Demand3 WARP

8/6/2019 Demand3 WARP

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CALIFORNIA INSTITUTE OF TECHNOLOGY

Division of the Humanities and Social Sciences

WARP and the Slutsky matrix

KC Border

Let X ⊂ R n be the consumption set. Samuelson [2] made the following observation.

If you demand x when y is in the budget set, it is because you prefer x to y. Thereforeyou should never demand y when x is in the budget set. (This of course implicitlyassumes a unique utility maximizer, or strict quasiconcavity of the utility.) This is notdeep. What is deep is that this is enough to deduce the negative semidefiniteness of theSlutsky matrix.

1 Definition (Samuelson’s Weak Axiom of Revealed Preference) For an ordinary demand function ξ : R

n

++ × R ++ → X , define the binary relation S on X by

x S y ⇐⇒ (∃( p,m) : x = ξ( p,m), y ∈ X, y = x, and p · y m).

That is, x is demanded when y is in the budget set, so x is revealed preferred to y.

The demand function ξ obeys Samuelson’s Weak Axiom of Revealed Preference (SWARP) if S is an asymmetric relation. That is, if for every x, y ∈ X ,

x S y =⇒ ¬y S x.

That is, if x is revealed preferred to y, then y is never revealed preferred to x.

The demand function ξ satisfies the budget exhaustion condition if for all ( p,m),

p · ξ( p,m) = m.

Under the budget exhaustion condition, we can rewrite the SWARP in the form thatSamuelson used. Let x0 and x1 belong to the range of ξ. That is, x0 = ξ( p0, m0) =ξ( p0, p0 · x0) and x1 = ξ( p1, m1) = ξ( p1, p1 · x1). Then p1 · x0 p1 · x1 and x0 = x1 implyx1 S x0, and ¬x0 S x1 implies p0 · x1 > p0 · x0. Thus, we can write Samuelson’s form:1

x0 = x1 and p1 · x0 p1 · x1 =⇒ p0 · x1 > p0 · x0.

2 Lemma Let ξ satisfy the budget exhaustion condition and SWARP. Let x0 = ξ( p0, m0)and x1 = ξ( p1, p1 · x0). Then

( p1 − p0) · (x1 − x0) 0.

1It may appear that this condition is weaker then than the one stated above, since it applies only

to x0 and x1 in the range of ξ, whereas the condition above applies to all x and y in X , which may

be larger than the range of ξ. However, any violation of SWARP as stated above would involve x and

y with x S y and y S x, which can only happen if both x and y belong to the range of ξ. Thus the

definitions are equivalent.

1



KC Border WARP and the Slutsky matrix 2

Proof : Cf. Samuelson [2, equation (70), p. 109]. By budget exhaustion

p

1

· x

1

= p

1

· x

0

. (1)Then either x1 = x0, in which case the lemmas is true, or else x1 S x0.

So assume x1 S x0. Then by SWARP, we have ¬x0 S x1, that is,

p0 · x1 > m0 = p0 · x0, (2)

where the second equality follows from budget exhaustion. Subtracting inequality (2)from equation (1) gives

( p1 − p0) · x1 ( p1 − p0) · x0,

which is the conclusion of the lemma.

This leads us to define the Slutsky compensated demand s in terms of the ordinarydemand function ξ via

s( p,ω) = ξ( p,p · ω).

Note that∂si( p,ω)

∂p j

=∂ξi( p,p · ω)

∂p j

+ ω j

∂ξi( p,p · ω)

∂m.

In particular, by setting ω = ξ( p,m) we may define

σi,j( p,m) =∂si

p,ξ( p,m)

∂p j

=∂ξi( p,m)

∂p j

+ ξ j( p,m)∂ξi( p,m)

∂m.

3 Theorem Let ξ : R n

++ × R ++ → R n

+ be differentiable and satisfy the budget exhaus-

tion condition and SWARP. Then for every ( p,m) ∈ R n

++ × R ++, and every v ∈ R n,

i

j

σi,j( p,m)viv j 0.

Proof : Fix ( p,m) ∈ R n

++ × R ++ and v ∈ R n. By homogeneity of degree 2 of the

quadratic form in v, without loss of generality we may scale v so that p ± v 0.Define the function x on [−1, 1] via

x(t) = s p + tv,ξ( p,m)

.

Note that this is differentiable, and x(0) = ξ( p,m).By Lemma 2

( p + tv − p) ·

x(t) − x(0)

= tv ·

x(t) − x(0) 0.

Dividing by t2 gives

v ·x(t) − x(0)

t 0.



KC Border WARP and the Slutsky matrix 3

Taking limits givesv · x(0) 0.

But

x

i(t) =

j

∂si

p + tv,x(0)

∂p j

v j.

Evaluating this at t = 0 gives

0 v · x(0) =

i

j

σi,j( p,m)viv j,

which completes the proof.

References

[1] Kihlstrom, R. E., A. Mas-Colell, and H. Sonnenschein. 1976. The demand theory of the weak axiom of revealed preference. Econometrica 44:971–978.

[2] Samuelson, P. A. 1965. Foundations of economic analysis . New York: Athenaeum.Reprint of edition published by Harvard University Press, 1947.

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