Utility

Marginal Rate of Substitution

The MRS is the slope of the indifference curve at a point

MRS=derivative of indifference curve

2x

1x

),( 21 xx

1x

2x

Interpretation of MRS

The MRS measures the rate at which the consumer is willing (i.e., indifferent) to substitute one good for the other.

If good 2 is measured in dollars, the MRS measures the consumer’s willingness to pay for an extra unit of good 1.

Assumptions on Preferences and the MRS

Monotonicity: MRS negative

(Strict) Convexity: MRS decreases as

increases

2x

1x1x

2x1x

Utility Function

Idea: assign a number to each consumption bundle, with higher numbers assigned to more-preferred bundles.

A utility function represents a preference relation :

),(),(),(),( 21212121 yyuxxuyyxx

),( 21 xxu

Utility Function: Does It Always Exist?

Q: Given a preference relation can we find a utility function that represents it?

A: If preferences are complete and transitive (plus a technical assumption called “continuity” is verified) we can. [Sufficient condition]

Is Transitivity Necessary?

Q: Is it necessary that preferences are transitive for the existence of a utility function that represents them?

A: Yes. Otherwise:

)1,1()1,4.0()5.0,1()1,1(

)1,1()1,4.0()5.0,1()1,1( uuuu

Utility is Just Ordinal

E.g.: (1,1) (1,0.5) (0.4,1) (0.8,0.8)

Can be represented in different ways:

Bundle

(1,1) 3 6 -1

(1,0.5) 2 4 -2

(0.4,1) 1 2 -3

(0.8,0.8) 1 2 -3

),( 21 xxu ),(2 21 xxu 4),( 21 xxu

~

In General

If :

1) represents

2) is a positive monotonic transformation

Then, also represents

),( 21 xxu )(uf

)),(( 21 xxuf

)]()([ 2121 ufufuu

Utility Functions and Indifference Curves

2x

1x

Utility function:

Indifference curves:

or:

2121 ),( xxxxu

uxx 21

12x

ux 2

3

5.12x

1x

Indifference Curves and Monotonic Transformations

2x

1x

Utility function:

Indifference curves:

or:

22

2121 ),( xxxxv

12

x

vx 4

925.1

2x

1x

vxx 22

21

2y

1y

Perfect Substitutes

2x

1x

Utility function:

Indifference curves:

2121 ),( bxaxxxu

12 xb

a

b

ux

Perfect Complements

2x

1x

Utility function:

),min(),( 2121 bxaxxxu

12 xb

ax

Cobb-Douglas Preferences

2x

1x

Utility function:

Indifference curves:

dc xxxxu 2121 ),(

d

c

d

x

ux

1

1

2

Cobb-Douglas

Most commonly used utility function in economics

Possible to assume without loss of generality that

Why?: apply to get

1 dc

dcuuf 1

)(

dc

d

dc

c

xxxxv 2121 ),(

Marginal Utility

Consider a consumer that is consuming

Q: By how much does his utility change as we increase by a very small amount his consumption of good 1?

A: Marginal utility of good 1:

),( 21 xx

1

2,1 )(

x

xxu

Marginal Utility and Units

The marginal utilities of good 1 and good 2 depend on the specific utility function we are using.

Consider:

Then:

))((),( 2,121 xxufxxv

1

2,12,1

1

2,1 )())(()(

x

xxu

u

xxuf

x

xxv

Marginal Utility and MRS

MRS only depends on preferences and not on their specific representation (utility function).

Marginal utilities can be used to compute the MRS between two goods.

Computing the MRS

Consider an indifference curve:

Q: What is the slope of this indifference curve (MRS)?

A:

uxxu ),( 21

2

2,1

1

2,1

1

22,1

)(

)(

)(

xxxu

xxxu

x

xxxMRS

The MRS and Monotonic Transformation

Consider a monotonic transformation

Q: What is the MRS in this case?

A:

)(2),( 2,121 xxuxxv

2

2,1

1

2,1

2

2,1

1

2,1

2,1)(

)(

)(2

)(2

)(

xxxu

xxxu

xxxu

xxxu

xxMRS