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Overview...

The setting

Budget sets

Revealed Preference

Axiomatic Approach

Consumption: Basics

The environment for the basic consumer optimisation problem.

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A method of analysis

Some treatments of micro-economics Some treatments of micro-economics handle consumer analysis first.handle consumer analysis first.

But we have gone through the theory of the But we have gone through the theory of the firm first for a good reason:firm first for a good reason:

We can learn a lot from the ideas and We can learn a lot from the ideas and techniques in the theory of the firm…techniques in the theory of the firm…

… …and reuse them.and reuse them.

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Reusing results from the firm

What could we learn from the way we analysed What could we learn from the way we analysed the firm....?the firm....?

How to set up the description of the environment.How to set up the description of the environment. How to model optimisation problems.How to model optimisation problems. How solutions may be carried over from one How solutions may be carried over from one

problem to the otherproblem to the other ...and more ....and more .

Begin with notation

Begin with notation

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Notation Quantities

xi •amount of commodity i

x = (x1, x2 , ..., xn) •commodity vector

•consumption setX

Prices

pi •price of commodity i

p = (p1 , p2 ,..., pn) •price vector

•incomey

x X denotes feasibility

x X denotes feasibility

a “basket of goods

a “basket of goods

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Things that shape the consumer's problem The set The set X and the number and the number yy are both are both

important.important. But they are associated with two distinct But they are associated with two distinct

types of constraint.types of constraint. We'll save We'll save yy for later and handle for later and handle X now. now. (And we haven't said anything yet about (And we haven't said anything yet about

objectives...) objectives...)

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The consumption set

The set The set X describes the basic entities of the describes the basic entities of the consumption problem.consumption problem.

Not a description of the consumer’s opportunities.Not a description of the consumer’s opportunities. That comes later.That comes later.

Use it to make clear the type of choice problem we Use it to make clear the type of choice problem we are dealing with; for example:are dealing with; for example: Discrete versus continuous choice (refrigerators vs. Discrete versus continuous choice (refrigerators vs.

contents of refrigerators)contents of refrigerators) Is negative consumption ruled out?Is negative consumption ruled out?

““xx X ” means “ ” means “xx belongs the set of logically belongs the set of logically feasible baskets.”feasible baskets.”

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The set X: standard assumptions

x1

Axes indicate quantities of the two goods x1 and x2.x2

Usually assume that X consists of the whole non-negative orthant.

Zero consumptions make good economic senseBut negative consumptions ruled out by definition

no points here…

no points here…

…or here…or here

Consumption goods are (theoretically) divisible…

…and indefinitely extendable…

But only in the ++ direction

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Rules out this case...

x1

Consumption set X consists of a countable number of points

x2

Conventional assumption does not allow for indivisible objects.

But suitably modified assumptions may be appropriate

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... and this

x1

Consumption set X has holes in itx2

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... and this

x1

Consumption set X has the restriction x1 < xx2

Conventional assumption does not allow for physical upper bounds

But there are several economic applications where this is relevant

ˉ

x̄

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Overview...

The setting

Budget sets

Revealed Preference

Axiomatic Approach

Consumption: Basics

Budget constraints: prices, incomes and resources

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The budget constraint

x1

Slope is determined by price ratio.

x2

Two important subcases determined by

1. … amount of money income y.

2. …vector of resources Rp1 – __p2

p1 – __p2

The budget constraint typically looks like this

“Distance out” of budget line fixed by income or resources

Let’s seeLet’s see

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Case 1: fixed nominal income

x1

x2

Budget constraint determined by the two end-points Examine the effect of changing p1 by “swinging” the boundary thus…

y . ._

_p2

y . ._

_p2

y . ._

_p1

y . ._

_p1

Budget constraint is

n

pixi ≤ yi=1

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x1

x2

Case 2: fixed resource endowment

R

n

y = piRi i=1

n

y = piRi i=1

Budget constraint determined by location of “resources” endowment R. Examine the effect of changing p1 by “swinging” the boundary thus…

Budget constraint is

n n

pixi ≤ piRi i=1 i=1

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Budget constraint: Key points

Slope of the budget constraint given by price ratio.Slope of the budget constraint given by price ratio. There is more than one way of specifying There is more than one way of specifying

“income”:“income”: Determined exogenously as an amount Determined exogenously as an amount yy.. Determined endogenously from resources.Determined endogenously from resources.

