Slutsky Decomposition
Econ 2100
Fall 2015
Lecture 7, September 23
Outline
1
Connections between Walrasian and Hicksian demand functions.
2
Slutsky Decomposition: Income and Substitution E¤ects
From Last Class
De…nition
Given a utility function u : Rn+ ! R, the Hicksian demand correspondence
h : Rn++ u(Rn+ ) ! Rn+ is de…ned by
h (p; v ) = arg minn p x subject to u(x)
x 2R+
v:
De…nition
Given a continuous utility function u : Rn+ ! R, the expenditure function
e : Rn++ u(Rn+ ) ! R+ is de…ned by
e(p; v ) = p x
for some x 2 h (p; v ).
Proposition
Suppose u is continuous and locally nonsatiated. If v > u(0n ), then:
x (p; w )
=
h (p; v (p; w ));
h (p; v )
=
x (p; e(p; v )):
Next a Proof of this proposition.
Walrasian and Hicksian Demand Are Equal
We want to show that x 2 x (p; w ) solves min p x subject to u(x)
We want to show that x 2 h (p; v ) solves max u(x) subject to p x
Proof.
u(x )
p x
Pick x 2 x (p; w ), and suppose x 2
= h (p; u(x )): x is a utility maximizer
but not expenditure minimizer.
p x0 < p x
w and u(x 0 ) u(x )
Then 9x 0 s.t.
By local nonsatiation, 9x " (close to x 0 ) s.t.
p x " < w and u(x ") > u(x 0 )
contradicting the fact that x is a utility maximizer. Therefore x must also be
a expenditure minimizer.
Since p x = w by full expenditure, we also have e (p; v (p; w )) = w .
Pick x 2 h (p; v ), and suppose x 2
= x (p; p x ): x an expenditure
minimizer but not a utility maximizer.
u(x 0 ) > u (x ) and p x 0 p x
x 0 with < 1. By continuity of u,
u( x 0 ) > u (x ) v for < 1; ! 1:
p x0 < p x0 p x
Therefore,
contradicting the fact that x is an expenditure minimizer. Therefore x must
also be a utility maximizer.
Then 9x 0 s.t.
Consider the bundle
Because u (x ) = v we have v (p; e(p; v )) = v .
Support Function
De…nition
Given a closed set K
Rn , the support function
K (p)
K
= inf p x:
: Rn ! R [ f 1g is
x2K
It can equal 1 since there might exists x 2 K such that p x becomes
unboundedly negative for a closed set.
for example: K = fx 2 R2 : x1 2 R; x2 2 [0; 1)g, and p = ( 1; 0).
if K is convex (closed and bounded in Rn ), the support function is …nite.
When a set K is convex, one can ‘recover’it using the support function:
given a p 2 Rn , fx 2 Rn : p x
K (p)g is an half space that contains K ;
furthermore, K is the intersection of all such half spaces (for all p).
If K is not convex, the intersection of all sets fx 2 Rn : p x
K (p)g is the
smallest closed convex set containing K (this is called the convex hull).
Theorem (Duality Theorem)
Let K be a nonempty closed set. There exists a unique x 2 K such that
p x = K (p) if and only if K is di¤erentiable at p. If so, r K (p) = x.
The support function is ‘linear’in p.
e(p; v ) is the support function of the set K = fx 2 Rn+ : u(x)
v g.
Properties of the Expenditure Function
Proposition
If u(x) is continuous, locally nonsatiated, and strictly quasiconcave, then e(p; v ) is
di¤erentiable in p.
Proof.
Immediate from the previous theorem (verify the assumptions hold).
Question 4 Problem Set 4
Using the result above and the fact that h (p; v ) = x (p; e(p; v )), provide
su¢ cient conditions for di¤erentiability with respect to p of the indirect utility
function and extend the su¢ cient conditions for continuity and di¤erentiability in p
of Walrasian demand to Hicksian demand.
Shephard’s Lemma
Proposition (Shephard’s Lemma)
Suppose u : Rn+ ! R is a continuous, locally nonsatiated, and strictly quasiconcave
utility function. Then, for all p 2 Rn++ and v 2 R,
h (p; v ) = rp e(p; v ):
Hicksian demand is the derivative of the expenditure function.
