Questions Economics

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Lecture 3: Last updated 9/03/2010

CHAPTER 11 (part 1)

SPECIAL TOPICS IN CONSUMER THEORY

Revealed Preferences:

So far we have approached demand theory by assuming the consumer has preferences satisfying

certain properties (complete, transitive, and strictly monotonic); then we have tried to deduce all the

observable properties of market demand that follow as a consequence (budget balances, symmetry,

and negative semidefinitenes of the Slutsky matrix). Thus we have begun by assuming something

about things we cannot observe, preferences, to make predictions about something we can observe,

consumer demand behavior. In his remarkable Foundations of Economic Analysis, Paul Samuelson

suggests an alternative approach. Why not start and finish with observable behavior? A consumers

observable market behavior can also be derived from few simple and sensible assumptions about

consumers observable choices, (rather than unobservable preferences). The basic idea is simple; ifconsumer buys one bundle instead of another affordable bundle, then the first bundle is considered to

be revealed preferred to the second. (Jehle, G.A., Reny, P.J., Advanced Microeconomic Theory,

2nd

Edition, Addison Wesley, pp. 86-87)

Then;

Is it possible to replace the utility maximization hypothesis with one based entirely on observable

quantities (consumed by the consumer)? OR Is it possible starting with a set of demand relations

which obey the symmetry and negative-semidefiniteness of the pure substitution terms to infer that

there exist some utility functions from which those demand functions are derivable? (Integrability)

Example: there are two goods

Y2

y1

x2 x1

y(y1,y2)

x(x1,x2)

x1

x2


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Note that x0

is revealed preferred to x1

and x1

is revealed preferred to x0. (can you see this in the

diagram above?)

Now we will try to see which of the results (homogeneity of the demand functions, negativity of the

pure substitution terms, symmetry of the Slusky terms ) implied by utility maximization are also

implied by the weak axiom of revealed preferences.

Proposition 1: demand relations ( )MPPxxn

M

i,.,..........11= are homogenous of degree zero in prices

and income.

Proof:

Proof must be read from the book, Silberberg pp 318.

Proposition 2: weak axiom implies that the relations ( )MPPxxn

M,,111 = are single valued (so, they

are functions).

Proof:


.

** So, the single valuedness and homogeneity of degree zero of demand functions are implied by

the weak axiom.

Weak axiom also implies that Hicks Slutsky substitutution terms are negative.

Proposition 3: The matrix Sij is negative semidefinite under the weak axiom of revealed

preferences.


Defn: Negative semidefinite matrix.

A square matrix aij is said to be negative semidefinite if

0jiji hah for all hi, hj.

(Meaning of Pdx = 0? This is what is implied by utility is held constant.

( )22112211

dxPdxPdxUdxUdU +=+= . So 0=dU , implies = 0Pdx , since 0 ,


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dx would be interpretable as pure substitution effect (so on the same indifference curve).

Hence dpdx 0 ==> own substitution effect is negative. )

Note that all 3 of the axioms must be proved by use of the weak axiom of the revealed

preferences.

However, the weak axiom does not imply the symmetry of the Slutsky terms.

Consider the choices of the consumer below,

( ) ( )

( ) ( )

=

=

==

==

5,

2

11,2

2

11,1,4

2,3,1'2,1,3

2,2,22,2,2

22

1

00

Px

Px

Px

Is this a consistent consumer?

To solve the problem, derive all the expenditure levels

P0x0=12 P1x0=12 P2x0=17

P0x1=12 P1x1=10 P2x1=17.5

P0x

2=13 P

1x

2=10 P

2x

2=17

Then, you will see that

(1)0 0 0 1 0 1 1

1 1 1 0 0 1

12 so x is revealed preferred to x , then, when x is chosen

P 10 12

P x P x

x P x x x

= =

< ==> < ==> >

(2)211222

2212111

5.1717P

chosen.iswhen xthen,xtopreferredrevealedisx10

xxxPx

xPxP

>==>

i.e. 17

2000==

xPxP , x

0

is revealed preferred to x

2

and then when x

2

is chosen

But 0222 xPxP < we see that this is not the case.

On the contrary,

13122000= x

0.

So the inconsistent consumer.


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What is the reason for this?

