ec411 – microeconomicspersonal.lse.ac.uk/nava/materials/ec411 handout.pdf · textbooks the course...
TRANSCRIPT
EC411 – Microeconomics Teachers Responsible
Term 1: Dr. Francesco Nava Office: 32L 3.20 Secretary: 32L 1.17 Term 2: Prof. Martin Pesendorfer Office: 32L 4.19 Secretary: 32L 1.17 Time: Wednesdays 10:00-12:00 Location: Old Theatre Examination: A 2 Hours Exam in Lent Term and a 2 Hours Exam in Summer Term. Aims and Objectives
This unit is a graduate level introduction to Microeconomics. The objective is to provide students with a firm grounding in the analytic methods of microeconomic theory used by economists working in research, government, and business. Special emphasis will be placed on the design and the solution of simple economic models and on ensuring that students become familiar with basic optimisation and equilibrium techniques. Course Summary
The first part of the course focuses on classical theories of consumer and producer behaviour and on the theory of general equilibrium. We begin with a careful analysis of the optimisation problems for price-taking consumers and firms. We then analyse market interaction and the formation of prices in the framework of perfect competition. We conclude with introductions to decision making under uncertainty and game theory. The second part of the course focuses on models of imperfect competition and information economics. We begin with an analysis of models of monopoly, oligopoly, and bargaining. Then, we study markets with imperfect and incomplete information including search, adverse selection, auctions, signalling, screening, and moral hazard. Special emphasis will be given to economic applications. Teaching and learning methods
Our approach will emphasise analytical methods of reasoning and thus problem-solving exercises will be a crucial learning tool. Problem sets will be provided and answers to selected problems will be discussed during classes.
Textbooks The course will draw mainly on the textbook: Riley, Essential Microeconomics, Cambridge University Press, 2012.
The following textbooks can also be useful as additional readings: Mas-Colell, Whinston, Green, Microeconomic Theory, Oxford University Press, 1995. Varian, Microeconomic Analysis, 3rd Edition, Norton, 1992 Course Outline: Term 1 A. Decision Theory, Consumer Theory and Welfare
(1) Riley, Essential Microeconomics, 2012, Chapter 2, 43-78. (2) Spiegler, Bounded Rationality and Industrial Organization, Appendix, 202-212. (3) Caplin, Schotter, The Foundations of Positive and Normative Economics, 2008, Chap 8.
Optional Readings: (a) Gul and Pesendorfer, The Case for Mindless Economics, 2005. (b) Riley, Essential Microeconomics, 2012, Chapter 1.
B. Producer Theory
(1) Riley, Essential Microeconomics, 2012, Chapter 4, 106-130.
C. Competitive Markets and Equilibrium
(1) Riley, Essential Microeconomics, 2012, Chapter 3, 85-105. (2) Riley, Essential Microeconomics, 2012, Chapter 5, 139-167.
D. Choice under Uncertainty
(1) Riley, Essential Microeconomics, 2012, Chapter 7, 218-250. E. Game Theory:
(1) Riley, Essential Microeconomics, 2012, Chapter 9, 303-338. (2) Shy, Industrial Organization, MIT Press, 1995, Chapter 2, 11‐42.
Optional Readings: (a) Kreps, A Course in Microeconomic Theory, Chapters 11‐14.
Course Outline: Term 2 A. Monopoly: Price Discrimination
1) J. Riley, Essential Microeconomics, 2012, Chapter 4.5, 130-138.
2) H. Gravelle and R. Rees, Microeconomics, Longman, 2004, Chapter 9. 3) R. Schmalensee, 'Output and Welfare Implications of Monopolistic Third Degree Price
Discrimination', American Economic Review, 1981.
4) E. P. Lazaer, ‘Retail Pricing and Clearance Sales’ American Economic Review, 1986.
B. Oligopoly Theory, Repeated Games and Bargaining
5) J. Riley, Essential Microeconomics, 2012, Chapter 9, 303-346.
6) R. Porter, ‘The Role of Information in U.S. Offshore Oil and Gas Lease Auctions,’
Econometrica, 1995.
7) D. Fudenberg and J. Tirole, Game Theory, MIT Press, 1991, Chapter 4, Section 4.4.
8) A. Mas-Colell, M. Whinston, and J. Green, Microeconomic Theory, Oxford University Press, 1995, Chapter 9, Appendix A, 296-299.
9) C. d'Aspremont, J. Jaskold Gabszewicz and J.-F. Thisse, 'On Hotelling's "Stability in Competition"' Econometrica, 1979.
C. Search and Adverse Selection 10) “Search” (unpublished note for lecture). 11) G. Akerlof, 'The Market for Lemons”, Quality Uncertainty and the Market Mechanism',
Quarterly Journal of Economics, 1970. D. Games with Incomplete Information and Auctions
12) J. Riley, Essential Microeconomics, 2012, Chapter 10.1, 347-350.
13) J. Riley, Essential Microeconomics, 2012, Chapter 12.3, 456-462. 14) M. H. Bazerman and W. F. Samuelson, I Won the Auction But Don’t Want the Prize,
Journal of Conflict Resolution, 1983.
15) A. Mas-Colell, M. Whinston, and J. Green, Microeconomic Theory, Oxford University Press, 1995, Chapter 23D, 883-891.
16) R. Myerson, `Optimal Auction Design,’’ Mathematics of Operations Research, 1981, 58-73.
17) R. Porter, ‘The Role of Information in U.S. Offshore Oil and Gas Lease Auctions,’ Econometrica, 1995.
E. Signalling, Moral Hazard and Screening
18) M. Spence, 'Job Market Signalling', Quarterly Journal of Economics, 1973. 19) B. Salanie, The Economics of Contracts. A Primer, MIT Press, 1997, Chapter 5.
20) M. Rothschild and J. Stiglitz, ‘Equilibrum in Competitive Insurance Markets: An Essay on the
economics of Imperfect Information’, Quarterly Journal of Economics, 1976.
Microeconom
icTheoryEC411
EC
411
Slid
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Nav
a
LSE
Mic
hae
lmas
Ter
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Nava
(LSE)
Microeconomic
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EC411
MichaelmasTerm
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Introdu
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Objectives
Th
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ich
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asT
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cou
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cove
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Dec
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elof
pref
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Cla
ssic
alth
eori
esof
con
sum
eran
dpr
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uce
rb
ehav
iou
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Mar
kets
,co
mp
etit
ion
and
the
com
pet
itiv
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briu
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Ch
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der
un
cert
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Gam
eth
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.
Th
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ive
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topr
ovid
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ents
wit
ha
firm
grou
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ing
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ean
alyt
icm
eth
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sof
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roec
onom
icth
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.
Alt
hou
ghth
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tere
dar
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dcl
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cal
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ofec
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Nava
(LSE)
Microeconomic
Theory
EC411
MichaelmasTerm
2/6
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Inform
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Lec
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Nava
(LSE)
Microeconomic
Theory
EC411
MichaelmasTerm
3/6
CourseProgram
1D
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Th
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&C
onsu
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Th
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:R
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Ch
apte
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and
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Ch
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Nava
(LSE)
Microeconomic
Theory
EC411
MichaelmasTerm
4/6
Not
es
Not
es
CourseRequirements
Cla
sses
star
tin
wee
k2
and
follo
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wit
hso
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Th
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ged
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iou
sye
ars.
Nava
(LSE)
Microeconomic
Theory
EC411
MichaelmasTerm
5/6
Materials
Ma
inT
extb
oo
k
Rile
y,E
ssen
tial
Mic
roec
onom
ics,
Cam
brid
geU
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len
tary
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dA
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inst
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and
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Mic
roec
onom
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ry,
1995
.[M
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Var
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econ
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An
alys
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3rd
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1992
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Mic
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onom
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Pri
nce
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,19
90.
[K]
Nava
(LSE)
Microeconomic
Theory
EC411
MichaelmasTerm
6/6
Not
es
Not
es
Pre
fere
nces
and
Con
sum
ptio
nE
C41
1S
lides
1
Nav
a
LS
E
Mic
hae
lmas
Ter
m
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m1
/9
0
Roa
dmap
:C
onsu
mpt
ion
Bas
icd
ecis
ion
theo
ry:
defi
nit
ion
pref
eren
ces,
uti
lity
repr
esen
tati
on.
Con
sum
ers’
opti
miz
atio
npr
oble
m:
uti
lity
max
imiz
atio
n,
exp
end
itu
rem
inim
izat
ion
,d
ual
ity.
Dem
and
,su
bst
itu
tion
and
inco
me
effec
ts.
Ind
irec
tu
tilit
yfu
nct
ion
and
exp
end
itu
refu
nct
ion
.
Ap
plic
atio
ns:
the
Slu
tsky
equ
atio
n,
the
Lab
our
mar
ket.
Cri
tici
sman
dal
tern
ativ
eth
eori
es.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m2
/9
0
Not
es
Not
es
SimpleDecisionTheory
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m3
/9
0
Sim
ple
Dec
isio
nT
heor
y
Ou
rpr
imit
ive
isa
pref
eren
cere
lati
onov
era
set
ofal
tern
ativ
es.
X⊆
Rn +
isth
ese
tof
alte
rnat
ives
(e.g
.co
nsu
mp
tion
set)
.
Th
eel
emen
tsof
Xar
ech
oice
s(e
.g.
con
sum
pti
onb
un
dle
s).
Aty
pic
alel
emen
tis
ave
ctor
x=
(x1,.
..,x
n).
%is
apr
efer
ence
rela
tion
over
X.
Wh
enx%
y,
the
con
sum
erw
eak
lyp
refe
rsx
toy
.
Exa
mp
le:
X=(
0,0),(1
,0),(0
,1),(1
,1)
;
(1,1)%
(1,0),
(1,1)%
(0,0),
(1,0)%
(0,1),
(0,1)%
(0,0).
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m4
/9
0
Not
es
Not
es
Str
ict
Pre
fere
nces
&In
diff
eren
ce
Fro
m%
,w
ed
eriv
etw
oad
dit
ion
alre
lati
ons
onX
:
Str
ict
Pre
fere
nce
():
x
yif
and
only
if
x%
yan
dn
oty%
x.
Ind
iffer
ence
(∼):
x∼
yif
and
only
if
x%
yan
dy%
x.
Inth
epr
evio
us
exam
ple
,(1
,1)
(1,0).
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m5
/9
0
Axi
oms
onC
onsu
mer
Beh
avio
r
Tw
oas
sum
pti
ons
onco
nsu
mer
beh
avio
rar
em
ain
tain
edth
rou
ghou
t:
1C
om
ple
ten
ess:
Ifx
,y∈
X,
eith
erx%
y,
ory%
x,
orb
oth
.
Inth
epr
evio
us
exam
ple
,pr
efer
ence
sar
en
otco
mp
lete
.
2T
ran
siti
vity
:F
orx
,y,z∈
X,
ifx%
yan
dy%
z,
then
x%
z.
We
say
that
apr
efer
ence
rela
tion
isra
tio
na
lif
itis
tran
siti
vean
dco
mp
lete
.
Exe
rcis
e:
Sh
owth
attr
ansi
tivi
tyex
ten
ds
tost
rict
pref
eren
cean
din
diff
eren
ce.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m6
/9
0
Not
es
Not
es
Oth
erD
imen
sion
s
Pre
fere
nce
sca
nb
ed
efin
edov
eral
tern
ativ
esth
atd
on
otb
elon
gto
Rn +
.
For
inst
ance
,co
nsi
der
:
pref
eren
ces
over
loca
tion
s:V
enic
e%
Rom
e,R
ome%
Par
is,
Par
is%
Ven
ice
pref
eren
ces
over
can
did
ates
inan
elec
tion
:
X%
Y%
ZY
%Z%
XZ%
X%
Y
Asi
de:
Tra
nsi
tivi
tyn
olo
nge
rh
old
sth
ough
wh
enw
etr
yto
aggr
egat
eth
ese
pref
eren
ces
X%
AY
%A
Z%
AX
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m7
/9
0
Uti
lity
Fun
ctio
n
Defi
niti
on
Au
tilit
yfu
nct
ion
isa
fun
ctio
nU
:X→
Rsu
chth
at,
for
any
x,y∈
X,
U(x)≥
U(y)
ifan
don
lyif
x%
y.
Ifa
pref
eren
cere
lati
onca
nb
ere
pres
ente
dby
au
tilit
yfu
nct
ion
,th
enth
epr
efer
ence
rela
tion
isra
tion
al.
How
ever
,n
otan
yra
tion
alpr
efer
ence
rela
tion
can
be
repr
esen
ted
bya
uti
lity
fun
ctio
n.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m8
/9
0
Not
es
Not
es
Exa
mpl
e:L
exic
ogra
phic
Pre
fere
nces
Su
pp
ose
that
X=
R2 +
and
that
x%
yif
and
only
if
eith
er“x
1>
y 1”
or“x
1=
y 1an
dx 2≥
y 2”.
No
uti
lity
fun
ctio
nca
nre
pres
ent
thes
era
tion
alpr
efer
ence
s.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m9
/9
0
Rep
rese
ntat
ion
The
orem
Tw
oad
dit
ion
alre
gula
rity
axio
ms
imp
lya
des
irab
lere
pres
enta
tion
theo
rem
:
Co
nti
nu
ity:
For
any
y∈
X,
the
setsx∈
X:x
%y
and
x∈
X:y
%x
are
clos
ed.
Mo
no
ton
icit
y:F
oran
yx
,y∈
X,
ifx i
>y i
,i=
1,..
.,n
,th
enx
y.
The
orem
Su
pp
ose
that
X=
Rn +
and
that
pref
eren
ces
are
com
ple
te,
tran
siti
ve,
con
tin
uou
s,an
dm
onot
onic
.T
hen
ther
eex
ists
aco
nti
nu
ous
fun
ctio
nU
:X→
Rsu
chth
at,
for
any
x,y∈
X,
U(x)≥
U(y)
ifan
don
lyif
x%
y.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m1
0/
90
Not
es
Not
es
Exa
mpl
e:H
owto
Der
ive
aU
tilit
yF
unct
ion
Con
sid
eran
yp
oin
tx
inth
eco
nsu
mp
tion
set.
Fin
dth
esc
alar
k(x),
for
wh
ich
y=
(k(x),
...,
k(x))∈
Xsa
tisfi
es
y∼
x.
Th
en,
set
U(x)=
k(x)
and
not
eth
atx
y=⇒
k(x)>
k(y).
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m1
1/
90
Mon
oton
icT
rans
form
atio
ns&
Con
vexi
ty
Con
sid
era
stri
ctly
incr
easi
ng
fun
ctio
ng
:R→
R.
IfU(·)
isa
uti
lity
fun
ctio
nth
atre
pres
ents
apr
efer
ence
rela
tion
%,
the
mon
oton
ictr
ansf
orm
atio
ng(U
(·))
also
repr
esen
ts%
.
Uti
lity
isan
ord
inal
con
cep
t.A
uti
lity
fun
ctio
nis
an
um
eric
alre
pres
enta
tion
ofa
pref
eren
cere
lati
on.
Occ
asio
nal
lya
fin
alax
iom
isim
pos
edon
pref
eren
ces
toen
sure
that
the
“wea
kly
pref
erre
dse
ts”
are
con
vex.
Co
nve
xity
:(i
)X
isco
nve
x.(i
i)F
orx
,y,z∈
X,
ifx%
zan
dy%
z,
then
λx+(1−
λ)
y%
zfo
ran
yλ∈[0
,1].
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m1
2/
90
Not
es
Not
es
Consumer
Theory
Uti
lity
Max
imiz
atio
n
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m1
3/
90
The
Uti
lity
Max
imiz
atio
nP
robl
em
No
tati
on
:
pi
isth
epr
ice
ofgo
od
i;
p=
(p1,.
..,p
n)
isth
epr
ice
vect
or;
mis
the
inco
me
ofth
eco
nsu
mer
.
Aco
nsu
mer
cho
oses
(x1,.
..,x
n)
tom
axim
ize
U(x
1,.
..,x
n)
sub
ject
top
1x 1
+..
.+pnx n≤
m.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m1
4/
90
Not
es
Not
es
Lo
cal
Non
Sat
iati
on
Giv
enx
,y∈
Rn,
let
d(x
,y)
den
ote
the
dis
tan
ceb
etw
een
xan
dy
.
Lo
cal
No
nS
ati
ati
on
:F
oran
yx∈
Xan
dε>
0,th
ere
exis
tsy∈
Xsu
chth
atd(x
,y)<
εan
dU(x)<
U(y).
Exe
rcis
e:S
how
that
LN
Sim
plie
spx∗=
mif
x∗
solv
esco
nsu
mer
prob
lem
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m1
5/
90
Sol
ving
the
Con
sum
er’s
Pro
blem
By
LN
Sw
ere
pla
cepx≤
mw
ith
the
bu
dge
tlin
epx
=m
.
Th
eL
agra
ngi
anfo
rth
eco
nsu
mer
’spr
oble
msa
tisfi
es
L=
U(x)+
λ(m−
px).
FO
Cfo
ran
inte
rior
,lo
cal
max
imu
msa
tisf
y
m−
px=
0,
∂U ∂xi−
λpi=
0fo
ran
yi=
1,..
.,n
.
Wit
htw
ogo
od
s,F
OC
imp
lyth
at
MR
S=−
∂U/
∂x1
∂U/
∂x2=−
p1
p2.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m1
6/
90
Not
es
Not
es
Indi
ffer
ence
Cur
ves
Defi
ne
anin
diff
eren
cecu
rve
asth
ese
tof
bu
nd
les
for
wh
ich
uti
lity
iseq
ual
toso
me
con
stan
t.
Ifth
ere
are
only
two
com
mo
dit
ies,
a(l
evel
k)
ind
iffer
ence
curv
esa
tisfi
es
U(x
1,x
2)=
k.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m1
7/
90
Mar
gina
lR
ates
ofS
ubst
itut
ion
Con
sid
era
two
com
mo
dit
ies
envi
ron
men
tan
dan
ind
iffer
ence
curv
e
U(x
1,x
2)=
k.
Tot
ally
diff
eren
tiat
ing
this
curv
e,yi
eld
s
∂U ∂x1
dx 1
+∂U ∂x
2d
x 2=
0
wh
ich
intu
rnim
plie
sth
at dx 2
dx 1
=−
∂U/
∂x1
∂U/
∂x2≡
MR
S.
We
ofte
nre
fer
dx 2
/d
x 1as
the
Mar
gin
al
Ra
teo
fS
ub
stit
uti
on
(MR
S)
bet
wee
ngo
od
1an
dgo
od
2.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m1
8/
90
Not
es
Not
es
Exa
mpl
es
Cob
b-D
ougl
as:
U(x
1,x
2)=
xα 1x
β 2=⇒
MR
S=−
αx 2
βx 1
.
Per
fect
Su
bst
itu
tes:
U(x
1,x
2)=
ax1+
bx2
=⇒
MR
S=−
a b.
Per
fect
Com
ple
men
ts
U(x
1,x
2)=
mina
x 1,b
x 2
Not
Diff
eren
tiab
le.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m1
9/
90
Fir
stO
rder
Con
diti
ons
Op
tim
alit
yin
the
con
sum
er’s
prob
lem
req
uir
esth
esl
ope
ofth
eb
ud
get
line
tob
eeq
ual
toth
esl
ope
ofth
ein
diff
eren
cecu
rve.
Rec
all
the
FO
Cof
the
con
sum
er’s
prob
lem
,
m−
px=
0&
∂U ∂xi−
λpi=
0fo
ran
yi=
1,..
.,n
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m2
0/
90
Not
es
Not
es
Sec
ond
Ord
erC
ondi
tion
s
SO
Cgu
aran
tee
that
aso
luti
onto
FO
Cis
loca
lm
ax:
Ifth
eu
tilit
yfu
nct
ion
isq
uas
icon
cave
,K
KT
FO
Cal
soim
ply
glob
alco
nst
rain
edm
axim
um
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m2
1/
90
Mar
shal
lian
Dem
ands
Sol
vin
gth
eu
tilit
ym
axim
izat
ion
prob
lem
,ob
tain
the
Mar
shal
lian
dem
and
s,
x i(p
,m),
for
any
i=
1,..
.,n
.
Den
ote
the
vect
orof
Mar
shal
lian
dem
and
sby
x(p
,m).
Mar
shal
lian
dem
and
sar
eh
omog
eneo
us
ofd
egre
eze
roin
(p,m
),th
atis
,
x(p
,m)=
x(t
p,t
m),
for
any
t>
0.
Defi
niti
on
Afu
nct
ion
f:R
n +→
Ris
hom
ogen
ous
ofd
egre
ek
ifan
don
lyif
,fo
ran
yt>
0an
dan
yx∈
Rn +
,f(t
x)=
tkf(x).
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m2
2/
90
Not
es
Not
es
Exa
mpl
e:C
obb-
Dou
glas
Con
sid
era
uti
lity
fun
ctio
n,
U(x
1,x
2)=
xα 1x
β 2.
Op
tim
alit
yre
qu
ires
that
:
MR
S=
αx 2
βx 1
=p
1
p2
=⇒
βp
1x 1
α=
p2x 2
.
Su
bst
itu
tin
gin
the
bu
dge
tlin
e,on
eob
tain
sth
at
p1x 1
( 1+
β α
) =m
.
Mar
shal
lian
dem
and
sh
ence
,sa
tisf
y
x 1(p
,m)=
( α
α+
β
) m p1
x 2(p
,m)=
( β
α+
β
) m p2
.
Th
efr
acti
onof
inco
me
spen
ton
x 1is
α
α+
βan
don
x 2is
β
α+
β.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m2
3/
90
Exa
mpl
es:
Per
fect
Com
plem
ents
&P
erfe
ctS
ubst
itut
es
Per
fect
Co
mp
lem
ents
:U(x
1,x
2)=
minx
1,x
2.
At
anop
tim
alch
oice
,x 1
=x 2
.
Th
us,
p1x 2
+p
2x 2
=m
imp
lies
x 2(p
,m)=
x 1(p
,m)=
m
p1+
p2
.
Per
fect
Su
bst
itu
tes:
U(x
1,x
2)=
x 1+
x 2.
Ifp
1<
p2,
then
x 1(p
,m)=
m p1
and
x 2(p
,m)=
0.
Ifp
1>
p2,
then
x 1(p
,m)=
0an
dx 2(p
,m)=
m p2
.
Ifp
1=
p2,
then
any
x(p
,m)
onth
eb
ud
get
line.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m2
4/
90
Not
es
Not
es
Exa
mpl
e:Q
uasi
-Lin
ear
Uti
lity
Con
sid
era
qu
asi-
linea
ru
tilit
yfu
nct
ion
:
U(x
1,x
2)=
2√
x 1+
x 2
Op
tim
alit
yre
qu
ires
that
:
MR
S=
(x1)−
1/
2=
p1
p2
.
Th
us
atan
inte
rior
opti
mu
mM
arsh
allia
nd
eman
ds
sati
sfy
x 1(p
,m)=
( p2
p1
) 2&
x 2(p
,m)=
m p2−
p2
p1
.
Ifm p
2−
p2
p1<
0,x 2
wou
ldb
ecom
en
egat
ive
wh
ich
isim
pos
sib
le.
Ifso
,th
efo
llow
ing
bou
nd
ary
solu
tion
app
lies
x 1(p
,m)=
m p1
&x 2(p
,m)=
0.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m2
5/
90
Pri
ceC
hang
es:
Ord
inar
yG
oo
ds
Ago
od
issa
idto
be:
Ord
inar
yif
its
Mar
shal
lian
dem
and
incr
ease
sas
its
pric
ed
ecre
ases
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m2
6/
90
Not
es
Not
es
Pri
ceC
hang
es:
Dem
and
Cur
ve
Dem
an
dcu
rves
map
equ
ilibr
ium
dem
and
sto
pric
es.
Ifa
goo
dis
ord
inar
yth
ed
eman
dcu
rve
isd
ecre
asin
g.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m2
7/
90
Pri
ceC
hang
es:
Giff
enG
oo
ds
Ago
od
issa
idto
be:
Giff
enif
its
Mar
shal
lian
dem
and
dec
reas
esas
its
pric
ed
ecre
ases
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m2
8/
90
Not
es
Not
es
Inco
me
Cha
nges
:N
orm
alG
oo
ds
Ago
od
issa
idto
be:
No
rma
lif
its
Mar
shal
lian
dem
and
incr
ease
sas
inco
me
incr
ease
s.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m2
9/
90
Inco
me
Cha
nges
:T
heE
ngle
Cur
ve
En
gle
curv
esm
apeq
uili
briu
md
eman
ds
toin
com
ele
vels
.
Ifa
goo
dis
nor
mal
the
En
gle
curv
eis
incr
easi
ng.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m3
0/
90
Not
es
Not
es
Inco
me
Cha
nges
:N
orm
alG
oo
ds
Ago
od
issa
idto
be:
Infe
rio
rif
its
Mar
shal
lian
dem
and
dec
reas
esas
inco
me
incr
ease
s.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m3
1/
90
The
Indi
rect
Uti
lity
Fun
ctio
n
Are
we
hap
pyab
out
chan
ges
inpr
ices
and
inco
me?
Can
pref
eren
ces
be
defi
ned
dir
ectl
yon
choi
ceor
bu
dge
tse
ts?
Th
ein
dir
ect
uti
lity
fun
ctio
nis
obta
ined
plu
ggin
gth
eM
arsh
allia
nd
eman
ds
inth
eu
tilit
yfu
nct
ion
v(p
,m)=
U(x(p
,m))
Th
ein
dir
ect
uti
lity
fun
ctio
nd
epen
ds
onth
ep
arti
cula
rre
pres
enta
tion
.
We
can
thin
kof
itas
uti
lity
fun
ctio
nd
efin
edov
erpr
ices
and
inco
me
pai
rs.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m3
2/
90
Not
es
Not
es
Pro
per
ties
ofth
eIn
dire
ctU
tilit
yF
unct
ion
Lem
ma
Th
ein
dir
ect
uti
lity
fun
ctio
nv(p
,m)
is:
(i)
non
-in
crea
sin
gin
pan
dn
on-d
ecre
asin
gin
m;
(ii)
v(p
,m)
ish
omog
eneo
us
ofd
egre
eze
roin
(p,m
);
(iii)
v(p
,m)
isq
uas
icon
vex
inp
;
(iv)
v(p
,m)
isco
nti
nu
ous
ifU(x)
isco
nti
nu
ous.
Pro
of
(i)
Wh
enm
incr
ease
sor
pi
dec
reas
es,
the
set
ofb
un
dle
sth
atsa
tisf
yth
eb
ud
get
con
stra
int
bec
omes
larg
er.
Ob
serv
atio
n:
Un
der
LN
Sv(p
,m)
isst
rict
lyin
crea
sin
gin
m,
bu
tn
otn
eces
sari
lyst
rict
lyd
ecre
asin
gin
ever
ypi
(eg
per
fect
sub
stit
ute
s).
(ii)
Mar
shal
lian
dem
and
sar
eh
omog
eneo
us
ofd
egre
eze
roin
(p,m
).
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m3
3/
90
Pro
per
ties
ofth
eIn
dire
ctU
tilit
yF
unct
ion
(iii)
Con
sid
er:
pric
esp
,p
,an
dp∗=
λp+(1−
λ)
p;
any
bu
nd
lex∗
such
that
p∗ x∗≤
m.
Itis
imp
ossi
ble
that
px∗>
man
dp
x∗>
msi
nce
:
λpx∗>
λm
and
(1−
λ)
px∗>
(1−
λ)
m⇒
p∗ x∗>
m.
Th
us,
eith
erpx∗≤
mor
px∗≤
m,
wh
ich
intu
rnim
plie
sth
at
v(p∗ ,
m)≤
maxv
(p,m
),v(p
,m)
and
that
low
erco
nto
ur
sets
are
con
vex.
Exe
rcis
es:
Dra
wv
’sin
diff
eren
cecu
rves
tosh
owth
ela
stp
art
ofth
ear
gum
ent.
Ass
ign
edfo
rcl
asse
s:pr
ove
(iv)
.N
ava
(LS
E)
Pre
fere
nce
sa
nd
Co
nsu
mp
tio
nM
ich
ael
ma
sT
erm
34
/9
0
Not
es
Not
es
Consumer
Theory
Exp
endi
ture
Min
imiz
atio
n
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m3
5/
90
The
Exp
endi
ture
Min
imiz
atio
nP
robl
em
Su
pp
ose
that
the
con
sum
erch
oos
esx
tom
inim
ize
px
sub
ject
toU(x)≥
u.
Th
isd
ual
prob
lem
reve
rses
the
role
ofob
ject
ive
fun
ctio
nan
dco
nst
rain
t.
Ass
um
eth
atU
isco
nti
nu
ous
wit
hL
NS
pref
eren
ces.
Ifth
eco
nst
rain
tse
tis
not
emp
ty,
aso
luti
onex
ists
.
Th
isis
the
du
alpr
oble
mof
our
orig
inal
prob
lem
,si
nce
ith
asth
esa
me
solu
tion
toa
corr
esp
ond
ing
uti
lity
max
imiz
atio
npr
oble
m,
wh
enw
ealt
his
pos
itiv
ean
du
tilit
yis
abov
e0
uti
lity.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m3
6/
90
Not
es
Not
es
Hic
ksia
nD
eman
ds
Th
eso
luti
ons
toex
pen
dit
ure
min
imiz
atio
npr
oble
mar
ekn
own
asth
eH
icks
ian
dem
and
s,
hi(
p,u),
for
i=
1,..
.,n
.
Den
ote
the
vect
orof
Hic
ksia
nd
eman
ds
byh(p
,u).
Hic
ksia
nd
eman
ds
are:
hom
ogen
ous
ofd
egre
e0
inp
;an
dsi
ngl
eva
lued
prov
ided
that
pref
eren
ces
sati
sfy
the
con
vexi
tyax
iom
.
Th
ese
are
also
know
nas
com
pen
sate
dd
eman
ds,
asin
com
ech
ange
sto
com
pen
sate
for
chan
ges
inpr
ices
soth
atu
tilit
yre
mai
ns
con
stan
tat
u.
Hic
ksia
nd
eman
dar
en
ot
ob
serv
ab
le,
only
Mar
shal
lian
dem
and
sar
e.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m3
7/
90
The
Exp
endi
ture
Fun
ctio
n&
Its
Pro
per
ties
Th
eex
pen
dit
ure
fun
ctio
nis e(p
,u)=
n ∑ i=1
pih
i(p
,u).
IfU(x)
isco
nti
nu
ous,
U(h(p
,u))
=u
.
Lem
ma
Th
eex
pen
dit
ure
fun
ctio
nis
:
(i)
non
-dec
reas
ing
inpi,
for
any
i=
1,..
.,n
;
(ii)
hom
ogen
eou
sof
deg
ree
1in
p;
(iii)
con
cave
inp
;
(iv)
con
tin
uou
sin
p.
Mor
eove
r,∂e(p
,u)
∂pi
=hi(
p,u),
for
any
i=
1,..
.,n
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m3
8/
90
Not
es
Not
es
Consumer
Theory
Dua
lity
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m3
9/
90
Dua
lity:
Exp
endi
ture
Min
imiz
atio
n&
Uti
lity
Max
imiz
atio
n
Bot
hth
eex
pen
dit
ure
fun
ctio
nan
dth
ein
dir
ect
uti
lity
fun
ctio
nca
ptu
reve
ryim
por
tan
tpr
oper
ties
ofth
eco
nsu
mer
prob
lem
.
The
orem
Su
pp
ose
that
:
pref
eren
ces
sati
sfy
LN
San
dU(·)
isco
nti
nu
ous;
pric
esar
est
rict
lyp
osit
ive
p
0.
Th
en:
1e(p
,v(p
,m))
=m
;
2v(p
,e(p
,u))
=u
;
3x i(p
,e(p
,u))
=hi(
p,u),
for
i=
1,..
.,n
;
4hi(
p,v(p
,m))
=x i(p
,m),
for
i=
1,..
.,n
.
Th
ela
sttw
opr
oper
ties
tie
obse
rvab
les
tou
nob
serv
able
s.N
ava
(LS
E)
Pre
fere
nce
sa
nd
Co
nsu
mp
tio
nM
ich
ael
ma
sT
erm
40
/9
0
Not
es
Not
es
Gra
phic
alR
epre
sent
atio
nof
Dua
lity
As
pric
esva
ry,
the
Hic
ksia
nd
eman
dgi
ves
us
the
dem
and
that
wou
ldar
ise
ifex
pen
dit
ure
was
chan
ged
toke
epu
tilit
yco
nst
ant.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m4
1/
90
Pro
ofof
Dua
lity
Par
t3
Su
pp
ose
that
h(p
,u)
do
esn
ot
max
imiz
eu
tilit
ysu
bje
ctto
px≤
e(p
,u).
Let
xb
eth
eu
tilit
ym
axim
izin
gb
un
dle
.T
hu
s,
U(x)>
U(h(p
,u) )
=u
.
Ob
serv
eth
ate(p
,u)=
ph(p
,u)=
px
,
wh
ere
the
firs
teq
ual
ity
hol
ds
byd
efin
itio
nan
dth
ese
con
dby
LN
S.
Con
sid
era
bu
nd
lex−
ε,w
her
eε
isa
smal
lp
osit
ive
vect
or.
By
con
tin
uit
y,w
eh
ave
that
U(x−
ε )>
u(x
isin
teri
orw
log)
.
How
ever
,th
isco
ntr
adic
tsth
eh
ypot
hes
isth
ate(p
,u)
isth
em
inim
um
exp
end
itu
reth
atac
hie
ves
uti
lity
grea
ter
than
oreq
ual
tou
,si
nce
p(x−
ε )=
e(p
,u)−
pε<
e(p
,u).
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m4
2/
90
Not
es
Not
es
Roy
’sId
enti
ty
Rec
all
that
par
t2
du
alit
yre
sult
esta
blis
hes
that
v(p
,e(p
,u))
=u
.
Diff
eren
tiat
ing
wit
hre
spec
tto
pi,
obta
inth
at
∂v(p
,e(p
,u) )
∂pi
+∂v
(p,e(p
,u) )
∂mhi(
p,u)=
0.
Exp
loit
ing
the
du
alit
yth
eore
m,
this
sim
plifi
esto
∂v(p
,m)
∂pi
+∂v
(p,m
)
∂mx i(p
,m)=
0.
Th
isis
know
nas
Ro
y’s
Iden
tity
and
req
uir
esth
at
x i(p
,m)=−
∂v(p
,m)
/∂p
i
∂v(p
,m)
/∂m
.
As
vis
not
inva
rian
tto
uti
lity
tran
sfor
mat
ion
s,th
ed
eriv
ativ
eof
vw
.r.t
.p
isn
orm
aliz
edby
the
mar
gin
alu
tilit
yof
wea
lth
tod
eriv
ed
eman
ds.
Th
isis
agr
eat
tool
,b
ecau
sew
eca
nu
seth
ed
eriv
ativ
esof
the
ind
irec
tu
tilit
yfu
nct
ion
tofi
nd
dem
and
s.N
ava
(LS
E)
Pre
fere
nce
sa
nd
Co
nsu
mp
tio
nM
ich
ael
ma
sT
erm
43
/9
0
Exa
mpl
e:C
obb-
Dou
glas
Th
ein
dir
ect
uti
lity
ofa
spec
ific
Cob
b-D
ougl
asu
tilit
yfu
nct
ion
is
v(p
,m)=
m2
p1p
2.
Th
us,
Ro
y’s
Iden
tity
inth
isca
sere
qu
ires
x 1(p
,m)=−
∂v(p
,m)
/∂p
1
∂v(p
,m)
/∂m
=−−( 1 p
1
) 2 m2
p2
2m p1p
2
=1 2
m p1
.
Not
eth
at,
byid
enti
ty2,
we
also
hav
eth
at
e(p
,u)2
p1p
2=
u=⇒
e(p
,u)=√
up
1p
2
We
can
der
ive
the
exp
end
itu
refu
nct
ion
from
the
ind
irec
tu
tilit
yfu
nct
ion
and
vice
vers
a.N
ava
(LS
E)
Pre
fere
nce
sa
nd
Co
nsu
mp
tio
nM
ich
ael
ma
sT
erm
44
/9
0
Not
es
Not
es
Consumer
Theory
App
licat
ions
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m4
5/
90
Tow
ards
Slu
tsky
Equ
atio
n
Rec
all
that
Mar
shal
lian
dem
and
curv
esca
nb
eu
pwar
dsl
opin
g.
Do
esu
tilit
ym
axim
izat
ion
imp
ose
any
rest
rict
ion
son
con
sum
erb
ehav
ior?
Par
t3
du
alit
yre
sult
esta
blis
hes
that
x i(p
,e(p
,u))
=hi(
p,u).
Diff
eren
tiat
ing
wit
hre
spec
tto
pj
we
obta
in
∂xi(
p,e(p
,u))
∂pj
+∂x
i(p
,e(p
,u))
∂mhj(
p,u)=
∂hi(
p,u)
∂pj
.
By
the
du
alit
yth
eore
m,
this
imp
lies
that
∂xi(
p,m
)
∂pj
+∂x
i(p
,m)
∂mx j(p
,m)
︸︷︷
︸θ ij(p
,m)
=∂h
i(p
,v(p
,m))
∂pj
︸︷︷
︸σ ij(p
,m)
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m4
6/
90
Not
es
Not
es
The
Sub
stit
utio
nM
atri
x
Defi
ne
the
sub
stit
uti
on
ma
trix
asth
em
atri
xθ(
m,p)
sati
sfyi
ng
θ(m
,p)=
θ 11(p
,m)
...
θ 1n(p
,m)
...
...
...
θ n1(p
,m)
...
θ nn(p
,m)
w
her
eas
bef
ore θ i
j(p
,m)=
∂xi(
p,m
)
∂pj
+∂x
i(p
,m)
∂mx j(p
,m).
Th
em
atri
xis
:
neg
ativ
ese
mid
efin
ite
(by
the
con
cavi
tyof
the
exp
end
itu
refu
nct
ion
);
sym
met
ric
(sin
cecr
oss-
pric
eeff
ects
coin
cid
e–
not
intu
itiv
e);
wit
hn
on-p
osit
ive
dia
gon
alte
rms.
Itd
epen
ds
only
onte
rms
that
are,
atle
ast
inpr
inci
ple
,ob
serv
able
.N
ava
(LS
E)
Pre
fere
nce
sa
nd
Co
nsu
mp
tio
nM
ich
ael
ma
sT
erm
47
/9
0
Slu
tsky
Equ
atio
n
Th
eS
luts
ky
equ
ati
on
dec
omp
oses
the
effec
tof
ach
ange
inth
epr
ice:
∂xi(
p,m
)
∂pj
=σ ij(
p,m
)−
∂xi(
p,m
)
∂mx j(p
,m).
Tw
oeff
ects
dis
cip
line
how
dem
and
resp
ond
sto
ach
ange
inth
epr
ice:
1th
esu
bst
itu
tio
neff
ect,
σ ij(
p,m
);
2th
ein
com
eeff
ect,−
∂xi(
p,m
)
∂mx j(p
,m).
Sin
ceσ ii(
p,m
)≤
0,th
eS
luts
kyeq
uat
ion
imp
lies
that
Giff
engo
od
s(∂
x i/
∂pi>
0)m
ust
alw
ays
be
infe
rior
(∂x i
/∂m
<0)
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m4
8/
90
Not
es
Not
es
Som
eIn
tuit
ion:
Slu
tsky
Equ
atio
n
By
the
enve
lop
eth
eore
mdP
e≡
de
/d
p=
h.
Th
eref
ore,
dph=
d2 pe
.
As
eis
con
tin
uou
s,d
iffer
enti
able
and
con
cave
,it
sH
essi
anis
sym
met
ric
and
neg
ativ
ese
mi-
defi
nit
e.
Th
eref
ore
com
pen
sate
do
wn
pri
ceeff
ects
are
no
n-p
osi
tive
,an
do
ther
effec
tsar
esy
mm
etri
c.
Slu
tsky
equ
atio
nim
plie
sth
ata
chan
gein
com
pen
sate
dd
eman
dh
iseq
ual
toth
ech
ange
inth
eu
nco
mp
ensa
ted
dem
and
xd
ue
topr
ices
only
,p
lus
the
chan
gein
un
com
pen
sate
dd
eman
dx
du
eto
the
chan
gein
exp
end
itu
rem
ult
iplie
dby
the
chan
gein
exp
end
itu
re.
Th
eR
HS
ofth
eS
luts
kyeq
uat
ion
give
su
sco
mp
uta
ble
rela
tion
ship
.
Th
eory
allo
wed
us
tod
eriv
ete
stab
lepr
edic
tion
sab
out
dph
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m4
9/
90
Gra
phic
alIn
tuit
ion:
Slu
tsky
Equ
atio
n
Con
sid
era
nor
mal
goo
di
and
anin
crea
seto
pi:
Ob
serv
eth
ath
isfl
atte
rth
anx
,as
∂xi/
∂pi≤
∂hi/
∂pi≤
0.
Th
ecu
rves
inte
rsec
tat
the
init
ial
pric
eas
du
alit
yim
plie
sx
0 i=
h0 i.
Wh
enth
epr
ice
incr
ease
s,in
com
ed
ecre
ases
furt
her
.
Th
us,
low
erin
com
ere
du
ces
con
sum
pti
onof
x1 i
furt
her
.
As
∂xi/
∂mis
pos
itiv
efo
rn
orm
algo
od
s,hi
resp
ond
sle
ssth
anx i
.
Inot
her
wor
ds,
hi
isco
mp
ensa
ted
byin
crea
sin
gw
ealt
hto
keep
the
dec
isio
nm
aker
atth
esa
me
uti
lity,
sod
eman
dd
oes
not
fall
asm
uch
.
