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Price Theory

Lecture 2: Utility

Topics for today’s lecture . . .

1. Preferences & consumption

2. Utility functions

3. Indifference curves

4. Special utility functions

Preferences & consumption

Consumption

Understanding consumer behaviour begins with describing the available alternatives.

Each of us consumes many different goods and services, including:

• Food

• Clothing

• Entertainment

• Utilities

• Housing

• Transport

Different goods and services are not typically alternatives in and of themselves: For example,

eating a meal does not prevent you from wearing a shirt.

Rather, when considering the goods and services we consume, an alternative must specify a

complete ‘plan’ of consumption.

Definition: Consumption basket

A combination of goods and services that an individual might consume.

A basket describes both which goods and services an individual consumes, and the quantities

consumed.

Food & clothing

00y

0 x

We will confine our attention to baskets

that contain two goods.

In this example,

• x represents number of meals, and,

• y represents number of items of clothing.

Each point on the graph describes a

different basket.

Food & clothing

0 x

Basket A contains 10 meals, and 8 items

of clothing.

Basket B contains 15 meals, and 6 items

of clothing.

Basket C contains 20 meals, and does

not contain any clothing.

Definition: More is better

An assumption that having more of a good is always better for a consumer.

The more is better assumption (also called monotonicity) provides a great deal of structure

to consumer preferences.

More is better & preferences

00y

0 x

More is better implies basket A is

preferred over the six remaining baskets:

• A has more of good y than B and C (and the same amount of good x ).

• A has more of good x than D and E (and the same amount of good y ).

• A has more of good x and good y than F and G .

Exercise: More is better

00y

0 x

1. Using the more is better assumption,

determine how basket B is ranked

with respect to the six remaining

baskets.

2. Does the assumption allow you to

construct complete preferences for

the seven baskets illustrated here?

Briefly explain.

Exercise solutions

00y

0 x

1. More is better gives us,

• B � C ,

• B � G ,

• A � B . Basket B cannot be ranked against D ,

E and F , using the more is better

assumption.

2. No. For example, more is better does

not tell us how a consumer ranks B

and F . Basket B has more of x , but

less of y , than F .

Utility functions

The problem with preferences . . .

Describing a consumer’s preferences over consumption baskets can be difficult.

• In lecture 1 we saw that the number of rankings needed to complete the preferences grows much faster than the number of alternatives.

To resolve this problem we will represent a consumer’s preferences with a utility function.

Note: Only rational preferences can be represented by a utility function.

Utility functions

A utility function assigns a number to every possible basket of goods and services.

For a basket A, we write U(A) to indicate the utility that a consumer derives from consuming

the basket.

Bigger is better: A � B implies U(A) > U(B), and vice versa.

• The sizes of any two numbers can be compared, therefore a utility function can only represent complete preferences.

• The ‘greater than’ relation is transitive, therefore a utility function can only represent transitive preferences.

Utility functions with a single good

U(x) = √

0 x

Consider a utility function U(x) = √

with a single good x , representing the

quantity of hamburgers consumed.

To evaluate the consumer’s utility,

substitute the number of hamburgers into

the function.

• When 1 hamburger is eaten, the consumer’s utility is U(1) = 1.

• When 4 hamburgers are eaten, the consumer’s utility is U(4) = 2.

Definition: Marginal utility

The rate at which total utility changes as the level of consumption of a good rises, holding

constant the level of consumption of all other goods.

The marginal utility of a good is the partial derivative (slope) of the utility function, with

respect to the good in question.

Marginal utility

1 2

1 4

U(x) = √

0 x

MUx = 1

2 √

0 x

For a single good utility function, the

marginal utility of x is written,

MUx = dU(x)

dx .

Note: You will always be provided with

marginal utilities in this course.

• When 1 hamburger is eaten, the marginal utility (slope) is 1/2.

• When 4 hamburgers are eaten, the marginal utility (slope) is 1/4.

Marginal utility & more is better

00 U(x) = √

0 x

MUx = 1

2 √

0 x

The following statements are equivalent:

• The utility function is upward sloping (in the direction of an increase in x ).

• The marginal utility of good x is positive.

• The more is better assumption is satisfied for good x .

Definition: Principle of diminishing marginal utility

The principle that as consumption of a good increases, the marginal utility of that good will

begin to fall.

The principle of diminishing marginal utility states that each unit of a good consumed,

creates less additional benefit for the consumer than the previous unit.

The principle of diminishing marginal utility

00 U(x) = √

0 x

MUx = 1

2 √

0 x

The following statements are equivalent:

• The utility function gets flatter as x increases.