There are di¤rent ways to prove Shephard’s Lemma:
Use the duality theorem.
Use the envelope theorem:
let (x ; q) = p x , (x (q); q) = e(p; v ), and F (x ; q) = u(x ) v ,
then:
>
Dq (x (q); q) = Dq (x ; q)jx =x (q);q=q [ (q)] Dq F (x ; q)jx =x
becomes
rp e(p; v ) = h (p; v ) 0
since F does not depend on p.
Use a brute force and ignorance (b…) argument (next slide).
(q);q=q
Shephard’s Lemma: Proof
Proof.
We want to show that h (p; v ) = rp e(p; v ).
Using the de…nition, and then the chain rule
rp e(p; v ) = rp [p h (p; u)] = h (p; u) + [p Dp h (p; u)] >
The …rst order conditions of the minimization problem say
p = rp u (h (p; u))
Therefore
rp e(p; v ) = h (p; u) + [rp u (h (p; u)) Dp h (p; u)] >
At an optimum, the constraint must bind and so
u (h (p; u)) = v
Thus
and therefore
as desired.
rp u (h (p; u)) Dp h (p; u) = 0
h (p; v ) = rp e(p; v ):
Roy’s Identity
Proposition (Roy’s identity)
Suppose u is continuous, locally nonsatiated, and strictly quasiconcave and v is
di¤erentiable at (p; w ) 6= 0. Then
1
x (p; w ) = @v (p;w ) rp v (p; w );
@w
This can be also written as
xk (p; w ) =
@v (p;w )
@p k
@v (p;w )
@w
;
for all k
Walrasian demand equals the derivative of the indirect utility function
multiplied by a “correction term”.
This correction normalizes by the marginal utility of wealth.
There are di¤rent ways to prove Roy’s Identity
Use the envelope theorem (last class).
Use the chain rule and the …rst order conditions.
Use a b… argument (next slide).
Roy’s Identity
Proof.
We want to show that
xk (p; w ) =
@v (p;w )
@p k
@v (p;w )
@w
Fix some p; w and let u = v (p; w ).
The following identity holds for all p
v (p; e (p; u)) = u
di¤erentiating w.r.t pk we get
@v @e
@v
+
=0
@pk
@w @pk
and therefore
@v
@v
hk = 0
+
@pk
@w
Let xk = xk (p; w ) and evaluate the previous equality at p; w :
@v
@v
+
xk = 0
@pk
@w
Solve for xi to get the result.
Slutsky Matrix
De…nition
The Slutsky matrix, denoted Dp h (p; v ), is the n n matrix of derivative of the
Hicksian demand function with respect to price (its …rst n dimensions).
Notice this says “function”, so the Slutsky matrix is de…ned only when
Hicksian demand is unique.
Proposition
Suppose u : Rn+ ! R is continuous, locally nonsatiated, strictly quasiconcave and
h ( ) is continuously di¤erentiable at (p; v ). Then:
1
The Slutsky matrix is the Hessian of the expenditure function:
2
Dp h (p; v ) = Dpp
e(p; v );
2
The Slutsky matrix is symmetric and negative semide…nite;
3
Dp h (p; v )p = 0n .
Proof.
Question 5, Problem Set 4.
Slutsky Decomposition
Proposition
If u is continuous, locally nonsatiated, and strictly quasiconcave, and h is
di¤erentiable, then: for all (p; w )
Dp h (p; v (p; w )) = Dp x (p; w ) + Dw x (p; w )x (p; w )>
or
@hj (p; v (p; w ))
@xj (p; w ) @xj (p; w )
=
+
xk (p; w );
@pk
@pk
@w
for all j; k
Proof.
Remember that
h (p; v ) = x (p; e(p; v )):
Take the equality for good j and di¤erentiate with respect to pk
@xj
@xj @e
@xj
@xj
@hj
=
+
=
+
h
@pk
@pk
@w @pk
@pk
@w k
Evaluate this for some p; w and u = v (p; w ) so that h ( ) = x ( )
@hj
@xj
@xj
=
+
x
@pk
@pk
@w k
Slutsky Equation
Hicksian decomposition of demand
Rearranging from the previous proposition:
@xj (p; w )
@hj (p; v (p; w ))
=
@pk
@pk
|
{z
}
substitution e¤ect
This is also known as the Slutsky equation:
@xj (p; w )
xk (p; w )
| @w {z
}
income e¤ect
it connects the derivatives of compensated and uncompensated demands.