Th
eco
nve
rse
hol
ds
for
infe
rior
goo
ds.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m5
0/
90
Not
es
Not
es
Gra
phic
alIn
tuit
ion:
Slu
tsky
Equ
atio
n
Hic
ksia
nis
com
pen
sate
dby
incr
easi
ng
wea
lth
toke
epu
tilit
yco
nst
ant.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m5
1/
90
Com
para
tive
Sta
tics
wit
h2
Go
ods
Diff
eren
tiat
ing
the
FO
C,
we
obta
inth
at:
0−
p1−
p2
−p
1U
11
U1
2
−p
2U
21
U2
2
dλ
/d
p1
dx 1
/d
p1
dx 2
/d
p1
= x 1 λ 0
0
−p
1−
p2
−p
1U
11
U1
2
−p
2U
21
U2
2
dλ
/d
md
x 1/
dm
dx 2
/d
m
= −
1 0 0
T
hes
eco
nd
itio
ns
are
solv
edto
der
ive
inco
me
and
sub
stit
uti
oneff
ects
:d
x 1 dm
=p
2U
12−
p1U
22
∣ ∣ H∣ ∣d
x 1d
p1=
−λ
p2 2 ∣ ∣ H∣ ∣
sub
stit
uti
on
effec
t
−x 1
p2U
12−
p1U
22
∣ ∣ H∣ ∣in
com
eeff
ect
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m5
2/
90
Not
es
Not
es
Lab
our
Sup
ply
Defi
ne
Las
leis
ure
,T
asto
tal
avai
lab
leti
me,
Cas
con
sum
pti
on,
was
the
wag
e,an
dp
asth
epr
ice
ofco
nsu
mp
tion
.
Aco
nsu
mer
cho
oses
(C,L)
tom
axim
ize
U(C
,L)
sub
ject
top
C=
w(T−
L)
Th
epr
oble
mis
ast
and
ard
uti
lity
max
imiz
atio
npr
oble
min
wh
ich
inco
me
mis
sim
ply
rep
lace
dby
wT
.
We
can
rew
rite
the
con
stra
int
asp
C+
wL=
wT
,an
dob
tain
dem
and
s
C(p
,w,
wT
Inco
me)
and
L(p
,w,
wT
Inco
me)
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m5
3/
90
Lab
our
Sup
ply
Th
ew
age
affec
tsd
eman
db
oth
asa
pric
ean
das
ad
eter
min
ant
ofin
com
e
dL(p
,w,w
T)
dw
=∂L ∂w
Sta
nd
ard
+T
∂L ∂mN
ewT
erm
.
By
the
Slu
tsky
dec
omp
osit
ion
,w
eh
ave
that
∂L ∂w=
σS
ub
stit
uti
on−
L∂L ∂m
Inco
me.
Th
us,
even
ifle
isu
reis
nor
mal
,it
sd
eman
dca
nin
crea
sein
wag
es
dL(r
,w,w
T)
dw
=σ+(T−
L)
∂L ∂m≥
0.
and
ther
efor
ela
bou
rsu
pp
ly,
T−
L,
can
be
dow
nw
ard
slop
ing.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m5
4/
90
Not
es
Not
es
Welfare
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m5
5/
90
Wel
fare
Bac
kto
stan
dar
du
tilit
yth
eory
.
We
wan
tto
use
the
tool
sd
evel
oped
(exp
end
itu
refu
nct
ion
,in
dir
ect
uti
lity,
con
dit
ion
ald
eman
d)
toas
sess
how
the
wel
fare
ofth
eco
nsu
mer
chan
ges
wh
enp
aram
eter
ssu
chas
pric
esch
ange
.
Con
sum
ers
surp
lus
ison
ew
ayto
asse
ssw
elfa
re.
Bu
tw
ew
ant
toth
ink
ofit
mor
ege
ner
ally
,an
dst
illin
clu
de
CS
.
Ap
ossi
ble
appr
oach
wou
ldb
eto
thin
kof
diff
eren
ces
inin
dir
ect
uti
lity.
Bu
tth
atm
ayn
oth
elp
mu
chas
wel
l.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m5
6/
90
Not
es
Not
es
Con
sum
erS
urpl
us
Con
sid
erth
em
arke
tfo
rgo
od
1,w
her
ex 1(p
,m)
isd
eman
dfo
rgo
od
1.
Su
pp
ose
that
the
only
pric
eof
goo
d1
dec
reas
esfr
omp
0 1to
p1 1.
Ref
erto
the
area
un
der
the
dem
and
curv
eas
the
con
sum
ersu
rplu
sC
S.
Itm
easu
res
the
“will
ingn
ess
top
ay”
ofa
con
sum
er.
CS
mea
sure
sth
en
orm
aliz
edch
ange
inp
urc
has
ing
pow
erof
the
con
sum
er
CS=∫ p0 1
p1 1
∂e(p
,v(p
,m))
∂p1
dp
1=∫ p0 1
p1 1
x 1(p
,m)d
p1=∫ p1 1
p0 1
∂v(p
,m)/
∂p1
∂v(p
,m)
/∂m
dp
1.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m5
7/
90
Alt
erna
tive
Cri
teri
a:A
Per
fect
Sub
stit
utes
Exa
mpl
e
Con
sid
era
uti
lity
map
u(x
1,x
2)=
x 1+
x 2an
din
com
em
=1.
Apr
ice
chan
gefr
om(2
,1)
to(1
/2,
1)
incr
ease
su
tilit
yfr
om1
to2.
How
mu
chis
this
pric
ech
ange
wor
thto
the
con
sum
er?
Cri
teri
on
1:
At
the
old
pric
es,
how
mu
chsh
ould
inco
me
grow
for
the
con
sum
erto
ach
ieve
uti
lity
2.
At
pric
es(2
,1)
the
con
sum
ern
eed
san
inco
me
of2
toat
tain
the
targ
etu
tilit
yof
2.If
so,
the
chan
geis
wor
th1.
Cri
teri
on
2:
At
the
new
pric
es,
how
mu
chsh
ould
inco
me
fall
for
the
con
sum
erto
ach
ieve
uti
lity
1.
At(1
/2,
1)
the
con
sum
ern
eed
san
inco
me
of1
/2
toat
tain
the
init
ial
uti
lity
of1.
Ifso
,th
ech
ange
isw
orth
1/2.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m5
8/
90
Not
es
Not
es
Alt
erna
tive
Cri
teri
a:M
onet
ary
Mea
sure
sof
Wel
fare
Su
pp
ose
that
pric
esch
ange
from
ave
ctor
p0
toa
vect
orp
1.
Let
:
u0=
v(p
0,m
)d
enot
eth
ein
itia
lu
tilit
y;
u1=
v(p
1,m
)d
enot
eth
efi
nal
uti
lity.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m5
9/
90
Alt
erna
tive
Cri
teri
a:E
quiv
alen
tV
aria
tion
s
Th
eeq
uiv
ale
nt
vari
ati
on
isd
efin
edas
EV
=e(p
0,u
1)−
m=
e(p
0,u
1)−
e(p
0,u
0).
Itis
the
chan
gein
inco
me
that
the
con
sum
ern
eed
sat
the
old
pric
esp
0to
be
asw
ell
offas
atth
en
ewpr
ices
p1,
u1=
v(p
0,m
+E
V).
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m6
0/
90
Not
es
Not
es
Alt
erna
tive
Cri
teri
a:C
omp
ensa
ting
Var
iati
ons
Th
eco
mp
ensa
tin
gva
ria
tio
nis
defi
ned
as
CV
=m−
e(p
1,u
0)=
e(p
1,u
1)−
e(p
1,u
0).
Itis
the
chan
gein
com
eth
atth
eco
nsu
mer
nee
ds
atth
en
ewpr
ices
p1
tob
eas
wel
loff
asat
the
old
pric
esp
0,
u0=
v(p
1,m−
CV).
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m6
1/
90
Alt
erna
tive
Cri
teri
a:C
Vvs
EV
Ifu
1>
u0,
EV
>0
and
CV
>0.
How
ever
,in
gen
eral
EV6=
CV
.
EV
and
CV
use
diff
eren
tre
fere
nce
poi
nts
:
EV
com
pen
sate
sfo
rm
ain
tain
ing
the
stat
us
qu
o;
CV
com
pen
sate
sfo
rch
angi
ng
the
stat
us
qu
o.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m6
2/
90
Not
es
Not
es
Exa
mpl
e:C
obb-
Dou
glas
Rec
all
that
for
Cob
b-D
ougl
aspr
efer
ence
sw
eh
ad:
v(p
,m)=
m2
p1p
2&
e(p
,u)=√
up
1p
2.
Aga
inco
nsi
der
anex
amp
lew
ith
:m
=1,
p0=
(1,1),
p1=
(1/
2,1).
Ifso
,fi
nd
that
:
u0=
1,u
1=
2;
EV
=e(p
0,u
1)−
m=√
2−
1=
0.41
;
CV
=m−
e(p
1,u
0)=
1−√
1/
2=
0.29
;
CS=
[log(1)−
log(1
/2)]
/2=
0.35
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m6
3/
90
Com
pari
ngW
elfa
reC
rite
ria
Su
pp
ose
that
p1 1<
p0 1
and
p1 i=
p0 i=
pi,
i=
2,..
.,n
.
Let
pt=
(pt 1,p
2,.
.,pn),
for
t∈0
,1.
Ifso
,
EV
=e(p
0,u
1)−
e(p
1,u
1)=∫ p0 1
p1 1
∂e(p
,u1)
∂p1
dp
1=∫ p0 1
p1 1
h1(p
,u1)d
p1,
CV
=e(p
0,u
0)−
e(p
1,u
0)=∫ p0 1
p1 1
∂e(p
,u0)
∂p1
dp
1=∫ p0 1
p1 1
h1(p
,u0)d
p1.
Su
pp
ose
that
goo
d1
isa
nor
mal
goo
dan
dL
NS
hol
ds.
Sin
cego
od
1is
nor
mal
and
x 1(p
,m)=
h1(p
,v(p
,m))
,w
eh
ave
∂x1
∂m=
∂h1
∂u
∂v ∂m≥
0.
LN
Sth
enim
plie
s∂v
/∂m
>0,
and
con
seq
uen
tly
∂h1/
∂u≥
0.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m6
4/
90
Not
es
Not
es
Com
pari
ngW
elfa
reC
rite
ria
Rec
all
that
h1(t
,p0 −
1,v(t
,p0 −
1,m
))=
x 1(t
,p0 −
1,m
).
So,
for
p1 1≤
t≤
p0 1,
we
obta
inth
at
u0≤
v(t
,p0 −
1,m
)≤
u1.
Con
seq
uen
tly
sin
ce∂h
1/
∂u≥
0,w
ekn
owth
at
h1(t
,p0 −
1,u
0)≤
x 1(t
,p0 −
1,m
),
x 1(t
,p0 −
1,m
)≤
h1(t
,p0 −
1,u
1).
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m6
5/
90
Com
pari
ngW
elfa
reC
rite
ria
For
nor
mal
goo
ds:
CV≤
CS≤
EV
.
For
infe
rior
goo
ds:
EV≤
CS≤
CV
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m6
6/
90
Not
es
Not
es
Exa
mpl
e:C
ompa
ring
Wel
fare
Cri
teri
a
Let
p−i
be
the
pric
eve
ctor
that
excl
ud
espi,
and
con
sid
er
v(p
,m)=
a(p)+
b(p−i)
m.
Mar
shal
lian
dem
and
ofgo
od
iis
ind
epen
den
tof
m,
x i(p
,m)=−
∂a(p)/
∂pi
b(p−i)
.
Th
eex
pen
dit
ure
fun
ctio
nsa
tisfi
es
e(p
,u)=−
a(p)
b(p−i)
+u
b(p−i)⇒
hi(
p,u)=−
∂a(p)/
∂pi
b(p−i)
.
Hen
ce,
sin
cehi(
p,u)=
x i(p
,m),
EV
=C
V=
CS
for
chan
ges
inpi.
Intu
itio
n:
Th
em
argi
nal
uti
lity
ofin
com
eis
ind
epen
den
tof
pi.
Hen
cech
ange
sar
ein
dep
end
ent
ofth
eb
ase
pric
epi.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m6
7/
90
Exa
mpl
e:Q
uasi
-Lin
ear
Uti
lity
Con
sid
erth
eu
tilit
yfu
nct
ion
:u(x
1,x
2)=
2√
x 1+
x 2.
At
anin
teri
orso
luti
onM
arsh
allia
nd
eman
ds
sati
sfy:
x 1(p
,m)=
( p2
p1
) 2&
x 2(p
,m)=
m p2−
p2
p1
.
Th
ein
dir
ect
con
seq
uen
tly
sati
sfies
:
v(p
,m)=
m p2+
p2
p1⇒
b(p−
1)=
1 p2
.
At
anin
teri
orso
luti
onE
V=
CV
=C
Sfo
rch
ange
sin
p1.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m6
8/
90
Not
es
Not
es
Ext
ra:
Fin
alC
onsi
dera
tion
son
Wel
fare
Rem
ark
:If
two
alte
rnat
ive
fin
alpr
ices
p1
and
p2
are
such
that
v(p
1,m
)=
v(p
2,m
)=
u1.
Itm
ust
be
that
EV
1=
EV
2,
sin
ce
v(p
0,m
+E
V1)=
v(p
0,m
+E
V2)=
u1.
Bec
ause
the
fin
alu
tilit
yan
dpr
ices
coin
cid
e,so
do
EV
.
Itis
no
tn
eces
sari
lytr
ue
how
ever
that
CV
1=
CV
2.
Sin
cepr
ices
diff
er,
som
ayC
V’s
,
v(p
1,m−
CV
1)=
v(p
2,m−
CV
2)=
u0.
We
nee
dto
know
the
exp
end
itu
refu
nct
ion
wh
ich
we
can
reco
ver
from
x.
Th
ou
gh
ts:
Wh
yd
ow
em
easu
rew
elfa
reth
rou
ghu
tilit
y?W
hat
abou
tw
ell
bei
ng
orh
app
ines
s?C
anit
be
pat
ron
izin
g?N
ava
(LS
E)
Pre
fere
nce
sa
nd
Co
nsu
mp
tio
nM
ich
ael
ma
sT
erm
69
/9
0
Advancesin
DecisionTheory
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m7
0/
90
Not
es
Not
es
Adv
ance
sin
Dec
isio
nT
heor
y
Th
eu
tilit
yre
pres
enta
tion
isn
eutr
alw
ith
resp
ect
toth
ew
orki
ng
ofth
eh
um
anm
ind
.
Th
em
otiv
es,
the
reas
onin
g,an
dth
eem
otio
ns
ofth
eco
nsu
mer
are:
1Im
plic
it;
2R
elev
ant
isso
far
asth
eyap
pea
rin
actu
alch
oice
s.
On
the
con
trar
y,b
ehav
iora
lec
onom
ics
atte
mp
tsto
mo
del
exp
licit
lyp
sych
olog
ical
asp
ects
ofd
ecis
ion
s.
Inp
arti
cula
r,it
con
sid
ers
seve
ral
rele
van
td
epar
ture
sfr
omou
rst
and
ard
axio
mat
icm
od
elof
beh
avio
r.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m7
1/
90
Ref
eren
ceD
epen
dent
Pre
fere
nces
Th
een
do
wm
ent
effec
t:co
nsu
mer
sva
lue
mor
eco
mm
od
itie
sth
eyow
nre
lati
veto
com
mo
dit
ies
they
do
not
own
.
Mor
ege
ner
ally
,th
epr
efer
ence
sof
con
sum
erca
nd
epen
don
refe
ren
cep
oin
tssu
chas
exp
ecta
tion
sor
targ
ets.
For
exam
ple
,a
con
sum
erm
ayb
eex
-an
tein
diff
eren
tb
etw
een
free
trip
toX
and
1,00
0£.
How
ever
,if
the
con
sum
erex
pec
tsto
trav
elto
X,
he
may
be
un
will
ing
togi
veu
pth
etr
ipfo
r1,
050£
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m7
2/
90
Not
es
Not
es
Exp
erim
enta
lE
vide
nce:
Mug
s?
Ran
dom
stu
den
tsw
ere
give
nC
orn
ell
Un
iver
sity
mu
gsw
hic
hw
ere
for
sale
for
6$at
the
cam
pu
sst
ore.
Th
ere
mai
nin
gst
ud
ents
wer
egi
ven
not
hin
g.
Aft
erth
isth
etw
ogr
oup
sw
ere
aske
dre
spec
tive
ly:
How
mu
chw
ould
you
will
ing
tose
llfo
r?
How
mu
chw
ould
you
be
will
ing
tob
uy
the
mu
gfo
r?
Th
ere
por
ted
selli
ng
pric
ew
as5.
25$.
Th
ere
por
ted
bu
yin
gpr
ice
only
2.75
$.
Imp
licat
ion
s:
loss
vsga
ineff
ect;
less
trad
eth
anst
and
ard
theo
ry.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m7
3/
90
Pro
spec
tT
heor
y:K
ahne
man
and
Tve
rsky
(197
9)
Pre
fere
nce
sd
epen
dn
otju
ston
outc
omes
,b
ut
also
onh
owou
tcom
esd
iffer
from
som
ere
fere
nce
poi
nt.
Bia
ses
inco
gnit
ion
and
per
cep
tion
also
affec
tpr
efer
ence
s.
Wh
enw
ere
spon
dto
attr
ibu
tes
such
asbr
igh
tnes
s,lo
ud
nes
s,or
tem
per
atu
re,
the
pas
tan
dpr
esen
tco
nte
xtof
exp
erie
nce
defi
nes
anad
apta
tion
leve
l,or
refe
ren
cep
oin
t,an
dst
imu
liar
ep
erce
ived
inre
lati
onto
this
refe
ren
cep
oin
t.T
hu
s,an
obje
ctat
agi
ven
tem
per
atu
rem
ayb
eex
per
ien
ced
ash
otor
cold
toth
eto
uch
dep
end
ing
onth
ete
mp
erat
ure
tow
hic
hon
eh
asad
apte
d.
(Kah
nem
anan
dT
vers
ky19
79)
Not
only
are
sen
sory
per
cep
tion
sre
fere
nce
-dep
end
ent
bu
tso
are
jud
gmen
tsan
dev
alu
atio
ns
abou
tou
tcom
esin
one’
slif
e.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m7
4/
90
Not
es
Not
es
Pro
spec
tT
heor
y:K
ahne
man
and
Tve
rsky
(197
9)
Do
Ih
ave
ago
od
inco
me?
Dep
end
son
the
stan
dar
ds
ofm
yso
ciet
y.
Sh
ould
Ib
eh
appy
wit
ha
5%ra
ise?
Dep
end
son
wh
atI
was
exp
ecti
ng.
Kah
nem
anan
dT
vers
ky(1
979)
intr
od
uce
fou
rm
ain
feat
ure
s:
Ref
eren
ceD
epen
den
ce
Los
sA
vers
ion
Dim
inis
hin
gS
ensi
tivi
ty
Non
linea
rP
rob
abili
tyW
eigh
tin
g
Kos
zegi
&R
abin
2006
-200
8ex
pan
dth
ism
od
elby
end
ogen
izin
gre
fere
nce
poi
nts
wit
hra
tion
alex
pec
tati
ons.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m7
5/
90
Ref
eren
ceD
epen
denc
e:K
osze
gi&
Rab
inS
tyle
KR
assu
me
that
the
pref
eren
ces
are
det
erm
ined
bya
map
u(x|r)
wh
ere
xd
enot
esa
con
sum
pti
onb
un
dle
and
rre
fere
nce
po
int.
Pre
fere
nce
sof
ap
laye
rin
par
ticu
lar
sati
sfy
u(x|r)=
v(x)+
g(x|r).
wh
ere
v(x)
isth
e“c
onsu
mp
tion
uti
lity”
and
g(x|r)
the
“gai
n-l
oss
uti
lity”
.
Con
sum
pti
onu
tilit
yv(x)
sati
sfies
v′>
0an
dv′′<
0,an
dth
us
exh
ibit
sd
ecre
asin
gm
argi
nal
uti
lity.
Uti
lity
isof
ten
also
add
itiv
ely
sep
arab
leac
ross
dim
ensi
ons
v(x)=
∑n i=
1v i(x
i)an
dg(x|r)=
∑n i=
1g i(x
i|ri)
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m7
6/
90
Not
es
Not
es
Ref
eren
ceD
epen
denc
e:K
osze
gi&
Rab
inS
tyle
Th
ega
in-l
oss
fun
ctio
nis
ofte
na
“un
iver
sal
gain
-los
sfu
nct
ion
”sa
tisf
yin
g
g i(x
i|ri)=
µ(v
i(x i)−
v i(r
i)).
Th
eex
pres
sion
says
that
gain
san
dlo
sses
are
afu
nct
ion
only
ofth
ech
ange
inco
nsu
mp
tion
uti
lity
rela
tive
toth
ere
fere
nce
poi
nt.
Th
efo
llow
ing
assu
mp
tion
sar
eim
pos
edon
un
iver
sal
gain
-los
sfu
nct
ion
s:
1µ(v)
isco
nti
nu
ous
for
any
van
dtw
ice
diff
eren
tiab
lefo
rv6=
0.
2µ(v)
isst
rict
lyin
crea
sin
g,an
dµ(0)=
0.
3µ(y)+
µ(−
y)<
µ(v)+
µ(−
v)
for
y>
v>
0.
4µ′′ (
v)≤
0fo
rv>
0,an
dµ′′ (
v)≥
0fo
rv<
0.
5µ′ −(0)/
µ′ +(0)>
1fo
r
µ′ +(0)=
limv→
0µ′ (|v|)
and
µ′ −(0)=
limv→
0µ′ (−|v|)
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m7
7/
90
Ref
eren
ceD
epen
denc
e:G
ain-
Los
sF
unct
ion
Lo
cal
con
cavi
tyat
0is
cap
ture
dby
assu
mp
tion
5.
Los
sav
ersi
onon
larg
est
akes
com
esfr
omth
e3.
Los
sav
ersi
onon
smal
lst
akes
com
esfr
omth
e5.
Dim
inis
hin
gse
nsi
tivi
tyco
mes
from
4.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m7
8/
90
Not
es
Not
es
Ref
eren
ceD
epen
denc
e:E
xam
ple
Ap
erso
nh
asu
tilit
yov
ertw
ogo
od
sm
ugs
,x 1
,an
dm
oney
,x 2
.
Let
v 1(x
1)=
6x1,
let
v 2(x
2)=
x 2,
and
let
µb
ep
iece
wis
e-lin
ear
µ(v)=
vv≥
02v
v<
0
Fir
st,
con
sid
era
mu
gles
sp
erso
nn
otex
pec
tin
gto
bu
ya
mu
g,r=
(0,0).
Ifsh
eb
uys
one
mu
gat
apr
ice
p,
she
gets
:
u(1
,−p|0
,0)=
v1(1)+
v2(−
p)+
µ(v
1(1)−
v1(0))
+µ(v
2(−
p)−
v2(0))
=6−
p+
µ(6)+
µ(−
p)=
6−
p+
6−
2p=
12−
3p.
Ifsh
ed
oes
n’t
bu
yth
ensh
ege
ts:
u(0
,0|0
,0)=
v1(0)+
v2(0)+
µ(v
1(0)−
v1(0))
+µ(v
2(0)−
v2(0))
=0.
So
if12−
3p>
0sh
ew
illb
uy.
Her
will
ingn
ess
top
ayfo
rth
em
ug
is4.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m7
9/
90
Ref
eren
ceD
epen
denc
e:K
osze
gi&
Rab
inS
tyle
Nex
t,co
nsi
der
ap
erso
nw
ith
am
ug
not
exp
ecti
ng
tose
llit
r=
(1,0).
Ifsh
ese
llsh
erm
ug
ata
pric
eq
,sh
ege
ts:
u(0
,q|1
,0)=
v1(0)+
v2(q)+
µ(v
1(0)−
v1(1))
+µ(v
2(q)−
v2(0))
=q+
µ(−
6)+
µ(q)=
2q−
12.
Ifsh
ed
oes
n’t
sell
then
she
gets
:
u(1
,0|1
,0)=
v1(1)+
v2(0)+
µ(v
1(1)−
v1(1))
+µ(v
2(0)−
v2(0))
=6.
So
if2q−
12≥
6sh
ew
illse
ll.H
erw
illin
gnes
sto
acce
pt
for
the
mu
gis
9.
Th
een
dow
men
teff
ect
app
lies
her
eas
aco
nsu
mer
wh
oow
ns
the
mu
gva
lues
it9
wh
erea
son
ew
ho
do
esn
otow
nit
valu
esit
only
4.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m8
0/
90
Not
es
Not
es
Ref
eren
ceD
epen
denc
e:P
erso
nal
Equ
ilibr
ium
An
app
aren
tpr
oble
mw
ith
such
pref
eren
ces
isth
at:
Act
ion
sar
ech
osen
bas
edon
exp
ecta
tion
s(a
per
son
exp
ecti
ng
tob
uy
will
pay
mor
eth
anon
ew
ho
do
esn
ot).
Exp
ecta
tion
sar
eco
nst
rain
edby
acti
ons
(exp
ecta
tion
sar
era
tion
al).
We
nee
da
solu
tion
con
cep
tto
get
arou
nd
this
circ
ula
rity
.S
olu
tion
:
Age
nts
corr
ectl
yp
erce
ive
the
envi
ron
men
tan
dth
eir
resp
onse
s.
Ref
eren
cep
oin
tsge
ner
ated
byth
ese
bel
iefs
max
imiz
eex
pec
ted
uti
lity.
Defi
niti
on
Aco
nsu
mp
tion
bu
nd
lex
isa
per
son
al
equ
ilib
riu
mif
u(x|x)≥
u(x′ |x
)fo
ran
yot
her
con
sum
pti
onb
un
dle
x′ .
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m8
1/
90
Ref
eren
ceD
epen
denc
e:P
erso
nal
Equ
ilibr
ium
Exa
mpl
e
Bot
ond
Go
esS
ho
eS
hop
pin
g:
Bot
ond
likes
togo
sho
esh
opp
ing.
Bot
ond
’sco
nsu
mp
tion
uti
lity
for
mon
eyx 1
and
sho
esx 2
isgi
ven
by
v 1(x
1)=
x 1an
dv 2(x
2)=
40x 2
.
Bot
ond
’su
niv
ersa
lga
inlo
ssfu
nct
ion
is
µ(v)=
ηv
for
v≥
0an
dµ(v)=
λη
vfo
rv<
0.
Bot
ond
know
sth
epr
ice
ofth
esh
oes
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m8
2/
90
Not
es
Not
es
Ref
eren
ceD
epen
denc
e:P
erso
nal
Equ
ilibr
ium
Exa
mpl
e
Wh
enis
ita
per
son
aleq
uili
briu
mfo
rB
oton
dto
bu
ysh
oes
?
Ifso
,th
ere
fere
nce
poi
nt
isr=
(−p
,1),
and
ifB
oton
db
uys
u(−
p,1|−
p,1)=−
p+
40+
µ(0)+
µ(0)=−
p+
40
IfB
oton
dd
oes
not
bu
y
u(0
,0|−
p,1)=
µ(−
40)+
µ(p)=−
40λ
η+
pη
Bot
ond
bu
ysif
the
uti
lity
ofb
uyi
ng
isgr
eate
rth
anif
he
do
esn
ot.
Th
ein
diff
eren
cep
oin
tis
wh
enth
eyar
eeq
ual
u(−
p,1|−
p,1)=
u(0
,0|−
p,1)⇔
p=
401+
λη
1+
η.
Th
ere
isP
Ein
wh
ich
Bot
ond
exp
ects
tob
uy
and
bu
ysif
p≤
p.
Ifλ=
3(l
osse
sar
e3
tim
esw
orse
than
gain
s)an
dη=
1(g
ain
sar
eas
imp
orta
nt
asle
vels
)th
em
ost
Bot
ond
will
pay
is$8
0.N
ava
(LS
E)
Pre
fere
nce
sa
nd
Co
nsu
mp
tio
nM
ich
ael
ma
sT
erm
83
/9
0
Ref
eren
ceD
epen
denc
e:P
erso
nal
Equ
ilibr
ium
Exa
mpl
e
Wh
enca
nit
be
ap
erso
nal
equ
ilibr
ium
for
Bot
ond
not
bu
yth
esh
oes
?
Ifso
,th
ere
fere
nce
poi
nt
isr=
(0,0),
and
ifB
oton
db
uys
u(−
p,1|0
,0)=−
p+
40+
µ(−
p)+
µ(4
0)=−(1
+λ
η)p
+(1
+η)4
0
IfB
oton
dd
oes
n’t
bu
y
u(0
,0|0
,0)=
0+
0+
µ(0)+
µ(0)=
0
Th
ein
diff
eren
cep
oin
tis
wh
enth
eyar
eeq
ual
u(−
p,1|0
,0)=
u(0
,0|0
,0)⇔
p=
401+
η
1+
λη
.
Th
ere
isP
Ein
wh
ich
Bot
ond
exp
ects
ton
otb
uy
and
do
esn
otif
p≥
p.
Ifλ=
3an
dη=
1,B
oton
dw
illn
otb
eab
leto
resi
stb
uyi
ng
the
sho
esfo
ran
ypr
ice
low
erth
an$2
0.N
ava
(LS
E)
Pre
fere
nce
sa
nd
Co
nsu
mp
tio
nM
ich
ael
ma
sT
erm
84
/9
0
Not
es
Not
es
Ref
eren
ceD
epen
denc
e:P
erso
nal
Equ
ilibr
ium
Exa
mpl
e
Wh
enp∈[p
,p],
Bot
ond
may
bu
yor
not
dep
end
ing
onh
isex
pec
tati
ons.
Ifh
eex
pec
tsto
bu
yh
ew
ill,
and
ifh
ed
oes
not
that
he
will
not
.
Defi
niti
on
LetX
be
the
set
ofp
erso
nal
equ
ilibr
ia.
Th
enx∗
isa
pre
ferr
edp
erso
na
leq
uili
bri
um
ifit
isa
PE
and
if
u(x∗ |
x∗ )≥
u(x|x)
for
all
x∈X
.
To
fin
dth
eP
PE
wh
enp∈[p
,p],
calc
ula
tep
ayoff
sin
the
two
PE
asa
fun
ctio
nof
the
pric
ep
.
Inth
eb
uy
PE
Bot
ond
’sp
ayoff
amou
nts
tou(−
p,1|−
p,1)=−
p+
40,
wh
ilein
the
no
bu
yP
Eto
u(0
,0|0
,0)=
0.
So
wh
enp>
40th
enn
otb
uyi
ng
isth
eon
lyP
PE
;w
hen
p<
40b
uyi
ng
isth
eon
lyP
PE
;an
dw
hen
p=
40b
oth
are
PP
E.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m8
5/
90
Ref
eren
ceD
epen
denc
e:K
osze
gi&
Rab
inS
tyle
Ref
eren
cep
oin
tr
isd
eriv
eden
dog
enou
sly
and
isb
ased
on”p
sych
olog
ical
prin
cip
les”
.
Alt
ern
ativ
eth
eori
esal
soex
pla
inh
owco
nsu
mer
sd
evel
open
dog
enou
sly
refe
ren
cep
oin
tsb
ased
onex
pec
tati
ons
abou
tfu
ture
outc
omes
.
Gu
lan
dP
esen
dor
fer
show
that
refe
ren
ced
epen
den
ceis
equ
ival
ent
topr
efer
ence
sfa
ilin
gtr
ansi
tivi
ty.
Ad
ecis
ion
mak
erw
ith
the
abov
eu
tilit
yre
pres
enta
tion
issi
mp
lya
dec
isio
nm
aker
wh
ose
beh
avio
rd
oes
not
exh
ibit
tran
siti
vity
for
wh
ich
ever
reas
on.
Th
e”p
sych
olog
ical
prin
cip
les”
are
void
ofem
pir
ical
con
ten
t.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m8
6/
90
Not
es
Not
es
Mul
ti-s
elve
sM
ode
l
Mu
lti-
selv
esm
od
els
cap
ture
pref
eren
ces
are
not
stat
ion
ary.
Now
con
sum
ers
cho
ose
sub
sets
ofco
nsu
mp
tion
bu
nd
les
(men
us)
,kn
owin
gth
atin
the
futu
reth
eyw
illh
ave
toch
oos
ean
bu
nd
lefr
omth
em
enu
.
Defi
ne
pref
eren
ces
over
any
two
men
us
asfo
llow
sif
A,B⊆
X,
A%
Bm
ean
sse
tA
ispr
efer
red
tose
tB
.
Axi
om
A1
:F
orev
ery
A,B⊆
Xei
ther
A∪
B∼
Aor
A∪
B∼
B.
The
orem
Apr
efer
ence
rela
tion
over
men
us
sati
sfies
A1
ifan
don
lyif
ther
eex
ist
two
fun
ctio
ns
u:X→
Ran
dv
:X→
Rsu
chth
atfo
rev
ery
A,B⊆
X
A%
Bif
and
only
ifm
axx∈a
rgm
ax z∈A
v(z)u(x)≥
max
x∈a
rgm
ax z∈B
v(z)u(x).
Her
ev
repr
esen
tspr
efer
ence
wh
ench
oos
ing
from
am
enu
,u
pref
eren
ces
wh
ench
oos
ing
bet
wee
nm
enu
s.N
ava
(LS
E)
Pre
fere
nce
sa
nd
Co
nsu
mp
tio
nM
ich
ael
ma
sT
erm
87
/9
0
Tem
ptat
ion
and
Sel
f-C
ontr
ol:
Gul
&P
esen
dorf
erS
tyle
Ifco
nsu
mer
sch
oos
em
enu
s,a
con
sum
erw
ho
isaf
raid
ofte
mp
tati
onm
aypr
efer
asm
alle
rm
enu
sto
ab
igge
ron
es.
Eg:
Bro
ccol
ian
dC
ho
cola
te.
Are
pres
enta
tion
theo
rem
byG
ul
and
Pes
end
orfe
rsh
ows
wh
yu
nd
erap
prop
riat
eas
sum
pti
ons
the
uti
lity
W(A
)of
ase
tA⊂
Xis
:
W(A
)=
max
x∈A
(u(x)−
c(x
,A))
wh
ere
u(x)
isth
eco
nsu
mp
tion
uti
lity,
and
c(x
,A)
the
tem
pta
tion
cost
.
Fu
rth
erm
ore,
tem
pta
tion
cost
sca
nb
ere
pres
ente
dby
the
fun
ctio
nal
c(x
,A)=
max
y∈A
V(y)−
V(x)
Th
isis
inte
rpre
ted
asth
eco
stof
cho
osin
gx
over
the
mos
tte
mp
tin
gel
emen
tof
A.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m8
8/
90
Not
es
Not
es
Tem
ptat
ion
and
Sel
f-C
ontr
ol:
Gul
&P
esen
dorf
erS
tyle
Pre
fere
nce
sov
erm
enu
ssa
tisf
yse
tb
etw
een
nes
sif
A%
Bim
plie
sA%
A∪
B%
B.
Set
bet
wee
nn
ess
isn
eces
sari
lysa
tisfi
esif
pref
eren
ces
are
repr
esen
ted
byW
.
Infa
ct,
GP
show
that
set
bet
wee
nn
ess,
ind
epen
den
cean
dco
nti
nu
ity
are
nec
essa
ryan
dsu
ffici
ent
for
pref
eren
ces
tob
ere
pres
ente
dby
W.
Lim
itat
ion
s:
Th
efu
nct
ion
su(x)
and
c(x
,A)
are
only
abst
ract
con
stru
ctio
ns.
On
lyth
epr
efer
ence
sov
ersu
bse
ts(A
%B
)h
ave
emp
iric
alco
nte
nt.
Ifu(x)
and
c(x
,A)
hav
en
oem
pir
ical
orp
sych
olog
ical
con
ten
t,h
owu
sefu
lis
this
repr
esen
tati
onfo
rd
ynam
icm
od
els
wh
ere
choi
ces
from
the
set
Aar
eac
tual
lym
ade?
Th
em
od
elis
sile
nt
wit
hre
spec
tto
wh
eth
erth
eco
nsu
mer
resi
sts
tem
pta
tion
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m8
9/
90
Whe
reto
find
mor
eon
this
?
Th
ege
ner
ald
ebat
e:m
ind
less
orm
ind
ful
econ
omic
s?
Rea
din
glis
t:
Sp
iegl
erb
ook
app
end
ix.
Kos
zegi
&R
abin
2006
,20
07,
2008
.
Gu
l&
Pes
end
orfe
r20
01,
2004
.
Na
va(L
SE
)P
refe
ren
ces
an
dC
on
sum
pti
on
Mic
ha
elm
as
Ter
m9
0/
90
Not
es
Not
es
Pro
duct
ion
and
Fir
ms
EC
411
Slid
es2
Nav
a
LSE
Mic
hae
lmas
Ter
m
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
1/65
Cla
ssic
alT
heor
yof
Pro
duct
ion
Ou
ru
nit
ofan
alys
isin
term
sof
beh
avio
ris
aF
irm
.
We
won
’td
iscu
ssth
eow
ner
ship
ofa
firm
,or
its
man
agem
ent
and
orga
niz
atio
n.
We
only
ask
how
toop
erat
eth
efi
rmto
max
imiz
epr
ofits
.
Th
efi
rmis
trea
ted
asa
bla
ckb
oxw
ith
no
own
ers,
wor
kers
orm
anag
ers.
Just
atr
ansi
tion
box
that
tran
sfor
ms
inp
uts
into
oup
uts
.
Th
em
ain
ingr
edie
nts
ofth
ecl
assi
cal
theo
ryof
pro
du
ctio
nar
e:
Tec
hn
olo
gy
Co
nst
rain
ts(L
imit
sto
pro
du
ctio
nd
ue
tote
chn
olog
ical
orle
gal
con
stra
ints
);
Pro
fit
Ma
xim
izin
gB
eha
vio
r(F
irm
sm
axim
ize
ap
arti
cula
rob
ject
ive
fun
ctio
n,
profi
ts);
Fre
eE
ntr
y&
Pri
ceT
ak
ing
Beh
avi
or.
Th
isth
eory
imp
oses
littl
ed
isci
plin
eon
tech
nol
ogy,
bu
tlo
tson
pay
offs.
Fir
ms
cou
ldm
axim
ize
sale
s,su
rviv
al,
wel
fare
ofw
orke
rsan
d/o
rm
anag
ers.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
2/65
Not
es
Not
es
Roa
dmap
:P
rodu
ctio
n
Th
eai
mof
the
sect
ion
isto
intr
od
uce
asi
mp
lem
od
elth
atal
low
su
sto
pred
ict
the
beh
avio
rof
firm
sin
the
mar
ket.
Th
ean
alys
ispr
oce
eds
asfo
llow
s:
Bas
icd
efin
itio
ns,
voca
bu
lary
and
tech
nol
ogic
alco
nst
rain
ts.
Fir
ms’
opti
miz
atio
npr
oble
m:
profi
tm
axim
izat
ion
,co
stm
inim
izat
ion
,d
ual
ity.
Su
pp
ly,
profi
tfu
nct
ion
and
cost
fun
ctio
n.
Th
esh
ort
run
and
the
lon
gru
n.
Par
tial
equ
ilibr
ium
and
taxe
s.
Cri
tici
sman
dal
tern
ativ
eth
eori
es.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
3/65
Tec
hnol
ogy
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
4/65
Not
es
Not
es
The
Tec
hnol
ogy
Defi
niti
on
Ap
rod
uct
ion
pla
nz=
(z1,.
..,z
m)
isa
vect
orof
net
outp
uts
.
For
agi
ven
pro
du
ctio
np
lan
z,
we
say
that
:
z iis
anin
pu
tif
z i<
0;
z iis
ano
utp
ut
ifz i>
0.
Th
ep
rod
uct
ion
po
ssib
ility
set,
Y,
isth
ese
tof
tech
nol
ogic
ally
feas
ible
pro
du
ctio
np
lan
s.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
5/65
Exa
mpl
es:
Sin
gle
Out
put
Wit
hon
lyon
eou
tpu
t,w
eca
nw
rite
apr
od
uct
ion
pla
nas
(y,−
x)
wh
ere:
yd
enot
esth
eam
oun
tof
outp
ut
and
x=
(x1,.
..x n)
isa
non
neg
ativ
eve
ctor
ofn
inp
uts
.
For
the
case
,of
one
inp
ut
we
hav
e:
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
6/65
Not
es
Not
es
Pro
per
ties
ofY
and
Iso
quan
ts
We
usu
ally
assu
me
Yto
be:
non
-em
pty
,cl
osed
,an
dth
atca
nd
on
oth
ing.
Alt
hou
ghsu
nk
cost
sor
ap
osit
ive
leve
lof
inp
uts
inp
lace
may
chan
geth
is.
Res
tric
tth
ean
alys
isto
the
case
ofo
ne
outp
ut.
Th
eIn
pu
tR
equ
irem
ent
Set
isd
efin
edas
V(y)=x∈
Rn|
(y,−
x)∈
Y.
An
Iso
qu
an
tis
defi
ned
as
Q(y)= x∈
Rn|
x∈
V(y)
and
x/∈
V(y′ )
ify′>
y .
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
7/65
Exa
mpl
e:Is
oqu
ants
IfY
= (y,
−x 1
,−x 2)|
y≤√
x 1+√
x 2 ,
we
hav
eth
at
V(y)=(
x 1,x
2)|
y≤√
x 1+√
x 2,
Q(y)=(
x 1,x
2)|
y=√
x 1+√
x 2.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
8/65
Not
es
Not
es
The
Pro
duct
ion
Fun
ctio
n
Defi
niti
on
Ap
rod
uct
ion
fun
ctio
nis
am
apy=
f(x)
wh
ere
f(x)
isth
em
axim
um
outp
ut
obta
inab
lefr
omx
inY
.
Th
ere
are
clos
eco
nn
ecti
ons
bet
wee
nf
and
the
prop
erti
esof
Y.
We
can
eith
erm
ake
rest
rict
ion
son
Yor
onf
.
