Download Microeconomics II Lecture 3 Constrained Envelope Theorem

Leonardo Felli
23 October, 2002
Microeconomics II
Lecture 3
Constrained Envelope Theorem
Consider the problem:
max f (x)
x
s.t. g(x, a) = 0
The Lagrangian is:
L(x, λ, a) = f (x) − λ g(x, a)
Necessary FOC are:
0
∗
f (x ) − λ
∗ ∂g(x
∗
, a)
=0
∂x
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 x∗(a)
0 ∗
∗ ∂g(x , a)
= f (x ) − λ
−
da
∂x
da
dλ∗(a)
∂g(x∗, a)
∗
∗
− λ (a)
−g(x (a), a)
da
∂a
∗
∂g(x , a)
= −λ∗(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 observation 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
X
pi
i=1
∂xi
=0
∂pj
or, more interestingly, with respect to m gives:
L
X
i=1
pi
∂xi
=1
∂m
There does not exist a clear cut comparative-static
property with the exception of:
0≥
L
X
i=1
pi
∂xi
= −xh(p, m)
∂ph
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 Marshallian demand:
∂xl
∂m
In the two commodities graph the set of tangency
points for different values of m is known as the income expansion path.
In the commodity income graph the set of optimal
choices of the quantity of the commodity is known as
Engel curve.
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We shall classify commodities with respect to the effect of changes in income in:
• normal goods:
∂xl
>0
∂m
• neutral goods:
∂xl
=0
∂m
• inferior goods:
∂xl
<0
∂m
Notice that for every level of income m at least one
of the L commodities is normal:
L
X
l=1
∂xl
pl
=1
∂m
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If the Engel curve is convex we are facing a luxury
good in other case a necessity.
x(p, m)
6
luxury
necessity
-
m
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Expenditure Minimization Problem
The dual problem of the consumer’s utility maximization problem is the expenditure minimization
problem:
min p x
{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 0 < U 00
(denote x0 and x00 the corresponding solution to the
e.m.p.) such that p x0 ≥ p x00 > 0.
If the latter inequality is strict we have an immediate
contradiction of x0 solving e.m.p.;
if on the other hand p x0 = p x00 > 0 then by continuity and strict monotonicity of u(·) there exists
α ∈ (0, 1) close enough to 1 such that u(α x00) > U 0
and p x0 > p αx00 which contradicts x0 solving e.m.p..
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(2.2): consider p0 and p00 such that p00l ≥ p0l but p00k =
p0k for every k 6= l.
Let x00 and x0 be the solutions to the e.m.p. with p00
and p0 respectively.
Then by definition of e(p, U )
e(p00, U ) = p00 x00 ≥ p0 x00 ≥ p0 x0 = e(p0, 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 p00 = t p + (1 − t) p0 for t ∈ [0, 1]. Let x00
be the solution to e.m.p. for p00. Then
e(p00, U ) = p00 x00 = t p x00 + (1 − t) p0 x00
≥ t e(p, U ) + (1 − t) e(p0, U )
since u(x00) ≥ U and by definition of e(p, U ).
Properties of the Hicksian demand functions:
h(p, U )
1. Shephard’s Lemma.
∂e(p, U )
= hl (p, U )
∂pl
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 homgeneity of degree r:
F (k x) = k r F (x)
∀k > 0
with respect to xl we obtain:
∂F (k x)
r ∂F (x)
k
=k
∂xl
∂xl
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The latter equation corresponds to the definition of
homogeneity of degree (r − 1):
∂F (k x)
(r−1) ∂F (x)
=k
.
∂xl
∂xl
Euler Theorem. If a function F (x) is homogeneous of degree r in x then:
r F (x) = ∇F (x) x
Proof: Differentiating with respect to k the identity:
F (k x) = k r 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 (Substitution matrix) with respect to p

∂h1
∂h1
·
·
·
∂pL
 ∂p1
.. . . . ..
S=

∂hL
∂hL
·
·
·
∂p1
∂p




L
is negative semi-definite and symmetric. (Main diagonal 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 concavity of e(p, U ) and the observation that S is the
Hessian of the function e(p, U ).
Microeconomics II
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 ∂xl ∂xl ∂e
=
+
∂pi ∂pi ∂m ∂pi
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Notice that by Shephard’s lemma:
∂e
= hi(p, U ) = xi[p, e(p, U )]
∂pi
then
∂hl ∂xl ∂xl
=
+
xi .
∂pi ∂pi ∂m
or
∂xl ∂hl ∂xl
=
−
xi .
∂pi ∂pi ∂m
Own price effect gives Slutsky equation:
∂xl ∂hl ∂xl
=
−
xl .
∂pl ∂pl ∂m
18
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Slutsky decomposition:
∂xl ∂hl ∂xl
=
−
xi .
∂pi ∂pi ∂m
Slutsky equation:
∂xl ∂hl ∂xl
=
−
xl .
∂pl ∂pl ∂m
This latter equation corresponds to the distinction
between substitution and income effect:
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Substitution effect:
∂hl
∂pl
Income effect:
∂xl
xl
∂m
x1
6
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e
e
e
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e
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e
x
x
x
-
x2
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We know the sign of the substitution effect it is nonpositive.
The sign of the income effect depends on whether the
good is normal or inferior.
In the case that:
∂xl
>0
∂pl
we conclude that the good is Giffen.
This is not a realistic feature, inferior good with a
big income effect.