1

12. Consumer Theory

Econ 494

Spring 2013

2

Agenda

Shifting gears…Focus on the consumer, rather than the firm Axioms of rational choice Primal: Utility maximization Marshallian

demands Dual: Expenditure minimization Hicksian

demands Link between utility max and expenditure min. Welfare measures

Readings Silb. Ch 10; also p 53-55

3

Introduction

Consumers purchase goods, x1…xn, at prices, p1…pn.

Consumers have a budget or income (M) with which to purchase these goods/services

How do consumers decide what to buy? Buy goods to make them happy Derive satisfaction from consumption This satisfaction can be described by a utility function Consumers will maximize utility s.t. budget constraint

1

1,

1

, subject to n

n

n i ix x

i

Max U x x M p x

4

Axioms of rational choice

These are behavioral postulates no need to prove

Budget constraint is straightforward, but notion of a utility function is not.

We assert that consumer preferences must exhibit the following characteristics:

5

1. Completeness

Individuals are always able to choose between two bundles

Consumers can rank bundles

Binary preference comparison

One of the following must be true:A is preferred to BB is preferred to AA and B are equally preferred

A B

or B A

or A B

6

2. Transitivity

The rankings consumers assign to bundles must be consistent or transitive

We only require that consumers can compare 2 bundles at a time, but the pairwise comparisons must be linked

Transitivity: If A is preferred to B,And if B is preferred to C,Then A must be preferred to C.

A BA C

B C

7

Comment on transitivityAssuming transitivity is somewhat controversialThere is some evidence that indicates that

choices are not always transitive

3. Reflexivity

Bundle A is at least as preferred as itself

Almost obvious, requires only very weak logical behavior

A A

8

The 1st 3 axioms The first 3 axioms: Completeness Transitivity Reflexivity

These formalize the notion that the consumer’s choices are rational or logically consistent.These require that individuals can rank choices ordinal measure of satisfaction/utilityDoes not require individuals to assign some level or degree of satisfaction with these choices not cardinal measure

Utility function is only a ranking. If A is preferred to B, then U(A) > U(B)

9

Utility is an ordinal measure

Ordinality individuals can rank bundles

If U(A)=50 and U(B)=25, then A is preferred to B, but we cannot say that A is preferred twice as much as B.

Utility function is not unique – can take a monotonic transformationAny transformation that preserves rankingsU0(x) = x2 or U1(x) = 2ln(x) will work just as well

10

4. Continuity & differentiability

Mathematical assumptions about utility function

Continuity If A is strictly preferred to B, then there is a

bundle “close” to A that is also preferred to B

DifferentiabilityUtility function is twice differentiable

11

5. Non-satiation

The utility function is monotonically increasing in the consumption of each good

“More is preferred to less”

0ii

UU

x

12

6. Substitution

Consumers can make trade-offs among goods

Assume bundles are perfectly divisible

The maximum x2 a consumer will give up to get 1 unit of x1 is the amount that will leave her indifferent between old and new situation Indifference curves (analogous to isoquants)

Slope of indifference curve represents trade-offs person is willing to make

Indifference curve – locus of consumption bundles that yield same level of utility

13

Indifference curves slope down

An explicit function for an indifference curve:0

2 2 1( , )x x x U

0 01 2 1, ( , )U x x x U U

Substitute into utility function to get identity:

Differentiate identity wrt x1:2

1 2 1

0U U x

x x x

Rearrange terms:

2 1 1

1 2 2

0 since 0i

x U x UU

x U x U

Downward sloping indifference curves is implied by assumption of nonsatiation

14

Marginal rate of substitution

A negatively sloped indifference curve means consumers are willing to make trade-offs less of one good for more of the other

Marginal rate of substitution (MRS)2 1

1 2

x U

MRSx U

15

Indifference curves are convex

The marginal value of any good decreases as more of that good is consumed.

