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 consumer’s
observable market behavior can also be derived from few simple and sensible assumptions about
consumer’s observable choices, (rather than unobservable preferences). The basic idea is simple; if
consumer 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
x2
x(x1,x2)
y1
y(y1,y2)
Y2
x2 x1
x1
P1 x1  P2 x2  P1 y1  P2 y2  x is revealed to preferred to y
1
x  x1 , x 2 ............................x n 
y   y1 , y 2 .......................... y n 
P  P1 , P2 ...........................Pn 
The vector notation
 x P  xP  M
i i
will be used
x2
P10 x10  P20 x20
x0(x10,x20)
P10 x11  P20 x12  P10 x10  P20 x20
X1
x1
Although x0 is more expensive than x1 (in prices of x0) and he can afford x1, the consumer chooses
x0. Then it is said that x0 is revealed preferred to x1.
Weak Axiom of Revealed Preferences (WARP)
Assume x0 is revealed preferred to x1, i.e. at P0, x0 is chosen and P0x0 ≥ P0x1. Then x1 will never be
revealed preferred to x0.
The axiom can be formulated as,
If P0x0 ≥ P0x1,
then when x1 is chosen (at the price level P1),
P1 x1 < P1 x0
Contradicting Case
This case is against the theory.
X2
x0
X1
X1
M1
M0
2
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 xi  x1M P1 ..........., Pn , M  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 x1  x1M P1 , Pn , M  are single valued (so, they
are functions).
Proof:
Proof must be read from the book, Silberberg pp 319.
.
** 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.
Proof must be read from the book, Silberberg pp 320.
Defn: Negative semidefinite matrix.
A square matrix aij is said to be negative semidefinite if
 h a h
i
ij
j
 0 for all hi, hj.
(Meaning of Pdx = 0?
This is what is implied by “utility is held constant”.
dU  U 1 dx1  U 2 dx 2   P1 dx1  P2 dx 2  . So dU  0 , implies
 Pdx  0 , since   0 ,
dx would be interpretable as “pure substitution effect” (so on the same indifference curve).
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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,
x 0  2,2,2  P 0  2,2,2 
x 1  3,1,2  P '  1,3,2 
1

 1 
x 2   4,1,1  P 2   2,1 ,5 
2

 2 
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
P0x2=13
P1x2=10
P2x2=17
Then, you will see that
(1)
(2)
P 0 x 0  P 0 x1  12
so x 0 is revealed preferred to x1 , then, when x1 is chosen
P1 x1  P1 x 0  10  12  x 0  x1
P1 x1  P1 x 2  10 x 1 is revealed preferred to x 2 then, when x 2 is chosen.
P 2 x 2  P 2 x1  17  17.5  x1  x 2
we expect that x 0  x 2
i.e. P 0 x 0  P 0 x 2  17 , x0 is revealed preferred to x2 and then when x2 is chosen
But P 2 x 2  P 2 x 0 we see that this is not the case.
On the contrary,
P 0 x 0  12  P 0 x 2  13
x2 is revealed preferred to x0.
x2 > x0.
So the inconsistent consumer.
What is the reason for this?
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Actually the weak axiom does not imply that sij=sji (and the indifference curves are parallel). So we
need another stronger axiom  The 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. P1 x1  P1 x 2 , P 2 x 2  P 2 x3 ,..., P k 1 x k 1  P k 1 x k , then
P k x k  P k x 1 , that is x k is not revealed preferred to x1 .
(Guarantees that I-curves do not intersect.)
Now the main theorem:
Theorem: Individual D – functions xi  xiM Pi ,........Pn , M  that are consistent with the strong axiom
of revealed preferences, are derivable from utility analysis.
Such that there exists a class of utility functions F U x1 , x2 ,..., xn  where F is any monotonic
transformation, which, when maximized subject to the budget constraint  Pi xi  M , result in those
particular demand functions.
Application (Integrability)
Recall in Chapter 10 page 269
Max x1x2
s. t. P1 x1  P2 x 2  M  x1M 
M
M
, x 2M 
2 P1
2 P2
Question: if these are the demand curves, then can we derive the utility function behind these
demand curves?
Steps:
1. Check whether x1M , x 2M 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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tM

 x1M  t 0 x1M 
2tP1

 both of them are homogenous to degree zero.
tM
M
0 M 

 x2  t x2

2tP2
x1M 
a)
x 2M
b) P1 x1M  P2 x 2M  P1
c) S ij 
S11 
M
M
 P2
M
2 P1
2 P2
xiM
x M
 xj i
Pj
M
x1M
x M
M  M  1 
M

   2  0
 x1 1   2  
P1
M
2 P1  2 P1  2 P1 
4 P1
S 22 
x 2M
x M
M
 x 2M 2   2  0
P2
M
4 P2
x1M
x M
M 
 x 2M 1 

P2
M
4 P1 P2 
S12  S 21
M 
 ......................... 
4 P1 P2 
d) S12 
S 21
e) For negative semi definiteness
S11S 22  ( S12 ) 2  0
so x1M , x 2M exhibit all the properties.
2- Then let us find the utility function behind
Slope of Indifference Curve
dx2
MUx1
P
  MRS  
 1
dx1
MUx2
P2
dU  U 1 dx1  U 2 dx 2  0
dx 2
U
 1
dx1
U2
U1 P1

U 2 P2

dx2
P
 1
dx1
P2
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x1M 
P1 
M M
M
, x2 
2 x1
2 x2
M
M
, P2 
2 x1
2 x2
M
P1
2 x1 x1


P2 M
x2
2 x2
First order equation that have to be solved.
dx2
x
dx
dx
  2  2   1
dx1
x1
x2
x1
Integrate both sides;
nx 2  nx1  nF (U )
nx 2  nx1  nF (U )
F (U )  x1 x 2  utility function
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,
dx2
p
h (x , x )
  1   1 1 2 and
dx1
p2
h2 ( x1 , x2 )
h1 ( x1 , x2 )dx1  h2 ( x1 , x2 )dx2  0 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
x1M  x1M ( p1 , p2 , M )
(1)
x  x ( p1 , p2 , M )
(2)
p1 x1M  p2 x2M  M
(3)
M
2
M
2
Then if there exists a utility function behind, then s12  s21 must be. Let’s check!!
(1) 
x1M
x M
x M
p1  1 p2  1 M  0
p1
p2
M
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(3)  p1 (
x1M
x M
x M
x M
 x1 1 )  p2 ( 1  x2 1 )  0
p1
M
p2
M
 p1s11  p2 s12  0
(A)
Similarly we get
 p1s21  p2 s22  0
(B)
Now differentiating (3) w.r.t. p1 and then w.r.t. M we get the following two functions.
p1
x1
x
 p2 2   x1
p1
p1
(4)
p1
x1
x
 p2 2  1
M
M
(5)
Then multiply (5) by  x1 to equate it to (4);
x
x
x
x
 x1 p1 1  x1 p2 2  p1 1  p2 2
M
M
p1
p1
 p1s11  p2 s21  0
(C)
and similarly
 p1s21  p2 s22  0
(D)
Thus (A) and (C) gives s12  s21 .
Remark: So, for the two variable case, it is always possible to find a utility function U which
generates D-curves xiM  xiM ( p1 , p2 , M ) . If in addition s11 , s22  0, s11s22  s122  0, then this utility
function will have the usual convex I-curve of consumer theory.
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