Wit
h1
outp
ut
and
nin
pu
ts,
Yis
con
vex⇒
fis
con
cave
.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
9/65
Exa
mpl
e:C
obb-
Dou
glas
Pro
duct
ion
Fun
ctio
n
Con
sid
erth
efo
llow
ing
pro
du
ctio
nfu
nct
ion
,
y=
xα 1x
β 2.
Th
eeq
uat
ion
des
crib
ing
its
iso
qu
ant
for
y=
2is
2=
xα 1x
β 2=⇒
x 2=
( 2 xα 1
)1 β
.
Th
esl
ope
ofth
isis
oq
uan
tis dx 2
dx 1
=−
α β
( 2
xα+
β1
)1 β
.
Itex
pres
ses
the
trad
e-off
bet
wee
nx 1
and
x 2th
atke
eps
outp
ut
con
stan
t.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
10/65
Not
es
Not
es
Exa
mpl
e:L
eont
ief
Pro
duct
ion
Fun
ctio
n
Con
sid
erth
efo
llow
ing
pro
du
ctio
nfu
nct
ion
,
y=
mina
x 1,b
x 2.
Its
iso
qu
ants
are
non
-diff
eren
tiab
lean
dp
lott
edb
elow
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
11/65
Tec
hnic
alR
ate
ofS
ubst
itut
ion
Th
esl
ope
ofis
oq
uan
tsis
fou
nd
byto
tally
diff
eren
tiat
ing
the
pro
du
ctio
nfu
nct
ion
and
eval
uat
ing
such
der
ivat
ive
atd
y=
0.
Con
sid
era
diff
eren
tiab
lepr
od
uct
ion
fun
ctio
ny=
f(x
1,x
2).
Tot
ally
diff
eren
tiat
ing
the
pro
du
ctio
nfu
nct
ion
we
obta
in
dy=
∂f ∂x1
dx 1
+∂f ∂x
2d
x 2.
Set
tin
gd
y=
0,w
efi
nd
the
Tec
hn
ica
lR
ate
of
Su
bst
itu
tio
n(T
RS
),
dx 2
dx 1
=−(∂
f/
∂x1)
(∂f
/∂x
2)
.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
12/65
Not
es
Not
es
Tec
hnic
alR
ate
ofS
ubst
itut
ion
At
afi
xed
outp
ut
leve
ly
,th
eT
RS
expr
esse
sh
owm
uch
do
we
nee
dto
chan
gex 2
ifw
eh
adch
ange
dx 1
inor
der
toke
epou
tpu
tat
y.
Th
eT
RS
isth
era
tio
ofth
em
argi
nal
pro
du
cts,
MPi=
(∂f
/∂x
i).
Th
eT
RS
isal
sokn
own
asM
arg
ina
lR
ate
of
Tra
nsf
orm
ati
on
(MR
T).
For
the
Cob
bD
ougl
aspr
od
uct
ion
fun
ctio
n
TR
S=−
αx
α−1
1x
β 2
βx
α 1x
β−1
2
=−
αx 2
βx 1
.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
13/65
Ret
urns
toS
cale
Apr
od
uct
ion
fun
ctio
nf(x)
can
exh
ibit
:
Con
stan
tR
etu
rns
toS
cale
(CR
S),
if
f(t
x)=
tf(x)
for
any
t>
0.
Dec
reas
ing
Ret
urn
sto
Sca
le(D
RS
),if
f(t
x)<
tf(x)
for
any
t>
1an
dx
for
wh
ich
f(x)>
0.
Incr
easi
ng
Ret
urn
sto
Sca
le(I
RS
):
f(t
x)>
tf(x)
for
any
t>
1an
dx
for
wh
ich
f(x)>
0.
For
inst
ance
,co
nve
xity
ofY
and
0∈
Yto
geth
erim
ply
non
-in
crea
sin
gre
turn
sto
scal
e.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
14/65
Not
es
Not
es
Exa
mpl
e:L
inea
rT
echn
olog
y
Con
sid
era
linea
rpr
od
uct
ion
fun
ctio
n,
y=
ax1+
bx2.
Th
em
apd
isp
lays
CR
S,
sin
ce
f(t
x)=
atx 1
+b
tx2=
t(a
x 1+
bx2)=
tf(x).
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
15/65
Exa
mpl
e:C
obb-
Dou
glas
Tec
hnol
ogy
For
aC
obb
Dou
glas
pro
du
ctio
nfu
nct
ion
y=
xα 1x
β 2w
eh
ave,
f(t
x)=
(tx 1)α
(tx 2)β
=tα
+βx
α 1x
β 2.
Th
eref
ore,
the
pro
du
ctio
nfu
nct
ion
dis
pla
ys:
CR
Sw
hen
α+
β=
1,si
nce
f(t
x)=
tα+
βx
α 1x
β 2=
txα 1x
β 2=
tf(x).
DR
Sw
hen
α+
β<
1,si
nce
f(t
x)=
tα+
βx
α 1x
β 2<
txα 1x
β 2=
tf(x)
for
t>
1&
xα 1x
β 26=
0.
IRS
wh
enα+
β>
1,si
nce
f(t
x)=
tα+
βx
α 1x
β 2>
txα 1x
β 2=
tf(x)
for
t>
1&
xα 1x
β 26=
0.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
16/65
Not
es
Not
es
Hom
ogen
eity
&H
omot
etic
ity
Defi
niti
on
Afu
nct
ion
f:R
n +→
Ris
ho
mo
gen
ou
sof
deg
ree
kif
and
only
if
f(t
x)=
tkf(x)
for
any
t>
0&
x∈
Rn +
.
Exa
mp
le:
y=
xα 1x
β 2is
hom
ogen
ous
ofd
egre
eα+
β.
Defi
niti
on
Afu
nct
ion
f:R
n +→
Ris
ho
mo
thet
icif
ast
rict
lyin
crea
sin
gfu
nct
ion
g:R→
Ran
da
hom
ogen
ous
ofd
egre
e1
fun
ctio
nh
:Rn +→
Rex
ist
such
that
f(x)=
g(h(x))
for
any
x∈
Rn +
.
Exe
rcis
e:If
afu
nct
ion
ish
omot
het
ic,
TR
Sat
xeq
ual
sT
RS
attx
.H
int:
Ifa
map
ish
omog
eneo
us
ofd
egre
ek
,it
sp
arti
ald
eriv
ativ
esar
e?
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
17/65
Pro
fitM
axim
izat
ion
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
18/65
Not
es
Not
es
Intr
oto
Pro
fit
Max
imiz
atio
n
We
hav
ein
tro
du
ced
assu
mp
tion
sab
out
firm
s:
Fir
ms
are
profi
tm
axim
izin
g,pr
ice
taki
ng
enti
ties
.
Fir
ms
hav
ea
tech
nol
ogy
defi
ned
bypr
od
uct
ion
pos
sib
iliti
esse
tY
.
We
thin
kof
Yas
non
-em
pty
,cl
osed
,an
dth
atca
nd
on
oth
ing.
Th
eb
oun
dar
yof
Yca
nb
ed
efin
edas
the
pro
du
ctio
nfu
nct
ion
.
Th
ete
chn
olog
yis
wh
atw
em
ake
assu
mp
tion
son
.
We
hav
ein
tro
du
ced
con
cep
tsof
retu
rns
tosc
ale.
Now
we
turn
topr
ofit
max
imiz
atio
n:
Th
eh
igh
est
profi
tlin
eon
Yw
illd
eter
min
eth
eso
luti
on.
Bu
ta
solu
tion
may
not
nec
essa
rily
exis
t.
For
exam
ple
ifth
epr
od
uct
ion
fun
ctio
nex
hib
its
IRS
.G
RA
PH
.
For
exis
ten
cew
en
eed
Yto
be
eith
erst
rict
lyco
nve
xor
bou
nd
ed.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
19/65
Pro
fit
Max
imiz
atio
n
Fir
ms
are
pric
e-ta
kers
byp
erfe
ctco
mp
etit
ion
.
We
emp
loy
the
follo
win
gco
nve
nti
ons:
yd
enot
esou
tpu
t;
py
den
otes
the
pric
eof
outp
ut;
x=
(x1,.
..,x
n)
den
otes
ave
ctor
ofin
pu
ts;
w=
(w1,.
..,w
n)
den
otes
the
vect
orof
inp
ut
pric
es;
wx=
∑n i=
1wix
id
enot
esex
pen
dit
ure
onin
pu
ts.
Afi
rmch
oos
es(y
,−x)
tom
axim
ize
py
y−
wx
sub
ject
to(y
,−x)∈
Y.
Th
eh
igh
est
profi
tlin
ein
ters
ecti
ng
Yid
enti
fies
the
solu
tion
ifit
exis
ts.
Bu
tn
oso
luti
onm
ayex
ist
(e.g
.if
Yex
hib
its
IRS
).
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
20/65
Not
es
Not
es
Pro
fit
Max
imiz
atio
n
By
profi
tm
axim
izat
ion
,a
firm
pro
du
ces
the
max
imal
outp
ut
give
nin
pu
ts.
Th
us,
the
prob
lem
can
be
rest
ated
usi
ng
the
pro
du
ctio
nfu
nct
ion
f(x).
Afi
rmch
oos
es(y
,−x)
tom
axim
ize
py
y−
wx
sub
ject
toy=
f(x).
Hen
ce,
afi
rmch
oos
esx
tom
axim
ize
py
f(x)−
wx
So
wit
hsi
ngl
eou
tpu
tth
efi
rmes
sen
tial
lych
oos
eson
lyin
pu
ts.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
21/65
Inpu
tD
eman
ds&
Out
put
Sup
ply
Sol
vin
gth
epr
ofit
max
imiz
atio
npr
oble
m,
we
obta
in:
the
vect
orof
(un
con
dit
ion
al)
fact
or
or
inp
ut
dem
an
ds
x(p
y,w
)=
(x1(p
y,w
),x 2(p
y,w
),..
.,x n(p
y,w
));
the
sup
ply
fun
ctio
nfo
rou
tpu
t
y(p
y,w
)=
f(x(p
y,w
)).
Fac
tor
dem
and
san
dth
esu
pp
lyfu
nct
ion
are
hom
ogen
eou
sof
deg
ree
zero
in(p
y,w
),
x(p
y,w
)=
x(t
py
,tw)
and
y(t
py
,tw)=
y(p
y,w
).
Do
we
alw
ays
hav
ea
solu
tion
?N
o,fo
rex
amp
leC
RS
wit
hso
me
pric
es,
orIR
S.
We
nee
dY
tob
eb
oun
ded
from
abov
ean
dst
rict
lyco
nve
x.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
22/65
Not
es
Not
es
Fir
stO
rder
Nec
essa
ryC
ondi
tion
s
Rec
all
that
x(p
y,w
)m
axim
izes
py
f(x)−
wx
.
Th
isis
anco
nst
rain
edop
tim
izat
ion
prob
lem
wit
hlin
ear
non
-neg
ativ
ity
con
stra
ints
,an
da
pos
sib
lyco
nca
veob
ject
ive
fun
ctio
n.
Ifx(p
y,w
)is
anin
teri
orso
luti
on,
Fir
stO
rder
Co
nd
itio
ns
req
uir
e
py
∂f(x)
∂xi
∣ ∣ ∣ ∣ x=x(py,w
)
−wi=
0fo
ran
yi=
1,2,
...,
n.
FO
Cfo
rth
epr
ofit
max
imiz
atio
npr
oble
mre
qu
ire:
the
mar
gin
alpr
od
uct
ofan
inp
ut
toeq
ual
its
pric
en
orm
aliz
edby
the
outp
ut
pric
e;
the
mar
gin
alra
teof
tran
sfor
mat
ion
bet
wee
ntw
oin
pu
tsto
equ
alth
eir
pric
era
tio.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
23/65
Sec
ond
Ord
erS
uffici
ent
Con
diti
ons
Defi
ne
seco
nd
ord
erp
arti
ald
eriv
ativ
esas
follo
ws,
f ij(x)=
∂2f(x)
∂xi∂
x j.
Als
od
efin
eth
eH
essi
anm
atri
xas
soci
ated
toth
eop
tim
izat
ion
prob
lem
as
H(x)=
py
f 11(x)
..py
f 1n(x)
.. ..py
f n1(x)
..py
f nn(x)
.N
eces
sary
con
dit
ion
sal
sore
qu
ire
H(x)
tob
en
egat
ive
sem
idefi
nit
eat
the
solu
tion
x=
x(p
y,w
).
Iff
isco
nca
ve,
orY
isst
rict
lyco
nve
x,F
OC
are
suffi
cien
tfo
rx(p
y,w
)to
be
agl
obal
max
imu
m(n
otn
eces
sari
lyu
niq
ue)
,as
H(x)
isn
egat
ive
defi
nit
e.
SO
Con
lyd
isci
plin
ef
asth
ere
stof
the
obje
ctiv
efu
nct
ion
islin
ear.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
24/65
Not
es
Not
es
The
Pro
fit
Fun
ctio
n
Defi
ne
the
profi
tfu
nct
ion
π(p
y,w
)as
π(p
y,w
)=
py
f(x(p
y,w
))−
wx(p
y,w
).
Rec
all
that
this
isa
max
imu
mva
lue
fun
ctio
n.
Exa
mp
le:
Con
sid
erth
epr
od
uct
ion
fun
ctio
ny=√
x 1+√
x 2.
FO
Cre
qu
ire
that
:
1 2py(x
1)−
1 2−
w1=
0⇒
x 1(p
y,w
)=
( py
2w1
) 2 ;
1 2py(x
2)−
1 2−
w2=
0⇒
x 2(p
y,w
)=
( py
2w2
) 2 .
Th
eref
ore,
itfo
llow
sth
at:
y(p
y,w
)=
py
2w1+
py
2w2
&π(p
y,w
)=
p2 y 4
( 1 w1+
1 w2
) .
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
25/65
Non
-Inc
reas
ing
Inpu
tD
eman
ds
Fro
mth
epr
evio
us
exam
ple
we
can
con
clu
de
that
∂x1(p
y,w
)
∂w1
<0
&∂y
(py
,w)
∂py
>0.
Inth
ege
ner
alca
se,
wit
hn
vari
able
s,F
OC
req
uir
e
py
df(x)=
w.
By
tota
llyd
iffer
enti
atin
gF
OC
wrt
w1
we
obta
in
py
f 11
...
py
f 1n
...
...
py
f n1
...
py
f nn
dx 1
/d
w1
...
...
dx n
/d
w1
= 1 0 ..
. 0
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
26/65
Not
es
Not
es
Non
-Inc
reas
ing
Inpu
tD
eman
ds
∂x1
∂w1=
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣1py
f 12
...
py
f 1n
0py
f 22
...
py
f 2n
...
0py
f n2
...
py
f nn
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣|H|
=
∣ ∣ ∣ ∣ ∣ ∣py
f 22
...
py
f 2n
...
py
f n2
...
py
f nn
∣ ∣ ∣ ∣ ∣ ∣|H|
Sin
ceth
eH
essi
anof
fis
neg
ativ
ese
mid
efin
ite:
the
den
omin
ator
isei
ther
zero
orh
asth
esa
me
sign
as(−
1)n
.
the
nu
mer
ator
isei
ther
zero
orh
asth
esa
me
sign
as(−
1)n−1.
Hen
ce,
∂x1/
∂w1≤
0,if
Hw
ell
defi
ned
soth
at|H|6=
0.
Pro
fit
max
imiz
atio
n⇒
inp
ut
dem
and
sar
en
on-i
ncr
easi
ng
inow
npr
ice.
We
will
der
ive
this
exp
licit
lyla
ter.
Rem
emb
erth
atw
ith
con
sum
ers
itd
epen
ded
onin
com
ean
dsu
bst
itu
tion
effec
ts.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
27/65
Exa
mpl
es&
Cou
nter
exam
ples
Exe
rcis
e:D
eter
min
eth
esi
gnof
∂y/
∂py
.H
int:
FO
Cre
qu
ire
py∑
n j=1f ij
∂xj
∂py+
∂f ∂xi=
0.
Exa
mp
le:
y=√ m
inx
1,x
2.
Fir
stn
ote
that
tom
axim
ize
profi
tsx 1
=x 2
.H
ence
,x 1
isch
osen
tom
axim
ize
py√
x 1−(w
1+
w2)
x 1.
Th
eu
nco
nd
itio
nal
fact
ord
eman
ds
are
x 1(p
y,w
)=
x 2(p
y,w
)=
(py
2(w
1+
w2)
) 2 .
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
28/65
Not
es
Not
es
Exa
mpl
es&
Cou
nter
exam
ples
Exa
mp
le:
y=
minx
1,x
2.
Ifpy>
w1+
w2,
profi
tsar
eu
nb
oun
ded
.T
he
firm
’ssu
pp
lyis
per
fect
lyel
asti
cat
py=
w1+
w2
(CR
S).
Exa
mp
le:
y=
x 1+
x 2.
Ifpy>
minw
1,w
2,
profi
tsar
eu
nb
oun
ded
.T
he
firm
’ssu
pp
lyis
per
fect
lyel
asti
cat
py=
minw
1,w
2
(CR
S).
Exa
mp
le:
y=
(x1+
x 2)2
.T
he
max
imu
mpr
ofits
are
un
bou
nd
edan
dth
efi
rm’s
sup
ply
isn
otw
ell
defi
ned
(IR
S).
Th
epr
evio
us
exam
ple
ses
tab
lish
that
:
we
do
n’t
alw
ays
ha
vea
solu
tio
n;
we
can
no
ta
lway
su
sefi
rst
ord
erco
nd
itio
ns;
we
som
etim
esh
ave
mu
ltip
leso
luti
on
s.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
29/65
Pro
per
ties
ofth
eP
rofi
tF
unct
ion
Lem
ma
Th
epr
ofit
fun
ctio
nπ(p
y,w
)sa
tisfi
esth
efo
llow
ing
prop
erti
es
1π(p
y,w
)is
non
-dec
reas
ing
inpy
;
2π(p
y,w
)is
non
-in
crea
sin
gin
wi,
for
any
i=
1,..
.,n
;
3π(p
y,w
)is
hom
ogen
eou
sof
deg
ree
1in
(py
,w);
4π(p
y,w
)is
con
vex
in(p
y,w
);
5π(p
y,w
)is
con
tin
uou
sin
(py
,w).
Th
ese
are
pred
icti
ons
that
can
be
test
ed.
Th
ein
tuit
ion
for
the
firs
ttw
op
arts
isob
viou
s(p
lot
tosh
ow).
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
30/65
Not
es
Not
es
Pro
ving
Pro
per
ties
ofth
eP
rofi
tF
unct
ion
1.
Tak
ep′ y≥
py
.T
hen
,
π(p
y,w
)≤
p′ yy(p
y,w
)−
wx(p
y,w
)≤
π(p′ y,w
).
2.
Th
epr
oof
for
wi
isan
alog
ous
and
left
asan
exer
cise
.
3.
By
defi
nit
ion
,fo
ran
y(y′ ,−
x′ )∈
Yw
eh
ave
py
y(p
y,w
)−
wx(p
y,w
)≥
py
y′ −
wx′ .
Hen
ce,
for
t>
0an
dfo
ran
y(y′ ,−
x′ )∈
Y,
tpy
y(p
y,w
)−
twx(p
y,w
)≥
tpy
y′ −
twx′ .
Th
us,(y(p
y,w
),−
x(p
y,w
))is
opti
mal
at(t
py
,tw)
and
π(t
py
,tw)=
tπ(p
y,w
).
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
31/65
Pro
ving
Pro
per
ties
ofth
eP
rofi
tF
unct
ion
4.
Con
sid
ertw
opr
ice
vect
ors,(p
y,w
)an
d( p′ y,
w′) .
For
som
eλ∈(0
,1)
defi
ne:
(p′′ y
,w′′ )
=λ(p
y,w
)+(1−
λ)( p′ y,w′) .
Wit
hth
ese
defi
nit
ion
s,n
ote
that
:
(1)
π(p
y,w
)≥
py
y(p′′ y
,w′′ )−
wx(p′′ y
,w′′ )
;
(2)
π(p′ y,w′ )≥
p′ yy(p′′ y
,w′′ )−
w′ x(p′′ y
,w′′ )
.
Mu
ltip
lyin
g(1
)by
λan
d(2
)by
(1−
λ),
and
sum
min
g
λπ(p
y,w
)+(1−
λ)π
(p′ y,w′ )≥
p′′ y
y(p′′ y
,w′′ )−
w′′ x
(p′′ y
,w′′ )
,
wh
ich
imp
lies
λπ(p
y,w
)+(1−
λ)π
(p′ y,w′ )≥
π(p′′ y
,w′′ )
.
5.
Exe
rcis
e(o
pti
onal
).
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
32/65
Not
es
Not
es
Hot
ellin
g’s
Lem
ma
Th
epr
ofit
fun
ctio
nob
tain
sby
plu
ggin
gy(p
y,w
)an
dx(p
y,w
)in
profi
ts,
π(p
y,w
)=
py
y(p
y,w
)−
wx(p
y,w
).
Can
we
reve
rse
the
oper
atio
nto
reco
ver
y(p
y,w
)an
dx(p
y,w
)?
Th
eE
nve
lop
eT
heo
rem
then
imp
lies
the
follo
win
gre
sult
.
Lem
ma
Hot
ellin
g’s
Lem
ma
stat
esth
at
∂π(p
y,w
)
∂py
=y(p
y,w
)&
∂π(p
y,w
)
∂wi
=−
x i(p
y,w
),fo
ri=
1,..
.,n
.
IfY
isw
ell
beh
aved
,pr
ice
taki
ng
beh
avio
ran
dfi
xed
pric
esim
ply
ad
ual
des
crip
tion
ofte
chn
olog
y.
Su
cha
des
crip
tion
has
agr
eat
virt
ue
inap
plic
atio
ns,
asw
eca
nco
mp
ute
sup
ply
dir
ectl
yfr
omth
een
velo
pe
theo
rem
.Nava
(LSE)
ProductionandFirms
MichaelmasTerm
33/65
Usi
ngH
otel
ling’
sL
emm
a
Exa
mp
le:
Con
sid
erth
epr
od
uct
ion
fun
ctio
ny=√
x 1+√
x 2.
Imm
edia
tely
obta
inth
at
π(p
y,w
)=
p2 y 4
( 1 w1+
1 w2
) ,y(p
y,w
)=
∂π(p
y,w
)
∂py
=py
2w1+
py
2w2
,..
.
By
Hot
ellin
g’s
Lem
ma
and
the
con
vexi
tyof
the
profi
tfu
nct
ion
we
hav
e
∂y(p
y,w
)
∂py
=∂2
π(p
y,w
)
∂p2 y
≥0,
∂xi(
py
,w)
∂wi
=−
∂2π(p
y,w
)
∂w2 i
≤0.
Th
isis
asi
mp
leS
lutz
ky
equ
ati
on
wit
hte
sta
ble
pre
dic
tio
ns.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
34/65
Not
es
Not
es
Impl
icat
ions
ofH
otel
ling’
sL
emm
a
Th
us,
byco
nve
xity
ofth
epr
ofit
fun
ctio
nit
follo
ws
that
:
ow
n-p
rice
effec
tsar
ep
osi
tive
for
ou
tpu
t;
ow
n-p
rice
effec
tsar
en
ega
tive
for
inp
uts
.
As
dia
gon
alte
rms
ofa
pos
itiv
ese
mid
efin
ite
mat
rix,
are
pos
itiv
e.
Th
isis
know
nas
the
law
of
sup
plie
s:q
uan
titi
esre
spon
din
the
sam
ed
irec
tion
aspr
ice
chan
ges.
Gen
eral
lyit
can
be
wri
tten
as
(p−
p′ )(y−
y′ )=
(py−
py′ )+(p′ y′ −
p′ y′ )≥
0.
Th
ere
are
no
wea
lth
effec
ts,
just
sub
stit
uti
oneff
ects
sin
cepr
ices
hav
en
oeff
ect
onco
nst
rain
tsn
ow.
An
oth
erte
stab
leim
plic
atio
nis
that
∂xj(
py
,w)
∂wi
=∂x
i(py
,w)
∂wj
=−
∂2π(p
y,w
)
∂wi∂
wj
.
Cro
ss-p
rice
effec
tsar
esy
mm
etri
c(n
otve
ryin
tuit
ive,
bu
tte
stab
le).
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
35/65
Cos
tM
inim
izat
ion
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
36/65
Not
es
Not
es
The
Cos
tM
inim
izat
ion
Pro
blem
Su
pp
ose
that
afi
rmm
ust
pro
du
cey
un
its
ofou
tpu
t.
Now
,th
efi
rmch
oos
esin
pu
tsx
tom
inim
ize
cost
s,
wx
sub
ject
toy=
f(x).
Th
eso
luti
ons
are
the
con
dit
ion
al
fact
or
or
inp
ut
dem
an
ds
x(w
,y)=
x 1(w
,y)
x 2(w
,y)
...
x n(w
,y)
.T
he
cost
fun
ctio
nth
us
sati
sfies
c(w
,y)=
wx(w
,y)=
w1x 1(w
,y)+
...+
wnx n(w
,y).
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
37/65
Cos
tM
inim
izat
ion:
Sol
utio
n
Th
eL
agra
ngi
anof
this
prog
ram
sati
sfies
L=
wx+
λ(y−
f(x) )
.
Ifx(w
,y)
isan
inte
rior
solu
tion
and
f i=
∂f(x)/
∂xi,
the
FO
Cre
qu
ire
that
∂L ∂λ=
y−
f(x(w
,y))
=0,
∂L ∂xi=
wi−
λf i(x(w
,y))
=0,
for
any
i=
1,..
.,n
.
Su
chco
nd
itio
ns
sim
plif
yto
wi
wj=
f i f j,
for
any
i,j=
1,..
.,n
.
At
aso
luti
on,
the
mu
ltip
lier
char
acte
rize
sth
em
argi
nal
cost
ofpr
od
uct
ion
(i.e
.m
argi
nal
valu
eof
rela
xin
gte
chco
nst
rain
t).
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
38/65
Not
es
Not
es
Cos
tM
inim
izat
ion:
Plo
t
For
the
case
oftw
ofa
ctor
s,−
w1/
w2
isth
esl
ope
ofth
eis
oco
stcu
rve
w1x 1
+w2x 2
=k
.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
39/65
Cos
tM
inim
izat
ion:
Intu
itio
n
Wh
ysh
ould
we
use
cost
min
imiz
atio
n?
ital
way
sh
old
slo
cally
even
ifth
epr
ofit
fun
ctio
nis
not
wel
lb
ehav
ed;
wh
enY
isco
nve
x,th
ere
isa
one-
to-o
ne
map
pin
gw
ith
profi
tm
ax;
inot
her
wor
ds,
profi
tm
axim
plie
sco
stm
inso
they
do
not
give
us
mor
ein
form
atio
n;
iffi
rms
are
only
pric
eta
kers
inth
ein
pu
tm
arke
tth
enw
orks
wh
erea
spr
ofit
max
do
esn
ot.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
40/65
Not
es
Not
es
Cos
tM
inim
izat
ion:
Suffi
cien
tC
ondi
tion
s
Con
sid
era
crit
ical
poi
nt
ofth
eL
agra
nge
an,(x∗ ,
λ∗ )
,
f(x∗ )
=y
and
λ∗ f
i(x∗ )
=wi
for
any
i=
1,..
.,n
.
Iff(x)
isco
nca
ve,
x∗
isa
glob
alm
inim
um
.
For
loca
lS
OC
,in
stea
d,
stu
dy
Bor
der
edH
essi
anof
the
prob
lem
,
H=
0−
f 1..
.−
f n−
f 1−
λf11
...−
λf1n
...
...
...
...
−f n−
λfn1
...−
λfnn
.If
at(x∗ ,
λ∗ )
the
larg
est
n−
1le
adin
gpr
inci
pal
min
ors
are
neg
ativ
e,x∗
isa
stri
ctlo
cal
min
imu
m.
[SO
Cfo
ra
min
imiz
atio
npr
oble
mre
qu
ire
the
larg
est
n−
ele
adin
gm
inor
sto
hav
eth
esi
gnof
(−1)e
,w
her
ee
isth
en
um
ber
ofb
ind
ing
con
stra
ints
].Nava
(LSE)
ProductionandFirms
MichaelmasTerm
41/65
Cos
tM
inim
izat
ion:
Exa
mpl
eI
Aga
in,
con
sid
erth
epr
od
uct
ion
fun
ctio
nf(x
1,x
2)=√
x 1+√
x 2.
FO
Cof
the
cost
min
imiz
atio
npr
oble
mre
qu
ire:
wi−
λx−1
/2
i 2=
0fo
ri∈1
,2.
Th
eref
ore,
we
obta
inth
at:
w1
w2=
√ x 2 x 1=⇒√
x 2=√
x 1w1
w2
=⇒
y=√
x 1
( 1+
w1
w2
) .
Con
seq
uen
tly,
con
dit
ion
alin
pu
td
eman
ds,
and
cost
ssa
tisf
y:
x i(w
,y)=
(wj
w2+
w1
) 2 y2
for
j6=
i∈1
,2;
c(w
,y)=
w1w2
w2+
w1
y2.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
42/65
Not
es
Not
es
Cos
tM
inim
izat
ion:
Exa
mpl
eII
Nex
t,co
nsi
der
the
pro
du
ctio
nfu
nct
ion
f(x
1,x
2)=
mina
x 1,x
2.
Op
tim
alit
yim
plie
sth
atax
1=
x 2=
y.
Con
seq
uen
tly,
con
dit
ion
alin
pu
td
eman
ds,
and
cost
ssa
tisf
y:
x 1(w
,y)=
y a;
x 2(w
,y)=
y;
c(w
,y)=( w 1 a
+w2
) y.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
43/65
Cos
tM
inim
izat
ion:
Exa
mpl
eII
I
Fin
ally
,co
nsi
der
the
pro
du
ctio
nfu
nct
ion
f(x
1,x
2)=
ax1+
x 2.
By
sub
stit
uti
ng
the
con
stra
int
wit
heq
ual
ity,
we
may
cho
ose
x 1to
min
imiz
e
w1x 1
+w2(y−
ax1)=
(w1−
w2a )
x 1+
w2y
.
Con
seq
uen
tly,
con
dit
ion
alin
pu
td
eman
ds
sati
sfy:
ifw1<
aw2
x 1(w
,y)=
y/
a&
x 2(w
,y)=
0if
w1>
aw2
x 1(w
,y)=
0&
x 2(w
,y)=
yif
w1=
aw2
ax1(w
,y)+
x 2(w
,y)=
y
wh
ileco
sts
ofpr
od
uct
ion
sati
sfy:
c(w
,y)=
minw
1 a,w
2y
.
Th
ese
exam
ple
sill
ust
rate
that
FO
Ca
pp
roa
chis
no
tid
eal.
Alt
hou
gha
solu
tion
exis
tsby
con
tin
uit
y,so
met
imes
we
can
not
use
FO
Cto
fin
dit
.Nava
(LSE)
ProductionandFirms
MichaelmasTerm
44/65
Not
es
Not
es
Pro
per
ties
ofth
eC
ost
Fun
ctio
n
Lem
ma
Th
eco
stfu
nct
ion
c(y
,w)
sati
sfies
the
follo
win
gpr
oper
ties
:
1c(w
,y)
isn
on-d
ecre
asin
gin
wi,
for
i=
1,..
.,n
;
2c(w
,y)
ish
omog
eneo
us
ofd
egre
e1
inw
;
3c(w
,y)
isco
nca
vein
w;
4c(w
,y)
isco
nti
nu
ous
inw
.
Pro
of:
Exe
rcis
e.
Th
eE
nve
lop
eT
heo
rem
then
imp
lies
the
follo
win
gre
sult
.
Lem
ma
Sh
eph
ard
’sL
emm
ast
ates
that
∂c(w
,y)
∂wi
=x i(w
,y)
for
any
i=
1,..
.,n
.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
45/65
Pro
per
ties
ofth
eC
ost
Fun
ctio
n
Th
eco
nca
vity
ofth
eco
stfu
nct
ion
agai
nim
plie
sth
at
∂xi(
w,y)
∂wi≤
0fo
ran
yi=
1,..
.,n
.
By
revi
siti
ng
agai
nth
epr
oper
ties
ofth
eim
plie
dd
eman
ds
we
can
fin
du
sual
con
clu
sion
sab
out
hom
ogen
eity
and
pric
eeff
ects
:
ow
n-p
rice
effec
tsar
en
ega
tive
for
inp
uts
;
cro
ss-p
rice
effec
tsar
esy
mm
etri
c.
To
con
clu
de:
Som
etim
esco
stfu
nct
ion
sch
arac
teri
zeco
mp
lete
lyte
chn
olog
y(w
hen
tech
nol
ogy
isco
nve
xan
dm
onot
one)
.C
ost
fun
ctio
ns
are
anim
por
tan
tto
ol,
and
are
gen
eral
lyob
serv
able
.T
he
prop
erti
esd
eriv
edh
ave
test
able
imp
licat
ion
san
dw
ere
der
ived
from
very
few
assu
mp
tion
s.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
46/65
Not
es
Not
es
Sho
rtvs
Lon
gR
unP
rodu
ctio
n
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
47/65
Sho
rtR
unvs
Lon
gR
unP
rodu
ctio
n
Giv
enan
yco
stfu
nct
ion
byc(y),
itis
pos
sib
leto
rest
ate
the
profi
tm
axim
izat
ion
prob
lem
asth
efi
rmch
oos
ing
yto
max
imiz
e
py
y−
c(y)
FO
Cfo
rth
ispr
oble
mre
qu
ire
py−
dc(y)
dy
=0
=⇒
Pri
ce=
Mar
gin
alC
ost.
So
far
we
hav
eon
lyfo
cuse
don
lon
gru
n(a
sal
lin
pu
tsw
ere
vari
able
).
Now
dec
omp
ose
tota
lco
stin
tova
ria
ble
an
dfi
xed
cost
s:
c(y)
=v(y)
+F
dep
end
ent
ony
ind
epen
den
tof
y
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
48/65
Not
es
Not
es
Fix
edC
osts
and
the
Sho
rtR
un
Som
eco
sts
are
fixe
d:
bec
ause
ofth
ep
hys
ical
nat
ure
ofin
pu
ts;
bec
ause
som
ein
pu
tsca
nn
otb
ead
just
edin
the
shor
tru
n.
Fix
edco
sts:
are
sun
kan
din
curr
edev
enw
hen
outp
ut
isze
ro;
are
ign
ore
din
the
sho
rtru
nm
axim
izat
ion
prob
lem
.
Ave
rage
Cos
tis
defi
ned
as
AC(y)≡
AV
C(y)+
AF
C(y)≡
v(y)
y+
F y.
IfA
C(y)
isin
crea
sin
gin
yw
eh
ave
dis
eco
no
mie
so
fsc
ale
.If
AC(y)
isd
ecre
asin
gin
yw
eh
ave
eco
no
mie
so
fsc
ale
.Nava
(LSE)
ProductionandFirms
MichaelmasTerm
49/65
Min
imum
Ave
rage
Cos
t
Fir
stco
nsi
der
min
imiz
ing
the
aver
age
cost
ofpr
od
uct
ion
min
yA
C(y)=
min
yv(y)
y+
F y.
FO
Cfo
rA
Cm
inim
izat
ion
sim
ply
req
uir
e
dA
C(y)
dy
=1 y
( dc(y)
dy−
AC(y)) =
0.
Th
eav
erag
eco
stis
min
imiz
edw
hen
itco
inci
des
wit
hth
em
argi
nal
cost
dc(y)
dy
=A
C(y).
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
50/65
Not
es
Not
es
The
Sho
rtR
unP
robl
em
Let
Fb
esu
nk
inth
esh
ort
run
.
Pro
fits
can
be
wri
tten
as
py
y−
c(y)=
y(p
y−
AV
C(y))−
F
Afi
rmse
llson
lyif
shor
tte
rmpr
ofits
are
pos
itiv
e
y(p
y−
AV
C(y))≥
0.
Pro
fit
max
imiz
atio
nre
qu
ires
the
firm
:
tose
ty=
0,if
py<
min
yv(y)/
y;
toch
oos
ey
soth
atpy=
c′ (
y)
oth
erw
ise.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
51/65
The
Sho
rtR
unS
uppl
yC
urve
Th
esh
ort
run
sup
ply
curv
eof
afi
rmis
plo
tted
her
e
AC
dec
reas
esb
efor
eit
sm
inim
um
(th
ein
ters
ecti
onw
ith
MC
)an
dth
enin
crea
ses.
AC
dec
reas
esat
firs
tb
ecau
seb
oth
the
AF
Can
dth
eA
VC
dec
reas
e.
Bu
tA
Cin
crea
ses
once
cap
acit
yis
reac
hed
asA
VC
incr
ease
s.Nava
(LSE)
ProductionandFirms
MichaelmasTerm
52/65
Not
es
Not
es
The
Sho
rtR
unA
ggre
gate
Sup
ply
Con
sid
era
mar
ket
wit
hm
firm
s.
Let
y i(p
y)
den
ote
the
shor
tru
nsu
pp
lyof
firm
i.
Th
esh
ort
run
ind
ust
ry(o
ra
gg
reg
ate
)su
pp
lyis
det
erm
ined
asfo
llow
s:
S(p
y)=
∑m i=
1y i(p
y).
Agg
rega
tion
isst
raig
htf
orw
ard
her
e.
Itis
,as
ifon
eb
igfi
rmh
adm
ade
anop
tim
alsu
pp
lyd
ecis
ion
.
Pro
du
ctio
nis
effici
entl
yal
loca
ted
amon
gfi
rms
asit
min
imiz
esco
sts!
All
ind
ivid
ual
prop
erti
esar
esa
tisfi
edat
the
aggr
egat
ele
vel.
Th
ese
pow
erfu
lre
sult
sar
ed
ue
topr
ice
taki
ng
beh
avio
r.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
53/65
The
Lon
gR
unA
ggre
gate
Sup
ply
Inth
elo
ng
run
,fi
rms
can
ente
ran
dex
itth
ein
du
stry
.
Wit
hn
ob
arri
ers
toen
try,
the
lon
gru
nin
du
stry
sup
ply
isp
erfe
ctly
ela
stic
ata
pric
eeq
ual
toth
em
inim
um
ave
rag
eco
st.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
54/65
Not
es
Not
es
Mar
kets
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
55/65
Pro
duce
rS
urpl
us
Con
sid
erth
em
arke
tfo
ron
eco
mm
od
ity,
wh
ere
yi (
py)
isfi
rmi’
ssu
pp
ly.
Su
pp
ose
that
the
pric
eof
outp
ut
incr
ease
sfr
omp0 y
top1 y.
By
Hot
ellin
g’s
Lem
ma,
the
chan
gein
profi
tsfo
rfi
rmi
is
∆i=∫ p1 y
p0 y
dπi (
py)
dpy
dpy=∫ p1 y
p0 y
yi (
py)d
py
.
We
refe
rto
∆i
asth
ep
rod
uce
rsu
rplu
s.It
iden
tifi
esth
ear
eau
nd
erth
esu
pp
lycu
rve
and
mea
sure
sth
e“w
illin
gnes
sto
pay
”of
apr
od
uce
r.
IfS(p
y)
isth
eag
greg
ate
sup
ply
,it
follo
ws
that
m ∑ i=1
∆i=∫ p1 y
p0 y
S(p
y)d
py
.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
56/65
Not
es
Not
es
Par
tial
Equ
ilibr
ium
&E
last
icit
ies
Let
D(p
y)
be
the
ag
gre
ga
ted
ema
nd
for
this
com
mo
dit
y.
Am
arke
tcl
eari
ng
pri
ceis
apr
ice
that
solv
es
S(p
y)=
D(p
y).
Ap
arti
al
equ
ilib
riu
min
this
mar
ket
con
sist
ofa
mar
ket
clea
rin
gpr
ice
and
ofth
eco
rres
pon
din
gop
tim
ald
eman
dan
dsu
pp
lyd
ecis
ion
s.
Th
ere
spon
sive
nes
sof
dem
and
and
sup
ply
topr
ice
chan
ges
ism
easu
red
by:
the
ela
stic
ity
of
dem
an
d,
εD(p
y)=
dD(p
y)
dpy
py
D(p
y)=
%ch
ange
ind
eman
d
%ch
ange
inpr
ice
;
the
ela
stic
ity
of
sup
ply
,
εS(p
y)=
dS(p
y)
dpy
py
S(p
y)=
%ch
ange
insu
pp
ly
%ch
ange
inpr
ice
.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
57/65
Tax
es
Con
sid
era
mar
ket
inw
hic
h:
pS
den
otes
the
pric
ere
ceiv
edby
pro
du
cers
;
pD
den
otes
pric
ep
aid
byco
nsu
mer
s;
td
enot
esth
eta
xra
te.
Th
isse
tup
cap
ture
sb
oth
:
qu
an
tity
taxe
s,pD=
pS+
t;
valu
eta
xes
pD=
pS(1
+t )
.
Eq
uili
briu
mst
illre
qu
ires
mar
ket
clea
rin
g(a
llu
nit
ssu
pp
lied
are
sold
).
Th
eref
ore
we
mu
sth
ave
that
D(p
D)=
S(p
S).
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
58/65
Not
es
Not
es
Qua
ntit
yT
axes
Th
ep
lot
dep
icts
the
equ
ilibr
ium
ina
mar
ket
wit
hq
uan
tity
taxe
s.
Th
eam
oun
tth
atco
nsu
mer
san
dpr
od
uce
rsw
ould
wan
tto
pay
for
rem
ovin
gth
eta
xis
exce
eds
the
reve
nu
era
ised
bec
ause
ofth
ed
ead
wei
gh
tlo
ss.
Th
isis
the
clas
sic
argu
men
tfo
rth
ein
effici
ency
ofta
xes.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
59/65
Tow
ards
Gen
eral
Equ
ilibr
ium
Ou
rbr
ief
intr
od
uct
ion
toeq
uili
briu
man
alys
isw
asco
nst
rain
edto
asi
ngl
em
arke
t,as
inp
ut
pric
esw
ere
kep
tco
nst
ant
thro
ugh
out.
