
JY,
So y, = 4y, . Therefore the planner makes yr = W5 and y, = 4Wl5
one possible method of determining a sociar preference relation is the Borda
count, also know as rank-order voting. Each voter is asked to rank an ofthe
alternatives' If there are 10 arternati'ies, you give your first choice a 1, your
se-cond choice a2, and so on. The voters's"or.. roi each arternative are than
added over all voters. The total score for an alternative is caled its Borda
count' For any two alternatives, x and y, if the Borda count of x is smaller
than or the same as the Borda count io, y, then x i" ;*"irly at least as
good as" y. Suppose that there are a finite number of alternatives to choose
from and that every individuar has complete, reflexive, ana transitive
preferences. For the time being, let us arso suppo"" tt"i individuals are
never indifferent between any iwo different alternatives but arways prefer
one to the other.
2.
(a)
(b)
(c)
5 lh. social preference ordering defined in this way complete?
Reflexive? Transitiv&
If everyone prefers x to y wilr the Borda count rank x as socia,y
preferred to y? Explain your answer.
Suppose that there are two voters and three candidates, x, y and, z.
Suppose that Voter 1 rank the candidates, x first, z second, and y
third. suppose that voter 2 ranks the candidates. y first, x second,
and z third. What is the Borda count for x? For y? ' . For
?.1
-
'Now suppose that it is discoverest that candidate z once
lifted a-beagle by the ears. voter 1 who has rather rrrg.-."." rrimself,
is appalled and changes his ranking to x first, y
"""orrf,,,
irrira. vot..
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2, who picks up his.own children by the ears, is favorably impressed
and changes his rankinE to y frst, i
"."o.ra,
x third. Now what is the
Borda count for x?_ Fo, yZ fii,
Does the social preference reration defined by the Borda count have
the property that social preferences u"i*.." x and y depend only on
how people rank x versus y and not on how they rank other
alternatives? Explain.
3' Suppose the utility possibility frontier for two individuals is given by
uA + 2u B = 200 . on the graph below, prot the utility r-.r,r.r.
(a) In order to maximize a "Nietzschean sociar welfare function,,
w(uo,u,)=*u*{uo,u"}, o, the utility pr*inilitv frontier
"trorr.,
above, one would set U A equals and U, equal to
Rawlsian criterion, W (U o, U
"
)=
function is maximized on
where U A equals
Suppose that socia,l wetfare is given uv (Uo,Uu)=UlrUil, In this
fjl*Il,l*ll^"- T:* utitity possibility frontier, social welfare is
maximized where Uo eqrals"_'_
"".i r, equals
A parent has two ch,dren named A and B and she loves both of them
equally. She has a total of $1,000 to give to them.
The parent's utility function is U(a,b)=..f-1Jb, where a is
amount of money she gives to A and b is the amount of money
gives to B. How will shelhoose to ai"ia. the monev?
Suppose that her utility function is U(a,b)=_!_|. rfo* witl she
(d)
4.
(b)
(c)
(a)
(b)
If instead we use a
the social welfare
possibility frontier
choose to divide the money?
min {uo,u"}, then
the above utility
and U B equals
the
she
ab
(c)
(d)
(e)
(0
Suppose that her utility function is U(a, b) = log a + log b. How will
she
Suppose that her utility function is U(a, b) = min {a, b}. How will
Suppose that her utility function is U(a, b) = max{a,b;. How will she
Suppose that her utility function is (u,b)=ur*b2. Ho* will she
choose to divide the money between her.children? Explain why she
doesn't set her marginal rate of
""["tit..tio, equar to 1 in this case.
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Ip th1 previous problem, suppose that A is a much more efficient shopper
than B so that A is able to get twice as much consumption gooas as B can
for every dollar that he spends. Let a be the amount oi consumption goods
that A gets and b the amount that B gets. we wilr measure consumption
goods so that one unit of consumption goods costs $1 for A and $2 for B.
Thus the parent's budget constraint is a i 2b = 1,000.
If the mohter's utility_-function U(a, b) = a + b, which child will get
more money?- which child will consume more goods? -=-'-.
If the mother's utility function is U(a, b) = a x b, which child will get
(a)
(b)
(c)
more money?
If the mother's utility function is
more money?
more?
Suppose that a
spaghetti exactly
Which child will get to consume mor"l
U(a, b) =
Which
I - I , which child will get
ab
child will get to consume
6.
