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Price Theory

Lecture 3: Consumer Choice

Topics for today’s lecture . . .

1. The budget constraint

2. Optimal choice

3. Corner solutions

4. Transfers and vouchers

The budget constraint

Another way of thinking about wealth

It is common to think about an individual’s wealth in terms of the monetary value of her/his

assets.

On the other hand, we recognise that an individual’s welfare (utility) is derived from her/his

consumption of goods and services (amongst other things).

It will be useful to describe a consumer’s wealth in terms of the consumption possibilities that

her/his wealth creates.

The cost of a consumption basket

0 x

Suppose that:

• The price of food (good x) is Px = $20 per meal.

• The price of clothing (good y) is Py = $100 per item.

Basket A, containing xA = 14 meals and

yA = 5 items of clothing, costs,

PxxA +PyyA = 20 × 14 + 100 × 5

= $780.

Definition: Budget constraint

The set of baskets that a consumer can purchase given her/his income and the prevailing

prices.

The budget constraint describes the alternative consumption possibilities that are available to

a consumer.

Affordable consumption baskets

0 x50

Suppose that the consumer’s income is

I = $1000:

• If the consumer spends all of her/his income on food then,

x = I

Px =

1000

20 = 50 meals.

• If the consumer spend all of her/his income on clothing then,

y = I

Py =

1000

100 = 10 items.

The budget line

BL1

0 x50

The equation of the budget constraint is,

Pxx +Pyy ≤ I.

The set of baskets that are only just

affordable is called the budget line.

Baskets that lie below the budget line

(such as A) are affordable.

Baskets that lie above the budget line

(such as B, which costs $1400) are

unaffordable.

The trade-offs along the budget line

BL1

0 x50

The slope of the budget line is the

negative of the ratio of prices,

slope = − Px Py

= − 20

100 = −

5 .

The ratio Px/Py is the number of units of

clothing (good y) that must be given up

in order to obtain an additional unit of

food (good x).

Quiz 1

Each week Sarah spends a total of $100 on coffee and cake. The price of coffee is $4 per

cup, and the price of cake is $12 per slice. If Sarah purchases 7 slices of cake, how many cups

of coffee does she purchase?

(a) 1.

(b) 4.

(d) 25.

Quiz 2

Each week Sarah spends a total of $100 on coffee and cake. The price of coffee is $4 per

cup, and the price of cake is $12 per slice. If Sarah wants to increase by 1 the number of

slices of cake she purchases, without altering the total amount she spends on coffee and

cake, how many cups of coffee must she give up?

(a) 1/3.

(b) 1.

(d) 12.

A fall in income

BL1BL2

0 x25

Suppose that the consumer’s income falls

from $1000 to $500.

Px =

500

20 = 25 meals.

Py =

500

100 = 5 items.

The slope of the budget line does not

change.

The shaded area represents the

consumption possibilities lost when the

consumer’s income falls.

An increase in the price of food

BL2 BL1

0 x50

Suppose that the price of food increases

from $20 to $40 per meal.

Px =

1000

40 = 25 meals.

The y-intercept does not change.

The budget line is steeper

(−Px/Py = −2/5).

The consumer has fewer available

consumption baskets.

Exercise: Inflation

BL1

0 x50

Suppose that Px = $20 per meal,

Py = $100 per item, and I = $1000.

1. On a graph, illustrate how the

consumer’s budget set changes if the

prices of food and clothing both

increase by 10%.

2. What happens to the budget set if the

consumer’s income also increases by

10%.

Exercise solutions

BL1BL2

0 x45

1. The new prices are:

• Px = 1.1 × 20 = $22 meals.

• Py = 1.1 × 100 = $110 items. The corresponding budget line

intercepts are,

Px =

1000

22 = 45.45 meals.

Py =

1000

110 = 9.09 items.

The new budget line connects these

points.

Exercise solutions

BL1 = BL3BL2

0 x45

2. The new income is:

• I = 1.1 × 1000 = $1100. The corresponding budget line

intercepts are,

Px =

1100

22 = 50 meals.

Py =

1100

110 = 10 items.

This is the original budget line.

Optimal choice

Constrained optimisation

BL1 x

A consumer’s objective is to achieve the

highest possible level of utility (reach the

highest available indifference curve).

The budget constraint limits the set of

baskets from which the consumer may

choose.

The optimal choice is the basket that

maximises her/his utility, within her/his

budget constraint.