How
ever
,w
hen
apr
ice
chan
ges,
pric
esin
oth
erm
arke
tsm
ayva
ryto
oas
con
sum
ers
and
firm
sre
adju
stco
nsu
mp
tion
and
pro
du
ctio
nd
ecis
ion
s.
Mor
ege
ner
ally
,w
eca
rrie
dou
ta
par
tia
leq
uili
bri
um
anal
ysis
.
Th
isca
nb
em
isle
adin
g,as
com
par
ativ
est
atic
sn
egle
ctch
ange
sin
oth
erm
arke
tsw
hic
hm
ayh
ave
afe
edb
ack
effec
ton
the
mar
ket
con
sid
ered
.
Nex
tw
eso
lve
for
ag
ener
al
equ
ilib
riu
mw
ith
all
the
mar
kets
aton
ce.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
60/65
Not
es
Not
es
New
The
orie
sof
the
Fir
m
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
61/65
Dep
artu
res
from
the
Cla
ssic
alT
heor
yof
the
Fir
m
Sev
eral
limit
atio
ns
ofth
ecl
assi
cal
theo
ryh
ave
bee
nad
dre
ssed
:
Pro
fit
max
imiz
atio
nm
ayn
otb
eth
etr
ue
ob
ject
ive
of
am
an
ag
er.
Pro
per
tyri
gh
tsa
nd
firm
ow
ner
ship
may
affec
tth
eeq
uili
briu
min
am
arke
tw
ith
fric
tion
s.F
orin
stan
ce,
asu
pp
lier
may
pref
era
mer
ger
wit
ha
clie
nt
tob
arga
inin
gin
the
mar
ket
over
pric
esan
dq
uan
titi
es,
asth
em
erge
rm
ayso
lve
ince
nti
vepr
oble
ms.
Ake
ypr
edic
tion
ofW
illia
mso
n’s
theo
ry(v
alid
ated
emp
iric
ally
)is
that
firm
sar
em
ore
likel
yto
mer
geif
wh
atth
eypr
od
uce
orth
eir
asse
tsar
esp
ecifi
cto
each
oth
eran
dn
otto
the
rest
ofth
em
arke
t.
No
n-p
rofi
tor
gan
izat
ion
sm
ayb
esu
cces
sfu
lw
ith
out
targ
etin
gpr
ofits
.L
ike
con
sum
ers
thei
rob
ject
ives
are
pin
ned
dow
nby
pref
eren
ces,
and
tech
niq
ues
exte
nd
toco
ver
such
scen
ario
s.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
62/65
Not
es
Not
es
Man
ager
sO
bjec
tive
s
Th
eob
ject
ives
ofm
anag
erm
ayd
iffer
from
profi
tsfo
rse
vera
lre
ason
s.
Exa
mp
le1
:M
an
ag
ers
may
dis
like
effo
rtan
dca
reon
lyab
out
thei
rw
age.
Ifso
,in
cen
tive
s(b
onu
ses)
may
be
req
uir
edto
ind
uce
the
opti
mal
effor
td
ecis
ion
.B
ut
wh
enin
cen
tive
sar
eco
stly
topr
ovid
e,th
efi
rmm
igh
tpr
efer
ind
uce
asu
bop
tim
aleff
ort
dec
isio
n.
You
will
dis
cuss
this
inLT
.
Exa
mp
le2
:M
an
ag
ers
may
ha
veca
reer
con
cern
s.If
man
ager
sar
ep
aid
mor
ein
the
mar
ket
wh
enth
eyar
ep
erce
ived
tob
eta
len
ted
.T
hey
may
care
mor
eab
out
app
eari
ng
tale
nte
dan
des
tab
lish
ing
are
pu
tati
on,
rath
erth
anab
out
firm
profi
ts.
Rep
uta
tion
alco
nce
rns
can
be
ben
efici
alan
dca
nin
du
cem
anag
ers
toex
ert
effor
t.H
owev
er,
they
are
occ
asio
nal
lyh
arm
ful
wh
enm
anag
ers
take
dec
isio
ns
inth
eir
bes
tin
tere
stra
ther
than
the
firm
’s.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
63/65
Ext
ra:
Car
eer
Con
cern
sI
Man
ager
sga
inh
igh
ersa
lari
esw
hen
thei
rre
pu
tati
onis
hig
h.
Th
est
ate
ofth
ew
orld
wis
eith
ergo
od
orb
ad,
w∈G
,B.
Th
est
ate
isB
wit
hpr
obab
ility
p>
0.5.
Th
eco
rrec
tac
tion
for
the
man
ager
tota
keis
:
toin
vest
inst
ate
G;
not
toin
vest
inst
ate
B.
Th
em
anag
erre
ceiv
esa
sign
als∈G
,B
abou
tw
,P
r(s=
w|w
)=
q.
Th
esi
gnal
ispr
ivat
ein
form
atio
n(i
tis
wh
yth
em
anag
eris
hir
ed).
For
ago
od
man
ager
q=
1,fo
rev
ery
oth
erm
anag
erq=
0.5.
Th
em
anag
erkn
ows
his
tale
nt,
wh
ileth
em
arke
tb
elie
ves
that
he
isgo
od
wit
hpr
obab
ility
α.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
64/65
Not
es
Not
es
Ext
ra:
Car
eer
Con
cern
sII
Pay
offs
are
rest
rict
edso
that
:
inve
stm
ent
ispr
ofit
max
imiz
ing
only
ifth
em
anag
eris
goo
dan
dsi
gnal
iss=
G;
the
man
ager
’spr
efer
ence
sar
efu
llyd
eter
min
edby
the
up
dat
edb
elie
fα′
abou
th
ista
len
t.
Ifm
anag
ers
max
imiz
epr
ofits
the
mar
ket
per
ceiv
esth
em
anag
er:
asgo
od
ifan
inve
stm
ent
isob
serv
ed,
α′=
1;
asle
sslik
ely
tob
ego
od
ifn
oin
vest
men
tis
obse
rved
,α′<
α.
Ifm
anag
erm
axim
ize
rep
uta
tion
s:
bad
man
ager
skn
owth
atth
ose
wh
oin
vest
are
bel
ieve
dto
be
goo
d;
itis
not
aneq
uili
briu
mfo
rth
emto
max
imiz
efi
rmpr
ofits
;
bad
man
ager
sto
inve
stm
ore
than
they
shou
ldto
show
off.
Nava
(LSE)
ProductionandFirms
MichaelmasTerm
65/65
Not
es
Not
es
Gen
eral
Equ
ilibr
ium
EC
411
Slid
es3
Nav
a
LSE
Mic
hae
lmas
Ter
m
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
1/35
Roa
dmap
:G
ener
alE
quili
briu
m&
Mar
kets
Th
eai
mis
toin
tro
du
cea
gen
eral
mo
del
ofb
ehav
ior
ofco
nsu
mer
san
dfi
rms
that
cap
ture
sth
eco
nse
qu
ence
sof
spill
over
sac
ross
mar
kets
.
Sin
cem
arke
tsar
ein
terd
epen
den
t,ex
ten
din
gth
eth
eory
ofp
arti
aleq
uili
briu
mto
gen
eral
equ
ilibr
ium
isa
nat
ura
lst
ep.
Th
ean
alys
ispr
oce
eds
asfo
llow
s:
Exc
han
geE
con
omie
s.
Defi
nit
ion
ofa
Wal
rasi
anE
qu
ilibr
ium
.
Pro
per
ties
ofW
alra
sian
Eq
uili
bria
.
Effi
cien
cy&
Par
eto
Effi
cien
cy.
Wel
fare
Th
eore
ms.
Gen
eral
Eq
uili
briu
mw
ith
Pro
du
ctio
n.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
2/35
Not
es
Not
es
Exc
hang
eE
cono
mie
s
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
3/35
Exc
hang
eE
cono
mie
s
Fir
stw
eab
stra
ctfr
ompr
od
uct
ion
and
stu
dy
how
goo
ds
are
trad
edby
con
sum
ers
end
owed
ofa
fixe
dam
oun
ts.
Con
sid
eran
exch
an
ge
eco
no
my
:
wit
hn
com
mo
dit
ies
and
kco
nsu
mer
s;
inw
hic
hea
chco
nsu
mer
jis
des
crib
edby
:
au
tilit
yfu
nct
ion
Uj (
xj )
=Uj( x
j 1,.
..,x
j n
) ;
anin
itia
len
dow
men
tωj=
( ωj 1,.
..,ω
j n
) .
Aco
nsu
mp
tio
nb
un
dle
for
con
sum
erj
isxj=
( xj 1,.
..,x
j n
) .
An
allo
cati
on
x=
(x1,.
..,x
k)
assi
gns
ab
un
dle
toea
chco
nsu
mer
.
An
allo
cati
onx
isfe
asi
ble
if∑
k j=1xj≤
∑k j=
1ω
j .
Afe
asib
leal
loca
tion
obta
ins
byre
dis
trib
uti
ng
ord
estr
oyin
gen
dow
men
ts.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
4/35
Not
es
Not
es
Wal
rasi
anE
quili
briu
m
Defi
niti
on
AW
alr
asi
an
equ
ilib
riu
m(x
,p)
con
sist
sof
anal
loca
tion
x=
(x1,.
..,x
k)
and
ofa
pric
eve
ctor
p=
(p1,.
..,p
n)
such
that
;
1x
isfe
asib
le;
2xj
max
imiz
esU
j (xj )
sub
ject
top
xj≤
pω
jfo
ral
lj=
1,..
.,k
.
Wal
rasi
aneq
uili
bria
are
also
refe
rred
toas
com
pet
itiv
eor
gen
eral
equ
ilibr
ia.
Aco
mp
etit
ive
equ
ilibr
ium
con
sist
sof
anal
loca
tion
and
ofa
pric
eve
ctor
st:
con
sum
ers
choi
ces
are
opti
mal
wh
enta
kin
gas
give
nsu
chpr
ices
;
the
imp
lied
dem
and
sar
efe
asib
le(i
.e.
all
mar
kets
clea
r).
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
5/35
Intr
odu
ctor
yE
xam
ple:
Mar
ket
Cle
arin
g
Con
sid
era
sim
ple
econ
omy
wit
h:
k=
2an
dn=
2;
ω1=
(3,1)
and
U1(x
1)=
minx
1 1,x
1 2;
ω2=
(1,3)
and
U2(x
2)=
x2 1x2 2.
Sol
vin
gth
eco
nsu
mer
s’pr
oble
ms
we
obta
inth
at
x1 1(p
,pω
1)=
3p1+
p2
p1+
p2
x1 2(p
,pω
1)=
3p1+
p2
p1+
p2
x2 1(p
,pω
2)=
p1+
3p2
2p1
x2 2(p
,pω
2)=
p1+
3p2
2p2
Eq
uili
briu
mco
nd
itio
ns
req
uir
eth
etw
om
arke
tsto
clea
r:
x1 1+
x2 1=
3p1+
p2
p1+
p2+
p1+
3p2
2p1
=4;
x1 2+
x2 2=
3p1+
p2
p1+
p2+
p1+
3p2
2p2
=4.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
6/35
Not
es
Not
es
Intr
odu
ctor
yE
xam
ple:
Equ
ilibr
ium
Fro
mth
eeq
uili
briu
mco
nd
itio
nfo
rgo
od
1ob
tain
(p1+
3p2)(p
2−
p1)
2p1(p
1+
p2)
=0⇒
p1=
p2.
Su
bst
itu
tin
gin
the
con
dit
ion
for
the
mar
ket
for
goo
d2
2+
2=
4.
Key
obse
rvat
ion
s:
Eq
uili
briu
min
the
mar
ket
for
goo
d1
imp
lies
Eq
uili
briu
min
the
mar
ket
for
goo
d2.
Th
eeq
uili
briu
mpr
ice
vect
oris
not
un
iqu
e.O
nly
the
pric
era
tio
isp
inn
edd
own
,p1/
p2=
1.
Th
eeq
uili
briu
mal
loca
tion
sati
sfies
:x1 1=
2,x1 2=
2,x2 1=
2,an
dx2 2=
2.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
7/35
Intr
odu
ctor
yE
xam
ple:
Edg
ewor
thB
ox
Au
sefu
lex
erci
seis
top
lot
the
exam
ple
inan
dE
dge
wor
thB
ox.
Pro
ceed
by:
plo
ttin
gth
eE
dge
wor
thb
oxan
dth
efe
asib
leal
loca
tion
s;
fin
din
gth
een
dow
men
t,an
dd
raw
ing
bu
dge
tco
nst
rain
ts;
plo
ttin
gth
eeq
uili
briu
mit
self
;
show
ing
that
p1/
p2>
1ca
nn
otb
ean
equ
ilibr
ium
,si
nce
con
sum
er1
dem
and
sof
goo
d2
mor
eth
an2
wan
tsto
give
.
Sh
owm
ore
gen
eral
lyth
atan
equ
ilibr
ium
isa
pric
era
tio
for
wh
ich
con
sum
ers’
sd
eman
ds
are
opti
mal
and
mar
kets
clea
r.
At
anin
teri
orso
luti
onco
nsu
mer
seq
ual
ize
MR
S.
Bu
tb
oun
dar
yso
luti
ons,
mu
ltip
licit
yof
equ
ilibr
iaan
dn
on-e
xist
ence
are
oth
erp
ossi
ble
ph
enom
ena.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
8/35
Not
es
Not
es
Intr
odu
ctor
yE
xam
ple:
Edg
ewor
thB
ox
Th
ep
lot
ofE
dge
wor
thB
oxin
this
exam
ple
thu
sam
oun
tsto
:
Th
eso
luti
onis
inte
rior
and
equ
aliz
esth
eM
RS
ofth
etw
oco
nsu
mer
s.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
9/35
Nor
mal
izin
gP
rice
s
Rec
all
that
:
xj i(p
,pω
j )d
enot
esj’
sd
eman
dfo
rgo
od
i;
xj (
p,p
ωj )
den
otes
vect
orof
j’s
dem
and
s;
xj i(p
,pω
j )is
hom
ogen
eou
sof
deg
ree
0in
p.
Fac
t
Bec
ause
dem
and
sar
eh
omog
eneo
us
ofd
egre
e0
ifp
isan
equ
ilibr
ium
pric
eve
ctor
,tp
,t>
0,is
also
aneq
uili
briu
mpr
ice
vect
or.
Hen
ce,
we
can
nor
mal
ize
one
pric
e.
Wit
hou
tlo
ss,
set
pi=
1fo
ron
ear
bit
rary
goo
di.
Cal
lgo
od
ith
en
um
era
ire
go
od
.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
10/35
Not
es
Not
es
Exc
ess
Dem
and
&W
alra
sL
aw
Defi
ne
the
vect
orof
j’s
exce
ssd
ema
nd
sas
zj (
p,p
ωj )=
xj (
p,p
ωj )−
ωj .
Defi
ne
the
vect
orof
ag
gre
ga
teex
cess
dem
an
ds
as
z(p)=
∑k j=
1zj (
p,p
ωj )
.
Defi
ne
the
ag
gre
ga
teex
cess
dem
an
dfo
rg
oo
di
as
z i(p)=
∑k j=
1
( xj i(p
,pω
j )−
ωj i) .
Lem
ma
(Wal
ras
Law
)
IfL
NS
hol
ds,
the
valu
eof
exce
ssd
eman
dis
equ
alto
zero
,piz
i(p)=
0.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
11/35
Wal
ras
Law
:P
roof
Pro
of:
As
LN
S(L
oca
lN
onS
atia
tion
)h
old
s,
pzj (
p,p
ωj )=
p( x
j (p
,pω
j )−
ωj) =
0,fo
ran
yj=
1,..
.,k
.
Su
mm
ing
over
all
pla
yers
then
imp
lies
that
pz(p)=
p∑
k j=1
zj (
p,p
ωj )=
∑k j=
1p
zj (
p,p
ωj )=
0.
Th
ela
tter
can
be
rest
ated
inte
rms
ofgo
od
sas
follo
ws
pz(p)=
∑n i=
1pi∑
k j=1
( xj i(p
,pω
j )−
ωj i) =
∑n i=
1piz
i(p)=
0.
Ob
serv
eth
atpiz
i(p)≤
0fo
ral
li
sin
ce:
pric
esar
en
on-n
egat
ive,
pi≥
0;
exce
ssd
eman
ds
are
non
-pos
itiv
eby
feas
ibili
tyz i(p)≤
0.
Bu
tth
ela
tter
than
imp
lies
that
piz
i(p)=
0fo
ral
li!
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
12/35
Not
es
Not
es
Som
eR
emar
ks:
Ass
umpt
ions
Com
pet
itiv
eeq
uili
bria
are
red
uce
dd
escr
ipti
ons
ofan
econ
omy
inw
hic
hin
stit
uti
ons
ofex
chan
gere
mai
nu
nm
od
elle
d(a
mar
ket
can
be
con
ceiv
edas
anau
ctio
nee
r,as
ace
ntr
alm
arke
tp
lace
,or
asa
mar
ket
mak
er).
We
do
not
exp
lain
how
con
sum
ers
trad
e.A
gen
tsac
tas
pric
e-ta
kin
gu
tilit
y-m
axim
izin
gm
ach
ines
,m
arke
tsar
eco
nce
ived
asfr
icti
onle
ssin
stit
uti
ons
inw
hic
hex
chan
geta
kes
pla
ceat
once
(in
visi
ble
ha
nd
).
Som
est
ron
gas
sum
pti
onw
ere
invo
ked
tod
eriv
eou
rre
sult
s.
Th
ese
req
uir
edm
arke
tsto
be
fric
tion
less
,an
dre
qu
ired
con
sum
ers:
tob
eab
leto
bu
yan
dse
llan
yam
oun
tof
goo
ds
atcu
rren
tpr
ices
;
tob
eaw
are
ofth
ecu
rren
tpr
ices
;
tota
kepr
ices
asgi
ven
;
tofa
ceth
esa
me
pric
es;
tom
axim
ize
uti
lity.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
13/35
Som
eR
emar
ks:
Lim
itat
ions
&E
vide
nce
Lim
itat
ion
sar
ise
bec
ause
the
mo
del
can
not
cap
ture
envi
ron
men
tsin
wh
ich
:
goo
ds
trad
eat
diff
eren
tpr
ices
atd
iffer
ent
loca
tion
s;
con
sum
ers
ign
ore,
bu
td
ynam
ical
lyfo
reca
stof
futu
repr
ices
;
con
sum
ers
can
not
bu
yor
sell
qu
anti
ties
wit
hou
taff
ecti
ng
you
rpr
ices
.
Sta
teco
nti
ng
ent
com
mo
dit
ies
can
byp
ass
som
eof
thes
elim
itat
ion
s.
Evi
den
ce:
Inco
ntr
olle
dex
per
imen
ts,
sub
ject
sw
ere
ind
uce
dto
hav
ed
iffer
ent
pref
eren
ces,
and
pu
tin
toan
exch
ange
econ
omy,
bu
yin
gan
dse
llin
gco
mm
od
itie
s.A
mid
dle
man
wh
oig
nor
edth
eag
greg
ate
dem
and
and
sup
ply
,w
asch
osen
tocl
ear
mar
kets
bu
yin
gfr
omse
llers
,w
hile
selli
ng
tob
uye
rs.
Th
isop
erat
ion
was
rep
eate
dn
um
erou
sti
mes
toal
low
lear
nin
gab
out
the
envi
ron
men
t.S
ub
ject
sle
arn
tq
uic
kly
the
pric
esat
wh
ich
mar
kets
wou
ldcl
ear,
and
thes
epr
ices
wer
ecl
ose
toeq
uili
briu
mpr
ices
.T
he
evid
ence
sup
por
ted
the
Wal
rasi
anE
qu
ilibr
ium
asan
appr
opri
ate
solu
tion
con
cep
tfo
rla
rge
cen
tral
ized
mar
kets
(Plo
tt19
86).
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
14/35
Not
es
Not
es
Effi
cien
cyan
dW
elfa
re
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
15/35
Par
eto
Effi
cien
cy
Aco
mm
onju
stifi
cati
onfo
rm
arke
tsis
that
mar
kets
atta
ineffi
cien
cy.
Th
en
ext
defi
nit
ion
iscr
uci
alfo
rth
ew
elfa
repr
oper
ties
ofeq
uili
bria
.
Defi
niti
on
An
allo
cati
onx=
(x1,.
..,x
k)
isP
aret
oeffi
cien
tif
and
only
if:
1it
isfe
asib
le;
2th
ere
isn
oot
her
feas
ible
allo
cati
onx=
(x1,.
..,x
k)
such
that
:
(a)
Uj (
xj )≥
Uj (
xj )
for
any
j∈1
,...
,k;
(b)
Uj (
xj )>
Uj (
xj )
for
som
ej∈1
,...
,k.
Com
men
ts:
ther
ear
en
opr
ices
invo
lved
inth
ed
efin
itio
nof
Par
eto
effici
ency
;
inan
EB
:fi
nd
the
ineffi
cien
t,effi
cien
t,b
oun
dar
yeffi
cien
tal
loca
tion
s.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
16/35
Not
es
Not
es
The
Par
eto
Pro
blem
&T
heC
ore
AP
aret
oeffi
cien
t(P
E)
allo
cati
on(x
1,x
2)
max
imiz
esU
1(x
1)
sub
ject
to:
U2(x
2)=
U2(x
2)
and
x1+
x2=
ω1+
ω2.
Th
eco
ntr
act
curv
eid
enti
fies
all
PE
allo
cati
ons,
wh
ileth
eco
reid
enti
fies
thos
eP
Eal
loca
tion
sin
wh
ich
pla
yers
are
bet
ter
offth
anat
the
end
owm
ent.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
17/35
Opt
imal
ity
inth
eP
aret
oP
robl
em
If(x
1,x
2)
isin
teri
or,
FO
Cfo
rth
eP
aret
opr
oble
mim
ply
MR
S1=
MR
S2
&x1+
x2=
ω1+
ω2.
Exa
mp
le:
Con
sid
eran
dec
onom
yw
ith
:
k=
2an
dn=
2;
ω1=
(3,1)
and
U1(x
1)=
x1 1x1 2;
ω2=
(1,3)
and
U2(x
2)=
x2 1x2 2.
FO
Cfo
rth
eP
aret
opr
oble
mre
qu
ire:
x1 2/
x1 1=
x2 2/
x2 1,
x1 1+
x2 1=
4,an
dx1 2+
x2 2=
4.
Itfo
llow
sim
med
iate
lyth
at
x1 2/
x1 1=
(4−
x1 2)/
(4−
x1 1)⇒
x1 2=
x1 1.
Th
eco
ntr
act
curv
eco
inci
des
wit
hth
e45
olin
e.Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
18/35
Not
es
Not
es
The
Fir
stW
elfa
reT
heor
em
The
orem
(The
Fir
stW
elfa
reT
heor
em)
IfL
NS
hol
ds,
any
com
pet
itiv
eeq
uili
briu
mal
loca
tion
isP
aret
oeffi
cien
t.
Pro
of:
Let
(x,p)
be
aco
mp
etit
ive
equ
ilibr
ium
.
By
defi
nit
ion
ofco
mp
etit
ive
equ
ilibr
ium
,x
isfe
asib
le.
Su
pp
ose
how
ever
that
xis
not
Par
eto
effici
ent.
Ifso
,th
ere
exis
tsan
oth
erfe
asib
leal
loca
tion
xsu
chth
at
(a)
Uj (
xj )≥
Uj (
xj )
for
any
j∈1
,...
,k;
(b)
Uj (
xj )>
Uj (
xj )
for
som
ej∈1
,...
,k.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
19/35
The
Fir
stW
elfa
reT
heor
em
Pro
of
Co
nti
nu
ed:
Rec
all
that
xsa
tisfi
es
(a)
Uj (
xj )≥
Uj (
xj )
for
any
j∈1
,...
,k;
(b)
Uj (
xj )>
Uj (
xj )
for
som
ej∈1
,...
,k.
Fir
st,
obse
rve
that
(a)
imp
lies
that
pxj≥
pω
jfo
ran
yj.
Ifp
xj<
pω
jfo
rso
me
j,by
LN
Sth
ere
exis
tsxj
such
that
pxj<
pω
jan
dU
j (xj )>
Uj (
xj )≥
Uj (
xj )
.
Bu
tth
isis
imp
ossi
ble
asxj
max
imiz
esu
tilit
y.
Nex
tob
serv
eth
at(b
)im
plie
sth
atp
xj>
pω
jfo
rso
me
j.
Th
isfo
llow
s,as
xj
max
imiz
esU
j (xj )
sub
ject
top
xj≤
pω
j .
Hen
ce,
∑k j=
1p
xj>
∑k j=
1p
ωj ,
con
trad
icti
ng
the
feas
ibili
tyof
x.
QE
D
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
20/35
Not
es
Not
es
Fir
stW
elfa
reT
heor
em:
Plo
t
Key
assu
mp
tion
s:
Pri
ceta
kin
gb
ehav
ior
[age
nts
take
pric
esas
agi
ven
]
No
exte
rnal
itie
s[u
tilit
yon
lyd
epen
ds
onco
nsu
mp
tion
]
Com
ple
tem
arke
ts[a
ny
com
mo
dit
yca
nb
etr
aded
]
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
21/35
Com
men
tson
the
Fir
stW
elfa
reT
heor
em
Inte
rior
com
pet
itiv
eeq
uili
bria
are
alw
ays
Par
eto
effici
ent
asco
nsu
mer
sse
tth
eir
MR
Seq
ual
toth
epr
ice
rati
o.T
her
efor
e,M
RS
coin
cid
eac
ross
pla
yers
and
effici
ency
obta
ins.
Gen
eral
Com
men
tson
Par
eto
Effi
cien
cy:
Alt
hou
ghP
aret
oE
ffici
ency
isa
mea
nin
gfu
lco
nce
pt
toev
alu
ate
wel
fare
itco
mp
lete
lyab
stra
cts
from
fair
nes
s.A
dic
tato
rco
nsu
min
gal
lgo
od
sw
ould
yiel
dP
aret
oeffi
cien
cy.
Oth
erw
elfa
recr
iter
iah
ave
bee
nco
nte
mp
late
dsu
chas
max
-min
uti
lity,
oru
tilit
yag
greg
atio
n.
Th
eJu
ngl
eE
con
omic
s:in
stea
dof
pric
es,
itis
pos
sib
leto
con
sid
erp
ower
rela
tion
s.A
neq
uili
briu
mal
loca
tion
issu
chth
atn
oon
eca
nb
eb
ette
roff
byta
kin
gso
met
hin
gav
aila
ble
toh
im.
Th
ism
od
elal
soh
asan
equ
ilibr
ium
wh
ich
isP
aret
oeffi
cien
t.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
22/35
Not
es
Not
es
Exa
mpl
eE
xter
nalit
ies
Con
sid
eran
econ
omy
wit
h:
k=
2an
dn=
2;
ω1=
(3,1)
and
U1(x
1)=
minx
1 1,x
1 2−
2x2 1;
ω2=
(1,3)
and
U2(x
2)=
x2 1x2 2−
2x1 2.
Th
eeq
uili
briu
mal
loca
tion
agai
nsa
tisfi
es:
x1 1=
2,x1 2=
2,x2 1=
2,x2 2=
2.
Bot
hco
nsu
mer
sar
ew
orse
offaf
ter
trad
e.
Th
eeq
uili
briu
mis
Par
eto
dom
inat
ed.
Th
ein
itia
lu
tilit
ies
are
U1=−
1,U
2=
1.
Eq
uili
briu
mu
tilit
ies
are
U1=−
2,U
2=
0.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
23/35
The
Sec
ond
Wel
fare
The
orem
(Eas
y)
The
orem
(The
Sec
ond
The
orem
ofW
elfa
re–
Eas
y)
Con
sid
era
Par
eto
effici
ent
allo
cati
onx
.
Su
pp
ose
that
aco
mp
etit
ive
equ
ilibr
ium
(p,x)
exis
tsw
hen
ω=
x.
Ifso
,(p
,x)
isa
com
pet
itiv
eeq
uili
briu
m.
Pro
of:
Ob
serv
eth
atfo
ran
yco
nsu
mer
j=
1,..
.,k
:
sin
cexj
isin
the
bu
dge
tco
nst
rain
tof
j,U
j (xj )≥
Uj (
xj )
;
sin
cex
isP
aret
oeffi
cien
t,U
j (xj )=
Uj (
xj )
.
Hen
ce,
xj
isop
tim
algi
ven
p.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
24/35
Not
es
Not
es
The
Sec
ond
Wel
fare
The
orem
(Har
d)
The
orem
(The
Sec
ond
The
orem
ofW
elfa
re–
Har
d)
Con
sid
era
Par
eto
effici
ent
allo
cati
onx
and
sup
pos
eth
at:
1T
he
allo
cati
onx
isin
teri
or.
2T
he
pref
eren
ces
ofev
ery
con
sum
erar
e
con
vex,
con
tin
uou
s,st
rict
lym
onot
onic
.
Th
en,
xis
aco
mp
etit
ive
equ
ilibr
ium
allo
cati
onw
hen
ω=
x.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
25/35
Con
vexi
ty
Th
eim
por
tan
ceof
con
vex
pref
eren
ces
isd
etai
led
byth
ep
lot
bel
ow:
Con
sum
er2
do
esn
otm
axim
ize
uti
lity
atω
.
Ifpr
efer
ence
sar
en
otco
nve
x,ex
iste
nce
can
not
guar
ante
ed.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
26/35
Not
es
Not
es
Com
men
tson
the
Sec
ond
Wel
fare
The
orem
Th
ese
con
dw
elfa
rest
ates
that
any
effici
ent
allo
cati
onca
nb
ed
ecen
tral
ized
into
aco
mp
etit
ive
equ
ilibr
ium
iftr
ansf
ers
are
feas
ible
.
To
dec
entr
aliz
ean
effici
ent
allo
cati
onp
lan
ner
wou
ldn
eed
:
info
rmat
ion
abou
tpr
efer
ence
san
den
dow
men
ts;
pow
erto
enac
ttr
ansf
ers;
tom
ake
sure
that
all
pric
esar
ekn
own
.
Th
isex
erci
sew
illfa
ilh
owev
erif
ther
ear
e:ex
tern
alit
ies,
mar
ket
pow
er,
pu
blic
goo
ds,
orin
com
ple
tein
form
atio
n.
Exi
sten
ceof
aco
mp
etit
ive
equ
ilibr
ium
ispr
oven
via
fixe
dp
oin
tth
eore
ms,
ifpr
efer
ence
sar
eco
nti
nu
ous,
con
vex
and
loca
llyn
on-s
atia
ted
.
GE
isa
clos
edin
terr
elat
edsy
stem
,as
opp
osed
top
arti
aleq
uili
briu
m.
Itis
suit
able
toad
dre
sspr
oble
ms
wh
ich
rela
teto
the
wh
ole
econ
omy.
Its
bea
uty
lies
inth
eam
bit
iou
sre
sult
sob
tain
edw
ith
few
free
par
amet
ers.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
27/35
Pro
duct
ion
Eco
nom
ies
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
28/35
Not
es
Not
es
Gen
eral
Equ
ilibr
ium
wit
hP
rodu
ctio
n
Con
sid
eran
econ
omy
wit
h:
nco
mm
od
itie
s;
kco
nsu
mer
s;
mfi
rms.
Su
pp
ose
that
each
firm
max
imiz
espr
ofits
.
Let
πi(
p)
be
the
profi
tsof
firm
i.
Let
αj i
be
con
sum
erj’
ssh
are
ofth
epr
ofits
offi
rmi.
Con
sum
erj’
sb
ud
get
con
stra
int
bec
omes
pxj≤
pω
j+
∑m i=
1αj iπ
i(p)
All
the
mai
nre
sult
sd
eriv
edfo
rex
chan
geec
onom
ies
exte
nd
topr
od
uct
ion
econ
omie
sw
ith
min
orm
od
ifica
tion
s.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
29/35
Exa
mpl
e:P
rodu
ctio
nI
Con
sid
erth
efo
llow
ing
econ
omy:
1T
wo
firm
s,1
and
2,pr
od
uce
goo
ds
y 1an
dy 2
usi
ng
the
sam
ein
pu
tx
.P
rod
uct
ion
fun
ctio
ns
sati
sfy
y 1=√
x 1an
dy 2
=√
x 2,
wh
ere
x iis
the
amou
nt
ofx
use
dfo
rpr
od
uci
ng
y i.
2T
wo
con
sum
ers,
Aan
dB
,h
ave
uti
lity
fun
ctio
ns
UA=
yA 1
yA 2
and
UB=
yB 1
yB 2
.
Con
sum
ers
der
ive
no
uti
lity
from
inp
ut
x.
3T
he
tota
len
dow
men
tof
y 1an
dy 2
isze
ro.
Con
sum
erA
own
sfi
rm1
and
5u
nit
sof
x.
Con
sum
erB
own
sfi
rm2
and
3u
nit
sof
x.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
30/35
Not
es
Not
es
Exa
mpl
e:P
rodu
ctio
nII
Let
pi
den
ote
the
pric
eof
outp
ut
y i,
let
rd
enot
eth
epr
ice
ofin
pu
tx
.
Nor
mal
ize
p2=
1.
Fir
mpr
ofit
max
imiz
atio
nF
OC
req
uir
eth
at
p1
2√
x 1−
r=
0an
d1
2√
x 2−
r=
0.
Su
chco
nd
itio
ns
pin
dow
nfa
ctor
dem
and
s,
xs 1=
( p1
2r
) 2an
dxs 2=
( 1 2r
) 2 ,
outp
ut
sup
plie
s,
ys 1=
p1
2ran
dys 2=
1 2r,
and
con
seq
uen
tly
profi
tfu
nct
ion
s,
π1=
(p1)2
4ran
dπ2=
1 4r.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
31/35
Exa
mpl
e:P
rodu
ctio
nII
I
Th
efi
rms’
prob
lem
sim
ply
that
:
the
inco
me
ofco
nsu
mer
Ais
mA=
5r+(p
1)2
/4r
;
the
inco
me
ofco
nsu
mer
Bis
mB=
3r+
1/
4r.
Fro
mF
OC
ofth
eco
nsu
mer
s’pr
oble
ms
fin
dou
tpu
td
eman
ds.
For
outp
ut
1,
yA 1=
1 2p1
( (p 1)2
4r+
5r
) and
yB 1=
1 2p1
( 1 4r+
3r
) .
Th
enby
Wal
ras
law
,th
eeq
uili
briu
mm
arke
tcl
eari
ng
con
dit
ion
sar
e
y 1m
arke
t:1 2p1
( 1+(p
1)2
4r+
8r
) =p1
2r
xm
arke
t:( p
1
2r
) 2 +
( 1 2r
) 2 =8
Th
eco
mp
etit
ive
equ
ilibr
ium
sati
sfies
p1=
1,r=
1/
4,yA 1=
yA 2=
9/
8,an
dyB 1=
yB 2=
7/
8.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
32/35
Not
es
Not
es
Par
eto
Effi
cien
cyI
Ext
end
the
abov
eex
amp
leto
gen
eral
pro
du
ctio
nfu
nct
ion
s
y 1=
f 1(x
1)
and
y 2=
f 2(x
2).
Su
pp
ose
that
tota
lin
pu
ten
dow
men
tis
x 1+
x 2=
γ.
By
allo
cati
ng
the
inp
ut
xb
etw
een
the
two
pro
du
ctio
npr
oce
sses
,ob
tain
the
pro
du
ctio
np
oss
ibili
tyfr
on
tier
y 2=
T(y
1).
For
mal
ly,
iff−1
1(y
1)
den
otes
the
inve
rse
off 1(x
1),
we
hav
eth
at
T(y
1)=
f 2( γ−
f−1
1(y
1)) .
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
33/35
Par
eto
Effi
cien
cyII
Rec
all
that
the
mar
gin
alra
teof
tran
sfor
mat
ion
,M
RT
,is
dT
dy 1
=−
∂f2/
∂x
∂f1/
∂x.
Par
eto
effici
ent
allo
cati
ons
can
now
be
fou
nd
bych
oos
ing(y
A 1,y
A 2,y
B 1,y
B 2)
soto
max
imiz
e
UA(y
A 1,y
A 2)
sub
ject
to
UB(y
B 1,y
B 2)=
uan
dyA 2+
yB 2=
T( y
A 1+
yB 1
) .
FO
Cim
med
iate
lyyi
eld
that
MR
SA=
MR
SB=
MR
T.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
34/35
Not
es
Not
es
Con
clud
ing
Com
men
ts
GE
isa
par
sim
onio
us
and
flex
ible
theo
ry,
wh
ich
relie
son
pric
eta
kin
g,in
div
idu
alm
axim
izat
ion
,an
dm
arke
tcl
eari
ng
tod
eriv
eim
plic
atio
ns
abou
tm
arke
tou
tcom
es.
Itca
nac
com
mo
dat
eti
me,
loca
tion
and
stat
eco
nti
nge
nt
pric
ing
once
com
mo
dit
ysp
aces
are
enla
rged
(an
emp
iric
ally
rele
van
tex
ten
sion
).
Con
clu
sion
son
wel
fare
furt
her
req
uir
eth
eab
sen
ceof
exte
rnal
itie
san
dpr
ivat
ein
form
atio
n,
and
the
com
ple
ten
ess
ofm
arke
ts.
Mem
ora
nd
um
:
Hom
ewor
kw
illb
ep
oste
dat
the
beg
inn
ing
ofw
eek
6M
T,
and
has
tob
eh
and
edin
clas
sd
uri
ng
wee
k7
MT
.
Nava
(LSE)
Gen
eralEquilibrium
MichaelmasTerm
35/35
Not
es
Not
es
Uncertainty
EC
411
Slid
es4
Nav
a
LSE
Mic
hae
lmas
Ter
m
Nava
(LSE)
Uncertainty
MichaelmasTerm
1/35
Roadm
ap:Choiceun
derUncertainty
We
intr
od
uce
the
clas
sica
lm
od
elof
dec
isio
nm
akin
gu
nd
eru
nce
rtai
nty
.
Th
ism
od
elis
anes
sen
tial
ben
chm
ark
toan
alyz
eb
ehav
ior
inn
um
erou
sm
arke
ts(fi
nan
ce,
insu
ran
ce,
bet
tin
g)an
din
any
typ
eof
gam
e.
Th
ean
alys
ispr
oce
eds
asfo
llow
s:
Lot
teri
es&
Exp
ecte
dU
tilit
y.
Axi
oms
onP
refe
ren
ces.
Rep
rese
nta
tion
Th
eore
m.
Ap
plic
atio
ns
toIn
sura
nce
.
Att
itu
des
tow
ard
sR
isk.
Exp
erim
enta
lE
vid
ence
.
New
Th
eori
es.
Nava
(LSE)
Uncertainty
MichaelmasTerm
2/35
Not
es
Not
es
DecisionTheory:
Uncertainty
Nava
(LSE)
Uncertainty
MichaelmasTerm
3/35
Uncertainty
Th
ere
areN
pos
sib
leo
utc
om
es(o
rst
ates
).
Den
ote
the
set
of
po
ssib
leou
tcom
esby
X=
(x1,.
..,x
N)
.
Asi
mp
lelo
tter
yL
isa
prob
abili
tyd
istr
ibu
tion
(p1,.
..,p
N)
over
outc
omes
inw
hic
h:
pi
den
otes
the
prob
abili
tyof
any
outc
omei∈1
,...
,N.
Aco
mp
ou
nd
lott
ery( α
1L1,.
..,α
kLk) is
apr
obab
ility
dis
trib
uti
onov
ersi
mp
lelo
tter
ies
inw
hic
h:
Lj
den
otes
asi
mp
lelo
tter
yfo
ran
yj∈1
,...
,k;
αj
den
otes
the
prob
abili
tyof
lott
eryLj .
Nava
(LSE)
Uncertainty
MichaelmasTerm
4/35
Not
es
Not
es
From
Com
pound
toSim
pleLotteries
Con
sid
eran
yco
mp
oun
dlo
tter
y( α
1L1,.
..,α
kLk) .
Let
pj i
isth
epr
obab
ility
ofou
tcom
ei
inlo
tter
yLj .
Ifso
,th
ere
exis
tsa
corr
esp
ond
ing
red
uce
dsi
mp
lelo
tter
y(p
1,.
..,p
N)
st:
pi=
∑k j=
1αj p
j i,fo
ran
yi=
1,..
.,N
.
Exa
mp
le:
Con
sid
erth
efo
llow
ing
scen
ario
:
X=
($1,$
2),
L1=
(1/
2,1
/2),
L2=
(1,0).
For
aco
mp
oun
dlo
tter
y(0
.5L1,0
.5L2),
Th
eco
rres
pon
din
gre
du
ced
lott
ery
is(0
.75,
0.25).
Nava
(LSE)
Uncertainty
MichaelmasTerm
5/35
Axiom
s:Com
pleteness,Transitivity,Continu
ity
Ass
um
eth
atd
ecis
ion
mak
ers
rega
rdan
yco
mp
oun
dlo
tter
yas
equ
ival
ent
toit
sre
du
ced
sim
ple
lott
ery.
Th
epr
efer
ence
sof
the
dec
isio
nm
aker
are
defi
ned
over
the
set
ofsi
mp
lelo
tter
iesL
,an
dd
enot
edby
.
Pre
fere
nce
ssa
tisf
yco
nti
nu
ity
iffo
ran
yL
,L′ ,L′′∈L
,th
efo
llow
ing
two
sets
are
clos
ed:
α∈[0
,1]
:(αL′ ,(1−
α)L′′ )
L
andα∈[0
,1]
:L
(αL′ ,(1−
α)L′′ ).
Th
efo
llow
ing
two
axio
ms
onpr
efer
ence
sar
ein
voke
dth
rou
ghou
t:
[A1
]
isco
mp
lete
and
tran
siti
ve.
[A2
]
sati
sfies
con
tin
uit
y.
