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Part IIa: Paper 1General Equilibrium and
Welfare Economics
Dr Snje Reiche
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O utline
Robinson Crusoe Model Social optimal outcome Competitive equilibrium Books: Cowell Ch 6.3, Varian Ch 32
Model of Exchange: Edgeworth BoxAnalysis Books: Cowell Ch. 7.2.3-7.3.2 Varian Ch 31-31.7
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Robinson Crusoe Model
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Robinson Crusoe ModelO ne individual on an island who both consumes and produces.
Simplest model because only one consumer and one producer.
As a consumer : consumes leisure and eats coconuts
As a producer: uses labour to produce coconuts withsome given technology
Distinguish Social O ptimumfrom
Competitive Equilibrium
Recognises interdependence betweenconsumption and production choices,
and maximises social welfare.Choices madeseparately subjectto prices for leisure andcoconuts
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Social O ptimum(recognising both consumer and producer)
z H f cto subject zcu zc !,max, ,a L u c z f H z cP! -
First-order conditions:
P!x
xc
zcuc ,: z H f z
zcu z !x
x ',: P
z H f
c
zcu z
zcu
!
x
xx
x'
,
,
Marginal Rateof
Substitution
MarginalProduct of
Labour:Marginal Rateof
transformationof leisure into
coconuts
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C
Labour: H
- z
H
f( H - z)
Social IndifferenceCurve
MRS = MRT
This is the optimal choice
but this tells us nothing about the behaviour of markets
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Competitive Equilibrium
As a consumer: z H w pcto subject zcu zc
! T ,max,
,a L u c z H z pcP T ! -
because R.C. owns the firm
Income fromworking
Key assumption is price taking behaviour.
So, choice of c and z will not change p and w
p
c
zcuc P!
x
x ,:
w
z
zcu z P!
x
x ,:
pw
c
zcu z
zcu
!
x
xx
x
,
,
Marginal Rateof Substitution
Real wage:financial rate of
transformation
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As a producer: w L L p f
L!T max
L = H - zAgain, price taking behaviour.So, choice of L does not change p and w
0!
x
xw
L
L f p
Marginal Rate of Transformation
Real wage:financial rate of transformation
pw
L
L f !x
x
Also, firms hire labour until the marginal product of labour is equal to the real wage.
Note also that only relative price is important
Consumer: MRS = w/pProducer: MRT = w/p
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H ave satisfied optimality , but is this a competitive equilibrium?
NO , because not necessarily feasible.
C
Labour: H - z
H
f( H - z)
pw /
pw
MRT !
pw
MRS !
Excessdemand
Excess Supply
pT
wage too low
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C
Labour: H - z H
f( H - z)
pw /
L*
H igher wage, lower price
Competitive equilibrium is defined by price ratio
which clears markets, given that agents optimise
c*Choices in
Competitive Equilibrium
are the same as inSocial O ptimum
Competitive Equilibrium
is efficient
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O ptimality: (1) c and z are chosen to maximise u( c, z)subject to
(2) L is chosen to maximise
z H w pc e T w L L p f !T
Feasibility:
Assumptions: (1) Price-taking behaviour (2) No externalities
(3) No uncertainty
( )c f L!
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Exchange and Edgeworth Box
B y
B x
A y
A x eeee Endowments
yand xGoods
Band AC onsumers
,,,
O ptimality: Utility maximisation
Feasibility: B A B
y A y
B A B x
A x
y yee
x xee
!
!
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O ptimisation by A
A y y
A x x
A y
A x
A A
y x
e pe p y p x pto subject
y xu A A
!
),(max,
][),( A y A
x y A
y x A
x A
y p x p pe pe y xu L ! % P
y A
A A
x A
A A
p y
y xu p
x y xu
PP !x
x!x
x ),(;
),(First order conditions:
y
x
A
A A
A
A A
p p
y y xu
x y xu
!
xx
xx
),(
),(
similarly for B
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B xy
y
x A xy MRS p
p MRS !!So, optimality :
But not necessarily feasible :
Use Edgeworth Box
O A
O B
x
y
I A
I B
e
B x
A x ee
B
y A
y ee
Trade can lead toincreased utility for both individuals
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What happens in the market?
O A
O B
x
y
I A
I B
e
B A
Excesssupply
Excessdemand
y
x
p p
price of x is too low
Note: change in p x willhave a wealth effect:changes the value of
endowments
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O A
O B
x
y
e
A= B
y
x
p p
Competitive Equilibrium
Price Adjustment: B x A x B A x ee x x XD !
q
o"
x x
x x
p XD
p XD
,0
,0
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O A
O B
x
y
Contract Curve and the Core
1e 2e
3e
Contract Curve:Locus of points of
tangency between MRS
Need price vector that passes through
endowment and is tangential to indifferencecurves on the contract curve.
Core: set of allocations on thecontract curve thatare better than e2for both A and B
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Competitive equilibrium: a price vector and allocationwhich is optimal and feasible .
Individuals and firms are price-takers, so possible tooptimise, but the market does not clear, so noequilibrium.
Shown in R.C. model and model of exchange.
Core: set of allocations that cannot be blocked by anycoalition. As number of people increases, core shrinks.In the limit the core is equal to the set of equilibria.
Summary