Diminishing MRS ¶2x1 / ¶x22 > 0

Differentiate MRS wrt

1

2 1 1 2 1

1 2 1 2 1

22 1 1 2 1

21 2 1 2 1

2 211 2 12 1 2 22 13

2

( , ( ))

( , ( ))

( , ( ))

( , ( ))

12 0

x U x x x

x U x x x

x U x x x

x x U x x x

U U U U U U UU

See notes #8 slides 25-26 for derivation

16

Graphical illustration

x2

x1

•

•b

a

U0

1 1

2 2

a b

a b

U U

U U

At point a, consumer is willing to give up more x2 for a unit of x1 than at point b.

17

Indifference curves cannot cross

x2

x1

U1

U2

B

AC

••

•

A and B are on same indifference curve U(A) = U(B) = U1

A and C are on same indifference curve

U(A) = U(C) = U2

By transitivity, it must be true that U(B) = U(C)

But…C includes more of both goods than B. By nonsatiation: U(B) < U(C)

Contradiction curves cannot cross

18

Indifference maps

x2

x1

•

•

U1

U2B

AC

U2 > U1

We can fully describe a utility function, and hence an individuals preferences in (x2, x1) space by an indifference map.

19

Utility maximization

Suppose a consumer gets utility from 2 goods and has income M. The indirect utility function is:

1 21 2 1 2 1 1 2 2

,

1 2 1 2 1 1 2 2

Lagrangian

( , , ) ( , ) . .

( , , ) ( , )

x xV p p M Max U x x s t M p x p x

x x U x x M p x p x

L

Assume person spends all her money.

V (p1, p2, M) is quasi-convex in (p1, p2)

we are not going to prove this

20

FONC

1 21 2 1 2 1 1 2 2

,

1 2 1 2 1 1 2 2

1 1 1 1 1

2 2 2 2 2

1 1 2 2

Lagrangian

FON

( , , ) ( , ) . .

( , , ) ( , )

0price ratio (relative prices)

0

C

0

x xV p p M Max U x x s t M p x p x

x x U x x M p x p x

U p U pMRS

U p U p

M p x p x

L

L

L

L

– Slope of indifference curve

– Slope of budget line

21

Tangency of indifference curve and budget line

22

SOSC

1 1 1

2 2 2

1 1 2 2

FONC

1,0

20

0

ii

U p

U

Up

p

M p x p x

i

L

L

L

11 12 1 11 12 1

21 22 2 12 22 2

1 2 1 2

2 21 2 21 1 22 2 11

2 21 2 21 1 22 2 11

2

0

0 0

2

SO

0

20

SC

U U p

BH U U p

p p

BH p p U p U p U

U U U U U U UBH

L L L

L L L

L L

1,2ii

Up i

Strict quasi-concavity of the utility function assures us that the

indifference curves will be strictly convex

We do NOT know sign of U11 and U22 !! Diminishing marginal utility not implied

23

Marshallian demand functions

By the IFT, if the SOSC are satisfied then the FONC can, in principle, be solved simultaneously for the explicit choice functions (Silb., p. 262, lays the IFT conditions out more formally):

xim(p1, p2, M) i=1,2

lm(p1, p2, M)

xim(p1, p2, M) are utility maximizing demands

also referred to as Marshallian demand functionsor money-income-held-constant demands

24

Elasticities

0 good i

Income e

infe

las

s

0 good

rior

n

ticit

is ormal

y

mi

iMi

ix M

iM x

0 good

Cross-p

and

rice elastic

complements

substit

are

0 good and ar ute es

itym

jiij

j i

p i jx

i jp x

1 good is

= 1 good is

elastic

unit elast

> 1 good

i

is

> 0 good i

Own-price elasticity

c

inelastic

Giffen goo ds a

mi i

iii i

i

ix p

ip x

i

25

Marshallian demands are HOD(0) in prices and income

For prices & income (p1, p2, M), the FONC:

1 11 1 2 2

2 2

and 0U p

M p x p xU p

For prices & income (tp1, tp2, tM), the FONC:

1 11 1 2 2

2 2

1 11 1 2 2

2 2

and 0

and 0

U pM p

tt t tx p x

U p

U pM x p

t

p xU p

Since the FONC are the same for both sets of prices, the solutions must also be the same. Therefore, xi

m(p1, p2, M) = xim(tp1, tp2, tM)

26

Comment on homogeneity

HOD(0) implies that “relative” price matterConsumption choices will not change with

inflation if all prices and wages are increased at the same rate

Consumption opportunities do not change if prices and income change by same proportion.