A1
and
A2
imp
lyth
atpr
efer
ence
sca
nb
ere
pres
ente
dby
au
tilit
yfu
nct
ion
.Nava
(LSE)
Uncertainty
MichaelmasTerm
6/35
Not
es
Not
es
Axiom
s:Independence
Ath
ird
and
fin
alax
iom
know
nas
the
ind
epen
den
cea
xio
mis
nee
ded
tod
eriv
ea
clas
sica
lan
dfu
nd
amen
tal
repr
esen
tati
onth
eore
m:
[A3
]
sati
sfies
ind
epen
den
ce.
Pre
fere
nce
ssa
tisf
yin
dep
end
ence
iffo
ran
yL
,L′ ,L′′∈L
and
α∈[0
,1]:
L
L′
ifan
don
lyif
(αL
,(1−
α)L′′ )
(αL′ ,(1−
α)L′′ )
.
Exa
mp
le:L=
100
$,L′=
10$,
L′′=
Tri
pto
Par
is.
A3
imp
lies
that
L
L′
ifan
don
lyif
αL⊕(1−
α)[
Tri
p]
αL′ ⊕
(1−
α)[
Tri
p].
Th
eu
tilit
yof
from
the
trip
can
cels
out.
Ad
ecis
ion
mak
ersh
ould
thu
ske
eppr
efer
rin
gm
ore
mon
eyto
less
rega
rdle
ssof
wh
eth
ersh
ew
ins
the
trip
.
Nava
(LSE)
Uncertainty
MichaelmasTerm
7/35
RepresentationTheorem
Theorem
(RepresentationTheorem
)
IfA1-A3hold,thereexists
avector
ofutilities(u
1,.
..,u
N)such
that,for
anytwolotteriesL=
(p1,.
..,p
N)andL′=
(p′ 1,
...,p′ N),
L
L′ifandon
lyif
∑N i=
1piu
i≥
∑N i=
1p′ iu
i.
Furthermore,
anyother
vector
ofutilities(u
1,.
..,u
N)represents
thesame
preferencesifandon
lyifforsomeconstants
b>
0anda:
ui=
a+
bui.
Th
eke
yfe
atu
res
ofth
ere
pres
enta
tion
imp
lyth
at:
pref
eren
ces
are
linea
rin
prob
abili
ties
;
the
uti
lity
ofa
lott
ery
isth
eex
pec
ted
uti
lity
ofth
eou
tcom
es;
only
affin
etr
ansf
orm
atio
ns
ofu
tilit
yre
pres
ent
the
sam
epr
efer
ence
s.Nava
(LSE)
Uncertainty
MichaelmasTerm
8/35
Not
es
Not
es
Intuition:
IndependenceAxiom
Con
sid
erth
efo
llow
ing
scen
ario
:
X=
(x1,x
2,x
3),
L=
(1,0
,0),
L′=
(0,1
,0),
L′′=
(0,0
,1).
Not
eth
atth
ere
pres
enta
tion
theo
rem
imp
lies
(αL
,(1−
α)L′′ )
(αL′ ,(1−
α)L′′ )
iffαu1+(1−
α)u
3≥
αu2+(1−
α)u
3.
Th
isin
turn
imm
edia
tely
imp
lies
the
ind
epen
den
ceax
iom
,as
αu1+(1−
α)u
3≥
αu2+(1−
α)u
3iff
u1≥
u2
iffL
L′ .
Nava
(LSE)
Uncertainty
MichaelmasTerm
9/35
RiskAttitudes
Nava
(LSE)
Uncertainty
MichaelmasTerm
10/35
Not
es
Not
es
Attitud
estowards
Risk
Now
assu
me
that
outc
omes
inX
are
qu
anti
fiab
leso
that
X=
R+
.
Th
isis
ago
od
assu
mp
tion
for
mon
ey,
con
sum
pti
on,
wea
lth
...
Alo
tter
yn
owco
nsi
sts
ofa
cum
ula
tive
dis
trib
uti
on
F:R
+→
[0,1],
wh
ereF(x)
isth
epr
obab
ility
that
the
outc
ome
issm
alle
ror
equ
alto
x.
Th
epr
efer
ence
sh
ave
anex
pec
ted
uti
lity
repr
esen
tati
on
U(F
)=∫ X
u(x)dF(x),
wh
ereu(x)
isth
eu
tilit
yof
xw
ith
cert
ain
ty.
Ifth
ed
ensi
tyf(·)
isw
ell
defi
ned
,th
en
U(F
)=∫ X
u(x)f(x)dx
.
Ass
um
eth
atu(·)
isst
rict
lyin
crea
sin
ga
nd
con
tin
uo
us.
Nava
(LSE)
Uncertainty
MichaelmasTerm
11/35
Attitud
estowards
Risk
Ad
ecis
ion
mak
eris
risk
ave
rse
ifan
don
lyif
,fo
ran
ylo
tter
yF
,
∫ Xu(x)dF(x)≤
u
( ∫ XxdF(x)) .
Ad
ecis
ion
mak
eris
risk
neu
tra
lif
and
only
if,
for
any
lott
eryF
,
∫ Xu(x)dF(x)=
u
( ∫ XxdF(x)) .
Ad
ecis
ion
mak
eris
risk
lovi
ng
ifan
don
lyif
,fo
ran
ylo
tter
yF
,
∫ Xu(x)dF(x)≥
u
( ∫ XxdF(x)) .
Fact
Adecisionmaker
isrisk
averse
ifandon
lyifu(·)isconcave.
Adecisionmaker
isrisk
lovingifandon
lyifu(·)isconvex.
Nava
(LSE)
Uncertainty
MichaelmasTerm
12/35
Not
es
Not
es
Attitud
estowards
Risk
Ad
ecis
ion
mak
eris
stri
ctly
risk
ave
rse
ifan
don
lyif
,fo
ran
ylo
tter
yF
that
do
esn
otas
sign
prob
abili
tyeq
ual
toon
eto
the
outc
ome∫ X
xdF(x),
∫ Xu(x)dF(x)<
u
( ∫ XxdF(x))
Ad
ecis
ion
mak
eris
stri
ctly
risk
lovi
ng
ifan
don
lyif
,fo
ran
ylo
tter
yF
that
do
esn
otas
sign
prob
abili
tyeq
ual
toon
eto
the
outc
ome∫ X
xdF(x),
∫ Xu(x)dF(x)>
u
( ∫ XxdF(x))
Exa
mp
les:
stri
ctly
risk
aver
se,u(x)=√x
;
risk
neu
tral
,u(x)=
x;
stri
ctly
risk
lovi
ng,
u(x)=
x2.
Nava
(LSE)
Uncertainty
MichaelmasTerm
13/35
Attitud
estowards
Risk
Con
sid
era
lott
eryX
=a
,b
andL=p
,1−
p.
Ifso
,gr
aph
ical
lyob
serv
eth
at:
Nava
(LSE)
Uncertainty
MichaelmasTerm
14/35
Not
es
Not
es
Example:
Insurance
Con
sid
erth
efo
llow
ing
scen
ario
:
Md
enot
esth
ein
itia
lw
ealt
h;
Ld
enot
esa
loss
that
may
be
incu
rred
;
pd
enot
esth
epr
obab
ility
ofth
elo
ss;
Sd
enot
esth
eam
oun
tof
cove
rage
;
rd
enot
esu
nit
pric
eof
cove
rage
(un
itpr
emiu
m).
Tw
oou
tcom
esar
ep
ossi
ble
:
M−
rSM−
L+
S−
rSw
ith
prob
abili
ty1−
pw
ith
prob
abili
typ
.
Ari
sk-a
vers
eco
nsu
mer
cho
oses
S≤
Lto
max
imiz
eex
pec
ted
uti
lity,
(1−
p)u(M−
rS)+
pu(M−
L+(1−
r )S)
.
Th
eF
OC
imm
edia
tely
yiel
ds
r(1−
p)u′ (M−
rS)=
(1−
r )pu′ (M−
L+(1−
r )S)
.
Exe
rcis
e:S
how
that
,ifr=
p,
ari
skav
erse
con
sum
erch
oos
esS=
L.
Nava
(LSE)
Uncertainty
MichaelmasTerm
15/35
Example:
Insurance
Dem
and
for
insu
ran
ceca
nb
ed
eriv
edex
actl
yas
dem
and
for
oth
ergo
od
s:
Th
eco
nsu
mer
par
tial
lyin
sure
sw
hen
ever
the
pric
eex
ceed
sth
eri
skr>
p.
Nava
(LSE)
Uncertainty
MichaelmasTerm
16/35
Not
es
Not
es
Measuresof
RiskAversion
Intu
itiv
ely,
the
mor
eco
nca
veis
au
tilit
yfu
nct
ion
the
mor
eri
skav
erse
isth
ed
ecis
ion
mak
er.
Sin
ceu(·)
andbu(·),
b>
0,re
pres
ent
the
sam
epr
efer
ence
s,u′′ (·)
isn
ota
sati
sfac
tory
mea
sure
ofri
skav
ersi
on.
Defi
ne
the
coeffi
cien
to
fa
bso
lute
risk
ave
rsio
nas
:
r A(x)=−u′′ (x)
u′ (x)
.
Exa
mp
le:
Con
sid
erpr
efer
ence
su(x)=−e−ax.
Ifso
,r A(x)=−−a2e−ax
ae−ax
=a.
Nava
(LSE)
Uncertainty
MichaelmasTerm
17/35
Concave
Transform
ations
I
Con
sid
ertw
od
ecis
ion
mak
ers,
one
wit
hu
tilit
yu(·),
one
wit
hu
tilit
yv(·).
Bot
hu(·)
andv(·)
are
stri
ctly
incr
easi
ng.
Th
eref
ore
ther
eex
ists
φ:R→
Rsu
chth
at
u(x)=
φ(v(x) )
for
any
x∈
R+
.
Itfo
llow
sth
at
u′ (x)=
φ′ (v(x) )
v′ (x)
and
u′′ (x)=
φ′′(v(x) )( v′ (x
)) 2 +φ′ (v(x) )
v′′ (x).
By
elim
inat
ing
φ′ (v(x) )
from
the
two
expr
essi
ons
obta
in
u′′ (x)=
φ′′(v(x) )( v′ (x
)) 2 +u′ (x)
v′ (x)v′′ (x).
Nava
(LSE)
Uncertainty
MichaelmasTerm
18/35
Not
es
Not
es
Concave
Transform
ations
II
Th
epr
evio
us
expr
essi
onis
equ
ival
ent
to
φ′′(v(x) )
=u′′ (x)v′ (x)−
v′′ (x)u′ (x)
(v′ (x) )
3.
Hen
ce,
φ(·)
isco
nca
veif
and
only
if
−v′′ (x)
v′ (x)≤−u′′ (x)
u′ (x)
Th
us,u(·)
isa
con
cave
tra
nsf
orm
ati
on
ofv(·)
ifan
don
lyifu(·)
has
ala
rger
coeffi
cien
tof
abso
lute
risk
aver
sion
than
v(·).
Ab
solu
teri
skav
ersi
onis
ap
arti
al
ord
er(n
otal
lpr
efer
ence
sca
nb
eco
mp
ared
)ov
erpr
efer
ence
sth
atco
mp
ares
ran
kspr
efer
ence
sb
ased
onat
titu
des
tow
ard
sri
sk.
Nava
(LSE)
Uncertainty
MichaelmasTerm
19/35
RelativeRiskAversion
Ab
solu
teri
skav
ersi
onm
easu
res
risk
atti
tud
esto
war
ds
abso
lute
chan
ges
inw
ealt
hat
the
curr
ent
wea
lth
leve
l.
Ad
iffer
ent
mea
sure
risk
aver
sion
can
cap
ture
risk
atti
tud
esto
war
ds
per
cen
tage
chan
ges
inw
ealt
hat
the
curr
ent
wea
lth
leve
l.
Defi
ne
the
coeffi
cien
to
fre
lati
veri
ska
vers
ion
as:
r A(x)=−xu′′ (x)
u′ (x)
.
Ab
solu
teri
skav
ersi
onis
ap
arti
al
ord
er(n
otal
lpr
efer
ence
sca
nb
eco
mp
ared
)ov
erpr
efer
ence
sth
atco
mp
ares
ran
kspr
efer
ence
sb
ased
onri
skat
titu
des
for
per
cen
tage
chan
ges
inw
ealt
h.
Asi
de:
Dec
reas
ing
abso
lute
risk
aver
sion
has
bee
nu
sed
asa
just
ifica
tion
for
inco
me
ineq
ual
ity.
..
Nava
(LSE)
Uncertainty
MichaelmasTerm
20/35
Not
es
Not
es
Paradoxes
&Advances:
ChoiceunderUncertainty
Nava
(LSE)
Uncertainty
MichaelmasTerm
21/35
AllaisParadox
I
Con
sid
erou
tcom
ese
tX
=5
mill
ion
s,1
mill
ion
s,0
mill
ion
s .
Fir
stco
nsi
der
the
follo
win
gtw
olo
tter
ies:
Ou
tcom
e:5
M$
1M
$0
M$
Lot
tery
A:
0%10
0%0%
Lot
tery
B:
10%
89%
1%
Evi
den
cesu
gges
tsin
div
idu
als
cho
ose
Aov
erB
.
Su
chb
ehav
ior
req
uir
es
u(1)>
0.1u
(5)+
0.89u(1)+
0.01u(0),
⇒0.
11u(1)>
0.1u
(5)+
0.01u(0).
Nava
(LSE)
Uncertainty
MichaelmasTerm
22/35
Not
es
Not
es
AllaisParadox
II
Nex
tco
nsi
der
the
follo
win
gtw
olo
tter
ies:
Ou
tcom
e:5
M$
1M
$0
M$
Lot
tery
C:
0%11
%89
%L
otte
ryD
:10
%0%
90%
Evi
den
cesu
gges
tsin
div
idu
als
cho
ose
Dov
erC
.
Su
chb
ehav
ior
req
uir
es
0.11u(1)+
0.89u(0)<
0.1u
(5)+
0.9u
(0),
⇒0.
11u(1)<
0.1u
(5)+
0.01u(0).
Pre
fere
nce
svi
olat
eth
ein
dep
end
ence
axio
m(n
otlin
ear
inpr
obab
iliti
es)!
Inex
per
imen
tsin
div
idu
als
will
corr
ect
this
mis
take
over
tim
e.
Pos
sib
leex
pla
nat
ion
com
esfr
omre
gret
theo
ry(c
hoi
ces
are
mad
eto
avoi
dd
isap
poi
ntm
ent
from
anou
tcom
eth
atd
idn
otm
ater
ializ
e).
Nava
(LSE)
Uncertainty
MichaelmasTerm
23/35
Sub
jectiveProbabilities
We
hav
eas
sum
edth
ata
dec
isio
nm
aker
ran
ksex
plic
itan
dob
ject
ive
prob
abili
tyd
istr
ibu
tion
sov
erou
tcom
es.
How
ever
,pr
obab
iliti
esm
ayb
eim
plic
itan
dsu
bje
ctiv
e.
For
exam
ple
,th
ed
ecis
ion
mak
erm
ayb
eas
ked
toch
oos
eb
etw
een
:
(a)
100£
ifC
hel
sea
win
sth
eC
ham
pio
ns
Lea
gue;
(b)
100£
ifB
arce
lon
aw
ins
the
Ch
amp
ion
sL
eagu
e.
Su
bje
ctiv
epr
obab
iliti
esm
ayb
ere
veal
edby
the
choi
ces
ofd
ecis
ion
mak
ers.
If(b
)is
pref
erre
dto
(a),
we
may
con
clu
de
that
the
dec
isio
nm
aker
thin
ksth
atth
epr
obab
ility
ofB
arce
lon
aw
inn
ing
the
titl
eis
hig
her
than
the
prob
abili
tyof
Ch
else
aw
inn
ing
the
titl
e.
Nava
(LSE)
Uncertainty
MichaelmasTerm
24/35
Not
es
Not
es
EllsbergParadox
&AmbiguityAversion
Bu
tch
oice
sm
ayn
otco
nsi
sten
tw
ith
”pro
bab
ilist
ic”
asse
ssm
ents
.
Con
sid
eran
urn
con
tain
ing
100
bal
ls.
Som
eb
alls
are
red
,w
hile
the
rest
are
blu
e.
Th
ed
ecis
ion
mak
eris
no
tto
ldh
owm
any
are
red
and
how
man
yar
eb
lue.
Ext
ract
ab
all
from
this
urn
and
con
sid
erth
eb
ets:
(a)£
100
ifth
eb
all
isre
d;
(b)£
100
ifth
eb
all
isb
lue;
(c)£
99w
ith
prob
abili
ty50
%–
coin
toss
.
Man
yin
div
idu
als
wou
ldch
oos
e(c
).
How
ever
,ei
ther
Pr(red)≥
1/
2or
Pr(blue)≥
1/
2,re
gard
less
ofw
het
her
prob
abili
ties
are
obje
ctiv
eor
sub
ject
ive.
Nava
(LSE)
Uncertainty
MichaelmasTerm
25/35
Maxmin
Utility(G
ilboa
&Schmeidler)
Let
Sd
enot
eth
ese
tof
pos
sib
lest
ates
.
Ab
etis
afu
nct
ionx
:S−→
R+
.
Inp
arti
cula
r,x(s)
isth
em
onet
ary
outc
ome
inst
ates.
Th
ese
tof
”pos
sib
le”
prob
abili
tyd
istr
ibu
tion
sov
erth
ese
tof
stat
esisC
.
Un
der
reas
onab
leax
iom
s,pr
efer
ence
sov
erb
etsx
can
alw
ays
be
repr
esen
ted
bya
uti
lity
fun
ctio
n
min
F∈C∫ S
u(x(s) )
dF(s)
.
Th
isap
proa
chca
nex
pla
inE
llsb
erg’
sP
arad
ox.
Wh
enin
div
idu
als
are
pes
sim
isti
cab
out
thei
rb
elie
fs,
they
may
pref
eran
obje
ctiv
elo
tter
y.
Nava
(LSE)
Uncertainty
MichaelmasTerm
26/35
Not
es
Not
es
Lim
itsto
Risk-Aversion
Ris
k-av
ersi
onis
ast
and
ard
econ
omic
par
adig
m,
bu
tit
imp
oses
tom
uch
dis
cip
line
wh
enco
mp
arin
gsm
all
risk
sto
larg
eon
es.
Rab
in(2
000)
show
sth
at:
ifan
EU
max
imiz
erre
ject
sa
50-5
0ga
mb
leto
win
11or
lose
10re
gar
dle
sso
fh
isw
ealt
h,w
,h
ere
ject
sa
50-5
0ga
mb
leto
win
yor
lose
100,
for
any
valu
eof
y!!
To
show
this
obse
rve
that
con
cavi
tyim
plie
s:
u(w
)−
u(w−
10)<
+10u′ (w−
10)
u(w
)−
u(w
+11)<−
11u′ (w+
11)
Sin
ceh
etu
rns
dow
nth
ega
mb
le,
we
also
know
:
u(w
)>
0.5u
(w−
10)+
0.5u
(w+
11)
Com
bin
ing
all
thre
ein
equ
alit
ies
yiel
ds:
u′ (w+
11)−
u′ (w−
10)<−
1 11u′ (w−
10)
Nava
(LSE)
Uncertainty
MichaelmasTerm
27/35
Lim
itsto
Risk-Aversion
So
gain
ing
21ca
use
sth
em
argi
nal
uti
lity
ofm
oney
tofa
llby
9%.
Iter
atin
gth
islo
gic
show
sth
atsm
all
stak
esco
mp
oun
dq
uic
kly
and
cau
sem
argi
nal
uti
lity
top
lum
met
onla
rge
stak
es.
Acc
ord
ing
toE
U,
ther
eis
no
amou
nt
we
cou
ldoff
erh
imto
acce
pt
the
chan
ceof
losi
ng
100.
EU
was
not
des
ign
edto
exp
lain
smal
l-st
akes
gam
ble
san
dre
qu
ires
ind
ivid
ual
sto
be
loca
llyri
skn
eutr
al.
Evi
den
cesu
gges
tsth
atth
eva
stm
ajor
ity
ofin
div
idu
als
wou
ldre
ject
the
smal
lb
et,
Bar
ber
is,
Hu
ang
and
Th
aler
(200
6)an
dA
rrow
(197
1).
Aso
luti
onin
volv
eski
nki
ng
the
uti
lity
fun
ctio
nar
oun
da
refe
ren
cep
oin
t.
Nava
(LSE)
Uncertainty
MichaelmasTerm
28/35
Not
es
Not
es
ProspectTheory(K
ahneman
&Tversky)
Su
bje
ctiv
epr
obab
iliti
esar
eb
iase
d:
smal
lpr
obab
iliti
esar
eov
eres
tim
ated
,
larg
epr
obab
iliti
esar
eu
nd
eres
tim
ated
.
Th
eu
tilit
yof
anou
tcom
em
ayd
epen
don
are
fere
nce
po
intr.
Let
(x1,.
..,x
N)
den
ote
ase
tof
mon
etar
you
tcom
es.
Con
sid
era
lott
ery
over
outc
omes
(p1,.
..,p
N).
Let
di(p
1,.
..,p
N)
den
ote
the
dis
tort
edp
rob
ab
ility
ofou
tcom
ex i
.
Ifso
,u
nd
erp
lau
sib
leax
iom
spr
efer
ence
sca
nb
ere
pres
ente
das
,
∑N i=
1di(p
1,.
..,p
N)u(x
i|r)
.
Nava
(LSE)
Uncertainty
MichaelmasTerm
29/35
LossAversion
Ifr
isw
ealt
han
du(x
i|r)=
v(x
i−
r ),
uti
lity
isd
efin
edon
gain
s&
loss
es.
Nava
(LSE)
Uncertainty
MichaelmasTerm
30/35
Not
es
Not
es
LossAversion:
Example
Su
pp
ose
anag
ent
has
ap
iece
wis
e-lin
ear
valu
efu
nct
ion
:
v(x)=
xif
x≥
02x
ifx<
0
Let
the
refe
ren
cep
oin
tb
eh
erin
itia
lw
ealt
hr=
w.
Do
essh
eac
cep
ta
50/5
0ga
mb
leto
win
11or
lose
10?
Ifsh
etu
rns
itd
own
,sh
ege
ts:
v(w−
r)=
v(0)=
0
Ifsh
eac
cep
ts,
she
gets
:
v(w
+11−
r)+
v(w−
10−
r)
2=
v(1
1)+
v(−
10)
2=−
4.5
So
she
turn
sit
dow
n
Nava
(LSE)
Uncertainty
MichaelmasTerm
31/35
LossAversion:
Example
Wh
atis
the
smal
lest
valu
ey
for
wh
ich
she
acce
pts
a50
/50
gam
ble
tow
iny
orlo
se10
0?
Ifsh
etu
rns
itd
own
,sh
ege
ts:
v(w−
r)=
v(0)=
0
Ifsh
eac
cep
ts,
she
gets
:
v(w
+y−
r)+
v(w−
100−
r)
2=
v(y)+
v(−
100)
2=
y 2−
100
So
she
will
acce
pt
ify≥
200.
Los
sav
ersi
onal
sopr
ovid
esan
exp
lan
atio
nfo
rth
een
dow
men
teff
ect.
Nava
(LSE)
Uncertainty
MichaelmasTerm
32/35
Not
es
Not
es
Gam
bles
onGains
andLosses
Mos
tp
eop
lear
e:
risk
-ave
rse
wh
enth
inki
ng
only
ofga
ins
bu
t
risk
-lov
ing
wh
enth
inki
ng
oflo
sses
.
Let
(y,p)
be
aga
mb
leth
atgi
ves:
yw
ith
prob
abili
typ
and
0w
ith
prob
abili
ty(1−
p).
Kah
nem
an&
Tve
rsky
1979
show
sth
at:
(400
0,0.
8)
(300
0,1)
(−40
00,0
.8)
(−30
00,1)
20%
80%
92%
8%
(300
0,0.
9)
(600
0,0.
45)
(−30
00,0
.9)
(−60
00,0
.45)
86%
14%
8%92
%
Rej
ects
equ
alpr
opor
tion
sch
oos
ing
the
risk
yop
tion
.Nava
(LSE)
Uncertainty
MichaelmasTerm
33/35
Dim
inishing
Sensitivity
Th
ese
con
dfi
nd
ing
ofK
ahn
eman
&T
vers
ky19
79isdim
inishingsensitivity
.
Mar
gin
alse
nsi
tivi
tyto
chan
ges
from
the
refe
ren
cep
oin
tar
esm
alle
rth
efa
rth
erth
ech
ange
isfr
omth
ere
fere
nce
poi
nt.
KT
argu
eth
isis
also
anex
ten
sion
ofh
oww
ep
erce
ive
thin
gs.
For
exam
ple
,an
obje
ct10
1fe
etaw
ayis
ind
isti
ngu
ish
able
from
anob
ject
100
feet
away
,b
ut
anob
ject
2fe
etaw
ayis
easi
lyp
erce
ived
asve
ryd
iffer
ent
from
anob
ject
1fo
otaw
ay.
Lik
ewis
e,th
ere
lati
veim
pac
tof
rece
ivin
g(r
esp
.lo
sin
g)10
inst
ead
of0
isla
rger
than
the
imp
act
ofre
ceiv
ing
(res
p.
losi
ng)
1,01
0in
stea
dof
1,00
0.
Th
isis
not
just
risk
-ave
rsio
n:
peo
ple
will
be
risk
-ave
rse
inth
ed
omai
nof
gain
sb
ut
risk
-see
kin
gin
the
dom
ain
oflo
sses
.
Nava
(LSE)
Uncertainty
MichaelmasTerm
34/35
Not
es
Not
es
Properties
ofValue
Fun
ctions
Val
ue
Fu
nct
ion
isas
sum
edto
be:
1v(x)
isco
nti
nu
ous
for
anyx
and
twic
ed
iffer
enti
able
forx6=
0.
2v(x)
isst
rict
lyin
crea
sin
g,an
dv(0)=
0.
3v(y)+
v(−
y)<
v(x)+
v(−
x)
fory>
x>
0.
4v′′ (x)≤
0fo
rx>
0,an
dv′′ (x)≥
0fo
rx<
0.
5v′ −(0)/
v′ +(0)>
1.
Th
ese
theo
ries
are
con
ven
ien
tan
alyt
ical
lyan
dva
luab
legu
ides
tob
ehav
ior.
Bu
to
ccas
ion
ally
fail
topr
ovid
ea
un
ified
appr
oach
toca
ptu
real
lb
iase
sd
isp
laye
dby
hu
man
beh
avio
r.
Nava
(LSE)
Uncertainty
MichaelmasTerm
35/35
Not
es
Not
es
Sta
tic
Gam
esE
C41
1S
lides
5
Nav
a
LSE
Mic
hae
lmas
Ter
m
Nava
(LSE)
StaticGames
MichaelmasTerm
1/80
Roa
dmap
:G
ame
The
ory
–S
tati
cG
ames
Aga
me
isa
mu
lti-
per
son
dec
isio
npr
oble
m.
Str
ateg
icth
inki
ng
insu
chen
viro
nm
ents
req
uir
esp
laye
rsto
gues
sth
eb
ehav
ior
ofot
her
sto
dec
ide
onth
eb
est
cou
rse
ofac
tion
.
Th
ean
alys
isof
stat
icco
nsi
der
stw
od
iffer
ent
clas
ses
ofga
mes
:
Com
ple
teIn
form
atio
nS
tati
cG
ames
Inco
mp
lete
Info
rmat
ion
Sta
tic
Gam
es(e
xtra
)
Intr
od
uce
sfu
nd
amen
tal
not
ion
sof
stra
tegy
and
bes
tre
spon
se.
Diff
eren
tS
olu
tion
Con
cep
tsar
epr
esen
ted
:
Dom
inan
tS
trat
egy
Eq
uili
briu
m
Iter
ativ
eE
limin
atio
nof
Dom
inat
edS
trat
egie
s
Nas
hE
qu
ilibr
ium
Bay
esN
ash
Eq
uili
briu
m(e
xtra
).
Nava
(LSE)
StaticGames
MichaelmasTerm
2/80
Not
es
Not
es
StrategicForm
Gam
es:
Sta
tic
Com
plet
eIn
form
atio
nG
ames
Nava
(LSE)
StaticGames
MichaelmasTerm
3/80
Sum
mar
y
Gam
esof
Com
ple
teIn
form
atio
n:
Defi
nit
ion
s:
Gam
e:P
laye
rs,
Act
ion
s,P
ayoff
sS
trat
egy:
Pu
rean
dM
ixed
Bes
tR
esp
onse
Sol
uti
onC
once
pt:
Dom
inan
tS
trat
egy
Eq
uili
briu
mN
ash
Eq
uili
briu
m
Pro
per
ties
ofN
ash
Eq
uili
bria
:
Non
-Exi
sten
cean
dM
ult
iplic
ity
Ineffi
cien
cy
Exa
mp
les
Nava
(LSE)
StaticGames
MichaelmasTerm
4/80
Not
es
Not
es
Intr
odu
ctio
nto
Gam
es
An
yen
viro
nm
ent
inw
hic
hth
ech
oice
sof
anin
div
idu
alaff
ect
the
wel
lb
ein
gof
oth
ers
can
be
mo
del
edas
aga
me.
Wh
atp
ins
dow
na
spec
ific
gam
e:
Wh
op
arti
cip
ates
ina
gam
e[P
laye
rs]
Th
ech
oice
sth
atp
arti
cip
ants
hav
e[C
hoi
ces]
Th
ew
ell
bei
ng
ofin
div
idu
als
[Pay
offs]
Th
ein
form
atio
nth
atin
div
idu
als
hav
e[R
ule
sof
the
Gam
e]
Th
eti
min
gof
even
tsan
dd
ecis
ion
s[R
ule
sof
the
Gam
e]
Nava
(LSE)
StaticGames
MichaelmasTerm
5/80
Com
plet
eIn
form
atio
n(S
trat
egic
For
m)
Gam
e
Aco
mp
lete
info
rmat
ion
gam
eG
con
sist
sof
:
Ase
tof
pla
yers
:
Nof
size
n
An
acti
onse
tfo
rea
chp
laye
rin
the
gam
e:
Ai
for
pla
yer
i’s
An
acti
onpr
ofile
a=
(a1,a
2,.
..,a
n)
pic
ksan
acti
onfo
rea
chp
laye
r
Au
tilit
ym
apfo
rea
chp
laye
rm
app
ing
acti
onpr
ofile
sto
pay
offs:
ui(
a)
den
otes
pla
yer
i’s
pay
offof
acti
onpr
ofile
a
B\G
sm
s5,
21,
2m
0,0
3,5
Nava
(LSE)
StaticGames
MichaelmasTerm
6/80
Not
es
Not
es
Rep
rese
ntin
gS
imul
tane
ous
Mov
eC
ompl
ete
Info
Gam
es
Str
ateg
icF
orm
1\2
sm
s5,
21,
2m
0,0
3,5
Ext
ensi
veF
orm
Nava
(LSE)
StaticGames
MichaelmasTerm
7/80
Rep
rese
ntin
gS
imul
tane
ous
Mov
eC
ompl
ete
Info
Gam
es
Str
ateg
icF
orm
1\2
sm
s5,
21,
2m
0,0
3,5
Ext
ensi
veF
orm
Nava
(LSE)
StaticGames
MichaelmasTerm
8/80
Not
es
Not
es
Fea
sibl
eP
ayoff
san
dE
ffici
ency
Str
ateg
icF
orm
1\2
sm
s5,
21,
2m
0,0
3,5
Fea
sib
leP
ayoff
s
Effi
cien
tp
ayoff
sar
eon
the
nor
th-e
ast
bou
nd
ary.
Nava
(LSE)
StaticGames
MichaelmasTerm
9/80
Info
rmat
ion
and
Pur
eS
trat
egie
s
Ast
rate
gyin
aga
me:
isa
map
from
info
rmat
ion
into
acti
ons
itd
efin
esa
pla
nof
acti
onfo
ra
pla
yer
Ina
com
ple
tein
form
atio
nst
rate
gic
form
gam
e:
pla
yers
hav
en
opr
ivat
ein
form
atio
n
pla
yers
act
sim
ult
aneo
usl
y
Inth
isco
nte
xta
stra
tegy
isan
yel
emen
tof
the
set
ofac
tion
s
For
inst
ance
a(p
ure
)st
rate
gyfo
rp
laye
ri
issi
mp
lya i∈
Ai
Nava
(LSE)
StaticGames
MichaelmasTerm
10/80
Not
es
Not
es
Not
atio
n
Defi
ne
apr
ofile
ofac
tion
sch
osen
byal
lp
laye
rsot
her
than
iby
a−i:
a−i=
(a1,.
..,a
i−1,a
i+1,.
..,a
n)
Defi
ne
the
set
ofp
ossi
ble
acti
onpr
ofile
sfo
ral
lp
laye
rsot
her
that
ias
A−i=×
j∈N\i
Aj
Defi
ne
the
set
ofp
ossi
ble
acti
onpr
ofile
sfo
ral
lp
laye
rsas
A=×
j∈N
Aj
Nava
(LSE)
StaticGames
MichaelmasTerm
11/80
Bes
tR
esp
onse
s
Th
eb
est
resp
onse
corr
esp
ond
ence
ofp
laye
ri
isd
efin
edby
:
bi(
a−i)=
arg
max
ai∈
Aiui(
a i,a−i)
for
any
a−i∈
A−i
Th
us
a iis
ab
est
resp
onse
toa−i
–i.
e.a i∈
bi(
a−i)
–if
and
only
if:
ui(
a i,a−i)≥
ui(
a′ i,a−i)
for
any
a′ i∈
Ai
BR
iden
tifi
esth
eop
tim
alac
tion
for
ap
laye
rgi
ven
choi
ces
mad
eby
oth
ers
For
inst
ance
:B\G
sm
s5,
21,
2m
0,0
3,5
Nava
(LSE)
StaticGames
MichaelmasTerm
12/80
Not
es
Not
es
StrategicForm
Gam
es:
Dom
inan
ce
Nava
(LSE)
StaticGames
MichaelmasTerm
13/80
Str
ict
Dom
inan
ce
Str
ateg
ya i
stri
ctly
do
min
ate
sa′ i
if:
ui(
a i,a−i)>
ui(
a′ i,a−i)
for
any
a−i∈
A−i
a iis
stri
ctly
do
min
an
tif
itst
rict
lyd
omin
ates
any
oth
era′ i
a iis
stri
ctly
un
do
min
ate
dif
no
stra
tegy
stri
ctly
dom
inat
esa i
a iis
stri
ctly
do
min
ate
dif
ast
rate
gyst
rict
lyd
omin
ates
a i
Inth
efo
llow
ing
exam
ple
sis
stri
ctly
dom
inan
tfo
rB
:
B\G
sm
s5,
-2,
-m
0,-
1,-
Nava
(LSE)
StaticGames
MichaelmasTerm
14/80
Not
es
Not
es
Wea
kD
omin
ance
Str
ateg
ya i
wea
kly
do
min
ate
sa′ i
if:
ui(
a i,a−i)≥
ui(
a′ i,a−i)
for
any
a−i
ui(
a i,a−i)>
ui(
a′ i,a−i)
for
som
ea−i
a iis
wea
kly
do
min
an
tif
itw
eakl
yd
omin
ates
any
oth
era′ i
a iis
wea
kly
un
do
min
ate
dif
no
stra
tegy
wea
kly
dom
inat
esa i
a iis
wea
kly
do
min
ate
dif
ast
rate
gyw
eakl
yd
omin
ates
a i
Inth
efo
llow
ing
exam
ple
sis
wea
kly
dom
inan
tfo
rB
:
B\G
sm
s5,
-2,
-m
0,-
2,-
Nava
(LSE)
StaticGames
MichaelmasTerm
15/80
Dom
inan
tS
trat
egy
Equ
ilibr
ium
Defi
niti
ons
(Dom
inan
tS
trat
egy
Equ
ilibr
ium
DS
E)
Ast
rict
Dom
inan
tS
trat
egy
equ
ilibr
ium
ofa
gam
eG
con
sist
sof
ast
rate
gypr
ofile
asu
chth
atfo
ran
ya′ −i∈
A−i
and
i∈
N:
ui(
a i,a′ −i)>
ui(
a′ i,a′ −i)
for
any
a′ i∈
Ai
For
wea
kD
SE
chan
ge>
wit
h≥
...
apr
ofile
ais
aD
SE
iffa i∈
bi(
a′ −i)
for
any
a′ −i
and
i∈
N
Exa
mp
le(P
riso
ner
’sD
ilem
ma)
:
B\S
NC
N5,
50,
6C
6,0
1,1
Nava
(LSE)
StaticGames
MichaelmasTerm
16/80
Not
es
Not
es
Exa
mpl
e:T
heH
otel
ling
Gam
e
Th
ere
are
two
par
ties
are
inan
elec
tion
–”L
eft”
and
”Rig
ht”
.
Pol
itic
alvi
ewp
oin
tsar
ere
pres
ente
dby
an
um
ber
bet
wee
n−
1an
d1.
Vot
ervi
ews
are
un
ifor
mly
dis
trib
ute
don
[−1,
1].
Th
eH
ote
llin
gG
am
e:
Eac
hp
arty
cho
oses
ap
olic
yp
osit
ion
inth
ein
terv
al[−
1,1]:
Th
eR
igh
tp
arty
cho
oses
in[0
,1];
Th
eL
eft
par
tych
oos
esin
[−1,
0].
Vot
ers
cast
bal
lots
infa
vor
ofth
ep
arty
clos
est
toth
eir
idea
lp
oin
t.
Th
ep
arty
that
gets
am
ajor
ity
ofvo
tes
win
s.
Par
ties
care
only
abou
tw
inn
ing.
Wh
ere
will
each
par
typ
osit
ion
itse
lf?
Isth
ere
any
dom
inan
tst
rate
gy?
Nava
(LSE)
StaticGames
MichaelmasTerm
17/80
Iter
ativ
eE
limin
atio
nof
Dom
inat
edS
trat
egie
s
To
fin
dd
omin
ant
stra
tegi
esel
imin
ate
dom
inat
edst
rate
gies
from
the
gam
e
Ifn
eces
sary
rep
eat
the
pro
cess
top
ossi
bly
rule
out
mor
est
rate
gies
Con
sid
erth
efo
llow
ing
exam
ple
:
1\2
LC
R
T1,
02,
13,
0M
2,3
3,2
2,1
D0,
21,
22,
5
⇒
1\2
LC
R
T1,
02,
13,
0M
2,3
3,2
2,1
D0,
21,
22,
5
At
the
firs
tin
stan
ceon
lyD
isd
omin
ated
for
pla
yer
1
No
stra
tegy
isd
omin
ated
apr
iori
for
pla
yer
2
[Str
ateg
ies
ingr
een
inth
eta
ble
are
dom
inat
edan
dth
us
elim
inat
ed]
Nava
(LSE)
StaticGames
MichaelmasTerm
18/80
Not
es
Not
es
Iter
ativ
eE
limin
atio
nof
Dom
inat
edS
trat
egie
s
On
ceD
has
bee
nel
imin
ated
from
the
gam
e:
Str
ateg
yR
isd
omin
ated
for
pla
yer
2
No
stra
tegy
isd
omin
ated
for
pla
yer
1
1\2
LC
R
T1,
02,
13,
0M
2,3
3,2
2,1
D0,
21,
22,
5
⇒
1\2
LC
R
T1,
02,
13,
0M
2,3
3,2
2,1
D0,
21,
22,
5
On
ceR
has
bee
nel
imin
ated
from
the
gam
e:
Str
ateg
yT
isd
omin
ated
for
pla
yer
1
Afi
nal
iter
atio
nyi
eld
s(M
,L)
asth
eon
lysu
rviv
ing
stra
tegi
es
Nava
(LSE)
StaticGames
MichaelmasTerm
19/80
Com
mon
Kno
wle
dge
ofR
atio
nalit
y
Defi
niti
on
Th
efa
ctF
isa
com
mon
know
led
geif
Eve
ryp
laye
rkn
ows
F;
Eve
ryp
laye
rkn
ows
that
ever
yot
her
pla
yer
know
sF
;
Eve
ryp
laye
rkn
ows
that
ever
yp
laye
rkn
ows
that
ever
yp
laye
rkn
ows
Fan
dso
on.
The
orem
If,
amon
gp
laye
rs,
ther
eis
aco
mm
onkn
owle
dge
ofth
ega
me
and
ofth
efa
ctth
atal
lp
laye
rsar
era
tion
al,
then
the
outc
ome
ofth
ega
me
mu
stb
eam
ong
thos
eth
atsu
rviv
eit
erat
ive
elim
inat
ion
ofd
omin
ated
stra
tegi
es.
To
use
dom
inat
ing
stra
tegi
esw
en
eed
ever
yp
laye
rto
be
rati
onal
and
tokn
owow
np
ayoff
s.Nava
(LSE)
StaticGames
MichaelmasTerm
20/80
Not
es
Not
es
Dom
inan
ce:
Fin
alC
onsi
dera
tion
sI
Dom
inan
cean
dit
erat
ive
elim
inat
ion
are
ben
chm
arks
ofra
tion
alit
y.
Th
eb
enefi
tsof
stri
ctit
erat
ive
elim
inat
ion
are
that
:
the
ord
erof
elim
inat
ion
isir
rele
van
t;
ther
eis
no
nee
dto
know
the
oth
erp
laye
r’s
acti
on;
ital
lco
mes
from
rati
onal
ity.
Th
elim
itat
ion
sof
stri
ctit
erat
ive
elim
inat
ion
are
that
:
won
’tal
way
sre
ach
solu
tion
;
we
mu
stas
sum
eco
mm
onkn
owle
dge
ofra
tion
alit
yan
dof
the
gam
e;
itof
ten
lead
sto
ineffi
cien
tou
tcom
es.
Nava
(LSE)
StaticGames
MichaelmasTerm
21/80
Dom
inan
ce:
Fin
alC
onsi
dera
tion
sII
Th
isis
prob
lem
atic
intw
ow
ays:
rati
onal
ity
mu
sth
old
wit
hpr
obab
ility
1;
ther
em
ust
be
un
limit
ed“d
epth
”of
rati
onal
ity.