(d) If the mother's utility function U(a, b) = max{a, b}, which child will get
more money?_ which child will get to consume more?
(e) IF the mother's utility function is U(a, b) = min{a, b}, which child will
get more money? Which child will get to consume more?
Romeo loves Juliet and Juriet roves Romeo. Besides love, they consume only
one good, spaghetti. Romeo likes spaghetti, but he also liies Juliet to be
happy and he knows that spaghetti mikes her happy. Juliet likes spaghetti,
but she also likes Romeo to bi trappy and she kn'#s it ri.p"gr,"tti makes
Romeo happy. Romeo's utility function is U*(S*,Sr)=SiS]-" aNn Juhet,s
utility function i. Ur(Sr,S^)=SiSl*, where S, and S^ are the amount of
spaghetti for Romeo and the amount of spaghetti for Juliet respectively.
There is a total of 24 units of spaghetti to be divided between Romeo and
Juliet.
(a)
(b)
(c)
(d)
= 2/3. If Romeo got to allocate the 24
as he wanted to, how much would
units of
he give
If Juliet got to allocate the spaghetti exactly as she wanted to, how
much would she take for herielp _ How much would she give
Romeo?
himselP How much would he give Juliet?
What are the pareto optimal allocation?
When we had to allocate two goods between two people, we drew an
Edgeworth box with indifference curves in it. when we have just one
good to allocate between two people, all we need is an ,,Edgeworth
line" and instead of indifferen".
"rr*.", we wil just have indifference
dots. consider the Edgeworth rine below. Let the distance from reft to
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(e)
(0
right denote spaghetti for Romeo and the distance from right to left
denote spaghetti for Juliet.
On the Edgeworth line you drew above, show Romeo,s favorite point
and Juliet's favorite point.
Suppose that a = 113. If Romeo got to allocate the spaghetti, how
much would he choose for himselF If Juliet got"to allocate
the spaghetti, how much would she- choose for herselp
Label the Edgeworth tine betow, showing th.;;-;;;;i.b f.;.it.
points and the locus of pareto optimal points.
(g)
Y!r" a = l/3 at the_ pareto optimal allocations what do Romeo and
Juliet disagree about?
Hatfield and Mccoy hate each other but love corn whiskey. Because they
hate for each other to be happy, each wants the other to haive less whiskey.
Hatfield's utility function is Ur(Wr,Wr)=Wr-U(, and McCoy,s utility
function is Ur(Wr,Wr)=Wr-ffi, where wM is McCoy,s daily whiskey
consumption and WH is Hatfield,s daily whiskey
"orsrmption (both
measured in quarts). There are 4 quarts of whiskey to be allocated.
If Mccoy got to alrocate arl of the whiskey, how wourd he anocate it?
^,,_*^Il.Iatfield
got to a-llocate a_ll of ihe whiskey, now wouta fre
7.
(a)
(b)
(c)
allocate it?
If each oJ them gets 2 qrrarts of whiskey, what will the utility of each
of them be? _---.-- If a bear spilled 2 q";.t" ;;,rrrrl'rri;r.ey and
they divided the remaining 2 quarts equany between them, what
would the utility of each of th"m be?
-
If it is possible to
throw away some of
.the
whiskey, is it pareto optimar for them each to
consume 2 quarts of whiskey?
If it is possible to throw away some whiskey and they must consume
equal amounts of whiskey, how much should they throw away?
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Unit-III
Market Failure
(Varian/Workoutf CHAPTER g4
EXTERNALITIES
A large factory pumps its waste into a nearby lake. The lake is also used for
recreation by 1,000 people. Let X be the amount of waste that the firm
pumps into the lake. Let y, be the number of hours per day that person i
spends swimming and boating in the lake and let C, be the number of
dollars that person i spends on consumption goods. If the firm pumps X
units of waste into the lake, its profits will be 1200x - 100x , . consumers
have identical utility functions, f{X,q,X)=C, +qY-Y -XY, and identical
incomes. suppose that there are no restrictions on pumping waste into the
lake and there is no charge to consumers for using the lake. Also, suppose
that the factory and the consumers make their decisions independently. The
factory will maximize its profits with respect to X equal to zero.) when k = 6,
each consumer maximizes utility by choosing yi=1.5. (Set the derivative of
utility with respect to Yi equal to zero.) Notice from the utility functions that
when each person is spending 1.5 hours a day in the lake, she will be willing
to pay 1.5 dollars to reduce X by 1 unit. Since there are 1,000 pecjple, the
total amount that people will be willing to pay to reduce the amount of waste
by 1 unit is $1000 people, the total amount that people witt be willing to pay
to reduce the amount of waste by l unit is $1,500. If the amount of wasti is
reduced f4rom 6 to 5 units, the factory's profits will fall from $3600 to
$ssoo. Evidently the consumers could afford to bribe the factory to reduce
its waste production by 1 unit.