27

Marshallian demands are invariant to positive monotonic transformation

The demand functions that solve:

1 2

1 2 1 1 2 2,

( , ) . .x x

Max U x x s t M p x p x

Are identical to the demand functions that solve:

1 21 2 1 1 2 2

,

1 2 1 2

ˆ ( , ) . .

where

ˆ ( , ) ( , )

and ( ) 0, ( ) 0

x xMax U x x s t M p x p x

U x x F U x x

F U F U

¤

see Silb p. 53-55, 264-5

Read his discussion on why “diminishing marginal utility” has no meaning with ordinal utility.

28

Proof of invariance to monotonic transformation

1 2

1 2 1 1 2 2,

1 1

2 2

1 1 2 2

FO

( , )

N

. .

0

C

x xMax U x x s t M p x p x

U p

U p

M p x p x

1 2

1 2 1 1 2 2,

1 1 1 1 1

2 2 2 2 2

1 1 2 2

1

2

( , ) . .

FONC

ˆ

0

ˆ

ˆ

x xU

U

Max x x s t M p x p x

p F U p U p

p F U p U p

M p x p

U

x

1 2 1 2( , ) ( , )

and (

ˆ

) 0

x x F UU x x

F U

Since FONCs are identical, xim that solve FONCs must be identical.

Note that U1/U2 = p1/p2 holds if F ' >0 or F ' <0. But if F ' <0, then an increase in both x1 and x2 would decrease utility. Hence, F ' <0 would correspond with minimizing utility. If F ' >0, then U and Û will move in same direction, and U will achieve a maximum IFF Û does. (See Silb p. 264 Proposition 1)

29

Comment on monotonic transformation

This proposition highlights the ordinal nature of preferences.

A positive monotonic transformation preserves the ranking of all bundles

A positive monotonic transformation says nothing about the behavior of the individual

30

Envelope theorem

The indirect objective function is: 1 2 1 1 2 2 1 2( , , ) ( , , ), ( , , )m mV p p M U x p p M x p p M

Apply envelope theorem:

1 2

1 2 1 2

1 2

1 2

1 2 1 2

1 2

1 2

( , , )1 1 1 2( , , ) ( , , )1 1( , , )

( , , )2 2 1 2( , , ) ( , , )2 2( , , )

( , , )(

( , , ) 0

( , , ) 0

m

m m

m

m

m m

m

m

m

m mx x p p M

x x p p M p p Mp p M

m mx x p p M

x x p p M p p Mp p M

x x p p Mp

Vx x p p M

p p

Vx x p p M

p p

V

M M

L

L

L1 2

1 2

1 2

( , , ) 1 2( , , )

, , )

( , , ) 0m

m

mx x p p M

p p Mp M

p p M

31

Characteristics of indirect utility functionIndirect utility function is: Non-increasing in prices

Proof: ¶V / ¶pi < 0 from previous slide Non-decreasing in income

Proof: ¶V / ¶M > 0 from previous slide

Indirect utility function is HOD(0) in prices and income Proof: since Marshallian demands are HOD(0):

Indirect utility fctn is quasi-convex in prices (p1, p2) We will not prove this…

1 2 1 1 2 2 1 2

1 1 2 2 1 2 1 2

( , , ) ( , , ), ( , , )

( , , ), ( , , ) ( , , )

m m

m m

V p p Mt t t tU x p p M x p p M

U x p

t t t t t

p M x p p M V p p M

32

Roy’s identity

Solving for xi, and using lm = ¶V / ¶M :

This relationship is known as Roy’s identity

1 2

1 2 1 2

1 2

( , , ) 1 2( , , ) ( , , )( , , )

( , , )

From before, using Envelope m

0

Th :

m

m m

m

m mx x p p Mi i

x x p p M p p Mi ip p M

Vx x p p M

p p

L

m ii

V px

V M

33

Interpreting l

At any given consumption point, additional utility, Ui, can be gained by consuming “a little more” xi.