Pro
ble
mI:
Rat
ion
alit
ym
ust
hol
dw
ith
prob
abili
ty1.
Con
sid
erth
efo
llow
ing
gam
efo
rk
very
larg
e:
1\2
LR
U2,
53,
4M
0,−
k2,
k
Iter
ativ
eel
imin
atio
nle
ads
for
(U,L
).
Bu
tis
itlik
ely
pla
yer
2w
illn
ever
pla
yR
for
any
valu
eof
k?
Nava
(LSE)
StaticGames
MichaelmasTerm
22/80
Not
es
Not
es
Dom
inan
ce:
Fin
alC
onsi
dera
tion
sII
I
Nex
tco
nsi
der
the
follo
win
gga
me
for
kve
ryla
rge:
1\2
LR
U2,
53,
4M
0,−
k2,
kD
3,5−
k,4
Iter
ativ
eel
imin
atio
nle
ads
for
(D,L
).Is
this
reas
onab
le?
Pro
ble
mII
:U
nlim
ited
dep
thof
rati
onal
ity.
Con
sid
era
gam
ein
wh
ich
:
10p
laye
rsco
mp
ete;
each
pla
yer
wri
tes
an
um
ber
bet
wee
n1
&10
000.
Den
ote
byX
=2/
3of
the
aver
age
ofth
ep
laye
rsn
um
ber
s.
the
pla
yer
wh
ose
nu
mb
erY
iscl
oses
tto
Xw
ins.
Aw
eake
rn
otio
nof
equ
ilibr
ium
may
byp
ass
som
eof
thes
elim
itat
ion
s.Nava
(LSE)
StaticGames
MichaelmasTerm
23/80
StrategicForm
Gam
es:
Nas
hE
quili
briu
m
Nava
(LSE)
StaticGames
MichaelmasTerm
24/80
Not
es
Not
es
Nas
hE
quili
briu
m:
Intr
odu
ctio
n
Dom
inan
cew
asth
eap
prop
riat
eso
luti
onco
nce
pt
ifp
laye
rsh
adn
oin
form
atio
nor
bel
iefs
abou
tch
oice
sm
ade
byot
her
s
Th
ew
eake
rn
otio
nof
equ
ilibr
ium
that
will
be
intr
od
uce
dpr
esu
mes
that
:
pla
yers
hav
eco
rrec
tb
elie
fsab
out
choi
ces
mad
eby
oth
ers
pla
yers
choi
ces
are
opti
mal
give
nsu
chb
elie
fs
the
envi
ron
men
tis
com
mon
know
led
geam
ong
pla
yers
Su
chm
od
elal
low
sfo
rti
ghte
rpr
edic
tion
sw
hen
dom
inan
ceh
asn
ob
ite
Nava
(LSE)
StaticGames
MichaelmasTerm
25/80
Nas
hE
quili
briu
m
Defi
niti
on(N
ash
Equ
ilibr
ium
NE
)
A(p
ure
stra
tegy
)N
ash
equ
ilibr
ium
ofa
gam
eG
con
sist
sof
ast
rate
gypr
ofile
a=
(ai,
a−i)
such
that
for
any
i∈
N:
ui(
a)≥
ui(
a′ i,a−i)
for
any
a′ i∈
Ai
apr
ofile
ais
aN
Eiff
a i∈
bi(
a−i)
for
any
i∈
N
Pro
per
ties
:
Str
ateg
ypr
ofile
sar
ein
dep
end
ent
Str
ateg
ypr
ofile
sco
mm
onkn
owle
dge
Str
ateg
ies
max
imiz
eu
tilit
ygi
ven
bel
iefs
Nava
(LSE)
StaticGames
MichaelmasTerm
26/80
Not
es
Not
es
Exa
mpl
es,
Pro
per
ties
and
Lim
itat
ions
Gam
esm
ayh
ave
mor
eN
E’s
(Bat
tle
ofth
eS
exes
):
B\G
sm
s5,
21,
2m
0,0
3,5
Nas
heq
uili
bria
may
not
be
effici
ent
(Pri
son
er’s
Dile
mm
a):
B\S
NC
N5,
50,
6C
6,0
1,1
Pu
rest
rate
gyN
ash
equ
ilibr
iam
ayn
otex
ist
(Mat
chin
gP
enn
ies)
:
B\G
HT
H0,
22,
0T
2,0
0,2
Nava
(LSE)
StaticGames
MichaelmasTerm
27/80
Exa
mpl
es,
Pro
per
ties
and
Lim
itat
ions
Gam
esm
ayh
ave
mu
ltip
leN
E(B
attl
eof
the
Sex
es):
B\G
sm
s5,
21,
2m
0,0
3,5
Nas
heq
uili
bria
may
be
ineffi
cien
t(P
riso
ner
’sD
ilem
ma)
:
B\S
NC
N5,
50,
6C
6,0
1,1
Pu
rest
rate
gyN
ash
equ
ilibr
iam
ayn
otex
ist
(Mat
chin
gP
enn
ies)
:
B\G
HT
H0,
22,
0T
2,0
0,2
Nava
(LSE)
StaticGames
MichaelmasTerm
28/80
Not
es
Not
es
Exa
mpl
e:C
oor
dina
tion
Gam
es
Con
sid
era
gam
ep
laye
dby
two
hu
nte
rs,
for
k∈[1
,8]:
B\G
Sta
gH
are
Sta
g9,
90,
kH
are
k,0
k,k
Th
ere
are
two
NE
:
inon
eth
eh
un
ters
coor
din
ate
onh
un
tin
gth
est
ag;
inth
eot
her
they
split
and
each
hu
nts
ah
are
onh
isow
n.
Wh
enco
mp
ared
toth
eh
are-
NE
,th
est
ag-N
Een
tails
:
grea
ter
gain
sfr
omco
ord
inat
ion
;
grea
ter
risk
sof
mis
coor
din
atio
n.
Ifp
laye
rsco
mm
un
icat
e,th
eyar
elik
ely
toop
tfo
rP
aret
od
omin
atin
gN
E.
Nas
heq
uili
bria
can
be
view
edas
foca
lp
oin
ts.
Nava
(LSE)
StaticGames
MichaelmasTerm
29/80
Exa
mpl
e:T
hree
Pla
yers
Aga
me
wit
hm
ore
than
2p
laye
rs:
3L
R
1\2 T D
AB
1,0,
11,
0,0
0,1,
11,
2,0
AB
0,1,
10,
1,1
1,0,
12,
1,1
To
fin
dal
lN
Ech
eck
bes
tre
ply
map
s:
3L
R
1\2 T D
AB
1,0,
11,
0,0
0,1,
11,
2,0
AB
0,1,
10,
1,1
1,0,
12,
1,1
Nava
(LSE)
StaticGames
MichaelmasTerm
30/80
Not
es
Not
es
War
ofA
ttri
tion
Exa
mpl
e
Con
sid
era
gam
ew
ith
two
com
pet
itor
sin
volv
edin
afi
ght:
Th
ese
tof
pla
yers
isN
=1
,2.
Com
pet
itor
sch
oos
eh
owm
uch
effor
tto
pu
tin
afi
ght
Ai=
[0,∞
).
Th
eva
lue
ofw
inn
ing
the
figh
tis
for
com
pet
itor
i∈
Nis
v i.
Th
eh
igh
est
effor
tw
ins
the
figh
tan
dti
esar
ebr
oken
atra
nd
om.
For
each
com
pet
itor
the
cost
offi
ghti
ng
issi
mp
lym
ina
i,a j.
Th
ep
ayoff
ofco
mp
etit
ori
give
nth
eir
effor
tle
vels
thu
ssa
tisf
y:
ui(
a i,a
j)=
v i−
a jif
a i>
a jv i
/2−
a jif
a i=
a j−
a iif
a i<
a j
Nava
(LSE)
StaticGames
MichaelmasTerm
31/80
War
ofA
ttri
tion
Exa
mpl
e
As
pay
offs
amou
nt
to:
ui(
a i,a
j)=
v i−
a jif
a i>
a jv i
/2−
a jif
a i=
a j−
a iif
a i<
a j
Bes
tre
spon
sefu
nct
ion
ssa
tisf
y:
bi(
a j)=
a i
>a j
ifa j
<v i
a i=
0or
a i>
a jif
a j=
v ia i
=0
ifa j
>v i
All
Nas
hE
qu
ilibr
iaof
the
gam
esa
tisf
yon
eof
the
follo
win
g:
a 1=
0an
da 2≥
v 1
a 2=
0an
da 1≥
v 2
Nava
(LSE)
StaticGames
MichaelmasTerm
32/80
Not
es
Not
es
War
ofA
ttri
tion
Exa
mpl
e
Th
ees
iest
way
tofi
nd
the
NE
insu
chga
mes
isp
lott
ing
BR
s:
All
Nas
hE
qu
ilibr
iaof
the
gam
esa
tisf
yon
eof
the
follo
win
g:
a 1=
0an
da 2≥
v 1
a 2=
0an
da 1≥
v 2
Nava
(LSE)
StaticGames
MichaelmasTerm
33/80
Exa
mpl
e:L
obby
ing
Con
sid
era
gam
ew
ith
two
lob
byis
tsp
etit
ion
ing
over
two
vers
ion
sof
ab
ill:
Th
eva
lue
ofh
avin
gb
illi∈1
,2
appr
oved
isv i
for
ian
dze
rofo
rj.
Lob
byis
tsch
oos
eh
owm
any
reso
urc
esto
inve
sta i∈[0
,∞).
Th
epr
obab
ility
that
the
pol
icy
pref
erre
dby
lob
byis
ti
isap
prov
edis
pi=
a ia i+
a j
Th
ep
ayoff
oflo
bby
ist
ith
us
amou
nts
to
ui(
a)=
a ia i+
a jv i−
a i
Th
ep
ayoff
isco
nca
vean
dsi
ngl
ep
eake
dfo
ran
yva
lue
a j.
Nava
(LSE)
StaticGames
MichaelmasTerm
34/80
Not
es
Not
es
Exa
mpl
e:L
obby
ing
Tak
ing
firs
tor
der
con
dit
ion
syi
eld
s
a j
(ai+
a j)2
v i=
1⇔
a i+
a j=
(via
j)1
/2
Th
us,
bes
tre
spon
sefu
nct
ion
nec
essa
rily
sati
sfy
bi(
a j)=
(v1
/2
i−
a1/2
j)a
1/2
jif
a j<
v i0
ifa j≥
v i
Sol
vin
gth
etw
osh
ows
that
the
un
iqu
eN
En
eces
sari
lysa
tisfi
es
a i=
v j
[ v iv i+
v j
] 2
Nava
(LSE)
StaticGames
MichaelmasTerm
35/80
DS
Eim
plie
sN
E
Fac
t
An
yd
omin
ant
stra
tegy
equ
ilibr
ium
isa
Nas
heq
uili
briu
m
Pro
of.
Ifa
isa
DS
Eth
ena i∈
bi(
a′ −i)
for
any
a′ −i
and
i∈
N.
Wh
ich
imp
lies
ais
NE
sin
cea i∈
bi(
a−i)
for
any
i∈
N.
Nava
(LSE)
StaticGames
MichaelmasTerm
36/80
Not
es
Not
es
Nas
hE
quili
briu
m:
Fin
alC
onsi
dera
tion
sI
Th
em
ost
not
able
limit
atio
ns
ofth
eN
Eso
luti
onco
nce
pt
are
that
:
pla
yers
hav
eco
rrec
tb
elie
fsab
out
oth
erp
laye
rs’
stra
tegi
es;
pla
yers
are
rati
on
al
and
cho
ose
the
opti
mal
acti
ongi
ven
thes
eb
elie
fs;
NE
are
only
imm
un
eto
dev
iati
ons
byin
div
idu
alp
laye
rs,
not
grou
ps.
Th
efi
rst
req
uir
emen
tis
very
prob
lem
atic
,es
pec
ially
inga
mes
wit
hm
ore
then
one
equ
ilibr
ium
.
Nava
(LSE)
StaticGames
MichaelmasTerm
37/80
Nas
hE
quili
briu
m:
Fin
alC
onsi
dera
tion
sII
NE
isa
goo
dso
luti
onco
nce
pt
for:
nor
ms
–ev
eryb
od
ykn
ows
wh
atsi
de
tod
rive
onth
ero
ad;
prel
imin
ary
talk
s–
reac
hin
gto
anag
reem
ent
that
isse
lf-e
nfo
rcin
g;
stab
leso
luti
ons
pla
yed
over
tim
e–
the
gam
eis
pla
yed
man
yti
mes
;p
laye
rsch
oos
eth
eac
tion
the
bes
tb
ased
onp
ast
exp
erie
nce
(dis
rega
rdin
gan
yst
rate
gic
con
sid
erat
ion
);N
Eis
an
eces
sary
con
dit
ion
for
stab
ility
(bu
tn
otsu
ffici
ent
and
not
nec
essa
rily
easy
con
verg
eto
).
Nava
(LSE)
StaticGames
MichaelmasTerm
38/80
Not
es
Not
es
StrategicForm
Gam
es:
Mix
edS
trat
egie
s
Nava
(LSE)
StaticGames
MichaelmasTerm
39/80
Intr
odu
ctio
nto
Mix
edS
trat
egie
s
Apr
oble
mat
icas
pec
tof
the
solu
tion
con
cep
tsd
iscu
ssed
inth
efi
rst
two
lect
ure
sw
asth
ateq
uili
bria
did
not
alw
ays
exis
t.
Intu
itiv
ely,
exis
ten
cew
asn
otgu
aran
teed
bec
ause
pla
yers
had
no
dev
ice
toco
nce
alth
eir
beh
avio
rfr
omot
her
s.
For
mal
ly,
the
reas
ons
for
the
lack
ofex
iste
nce
wer
e:
Non
-con
vexi
ties
inth
ech
oice
sets
Dis
con
tin
uit
ies
ofth
eb
est
resp
onse
corr
esp
ond
ence
s
Nex
tw
ein
tro
du
cem
ixed
stra
tegi
esw
hic
hso
lve
bot
hpr
oble
ms
and
guar
ante
eex
iste
nce
ofat
leas
ta
Nas
heq
uili
briu
m.
Nava
(LSE)
StaticGames
MichaelmasTerm
40/80
Not
es
Not
es
Mix
edS
trat
egy
Defi
niti
on
Con
sid
erco
mp
lete
info
rmat
ion
stat
icga
me N
, Ai,
ui
i∈N
A
mix
edst
rate
gyfo
ri∈
Nis
apr
obab
ility
dis
trib
uti
onov
erac
tion
sin
Ai
Th
us
σ iis
am
ixed
stra
tegy
if:
σ i(a
i)≥
0fo
ran
ya i∈
Ai
∑ai∈
Aiσ i(a
i)=
1
Intu
itiv
ely
σ i(a
i)is
the
prob
abili
tyth
atp
laye
ri
cho
oses
top
lay
a i
E.G
.σ 1(B
)=
0.3
and
σ 1(C
)=
0.7
isa
mix
edst
rate
gyfo
r1
in:
1\2
BC
B2,
00,
2C
0,1
1,0
Nava
(LSE)
StaticGames
MichaelmasTerm
41/80
Sim
plex
Th
ese
tof
pos
sib
lepr
obab
ility
dis
trib
uti
ons
ona
set
Bis
:
calle
dsi
mp
lex
and
den
oted
by∆(B
)
Th
ese
tis
know
nto
be:
Clo
sed
Bou
nd
ed
Con
vex
Nava
(LSE)
StaticGames
MichaelmasTerm
42/80
Not
es
Not
es
Not
atio
n
Defi
ne
apr
ofile
ofst
rate
gies
for
all
pla
yers
oth
erth
ani
by:
σ−i=
(σ1,.
..,σ
i−1,σ
i+1,.
..,σ
N)
Defi
ne
apr
ofile
ofst
rate
gies
for
all
pla
yers
by:
σ=
(σ1,.
..,σ
N)
Nava
(LSE)
StaticGames
MichaelmasTerm
43/80
Pay
offs
from
Mix
edS
trat
egie
s
Mix
edst
rate
gyar
ech
osen
ind
epen
den
tly
byp
laye
rs.
Th
us,
Pr(
a|σ)=
∏j∈
NP
r(a j|σ)=
∏j∈
Nσ j(a
j)
Th
eex
pec
ted
uti
lity
ofp
laye
ri
from
am
ixed
stra
tegy
profi
leσ
is
ui(
σ)=
∑a∈A
Pr(
a|σ)u
i(a)=
=∑
a∈A
∏j∈
Nσ j(a
j)ui(
a)
Exa
mp
le:
Ifp
laye
rsfo
llow
σ 1(B
)=
σ 2(B
)=
0.3
inth
ega
me:
Pay
offs:
1\2
BC
B2,
00,
2C
0,1
1,0
Pro
bab
iliti
es:
1\2
BC
B9%
21%
C21
%49
%
Th
ep
ayoff
top
laye
r1
is:
u1(σ
1,σ
2)=
(.09)2
+(.
49)1
+(.
42)0
=0.
67
Nava
(LSE)
StaticGames
MichaelmasTerm
44/80
Not
es
Not
es
Bes
tR
eply
Cor
resp
onde
nces
Den
ote
the
bes
tre
ply
corr
esp
ond
ence
ofi
bybi(
σ−i)
Th
em
apis
defi
ned
by:
bi(
σ−i)=
arg
max
σ i∈
∆(A
i)ui(
σ i,σ−i)
For
inst
ance
con
sid
erth
ega
me: 1\2
sm
s5,
21,
2m
0,0
3,5
Ifσ 1(s)=
1th
enan
yσ 2(s)∈[0
,1]
sati
sfies
σ 2∈
b2(σ
1)
Ifσ 1(s)<
1th
enon
lyσ 2(s)=
0sa
tisfi
esσ 2∈
b2(σ
1)
Nava
(LSE)
StaticGames
MichaelmasTerm
45/80
Dom
inat
edS
trat
egie
s
Str
ateg
yσ i
wea
kly
do
min
ate
sa i
if:
ui(
σ i,a−i)≥
ui(
a i,a−i)
for
any
a−i
ui(
σ i,a−i)>
ui(
a i,a−i)
for
som
ea−i
a iis
wea
kly
un
do
min
ate
dif
no
stra
tegy
wea
kly
dom
inat
esit
Th
isal
low
su
sto
rule
out
mor
est
rate
gies
than
bef
ore,
eg:
1\2
LC
R
T6,
60,
20,
0B
0,0
0,2
6,6
σ 2(L)=
σ 2(R
)=
0.5
stri
ctly
dom
inat
esC
sin
ce:
u2(σ
2,a
1)=
3>
u2(C
,a1)=
2
Nava
(LSE)
StaticGames
MichaelmasTerm
46/80
Not
es
Not
es
Nas
hE
quili
briu
m
Defi
niti
on(N
ash
Equ
ilibr
ium
NE
)
AN
ash
equ
ilibr
ium
ofa
gam
eco
nsi
sts
ofa
stra
tegy
profi
leσ=
(σi,
σ−i)
such
that
for
any
i∈
N:
ui(
σ)≥
ui(
a i,σ−i)
for
any
a i∈
Ai
Imp
licit
toth
ed
efin
itio
nof
NE
are
the
follo
win
gas
sum
pti
ons:
Eac
hag
ent
cho
oses
his
mix
edst
rate
gyin
dep
end
entl
yof
oth
ers
Eac
hag
ent
kn
ow
sex
actl
yth
est
rate
gie
sth
eot
her
sad
opt
Eac
hag
ent
cho
oses
soto
ma
xim
ize
exp
ecte
du
tilit
ygi
ven
his
bel
iefs
Nava
(LSE)
StaticGames
MichaelmasTerm
47/80
Nas
hE
quili
briu
m:
Com
puta
tion
Hel
p
Ast
rate
gypr
ofile
σis
aN
ash
Eq
uili
briu
mif
and
only
if:
ui(
σ)=
ui(
a i,σ−i)
for
any
a isu
chth
atσ i(a
i)>
0
ui(
σ)≥
ui(
a i,σ−i)
for
any
a isu
chth
atσ i(a
i)=
0
Ifa i
isst
rict
lyd
omin
ated
,th
enσ i(a
i)=
0in
any
Nas
heq
uili
briu
m
Ifa i
isw
eakl
yd
omin
ated
,th
enσ i(a
i)>
0in
aN
ash
equ
ilibr
ium
only
ifan
ypr
ofile
ofac
tion
sa−i
for
wh
ich
a iis
stri
ctly
wor
seo
ccu
rsw
ith
zero
prob
abili
ty
Nava
(LSE)
StaticGames
MichaelmasTerm
48/80
Not
es
Not
es
Exa
mpl
es:
Cla
ssic
alG
ames
Gam
esm
ayh
ave
mor
eN
Es
(Bat
tle
ofth
eS
exes
):
1\2
sm
s5,
21,
1m
0,0
2,5
Th
ere
are
2P
NE
&a
mix
edN
Ein
wh
ich
σ 1(s)=
5/
6&
σ 2(s)=
1/
6:
u1(s
,σ2)=
5σ 2(s)+(1−
σ 2(s))
=2(1−
σ 2(s))
=u1(m
,σ2)
u2(m
,σ1)=
σ 1(s)+
5(1−
σ 1(s))
=2
σ 1(s)=
u2(s
,σ1)
Gam
esm
ayh
ave
only
mix
edN
E(M
atch
ing
Pen
nie
s):
B\G
HT
H0,
22,
0T
2,0
0,2
Th
ere
isa
un
iqu
eN
Ein
wh
ich
σ 1(H
)=
σ 2(H
)=
1/
2:
2(1−
σ i(H
))=
2σ i(H
)
Nava
(LSE)
StaticGames
MichaelmasTerm
49/80
Exa
mpl
e:A
Con
tinu
umof
NE
Gam
esm
ayh
ave
aco
nti
nu
um
ofN
E:
1\2
LR
C
T4,
10,
23,
2D
0,2
2,0
1,0
Th
ega
me
has
1P
NE
&a
con
tin
uu
mof
NE
s,n
amel
y:
σ 1(T
)=
2/
3&
σ 2(R
)=
1−
σ 2(L)−
σ 2(C
)
σ 2(L)=
1/
3−(2
/3)σ
2(C
)fo
rσ 2(C
)∈[0
,1/
2]
Th
ese
are
all
NE
’ssi
nce
:
u2(L
,σ1)=
σ 1(T
)+
2(1−
σ 1(T
))=
2σ 1(T
)=
u2(R
,σ1)=
u2(C
,σ1)
u1(T
,σ2)=
4σ 2(L)+
3σ 2(C
)=
2σ 2(R
)+
σ 2(C
)=
u1(D
,σ2)
Nava
(LSE)
StaticGames
MichaelmasTerm
50/80
Not
es
Not
es
Exa
mpl
e:T
heC
ase
ofK
itty
Gen
oves
e
Qu
een
s19
64.
Aw
oman
ism
urd
ered
.38
nei
ghb
ors
hea
r.N
oon
eca
lls91
1.
New
Yor
kT
ime
–M
arch
27th
,19
64:
ww
w.g
arys
turt
.fre
e-on
line.
co.u
k/T
he%
20ca
se%
20of
%20
Kit
ty%
20G
enov
ese.
htm
Mo
del
:
nid
enti
cal
nei
ghb
ors;
1in
div
idu
alco
stfo
rca
llin
g;
x>
1u
tilit
yof
ever
yp
laye
rif
som
eon
eca
llsth
ep
olic
e.
Inan
yP
NE
,ex
actl
yon
en
eigh
bor
calls
.Im
prob
able
wit
hou
tco
ord
inat
ion
.
Ina
mix
edN
E,
ever
yn
eigh
bor
calls
wit
hpr
obab
ility
p,
and
isin
diff
eren
tb
etw
een
calli
ng
and
not
calli
ng:
U(c
all
,p)=
x−
1=
x( 1−(1−
p)n−1) =
U(n
oca
ll,p),
Nava
(LSE)
StaticGames
MichaelmasTerm
51/80
Exa
mpl
e:T
heC
ase
ofK
itty
Gen
oves
e
Th
eref
ore,
the
prob
abili
tyth
ata
give
nn
eigh
bor
calls
911
is
p=
1−(1
/x)1
/(n−1) ,
wh
ileth
epr
obab
ility
that
atle
ast
one
ofth
en
eigh
bor
sca
lls91
1is
P=
1−(1−
p)n
=1−(1
/x)n
/(n−1)
For
n=
1,w
eh
ave
P=
1.B
ut
asn→
∞,
Pd
ecre
ases
to1−(1
/x).
For
x=
2,w
eh
ave
Nava
(LSE)
StaticGames
MichaelmasTerm
52/80
Not
es
Not
es
StrategicForm
Gam
es:
NE
Exi
sten
ceT
heor
ems
Nava
(LSE)
StaticGames
MichaelmasTerm
53/80
Nas
hE
quili
briu
mE
xist
ence
The
orem
(NE
Exi
sten
ce)
An
yga
me
wit
ha
fin
ite
nu
mb
erof
acti
ons
pos
sess
esa
Nas
heq
uili
briu
m.
The
orem
(PN
EE
xist
ence
)
An
yga
me
wit
hco
nve
xan
dco
mp
act
acti
onse
tsan
dw
ith
con
tin
uou
san
dq
uas
i-co
nca
vep
ayoff
fun
ctio
ns
pos
sess
esa
pu
rest
rate
gyN
ash
equ
ilibr
ium
.
Ass
um
pti
ons
inb
oth
theo
rem
sgu
aran
tee
that
bes
tre
spon
ses
are
up
per
-hem
icon
tin
uou
s,cl
osed
and
con
vex
valu
edon
con
vex
com
pac
tse
ts,
and
thu
sex
iste
nce
hol
ds
via
Kak
uta
ni
Fix
edP
oin
tT
heo
rem
.
Nava
(LSE)
StaticGames
MichaelmasTerm
54/80
Not
es
Not
es
Nas
hE
quili
briu
mE
xist
ence
:2x
2In
tuit
ion
Intu
itiv
ely
dra
wco
nti
nu
ous
BR
map
sin
the
char
t:
T
LR
B
Sh
owth
atB
Rm
aps
mu
stin
ters
ect
sin
ce:
one
map
mov
esfr
omth
ele
ftto
the
righ
tb
oun
dar
y;
wh
ileth
eot
her
from
the
top
toth
eb
otto
mb
oun
dar
y.
Usu
ally
BR
sin
ters
ect
ano
dd
nu
mb
erof
tim
es.
Nava
(LSE)
StaticGames
MichaelmasTerm
55/80
Kak
utan
iF
ixed
Poi
ntT
heor
em
Th
eB
Rco
rres
pon
den
ceis
ofp
laye
ri:
Up
per
-hem
ico
nti
nu
ou
sif
for
any
ase
qu
ence σ
k ∞ k
=1
ofst
rate
gy
profi
les
such
that
σk i∈
bi(
σk −i)
for
any
kw
eh
ave
that
σk→
σim
plie
sσ i∈
bi(
σ−i)
.
Co
nve
x-va
lued
ifσ i
,σ′ i∈
bi(
σ−i)
imp
lies
pσ i+(1−
p)σ′ i∈
bi(
σ−i)
for
any
p∈[0
,1]
The
orem
(Kak
utan
iF
ixed
Poi
ntT
heor
em)
Let
Xb
ea
non
-em
pty
,co
mp
act
and
con
vex
sub
set
ofR
n.
Let
b:X
⇒X
be
an
on-e
mp
ty,
con
vex-
valu
ed,
UH
Cco
rres
pon
den
ce.
Th
enb
has
afi
xed
poi
nt.
Nava
(LSE)
StaticGames
MichaelmasTerm
56/80
Not
es
Not
es
UH
CB
est
Res
pon
ses
Impl
yE
xist
ence
Aga
me
wit
hou
tP
NE
and
wit
ha
sin
gle
NE
:
1\2
BC
B2,
00,
2C
0,1
1,0
Nava
(LSE)
StaticGames
MichaelmasTerm
57/80
BayesianGam
es:
Sta
tic
Gam
esw
ith
Inco
mpl
ete
Info
rmat
ion
[No
tE
xam
ina
ble
]
Nava
(LSE)
StaticGames
MichaelmasTerm
58/80
Not
es
Not
es
Sum
mar
y
Gam
esof
Inco
mp
lete
Info
rmat
ion
:
Defi
nit
ion
s:
Inco
mp
lete
Info
rmat
ion
Gam
eIn
form
atio
nS
tru
ctu
rean
dB
elie
fsS
trat
egie
sB
est
Rep
lyM
ap
Sol
uti
onC
once
pts
inP
ure
Str
ateg
ies:
Dom
inan
tS
trat
egy
Eq
uili
briu
mB
ayes
Nas
hE
qu
ilibr
ium
Exa
mp
les
Mix
edS
trat
egie
s&
Bay
esN
ash
Eq
uili
bria
Nava
(LSE)
StaticGames
MichaelmasTerm
59/80
Bay
esia
nG
ames
An
stat
icin
com
ple
tein
form
atio
nga
me
con
sist
sof
:
Nth
ese
tof
pla
yers
inth
ega
me
Ai
pla
yer
i’s
acti
onse
t
Xi
pla
yer
i’s
set
ofp
ossi
ble
sign
als
Apr
ofile
ofsi
gnal
sx=
(x1,.
..,x
n)
isan
elem
ent
X=×
j∈N
Xj
fa
dis
trib
uti
onov
erth
ep
ossi
ble
sign
als
ui
:A×
X→
Rp
laye
ri’
su
tilit
yfu
nct
ion
,ui(
a|x)
Nava
(LSE)
StaticGames
MichaelmasTerm
60/80
Not
es
Not
es
Exa
mpl
e:B
ayes
ian
Gam
e
Con
sid
erth
efo
llow
ing
Bay
esia
nga
me:
Pla
yer
1ob
serv
eson
lyon
ep
ossi
ble
sign
al:
X1=C
Pla
yer
2’s
sign
alta
kes
one
oftw
ova
lues
:X
2=L
,R
Pro
bab
iliti
esar
esu
chth
at:
f(C
,L)=
0.6
Pay
offs
and
acti
onse
tsar
eas
des
crib
edin
the
mat
rix:
1\2
.Ly 2
z 21\2
.Ry 2
z 2y 1
1,2
0,1
y 11,
30,
4z 1
0,4
1,3
z 10,
11,
2
Nava
(LSE)
StaticGames
MichaelmasTerm
61/80
Info
rmat
ion
Str
uctu
re
Info
rmat
ion
stru
ctu
re:
Xi
den
otes
the
sign
alas
ara
nd
omva
riab
le
bel
ongs
toth
ese
tof
pos
sib
lesi
gnal
sX
i
x id
enot
esth
ere
aliz
atio
nof
the
ran
dom
vari
able
Xi
X−i=
(X1,.
..,X
i−1,X
i+1,.
..,X
n)
den
otes
apr
ofile
ofsi
gnal
sfo
ral
lp
laye
rsot
her
than
i
Pla
yer
iob
serv
eson
lyXi
Pla
yer
iig
nor
esX−i,
bu
tkn
ows
f
Wit
hsu
chin
form
atio
np
laye
ri
form
sb
elie
fsre
gard
ing
the
real
izat
ion
ofth
esi
gnal
sof
the
oth
erp
laye
rsx −
i
Nava
(LSE)
StaticGames
MichaelmasTerm
62/80
Not
es
Not
es
Bel
iefs
abou
tS
igna
ls:
Inde
pen
denc
e
Ifsi
gnal
sar
ein
dep
end
ent:
f(x)=
∏j∈
Nf j(x
j)
Th
isim
plie
sth
atth
esi
gnal
x iof
pla
yer
iis
ind
epen
den
tof
X−i
Bel
iefs
are
apr
obab
ility
dis
trib
uti
onov
erth
esi
gnal
sof
the
oth
erp
laye
rs
An
yp
laye
rfo
rms
bel
iefs
abou
tth
esi
gnal
sre
ceiv
edby
the
oth
erp
laye
rsby
usi
ng
Bay
esR
ule
Ind
epen
den
ceim
plie
sth
atco
nd
itio
nal
obse
rvin
gXi=
x ith
eb
elie
fsof
pla
yer
iar
e:f i(x−i|x
i)=
∏j∈
N\i
f j(x
j)=
f −i(
x −i)
Nava
(LSE)
StaticGames
MichaelmasTerm
63/80
Bel
iefs
abou
tS
igna
ls:
Inte
rdep
ende
nce
Inth
ege
ner
alsc
enar
iow
ith
inte
rdep
end
ent
sign
als,
pla
yers
form
bel
iefs
abou
tth
esi
gnal
sre
ceiv
edby
oth
ers
byu
sin
gB
ayes
Ru
le
Con
dit
ion
alob
serv
ing
Xi=
x ith
eb
elie
fsof
pla
yer
iar
e:
f i(x−i|x
i)=
Pr(
X−i=
x −i|X
i=
x i)=
=P
r(X−i=
x −i∩
Xi=
x i)
Pr(
Xi=
x i)
=
=P
r(X−i=
x −i∩
Xi=
x i)
∑y −
i∈X−iP
r(X−i=
y −i∩
Xi=
x i)=
=f(x−i,
x i)
∑y −
i∈X−if(y−i,
x i)
Bel
iefs
are
apr
obab
ility
dis
trib
uti
onov
erth
esi
gnal
sof
the
oth
erp
laye
rsNava
(LSE)
StaticGames
MichaelmasTerm
64/80
Not
es
Not
es
Str
ateg
ies
Str
ateg
yP
rofi
les:
Ast
rate
gyco
nsi
sts
ofa
map
from
avai
lab
lein
form
atio
nto
acti
ons:
αi
:Xi→
Ai
Ast
rate
gypr
ofile
con
sist
sof
ast
rate
gyfo
rev
ery
pla
yer:
α(X
)=
(α1(X
1),
...,
αN(X
N))
We
adop
tth
eu
sual
con
ven
tion
:
α−i(
X−i)=
(α1(X
1),
...,
αi−
1(X
i−1),
αi+
1(X
i+1),
...,
αN(X
N))
Nava
(LSE)
StaticGames
MichaelmasTerm
65/80
Exa
mpl
e:B
ayes
ian
Gam
e
Con
sid
erth
efo
llow
ing
gam
e:
Pla
yer
1ob
serv
eson
lyon
ep
ossi
ble
sign
al:
X1=C
Pla
yer
2’s
sign
alta
kes
one
oftw
ova
lues
:X
2=L
,R
Pro
bab
iliti
esar
esu
chth
at:
f(C
,L)=
0.6
Pay
offs
and
acti
onse
tsar
eas
des
crib
edin
the
mat
rix:
1\2
.Ly 2
z 21\2
.Ry 2
z 2y 1
1,2
0,1
y 11,
30,
4z 1
0,4
1,3
z 10,
11,
2
Ast
rate
gyfo
rp
laye
r1
isan
elem
ent
ofth
ese
tα1∈y
1,z
1
Ast
rate
gyfo
rp
laye
r2
isa
map
α2
: L
,R→y
2,z
2
Pla
yer
1ca
nn
otac
tu
pon
2’s
priv
ate
info
rmat
ion
Nava
(LSE)
StaticGames
MichaelmasTerm
66/80
Not
es
Not
es
Dom
inan
tS
trat
egy
Equ
ilibr
ium
Str
ateg
yαi
wea
kly
do
min
ate
sα′ i
iffo
ran
ya −
ian
dx∈
X:
ui(
αi(
x i),
a −i|x
)≥
ui(
α′ i(
x i),
a −i|x
)[s
tric
tso
mew
her
e]
Str
ateg
yαi
isd
om
ina
nt
ifit
dom
inat
esan
yot
her
stra
tegy
α′ i
Str
ateg
yαi
isu
nd
om
ina
ted
ifn
ost
rate
gyd
omin
ates
it
Defi
niti
ons
(Dom
inan
tS
trat
egy
Equ
ilibr
ium
DS
E)
AD
omin
ant
Str
ateg
yeq
uili
briu
mof
anin
com
ple
tein
form
atio
nga
me
isa
stra
tegy
profi
leα
that
for
any
i∈
N,
x∈
Xan
da −
i∈
A−i
sati
sfies
:
ui(
αi(
x i),
a −i|x
)≥
ui(
α′ i(
x i),
a −i|x
)fo
ran
yα′ i
:Xi→
Ai
I.e.
αi
isop
tim
alin
dep
end
entl
yof
wh
atot
her
skn
owan
dd
o
Nava
(LSE)
StaticGames
MichaelmasTerm
67/80
Inte
rim
Exp
ecte
dU
tilit
yan
dB
est
Rep
lyM
aps
Th
ein
teri
mst
ag
eo
ccu
rsju
staf
ter
ap
laye
rkn
ows
his
sign
alXi=
x i
Itis
wh
enst
rate
gies
are
chos
enin
aB
ayes
ian
gam
e
Th
ein
teri
mex
pec
ted
uti
lity
ofa
(pu
re)
stra
tegy
profi
leα
isd
efin
edby
:
Ui(
α|x
i)=
∑X−iui(
α(x)|
x)f(x−i|x
i):X
i→
R
Wit
hsu
chn
otat
ion
inm
ind
not
ice
that
:
Ui(
a i,α−i|x
i)=
∑X−iui(
a i,α−i(
x −i)|x)f(x−i|x
i)
Th
eb
est
rep
lyco
rres
pon
den
ceof
pla
yer
iis
defi
ned
by:
bi(
α−i|x
i)=
arg
max
ai∈
AiUi(
a i,α−i|x
i)
BR
map
sid
enti
fyw
hic
hac
tion
sar
eop
tim
algi
ven
the
sign
alan
dth
est
rate
gies
follo
wed
byot
her
sNava
(LSE)
StaticGames
MichaelmasTerm
68/80
Not
es
Not
es
Pur
eS
trat
egy
Bay
esN
ash
Equ
ilibr
ium
Defi
niti
ons
(Bay
esN
ash
Equ
ilibr
ium
BN
E)
Ap
ure
stra
tegy
Bay
esN
ash
equ
ilibr
ium
ofan
inco
mp
lete
info
rmat
ion
gam
eis
ast
rate
gypr
ofile
αsu
chth
atfo
ran
yi∈
Nan
dx i∈
Xi
sati
sfies
:
Ui(
α|x
i)≥
Ui(
a i,α−i|x
i)fo
ran
ya i∈
Ai
BN
Ere
qu
ires
inte
rim
op
tim
alit
y(i
.e.
do
you
rb
est
give
nw
hat
you
know
)
BN
Ere
qu
ires
αi(
x i)∈
bi(
α−i|x
i)fo
ran
yi∈
Nan
dx i∈
Xi
Nava
(LSE)
StaticGames
MichaelmasTerm
69/80
Exa
mpl
e:B
ayes
ian
Gam
e
Con
sid
erth
efo
llow
ing
Bay
esia
nga
me
wit
hf(C
,L)=
0.6:
1\2
.Ly 2
z 21\2
.Ry 2
z 2y 1
1,2
0,1
y 11,
30,
4z 1
0,4
1,3
z 10,
11,
2
Th
eb
est
rep
lym
aps
for
bot
hp
laye
rar
ech
arac
teri
zed
by:
b2(α
1|x
2)=
y 2if
x 2=
Lz 2
ifx 2
=R
b1(α
2)=
y 1if
α2(L)=
y 2z 1
ifα2(L)=
z 2
Th
ega
me
has
au
niq
ue
(pu
rest
rate
gy)
BN
Ein
wh
ich
:
α1=
y 1,
α2(L)=
y 2,
α2(R
)=
z 2
DO
NO
TA
NA
LYZ
EM
AT
RIC
ES
SE
PA
RA
TE
LY!!
!
Nava
(LSE)
StaticGames
MichaelmasTerm
70/80
Not
es
Not
es
Tra
nsfo
rmin
ga
Bay
esia
nG
ame
Exa
mpl
e
Con
sid
erth
efo
llow
ing
Bay
esia
nga
me
wit
hf(C
,L)=
p>
1/
2:
1\2
.LC
D1\2
.RC
DA
1,2
0,1
A1,
30,
4B
0,4
1,3
B0,
11,
2
Th
isis
equ
ival
ent
toth
efo
llow
ing
com
ple
tein
form
atio
nga
me:
1\2
CC
DC
CD
DD
A1,
3−
p1−
p,3−
2pp
,4−
2p0,
4−
3pB
0,1+
3pp
,1+
2p1−
p,2
+2p
1,2+
p
You
may
then
fin
dB
Rs
and
BN
Es
inth
ism
od
ified
tab
le.
Nava
(LSE)
StaticGames
MichaelmasTerm
71/80
Rel
atio
nshi
psb
etw
een
Equ
ilibr
ium
Con
cept
s
Ifα
isa
DS
Eth
enit
isa
BN
E.
