Economics 102 Lecture 4 Utility Rev

7/28/2019 Economics 102 Lecture 4 Utility Rev

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Lecture 4: Preferences: Utility

Utility

Utility functions and indifference

curves

Examples of utility functions

Marginal utility

Marginal utility and MRS



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Utility- a way to describe preferences

Originally, preferences were defined in terms of utility: tosay a bundle X was preferred to a bundle Y meant that the X bundle had a higher utility than the Y bundle.

Things are now the other way around, if we say that X ispreferred to Y, then we can represent these preferencesby utility where X was higher than that of Y.

Utility function – is a way of assigning a number to every

possible consumption bundle such that more preferredbundles get assigned larger numbers than less preferredbundles.

A utility function U(x) represents a

preference relation if and only if:

x’ x” U(x’) > U(x”)

x’ x” U(x’) < U(x”)

x’ ~ x” U(x’) = U(x”).

~f p

p



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Ordinal utility vs. Cardinal utility

Ordinal utility: The magnitude of the utility function isonly important insofar as it ranks the different consumption bundles; the size of the utility differencebetween any two consumption bundles doesn’t matter.

The only property that is important is how it ordersthe bundles.

Cardinal utility theory states that the size of the utilitydifference between two bundles of goods matters.

If all you need to make choices are rankings, thenordinal rankings are all that are necessary

There can be no unique way to assign utilitiesto bundles of goods. As long as it preserves thepreferences of the consumer, then any functioncan be a utility function.

A Monotonic transformation of a utilityfunction is a utility function that representsthe same preferences as the original utilityfunction.



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Monotonic transformation – a way of

transforming one set of numbers into

another set of numbers in a way that

preserves the order of the numbers

)()(implies

if ation,transformmonotonic)(

2121 u f u f uu

u f

Examples: multiplying by a number

adding any number

raising to an odd power

A monotonic function has a positive rate of change or a positive slope. Rate of change as u changes:

In a monotonic transformation, the numeratoralways has the same sign as the denominator.

12

12 )()(

uu

u f u f

u

f



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If f(u) is a monotonic transformation of a utilityfunction that represents some preferences, thenf(u(x1,x2)) is also a utility function that represents the same preferences.

Argument: To say that u(x1,x2) represents some particular preferences

means that u(x1,x2) >u(y1,y2) iff (x1,x2) is strictly preferredto (y1,y2)

If f(u) is a monotonic transformation then u(x1,x2) >u(y1,y2) iif f(u(x1,x2))> f(u(y1,y2)).

Therefore, f(u(x1,x2))> f(u(y1,y2)) iff (x1,x2) is strictlypreferred to (y1,y2), so the function f(u) represents thepreferences in the same way as the original utility functionu(x1,x2).

U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).

Define V = U2.

Then V(x1,x2) = x12x2

2 and

V(2,3) = 36 > V(4,1) = V(2,2) = 16so again

(2,3) (4,1) ~ (2,2).

V preserves the same order as U and so

represents the same preferences.

p

p



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U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).

Define W = 2U + 10.

Then W(x1,x2) = 2x1x2+10 so

W(2,3) = 22 > W(4,1) = W(2,2) = 18. Again,

(2,3) (4,1) ~ (2,2).

W preserves the same order as U and V andso represents the same preferences.

p

p

Utility function – can be seen as a way tolabel indifference curves

a utility function is a way of assigningnumbers to different indifference curves in away that higher indifference curves get assigned larger numbers.

a monotonic transformation is just arelabeling of indifference curves



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An indifference curve contains equally

preferred bundles.

Equal preference same utility level.

Therefore, all bundles in an indifference

curve have the same utility level.

Level set – the set of all (x1,x2) such that

u(x1,x2) is a constant. Indifference

curves are level sets of utility functions.

Indifference curves from utility:

To derive an indifference curve from a

utility function, just equate the utility

function to a constant, and then solve the

equation for one of the terms.



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For instance:

1

2

12

21

2121

,of functionaasfor solving

),(

x

k x

x x

x xk

x x x xu

For a monotonic transformation of the utility curve above,

the indifference curves would have exactly the same shape,

except that the labels would be different.

The function v describes the same preferences as u since

it orders all of the bundles the same way.

2

21

2

21

2

2

2

121

2

2

2

121

),(),(),(

),(

x xu x x x x x xv

x x x xv



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U 6U 4

(2,3) (2,2) ~ (4,1)

x1

x2 p

U(2,3) = 6

U(2,2) = 4U(4,1) = 4

3D plot of consumption & utility levels for 3 bundles

x1

x2

Utility



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1

U

U

Higher indifferencecurves contain

more preferred

bundles.