The exact specification can affect behaviour when The exact specification can affect behaviour when prices change.prices change. Take care when income is endogenous. Take care when income is endogenous. Value of income is determined by prices.Value of income is determined by prices.

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Overview...

The setting

Budget sets

Revealed Preference

Axiomatic Approach

Consumption: Basics

Deducing preference from market behaviour?

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A basic problem

In the case of the firm we have an observable In the case of the firm we have an observable constraint set (input requirement set)…constraint set (input requirement set)…

……and we can reasonably assume an obvious and we can reasonably assume an obvious objective function (profits)objective function (profits)

But, for the consumer it is more difficult.But, for the consumer it is more difficult. We have an observable constraint set (budget set)We have an observable constraint set (budget set)

…… But what objective function?But what objective function?

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The Axiomatic Approach

We could “invent” an objective function.We could “invent” an objective function. This is more reasonable than it may sound:This is more reasonable than it may sound:

It is the standard approach.It is the standard approach. See later in this presentation.See later in this presentation.

But some argue that we should only use what we But some argue that we should only use what we can observe:can observe: Test from market data? Test from market data? The “revealed preference” approach.The “revealed preference” approach. Deal with this now.Deal with this now.

Could we develop a coherent theory on this basis Could we develop a coherent theory on this basis alone?alone?

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Using observables only

Model the opportunities faced by a Model the opportunities faced by a consumer.consumer.

Observe the choices made.Observe the choices made. Introduce some minimal “consistency” Introduce some minimal “consistency”

axioms.axioms. Use them to derive testable predictions Use them to derive testable predictions

about consumer behaviourabout consumer behaviour

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x1

x2

“Revealed Preference”

x

Let market prices determine a person's budget constraint..

Suppose the person chooses bundle x...

Use this to introduce Revealed Preference

x′

x is revealed preferred to all these points.

x is revealed preferred to all these points.

For example x is revealed preferred to x′

For example x is revealed preferred to x′

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Axioms of Revealed PreferenceAxiom of Rational Choice

the consumer always makes a choice, and selects the most preferred bundle that is available.

Essential if observations are to have meaning

Weak Axiom of Revealed Preference (WARP)

If x RP x' then x' not-RP x.

If x was chosen when x' was available then x' can never be chosen whenever x is available

WARP is more powerful than might be thought

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WARP in the market

Suppose that x is chosen when prices are p. If x' is also affordable at p then:

Now suppose x' is chosen at prices p' This must mean that x is not affordable at p':

graphical interpretation

graphical interpretationOtherwise it would

violate WARP

Otherwise it would violate WARP

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x1

x2

WARP in action Take the original equilibrium

Now let the prices change...

WARP rules out some points as possible solutions

x

x′

x°Monday's

prices

Tuesday's

prices

Clearly WARP induces a kind of negative substitution effect

But could we extend this idea...?

Could we have chosen x° on Monday? x° violates WARP; x does not.

Could we have chosen x° on Monday? x° violates WARP; x does not.

Tuesday's choice:On Monday we could have afforded Tuesday’s bundle

Tuesday's choice:On Monday we could have afforded Tuesday’s bundle

Monday's choice:

Monday's choice:

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Trying to Extend WARP

x1

x2

x

x'

x''

Take the basic idea of revealed preference

Invoke revealed preference again

Invoke revealed preference yet again

Draw the “envelope”

Is this an “indifference curve”...?

No. Why?

x is revealed preferred to all these points.

x is revealed preferred to all these points.

x' is revealed preferred to all these points.

x' is revealed preferred to all these points.

x″ is revealed preferred to all these points.

x″ is revealed preferred to all these points.

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Limitations of WARPWARP rules out this pattern

...but not this

WARP does not rule out cycles of preference

You need an extra axiom to progress further on this:

the strong axiom of revealed preference.

x″′

x′x

x″

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Revealed Preference: is it useful?

You can get a lot from just a little:You can get a lot from just a little: You can even work out substitution effects.You can even work out substitution effects.

WARP provides a simple consistency test:WARP provides a simple consistency test: Useful when considering consumers en masse.Useful when considering consumers en masse. WARP will be used in this way later on.WARP will be used in this way later on.

You do not need any special assumptions You do not need any special assumptions about consumer's motives:about consumer's motives: But that's what we're going to try right now.But that's what we're going to try right now. It’s time to look at the mainstream It’s time to look at the mainstream modellingmodelling of of

preferences.preferences.

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Overview...