Infa
ctfo
ran
yac
tion
a ian
dsi
gnal
x i:
ui(
αi(
x i),
a −i|x
)≥
ui(
a i,a−i|x
)∀a−i,
x −i⇒
ui(
αi(
x i),
α−i(
x −i)|x)≥
ui(
a i,α−i(
x −i)|x)∀α−i,
x −i⇒
∑X−iui(
α(x)|
x)fi(
x −i|x
i)≥
∑X−iui(
a i,α−i(
x −i)|x)fi(
x −i|x
i)∀α−i⇒
Ui(
α|x
i)≥
Ui(
a i,α−i|x
i)∀α−i
Nava
(LSE)
StaticGames
MichaelmasTerm
72/80
Not
es
Not
es
BN
EE
xam
ple
I:E
xcha
nge
Ab
uye
ran
da
selle
rw
ant
totr
ade
anob
ject
:
Bu
yer’
sva
lue
for
the
obje
ctis
3$
Sel
ler’
sva
lue
isei
ther
0$or
2$b
ased
onth
esi
gnal
,X
S=L
,H
Bu
yer
can
offer
eith
er1$
or3$
top
urc
has
eth
eob
ject
Sel
ler
cho
ose
wh
eth
eror
not
tose
ll
B\S
.Lsa
len
osa
leB\S
.Hsa
len
osa
le3$
0,3
0,0
3$0,
30,
21$
2,1
0,0
1$2,
10,
2
Th
isga
me
for
any
prio
rf
has
aB
NE
inw
hic
h:
αS(L)=
sale
,αS(H
)=
no
sale
,αB=
1$
Sel
ling
isst
rict
lyd
omin
ant
for
S.L
Off
erin
g1$
isw
eakl
yd
omin
ant
for
the
bu
yer
Nava
(LSE)
StaticGames
MichaelmasTerm
73/80
BN
EE
xam
ple
II:
Ent
ryG
ame
Con
sid
erth
efo
llow
ing
mar
ket
gam
e:
Fir
mI
(th
ein
cum
ben
t)is
am
onop
olis
tin
am
arke
t
Fir
mE
(th
een
tran
t)is
con
sid
erin
gw
het
her
toen
ter
inth
em
arke
t
IfE
stay
sou
tof
the
mar
ket,
Eru
ns
apr
ofit
of1$
and
Ige
ts8$
IfE
ente
rs,
Ein
curs
aco
stof
1$an
dpr
ofits
ofb
oth
Ian
dE
are
3$
Ica
nd
eter
entr
yby
inve
stin
gat
cost0
,2
dep
end
ing
onty
peL
,H
IfI
inve
sts:
I’s
profi
tin
crea
ses
by1
ifh
eis
alon
e,d
ecre
ases
by1
ifh
eco
mp
etes
and
E’s
profi
td
ecre
ases
to0
ifh
eel
ects
toen
ter
E\I
.LIn
vest
Not
Inve
stE\I
.HIn
vest
Not
Inve
stIn
0,2
3,3
In0,
03,
3O
ut
1,9
1,8
Ou
t1,
71,
8
Nava
(LSE)
StaticGames
MichaelmasTerm
74/80
Not
es
Not
es
BN
EE
xam
ple
II:
Ent
ryG
ame
Let
πd
enot
eth
epr
obab
ility
that
firm
Iis
ofty
pe
Lan
dn
otic
e:
αI(H
)=
Not
Inve
stis
ast
rict
lyd
omin
ant
stra
tegy
for
I.H
For
any
valu
eof
π,
αI(L)=
Not
Inve
stan
dαE=
Inis
BN
E:
uI(N
ot,I
n|L)=
3>
2=
uI(I
nve
st,I
n|L)
UE(I
n,α
I(X
I))
=3>
1=
UE(O
ut,
αI(X
I))
For
πh
igh
enou
gh,
αI(L)=
Inve
stan
dαE=
Ou
tis
also
BN
E:
uI(I
nve
st,O
ut|
L)=
9>
8=
uI(N
ot,O
ut|
L)
UE(O
ut,
αI(X
I))
=1>
3(1−
π)=
UE(I
n,α
I(X
I))
E\I
.LIn
vest
Not
Inve
stE\I
.HIn
vest
Not
Inve
stIn
0,2
3,3
In0,
03,
3O
ut
1,9
1,8
Ou
t1,
71,
8
Nava
(LSE)
StaticGames
MichaelmasTerm
75/80
Mix
edS
trat
egie
sin
Bay
esia
nG
ames
Str
ateg
yP
rofi
les:
Am
ixed
stra
tegy
isa
map
from
info
rmat
ion
toa
prob
abili
tyd
istr
ibu
tion
over
acti
ons
Inp
arti
cula
rσ i(a
i|xi)
den
otes
the
prob
abili
tyth
ati
cho
oses
a iif
his
sign
alis
x i
Am
ixed
stra
tegy
profi
leco
nsi
sts
ofa
stra
tegy
for
ever
yp
laye
r:
σ(X
)=
(σ1(X
1),
...,
σ N(X
N))
As
usu
alσ−i(
X−i)
den
otes
the
profi
leof
stra
tegi
esof
all
pla
yers
,b
ut
i
Mix
edst
rate
gies
are
ind
epen
den
t(i
.e.
σ ica
nn
otd
epen
don
σ j)
Nava
(LSE)
StaticGames
MichaelmasTerm
76/80
Not
es
Not
es
Inte
rim
Pay
off&
Bay
esN
ash
Equ
ilibr
ium
Th
ein
teri
mex
pec
ted
pay
offof
mix
edst
rate
gypr
ofile
sσ
and(a
i,σ−i)
are:
Ui(
σ|x
i)=
∑ X−i
∑ a∈A
ui(
a|x)
∏ j∈N
σ j(a
j|xj)
f(x−i|x
i)
Ui(
a i,σ−i|x
i)=
∑ X−i
∑a−i∈
A−iu
i(a|
x)
∏ j6=iσ j
(aj|x
j)f(x−i|x
i)
Defi
niti
ons
(Bay
esN
ash
Equ
ilibr
ium
BN
E)
AB
ayes
Nas
heq
uili
briu
mof
aga
me
Γis
ast
rate
gypr
ofile
σsu
chth
atfo
ran
yi∈
Nan
dx i∈
Xi
sati
sfies
:
Ui(
σ|x
i)≥
Ui(
a i,σ−i|x
i)fo
ran
ya i∈
Ai
BN
Ere
qu
ires
inte
rim
op
tim
alit
y(i
.e.
do
you
rb
est
give
nw
hat
you
know
)
Nava
(LSE)
StaticGames
MichaelmasTerm
77/80
Com
puti
ngB
ayes
Nas
hE
quili
bria
Tes
tin
gfo
rB
NE
beh
avio
r:
σis
BN
Eif
only
if:
Ui(
σ|x
i)=
Ui(
a i,σ−i|x
i)fo
ran
ya i
s.t.
σ i(a
i|xi)>
0
Ui(
σ|x
i)≥
Ui(
a i,σ−i|x
i)fo
ran
ya i
s.t.
σ i(a
i|xi)=
0
Str
ictl
yd
omin
ated
stra
tegi
esar
en
ever
chos
enin
aB
NE
Wea
kly
dom
inat
edst
rate
gies
are
chos
enon
lyif
they
are
dom
inat
edw
ith
prob
abili
tyze
roin
equ
ilibr
ium
Th
isco
nd
itio
ns
can
be
use
dto
com
pu
teB
NE
(see
exam
ple
s)
Nava
(LSE)
StaticGames
MichaelmasTerm
78/80
Not
es
Not
es
Exa
mpl
e:M
ixed
BN
EI
Con
sid
erth
efo
llow
ing
exam
ple
for
f(1
,L)=
1/
2:
1\2
.LX
Y1\2
.RW
ZT
1,0
0,1
T0,
01,
1D
0,1
1,0
D1,
10,
0
All
BN
Es
for
this
gam
esa
tisf
y:
σ 1(T
)=
1/
2an
dσ 2(X|L)=
σ 2(W|R
)
Su
chga
mes
sati
sfy
all
BN
Eco
nd
itio
ns
sin
ce:
U1(T
,σ2)=
(1/
2)σ
2(X|L)+(1
/2)(
1−
σ 2(W|R
))=
=(1
/2)(
1−
σ 2(X|L))
+(1
/2)σ
2(W|R
)=
U1(D
,σ2)
u2(X
,σ1|L)=
σ 1(T
)=
1−
σ 1(T
)=
u2(Y
,σ1|L)
u2(W
,σ1|R
)=
(1−
σ 1(T
))=
σ 1(T
)=
u2(Z
,σ1|R
)
Nava
(LSE)
StaticGames
MichaelmasTerm
79/80
Exa
mpl
e:M
ixed
BN
EII
Con
sid
erth
efo
llow
ing
exam
ple
for
f(1
,L)=
q≤
2/
3:
1\2
.LX
Y1\2
.RW
ZT
0,0
0,2
T2,
20,
1D
2,0
1,1
D0,
03,
2
Th
em
ixed
BN
Efo
rth
isga
me
sati
sfy
σ 1(T
)=
2/
3an
d:
σ 2(X|L)=
0(d
omin
ance
)an
dσ 2(W|R
)=
3−
2q
5−
5q
Su
chst
rate
gies
sati
sfy
all
BN
Ere
qu
irem
ents
sin
ce:
U1(T
,σ2)=
2(1−
q)σ
2(W|R
)=
=q+
3(1−
q)(
1−
σ 2(W|R
))=
U1(D
,σ2)
u2(X
,σ1|L)=
0<
2σ 1(T
)+(1−
σ 1(T
))=
u2(Y
,σ1|L)
u2(W
,σ1|R
)=
2σ 1(T
)=
σ 1(T
)+
2(1−
σ 1(T
))=
u2(Z
,σ1|R
)
Nava
(LSE)
StaticGames
MichaelmasTerm
80/80
Not
es
Not
es
Dyn
amic
Gam
esE
C41
1S
lides
6
Nav
a
LSE
Mic
hae
lmas
Ter
m
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
1/68
Roa
dmap
:G
ame
The
ory
–D
ynam
icG
ames
Th
ean
alys
isco
nsi
der
sd
iffer
ent
clas
ses
ofga
mes
:
Ext
ensi
veF
orm
Gam
es
Infi
nit
ely
Rep
eate
dG
ames
Fin
itel
yR
epea
ted
Gam
es
Th
en
otio
ns
ofb
ehav
iora
lst
rate
gy,
info
rmat
ion
set,
and
sub
gam
ear
ein
tro
du
ced
.
Diff
eren
tS
olu
tion
Con
cep
tsar
epr
esen
ted
:
Nas
hE
qu
ilibr
ium
Su
bga
me
Per
fect
Eq
uili
briu
m
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
2/68
Not
es
Not
es
Extensive
Form
Gam
es
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
3/68
Sum
mar
y
Dyn
amic
Gam
es:
Defi
nit
ion
s:
Ext
ensi
veF
orm
Gam
eIn
form
atio
nS
ets
and
Bel
iefs
Beh
avio
ral
Str
ateg
yS
ub
gam
e
Sol
uti
onC
once
pts
:
Nas
hE
qu
ilibr
ium
Su
bga
me
Per
fect
Eq
uili
briu
mP
erfe
ctB
ayes
ian
Eq
uili
briu
m(e
xtra
)
Exa
mp
les
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
4/68
Not
es
Not
es
Dyn
amic
Gam
es
All
gam
esd
iscu
ssed
inpr
evio
us
not
esw
ere
stat
ic.
Th
atis
:
pla
yers
wer
eta
kin
gd
ecis
ion
ssi
mu
ltan
eou
sly,
or
wer
eu
nab
leto
obse
rve
the
choi
ces
mad
eby
oth
ers.
To
day
we
rela
xsu
chas
sum
pti
ons
bym
od
ellin
gth
eti
min
gof
dec
isio
ns.
Inco
mm
onin
stan
ces
the
rule
sof
the
gam
eex
plic
itly
defi
ne:
the
ord
erin
wh
ich
pla
yers
mov
e;
the
info
rmat
ion
avai
lab
leto
them
wh
enth
eyta
keth
eir
dec
isio
ns.
Aw
ayof
repr
esen
tin
gsu
chd
ynam
icga
mes
isin
thei
rE
xten
sive
For
m.
Th
efo
llow
ing
defi
nit
ion
sar
eh
elp
ful
tod
efin
esu
chn
otio
n.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
5/68
Bas
icG
raph
The
ory
Agr
aph
con
sist
sof
ase
tof
no
des
and
ofa
set
ofbr
anch
es.
Eac
hbr
anch
con
nec
tsa
pai
rof
no
des
.
Abr
anch
isid
enti
fied
byth
etw
on
od
esit
con
nec
ts.
Ap
ath
isa
set
ofbr
anch
es:
x k
,xk+1|k
=1,
...,
m
wh
ere
m>
1an
dev
ery
x kis
ad
iffer
ent
no
de
ofth
egr
aph
.
Atr
eeis
agr
aph
inw
hic
han
yp
air
ofn
od
esis
con
nec
ted
byex
actl
yon
ep
ath
.
Aro
oted
tree
isa
tree
inw
hic
ha
spec
ial
no
des
des
ign
ated
asth
ero
ot.
Ate
rmin
aln
od
eis
an
od
eco
nn
ecte
dby
only
one
bran
ch.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
6/68
Not
es
Not
es
An
Ext
ensi
veF
orm
Gam
e
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
7/68
Ext
ensi
veF
orm
Gam
es
An
exte
nsi
vefo
rmga
me
isa
root
edtr
eeto
geth
erw
ith
fun
ctio
ns
assi
gnin
gla
bel
sto
no
des
and
bran
ches
such
that
:
1.E
ach
non
-ter
min
aln
od
eh
asa
pla
yer-
lab
elinC
,1,.
..,n
1,.
..,n
are
the
pla
yers
inth
ega
me
No
des
assi
gned
tola
bel
Car
ech
ance
no
des
No
des
assi
gned
tola
bel
i6=
Car
ed
ecis
ion
no
des
con
trol
led
byi
2.E
ach
alte
rnat
ive
ata
chan
cen
od
eh
asa
lab
elsp
ecif
yin
git
spr
obab
ility
:
Ch
ance
prob
abili
ties
are
non
neg
ativ
ean
dad
dto
1
3.E
ach
no
de
con
trol
led
byp
laye
ri>
0h
asa
seco
nd
lab
elsp
ecif
yin
gi’
sin
form
atio
nst
ate
:
Th
us
no
des
lab
eled
i.s
are
con
trol
led
byi
wit
hin
form
atio
ns
Tw
on
od
esb
elon
gto
i.s
iffi
can
not
dis
tin
guis
hth
em
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
8/68
Not
es
Not
es
Ext
ensi
veF
orm
Gam
es
4.E
ach
alte
rnat
ive
ata
dec
isio
nn
od
eh
asm
ove
lab
el:
Iftw
on
od
esx
,yb
elon
gto
the
sam
ein
form
atio
nse
t,fo
ran
yal
tern
ativ
eat
xth
ere
mu
stb
eex
actl
yon
eal
tern
ativ
eat
yw
ith
the
sam
em
ove
lab
el
5.E
ach
term
inal
no
de
yh
asa
lab
elth
atsp
ecifi
esa
vect
orof
nn
um
ber
su
i(y)
i∈1
,...,n
such
that
:
Th
en
um
ber
ui(
y)
spec
ifies
the
pay
offto
iif
the
gam
een
ds
atn
od
ey
6.A
llp
laye
rsh
ave
per
fect
reca
llof
the
mov
esth
eych
ose
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
9/68
Per
fect
Rec
all
Wit
hp
erfe
ctre
call
info
rmat
ion
sets
1.2
and
1.3
can
not
coin
cid
e:
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
10/68
Not
es
Not
es
Wit
hout
Per
fect
Rec
all
Wit
hou
tp
erfe
ctre
call
assu
mp
tion
this
isp
ossi
ble
:
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
11/68
Per
fect
Info
rmat
ion
An
exte
nsi
vefo
rmga
me
has
per
fect
info
rmat
ion
ifn
otw
on
od
esb
elon
gto
the
sam
ein
form
atio
nst
ate
Wit
hP
erfe
ctIn
form
atio
nW
ith
out
Per
fect
Info
rmat
ion
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
12/68
Not
es
Not
es
Beh
avio
ral
Str
ateg
ies
Th
rou
ghou
tle
t:
Si
be
the
set
info
rmat
ion
stat
esof
pla
yer
i∈
N;
Ai.s
be
the
acti
onse
tof
pla
yer
iat
info
stat
es∈
Si.
Ab
ehav
iora
lst
rate
gyfo
rp
laye
ri
map
sin
form
atio
nst
ates
topr
obab
ility
dis
trib
uti
ons
over
acti
ons.
Inp
arti
cula
rσ i
.s(a
i.s)
isth
epr
obab
ility
that
pla
yer
iat
info
rmat
ion
stag
es
cho
oses
acti
ona i
.s∈
Ai.s.
Th
rou
ghou
td
enot
e:
ab
ehav
iora
lst
rate
gyof
pla
yer
iby
σ i=σ
i.s s∈S
i;
apr
ofile
ofb
ehav
iora
lst
rate
gyby
σ=σ
ii∈
N;
the
chan
cepr
obab
iliti
esby
π=π
0.s s∈S
0.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
13/68
Pro
babi
litie
sov
erT
erm
inal
No
des
For
any
term
inal
no
de
yan
dan
yb
ehav
iora
lst
rate
gypr
ofile
σ,
let
P(y|σ)
den
ote
the
prob
abili
tyth
atth
ega
me
end
sat
no
de
y.
E.g
.in
the
follo
win
gga
me
P(c|σ)=
π0.1(1)σ
1.1(R
)σ2.2(C
)=
1/
9:
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
14/68
Not
es
Not
es
Exp
ecte
dP
ayoff
s
IfΩ
den
otes
the
set
ofen
dn
otes
,th
eex
pec
ted
pay
offof
pla
yer
iis
:
Ui(
σ)=
Ui(
σ i,σ−i)=
∑y∈
ΩP(y|σ)u
i(y).
E.g
.in
the
follo
win
gga
me
U1(σ)=
4/
9an
dU2(σ)=
5/
9:
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
15/68
Exe
rcis
e
Dra
wth
eex
ten
sive
form
for
the
follo
win
gtw
op
laye
rga
me.
Fir
ms
1h
asto
dec
ide
wh
eth
erto
ente
r(i
n)
orst
ayou
t(o
ut)
ofa
mar
ket
inw
hic
hit
do
esn
otop
erat
e.
Fir
m2
isan
old
firm
inth
em
arke
tan
dd
ecid
esu
pon
entr
yof
Fir
m1,
wh
eth
erto
figh
t(F
)or
coop
erat
e(C
).
Fir
m2’
sco
sts
are
com
mon
know
led
ge.
Fir
m1’
sco
sts
are
ran
dom
and
on
lykn
own
tofi
rm1.
Fir
m1’
sco
sts
are
hig
h(H
)or
low
(L)
wit
heq
ual
chan
ce.
Den
ote
pay
offs
by(π
1,π
2),
wh
ere
πi
isth
ep
ayoff
ofF
irm
i:
(π1,π
2)=
(0,3)
ifa 1
.1=
out
(−3,
2)
ifa 1
.1=
in,
a 2.1=
F,
cost
are
H(2
,−3)
ifa 1
.1=
in,
a 2.1=
F,
cost
are
L(1
,1)
ifa 1
.1=
in,
a 2.1=
C,
cost
are
H(4
,1)
ifa 1
.1=
in,
a 2.1=
C,
cost
are
L
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
16/68
Not
es
Not
es
Ans
wer N
ava
(LSE)
Dyn
amic
Games
MichaelmasTerm
17/68
Extensive
Form
Gam
es:
Nas
hE
quili
briu
m
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
18/68
Not
es
Not
es
Nas
hE
quili
briu
m
Defi
niti
on(N
ash
Equ
ilibr
ium
–N
E)
AN
ash
Eq
uili
briu
mof
anex
ten
sive
form
gam
eis
any
profi
leof
beh
avio
ral
stra
tegi
essu
chth
at:
Ui(
σ)≥
Ui(
σ′ i,
σ−i)
for
any
σ′ i∈×
s∈Si∆(A
i.s)
Rec
all
that
σ′ i
isan
ym
app
ing
from
info
rmat
ion
stat
esto
prob
abili
tyd
istr
ibu
tion
sov
erav
aila
ble
acti
ons.
Th
ed
efin
itio
nof
NE
isas
inst
rate
gic
form
gam
es.
Wh
atd
iffer
sis
the
not
ion
ofst
rate
gyth
atis
now
defi
ned
for
ever
yin
form
ati
on
sta
te.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
19/68
Exa
mpl
e:N
Ein
the
Ext
ensi
veF
orm
Aga
me
may
hav
em
any
NE
s:
σ 1(A
)=
0,σ 2(C
)=
0,σ 3(E
)=
0;
σ 1(A
)=
σ 3(E
)/(1
+σ 3(E
)),
σ 2(C
)=
1/
2,σ 3(E
)∈[0
,1].
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
20/68
Not
es
Not
es
Exa
mpl
e:N
Ein
the
Ext
ensi
veF
orm
Pro
file
σ 1(A
)=
0,σ 2(C
)=
0,σ 3(E
)=
0is
NE
:
U1(B
,σ−1)=
1>
0=
U1(A
,σ−1)
U2(D
,σ−2)=
0≥
0=
U2(C
,σ−2)
U3(F
,σ−3)=
2≥
2=
U3(E
,σ−3)
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
21/68
Exa
mpl
e:N
Ein
the
Ext
ensi
veF
orm
Pro
file
σ 1(A
)=
σ 3(E
)/(1
+σ 3(E
)),
σ 2(C
)=
1/
2,σ 3(E
)∈[0
,1]
isN
E:
U1(A
,σ−1)=
σ 2(C
)=
(1−
σ 2(C
))=
U1(B
,σ−1)
U2(D
,σ−2)=
σ 1(A
)=
(1−
σ 1(A
))σ 3(E
)=
U2(C
,σ−2)
U3(F
,σ−3)=
(1−
σ 1(A
))(2−
σ 2(C
))=
U3(E
,σ−3)
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
22/68
Not
es
Not
es
Ult
imat
umG
ame
Th
isga
me
pos
sess
esth
ree
typ
esof
NE
,n
amel
yfo
ran
ya 2
.2,a
2.3
NE
1:σ 1
=[1]
&σ 2
=[A
,a2.2
,a2.3]
NE
2:σ 1
=[2]
&σ 2
=[R
,A,a
2.3]
NE
3:σ 1
=[3]
&σ 2
=[R
,R,A
]
Inth
eta
ble
,σ 1
and
σ 2d
enot
eth
eb
ehav
iora
l(p
ure
)st
rate
gies
ofth
etw
op
laye
rs. N
ava
(LSE)
Dyn
amic
Games
MichaelmasTerm
23/68
Ult
imat
umG
ame
Str
ateg
yσ 1
=[1],
σ 2=
[A,a
2.2
,a2.3]
isN
Esi
nce
:
U1(1
,σ2)=
3>
U1(a
1,σ
2)
for
any
a 1∈2
,3
U2(σ
2,1)=
1≥
U2(a
2,1)
for
any
a 2∈A
,R3
Str
ateg
yσ 1
=[2],
σ 2=
[R,A
,a2.3],
isN
Esi
nce
:
U1(2
,σ2)=
2>
U1(a
1,σ
2)
for
any
a 1∈1
,3
U2(σ
2,2)=
2≥
U2(a
2,2)
for
any
a 2∈A
,R3
Asi
mila
rar
gum
ent
wor
ksfo
rth
eot
her
prop
osed
NE
.
On
lyth
efi
rst
typ
eof
NE
how
ever
,in
volv
esth
reat
sth
atar
ecr
edib
le,
sin
cep
laye
r2
wou
ldn
ever
cred
ibly
reje
ctan
offer
wor
that
leas
t1
wh
enfa
ced
wit
han
outs
ide
opti
onof
0.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
24/68
Not
es
Not
es
AT
rans
form
atio
nto
Fin
dN
E
Th
eN
Eof
the
gam
eca
nb
efo
un
dby
look
ing
atN
Eof
the
stra
tegi
cfo
rm:
1/
2A
AA
RA
AA
RA
AA
RR
RA
RA
RA
RR
RR
R
13,
10,
03,
13,
10,
00,
03,
10,
02
2,2
2,2
0,0
2,2
0,0
2,2
0,0
0,0
31,
31,
31,
30,
01,
30,
00,
00,
0
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
25/68
Extensive
Form
Gam
es:
Sub
gam
eP
erfe
ctE
quili
briu
m
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
26/68
Not
es
Not
es
Sub
gam
eP
erfe
ctE
quili
briu
m
Asu
cces
sor
ofa
no
de
xis
an
od
eth
atca
nb
ere
ach
edfr
omx
for
anap
prop
riat
epr
ofile
ofac
tion
s.
Defi
niti
on(S
ubga
me)
Asu
bga
me
isa
sub
set
ofan
exte
nsi
vefo
rmga
me
such
that
:
1It
beg
ins
ata
sin
gle
no
de.
2It
con
tain
sal
lsu
cces
sors
.
3If
aga
me
con
tain
san
info
rmat
ion
set
wit
hm
ult
iple
no
des
then
eith
eral
lof
thes
en
od
esb
elon
gto
the
sub
set
orn
one
do
es.
Defi
niti
on(S
ubga
me
Per
fect
Equ
ilibr
ium
–S
PE
)
Asu
bga
me
per
fect
equ
ilibr
ium
isan
yN
Esu
chth
atfo
rev
ery
sub
gam
eth
ere
stri
ctio
nof
stra
tegi
esto
this
sub
gam
eis
also
aN
Eof
the
sub
gam
e.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
27/68
NE
but
not
SP
E
Th
efo
llow
ing
gam
eh
asa
con
tin
uu
mof
NE
bu
ton
lyon
eS
PE
:
σ 1.1(A
)=
1an
dσ 2
.1(C
)=
1is
un
iqu
eS
PE
σ 1.1(A
)=
0an
dσ 2
.1(C
)≤
2/
3ar
eal
lN
E,
bu
tn
one
SP
E
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
28/68
Not
es
Not
es
NE
but
not
SP
E
Str
ateg
yσ 1
.1(A
)=
1an
dσ 2
.1(C
)=
1is
the
un
iqu
eS
PE
sin
ce:
U1.1(A
,C)=
400>
300=
U1.1(C
,C)
U2.1(C
)=
400>
100=
U2.1(D
)
An
yst
rate
gyσ 1
.1(A
)=
0an
dσ 2
.1(C
)=
q≤
2/
3is
NE
sin
ce:
U1(C
,σ2)=
300≥
400q
+10
0(1−
q)=
U1(A
,σ2)
U2(C
,C)=
1000
=10
00=
U2(D
,C)
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
29/68
Com
puti
ngS
PE
–B
ackw
ard
Indu
ctio
n
Defi
nit
ion
ofS
PE
isd
eman
din
gb
ecau
seit
imp
oses
dis
cip
line
onb
ehav
ior
even
insu
bga
mes
that
one
exp
ects
not
tob
ere
ach
ed.
SP
Eh
owev
eris
easy
toco
mp
ute
inp
erfe
ctin
form
atio
nga
mes
.
Bac
kwar
d-i
nd
uct
ion
algo
rith
mpr
ovid
esa
sim
ple
way
:
At
ever
yn
od
ele
adin
gon
lyto
term
inal
no
des
pla
yers
pic
kac
tion
sth
atar
eop
tim
alfo
rth
emif
that
no
de
isre
ach
ed.
At
all
prec
edin
gn
od
esp
laye
rsp
ick
anac
tion
sth
atop
tim
alfo
rth
emif
that
no
de
isre
ach
edkn
owin
gh
owal
lth
eir
succ
esso
rsb
ehav
e.
An
dso
onu
nti
lth
ero
otof
the
tree
isre
ach
ed.
Ap
ure
stra
tegy
SP
Eex
ists
inan
yp
erfe
ctin
form
atio
nga
me.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
30/68
Not
es
Not
es
Exa
mpl
e:B
ackw
ard
Indu
ctio
n
SP
E:
σ 1.2=
σ 1.3=
[A],
σ 2.1=
[t],
any
σ 2.2
and
σ 1.1=
[T]
NE
bu
tn
otS
PE
:σ 1
.2=
σ 1.3=
[A],
σ 2.1=
[b],
any
σ 2.2
and
σ 1.1=
[B]
Aga
inN
Em
ayb
esu
pp
orte
dby
emp
tyth
reat
s.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
31/68
Exa
mpl
e:M
arke
tE
ntry
Con
sid
erth
efo
llow
ing
gam
eb
etw
een
two
pro
du
cers
.
Fir
m1
isth
ein
cum
ben
tan
dfi
rm2
isth
ep
oten
tial
entr
ant.
Ass
um
eP>
L>
0an
dM
>P>
F.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
32/68
Not
es
Not
es
Exa
mpl
e:M
arke
tE
ntry
To
fin
dan
SP
Ew
ith
succ
essf
ul
det
erre
nce
,n
otic
eth
at:
If1
do
esn
otin
vest
,it
pref
ers
toco
nce
de
give
nen
try
asP>
F.
Th
us
firm
2pr
efer
sto
ente
rif
1d
oes
not
inve
stas
P>
L.
If1
do
esin
vest
itpr
efer
sto
figh
tif
entr
yta
kes
pla
ce,
prov
ided
that
:
F>
P−
K
Ifso
firm
2pr
efer
sto
stay
out
if1
has
inve
sted
asL>
0.
Th
us
firm
1pr
efer
sto
inve
stan
dd
eter
entr
yif
:
M−
K>
P
An
SP
Eex
ists
inw
hic
hen
try
iseff
ecti
vely
det
erre
dif
the
cost
sati
sfies
:
M−
P>
K>
P−
F
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
33/68
Exa
mpl
e:C
onsi
sten
tof
Mon
etar
yP
olic
y
Fir
stst
ag
e:W
orke
rsch
oos
efo
ra
nom
inal
wag
e,w
.
Sec
on
dst
ag
e:T
he
cen
tral
ban
kch
oos
esan
infl
atio
nra
te,
π.
Th
ere
alw
age
isd
enot
edby
wr=
w/(1
+π).
Th
ed
eman
dfo
rw
orke
rsd
enot
edby
Q(w
r)
and
sati
sfies
Q(w
r)=
1−
wr.
Wor
ker’
su
tilit
yis
det
erm
ined
byth
eto
tal
wag
e,
U(π
,w)=
wrQ(w
r)=
wr(1−
wr).
Cen
tral
ban
k’s
obje
ctiv
efu
nct
ion
is
V(π
,w)=
αQ(w
r)−
π2/
2.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
34/68
Not
es
Not
es
Exa
mpl
e:C
onsi
sten
tof
Mon
etar
yP
olic
y
SP
Eof
this
gam
eca
nb
efo
un
du
sin
gb
ackw
ard
ind
uct
ion
.
Inth
ela
stst
age
give
nan
yw
age
w:
Th
eb
ank
solv
es
max
πV(π
,w)=
max
πα
( 1−
w
1+
π
) −π2 2
.
Fir
stor
der
con
dit
ion
sim
ply
that
αw
=π(1
+π)2
Th
eso
luti
onπ(w
)is
anin
crea
sin
gfu
nct
ion
ofα
w.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
35/68
Exa
mpl
e:C
onsi
sten
tof
Mon
etar
yP
olic
y
Inth
efi
rst
stag
e,th
ew
orke
rs:
Cal
cula
teπ(w
)an
dta
keit
asgi
ven
.
Sol
veth
efo
llow
ing
prob
lem
:
max
wU(π
(w),
w)=
max
ww
1+
π(w
)
( 1−
w
1+
π(w
)
) .
FO
Cim
ply
that
:
w∗ r=
w∗
1+
π(w∗ )
=1 2
.
Th
ela
stco
nd
itio
nre
du
ces
the
ban
k’s
FO
Cat
the
opti
mu
mto
:
α/
2=
π∗(1
+π∗ )
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
36/68
Not
es
Not
es
Exa
mpl
e:C
onsi
sten
tof
Mon
etar
yP
olic
y
Co
ncl
usi
on
s:
w∗ r=
Q(w∗ r)=
1/
2in
dep
end
ent
ofπ∗ ;
π∗>
0p
osit
ive
infl
atio
nif
α>
0.
Incl
ud
ing
emp
loym
ent
Qin
toth
eb
ank’
sob
ject
ive
fun
ctio
nh
asn
oeff
ect
onem
plo
ymen
tQ
orre
alw
ages
,b
ut
lead
sto
ap
osit
ive
infl
atio
nra
te.
Th
eb
ank
wan
tsto
surp
rise
wor
kers
wit
hp
osit
ive
infl
atio
n.
Bu
tw
orke
rsan
tici
pat
eth
eb
ank’
sre
spon
se,
take
pos
itiv
ein
flat
ion
into
acco
un
t,an
dn
eutr
aliz
ean
yeff
ect
ofth
eb
ank’
sp
olic
yon
real
wag
es.
Th
eb
ank
can
not
cred
ibly
com
mit
not
toch
ange
the
infl
atio
nra
te.
Th
isar
gum
ent
favo
rsre
stri
ctin
gce
ntr
alb
ank
pol
icy
toin
flat
ion
targ
etin
g,le
avin
gou
tan
yem
plo
ymen
tco
nsi
der
atio
n.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
37/68
Extensive
Form
Gam
es:
Con
clud
ing
Rem
arks
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
38/68
Not
es
Not
es
Lim
itat
ions
toN
Ean
dS
PE
Beh
avio
r
Th
eex
amp
les
show
edth
atN
Eb
ehav
ior
was
not
imm
un
eto
emp
tytr
eats
.
Ina
NE
any
pla
yer
wou
ld:
bel
ieve
wit
hce
rtai
nty
that
all
the
oth
erw
illco
mp
lyw
ith
the
equ
ilibr
ium
stra
tegy
;
not
con
sid
erth
ecr
edib
ility
(rat
ion
alit
y)of
opp
onen
ts’
beh
avio
r.
SP
Eb
ehav
ior
byp
asse
sth
ese
limit
atio
ns
byim
pos
ing
rati
onal
ity
onop
pon
ents
’b
ehav
ior,
ason
lycr
edib
leth
reat
sar
eb
elie
ved
ineq
uili
briu
m.
SP
Eb
ehav
ior
how
ever
,is
not
imm
un
eto
limit
atio
ns:
To
om
uch
dis
cip
line
oneq
uili
briu
mb
ehav
ior
(cen
tip
ede
gam
e);
Un
bo
un
ded
rati
on
alit
y(c
hes
s).
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
39/68
Lim
itat
ions
toS
PE
Beh
avio
r:C
enti
ped
eG
ame
Ros
enth
alh
igh
ligh
ted
ake
yS
PE
com
plic
atio
nin
the
cen
tip
ede
ga
me.
Con
sid
erth
efo
llow
ing
exte
nsi
vefo
rmga
me:
Inan
yS
PE
bot
hp
laye
rsch
oos
eou
t.
Ifw
eit
erat
eth
ega
me
asb
elow
,in
any
SP
Ep
laye
rsal
way
sch
oos
eou
t:
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
40/68
Not
es
Not
es
Lim
itat
ions
toS
PE
Beh
avio
r:C
enti
ped
eG
ame
Ifw
eco
nti
nu
eit
erat
ing
the
gam
e:
Inan
yS
PE
pla
yers
keep
cho
osin
gou
tin
ever
yp
erio
d.
How
ever
,su
cha
beh
avio
rap
pea
rsim
pla
usi
ble
,as
pla
yers
sacr
ifice
larg
efu
ture
mu
tual
gain
sin
ord
erto
com
ply
wit
hra
tion
alb
ehav
ior.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
41/68
Lim
itat
ions
toS
PE
Beh
avio
r:C
hess
Ch
ess
can
be
mo
del
edas
anex
ten
sive
form
gam
e:
1th
etr
eeis
fin
ite;
2th
ega
me
isze
rosu
mga
me
(on
ew
ins,
the
oth
erlo
ses)
;
3it
has
com
ple
tein
form
atio
n(i
nfo
rmat
ion
sets
con
tain
one
no
de)
.
Ifw
est
art
atth
een
dof
the
gam
eap
ply
bac
kwar
din
du
ctio
nat
each
step
bac
kwar
ds,
we
obta
inex
actl
yon
ep
air
ofp
ayoff
s(a
sth
ega
me
isze
rosu
m).
Ch
erm
elo
19
12
:
Ch
ess
isa
gam
ew
ith
ad
eter
min
isti
cou
tcom
e:
eith
erw
hit
eca
ngu
aran
tee
vict
ory;
orb
lack
can
guar
ante
evi
ctor
y;
orea
chp
laye
rca
ngu
aran
tee
ati
e.
Th
ish
igh
ligh
tth
ese
con
dm
ajor
limit
atio
nof
SP
Eb
ehav
ior.
Nam
ely
that
itre
qu
ires
and
un
bo
un
ded
leve
lof
rati
on
alit
y(c
omp
uti
ng
pow
er).
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
42/68
Not
es
Not
es
Ext
ra:
Dyn
amic
san
dU
ncer
tain
ty
Ifex
ten
sive
form
gam
esw
ith
out
per
fect
info
rmat
ion
,su
bga
me
per
fect
ion
can
not
be
imp
osed
atev
ery
no
de,
bu
ton
lyon
sub
gam
es.
Insu
chga
mes
afu
rth
ereq
uili
briu
mre
fin
emen
tm
ayh
elp
toh
igh
ligh
tth
ere
leva
nt
equ
ilibr
iaof
the
gam
eby
sele
ctin
gth
ose
wh
ich
sati
sfy
Bay
esru
le.
Defi
niti
on(P
erfe
ctB
ayes
ian
Equ
ilibr
ium
–P
BE
)
Ap
erfe
ctB
ayes
ian
equ
ilibr
ium
ofan
exte
nsi
vefo
rmga
me
con
sist
sof
apr
ofile
ofb
ehav
iora
lst
rate
gies
and
ofb
elie
fsat
each
info
rmat
ion
set
ofth
ega
me
such
that
:
1st
rate
gies
form
anS
PE
give
nth
eb
elie
fs;
2b
elie
fsar
eu
pd
ated
usi
ng
Bay
esru
leat
each
info
rmat
ion
set
reac
hed
wit
hp
osit
ive
prob
abili
ty.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
43/68
RepeatedGam
es:
Infi
nite
Hor
izon
Gam
es
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
44/68
Not
es
Not
es
Sum
mar
y
Rep
eate
dG
ames
:
Defi
nit
ion
s:
Fea
sib
leP
ayoff
sM
inm
axR
epea
ted
Gam
eS
tage
Gam
eT
rigg
erS
trat
egy
Mai
nR
esu
lt:
Fol
kT
heo
rem
Exa
mp
les:
Pri
son
er’s
Dile
mm
a
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
45/68
Fea
sibl
eP
ayoff
s
Q:
Wh
atp
ayoff
sar
efe
asib
lein
ast
rate
gic
form
gam
e?
A:
Apr
ofile
ofp
ayoff
sis
feas
ible
ina
stra
tegi
cfo
rmga
me
ifca
nb
eex
pres
sed
asa
wei
ghed
aver
age
ofp
ayoff
sin
the
gam
e.
Defi
niti
on(F
easi
ble
Pay
offs)
Apr
ofile
ofp
ayoff
sw
ii∈
Nis
feas
ible
ina
stra
tegi
cfo
rmga
me
N,
Ai,
ui
i∈N
ifth
ere
exis
tsa
dis
trib
uti
onov
erpr
ofile
sof
acti
ons
πsu
chth
at:
wi=
∑a∈A
π(a)u
i(a)
for
any
i∈
N
Un
feas
ible
pay
offs
can
not
be
outc
omes
ofth
ega
me.
Poi
nts
onth
en
orth
-eas
tb
oun
dar
yof
the
feas
ible
set
are
Par
eto
effici
ent.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
46/68
Not
es
Not
es
Min
max
Q:
Wh
at’s
the
wor
stp
ossi
ble
pay
offth
ata
pla
yer
can
ach
ieve
ifh
ech
oos
esac
cord
ing
toh
isb
est
resp
onse
fun
ctio
n?
A:
Th
em
inm
axp
ayoff
.
Defi
niti
on(M
inm
ax)
Th
e(p
ure
stra
tegy
)m
inm
axp
ayoff
ofp
laye
ri∈
Nin
ast
rate
gic
form
gam
e N
, Ai,
ui
i∈N
is:
ui=
min
a−i∈
A−i
max
ai∈
Ai
ui(
a i,a−i)
Mix
edst
rate
gym
inm
axp
ayoff
ssa
tisf
y:
vi=
min
σ−i
max σ i
ui(
σ i,σ−i)
Th
em
ixed
stra
tegy
min
max
isn
oth
igh
erth
anth
ep
ure
stra
tegy
min
max
.Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
47/68
Exa
mpl
e:P
riso
ner’
sD
ilem
ma
Min
max
Pay
off:(1
,1)
Fea
sib
leP
ayoff
s:b
lue
and
red
area
s.
Ind
ivid
ual
lyR
atio
nal
Pay
offs:
red
area
.
Sta
geG
ame
Pay
offs
1\2
ws
w2,
20,
3s
3,0
1,1
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
48/68
Not
es
Not
es
Exa
mpl
e:B
attl
eof
the
Sex
es
Min
max
Pay
off:(2
,2)
Fea
sib
leP
ayoff
s:b
lue
and
red
area
s.
Ind
ivid
ual
lyR
atio
nal
Pay
offs:
red
area
.
Sta
geG
ame
Pay
offs
1\2
ws
w3,
31,
0s
0,1
2,2
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
49/68
Rep
eate
dG
ame:
Tim
ing
Con
sid
eran
yst
rate
gic
form
gam
eG
= N
, Ai,
ui
i∈N
.
Cal
lG
the
stag
ega
me.
An
infi
nit
ely
rep
eate
dga
me
des
crib
esa
stra
tegi
cen
viro
nm
ent
inw
hic
hth
est
age
gam
eis
pla
yed
rep
eate
dly
byth
esa
me
pla
yers
infi
nit
ely
man
yti
mes
.
Rou
nd
1..
.R
oun
dt
...
...
...
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
50/68
Not
es
Not
es
Rep
eate
dG
ames
:P
ayoff
san
dD
isco
unti
ng
Th
eva
lue
top
laye
ri∈
Nof
ap
ayoff
stre
amu
i(1),
ui(
2),
...,
ui(
t),.
..
is:
(1−
δ)∑
∞ t=1δt−1ui(
t)
Th
ete
rm(1−
δ)am
oun
tsto
asi
mp
len
orm
aliz
atio
n
...
and
guar
ante
esth
ata
con
stan
tst
reamv
,v,.
..
has
valu
ev
.
Fu
ture
pay
offs
are
dis
cou
nte
dat
rate
δ.
An
infi
nit
ely
rep
eate
dga
me
can
be
use
dto
des
crib
est
rate
gic
envi
ron
men
tsin
wh
ich
ther
eis
no
cert
ain
tyof
afi
nal
stag
e.
