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EC 201 Lent Term

Week 1

General Competitive Equilibrium

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General

CompetitiveEquilibrium

How do competitive markets aggregate consumer andproducer choices into equilibrium prices and quantities?

How does activity on one competitive market affect activityon another? For example, how do our conclusions about the

effects of taxing a single-market economy carry over to two-market economies?

What properties do competitive equilibria have? Forexample, when are competitive markets efficient?2

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Example

In 2007, UK government increased excise duty on bottle of wine(paid by seller) by 4p whilst keeping tax on spirits constant

What should happen to wine and spirits prices?

Partial equilibrium analysis analyses each market in isolation. Itpredicts: no change in spirits price because tax on spiritsunchanged; wine price increases from p to p

3

pwine

Old Supply

New Supply

4p

Demandqwine

pp

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General-Equilibrium Effects

Because wine and spirits are substitutes, when tax increase causeswine price to rise to p, demand for spirits rises, increasing themarket price of spirits

Because spirits market price increases, demand for wine increases

(again since wine and spirits substitutes), causing its market priceto rise to p>p

4

pwineOld Supply

New Supply

4p

Old Demandqwine

New Demand

pp

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Where Does it End?Now that the price of wine is p, demand for spiritsrises, causing its price to rise, increasing demand forwine and causing its price to rise to p>p>p. And soon and so forth

By analysing equilibrium in the two marketssimultaneously, general equilibrium allows economiststo account for the interplay between different markets,solving for equilibrium prices in the two markets

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The simplest economy to analyse is one

where people merely exchange goods that

they already own

Our market model of exchange economies provides asimplified representation of markets from the London StockExchange to international trade to village bazaars to allocatingcourses to students at Harvard Business School

Like all models, it is too simple to incorporate all theimportant features of these or other trading environments. Yetit provides important insights into economic behaviour

6

Exchange Economies

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Overview

In an exchange economy, each consumer begins withan endowment of the various different goods in theeconomy

She trades her goods endowment for the bestallocation that she can afford given that endowmentand market prices

She consumes her allocation

The end

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Formal Description of Exchange

Economy

N2 Different Consumers (Traders) L2 Different Goods in Economy Consumer iendowedwith amount e1i0 of Good 1,

e2i0 of Good 2, etc. We mostly use two goods in thiscourse

Consumer ihas utility function ui

NB We use superscripts for people and subscripts for goodsWe also use ei to refer to the vector of Consumer is endowment of

goods, i.e. ei=(e1i,e2i)

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Exchange Economies without

Externalities

Absent production (firms), the N consumers simplyexchange (trade) their endowments of the L goods

in the economy

(Until further notice), we assume that eachconsumer cares only about her own privateconsumption; when Consumer i consumes the

bundle of goods (x1i,x2i), her utility ui depends solely

upon (x1i,x2i): ui(x1i,x2i). (In particular, Person is utilitydoes not depend upon Person js consumption, nor doesPerson i care about money except insofar as it allowsher to consume more, e.g., there is no saving)

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Competitive Budget Sets Consumers budget sets depend on fixed market

prices for the L goods in the economy, p1,p2,...0

We write p (without subscript) to mean the pricevector: p=(p1,p2,...)

Each consumer faces the same, fixedmarket prices!

That is, consumers take market prices as given anddo not enjoy quantity discounts or other forms of

non-linear prices, nor can they bargain over or

otherwise affect prices

Budget sets also depend upon endowments

Different consumers may have differentendowments! 10

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Describing Budget Sets

Consumer ican afford to consume any bundle of

goods (x1i,x2i)0such that

p1x1i+p2x2ip1e1i+p2e2i

Cost of bundle (x1i,x2i)at market pricesValue of endowment(e1i,e2i)at market prices

We can think of this as consumer first selling all herendowment at market prices and using the money she

raises to buy consumption goods (e.g., poultry farmersells all chickens at market price before buying a

consumption bundle consisting of chickens and goats)

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Goods-EndowmentBudget Set

ei Budget Set

Good 1

Good 2

e1i

e2i

Slope = -p1/p2

12

e2i+(p1/p2)e1i

e1i+(p2/p1)e2i

Goods endowment

As usual, you can work out intercepts on two axes by

calculating how much of each good the consumer can afford

when she buys only that good

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Features of Budget Set

Budget set depends upon the relative prices p1/p2 butnot upon the absolute prices p1and p2

Each consumers budget set passes through her goodsendowment: regardless of what it is, the consumer canalways afford to consume her endowment. She can

does this either by selling endowment and buying it

back (as prices being fixed and common to all

consumers means that buying price = selling price) orsimply by not trading at all

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Money as Unit of Account

In economy without money as such, we use moneysimply as a unit of account

Prices in general equilibrium are really relative prices,how much one good costs in terms of another

For example, consider economy with only two goods,chickens and goats. Suppose that each chicken costs kgoats. Someone endowed with cchickens andggoatshas an endowment with value ofg+kcgoats.

