Why? Theory – observable characteristics – testing – for future
5.1 Relative prices and real income
z Money is a veil. Economists often prefer to measure important economic
variables in real, rather than nominal terms. z Relative prices
pi
pjw pj
z Relative income
z Homogeneity and budget balancedness
x(P,w)=x(tP,tw)∀t>0⇒x(P,w)=x(
p1p2w,,...,1,) pnpnpn
Px(P,w)=w∀(P,w)
5.2 The Slutsky equation
z What do we expect from the theory of consumer? --- The law of demand
Classical theory: the principle of diminishing marginal utility
z However, Giffen good! It is a product for which, among a poor population, a rise
in price will lead people to buy even more of the product. Possible examples: the potato in the Irish famine of 1845-1849
[Figure 1.19]
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z Normal good and inferior good
∂xi(P,w)
>0 ∂w∂x(P,w)
The “bad” is … i<0
∂wThe “good” is the product for which demand increases in income.
A Giffen good must be inferior, OR/AND an inferior good must be a Giffen good ??
z Income effect and substitution effect TE=SE+IE
Total effect: (price effect) measures the change of quantity demanded for a good
in response to a change in its price.
Substitution effect: measures the effect of relative price change, in which the
consumer substitutes the relatively cheaper good for the relatively expensive good, and remains indifferent after the price change. Income effect:
[Figure 1.20]
z Theorem 5.1 The Slutsky equation:
∂xi(P,w)∂xih(P,u∗)∂x(P,w)
=−xj(P,w)i
∂pj∂pj∂w
Proof: from last chapter we know: xih(P,u*)=xi(P,e(P,u*))
∂xih(P,u∗)∂xi(P,e(P,u*))∂xi(P,e(P,u*))∂e(P,u*)
=+
∂pj∂pj∂w∂pj
e(P,u*)=e(P,v(P,w))=w
Shephard’s Lemma:
∂e(P,u*)h
=xhj(P,u*)=xj(P,v(P,w)) ∂pj
xhj(P,v(P,w))=xj(P,w)
2
∂xih(P,u∗)∂xi(P,w)∂xi(P,w)⇒=+xj(P,w)
∂pj∂pj∂w∂xi(P,w)∂xih(P,u∗)∂x(P,w)
⇒=−xj(P,w)i
∂pj∂pj∂w
z Theorem 5.2 The law of demand:
∂xih(P,u)
1. Hicksian demand: ≤0 (Negative Own-Substitution Terms)
∂pi
2. Marshallian demand: A decrease in the own price of a normal good will cause quantity demanded to increase. If an own price decrease causes a decrease in quantity demanded, the good must be inferior.
Proof: using Shephard’s Lemma and the fact that the expenditure is a concave function. Hence…
Classical theory vs. Modern theory
z Theorem 5.3 Symmetric, negative semidefinite (Hicksian) substitution
matrix
⎛∂x1h(P,u)∂x1h(P,u)⎞
⎜⎟...
∂pn⎟⎜∂p1
⎟ :::σ(P,u)≡⎜
hh⎜∂xn(P,u)∂xn(P,u)⎟
...⎜⎟⎜∂p1∂pn⎟
⎝⎠
Proof: 1) The order of differentiation of the expenditure function makes no
h
∂e2(P,u)∂e2(P,u)∂xih(P,u)∂xj(P,u)
difference! = ⇒=
∂pi∂pj∂pj∂pi∂pj∂pi
2) σ(P,u) is simply the Hessian matrix of second-order price partials of
the expenditure function.
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z Slutsky matrix
∂x1(P,w)∂x1(P,w)⎛∂x1(P,w)∂x1(P,w)⎞
++x1(P,w)...xn(P,w)⎟⎜
∂w∂pn∂w⎜∂p1⎟:::⎜⎟
∂xn(P,w)∂xn(P,w)⎜∂xn(P,w)∂xn(P,w)⎟
+(,)...+(,)xPwxPw1n⎜∂p⎟∂w∂pn∂w1⎝⎠
5.3 Some Elasticity Relations
z A general formula for elasticity: exy=
∆x%∂xy∂lnx
== ∆y%∂yx∂lny
z Income Elasticity ηi≡
∂xi(P,w)w
∂wxi(P,w)∂xi(P,w)pj
∂pjxi(P,w)
z Price Elasticity εij≡
z Aggregation in consumer demand, where income share
si≡
pixi(P,w)
w
si≥0and
∑s
in
i
=1
Engel aggregation: ∑i=1siηi=1 Cournot aggregation:
5.4 Integrability
z We ask a question: if we observe a demand function x(P,w) that has these
∑
n
i=1iij
sε=−sj
properties (homogeneous of degree zero, satisfy Walras’ law, and have a symmetric and negative semidefinite Slutsky matrix), can we find preferences that rationalize x(P,w)?