Utility

x2

x1

U 6

U 4

U 2x1

x2



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U 6

U 5

U 4

U 3U 2

U 1

x1

x2

Utility

x1

x2



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1

x1

x2

x1

x2



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1

x1

x2

x1

x2



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1

x1

x2

x1



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1

x1

x1



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1

x1

x1



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1

x1

x1



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1

x1

x1



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1

x1

Existence of utility functions- Given a preferenceordering, can we always find a utility functionthat will order bundles of goods in the same wayas those preferences?

Ans: If we rule out perverse cases such asintransitive preferences, we will typically be ableto find a utility function to represent preferences.Nearly any kind of “reasonable” preferences canbe represented by a utility function



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2

How to construct indifference curves from utility

functions : two ways can be thought of:

Mathematical: Given indifference curves, we want to

find a function that is constant along each

indifference curve and that assigns higher values to

higher indifference curves.

Intuitive: given preferences, we try to think about

what the consumer is trying to maximize.

Perfect Substitutes:

Preferences for perfect substitutes can be represented by autility function of the form:

a and b are some positive numbers that measure the “value” of goods 1 and 2 to the consumer.

This is a utility function because (a) it is constant along anindifference curve and (b) it assigns a higher label to morepreferred bundles.

Any monotonic transformation of this function would also work.

b

a

ba

bxax x xu

-: bygivenisslope

0,

),( 2121



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2

Consider

V(x1,x2) = x1 + x2.

What do the indifference curves for this

“perfect substitution” utility function look like?

5

5

9

9

13

13

x1

x2

x1 + x2 = 5

x1 + x2 = 9

x1 + x2 = 13

All are linear and parallel.

V(x1,x2) = x1 + x2.



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Perfect Complements

In general a perfect complements utility functionis given by:

The minimum number of the good dictates orserves as a constraint to the consumption of theother.

For goods that are complementary on a one toone basis, a and b are equal to 1.

For goods that are consumed in a proportion that is different from one, then a and b are unequal.

consumedisgoodhein which ts proportiontheindicateand ,0,

},min{),( 2121

ba

bxax x xu

Consider

W(x1,x2) = min{x1,x2}.

What do the indifference curves for this“perfect complementarity” utility function

look like?



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2

x2

x1

45o

min{x1,x2} = 8

3 5 8

3

5

8

min{x1,x2} = 5

min{x1,x2} = 3

All are right-angled with vertices on a ray

from the origin.

W(x1,x2) = min{x1,x2}

Quasilinear utility- or partly linear utility. The utility functionis linear in good 2 but possibly non-linear in good 1

U(x1,x2) = f(x1) + x2

E.g. U(x1,x2) = 2x11/2 + x2.

Equation for such indifference curves takes the form:

k is a different constant for each curve

To label the indifference curve, k could be used. Solving for k andsetting it equal to utility, we have:

Indifference curves are vertical translates of one another

)( 12 xvk x

2121 )(),( x xvk x xu



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2

x2

x1

Each curve is a vertically shifted copy of the others.

Cobb-Douglas preferences

Cobb-Douglas utility function takes the form:

E.g. U(x1,x2) = x11/2 x21/2 (a = b = 1/2)V(x1,x2) = x1 x2

3 (a = 1, b = 3)

C-D indifference curves look just like the nice convexindifference curves that were referred to as “well-behaved indifference curves”

C-D indifference curves are the standard examplesused of well behaved indifference curves.

0,

),( 2121

d c

x x x xud c



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2

x2

x1

All curves are hyperbolic,

asymptoting to, but never

touching any axis.

Some monotonic transformations of the

Cobb-Douglas utility function:

natural log transformation:

transform that makes the sum of the

exponents =1

212121 lnln)ln(),( xd xc x x x xvd c

aa

d c

d

d c

c

d c

x x x xv

x x

x x x xv

12121

21

2121

),(

,dc

c anumber newadefine

.

: power d)1/(ctheutility toraising

),(



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2

Marginal utility= rate of change in utility as onegood is increased. It measures the rate of change inutility associated with a small change in the amount of good 1, holding the amount of good 2 constant.

The marginal utility with respect to good 2 isdefined in a similar way

The change in utility given a change in good 1 istherefore, and is of the same form for good 2:

1

21211

1

1

),(),(

x

x xu x x xu

x

U MU

11 x MU U

Marginal means “incremental”.