The setting

Budget sets

Revealed Preference

Axiomatic Approach

Consumption: Basics

Standard approach to modelling preferences

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The Axiomatic Approach

Useful for setting out a priori what we mean Useful for setting out a priori what we mean by consumer preferences. by consumer preferences.

But, be careful... But, be careful... ...axioms can't be “right” or “wrong,”......axioms can't be “right” or “wrong,”... ... although they could be inappropriate or ... although they could be inappropriate or

over-restrictive.over-restrictive. That depends on what you want to model.That depends on what you want to model. Let's start with the basic relation...Let's start with the basic relation...

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The (weak) preference relation

The basic weak-preference relation:

x x'

"Basket x is regarded as at least as good as basket x' ..."

…and the strict preference relation…

x x'

“ x x' ” and not “ x' x. ”

From this we can derive the

indifference relation.

x x'

“ x x' ” and “ x' x. ”

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Fundamental preference axioms

CompletenessCompleteness

TransitivityTransitivity

ContinuityContinuity

GreedGreed

(Strict) Quasi-concavity(Strict) Quasi-concavity

SmoothnessSmoothness

For every x, x' X either x<x' is true, or x'<x is true, or both statements are true

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Fundamental preference axioms

CompletenessCompleteness

TransitivityTransitivity

ContinuityContinuity

Greed: local non-satiationGreed: local non-satiation

(Strict) Quasi-concavity(Strict) Quasi-concavity

SmoothnessSmoothness

For all x, x' , x″ X if x x' and x‘ x″ then x x’’

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Fundamental preference axioms

CompletenessCompleteness

TransitivityTransitivity

ContinuityContinuity

GreedGreed

(Strict) Quasi-concavity(Strict) Quasi-concavity

SmoothnessSmoothness

For all x' X the not-better-than-x' set and the not-worse-than-x' set are closed in X

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x1

x2

Better than x ?

Better than x ?

Continuity: an example Take consumption bundle x°. Construct two other bundles, xL with Less than x°, xM with More

There is a set of points like xL, and a set like xM

Draw a path joining xL , xM.

If there’s no “jump”…

“No jump” means no inverse preference, that is to say, a sequence

then

The indifference curve

The indifference curve

x°

Worse than x?

Worse than x?

xL

xM

but what about the boundary points between the two?

but what about the boundary points between the two?

do we jump straight from a point marked “better” to one marked “worse"?

do we jump straight from a point marked “better” to one marked “worse"?

n nx ylim limn n

n nx y

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Axioms 1 to 3 are crucial ...

The utilityfunction

completeness

transitivity

continuity

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A continuous utility function then represents preferences...

U(x) U(x')x x'

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Notes on utility That is ,U measures all the objects of choice That is ,U measures all the objects of choice

on a numerical scale, and a higher measure on a numerical scale, and a higher measure on the scale means the consumer like the on the scale means the consumer like the object more. It’s typical to refer to a object more. It’s typical to refer to a function . Utility is a fn. for the consumer .function . Utility is a fn. for the consumer .

Why would we want to a know whether Why would we want to a know whether preference has a numerical representation ? preference has a numerical representation ? Essential, it is convenient in application Essential, it is convenient in application

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to work with utility fn. It is relativelyto work with utility fn. It is relatively

easy to specify a consumer’s easy to specify a consumer’s

preference by writing down a utilitypreference by writing down a utility

function. function. And we can turn a choice And we can turn a choice

problem into a numerical maximization problem into a numerical maximization

problem.problem. That is, if preference has That is, if preference has

numerical representation U, then the numerical representation U, then the

““best” alternatives out of a set best” alternatives out of a set

according to are precisely those according to are precisely those A X

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element of A that has the element of A that has the maximum utility. If we are lucky maximum utility. If we are lucky enough to know that the utility enough to know that the utility function U, and the set A fromfunction U, and the set A fromwhich choice is made are “nicely which choice is made are “nicely behaved” (eg. U is differentiable and behaved” (eg. U is differentiable and A is a convex compact set), then we A is a convex compact set), then we can think of applying the technique can think of applying the technique

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of optimization theory to solve this of optimization theory to solve this

choice problem.choice problem.

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Tricks with utility functions

UU-functions represent preference -functions represent preference orderingsorderings..

So the utility scales don’t matter.So the utility scales don’t matter. And you can transform the And you can transform the UU-function in -function in

any (monotonic) way you want...any (monotonic) way you want...

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Irrelevance of cardinalisation So take any utility function...