The picturesque village of Horsehead, Massachusetts, lies on a bay that is
inhabited by the delectable crustacean, homarus Americans, also known as
the lobster. The town council of Horsehead. issues permits to trap lobsters
and is tryrng to determine how many permits to issue. The economics of the
situation is this:
C
(i)
(ii)
It costs $ZOOO dollars a month to operate a 1obster boat.
If there are x boats operating in Horsehead Bay, the total revenue
from the lobster catch per month witl be f(x)=5169(10-*')
If the permits are free of charge, how many boats will trap lobsters in
Horsehead, Massachusetts?
What number of boats maximizes total profits?
If Horsehead, Massachusetts, wants to restrict the number of boats to
the number that maximizes total profits, how much should it charge
per month for a lobstering permit? _
(a)
(b)
(c)
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Suppose ttrat a honey farm is located next to an apple orchard and each acts
as a competitive firm. Let the ;;;;';;nor,"v p.ii;;;;:,;.asured by H.
The cost functions of the two firms are cr(H)=Hrll6c and
co(d) = 4 11 00-H. The price of honey is $2 and the price of apptes is g3.
(a)
i::B fftr":ff'#ierate
independently,. the equitibrium amount of
Suppose that the holev and apple firms merged. what would be the
profit-maximizins orrtput of frorif f*'it"
"o*Uired firm?
What would be tf,e p."nt__""i*ii.rj.*orr, of
"ppl."?__.
What is the sociallv efficient output of honey?
firms stayed sepa.aie. rro* -r"n.,rlo,ria rror.v p."d*ti", hr*i:ri:
subsidized to induce an efficieni;";;;;
(b)
(c)
4.
it#,:.T:Xi":::';11'"rnia,
popuration1,001, there is not much to do except
y:*:r"*,",,fi i:T:,il*xril;,S"g.i*Iffi #ii#i;ttT*JI"",,"T;
nolse' and porlution caused uy trum'". a typicai .."ia.r,tlirtility function is
(md,h)=rn +r6d-d2 -eitioi,'^;;.;" m is the resident,s daily
;:i[,::TX]T,:',?F [?ii ;l,li,,,i; ji#ff: _or
h ours p er day that he, him s err,
day) done uy att oirrer residenrs o1B1 6r.i1F-lmeasured
in person-hours per
t.:h .
Every-person i, et C;;;;#
vqr'buretor' The price of Big rvl."* ir'$-i
ca-lcutatioris
"i-pr.,'"rppose it
"o*,.
,,oiuX! ilTfr: f;-#
p"' J'y. r" L.p
(a) If- an individual believes that the
:ff :: JT
.",
""" ; il;; ;n;:. J,T" :il# T ;i, i#:S #: j:nf ; :
(b)
lH:fff*;*."."""*S ""
d, then what is the total amount h or
(c)
(d)
(e)
(0
What will be the utility of each resident?
[':::7:"".?J.:l'ffif *tT,iiir' *n',*il * ut,ity lever or a
Suppose that the *:ld:lp decided. a o.."r * restricting the total
number of hours that anyone is attowJJ to arwe.
^""#
,,"1, driving
,T."tll #i,*.lffi
b e,atl owe'J ;i; ; ;#iX.,
"
t o * u,,r
-L.'iill' u t,rity or
The same objective
,could be achieved yth , tax on driving. How
much would the tax rru". io U.;Ir.liffi of driving?