The optimal consumption basket lies on the budget line

BL1

0 x

Take any basket A that lies below the

budget line.

The more is better assumption implies

that any basket above and to the right of

A delivers a higher level of utility.

Therefore, A cannot be optimal as there

must exist a basket B on the budget line

that is preferred to A.

At the optimum, the indifference curve does not cross the budget line

BL1

0 x

Take any basket A lying on the budget

line, at a point at which an indifference

crosses the budget line.

There must exist a basket B, on the same

indifference curve as A, that lies below

the budget line.

Therefore, A cannot be optimal as the

basket C is preferred over B, and the

consumer is indifferent between B and A.

At the optimum, the indifference curve is tangent to the budget line

BL1

0 x

The optimum must lie on an indifference

curve that touches, but does not cross,

the budget line.

Diminishing marginal rate of substitution

implies that we can find the optimal

basket A where the indifference curve is

tangent to the budget line.

At most one consumption basket can

satisfy these conditions.

Two conditions for an interior optimum

• The optimal consumption basket lies on the budget line, therefore,

Pxx +Pyy = I.

• At the optimum the slope of the indifference curve is equal to the slope of the budget line, therefore,

MUx MUy

= Px Py .

Note: These conditions are predicated on the more is better assumption holding for both

goods, and on a diminishing marginal rate of substitution.

Exercise: Finding the optimum consumption basket

The utility Sarah derives from coffee (good x) and cake (good y), is given by the function

U(x,y) = √ xy. The corresponding marginal utilities are,

MUx =

√ y

2 √ x

and MUy =

√ x

2 √ y .

The price of coffee is Px = $4 per cup, the price of cake is Py = $12 per slice, and Sarah’s

income is I = $120.

1. Derive the equation for Sarah’s budget line.

2. Derive the expression for Sarah’s marginal rate of substitution and use it to construct

Sarah’s tangency condition.

3. Solve the two equations simultaneously to find Sarah’s optimal consumption basket.

Exercise solutions

1. The equation for Sarah’s budget line can be found by substituting prices and income into

the equation,

Pxx +Pyy = I yielding 4x + 12y = 120.

2. Sarah’s marginal rate of substitution can be found by substituting for the marginal utilities,

MRSxy = MUx MUy

√ y

2 √ x ×

2 √ y

√ x

= y

x .

The tangency condition equates the marginal rate of substitution with the ratio of prices,

MUx MUy

= Px Py

or y

x =

12 .

Exercise solutions

3. An interior optimum satisfies the budget line equation and the tangency condition

simultaneously. Begin by rearranging the tangency condition to solve for y (solving for x

would also work), y

x =

12 implying y =

3 .

Now substitute for y into the budget line equation,

4x + 12 × (x

) = 120 ⇒ 8x = 120 ⇒ x = 15.

Now substituting for x into the tangency condition,

y = 15

3 = 5.

Sarah’s optimal consumption basket contains 15 cups of coffee and 5 slices of cake.

Bang for your buck

An alternative way of stating the tangency condition is,

MUx Px

= MUy Py

In words, at an interior optimum the marginal utility per dollar must be equal for all goods.

• The bang for your buck condition must hold for every pair of goods an individual consumes, regardless of the number of goods in her/his consumption basket.

Intuitively, if the marginal utility per dollar is higher for good x than for good y, the consumer

can increase her/his utility by reducing spending on y, and increasing spending on x.

Exercise: Perfect complements

0 x

Suppose left shoes (good x), and right

shoes (good y), are perfect complements

with,

U(x,y) = min{x,y}.

Moreover, Px = $3, Py = $2, and the

consumer’s income is I = $35.

1. What is the optimal consumption

basket?

2. What is the consumer’s utility at the

optimum?

Exercise solutions

0 x

1. A left shoe only creates utility for the

consumer when paired with a right

shoe.

• The cost of a pair of shoes is $5.

• The consumer can afford 7 pairs. Therefore the optimum has x = 7 left

shoes, and y = 7 right shoes.

2. U(7, 7) = min{7, 7} = 7.

Corner solutions

When the tangency condition fails

U1 U2 U3

BL1

0 x

It is possible that no basket on the budget

line satisfies the tangency condition.

The highest indifference curve this

consumer can reach is U3, by choosing

basket A.

This is called a corner solution because it

lies at a corner of the budget set.

