8/8/2019 GWLecture2

1/19

Part IIa: Paper 1General Equilibrium and

Welfare Economics

Dr Snje Reiche

8/8/2019 GWLecture2

2/19

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

8/8/2019 GWLecture2

3/19

8/8/2019 GWLecture2

4/19

Robinson Crusoe Model

8/8/2019 GWLecture2

5/19

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

8/8/2019 GWLecture2

6/19

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

8/8/2019 GWLecture2

7/19

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

8/8/2019 GWLecture2

8/19

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

8/8/2019 GWLecture2

9/19

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

8/8/2019 GWLecture2

10/19

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

8/8/2019 GWLecture2

11/19

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

8/8/2019 GWLecture2

12/19

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!

8/8/2019 GWLecture2

13/19

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

!

!

8/8/2019 GWLecture2

14/19

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

8/8/2019 GWLecture2

15/19

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

8/8/2019 GWLecture2

16/19

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

8/8/2019 GWLecture2

17/19

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

8/8/2019 GWLecture2

18/19

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

8/8/2019 GWLecture2

19/19

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