Leonardo Felli 23 October, 2002
Microeconomics II Lecture 3
Constrained Envelope Theorem
Consider the problem:
maxx
f (x)
s.t. g(x, a) = 0
The Lagrangian is:
L(x, λ, a) = f (x)− λ g(x, a)
Necessary FOC are:
f ′(x∗)− λ∗∂g(x∗, a)
∂x= 0
g(x∗(a), a) = 0
1
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Substituting x∗(a) and λ∗(a) in the Lagrangian we
get:
L(a) = f (x∗(a))− λ∗(a) g(x∗(a), a)
Differentiating we get:
dL(a)
d a=
[f ′(x∗)− λ∗
∂g(x∗, a)
∂x
]d x∗(a)
d a−
−g(x∗(a), a)dλ∗(a)
da− λ∗(a)
∂g(x∗, a)
∂a
= −λ∗(a)∂g(x∗, a)
∂a
by the necessary FOC.
In other words — to the first order — only the direct
effect of a on the Lagrangian function matters.
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3. Roy’s identity:
xi(p, m) = −∂V/∂pi
∂V/∂m
By the constrained envelope theorem and the obser-
vation that:
V (p, m) = u(x(p, m))− λ(p, m) [p x(p, m)−m]
we shall obtain:
∂V/∂pi = −λ(p, m) xi(p, m) ≤ 0
and
∂V/∂m = λ(p, m) ≥ 0
which is the marginal utility of income.
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(Notice that the sign of the two inequalities above
prove property 1 of the indirect utility function V (p, m).)
We conclude the proof substituting
∂V/∂m = λ(p, m)
into
∂V/∂pi = −λ(p, m) xi(p, m)
and solving for xi(p, m).
4. Adding up results. From the identity:
p x(p, m) = m ∀p, ∀m
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Differentiation with respect to pj gives:
xj(p, m) +
L∑i=1
pi∂xi
∂pj= 0
or, more interestingly, with respect to m gives:
L∑i=1
pi∂xi
∂m= 1
There does not exist a clear cut comparative-static
property with the exception of:
0 ≥L∑
i=1
pi∂xi
∂ph= −xh(p, m)
which means that at least one of the Marshallian
demand function has to be downward sloping in ph
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Effect of a change in income on the level of the Mar-
shallian demand:∂xl
∂m
In the two commodities graph the set of tangency
points for different values of m is known as the in-
come expansion path.
In the commodity income graph the set of optimal
choices of the quantity of the commodity is known as
Engel curve.
Microeconomics II 7
We shall classify commodities with respect to the ef-
fect of changes in income in:
• normal goods:∂xl
∂m> 0
• neutral goods:∂xl
∂m= 0
• inferior goods:∂xl
∂m< 0
Notice that for every level of income m at least one
of the L commodities is normal:
L∑l=1
pl∂xl
∂m= 1
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If the Engel curve is convex we are facing a luxury
good in other case a necessity.
-
6x(p, m)
m
luxury
necessity
Microeconomics II 9
Expenditure Minimization Problem
The dual problem of the consumer’s utility maxi-
mization problem is the expenditure minimization
problem:
min{x}
p x
s.t. u(x) ≥ U
Define the solution as:
x = h(p, U) =
h1(p1, . . . , pL, U)
...
hL(p1, . . . , pL, U)
the Hicksian (compensated) demand functions.
We shall also define:
e(p, U) = p h(p, U)
as the expenditure function.
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Properties of the expenditure function:
1. Continuous in p and U .
2. ∂e∂U > 0 (2.1) and ∂e
∂pl≥ 0 (2.2) for every l =
1, . . . , L.
Proof: (2.1): Suppose not: there exist U ′ < U ′′
(denote x′ and x′′ the corresponding solution to the
e.m.p.) such that p x′ ≥ p x′′ > 0.
If the latter inequality is strict we have an immediate
contradiction of x′ solving e.m.p.;
if on the other hand p x′ = p x′′ > 0 then by con-
tinuity and strict monotonicity of u(·) there exists
α ∈ (0, 1) close enough to 1 such that u(α x′′) > U ′
and p x′ > p αx′′ which contradicts x′ solving e.m.p..
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(2.2): consider p′ and p′′ such that p′′l ≥ p′l but p′′k =
p′k for every k 6= l.
Let x′′ and x′ be the solutions to the e.m.p. with p′′
and p′ respectively.
