Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Consumer: Aggregation

MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis

Frank Cowell Frank Cowell

Almost essential

Firm: Optimisation

Consumption: Basics

Almost essential

Firm: Optimisation

Consumption: Basics

PrerequisitesPrerequisites

October 2006October 2006

Frank C

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Frank C

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icroeconomics

Microeconom

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Use of consumer models is often simplified... (1) We usually suppose that a many-dimensioned (1) We usually suppose that a many-dimensioned

commodity space can be represented appropriately commodity space can be represented appropriately in terms of just a few commodities. in terms of just a few commodities. Requires Requires aggregation over goodsaggregation over goods

(2) We often assume that there is a “representative (2) We often assume that there is a “representative consumer.”consumer.” Requires Requires aggregation over consumersaggregation over consumers

We can use economic analysis to see whether and We can use economic analysis to see whether and when these two simplifications are appropriatewhen these two simplifications are appropriate

Frank C

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Frank C

owell: M

icroeconomics

Microeconom

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Aggregation over goods: the issue

Why Why n n goods?goods? What determines the boundaries between goods?What determines the boundaries between goods? Diagrams all with 2 goods.Diagrams all with 2 goods.

Is this valid?Is this valid? What assumptions are we making?What assumptions are we making?

Is it legitimate to simplify the Is it legitimate to simplify the nn-commodity -commodity problem to, say, a 2-commodity problem?problem to, say, a 2-commodity problem?

Frank C

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Frank C

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icroeconomics

Microeconom

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Aggregation over goods: the model

Use the standard preference model with Use the standard preference model with nn goods. goods. Find an aggregate Find an aggregate xx and a functionand a functionUU((xx11,,x x ) )

that yield the same behaviour as that yield the same behaviour as UU((xx11, x, x22,,xx33,..., ,...,

xxnn))

Then we can say that Then we can say that UU((•,••,•) also exactly ) also exactly represents the consumer’s preferences.represents the consumer’s preferences.

The aggregation problem is then solved.The aggregation problem is then solved.

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Aggregation over goods: result

The “composite commodity” theorem: The “composite commodity” theorem: You can always aggregate over goods 2,3,..,You can always aggregate over goods 2,3,..,nn if if

relative prices of goods 2,3,..,relative prices of goods 2,3,..,nn stay constant. stay constant. UU((•, •, …,••, •, …,•) and) andUU((•,••,•) ) then represent the same then represent the same

preferencespreferences Clearly this can be done for any arbitrary group of Clearly this can be done for any arbitrary group of

commodities. commodities. You just need the condition on relative pricesYou just need the condition on relative prices

Frank C

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Frank C

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icroeconomics

Microeconom

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Aggregation over consumers

We need to model the behaviour of We need to model the behaviour of nnhh consumers. consumers. Consumer Consumer hh has utility function has utility function UUhh and income and income yyhh . . From this get demand for good From this get demand for good ii in usual way, in usual way,

given prices given prices pp.. DDhihi((pp, , yyhh). ).

If all goods are “private” we can easily get total If all goods are “private” we can easily get total demand for demand for ii . . Just add up over the Just add up over the DDhihi

Let’s look at the simple mechanics.Let’s look at the simple mechanics.Alf and BillAlf and Bill

Frank C

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Frank C

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icroeconomics

Microeconom

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Aggregation of consumer demand

Alf Bill The Market

p1

x1a x1

b x1

p1p1

Alf’s demand curve for good 1. Bill’s demand curve for good 1. Pick any price Sum of consumers’ demand Repeat to get the market demand curve

Frank C

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Frank C

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icroeconomics

Microeconom

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Aggregation over consumers: the issues Demand for good Demand for good i i by each consumer by each consumer hh depends depends

on prices on prices pp and income and income yyhh.. Aggregation problems could arise as with firms.Aggregation problems could arise as with firms. But main issue is: will the mass of consumers But main issue is: will the mass of consumers

behave in the same way as a single consumer?behave in the same way as a single consumer? In general market demand will depend on the In general market demand will depend on the

distributiondistribution of incomes of incomes yyhh.. Can we write average demand asCan we write average demand asDDii((pp,,y y ), say?), say?

For exampleFor exampley y could be average income in the could be average income in the market.market.

Just take the mean over the consumersJust take the mean over the consumers We can do this only in special cases…We can do this only in special cases…

Graphical illustration

Graphical illustration

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Frank C

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icroeconomics

Microeconom

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Aggregable demand functionsxi

h

aih(p)

bi(p)

Dhi (p,yh )

yh

h’s demand for good i.

Plot demand against income.

Must be linear in income.

“subsistence minimum” demand

Intercept could differ amongst the h.

Relation of demand to income

Slope must be the same for all h.

Frank C

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Frank C

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icroeconomics

Microeconom

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Aggregability

Aggregable demands require restrictions on Aggregable demands require restrictions on income effects.income effects. In our case average demand depends on average In our case average demand depends on average

income.income. Must have demand that is linearMust have demand that is linear

in income, with the same slope for all.in income, with the same slope for all.

Implies restrictions on preferencesImplies restrictions on preferences But what could happen if this condition were not But what could happen if this condition were not

satisfied? satisfied? Let's consider an example...Let's consider an example... Alf and Bill

again

Alf and Bill again

Frank C

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Frank C

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icroeconomics

Microeconom

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A consumer (Alf)

x1

x2

Preferences look like this.

Prices then change thus.

Initial prices determine budget constraint.

Equilibrium moves from A to A'

A'A' AA

Frank C

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Frank C

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icroeconomics

Microeconom

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B'B'BB

Another consumer (Bill)

x1

x2

Different preferences, (and resources) but....

Prices then change the same way as for Alf

Faces the same initial prices.

Equilibrium moves from B to B'

Frank C

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Frank C

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icroeconomics

Microeconom

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Alf and Bill combined

x1

x2

Initial equilibrium

In the aggregate WARP does not hold (!)

New equilibrium.

•

A'&B'

A&B •

Each consumer’s behaviour is conventional.

Each individual satisfies WARP.

But the joint behaviour does not satisfy WARP.

This is because Alf and Bill are “too dissimilar” in their preferences.

Frank C

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Frank C

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icroeconomics

Microeconom

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What next?

Integrate production and consumption Integrate production and consumption decisions.decisions.

Examine behaviour in Examine behaviour in general equilibriumgeneral equilibrium