• The marginal utility of x is downward sloping (diminishing).

• The principal of diminishing marginal utility is satisfied for good x .

Diminishing total utility

00U

0 x

Positive MU Negative MU

For many goods more is not always

better:

• Food.

• Alcohol.

• Pharmaceuticals.

Past a certain point, increased

consumption reduces welfare.

Does this matter?

Probably not: An optimising consumer

will choose to consume a quantity with a

non-negative marginal utility.

Indifference curves

Utility functions with two goods

U(x, y ) = √

This graph illustrates a total utility

function with two goods:

• Food (good x ).

• Clothing (good y ).

Each point on the horizontal plane

corresponds to a consumption basket.

The utility created by this consumption

basket is represented by the hight of the

function.

For this consumer U(12, 3) = 6.

The marginal utility of x

U(x, y ) = √

Treat the quantity of good y as fixed.

The marginal utility of x is the slope of

the utility function in the direction of an

increase in x .

This is the partial derivative of the utility

function with respect to x ,

MUx = ∂U(x, y )

∂x .

The marginal utility of y

U(x, y ) = √

Treat the quantity of good x as fixed.

The marginal utility of y is the slope of

the utility function in the direction of an

increase in y .

This is the partial derivative of the utility

function with respect to y ,

MUy = ∂U(x, y )

∂y .

Quiz 1

Suppose that a consumer’s preferences over food (good x ) and clothing (good y ) are

represented by the utility function U(x, y ) = 3 √

x + 2y . The associated marginal utilities are,

MUx = 3

2 √

x and MUy = 2.

The more is better assumption is satisfied for,

(a) both food and clothing.

(b) food, but not clothing.

(d) neither food nor clothing.

Quiz 2

Suppose that a consumer’s preferences over food (good x ) and clothing (good y ) are

represented by the utility function U(x, y ) = 3 √

x + 2y . The associated marginal utilities are,

MUx = 3

2 √

x and MUy = 2.

The principle of diminishing marginal utility is satisfied for,

(a) both food and clothing.

(b) food, but not clothing.

(d) neither food nor clothing.

Definition: Indifference curves

A curve connecting a set of consumption baskets that yield the same level of utility to the

consumer.

Each level of utility has a corresponding indifference curve.

Indifference curves

U = 2

U = 4

U = 6

U(x, y ) = √

An indifference curve is a line indicating

all points on the utility function at the

same level.

Higher indifference curves correspond to

higher levels of utility.

We can only illustrate a few indifference

curves at a time.

Flattening the utility function U(x, y ) = √

U = 4

U = 2

U = 6

0 x

Consider the indifference curve

corresponding to utility U = 4.

The indifference curves for lower levels of

utility lie below and to the left.

The indifference curves for higher levels of

utility lie above and to the right.

Exercise: Indifference curves

Suppose that a consumer’s preferences over food (good x ) and clothing (good y ) are

represented by the utility function U(x, y ) = 3 √

x + 2y . The associated marginal utilities are,

MUx = 3

2 √

x and MUy = 2.

1. Will the indifference curves for this utility function intersect the x -axis? Briefly explain.

(Hint: Consider the indifference curve corresponding to a utility of 12.)

2. Will the indifference curves for this utility function intersect the y -axis? Briefly explain.

Exercise solutions

1. Yes. Along the indifference curve corresponding to a utility of 12,

3 √

x + 2y = 12.

Along the x -axis y = 0. Substituting for y and solving for x , we get x = 16. Therefore,

the x -intercept of this indifference curve occurs at x = 16.

2. Yes. Along the y -axis x = 0. Substituting for x and solving for y , we get y = 6.

Therefore, the y -intercept of this indifference curve occurs at y = 6.

1. Indifference curves slope down

less

preferred

0 x

The more is better assumption allows us

to derive three properties of indifference

curve.

Pick any basket A that lies on the

indifference curve U1.

The consumer prefers A over any basket

that is below and to the left.

The consumer prefers any basket that is

above and to the right, over A.

1. Indifference curves slope down (another way)

0 x

The more is better assumption implies

that both goods have positive marginal

utilities (MUx > 0 and MUy > 0).

Consuming more of good y makes the

consumer better off, raising the consumer

to a higher level of utility.

To return to the original indifference

curve, the increase must be offset by a

decrease in the consumption of good x .

2. Indifference curves do not intersect

U1 U2

0 x

More is better tells us that B � A.

Therefore, the utility of every basket on

U2, must be higher than the utility of

every basket on U1.

But C lies on both indifference curves,

implying that U1 = U2.

