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Part IIa: Paper 1General Equilibrium and Welfare

Economics

Dr Snje Reiche

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Outline

Summarising competitive equilibrium

Cowell 6.4

Varian 32.9-32.13

Questions remaining

Excess demand functions and defining

equilibrium

Properties of excess demand functions

Cowell 7.4-7.4.2

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Competitive Equilibrium in a

Closed Economy

x

y

y

x

p

p

Agg I.C.

PPF

MRTp

pMRS

y

x !!

sd

xx !

sdyy !

Contract

Curve

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Competitive Equilibrium in an

Open Economy

x

y

y

x

p

p

y

x

ppMRS

d

x

d

y

y

x

p

pMRT

sy

s

x Import

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Questions to Address

Existence: Under what circumstances can we be

sure that an equilibrium exists?

Uniqueness:Will an economy have only oneequilibrium?

Stability: Will the economy somehow tend to or

revert to this equilibrium?

Price Determination: And will this determine the

price system for us?

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Excess Demand Function

d sx xE p x p x p R|

Aggregate

demand forxgiven prices

Aggregate

supply forxgiven prices

Economys

natural resourceofx

This function gives excess demand for good x as a

function of the price vectorp. xE p

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Aggregation of Consumer Demand

,d d

i ii x p x p m

!

Partial equilibrium approach

General equilibrium approach

d dii

x p x p!

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x

px px

x

Individual 1 Individual 2

x

px MarketDemand

Convenientlydownward

sloping!

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Aggregation of Supply s sf

f

x p x p!

x

px px

x

Firm 1 Firm 2

x

px MarketSupply

Conveniently

upward sloping!

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xR

px

Resource Stock

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d sx xp x p x p R|

Putting these three elements together ...

xp

xE p

Excesssupply

Excessdemand

0x p !

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Competitive EquilibriumEquilibrium given by price vectorp*which

satisfies 3 conditions

* 0

* 0

* * 0

E p

p

E p p

e

u

!g

foreach good

So, if stronglymonotone preferences,

and

equilibrium given by

* 0E p !

No excess demand

Prices non-negative

If excess supply,

price must be zero

* 0p "

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Properties of Excess Demand

Functions(these properties hold in and out of equilibrium)

(1) Homogenous of degree zero

0x xE t p E p t ! "g

so, we can normalise prices aribitrarily:

E.g. divide by labourp

or, divide by so that prices sum to 1.h

hp

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(2) Walrass Law (local non-satiation)

0h hh

p p !

Once we know excess demand in H-1 markets, we

know excess demand in the Hth market

xp

xE p

Excesssupply

Excessdemand

Diagram for two goods and normalised prices

1

Individuals spend all their income because of non satiation.

Income is determined by resources and profits.

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ExistenceIs there a set of prices such that ? 0

x

E p !

Well-behaved economy

xp

x

E p

Excesssupply

Excessdemand

1

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Existence requires each excessdemand

function tobe continuous

production function concave

indifference curves continuous and strictly convex

Problems

Discontinuous excess demand

xp

x p

xcesssupply

xcessdemand

1

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Problem of non-concavity of production

labour

sx

more generally, non-concave if

increasing returns to scale

s

x

One firm

Many firms

labour

BUT:

if increasing returns to

scale, then only a

small number of

firms, so non-

concavity remains

p

w

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Non-Convexity of indifference curves

x

y

x

px

Discontinuousdemandfunction

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Summary

Defined excess demand functions and showed

requirements for competitive equilibrium using these

functions

Properties of excess demand functions

Homogenous of degree 0

Walrass Law

Using excess demand functions to consider

existence