Budget constrain

The budget constraint and choice

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 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


• 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


Meals

Cinema

maxmealx

maxmeal

meal

Ix

p Maximum amount of

meals you can buy

Diagram in “consumption space”


maxcin.

cin.

Ix

p

Maximum amount of cinema tickets you can buy

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Budget constraint


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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 constraintAny 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).

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F

E


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Budget constraint

Budget set


• 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


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Effect of a fall in income (I)


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Effect of a rise in income (I)


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Increase in the price of cinema tickets


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Decrease in the price of cinema tickets

Lets think about these things

• Budget constraint equation• Slope of budget line – Px / Py• budget line is of 45 ° if???• Shifts in line can be caused by???• Shows purchasing power of indivisuals• Budget line and what it represents

Questions

A budget constrainta.shows the prices that a consumer chooses to pay

for products he consumes.b.shows the purchases made by consumers.c. shows the consumption bundles that a consumer

can afford.d.represents the consumption bundles that give a

consumer equal satisfaction.

A consumer that doesn't spend all of her incomea.would be at a point outside of her budget

constraint.b.would be at a point inside her budget constraint.c. must not be consuming positive quantities of all

goods.d.must be consuming at a point where her budget

constraint touches one of the axes.

The following diagram shows a budget constraint for a particular consumer.If the price of x is $10, what is the price of y?a.$15b.$25c. $35d.$70

Which of the graphs in the figure reflects a decrease in the price of good X only?

a. graph (a)b.graph (b)c. graph (c)d.graph (d)

Which of the graphs in the figure reflects an increase in the price of good Y only?

a. graph (a)b. graph (b)c. graph (c)d. graph (d)

Which of the graphs in the figure could reflect a decrease in the prices of both goods?

a. graph (a)b. graph (b)c. graph (c)d. graph (d)

• A halving of the prices good A and good B has the same effect on the budget line as doubling the income. Is this sentence true or false? Show it by using the equation of the budget line.

• What about the opposite? Doubling the prices good A and good B has the same effect on the budget line as halving of the income.

Suppose the price of pizza is $10, the price of cola is $1, and the consumer’s income is $50. In addition, suppose the consumer’s budget constraint measures pizza on the horizontal axis and cola on the vertical axis.

1) If the price of cola doubles to $2, then thea. budget constraint intersects the vertical axis at 25 colas.b. slope of the budget constraint rises to -2.c. budget constraint intersects the vertical axis at 100 colas.d. budget constraint shifts inward in a parallel fashion.

2) If the consumer's income rises to $60, then the budget line for pizza and cola would

a. now intersect the horizontal axis at 6 pizzas and the vertical axis at 60 colas.

b. not change.c. now intersect the horizontal axis at 4 pizzas and the vertical axis at 16

colas.d. rotate outward along the cola axis.

Answer the following questions based on the table. A consumer is able to consume the following bundles of rice and beans when the price of rice is $2 and the price of beans is $3.

RICE BEANS12 06 40 8

a. How much is this consumer's income?b. Draw a budget constraint given this information. Label it B.c. Construct a new budget constraint showing the change if the price of

rice falls $1. Label this C.d. Given the original prices for rice ($2) and beans ($3), construct a new

budget constraint if this consumer's income increased to $48. Label this D.

Draw a budget constraint that is consistent with the following prices and income.Income = 200PY = 50

PX = 25

a. Demonstrate how your original budget constraint would change if income increases to 500.

b. Demonstrate how your original budget constraint would change if PY decreases to 20.

c. Demonstrate how your original budget constraint would change if PX increases to 40.

Evaluate the following statement, "Warren Buffet is the second richest person in the world. He doesn't face any constraint on his ability to purchase commodities he wants."



The optimal consumer choice

Income and substitution effects


• 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


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Which is the best bundle ?

F

Optimal bundleC

D

E

B

A


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The budget constraint is tangent to the

indifference curve at F

F

Definition of the MRS at F !!!


• 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


• 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


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Example of optimal choice with concave preferences

F

G

The optimal solution is a “corner solution”






• 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


• 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

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

A


• 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 ?

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A

B

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


• 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)

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A

B

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

Im


• 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

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A

B

The substitution effect

Im


• 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

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A

B

The income effect

Im


• 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

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A

B

The overall effect

Im


• 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)

Economy & Finance

Budget constrain