Alternatively, that same endowment is worthg/k+cchickens. Whether prices are quoted in chicken orgoat units does not affect economic behaviour

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Uncompensated Demand In consumer theory, Consumer is uncompensated

demand (x1i

(p1,p2;mi

), x2i

(p1,p2;mi

)) maximises her utilityui(x1,x2) subject to budget constraint defined by prices(p1,p2) and wealth mi: (x1i(p1,p2;mi), x2i(p1,p2;mi)) solvesmaxui(x1,x2) subject to (s.t.) p1x1+p2x2 mi

In exchange economy, consumers uncompensateddemand (x1i(p1,p2;e1i, e2i), x2i(p1,p2;e1i, e2i)) maximisesutility ui(x1,x2) subject to budget constraint defined byprices (p1,p2) and goods endowment ei:

(x1i(p1,p2;e1i, e2i), x2i(p1,p2;e1i, e2i)) solves

maxui(x1,x2) s.t. p1x1+p2x2 p1e1i+p2e2i

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Demand with Goods Endowment

ei Budget Set

Good 1

Good 2

e1i

e2i

Slope = -p1/p2

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x1i(p;ei)

x2i(p;ei)

Indifference curves

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Excess Demand

We call xji(p;ei)-ejiConsumer is excess demandforgood j=1,2, the difference between what she

consumes and her endowment

If xji(p;ei)-eji>0, then Consumer iis a net consumerof Goodj

If xji(p;ei)-eji

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Excess Demand with Goods

Endowment

ei

Good 1

Good 2

e1i

e2i

Slope = -p1/p2

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x1i(p;ei)

x2i(p;ei)

is Excess Demand Good 2

is Excess Supply Good 1

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Change in Utility from Price

Change

ei

Good 1

Good 2

e1i

e2i

new prices

19

When i supplies Good 1, she benefits when its price rises:

at new prices she can afford old bundle (and hence utility)

and may be able to afford a bundle with higher utility

old prices

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Change in Utility from Price

Change

ei

Good 1

Good 2

new prices

20

When i supplies Good 1, she may even benefit when its

price falls if that leads her to supply Good 2 instead

old pricese2i

e1i

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Change in Utility from Price

Change

ei

Good 1

Good 2

e1i

e2inew prices

21

However, the more usual case is that the utility of a

supplier of Good 1 falls when its price falls

old prices

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Effect of Price Change on Demand

with Goods Endowment

Notice that with a goods endowment, demand for agood can rise when its price rises even when it is a

normal good (unlike consumer theory with money

endowment, where only inferior goods can be Giffen)

In our first example of a price change, both goods arenormal, yet when the price of Good 1 rises, the

consumer consumes more of it. The reason is that a

supplier of Good 1 becomes wealthier when its pricerises, so the income effect works in the opposite

direction than you are accustomed

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Market Excess Demand

The market excess demand for good j=1,2 at prices p is

the sum of all N consumers excess demands forthat good:i=1,2,...,N(xji(p;ei)-eji)

When i=1,2,...,N(xji(p;ei)-eji)>0, then there is excessdemandfor Goodj at prices p

When i=1,2,...,N(xji(p;ei)-eji)

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Trading in Exchange Economy

In an exchange economy, Consumer ibegins byowning her endowment and ends by owning anallocation yi=(y1i,y2i)0 that specifies the (non-negative)quantity that she consumes of each of the goods

An allocation for all N consumers in the economy

(y11,y21; y12,y22;...;y1N,y2N)

specifies each consumers consumption of each good

NB Consumer is demand (x1i(p;ei), x2i(p;ei)) is a function ofprices and endowment, whereas her allocation (y1i,y2i) is aconstant!