5. Tom and Jerry
."*"i#",,"1H,.T,il1T*ltii;"11:1,"p:ig, a totar or 80 hours a week
;",i#il"':ljl,T i.?T: #[ :,lT^
jtj. :: *;, :.:; *r,; ;::ffi "
E::
;ji:T#;::lJ.'n!:.#)":,::y*"."i,'f ,ilfffit"::ii::fr ;"i':
;:;#"?:ffii:
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played while he is in their room. Jeery hates all kinds of music. His utility
function is U(cr,u)=cl-Mzll2. Every week, Tom and Jerry each get two
doz92 chocolate chip cookies sent from home. They have no other source of
cookies. we can describe this situation with a box that looks like an
Edgeworth box. The box has cookies on the horizontaf axis and hours of
music on the vertical axis. Since cookies are private goods, the number of
cookies that Tom consumes per week plus the rrumbeittrat Jerry consumes
per week must equal 48. But music in their room is a public good. Each
must consume the same number of hours of musi, wheiher he rikes it or
not. In the box let the height of a point represent the total number of hours
o_f music played in their room per week. Let the distance of the point from
the left side of the box be "cookies for Tom,, and the d.istance of the point
from the right side of the box be ,.cookies
for Jerry.,,
(a) suppose lternatively, that the dorm's policy is ,,rock-n-roll
is good for
the soul." You don't need your .o-*"t.'"-permission to play music.
Then the initial endowment is one in which rom plays music'for a1l of
the 80 hours per week that they are I the roor,, to!.trrer and where
each consumes 2 dozen cookies per week. Mark this endowment point
in the box above and label it B. Use red ink to sketch the indifference
curve for Tom that passes through this point, and use brue ink to
sketch the indifference curye for Jirry thai passes through this point.
Given the available resources, can both rom and .lerry bJ made better
6. A clothing store-_and a jewelry store are rocated side by side in a small
shopping mall. The number of customers who come to ihe shopping marl
intending to shop at either store depends on the amount or money tr[t tne
store spends on advertising per day. Each store also attracts some
customers who came to shop at the neighboring store. If the clothing store
spends $a per day on advertising, and the jeweler spends $x, on
advertising per day, then the total profits per day of the clothing store are
lL(r",*,)=(OO+xr)x" -24,and the total profits per day of the jewetry store
.r" fl"(x.,xr)=(t0s+x r)x,-2x1. (In each case, these are profits-net of all
costs, including advertising.)
off than they are at point B>
jeweler are
-
(a) If each store believes that the other store's amount of advertising is
independent of its own advertising expenditure, then we can find the
equilibrium amount
-
of advertising ior each store by solving tow
equations in two unknowns. one of these equations says trrit ttre
derivative of the clothing store's profits wit-h respect io its own
advertising is zero. The other equation requires that the derivative of
the jeweller's profits with respect to its own ad.vertising is zero. These
two equations are written as _ and _. The
equilibrium amounts of rd,r.rU"irg *" _
=-;d
----_.-..-
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(b) The extra prolit that the jewerer wourd get from an extra dorar,s worth
of advertising by the. ciothing .to." is app.oximately equal to the
derivative of the jeweller's p.#t" *iit-i."p""t to the
"totti.rg
store,s
advertising expenditure. Wf,"r, tfr.
-t*o
stores are Jomg the
equilibrium amount of
.
advertising that you calculated above, a
dollar's worth of advertising by tri.
"iotrri.rg
store *""ia give the
of adve-rtising by the jeweler *;"ld gi* the clothing store an extra
profit of about
suppose that the owner of the crothing store knows the profit
functions of both stores. She reasons to hlrserf ." io,o*".^suppose
that I can decide how much. advertising 1 will do before ihe i.*.re,
decides what he is qoing to do. Wt.n J-tett him what I am aoing, he
will have to adjust- hiJ behavior .cco.airrgry. I can ca-rcurate his
(c)
reaction function to my choice of t, by setting the
p.olt" with respect to his own advertising equal to
for his amount of advertisi"g
"" ;-i;"tion ot my
When I do this, I find that x.,
derivative of his
zero and solving
own advertising.
substitute this value of X; into my profit function and. then choose
IfI
tu
and he wili choose x, : In this
case my profit will be and his profits wilt be
::1lT:-:1,1,-ln:"l"llig store and the jewetry store have the same
H:T*i,"-,i:":.":,1:lTe"butareo.Jne;d;,"*ilffi #1l"Jff ;#:
li:":f "-:::: -
rtrI"-'li
"1"* j". ". i"-"-#r;#'i;":I'J'Ti ;f." J:
stores' profits. The single firm would choose
(d)
and
7.