-- This integrability problem has a long tradition, beginning with Antonelli 1886.
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z Why we ask such question?
1. On a theoretical level, the properties of demand (homogeneity of degree one,
satisfaction of Walras’ law, and a symmetric and negative semidefinite substitution matrix) are not only necessary consequences of the preference-based demand theory, but these are also all of the consequences. As long as consumer demand satisfies these properties, there is some rational preference relation that could have generated this demand.
2. The result completes our study of the relation between the preference-based
theory of demand and the choice-based theory of demand. In general, demand satisfying the weak axiom cannot be rationalized by preferences, because the substitution matrix could be not symmetric. Hence, the result here shows us, that the only thing added to the properties of demand by the rational preference hypothesis, beyond what is implied by the weak axiom, homogeneity of degree one, and Walras’ law, is symmetry of the substitution matrix.
3. On a practical level, the result tells us how and when we can recover the
information of preferences from observation of the consumer’s demand behavior. 4. In empirical studies, it allows us to begin by specifying a tractable demand
function and then check whether it satisfies the necessary and sufficient conditions. We need not derive the utility function. z
Two steps to recover preferences from demand:
1. recovering preferences from the expenditure function e(P,u)
Theorem 5.4 Constructing a utility function from an expenditure function
Let )e(P,w be a function satisfying all properties of an expenditure function, then we can construct a utility function by u(x)≡max{u≥0|x∈A(u)}, where
nA(u)≡{x∈R+|Px≥e(P,u)∀P>>0}
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2. recovering an expenditure function from x(P,w)
Theorem 5.5 Constructing an expenditure function from a demand function
The necessary and sufficient condition for the recovery of an underlying expenditure function is the symmetric and negative semidefiniteness of the Slutsky equation.
At first, we consider the case of two goods. Suppose p2=1 without loss of generality, pick an arbitrary point (p10,1,w0,u0).
We will now recover the value of the expenditure function e(p1,1,u0)∀p1>0 from following differential equation (to integrate):
de(p1)
=x1h(p1,u)=x1(p1,e(p1)) with the initial condition e(p10)=w0 dp1
For the general case of L commodities, we have a system of partial differential equations:
∂e(P)
=x1(P,e(P))∂p1
...
∂e(P)
=xL(P,e(P))∂pL
The existence of a solution to above system is guaranteed when L>2 if and only if
2
its Hessian matrix Dpe(P) is symmetric. ( Frobenius’ theorem)
In addition, if a solution exists, as long as S(P,w) is negative semidefinite, it will possess the properties of an expenditure function.
The intuition: by changing prices one at a time, we can decompose this problem into L subproblems where only one price changes at each step. Hence, the symmetry of Slutsky matrix implies that the value of expenditure function should not depend on the particular path from the original price to the new price.
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Example:
Suppose there are three goods, the demand function is as follows:
xi(p1,p2,p3,w)=
αiw
pi
,i=1,2,3, where αi>0,andα1+α2+α3=1
We want to find e(p1,p2,p3,u).
We have:
∂e(p1,p2,p3,u)αie(p1,p2,p3,u),=
pi∂pi⇒
∂lne(p1,p2,p3,u)αi=,pi∂pi
i=1,2,3
i=1,2,3
⎧ln(e(P,u))=α1ln(p1)+c1(p2,p3,u)
⎪
⇒⎨ln(e(P,u))=α2ln(p2)+c2(p1,p3,u)⎪ln(e(P,u))=αln(p)+c(p,p,u)
33312⎩
⇒ln(e(P,u))=α1ln(p1)+α2ln(p2)+α3ln(p3)+c(u)
α1α2α3
⇒e(P,u)=c(u)p1p2p3
Because the implied demand behavior is independent of strictly increasing transformation, we can choose c(u)=u.
Recover a utility function from this expenditure function.
n
u(x)≡max{u≥0|x∈A(u)}, where A(u)≡x∈R+|Px≥e(P,u)∀P>>0
{}α1α2α3Px≥e(P,u)=up1p2p3⇔u≤
Px
α1α2α3
p1p2p3
Hence u(x)=
p1x1+p2x2+p3x3
α1α2α3
p1p2p3
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5.5 Welfare analysis
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