It is not possible to have both U1 = U2

and U1 < U2, therefore we can conclude

that indifference curve cannot intersect.

3. Indifference curves are not thick

0 x

If an indifference curve is thick then we

can pick a basket A at the bottom edge

of the curve.

We can also pick a basket B that is above

and to the right of A, but still on U1.

More is better implies B � A, therefore A and B cannot lie on the same indifference

curve.

Quiz 3

0 x

Suppose E and F lie on the same

indifference curve, and that the more is

better assumption is satisfied.

Which basket(s) CANNOT lie on the

same indifference curve as E and F ?

(a) B only.

(b) B and C .

(d) B , C and D .

Definition: Marginal rate of substitution

The rate at which units of x are exchanged for units of y along the indifference curve.

The marginal rate of substitution of good x for good y (represented by MRSx,y), is the

negative of the slope of the indifference curve.

Deriving the marginal rate of substitution

The rate at which total utility changes with consumption of goods x and y is given by the

total differential,

dU(x, y ) = ∂U(x, y )

∂x dx +

∂U(x, y )

∂y dy = MUxdx + MUydy.

Along an indifference curve utility is constant, and therefore dU(x, y ) = 0.

Substituting for dU(x, y ) and rearranging, gives us,

MRSx,y = − dy

dx =

MUx MUy

Definition: Diminishing marginal rate of substitution

A feature of consumer preferences for which the marginal rate of substitution of one good for

another good diminishes as the consumption of the first good increases along an indifference

curve.

Diminishing marginal rate of substitution of x for y implies that indifference curves get flatter

as we move down along the curve.

Diminishing marginal rate of substitution

0 x

This indifference curve displays a

diminishing marginal rate of substitution.

• Basket A contains a lot of clothing (good y ), but little food (good x ).

• Basket B contains a lot of food, but little clothing.

The diminishing MRSx,y assumption

implies a consumer would prefer a more

even mix of food and clothing, over either

extreme.

Exercise: Sketching indifference curves

Suppose that a consumer’s preferences over food (good x ) and clothing (good y ) are

represented by the utility function U(x, y ) = 3 √

x + 2y . The associated marginal utilities are,

MUx = 3

2 √

x and MUy = 2.

1. Derive an expression for the marginal rate of substitution.

2. Is the marginal rate of substitution diminishing? Briefly explain.

3. On a graph, draw two typical indifference curves. The graph should illustrate all of the

features that you have derived in previous questions and exercises.

Exercise solutions

1. The marginal rate of substitution is,

MRSx,y = MUx MUy

= 3

2 √

x ×

2 =

4 √

x .

2. Yes. As x increases and y decreases, the denominator of the MRSx,y increases, while the

numerator remains constant.

Exercise solutions

U = 12

U = 24

0 x

3. The graph illustrates the following

features:

• Positive marginal utilities.

• x - and y -axis intercepts.

• Diminishing MRSx,y.

Special utility functions

Perfect substitutes

U1 U2 U3

0 x

Two goods are perfect substitutes if the

marginal rate of substitution is constant.

The utility function for perfect substitutes

takes the form,

U(x, y ) = αx + βy,

where α and β are positive constants.

The corresponding marginal utilities are

MUx = α and MUy = β, thus the slope

of the indifference curves is −α/β.

Perfect complements

fixed

proportions

2 3

A B

0 x

Two goods are perfect complements

when a consumer wants to consume them

in fixed proportions.

The utility function for perfect

complements takes the form,

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

For example, if x is left shoes and y is

right shoes, A and B produce the same

utility because they both contain two

complete pairs.

Exercise: Interpreting indifference curves

U1 U2 U3

0 x

Suppose that Graham’s preferences over

baskets containing tea (good x ), and

coffee (good y ), are illustrated by the

indifference curves in the figure.

1. If U1 < U2 < U3, how would you

explain the shape of these indifference

curves?

2. Which of our assumptions concerning

consumer preferences have been

violated here?

Exercise solutions

U1 U2 U3

0 x

1. Holding consumption of coffee (good

y ) constant, increasing consumption

of tea (good x ) increases utility.

Holding consumption of tea constant,

increasing consumption of coffee

reduces Graham’s utility.

For Graham, coffee is a bad, reducing

his utility. (Graham dislikes coffee.)

2. The more is better assumption has

been violated for coffee (good y ).

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:

• Consumption basket

• More is better (monotonicity)

• Ordinal ranking

• Cardinal ranking

• Utility function

• Marginal utility

• Diminishing marginal utility

• Indifference curves

• Marginal rate of substitution

• Diminishing marginal rate of substitution

• Perfect substitutes

• Perfect complements