the budget constraint and choice the problem of limited resources and its effect on choice

The budget constraint and choice

The problem of limited resources and its effect on choice

The budget constraint and choice

Last week: We saw that preferences can be represented

by utility functions ... That indifference curves can be used to map a

utility function into “consumption space” But we still don’t know how consumers choose

amongst the different bundles... This week:

We introduce the concept of a budget, This is the 2nd half of consumer theory

The budget constraint and choice

The budget constraint

The optimal consumer choice

Income and substitution effects

The budget constraint

The basic concept is really straightforward:

The consumer has a limited income (I) to purchase different goods

Each type of good has a defined price (p) per unit

We assume that the consumer does not save and spends all his income This possibility will be examined later

The budget constraint

The general budget constraint for n goods is:

If we only look at 2 goods (Same simplification as last week), it can be expressed as:

n

i ii 1

I p x

1 1 2 2I p x p x

The budget constraint

Imagine the following “student entertainment budget” You have 50 € The price of a meal is 10 € The price of a cinema ticket is 5 €

Your budget constraint is:

50 5 tickets 10 meals 1 1 2 2I p x p x

The budget constraint

Meals

Cinema

maxmealx

maxmeal

meal

Ix

p Maximum amount of

meals you can buy

Diagram in “consumption space”

The budget constraint

maxcin.

cin.

Ix

p

Maximum amount of cinema tickets you can buy

maxcin.x Cinema

Meals

maxmealx

The budget constraint

maxcin.x Cinema

Meals

maxmealx

Budget constraint

The budget constraint

maxcin.x Cinema

Meals

maxmealx

1 1 2 2I p x p x

21 2

1 1

pIx x

p p

The budget constraint is

Dividing by p1 and rearranging:

cin.meal cin.

meal meal

pIx x

p p

slope

intercept

The budget constraint

Any bundle within the budget constraint is affordable , but not all the budget is spent (C,D).

Any bundle beyond the budget constraint cannot be afforded (H,G).

C

H

D

GAny bundle on the budget constraint is affordable and ensures all the budget is spent (E,F).

maxcin.x Cinema

Meals

maxmealx

F

E

The budget constraint

maxcin.x Cinema

Meals

maxmealx

Budget constraint

Budget set

The budget constraint

The position of the budget constraint depends on

The income of the agent (I)

The price of the two goods (p1 and p2)

21 2

1 1

pIx x

p p

The budget constraint

maxcin.x Cinema

Meals

maxmealx

Effect of a fall in income (I)

The budget constraint

maxcin.x Cinema

Meals

maxmealx

Increase in the price of cinema tickets

The budget constraint and choice

The budget constraint

The optimal consumer choice

Income and substitution effects

The optimal consumer choice

This requires bringing in the two elements of the theory The indifference curves, which show how

agents rank the different bundles The budget constraint, which shows which

bundles are affordable, and which are not

Both of these are defined over the “consumption space”, so they can be superposed easily

The optimal consumer choice

maxcin.x Cinema

Meals

maxmealx

Which is the best bundle ?

F

Optimal bundleC

D

E

B

A

The optimal consumer choice

maxcin.x Cinema

Meals

maxmealx

The budget constraint is tangent to the

indifference curve at F

F

Definition of the MRS at F !!!

The optimal consumer choice

The optimal bundle is on the tangency between the budget constraint and the indifference curve.

This means that for the optimal bundle the slope of the IC is equal to the slope of the budget constraint

MRS = ratio of prices

The optimal consumer choice

This condition gives a central result of consumer theory:

The optimal bundle is the one which equalises the marginal utility per € spent If you were to receive an extra € of income,

your marginal utility will be the same regardless of where you spend it

2 2

1

2

1 21

1mU pMRS

mU p

mU mU

p p

The optimal consumer choice

maxcin.x Cinema

Meals

maxmealx

Example of optimal choice with concave preferences

F

G

The optimal solution is a “corner solution”

The budget constraint and choice

The budget constraint

The optimal consumer choice

Income and substitution effects

Income and substitution effects

Consumer theory is used to understand how choice is affected by changes in the environment

These can be complex, and the theory helps to isolate these different effects

The separation of income and substitution effects is a good illustration of the concept of “ceteris paribus” Each variable is isolated and analysed

separately from the others

Income and substitution effects

maxcin.x Cinema

Meals

maxmealx

An increase in the price of cinema tickets has 2 effects :

A

1: A change in real income A previously affordable bundle

(A) is no longer affordable

2: A relative price change The slope of the budget

constraint changes, and meals become relatively cheaper

Income and substitution effects

maxcin.x Cinema

Meals

maxmealx

A

B

Effect of an increase in the price of cinema tickets on consumer choice

Fall in the consumption of cinema

Increase in the consumption of meals

Question: How can we separate the effect of the change in real income from the effect of the change in relative prices ?

Income and substitution effects

maxcin.x Cinema

Meals

maxmealx

A

B

In order to separate the 2 effects, we add an imaginary budget constraint

Parallel to the new budget constraint

Tangent to the original IC

There is only a single curve that satisfies these two requirements

This gives an imaginary optimal bundle (Im)

Im

Income and substitution effects

maxcin.x Cinema

Meals

maxmealx

A

B

The substitution effect From A to Im, real income is

held constant We are still on the same

indifference curve, so utility is the same

The change of bundle is due entirely to the change in relative price

This is the substitution effect

Im

Income and substitution effects

maxcin.x Cinema

Meals

maxmealx

A

B

The income effect From Im, to B, relative prices

are held constant The two budget constraints are

parallel, so the slope is the same The change of bundle is due

entirely to the fall in income. This is the income effect

Im

Income and substitution effects

maxcin.x Cinema

Meals

maxmealx

A

B

The overall effect By combining the two, one gets

the overall effect One can see that the interaction

is different for the two goods The 2 effects can work against

each other, or add up Depending on the relative

strength of the effects, this can lead to increases or falls in consumption

Im

Income and substitution effects

This type of approach is fundamental to micro-economic analysis Any price change is always accompanied by

income and substitution effects. So this helps understand the effects of

taxation, shocks to prices, taste changes, etc. Look at the complex effects of oil price

increases on consumption Price change ⇒ Complex change in bundle

Clearly, this will also help understand how demand curves are built (next week)