Actually the weak axiom does not imply that sij=sji (and the indifference curves are parallel). So we

need another stronger axiomThe Strong Axiom of the Revealed Preferences.

Strong Axiom of the Revealed Preferences

Let xi be purchased at Pi. then if x1>x2, x2>x3,.,.,xk-1>xk (x1>x2 notation meaning, x1 is revealed

preferred to x2), i.e. 1 1 1 2 2 2 2 3 1 1 1, ,...,

k k k k P x P x P x P x P x P x

, then

1xPxP

kkk< , that is

kx is not revealed preferred to 1x .

(Guarantees that I-curves do not intersect.)

Now the main theorem:

Theorem: Individual D functions ( )MPPxxni

M

ii,,........= that are consistent with the strong axiom

of revealed preferences, are derivable from utility analysis.

Such that there exists a class of utility functions ( )( )n

xxxUF ,...,,21

where F is any monotonic

transformation, which, when maximized subject to the budget constraint MxPii= , result in those

particular demand functions.

Application (Integrability)

Recall in Chapter 10 page 269

Max x1x2

s. t.2

2

1

122112

,2 P

Mx

P

MxMxPxP

MM====>=+

Question: if these are the demand curves, then can we derive the utility function behind these

demand curves?

Steps:

1. Check whether MM xx21 , are really the demand functions.

a. Homogenous of degree 0

b. Satisfy the budget constraint.

c. Negative Slutsky terms

d. Check S12 = S21

e. Check S11S22 (S12)2

= 0


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a)

===

===

MMM

MMM

xtxtP

tMx

xtxtP

tMx

2

0

2

2

2

1

0

1

1

1

2

2both of them are homogenous to degree zero.

b) MP

MPP

MPxPxPMM

=+=+

2

2

1

12211

22

c)M

xx

P

xS

M

i

j

j

M

i

ij

+

=

042

1

222

111

2

1

1

1

1

1

11


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2

2

1

1

2

2

1

1

2,

2

2,

2

x

MP

x

MP

x

Mx

x

Mx

MM

==

==

2

1

2

1

2

1

2

2

x

x

xM

xM

P

P==

First order equation that have to be solved.

1

1

2

2

1

2

1

2

x

dx

x

dx

x

x

dx

dx===>=

Integrate both sides;

functionutilityxxUF

UnFnxnx

UnFnxnx

==>=

=+

+=

21

12

12

)(

)(

)(

With this solution several things went right. For example slope of the indifference curve could be

expressed in terms of x1 and x2. But this might not be possible always with more than two

variables. In genereal, 2 1 1 1 2

1 2 2 1 2

( , )

( , )

dx p h x x

dx p h x x

= = and

1 1 2 1 2 1 2 2( , ) ( , ) 0h x x dx h x x dx+ = is a function not easy to solve always. But luckily for the two

variable case a solution always exists (by integrating factor method).

Now take

1 1 1 2

2 2 1 2

( , , ) (1)

( , , ) (2)

M M

M M

x x p p M

x x p p M

=

=

1 1 2 2 (3)M Mp x p x M+ =

Then if there exists a utility function behind, then12 21

s s= must be. Lets check!!

(1) 1 1 11 2

1 2

0

M M Mx x x

p p Mp p M

+ + =


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(3) 1 1 1 11 1 2 2

1 2

( ) ( ) 0M M M M

x x x xp x p x

p M p M

+ + + =

1 11 2 12

0p s p s+ = (A)

Similarly we get

1 21 2 22

0p s p s+ = (B)

Now differentiating (3) w.r.t.1

p and then w.r.t. Mwe get the following two functions.

1 2

1 2 1

1 1

x xp p x

p p

+ =

(4)

1 2

1 21

x xp p

M M

+ =

(5)

Then multiply (5) by1

x to equate it to (4);

1 2 1 2

1 1 1 2 1 2

1 1

x x x xx p x p p p

M M p p

= +

1 11 2 21

0p s p s+ = (C)

and similarly

1 21 2 22

0p s p s+ = (D)

Thus (A) and (C) gives 12 21s s= .

Remark: So, for the two variable case, it is always possible to find a utility function Uwhich

generates D-curves1 2

( , , )M Mi i

x x p p M= . If in addition11 22

, 0,s s < 2

11 22 120,s s s = then this utility

function will have the usual convex I-curve of consumer theory.

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