;h . rh.. r^ ;, y-1l:^t -,:,*uJltins
ac tual p ;J;, ;;i v.il ;".##
Il: 13* *:,r.,yi: l," r11.o.
r,is-h;, 1"o;, ;.,ii, ;, _ ;;: x ffi1.ffil:
woutd be when,n.J^..."r*.:
:ffi
',g;,:,""""",1i#L1oX""#i
How much would the a,.f p."ii," "i"
(a) Calculate his utility level
The cottagers on the shores of Lake Invidious are an unsavory bunch. There
are 100 0f them, and they live I a circle around the lake. Each cottager has
two neighbors' one on- his right anJ one on his left. There is only one
commodity, and they a]r consute it on their front r;rr; i" zurr view of their
tow neighbors. Each cottager rikes to consume the commodity but is very
envious of consumption bylhe neigfrtoi on his left. Cr.ior"tv, nobody cares
what the neighbor on his ,igtt r" aii"g. I"-f*; ;;;;".ffier has a utility
function lJ(c,l)=g-12 , where c is l .
consumption by his neighbor on the ,"r,.
t'" own consumption and / is
:Xfl:::Jn:each
consumer owns 1 unit of the consumption good
Bliss Point Studies
ott-4507622r
33 Ravindra N Jha
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(c)
(d)
(b) Suppose that each consumer consumes only 3A of a unlt. .-.
individuals be better off or worse ofp
what is^the best possible consumption if all are to consume the
amount?
Suppose that everybod.y around. the lake is consuming 1 urui
il;I-_jy?..,f:"1]: *rk: themselves both better o?r- .itrr..
redistributing
"on"rmption betwe., tfr.rri o. #ffi#i,, ;:H:;
away? 'ur6 'u
(e)
(0
How about a group of three people?
How large is the smallest group that courd cooperate to benefit a* ::
members?
suppose that Jim has a right to half their joint income and Tarr.:-:
has a rioht t^ fh- ^+L^* L^1. ^
*:"_l :,*,,_,.^ ^,1: 1,I_:l l ". 9 "nn, " "'
i";,h J','#'ii, #' l" lil .-,
bargains with each other about rr"ii-""ii""iir,# r'
"'rla"r','iff: "
_
COnSf lme Ffnrrr *"^l^ ^c ^ ^L:'':L
consume. How much of activity x *iii .li_';ti;"l;
"#L,=;
How much of activity y -iff iammy consu:i::
Therefore we can write
cr *% = I 000,00020x-1 (-rr,'
Because Jim and T^amm.f have quasilinear utility functions, ti:-:
utility possibilitv frontier inctua.s a straight line segme:-:
Furthermore, this segment can be found by maximizing the sur: :.
their utilities. Notice that
know from --A;-' r":.1:
8' Jim and rammy are partners in Business and I Life. As is an too commc:- -
this imperfect worrd, each has a little habit thal
"rrioy.
the other. .I.* :
habit, we will, call-activity X, and Tammy,s habit, ."tirrity y. Let x be --:_:
amount of activity X that Jim pursues .na y be the amount of activity y ---- --
Tammy pursues. Due to .
".ii"" of unfort."un"t" i.d""s, Jim and Ta-r:__
have a total of only$1,000,000 a year to spend. Jim,s utility functior- _:.
{4 =c, +500lrx-10y, where c, is the money he spends per year on go:,:i
other than his habit, x is the number of units of activity X that he consu...:-.
per year. Tammy,s utility function i" U, = c, +500 lny_-10x, where c, is --::
amount of money she spends on goods other than
""ii,orty
y, y is the nur::r:
of units of activity y trrat
"h" "Jr.r-es, and x is the number of unirs :
.
fi:tfY.XJl,T
Jim consumes. Activitv x
"".t. sid p.'
""i,. ectivity i cc.:u
(a)
(b)
Let us now choose x and y so as to maximiz" Ur(q,*,y)+Ur(a_,a.:
setting the partiar derivatives with respect to x and y equal to zero. ;:
find the maximum where x = ' o-.t ,,:
If we plug these
""-b.^
*"*irlto
th.
"q"...="
Bliss Point Studies
orl-45076221
34 Ravindra N lha
9811343411
=j{-

U, (cr, x,y) +Ur(cr, x,y) = 1,000,0
the utility possibility
find that
U, +U. =
+5001n x+5001n y_30x _110y, we
frontier is described by the equation
9. An airport is rocated next to a large tract of land owned by a housing
developer. The developer would rlt.-&"urila nouses t., irri" r..ra, but noise
from the airport reduces the value orin. land. The more pranes that fly, the
lower is the amount of profits trrritl. i"""loper makes. i-et x be the number
of planes that fly per day a"a r.l- v Le trr. number or t or"". that the
developer build. The airp^oit's total pron," are 4gX - X 2
, and the deveroper,s
total profits are 60y -y2 -xy. Leius consider the outcome under various
assumptions about institutional rules and about bargaining between the
airport and the developer.