Then by definition of e(p, U)
e(p′′, U) = p′′ x′′ ≥ p′ x′′ ≥ p′ x′ = e(p′, U).
3. Homogeneous of degree 1 in p.
Proof: The feasible set of the e.m.p. does not change
when prices are multiplied by the factor k > 0.
Hence ∀k > 0, minimizing (k p) x on this set leads
to the same answer. Let x∗ be the solution, then:
e(k p, U) = (k p) x∗ = k e(p, U).
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4. Concave (graphic intuition) in p.
Proof: let p′′ = t p + (1− t) p′ for t ∈ [0, 1]. Let x′′
be the solution to e.m.p. for p′′. Then
e(p′′, U) = p′′ x′′ = t p x′′ + (1− t) p′ x′′
≥ t e(p, U) + (1− t) e(p′, U)
since u(x′′) ≥ U and by definition of e(p, U).
Properties of the Hicksian demand func-
tions:
h(p, U)
1. Shephard’s Lemma.
∂e(p, U)
∂pl= hl(p, U)
Proof: by constrained envelope theorem.
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2. Homogeneity of degree 0 in p.
Proof: by Shephard’s lemma and the fact that the
following theorem.
Theorem. If a function F (x) is homogeneous of
degree r in x then (∂F/∂xl) is homogeneous of
degree (r − 1) in x for every l = 1, . . . , L.
Proof: Differentiating the identity that defines hom-
geneity of degree r:
F (k x) = kr F (x) ∀k > 0
with respect to xl we obtain:
k∂F (k x)
∂xl= kr∂F (x)
∂xl
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The latter equation corresponds to the definition of
homogeneity of degree (r − 1):
∂F (k x)
∂xl= k(r−1) ∂F (x)
∂xl.
Euler Theorem. If a function F (x) is homoge-
neous of degree r in x then:
r F (x) = ∇F (x) x
Proof: Differentiating with respect to k the identity:
F (k x) = kr F (x) ∀k > 0
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we obtain:
∇F (kx) x = rk(r−1) F (x)
for k = 1 we obtain:
∇F (x) x = r F (x).
3. The matrix of cross-partial derivatives (Substitu-
tion matrix) with respect to p
S =
∂h1∂p1
· · · ∂h1∂pL
... . . . ...∂hL∂p1
· · · ∂hL∂pL
is negative semi-definite and symmetric. (Main di-
agonal non-positive).
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Proof: Simmetry follows from Shephard’s lemma
and Young Theorem.
Indeed:
∂hl
∂pi=
∂
∂pi
(∂e(p, U)
∂pl
)=
∂
∂pl
(∂e(p, U)
∂pi
)=
∂hi
∂pl
While negative semi-definiteness follows from the con-
cavity of e(p, U) and the observation that S is the
Hessian of the function e(p, U).
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Identities:
V [p, e(p, U)] ≡ U
xl[p, e(p, U)] ≡ hl(p, U) ∀l
e[p, V (p, m)] ≡ m
hl[p, V (p, m)] ≡ xl(p, m) ∀l
Slutsky decomposition:
start from the identity
hl(p, U) ≡ xl[p, e(p, U)]
if the price pi changes the effect is:
∂hl
∂pi=
∂xl
∂pi+
∂xl
∂m
∂e
∂pi
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Notice that by Shephard’s lemma:
∂e
∂pi= hi(p, U) = xi[p, e(p, U)]
then∂hl
∂pi=
∂xl
∂pi+
∂xl
∂mxi.
or∂xl
∂pi=
∂hl
∂pi− ∂xl
∂mxi.
Own price effect gives Slutsky equation:
∂xl
∂pl=
∂hl
∂pl− ∂xl
∂mxl.
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Slutsky decomposition:
∂xl
∂pi=
∂hl
∂pi− ∂xl
∂mxi.
Slutsky equation:
∂xl
∂pl=
∂hl
∂pl− ∂xl
∂mxl.
This latter equation corresponds to the distinction
between substitution and income effect:
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Substitution effect:
∂hl
∂pl
Income effect:∂xl
∂mxl
x2
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Microeconomics II 21
We know the sign of the substitution effect it is non-
positive.
The sign of the income effect depends on whether the
good is normal or inferior.
In the case that:
∂xl
∂pl> 0
we conclude that the good is Giffen.
This is not a realistic feature, inferior good with a
big income effect.