The marginal utility of commodity i is the rate-of-

change of total utility as the quantity of

commodity i consumed changes; i.e.

MUU

xi

i



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2

So, if U(x1,x2) = x11/2 x2

2 then

2

21

1

2

2

2

2

21

1

1

1

2

2

1

x x x

U

MU

x x x

U MU

/

/

The magnitude of marginal utility dependson the magnitude of utility.

depends on the particular way that utility ismeasured

Transforms of utility functions would havedifferent values of marginal utility

Marginal utility can’t be derived fromconsumer’s choice behavior

Marginal utility can be used to calculate themarginal rate of substitution.



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2

A utility function can be used to measure the marginal

rate of substitution, MRS (the slope of the indifference

curve at a given bundle of goods, the rate at which a

consumer is just willing to substitute a small amount of

good 2 for good 1.)

To derive, consider a small change in goods 1 and 2 that

keeps utility constant (i.e., we stay at the same

indifference curve): Then we must have:

2

1

1

2

2211

MRS

:curveceindifferentheof slopefor thesolving0

MU

MU

x

x

x MU x MU U

The sign of the MRS is negative, meaning we have to give upone good for more of the other in order to keep the samelevel of utility.

While the magnitude of marginal utility depends on theparticular transform of the utility function that you choose,

the ratio of marginal utilities is independent of theparticular transformation of the utility function.

transforms just relabel indifference curves and we areconcerned with moving along an indifference curve when wederive the MRS.



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2

MRSx

x 2

1

MRS(1,8) = - 8/1 = -8

MRS(6,6) = - 6/6 = -1.

x1

x2

8

6

1 6

U = 8

U = 36

U(x1,x2) = x1x2;

A quasi-linear utility function is of the form U(x1,x2) = f(x1)+ x2.

MRS = - f (x1) does not depend upon x2 so the slope of indifference curves for a quasi-linear utility function is constant along any line for which x1 is constant. What does that make theindifference map for a quasi-linear utility function look like?

so

U

xf x

11( )

U

x2

1

MRSdx

dx

U x

U xf x 2

1

1

21

/

/( ).



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3

x2

x1

Each curve is a vertically shifted copy of the

others.

MRS is a constant

along any line for which x1

is

constant.

MRS =

- f(x1’)

MRS = -f(x1”)

x1’ x1”

Applying a monotonic transformation to a utility

function representing a preference relation simply

creates another utility function representing the

same preference relation.

What happens to marginal rates-of-substitution

when a monotonic transformation is applied?



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3

For U(x1,x2) = x1x2 the MRS = - x2/x1.

Create V = U2; i.e. V(x1,x2) = x12x2

2. What is the

MRS for V?

which is the same as the MRS for U.

MRSV x

V x

x x

x x

x

x

/

/

1

2

1 22

12

2

2

1

2

2

More generally, if V = f(U) where f is a strictly

increasing function, then

MRSV x

V x

f U U x

f U U x

/

/

( ) /

' ( ) /1

2

1

2

U x

U x

/

/.1

2

So MRS is unchanged by a positive monotonic

transformation.



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3

This gives a useful way to recognize

preferences that are represented by

different utility functions:

Given two utility functions, just compute MRS

If the same, then the two utility function has the

same indifference curves

If the direction of increasing preference is thesame for each utility function, then the

underlying preferences must be the same.

Mathematical derivation of the MRS

Marginal, a small change, means derivative. We use partialderivatives because marginal utility of a good is computedholding the other good constant.

Differentials

Consider a change (dx1, dx2) that keeps utility constant

First term measures the increase in utility from a small change dx1

the second term measures the increase in utility from a smallchange dx2. Solving for dx2/dx1:

0),(),(

2

2

211

1

21

dx x

x xudx x

x xudu

2

21

1

21

1

2

),(

),(

x

x xu

x

x xu

dx

dx



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Implicit function: Indifference curve as a function x2(x1), indicates the value of x2 that

for each value of x1 keeps us on a specific indifference curve. Thisfunction has to satisfy the identity:

Differentiating both sides of the identity with respect to x1 to get:

Solving for , to find:

k x x xu ))(,( 121

0)(),(),(

1

12

2

21

1

21

x

x x

x

x xu

x

x xu

1

12 )(

x

x x

2

21

1

21

1

12

),(

),(

)(

x

x xu

x

x xu

x

x x

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Economics 102 Lecture 4 Utility Rev

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