The marginal cost of this extra xi is pi.

The marginal utility per dollar spent on xi is Ui / pi

1 1 1

2 2 2

2

1

2

1

2

1 1 2

0

F NC

0

O

0

U p

U

U U

p pp

M p x p x

L

L

L

See Silb., p. 266-7

34

Use Envelope Theorem to Interpret l

The Lagrange multiplier is always interpreted as the marginal effect on the optimal value of the objective function as the constraint changes.

lm(p1, p2, M) is the marginal utility of income

1 2,,

( , , ) 0m m

m m

m

xx

Vp p M

M M

L

35

Comparative statics

Earlier, we used the IFT to solve the FONC:xi

m(p1, p2, M) i=1,2

lm(p1, p2, M)

If we substitute these explicit choice functions back into the FONC, we get the following identities:

1 1 1 2 2 1 2 1 2 1

2 1 1 2 2 1 2 1 2 2

1 1 1 2 2 2 1 2

( , , ), ( , , ) ( , , ) 0

( , , ), ( , , ) ( , , ) 0

( , , ) ( , , ) 0

m m m

m m m

m m

U x p p M x p p M p p M p

U x p p M x p p M p p M p

M p x p p M p x p p M

36

Comparative statics for p1

Differentiate the identities wrt p1, then express in matrix form:

1

1

1

111 12 1 1 1

21 22 2 2 1 2

1 2 1

11 12 1 1 1

21 22 2 2 1

1 2 1 1

0

0

mp

mp

m

p

m m

m

m m

x p

x p

p

U U p x p

U U p x p

p p p x

LL L L

L L L L

L L L L

37

Apply Cramer’s rule to solve

Signs are indeterminate If M increases, cannot use less of both goods – would

violate budget constraint

Find ¶x2m / ¶p1 on your own

Why is sign indeterminate? Parameters show up in the constraint.

212 12 1 1 22 1 2

1

12 12 1 22

10

10

mm m m

m

xp U x p U x p

p BH

xp U p U

M BH

38

Show: Cannot use less of both goods(when income increases)

Substitute solution back into budget constraint:

1 1 1 2 2 2 1 2, , , ,m mp x p p M p x p p M M

Differentiate wrt M:1 2

1 2 1 at least one comparative static must be 0m mx x

p pM M

39

Engel Aggregation

Convert to elasticity form:

1 21 2 1

m mx xp p

M M

1 1 2 2 1 share-weighted income elasticities

must sum to one.

share of budget spent on good

M M

j jj

s s

p xs j

M

1 21 21 2

1

1 2

1

1m mm m

m m

x xp p

M M

M M

x x

xM M x

1 1 2

1

21 2

2

1m m

m m

m mx p x pM M

M MM

x x

xM x

40

Cournot aggregation

1 11 2 21 1 j jk kj

s s s s s

1 21 11 1 2 2 1

1 11 2

1m mm m

m

m

m

p p p

M M

p x p x x

xp Mxp

x x

1 21 21 2 1

1

1 1 1

1 1

1 21

m m

mm

m m

m

p p p

M

x x

x

x xp p x

p p Mx M

1

1 21 1 2

1 1

Differentiate wrt

0m m

mx xp x p

p p

p

1 1 1 2 2 2 1 2, , ,

Budget identi y

,

tm mp x p p M p x p p M M

41

Cournot and Engel Aggregation

Cournot and Engel aggregation can be useful when estimating demand functionsCan be used as a restriction in a regressionOr, can test whether estimated coefficients are

consistent with this.

42

Slutsky (later we will prove)

Later we will prove:

1 1 11

1 1 000

h m mmx x x

xp p M

Find the sign, using comp. static results:

1 11

1

22 12 1 1 22 1 2 1 2 12 1 22

22

1

10 since 0

m mm

m m m m

m

x xx

p M

p U x p U x p x p U p UBH

p BHBH