Insu
chvi
ewδ
des
crib
esth
epr
obab
ility
that
the
gam
ed
oes
not
end
atth
en
ext
rou
nd
wh
ich
wou
ldre
sult
ina
pay
offof
0th
erea
fter
.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
51/68
Rep
eate
dG
ames
:P
erfe
ctIn
form
atio
nan
dS
trat
egie
s
To
day
we
rest
rict
atte
nti
onto
per
fect
info
rmat
ion
rep
eate
dga
mes
.
Insu
chga
mes
all
pla
yers
prio
rto
each
rou
nd
obse
rve
the
acti
ons
chos
enby
all
oth
erp
laye
rsat
prev
iou
sro
un
ds.
Let
a(s)
=a
1(s),
...,
a n(s)
den
ote
the
acti
onpr
ofile
pla
yed
atro
un
ds.
Ah
isto
ryof
pla
yu
pto
stag
et
thu
sco
nsi
sts
of:
h(t)=a(1),
a(2),
...,
a(t−
1)
Inth
isco
nte
xtst
rate
gies
map
his
tori
es(i
ein
form
atio
n)
toac
tion
s:
αi(
h(t))∈
Ai
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
52/68
Not
es
Not
es
Pri
sone
r’s
Dile
mm
aF
olk
The
orem
Con
sid
erth
epr
ison
er’s
dile
mm
ad
iscu
ssed
earl
ier:
1\2
ws
w2,
20,
3s
3,0
1,1
To
un
der
stan
dh
oweq
uili
briu
mb
ehav
ior
isaff
ecte
dby
rep
etit
ion
,le
t’s
show
wh
y(2
,2)
isS
PE
ofth
ein
fin
itel
yre
pea
ted
pris
oner
’sd
ilem
ma.
Fol
kth
eore
msh
ows
that
any
feas
ible
pay
offth
atyi
eld
sto
bot
hp
laye
rsat
leas
tth
eir
min
max
valu
eis
aS
PE
ofth
ein
fin
itel
yre
pea
ted
gam
eif
the
dis
cou
nt
fact
oris
suffi
cien
tly
hig
h.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
53/68
Gri
mT
rigg
erS
trat
egie
s
Con
sid
erth
efo
llow
ing
stra
tegy
(kn
own
asgr
imtr
igge
rst
rate
gy):
a i(t)=
wif
a(z)=
(w,w
)fo
ran
yz<
ts
oth
erw
ise
Su
cha
stra
tegy
can
be
grap
hic
ally
repr
esen
ted
asa
2-st
ate
auto
mat
on:
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
54/68
Not
es
Not
es
Gri
mT
rigg
erS
PE
Con
sid
erth
egr
imtr
igge
rst
rate
gy:
Ifal
lp
laye
rsfo
llow
such
stra
tegy
,n
op
laye
rca
nd
evia
tean
db
enefi
tat
any
give
nro
un
dpr
ovid
edth
atδ≥
1/
2.
Insu
bga
mes
inw
hic
ha(
t)=
(w,w
)n
op
laye
rb
enefi
tsfr
oma
dev
iati
onas
:
(1−
δ )( 3
+δ+
δ2+
δ3+
...) =
3−
2δ≤
2⇔
δ≥
1/
2
Insu
bga
mes
inw
hic
ha(
t)=
(s,s)
no
pla
yer
ben
efits
from
ad
evia
tion
as:
(1−
δ )( 0
+δ+
δ2+
δ3+
...) =
δ≤
1⇔
δ≤
1
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
55/68
Fol
kT
heor
em
The
orem
(SP
EF
olk
The
orem
)
Inan
ytw
o-p
erso
nin
fin
itel
yre
pea
ted
gam
e:
1F
oran
yd
isco
un
tfa
ctor
δ,th
ed
isco
un
ted
aver
age
pay
offof
each
pla
yer
inan
yS
PE
isat
leas
th
ism
inm
axva
lue
inth
est
age
gam
e.
2A
ny
feas
ible
pay
offpr
ofile
that
yiel
ds
toal
lp
laye
rsat
leas
tth
eir
min
max
valu
eis
the
dis
cou
nte
dav
erag
ep
ayoff
ofa
SP
Eif
the
dis
cou
nt
fact
orδ
issu
ffici
entl
ycl
ose
to1.
3If
the
stag
ega
me
has
aN
Ein
wh
ich
each
pla
yers
’p
ayoff
ish
ism
inm
axva
lue,
then
the
infi
nit
ely
rep
eate
dga
me
has
aS
PE
inw
hic
hev
ery
pla
yers
’d
isco
un
ted
aver
age
pay
offis
his
min
max
valu
e.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
56/68
Not
es
Not
es
Tes
ting
SP
Ein
Rep
eate
dG
ames
Defi
niti
on(O
ne-D
evia
tion
Pro
per
ty)
Ast
rate
gysa
tisfi
esth
eon
e-d
evia
tion
prop
erty
ifn
op
laye
rca
nin
crea
seh
isp
ayoff
bych
angi
ng
his
acti
onat
the
star
tof
any
sub
gam
ein
wh
ich
he
isth
efi
rst-
mov
ergi
ven
oth
erp
laye
rs’
stra
tegi
esan
dth
ere
stof
his
own
stra
tegy
.
Fac
t
Ast
rate
gypr
ofile
inan
exte
nsi
vega
me
wit
hp
erfe
ctin
form
atio
nan
din
fin
ite
hor
izon
isa
SP
Eif
and
only
ifit
sati
sfies
the
one-
dev
iati
onpr
oper
ty.
Th
isob
serv
atio
nca
nb
eu
sed
tote
stw
het
her
ast
rate
gypr
ofile
isa
SP
Eof
anin
fin
itel
yre
pea
ted
gam
eas
we
did
inth
eP
riso
ner
’sd
ilem
ma
exam
ple
.
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
57/68
Exa
mpl
e:D
iffer
ent
Equ
ilibr
ium
Beh
avio
r
Nex
tsh
oww
hy(1
.5,1
.5)
isal
soan
SP
Eof
the
rep
eate
dP
Das
δ→
1.
Con
sid
erth
efo
llow
ing
pai
rof
stra
tegi
es:
a(t)
=
(w,s)
ift
isev
ena
nd
a(z)
/∈(
s,s)
,(w
,w)
for
any
z<
t(s
,w)
ift
iso
dd
an
da(
z)
/∈(
s,s)
,(w
,w)
for
any
z<
t(s
,s)
oth
erw
ise
Ifδ≥
1/
2,n
op
laye
rca
npr
ofita
bly
dev
iate
from
the
stra
tegy
.
Wh
ena(
t)=
(s,w
),(w
,s),
no
pla
yer
ben
efits
from
ad
evia
tion
as:
1≤
(1−
δ )( 3
δ+
3δ3
+3
δ5+
...) =
3δ/
(1+
δ)
(1−
δ )( 2
+δ+
δ2..
.) =2−
δ≤
(1−
δ )( 3
+3
δ2+
3δ4
...) =
3/(1
+δ)
Wh
ena(
t)=
(s,s),
no
pla
yer
ben
efits
from
ad
evia
tion
as:
(1−
δ )( 0
+δ+
δ2+
δ3+
...) =
δ≤
1⇔
δ≤
1
Nava
(LSE)
Dyn
amic
Games
MichaelmasTerm
58/68
Not
es
Not
es
Exa
mpl
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Dyn
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MichaelmasTerm
59/68
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Nava
(LSE)
Dyn
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Games
MichaelmasTerm
60/68
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Dyn
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MichaelmasTerm
61/68
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Dyn
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MichaelmasTerm
62/68
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Dyn
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MichaelmasTerm
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Dyn
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Dyn
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MichaelmasTerm
65/68
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Dyn
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MichaelmasTerm
66/68
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(LSE)
Dyn
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Games
MichaelmasTerm
67/68
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(LSE)
Dyn
amic
Games
MichaelmasTerm
68/68
Not
es
Not
es
EC411ClassesSchedule
Week2MT–Consumption:
ProblemSet1Solutions
Week3MT–Consumption:
ProblemSet2Solutions
Week4MT–Production:
ProblemSet3Solutions
Week5MT–Production:
ProblemSet4Solutions
Week6MT–PartialEquilibrium:
ProblemSet5Solutions
Week7MT–GeneralEquilibrium:
ProblemSet6Solutions
Week8MT–Uncertainty:
ProblemSet7Solutions
Hand‐InHomework1Solutions
Week9MT–UncertaintyandGameTheory:
ProblemSet8Solutions
Week10MT–GameTheory:
ProblemSet9Solutions
Week11MT–GameTheory:
ProblemSet10Solutions
Optional:Hand‐InHomework2Solutions
PleasegooverALLmandatoryproblemsandsomeoftheadditionalproblems.
MT 2017/2018 Francesco Nava
Problem SetsEC411 —LSE
1
Problem Set 1: Consumption Francesco Nava
EC411 —LSE MT 2017/2018
1. The preferences of a consumer are represented by the utility function
u(x, y) = x+ y.
The price of a unit of good x is 2, the price of a unit of good y is p, and the income of the consumer
is M . Moreover, the consumer is given a lump-sum subsidy of S that can only be used to purchase
good y. Suppose that p 6= 2
(a) Define the utility maximization problem of the consumer.
(b) Find the optimal amounts of x and y as functions of p, M and S.
2. Answer the following questions:
(a) A consumer has a utility function
u(x1, x2) =√x21 + x
22.
Find the Marshallian demand functions for both goods.
(b) A consumer has a utility function
u(x1, x2) = minx1, 3 + x2.
Find the Marshallian demand functions for both goods.
(c) True or false? In order to aid the poor, the government introduces a scheme whereby the first 1kg
of butter a family buys is subsidised and the remaining amounts are taxed. Consider a family
which consumes butter and is made neither better off nor worse off as a result of this scheme. If
so, the total amount of tax it pays cannot exceed the subsidy it receives.
(d) True or false? During a war, food and clothing are rationed. In addition to a money price, a
certain number of ration points must be paid to obtain a good. Each consumer has an allocation
of ration points which may be used to purchase either good, and also has a fixed money income.
Suppose the money income of a consumer is raised and he buys more food and less clothing. If
follows that clothing is an inferior good.
3. Assume that Marshallian demands are single valued —so that a unique bundle x(p,m) is optimal for
all p >> 0 and all m > 0. Prove that the indirect utility v(p,m) is continuous if U(x) is continuous.
2
Extra Problem:
1. A consumer is paid each week 4 units of x1 and 4 units of x2 which she may consume directly or trade
with other customers at the going prices. In the first week she trades, and finally consumes 5 units of
x1 and 3 units of x2. In the second week prices change, and she finally consumes 6 units of x1 and 1
units of x2. Assuming that her tastes remain unchanged for these two weeks:
(a) Find the price ratio p1/p2 in the two weeks?
(b) In which week is the consumer better off?
(c) Is x1 an inferior good for this consumer?
(d) Is x1 a Giffen good for this consumer?
3
Problem Set 2: Consumption Francesco Nava
EC411 —LSE MT 2017/2018
1. Consider a strictly increasing and strictly concave map ϕ and a consumer with preferences
u(x1, x2) = x1 + ϕ(x2).
Show that the Marshallian and Hicksian curves for x2 coincide. What is the significance of this for
the estimation of the welfare changes due to an increase in the price of x2?
2. Consider a consumer with preferences
u(x1, x2) = x1minx2, 1.
Derive the expenditure function for this problem, and the Hicksian demand for each of the two goods.
Next state and use the relevant parts of the duality theorem to derive the value function and the
corresponding Marshallian demands.
3. The preferences of a consumer are represented by the utility function
u(x, y) = xy.
Let px denote the price of good x and py denote the price of good y.
(a) Establish that utility is quasi-concave.
(b) Solve the utility maximization problem if the income of the consumer equals 12. Derive the
Marshallian demand for each of the two goods.
Extra Problem:
1. The expenditure function of a consumer is denoted by e(p, u), where p is the price vector and u the
utility level. Show that if∂2e(p, u)
∂pj∂u> 0,
commodity j cannot be an inferior good.
4
Problem Set 3: Welfare & DT Advances Francesco Nava
EC411 —LSE MT 2017/2018
1. The preferences of a consumer are represented by the utility function
u(x, y) = xy.
Suppose that the income of the consumer equals 12, and that the price of good y equals 3, while the
price of good x increases from 1 to 2.
(a) Derive the income and substitution effect for such a change in prices.
(b) Derive the consumer surplus, the compensating variation, and the equivalent variation.
2. Consider a consumer with preferences
u(x1, x2) = x1 − 1/x2.
Suppose the price of x1 is held at 1 and the price of x2 rises from 1 to 4. Calculate the welfare change.
In this scenario which of our measures of welfare will be largest?
3. Consider a consumer with reference dependent preferences deciding whether to buy a car. His con-
sumption utility for money x1 and cars x2 is given by
v1(x1) = x1 and v2(x2) = 60x2.
whereas his universal gain loss function satisfies
µ(v) = 3v for v ≥ 0 and µ(v) = 5v for v < 0.
Find the personal equilibria of this economy for any possible price p. What is the preferred personal
equilibrium for any value of p?
Extra Problem:
1. Sketch the isoquants for the technologies defined by
(i) y =(x1/21 + x
1/22
)2;
(ii) y =(x21 + x
22
)1/2.
Comment on the economic significance of the differences between the technologies.
5
Problem Set 4: Theory of the Firm Francesco Nava
EC411 —LSE MT 2017/2018
1. Consider the following production function
y = xα1xβ2 .
for α+ β < 1 and α, β > 0.
(a) Derive the factor demands x1(p, w) and x2(p, w).
(b) Derive the supply function y(p, w).
(c) Find all of marginal price effects. Confirm the signs (and, where appropriate, relative magnitudes)
that were derived in lectures.
(d) Find the profit function π(p, w). Confirm its properties.
2. Consider the following production function
y = xα1xβ2 .
for any α, β > 0.
(a) Derive the conditional factor demands h1(w, y) and h2(w, y).
(b) Find all of the marginal price effects. Confirm the signs (and, where appropriate, relative mag-
nitudes) that were derived in lectures.
(c) Find the cost function c(w, y). Confirm its properties.
(d) Sketch c(w, y) as a function of y for each of the three cases:
(i) α+ β < 1; (ii) α+ β > 1; (iii) α+ β = 1.
Also sketch the marginal and the average costs as functions of y, for each of the three cases. Why
is it not necessary to assume α+ β < 1 for cost minimization?
(e) Confirm that Shephard’s Lemma holds.
3. Consider the problem of a beekeeper who sells honey at a price ph per jar. It costs maxh2, 1 toproduce h jars of honey. Find the supply of honey of the beekeeper. Now, suppose that he learns how
to produce wax. In particular, assume that for each jar of honey w grams of wax can be produced at
no additional cost. If pw denotes the price of wax, find the supply of wax of the beekeeper.
6
Extra Problem:
1. Consider the following production function
y = xα1xβ2 .
for α, β > 0.
(a) Sketch the production possibility set, Y , in each of the three cases:
(i) α+ β < 1; (ii) α+ β > 1; (iii) α+ β = 1.
For each case, draw the marginal product, average product and marginal rate of technical sub-
stitution as functions of x1.
(b) Is the production function homothetic and/or homogeneous? In what arguments? To what
degree? What returns to scale are there?
7
Problem Set 5: Partial Equilibrium Francesco Nava
EC411 —LSE MT 2017/2018
1. Suppose that every firm in the kitchen foil industry has a production function
Y = (A+ T )1/2L1/2,
where Y is the amount of foil, A is aluminium, T is tin, and L is labour. The production function
of tin and the production function of aluminium exhibit constant returns to scale. In particular, 1
unit of tin can be produced with 1 unit of labour and 1 unit of aluminium can be produced with β
units of labour, for some β ∈ (0, 2). All the markets are perfectly competitive. The supply of labouris perfectly elastic at a wage equal to 4.
(a) Find the cost function of each firm.
(b) For all values of β in the interval (0, 2), determine how much tin and how much aluminium will
be used in equilibrium for the production of one unit of foil.
(c) For all values of β in the interval (0, 2), find the equilibrium price of foil.
2. In a competitive industry there are 200 firms. The cost function for 100 of these firms amounts to
c1(y) =
10y if y ≤ 10y2 − 10y + 100 if y > 10
where y is the amount of output produced by the firm and c(y) is its total cost in pence. The cost
function for the remaining 100 firms amounts to
c2(y) =
12y if y ≤ 10y2 − 8y + 100 if y > 10
(a) Derive and sketch the industry’s supply curve.
(b) If the market demand curve is described by
yd(p) = 2400− 100p,
where yd is demand and p is the price in pence, find the price, the total output of the industry,
and the outputs of the individual firms in equilibrium.
(c) If the tax of 1 pence is imposed on sales of the good, what will happen to the price paid by
consumers to the price received by firms and to output? What would be the effects of an increase
8
in tax to 2 pence?
(d) Set out an analysis of the costs and benefits of the 2 pence tax.
3. Consider a competitive industry with downward-sloping demand and free-entry. The industry is
composed of identical firms with U-shaped average total cost curves. Initially we are in long-run
equilibrium.
(a) The government announces that a substantial fixed annual licence fee will be imposed on all firms
choosing to remain in the industry. How does each of the following change, (i) industry output,
(ii) output per firm, and (iii) numbers of firms active in the industry.
(b) Suppose that instead of the licence fee, the government imposes a per unit sales tax. Relative
to the original situation with no tax, what is the long-run effect on: (i) output per firm, and (ii)
number of firms active in the industry?
9
Problem Set 6: General Equilibrium Francesco Nava
EC411 —LSE MT 2017/2018
1. Consider a pure exchange economy with two commodities, 1 and 2, and two agents, A and B. Their
utility functions are
UA(xA) = log(xA1 ) + log(xA2 );
UB(xB) = log(xB1 ) + 2 log(xB2 ).
Agent A has an initial endowment of 1 unit of good 1 and 4 units of good 2. Agent B has 6 units of
good 1 and 3 units of good 2.
(a) Find the competitive equilibrium price ratio, and the equilibrium allocation of commodities.
(b) Repeat part (a), but assuming that B’s preferences satisfy
UB(xB, xA) = log(xB1 ) + 2 log(xB2 ) + 2 log(x
A1 ).
(c) Is the competitive equilibrium that you found in part (b) Pareto effi cient? Prove your answer.
What is the economic significance of the final term in B’s utility function in part (b) and what
effects does it have on the competitive equilibrium and on welfare?
2. Consider an economy with two consumption goods, 1 and 2, and one factor of production, 3, with
prices denoted by p1, p2, and p3, respectively. There are two types of agents, A and B. The utility
function of a type i agent, for i ∈ A,B satisfies
U i(xi) = minxi1, x
i2
where xij denotes i’s consumption of good j. Each type A agent owns a firm that produces good 1
using good 3 via a technology characterised by the profit function
πA(p) = p21/4p3.
Each type B agent owns a firm that produces good 2 using good 3 via a technology characterised by
the profit function
πB(p) = p22/p3.
The initial endowments of goods 1 and 2 are equal to 0 and the initial endowment of good 3 is equal
to 10 for both types of agents. The proportion of type A agents in this economy is 1/2. All the agents
10
are price-takers.
(a) Find the supply and the input demand functions of types A and B.
(b) Find the competitive equilibrium prices.
3. Consider a production economy with two goods (call them good 1 and 2), two consumers (call them
A and B), and one firm (call it F ). Consumer A is the sole owner of firm F , and is endowed with
10 units of good 1 and with none of good 2. Consumer B owns only 26 units of good 2. The firm
transforms combinations of good 1 and 2 into good 1, and its production function satisfies
f(xF)= 3
(xF1)1/3 (
xF2)1/3
.
The preferences of the two consumers respectively satisfy
UA(xA) = log(xA1 ) + 3 log(xA2 );
UB(xB) = xB1 + 2(xB2 )1/2.
(a) State and solve the profit maximization problem of the firm.
(b) State and solve the utility maximization problem of each consumer.
(c) Find the Walrasian equilibrium.
(d) Consider the economy you have analysed in parts (a), (b) and (c) of this exercise, but suppose
that there is no firm and that consumer A is only endowed of 10 units of good 1. Define and
characterize the contract curve and the core of this exchange economy.
Extra Problem:
1. Consider an exchange economy with two goods (call them good 1 and 2), two consumers (call them
A and B). Consumer A is endowed with 10 units of good 1 and with 5 units of good 2. Consumer B
owns only 5 units of good 2. The preferences of the two consumers respectively satisfy
UA(xA) = maxxA1 , xA2 ;
UB(xB) = xB1 + 2(xB2 )1/2.
(a) State and solve the utility maximization problem of each consumer.
(b) Does this economy possess a competitive equilibrium? If yes, find it. If not, prove why.
(c) Define and characterize the contract curve and the core of this economy.
(d) Does the second welfare theorem apply to this setting.
11
Problem Set 7: Uncertainty Francesco Nava
EC411 —LSE MT 2017/2018
1. A consumer who has an initial wealth of 36 pounds must choose between two options, X and Y . Option
X is a lottery which gives 0 with probability 1/4 and 28 pounds with probability 3/4; option Y instead
is worth 13 pounds with probability equal to 1. The consumer is an expected utility maximizer with
a utility function u(w), where w denotes the consumer’s wealth in pounds. Utility u(w) is continuous
and strictly increasing in w.
(a) Suppose that the consumer is risk-averse. Can you determine which option she will choose?
(b) Suppose that someone who knows the outcome of the lottery in option X is willing to sell the
information to the consumer. If the consumer is risk-neutral, how much would she be willing to
pay to know the outcome of the lottery before making her choice between X and Y ?
(c) Suppose that the consumer is risk-loving and, as in (b), she can buy information about the
outcome of the lottery prior to making her choice between X and Y . Will the consumer pay a
positive amount to know the outcome of the lottery? Explain carefully.
2. A decision maker is considering how much to invest in two assets. The first asset yields no return,
whereas the second asset is risky and has a return of 20% with probability 1/2, but makes a loss of
10% otherwise. The preferences of the decision maker over future monetary holdings x are determined
by the function u(x) = log(x). The price of the first asset is 1, whereas p denotes the price of the risky
asset. Derive the demand for the risky asset if the decision maker has M dollars to invest.
3. A farmer can grow wheat, or potatoes, or both. If the weather is good, an acre of land yields a profit
of £2, 000 if devoted to wheat, of £1, 000 if devoted to potatoes. Should the weather be bad, an acre
of wheat yields £1, 000 and of potatoes £1, 750. Good and bad weather are equally likely.
(a) Assuming that the farmer has utility function log(m) when his income is m, what proportion of
his land should he turn over to wheat?
(b) Suppose the farmer can buy an insurance policy which pays, for every £1 of premium, £2 if the
weather is bad, and nothing if the weather is good. How much insurance will he take out and
what proportion of his land will he devote to wheat? What would the answers be if the policy
paid only £1.50 to compensate for bad weather?
12
Problem Set 8: Uncertainty & Static Games Francesco Nava
EC411 —LSE MT 2017/2018
1. In a portfolio selection model: let W denote initial wealth; let I denote final wealth; let B denote
amount invested in a “risky asset”; let M denote amount of wealth held as money where money is a
“sure asset”with a zero return; let r denote the (random) return on the risky asset with E(r) > 0 —
meaning that when £B are placed in the risky asset, a random amount £(1 + r)B will be realised.
Suppose that the investor has a Bernoulli utility function
u(I) = βI − αI2,
for α, β > 0.
(a) Graph this utility function and show that it exhibits risk aversion. Interpret α and β.
(b) Write out the choice problem which determines the utility-maximising division of initial wealth
between the risky asset and money.
(c) Compute the optimal investment B and prove that it increases with β. Interpret the result.
2. Identify all the Nash Equilibria of the following games, and find dominant strategies:
(i)
1\2 a b
A 0, 1 2, 2
B 3, 0 0, 3
(ii)
1\2 a b
A 2, 1 2, 2
B 3, 1 1, 0
(iii)
1\2 a b
A 0, 2 2, 2
B 3, 0 0, 3
3. Consider the following Matching pennies game
1\2 a b
A 1, 0 0, 1
B 0, 1 1, 0
(a) Show that the game below has no Nash equilibrium in pure strategies.
(b) Suppose player 1 tosses a coin to decide which action to take; i.e. he has probability p of playing
A and probability (1−p) of playing B. Similarly, suppose that 2 plays a with probability q and bwith probability (1− q). Plot best responses and find values of p and q such that these strategiesform a Nash equilibrium.
(c) Can you see any property which this equilibrium shares with game (iii) of Problem 2?
13
Extra Problem:
1. Write down your preferences in the following two situations. Situation 1:
Outcome: 5 M$ 1 M$ 0 M$
Lottery A: 0% 100% 0%
Lottery B: 10% 85% 5%
Situation 2:Outcome: 5 M$ 1 M$ 0 M$
Lottery C: 0% 15% 85%
Lottery D: 10% 0% 90%
Having made your two choices, write down the restrictions which these imply on your utility function
of income if they are to be consistent. Are they consistent?
14
Problem Set 9: Static & Dynamic Games Francesco Nava
EC411 —LSE MT 2017/2018
1. Consider an auction with two buyers participating and a single object for sale. Suppose that each
buyer knows the values of all the other bidders. Order players so that values decrease, x1 > x2.
Consider a 2nd price sealed bid auction. In such auction: all players simultaneously submit a bid bi;
the object is awarded to the highest bidder; the winner pays the second highest submitted bid to the
auctioneer; the losers pay nothing. Suppose ties are broken in favour of player 1. That is: if b1 = b2
then 1 is awarded the object.
(a) Characterize the best response correspondence of each player.
(b) Characterize all the Nash equilibria for a given profile (x1, x2).
(c) Consider the Nash equilibrium in which both players bid their value —that is, bi = xi for i ∈ 1, 2.Is this a dominant strategy equilibrium?
2. Four patients have to undergo surgery and rehabilitation in one of two hospitals. Hospital A specializes
in surgery. But its elite surgery unit is small. The likelihood of successful surgery, pSA, depends on the
number of patients treated in the surgery unit, n, as follows:
pSA(n) =
17/20 if n = 1
15/20 if n = 2
11/20 if n = 3
7/20 if n = 4
.
The rehabilitation unit of hospital A is large, but conventional. The likelihood of successful rehabilita-
tion pRA = 1/2 is independent of the number of patients treated. Hospital B specializes in rehabilitation.
But its elite rehabilitation unit is small. The likelihood of successful rehabilitation, pRB, depends on
the number of patients treated by the unit, n, as follows:
pRB(n) =
1 if n = 1
16/20 if n = 2
14/20 if n = 3
12/20 if n = 4
The surgery unit of hospital B is large, but conventional. The likelihood of successful surgery pSB = 1/2
is independent of the number of patients treated. Surgery outcomes are independent of rehabilitation
outcomes. A patient’s payoff is 1 if both treatments are successful, and 0 otherwise. A patient is
15
classified as recovered, only if both treatments are successful. [Hint: Recall that if events A and B are
independent Pr(A ∩B) = Pr(A) Pr(B)].
(a) First suppose that the existing health regulations require all patients to undergo surgery and
rehabilitation at the same hospital. Patients can only choose in which of the two hospital to get
both treatments. Set this problem up as a strategic-form game. Find a Nash equilibrium of this
game. What is the average probability of recovery among the four patients in this equilibrium?
(b) In an attempt to increase the average recovery probability regulators decide to lift the ban on
having surgery and rehabilitation at different hospitals. Now patients are free to choose in which
hospital to get either treatment. Set this problem up as a strategic-form game. Find a Nash
equilibrium of this game. What is the average probability of recovery among the four patients in
this equilibrium?
(c) Still unsatisfied about the average recovery probability, regulators decide to try a third policy,
in which one of the four patients is randomly selected and sent to the two elite units (surgery in
A and rehabilitation in B), while the remaining three are sent to the larger conventional units
(surgery in B and rehabilitation in A). What is the average probability of recovery among the
four patients with this policy in place? Compare the average probabilities of recovery under the
three regulations. Give an intuitive explanation to the observed change in recovery probabilities.
3. Consider the following extensive form game:
(a) Find the unique Subgame Perfect equilibrium of this game.
(b) Find a pure strategy Nash equilibrium with payoffs (3, 5).
(c) Find a pure strategy Nash equilibrium with payoffs (4, 2).
16
Extra Problem:
1. Consider the following game
1\2 a b
A 6, 4 3, 5
B 5, 3 2, 2
.
(a) In the game shown below, identify the unique Nash Equilibrium in pure strategies.
(b) Now consider a “sequential”adaptation with the same payoffs as the game below: player 1 first
chooses a strategy, and then player 2, knowing 1’s choice, chooses his action. What are the pure
strategy Nash equilibria in this game? What are the pure strategy subgame perfect equilibria?
17
Problem Set 10: Dynamic & Repeated Games Francesco Nava
EC411 —LSE MT 2017/2018
1. Consider two firms trying to gain the status of market leader. Becoming market leader is worth v
dollars to each competitor. Each firm can decide how much effort to devote to this process. The effort
chosen by each firm belongs to the interval [0, 1]. Given the effort decisions of the two firms,eA and
eB, the likelihood of Firm A becoming market leader is given by eA(1− eB), whereas the likelihood ofFirm B becoming leader is given by the complementary probability. Effort is costly, and the monetary
cost of effort depends on the resources devoted by both competitors. In particular, for both firms the
effort cost coincides with ke2AeB. Suppose that v < k.
(a) Derive the best responses for each of the two firms in this game. Plot them and find all the pure
strategy Nash equilibria of this game.
(b) Prior to the beginning of the game Firm A can choose how much to invest in developing the
market. In particular, suppose that Firm A can set v to any number in the interval [0, k] at a
cost c(v) = v3/3k. What investment would the firm choose in a Subgame Perfect equilibrium, if
both firms exert effort at the competition stage?
2. Suppose that two players have to split a cake of size 10 according to an “I cut, you choose”protocol.
In particular, assume that Player 1 cuts the cake into two parts and that Player 2 gets to choose which
of the two slices to consume (while the other slice is consumed by Player 1). Suppose that the cake is
continuously divisible.
(a) First, assume that the cake is homogeneous, so that both players value all its parts alike, and care
only about the size of the slice that they get to consume. Set this problem up as an extensive
form game. Find the Subgame Perfect equilibria of this game. Write the behavioural strategy
for both players, and check that no deviation is profitable.
(b) Now suppose that 3/5 of the cake is chocolate flavoured, while 2/5 of the cake is vanilla flavoured.
Player 1 only likes chocolate, and only cares about the amount of chocolate in the slice he
consumes; while 2 likes all the parts of the cake alike, and cares only about the size of the slice
consumed. Also, assume that Player 1 can cut slices with any composition of chocolate and
vanilla. Find a Subgame Perfect equilibrium of this game. Write the behavioural strategy for
both players, and check that no deviation is profitable.
(c) Does the game in part (a) possess a pure-strategy Nash equilibrium in which a player receives a
slice different in size from the one that you have characterized in part (a)? Explain.
18
3. Consider the following asymmetric Prisoner’s Dilemma:
1\2 C D
C 3, 4 1, 6
D 4, 0 2, 2
(a) Find the minmax values of this game. Consider the infinitely repeated version of this game in
which all players discount the future at the same rate δ. The following is a "tit for tat" strategy:
any player chooses C provided that the other player never chose D; if at any round t a player
chooses D, then the other player chooses D in round t + 1 and continues playing D until the
player who first chose D reverts to C; if at any round t a player chooses C, then the other player
chooses C in round t+ 1. Write this strategy explicitly.
(b) Find the unique value for the common discount factor δ for which the strategy of part (a) sustains
always playing C as a SPE of the infinitely repeated game.
(c) Then, consider the following "trigger" strategy: any player chooses C provided that no player
ever played D; otherwise any player chooses D. Write the two incentive constraints that would
have to be satisfied for such a strategy to be a NE. Then, write the two additional incentive
constraints that would have to be satisfied for such a strategy to be a SPE. What is the lowest
discount rate for which such strategy satisfies all the constraints.
Extra Problem:
1. Consider the following game
1\2 L R
T 0, 0 4, 1
D 1, 4 3, 3
Find all the Nash equilibria of this static game. Next consider the infinite repetition of the game,
and a strategy prescribing to play (D,R) so long as no player has ever deviated and to play (T, L)
otherwise. Is this strategy a Nash equilibrium for the infinite repetition? Is it a Subgame Perfect
equilibrium?
19
Hand-In Homework Francesco Nava
EC411 —LSE MT 2017/2018
Please submit your homework to your class teacher before the end of week 7.
Question 1 (35 Marks)
Consider a consumer whose utility function is
u(x1, x2) = − exp(−ax1x2)
Can you take first order conditions to solve the utility maximization problem? Explain your argument. Next
solve the utility maximization problem, and derive Marshallian demands and the indirect utility function.
Given your calculations, state and use the duality theorem to find the expenditure function and Hicksian
demands.
Question 2 (40 Marks)
A price-taking firm has a production function given by
y = 3(x3)1/3 (maxx1, 8x2)1/3
where x1, x2 and x3 are inputs and y is output. Let w1, w2 and w3 denote input prices, and let p denote
the output price. Solve the profit maximization problem and the corresponding cost minimization problem.
While solving these problems motivate your approach.
Question 3 (25 Marks)
Consider a simple exchange economy with two goods and two consumers. The endowments of the two
consumers satisfy respectively ω1 = (5, 1) and ω2 = (1, 9), while their preferences satisfy:
U1(x1) = min2x11, x12 + 2 and U2(x2) = 2√x21x
22.
Find Marshallian demands for each of the two consumers. Explain your approach. Solve for the Walrasian
equilibrium of this economy.
1
Hand-In Homework 2 Francesco Nava
EC411 —LSE MT 2017/2018
Please submit your homework to your class teacher before the end of week 10.
Question 1 (50 Marks)
Consider an exchange economy with two goods (call them good 1 and 2) and two consumers (call
them A and B). The preferences of the two consumers respectively satisfy
uA(xA) = −(xA1 − 4)2 − (xA2 − 4
)2,
uB(xB) = minxB1 , x
B2
.
Consumer A owns only k ∈ (0, 16] units of good 2. Consumer B is endowed with 16 units of good 1
and with 16− k units of good 2.
1. State and solve the utility maximisation problem of each consumer. (12 Marks)
2. Find the competitive equilibria of this economy for all values of k ∈ (0, 16]. (14 Marks)
3. Define and characterise the contract curve and the core of this economy. (16 Marks)
4. Does the first welfare theorem apply to this problem? If so, argue why. If not, find a violation.
(8 Marks)
Question 2 (20 Marks)
Bob is considering whether to buy health insurance. If Bob is healthy, his wealth amounts to W .
However if he falls ill, it only amounts to V < W . The likelihood of eventually falling ill is 1/3. Bob’s
preferences depend only on realized wealth x, and his utility function is determined by
u(x) = log(x).
Assume that W = 30, that V = 10, and that the unit-price of insurance coverage is equal to r ∈[1/3,∞). Find the expected value of purchasing S units of coverage. Find Bob’s expected utility from
purchasing S units of coverage. Find Bob’s demand for coverage as a function of r.
1
Question 3 (30 Marks)
Consider a game played by two lobbyists 1 and 2 sponsoring two distinct versions of a bill. The value
of having version i ∈ 1, 2 approved equals vi to lobbyist i and equals 0 to lobbyist j, for some vi > 0.
Lobbyists choose how many resources, ri ∈ [0,∞), to invest to sway the parliament to support their
preferred version. The probability that the policy preferred by lobbyist i is approved is given by the
following function of the resources invested by the two lobbyists
pi(ri, rj) =
1/2 if ri = rj
ri/(ri + rj) if ri 6= rj.
Resources are however costly and the payoff of lobbyist i amounts to
ui(ri, rj) = pi(ri, rj)vi − rirj
The payoff is concave and single peaked in ri for any value rj . Find the best response for each of the
two lobbyists. Find the Nash equilibria of this game. Is there always an effi cient equilibrium?
2
On the Concavity of the Expenditure Function (EC411 —Nava)
Let h(p, u) denote the n× 1 vector of Hicksian demands.
The law of compensated demand states that for preferences that satisfy LNSand convexity, and for any two strictly positive 1 × n price vectors p, p >> 0,we have that
(p− p)︸ ︷︷ ︸1×n
· (h(p, u)− h(p, u))︸ ︷︷ ︸n×1
≤ 0.
This obtains because by the optimality of h(p, u) in the expenditure minimiza-tion problem, as it implies both
p · h(p, u) ≤ p · h(p, u) &
p · h(p, u) ≤ p · h(p, u).
By adding the two and rearranging the result the law of compensated demandsfollows.
Next we exploit the law of compensated demand to prove the concavity of theexpenditure function. Note that the differential version of the law of demandstates that
dp︸︷︷︸1×n
· dh(p, u)︸ ︷︷ ︸n×1
≤ 0.
Let Dph(p, u) denote the n×n Jacobian of h(p, u) with respect to prices. Recallthat that dh(p, u) = Dph(p, u) · dpT . Substituting we obtain that
dp︸︷︷︸1×n
·Dph(p, u)︸ ︷︷ ︸n×n
· dpT︸︷︷︸n×1
≤ 0,
and therefore that Dph(p, u) is negative semidefinite.
This immediately implies the concavity of the expenditure function since:
de(p, u)
dp= h(p, u) and
d2e(p, u)
dp2= Dph(p, u).
1
On the Convexity of the Production Set (EC411 —Nava)
Consider a firm producing 1 output with n inputs.
Let’s show that if the production possibility set Y is convex, the productionfunction f : Rn → R is concave.
Consider any two feasible production plans
(f(x′),−x′), (f(x′′),−x′′) ∈ Y ,
and for λ ∈ (0, 1), let
(y,−x) = λ(f(x′),−x′) + (1− λ)(f(x′′),−x′′).
As Y is convex (y,−x) ∈ Y . Moreover, by definition of the production function
f(x) ≥ y, for any (y,−x) ∈ Y .
This immediately delivers the result as the previous observations imply that
f(x) = f(λx′ + (1− λ)x′′) ≥ λf(x′) + (1− λ)f(x′′).
1
On Concavity and Returns to Scale (EC411 —Nava)
Let f(x) denote a concave production function for a vector of inputs x.
Concavity of the production function requires that for any p ∈ [0, 1] and for anytwo input vectors x, x′ we have
f(px+ (1− p)x′) ≥ pf(x) + (1− p)f(x′).
Further assume that f(0) = 0. If so, concavity implies that for any p ∈ [0, 1]
f(px) = f(px+ (1− p)0) ≥ pf(x) + (1− p)f(0) = pf(x).
This implies that the production function is subadditive as for any t ≥ 1,
tf(x) = tf
(1
ttx
)≥ t1
tf (tx) = f (tx) .
This implies decreasing or constant returns to scale. Strict concavity equiva-lently implies DRS.
1
On the Slope of the Supply Function (EC411 —Nava)
Let y(p, w) denote the supply function of a firm and let x(p, w) denote the vectorof input demands.
Recall that y(p, w) satisfies
y(p, w) = f(x(p, w)).
Thus the slope of the supply function with respect to the output price is equalto
∂y(p, w)
∂p=∑n
i=1
[∂f(x(p, w))
∂xi
∂xi(p, w)
∂p
]=
[∂x(p, w)
∂p
]T︸ ︷︷ ︸
1×n
[∂f(x(p, w))
∂x
]︸ ︷︷ ︸
n×1
.
Next recall that first order conditions for profit maximization imply that
p∂f(x(p, w))
∂x= w︸︷︷︸
n×1
.
Totally differentiating this equation with respect to the output price p we furtherobtain that
∂f(x(p, w))
∂x+ p
∂2f(x(p, w))
∂x2︸ ︷︷ ︸n×n
∂x(p, w)
∂p= 0.
Next exploit the previous equation to obtain that
∂y(p, w)
∂p=
[∂x(p, w)
∂p
]T [∂f(x(p, w))
∂x
]=
=
[∂x(p, w)
∂p
]T [−p∂
2f(x(p, w))
∂x2∂x(p, w)
∂p
]=
= −p[∂x(p, w)
∂p
]T [∂2f(x(p, w))
∂x2
] [∂x(p, w)
∂p
]> 0,
where the last inequality holds since the production function is concave, and thussince ∂2f(x(p,w))
∂x2 is negative definite. This establishes that the supply functionis upward sloping in the output price.
1
On the Continuity of the Cost Function (EC411 —Nava)
Let x(w, y) denote the n×1 vector of conditional factor demands, and let c(w, y)denote the cost function.
Suppose that w′ ≥ w, and observe that
c(w, y) = wx(w, y) ≤ w′x(w, y),
as w′ ≥ w. Moreover, as costs are non-decreasing in factor prices,
c(w, y) = wx(w, y) ≤ wx(w′, y) ≤ w′x(w′, y) = c(w′, y).
Finally by optimality of the cost function
c(w′, y) = w′x(w′, y) ≤ w′x(w, y).
These observations together imply that
c(w′, y) ∈ [wx(w, y), w′x(w, y)].
The cost function is then obviously continuous as
limw′→w
c(w′, y) ∈ limw′→w
[wx(w, y), w′x(w, y)] = c(w, y).
1
More on Walras Law (EC411 —Nava)
By Walras Law we know that LNS implies pz(p) = 0. We want to show that itfurther requires pizi(p) = 0 for any good i.
If pz(p) = 0, for any good i it must be that
pizi(p) = −∑
k 6=i pkzk(p).
Thus, suppose that for some good i we have that pizi(p) < 0. As prices areare non-negative, this requires pi > 0 and zi(p) < 0. If so,
∑k 6=i pkzk(p) > 0
by Walras Law, and therefore pkzk(p) > 0 for some k. But this would violatefeasibility in the market of good k, as it requires pk > 0 and zk(p) > 0. Thus,Walras Law implies that pizi(p) = 0 for any good i.
1