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G l C

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General Competitive

Equilibrium

A General Competitive (or Walrasian) Equilibrium consistsof a price vectorp= (p1,p2) and allocation (y11,y21;

y12,y22;...;y1N,y2N) satisfying the following properties:

1. Utility Maximisation: Each consumer i maximises utility

within her budget set as defined by the price vector pand her endowment eiby choosing (y1i,y2i); that is,

(y1i

,y2i

)=(x1i

(p;ei

), x2i

(p;ei

))2.Market Clearing: given the price vector p, supply equals

demand:i=1,2,...,N(x1i(p;ei)-e1i)= i=1,2,...,N(x2i(p;ei)-e2i) =025

Leon Walras

(1834-1910)

I l i i ilib i

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In a general competitive equilibriu

of an exchange economy1. Each consumer consumes the bundle of goods within

her budget set (as defined by her endowment and

equilibrium prices) that gives her the most utility

2. Market supply (total endowment) of each goodequals market demand at equilibrium prices. (Strictly

speaking, we merely need to require that the supply

of each good be at least as large as its demand: in a

general competitive equilibrium, there may be somegood whose supply exceeds its demand. You may

safely ignore this possibility.)

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Equilibrium

Last term you analysed Nash Equilibrium ingames: in a Nash Equilibrium, no player canimprove her payoffgiven how other playersbehave. In that sense, behaviour is stable

In a general competitive equilibrium, noconsumer can increase her utilitygiven herendowment and market prices

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Equilibrium Reasoning Like game theory, general-equilibrium theory

primarily addresses the question of whichcombinations of allocations and prices constitute

general competitive equilibrium and which do not; it

typically does not address how the economy reaches

a general competitive equilibrium

As general-equilibrium theorists, you should askyourself whether a particular allocation-price

combination is a general equilibrium rather than how

it came to be so

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Market Model

Our notion of general equilibrium is one of many

possible models of trading

It makes strong assumptions that all consumers takemarket prices as given

This assumption most appropriate when eachconsumer is small relative to the market, i.e. when

there are many traders

It also assumes that traders know all market pricesand are free to make any trade they wish subject totheir budget set

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Two-Good, Two-Person

Exchange Economy When the exchange economy consists of

only two consumers and two goods, we can

represent it in a two-dimensional diagramknown as the Edgeworth Box

30

Oxford Economist Francis Edgeworth

(1849-1926)

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Endowments

31

e=(eA,eB)

Person As Origin

e1Ae2A

e1BPerson Bs Origin

e2B

Good 1

G

oo

d

2

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Dimensions of the Edgeworth Box

The Edgeworth Box has length e1A+ e1B and heighte2A+ e2B

Each point in the box corresponds to an allocation,

yA

and yB

, that is non-wasteful:y1A +y1B= e1A +e1B andy2A +y2B= e2A +e2B

At the SW corner of the box, Person B gets all theresources in the economy; at the NE corner Person

A does. Movements to NE thus increase PersonAsshare of total resources

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As Utility Rises to NE

33

e

Person A Origin e1A

e2A

e1B Person B Origin

e2B

Two Indifference Curves

for Person A

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When more is better for Person A, higher indifferencecurves lie to the NE of lower indifference curves

Every possible allocation for Person A, yA=(y1A,y2A)0,lies on some indifference curve for Person A, i.e., shehas indifference curves through bundles that lie outside

the Edgeworth Box in the north or east direction (but

neither south of the Edgeworth box, where y2A

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Bs Utility Rises to SW

35

e

Person A Origin e1A

e2A

e1B Person B Origi

e2B

Two Indifference Curves for Person B

T di A f E d t

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Trading Away from Endowments

Can Raise Both Peoples Utilities

36

e

Person A e1A

e2A

e1B Person B

e2B

Mutually Beneficial Trades

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Prices in the Edgeworth Box

37

e

Person A e1A

e2A

e1B Person B

e2B

Slope=-p1/p2

Budget Line

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38

e

Person A e1A

e2A

e1B Person B

e2B

Slope=-p1/p2

Budget Line

Person As Budget Set Given Endowment and

Prices as Depicted by Budget Line

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As Much of Person Bs Budget

Set as Fits in the Slide

39

Person A e1A

e2A

e1B Person B

e2B

Slope=-p1/p2

Budget Line

P i Whi h Th E i

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Prices at Which There Exists

Excess Demand for Good x

40

e

Person A

Person B

Budget Line

x1A(p;eA)

x2B(p;eB)

x1B(p;eB)

x2A(p;eA)

x1A(p;eA)+x1B(p;eB)> e1A+e1B

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Because there is excess demand for Good 1 atprices p, the economy is not in a generalcompetitive equilibrium at these prices

For markets to clear, the price of Good 1 relativeto Good 2 must rise: the budget line must

become steeper

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Market-Clearing Price

42

e

A

B

Budget Line

x1A(p;eA)

x2A(p;eA)

x1B(p;eB)

x2B(p;eB)

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Two Steps to Finding a General

Competitive Equilibrium1. Find Consumers uncompensated demands as a

function of price (exactly as you did last term

when you take mi=p1e1i+p2e2i)

2. Find price at which Supply=Demand

Shortcut: Because only relative and not absoluteprices matter, you may set the price of a single

good equal to one

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Example

Two goods cherries cand damsons d Utilities uA(xcA,xdA)= xcAxdA and uB(xcB,xdB) = xcBxdB

Consider eA =(2,0), eB =(0,2)

Set the price of damsons = 1 and of cherries = p

You know from last term that consumers with suchCobb-Douglas preferences with equal exponents

for both goods spend half of income on each good.(If you dont, finding demand is good exercise.)