"Free to Choose wln fV.o Bargaining,,: Suppose that o bargains can be
struck between the airport-ana ihe-developer and that each can
decide on its own, revel tr activiiy.- w:o matter how many houses the
developer bu,ds, the number
"r
pi."." per day that maximizes profits
for the airport i" . . Given that the airport is landing
this number of planes, the numL". of hor"es that maximizes the
developer's profits is _.._.-._.---io,, profits of the airport w,r be
sum of their profits will be
"Strict Prohibition": suppose that a l0car ordinance makes it illegal to
land ptanes at the airptit u."*". tt J, i*po"" r.,
";r;.rr;ity on the
developer. Then no planeg *ill fl;.-Th;'der.lope. will buitd
houses and will have total profits"of _]--
"Lawyer's paradise":
-
Suppose that a law is passed that makes the
airport liabre for ar1 damales to irr" a"ulrop"rt property varues. Since
the developer,s profits .." oOy _rr_XV and his profits would be
60Y -y, if no planes were flown, the total amount of damages
awarded to the developer win be xy. itrereror. ii
-,-r*
.rrf,ort fires X
planes and the developer buifJ V-tor."", then the airport,s profits
after it has paid damages wilt be ori _x,_Xy. T;;;;""loper,s
profits incruding the amount he receives in payment otaamages will
be 60y-yr-Xy+Xy=60y_y2. To maximize his net profits, the
developer will choose to build
pranes"."r[;;];*"#H,]";o,**,X""l.ff ;;#:::.,1".*",ffi
will choose to land planes. Total profit" ,ittE Ji""loper will
be ana totaf pr"nt" of ,frl airport will be
The sum of their profits will be
(a)
(b)
(c)
10. This.problem continues the story of the
previous problem.
airport and the developer from the
Bliss Point Studies
oLt-45076221
35 Ravindra N Jha
9811343411
I :lilll:]:]illlilllliliilll:lil:liill'rril:ilili
-
-
-
:
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(a) "Merger" : Suppose that
What is the profit
the housing developer purchases the airport.
function for the new joint entity?
11.
To maximize joint profits, it should build
-
houses and let
planed land. Combined profit is now
Explain why each of the institutional rules proposed in the previous
problem fails to achieve an efficient outcome and hence has lower
combined prolits.
(b) "Dealing":' Suppose that the airport and the developer remain
independent. If the original situation was one of "free to choose",
could the developer increase his net profits by birbing the airport to
cut back one fight per day if the developer has to pay for all of the
airport's lost profits? . The developer decides to get the
airport to reduce its fights by paying for all lost profits coming from
the reduction of flights. To maximize his own net profits, how many
Iightsperdayshouldhepaytheairporttoe1iminate?-.
Every morning 6000 commuters must travel from East Potato to West
Potato. Commuters all try to minimize the time it takes to get to work. There
are two ways to make the trip. One way is to drive straight across town,
through the heart of Middle Potato. The other way is to take the Beltline
Freeway that circles the Potatoes. The Beltline Freeway is entirely
uncongested, but the drive is roundabout and it takes 45 minutes to get
from East Potato to West Potato by this means. The road through Middle
Potato is much shorter, and if it were un-congested, it would take only 230
minutes to travel from East Potato to West Potato by this means. But this
road can get congested. In fact, if the number of commuters who use this
road is N, then the number of minutes that it takes to drive from East Potato
to West Potato through Middle Potato is 20 + N/ 100.
Assuming that no tolls are charged for using either road, in
equilibrium how many commuters will use the road through Middle
Potato? What will be the total number of person-minutes
per day spent by commuters travelling from East Potato to West
Potato?
Suppose that a social planner controlled access to the road through
Middle Potato and set the number of persons permitted to travel this
way so as to minimize tlne total number of person-minutes per day
spent by commuters travelling from Potato to West Potato. Write an
expression for the total number of person-minutes per day spent by
commuters travelling from East Potato to West Potato as a function of
the number N of commuters permitted to travel on the Middle Potato
road. How many commuters per
(a)
(b)
day would the social
Potato?
planner allow to use to road through Middle
In this case, how long would it take commuters
Ravindra N Jha
9811343411
36
Bliss Point Studies
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llllillililllilll
lilllilli
ll:lillill:ll:ll

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