A corner solution is only a possibility if

indifference curves intersect one or both

axes.

Bang for your buck in a corner solution

U1 U2 U3

BL1

0 x

At basket A the marginal utility per dollar

is greater for good x than good y,

MUx Px

> MUy Py

Note: This inequality is reversed for a

corner solution on the y-axis.

The consumer would like to exchange

good x for good y, but cannot as she/he

cannot consume fewer than zero units of

good y.

Discussion: Corner solutions

How many different products do you think are sold in your local supermarket?

How many of these products have you, yourself, purchased?

How does marginal utility per dollar (bang for your buck) help explain your behaviour?

Exercise: Identifying a corner solution

The utility Oscar derives from coffee (good x) and cake (good y), is given by the function

U(x,y) = √ x +y/2. The corresponding marginal utilities are,

MUx = 1

2 √ x

and MUy = 1

2 .

The price of coffee is Px = $4 per cup, the price of cake is Py = $12 per slice, and Oscar’s

income is I = $20.

1. Derive the equation for Oscar’s budget line.

2. Derive the expression for Oscar’s marginal rate of substitution and use it to construct

Oscar’s tangency condition.

3. Solve the two equations simultaneously to find Oscar’s optimal consumption basket.

Exercise solutions

1. The equation for Oscar’s budget line can be found by substituting prices and income into

the equation,

Pxx +Pyy = I yielding 4x + 12y = 20.

2. Oscar’s marginal rate of substitution can be found by substituting for the marginal

utilities,

MRSxy = MUx MUy

= 1

2 √ x ×

1 =

1 √ x .

The tangency condition equates the marginal rate of substitution with the ratio of prices,

MUx MUy

= Px Py

or 1 √ x =

12 .

Exercise solutions

3. An interior optimum satisfies the budget line equation and the tangency condition

simultaneously. Begin by solving the tangency condition to solve for x (notice that y is

not present in the MRSxy in this example),

1 √ x =

12 ⇒

√ x = 3 ⇒ x = 9.

Now substitute for x into the budget line equation,

4 × 9 + 12y = 20 ⇒ 12y = −16 ⇒ y = − 4

3 .

It is not possible to consume a negative quantity therefore we must have a corner solution

in which y = 0. To find x substitute for y into the budget line equation,

4x + 12 × 0 = 20 ⇒ 4x = 20 ⇒ x = 5.

Corner solutions and perfect substitutes

BL1

0 x

The perfect substitutes utility function is

U(x,y) = αx + βy, where α and β are

positive constants.

The corresponding marginal utilities are

MUx = α and MUy = β.

An optimising consumer will exclusively

purchase good x if,

Px >

Py .

The optimum is basket A in the figure.

Corner solutions and perfect substitutes

BL1

0 x

An optimising consumer will exclusively

purchase good y if,

Px <

Py .

The optimum is basket A in the figure.

Every basket on the budget line is optimal

if, α

Px =

Py .

This is the only case in which perfect

substitutes has an interior solution.

Quiz 3

Jeff ’s preferences over pizza (good x), and hotdogs (good y), are represented by the perfect

substitutes utility function U(x,y) = 3x + 5y. The corresponding marginal utilities are

MUx = 3 and MUy = 5.

If the price of pizza is Px = $2 per slice, the price of hotdogs is Py = $3 each, and Jeff ’s

income is $60, then Jeff ’s optimal consumption basket contains,

(a) 30 slices of pizza, and no hotdogs.

(b) No slices of pizza, and 20 hotdogs.

(d) 30 slices of pizza, and 20 hotdogs.

Transfers and vouchers

Suppose that, as a part of its commitment to renewable energy, the Government wants to

increase the number of solar panels installed on household rooftops.

The government is considering two policies:

• Transfer $20,000 to each household.

• Provide each household with a voucher redeemable for $20,000 of solar panels. (A voucher cannot be redeemed for any other good.)

Which policy best implements the Government’s objective?

Which policy is best for households?

Definition: Composite good

A good that represents a consumer’s collective expenditures on every good except the

product under consideration.

By aggregating all but one good into a single composite good representing ‘everything else’,

we can reduce the economy to two dimensions.

Joey’s household

Joey has preferences over solar panels (good x) and the composite good (good y)

represented by the utility function U(x,y) = xy.

Joey’s marginal utilities are MUx = y and MUy = x.

Joey’s household income is I = $30,000.