And, for any monotone increasing φ, this represents the same preferences.

…and so do both of these

φ( U(x1, x2,..., xn) )

U(x1, x2,..., xn)

( U(x1, x2,..., xn) )

exp( U(x1, x2,..., xn) )

This transformation represents the same preferences...

log( U(x1, x2,..., xn) )

U is defined up to a monotonic transformation

Each of these forms will generate the same contours.

Let’s view this graphically.

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A utility function

0x2

x1

Take a slice at given utility level Project down to get contours

U(x1,x2)U(x1,x2)

The indifference curve

The indifference curve

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Another utility function

0x2

x1

Again take a slice… Project down … U*(x1,x2)U*(x1,x2)

The same indifference curve

The same indifference curve

By construction U* = φ(U)

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Assumptions to give the U-function shape CompletenessCompleteness

TransitivityTransitivity

ContinuityContinuity

GreedGreed

(Strict) Quasi-concavity(Strict) Quasi-concavity

SmoothnessSmoothness

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The greed axiom

x1

increa

sing

prefer

ence

Pick any consumption bundle in X.

x'

Gives a clear “North-East” direction of preference.

x2

What can happen if consumers are not greedy

increasing

preference

B

Increasing

preference

Increasingpreference

increasingpreference

Greed implies that these bundles are preferred to x'.

Bliss!

Greed: utility function is monotonic

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A key mathematical concept We’ve previously used the concept of concavity:We’ve previously used the concept of concavity:

Shape of the production function.Shape of the production function. But here simple concavity is inappropriate:But here simple concavity is inappropriate:

The The UU-function is defined only up to a monotonic transformation. -function is defined only up to a monotonic transformation. UU may be concave and may be concave and UU22 non-concave even though they represent non-concave even though they represent

the same preferences.the same preferences. So we use the concept of “quasi-concavity”:So we use the concept of “quasi-concavity”:

““Quasi-concave” is equivalently known as “concave contoured”.Quasi-concave” is equivalently known as “concave contoured”. A concave-contoured function has the same contours as a concave A concave-contoured function has the same contours as a concave

function (the above example).function (the above example). Somewhat confusingly, when you draw the IC in (Somewhat confusingly, when you draw the IC in (xx11,,xx22)-space, )-space,

common parlance describes these as “convex to the origin.”common parlance describes these as “convex to the origin.” It’s important to get your head round this:It’s important to get your head round this:

Some examples of ICs coming up…Some examples of ICs coming up…

Review ExampleReview

Example

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sometimes these assumptions can

be relaxed

sometimes these assumptions can

be relaxed

ICs are smooth

…and strictly concaved-contoured

I.e. strictly quasiconcave

Pick two points on the same indifference curve.

x1

x2

Draw the line joining them. Any interior point must line on a higher indifference curve

Conventionally shaped indifference curves

(-) Slope is the Marginal Rate of Substitution

U1(x) . —— .U2 (x) .

(-) Slope is the Marginal Rate of Substitution

U1(x) . —— .U2 (x) .

increa

sing

prefer

ence

C

A

B

Slope well-defined everywhere

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Other types of IC: Kinks

x1

x2

Strictly quasiconcave

C

A

B

But not everywhere smooth

MRS not defined here

MRS not defined here

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Other types of IC: not strictly quasiconcave

x2

Slope well-defined everywhere

Indifference curves with flat sections make sense

But may be a little harder to work with...

C

A

B

utility here lower than at A or B

utility here lower than at A or B

Not quasiconcave

Quasiconcave but not strictly quasiconcave

x1

Indifference curve follows axis here

Indifference curve follows axis here

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Summary: why preferences can be a problem Unlike firms there is no “obvious” objective Unlike firms there is no “obvious” objective

function.function. Unlike firms there is no observable Unlike firms there is no observable

objective function. objective function. And who is to say what constitutes a And who is to say what constitutes a

“good” assumption about preferences...?“good” assumption about preferences...?

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Review: basic concepts

Consumer’s environmentConsumer’s environment How budget sets workHow budget sets work WARP and its meaning WARP and its meaning Axioms that give you a utility functionAxioms that give you a utility function Axioms that determine its shapeAxioms that determine its shape

ReviewReview

ReviewReview

ReviewReview

ReviewReview

ReviewReview

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What next?

Setting up consumer’s optimisation Setting up consumer’s optimisation problemproblem

Comparison with that of the firmComparison with that of the firm Solution concepts.Solution concepts.