44

A demands x A(p;eA)=(1/2)(pe A+edA)/p

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A demands xc (p;e)=(1/2)(pec +ed )/p=(1/2)(2p)/p=1 and

xdA(p;eA)= (1/2)(pecA+edA)/1=(1/2)(2p)/1=p

B demands xcB(p;eB)=(1/2) (pecB+edB)/p=(1/2)(2)/p=1/p and

xdB(p;eB) =(1/2) (pecB+edB)/1=(1/2)(2)/1=1

Excess demand for cherries is zero whenxcA(p;eA)+ xcB(p;eB)-2=1+(1/p)-2=0

This happens when p=1

Thus, general competitive prices=(1,1) General competitive allocation: ycA= xcA(1,1;eA)=1,ydA= xdA(1,1;eA)=(1/2)(2)/1=1; ycB= xcB(1,1;eB)=(1/2)(2)/1=1and ydB=xdB(1,1;eB)=(1/2)(2)/1=145

C t d f t

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eA

B

xcA(p;eA)

xdA(p;eA)

xcB(p;eB)

xdB(p;eB)y

Consumers trade from e to y

at prices (1,1)

cherries

damsons

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Trade and Welfare

Notice that uA(eA)=uA(2,0)=0; absent trade (inautarky) A gets zero utility. After trade, uA(ycA,ydA)=uA(1,1)=1. Person A strictly benefits from trade. Sotoo does Person B.

In general, consumers must always be weakly better

off after trade than in autarky because they always

have the freedom not to trade. In most cases, like

here, consumers will be strictly better off with trade

Economists tend to believe that moving fromautarky to free trade strictly benefits all parties

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Another example

Let eA =(0,1), eB =(2,0)

Suppose uA=x1A and uB=min{x1B,x2B}

Check that (y1A

,y2A

)=(1,0), (y1B

,y2B

) =(1,1),p=(1,1) is General Competitive Equilibrium

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e

Slope=-p1/p2=-1

As indifference curves in red and Bs in blueA

B

y

Cl l A i i tilit ithi

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Clearly A maximises utility within

budget set by demanding yA

e

Slope=-p1/p2=-1

As indifference curves in red and Bs in blueA

B

y

As budget set

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B maximises utility within

budget set by demandingy

e

Slope=-p1/p2=-1

A

B

y

Bs budget set

Ch i E d m t

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Changing Endowments

Consider a new economy exactly like first one exceptthat As endowment increases: now eA =(0,2)

First, find demands: since A cares only for Good 1,xA(p;eA)= ((p1e1A+ p2e2A)/p1,0)

=(p2e2A

/p1,0)= (2p2/p1,0)

Since B has Leontief (aka Fixed-Proportions) utility,B choosesx1B= x2B, so

xB

(p;eB

) =((p1e1B

+p2e2B

)/(p1+p2),(p1e1B+p2e2B)/(p1+p2))

= (2p1/(p1+p2),2p1/(p1+p2))

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For the market for Good 1 to clear,x1A(p;eA)+ x1B(p;eB)e1A+e1B=0+2=2

So 2p2/p1+ 2p1/(p1+p2)2 Because relative prices are all that matter in general

equilibrium, we can always set one price equal to

one

Take p1=1

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This gives 2p2+ 2/(1+p2)2, or 2p2(1+p2)+ 22(1+p2)

2p220, which implies that p2=0 Thus p=(1,0). Substituting these prices into

demand givesxA(p;eA) =(0,0), xB(p;eB) =(2,2). Thusp=(1,0), yA=(0,0), yB=(2,2) is our general competitiveequilibrium.

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e

lope=-p1/p2=-

As indifference curves in red and Bs in blueA

B

y

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Comparing Static Exchange

Economies In the first economy (where eA =(0,1)), the generalcompetitive equilibrium hasyA =(1,0), and

uA(yA)=1+0=1.

In the second economy (where eA =(0,2)), the generalcompetitive equilibrium hasyA =(0,0), anduA(yA)=0+0=0.

As endowment has increased (with everything elsethe same), yet her utility falls!