The price of solar panels is Px = $1,000 per panel.

The budget constraint with a composite good

BL1

0 x30

30k

We will follow the convention that

Py = $1 when y is the composite good.

It follows that the vertical axis represents

both the quantity of good y, and total

expenditure on the composite good.

With I = $30,000 and Px = $1000, the

horizontal intercept of Joey’s budget line

is 30 solar panels.

Joey’s optimum with a $20,000 transfer

50k

BL2

25k

15k

30k

BL1

0 x

Absent a transfer or voucher, Joey’s

optimum (basket A) contains x = 15

solar panels, and y = 15,000 units of the

composite good. (You should check this.)

A transfer of $20,000 increases Joey’s

available income to $50,000.

Joey’s optimum with the transfer (basket

B) contains x = 25 solar panels, and

y = 25,000 units of the composite good.

(You should check this.)

The budget constraint with a $20,000 voucher

BL2

30k

BL1

0 x50

If Joey spends all his income on the

composite good, he can still purchase 20

solar panels with the voucher.

If he spends all his income on solar panels,

he can purchase 50.

The voucher shifts Joey’s budget line to

the right by the number of panels that

can be purchased with the voucher.

Joey’s optimum with a $20,000 voucher

50k

BL2

25k

15k

30k

BL1

0 x

The budget line with the voucher is the

same as with the transfer, except for a

‘kink’ at the level of Joey’s income.

Joey’s optimum with a transfer remains

feasible with a voucher.

Therefore, the transfer and voucher have

identical effects on Joey’s behaviour and

welfare.

Exercise: Vouchers and corner solutions

Cynthia has preferences represented by the utility function U(x,y) = 1000 √ x + 1 +y. Her

corresponding marginal utilities are MUx = 500/ √ x + 1 and MUy = 1. Cynthia’s income is

I = $30,000, and the price of solar panels is Px = $1000 per panel.

1. Show that, absent a transfer or voucher, Cynthia’s optimal consumption basket contains

x = 0 solar panels.

2. Show that a transfer of $20,000 does not affect of the number of solar panels Cynthia

purchases. How does the transfer alter Cynthia’s behaviour?

3. Show that a voucher of $20,000, redeemable for solar panels, will cause Cynthia to

purchase exactly x = 20 solar panels.

Exercise solutions

1. We need to show that Cynthia’s optimal consumption basket is a corner solution.

If x = 0 then Cynthia spends her entire income on the composite good. Given that

Py = $1, the top end of the budget line intersects the y-axis at y = 30,000 units of the

composite good.

For this basket the marginal utilities per dollar of goods x and y are,

MUx Px

= 500 √ x + 1

× 1

1000 =

2 , and

MUy Py

= 1

1 = 1.

This is consistent with a corner solution in which x = 0.

Exercise solutions

2. With a $20,000 transfer Cynthia’s available income increases to $50,000. Again we need

to verify a corner solution.

The y-axis intercept of the budget line is now y = 50,000 units of the composite good.

The marginal utilities per dollar of goods x and y do not change as they do not depend on

the value of y,

MUx Px

= 500 √ x + 1

× 1

1000 =

2 , and

MUy Py

= 1

1 = 1.

Again, this is consistent with a corner solution in which x = 0.

Exercise solutions

50k

BL2

30k

BL1

0 x

3. Cynthia’s optimal consumption basket

must lie on the budget line.

The basket with x = 20 solar panels

lies at the kink in the budget line.

The marginal utility per dollar of good

x is,

MUx Px

= 500 √ x + 1

× 1

1000 ≈ 0.11.

The marginal utility per dollar of good

y is MUy/Py = 1.

Transfers versus vouchers

50k

BL2

30k

BL1

0 x

A transfer provides a consumers with at

least as much utility as a voucher of the

same monetary value.

• A consumer will prefer a transfer over a voucher if she/he would like to

spend less than the value of the

voucher on the specified good.

If the Government’s objective is a level of

consumption (as opposed to maximising

consumer welfare), then vouchers perform

at least as well as transfers.

Questions?

Key concepts from today’s lecture

You can use these concepts (as search terms) to conduct further research into the topics

covered in today’s lecture:

• Budget constraint

• Ratio of prices

• Consumption possibilities

• Optimal choice

• Tangency condition

• Marginal utility per dollar

• Corner solution

• Composite good

• Transfers

• Vouchers