If one takes k = j, the following is the “own price” Slutsky equation
@xj (p; w )
@hj (p; v (p; w )) @xj (p; w )
=
x (p; w )
@pj
@pj
@w {z j
}
|
{z
} |
substitution e¤ect
income e¤ect
Normal, Inferior, and Gi¤en Goods
The own price Slutsky equation
@hj (p; v (p; w ))
@xj (p; w )
=
@pj
@pj
|
{z
}
substitution e¤ect
@xj (p; w )
xj (p; w ) :
| @w {z
}
income e¤ect
Since the Slutsky matrix is negative semide…nite, @hk =@pk
0;
the …rst term is always negative while the second can have either sign;
the substitution e¤ect always pushes the consumer to purchase less of a
commodity when its price increases.
Normal, Inferior, and Gi¤en Goods
A normal good has a positive income a¤ect.
An inferior good has a negative income e¤ect.
A Gi¤en good has a negative overall e¤ect (i.e. @xk =@pk > 0);
this can happen only if the income e¤ect is negative and overwhelms the
substitution e¤ect
@xk
@hk
xk <
0:
@w
@pk
Slutsky Equation and Elasticity
De…nition
"y ;q =
@y q
@q y
is called the elasticity of y with respect to q
Elasticity is a unit free measure (percentage change in y for a given percentage
change in q) that is often used to compare price e¤ects across di¤erent goods.
The own price Slutsky equation
@hj (p; v (p; w ))
@xj (p; w )
=
@pj
@pj
@xj (p; w )
xj (p; w )
@w
Multiply both sides by pj =xj ( ) and rewrite as
@xj (p; w )
@hj (p; v (p; w ))
pj
pj
=
@pj
xj (p; w )
@pj
xj (p; w )
or
"xj ;pj = "hj ;pj
"xj ;w
@xj (p; w )
xj (p; w )
w
pj
@w
xj (p; w )
w
pj xj (p; w )
w
The elasticty of Walrasian (uncompensated) demand is equal to the elasticity of
Hicksian (compensated) demand minus the income elasticity of demand
multiplied by the share that good has in the budget.
Gross Substitutes
The cross price Slutsky equation
@hj (p; v (p; w ))
@xj (p; w )
=
@pk
@pk
@xj (p; w )
xj (p; w )
@w
Intuitively, we think of two goods as substitutes if the demand for one increases
when the price of the other increases.
De…nition
We say good j is a gross substitute for k if
@xj (p;w )
@p k
0.
Unfortunately the de…nition of substitutes based on Walrasian
(uncompensated) demand is not very useful since it does not satisfy symmetry:
@x (p;w )
@xk (p;w )
we can have j@pk > 0
and
< 0.
@p j
There is a better de…nition that is based on Hicksian (compensated) demand.
Net Substitutes
De…nition
We say goods j and k are
@h (p;w )
net substitutes if j@pk
0
This de…nition is symmetric:
@h j (p;w )
@p k
and
net complements if
0 if and only if
@h k (p;w )
@p j
@h j (p;w )
@p k
0.
Proof.
)
@ @e(p;v
@hj (p; w )
@p j
=
@pk
@pk
2
@ e(p; v )
=
@pk @pj
@ 2 e(p; v )
@pj @pk
@hk (p; w )
=
@pj
=
by Shephard’s Lemma
by Young’s Theorem
by Shephard’s Lemma again
0.
Econ 2100
Fall 2015
Problem Set 4
Due 28 September, Monday, at the beginning of class
1. Suppose u is twice continuously di¤erentiable, locally nonsatiated, strictly quasiconcave, and that
there exists " > 0 such that xi (p; w) > 0 if and only if xi (p; w) > 0, for all (p; w) such that
k(p; w) (p; w)k < ". Also assume that H is nonsingular at (p; w). Prove that under these conditions
x is continuously di¤erentiable at (p; w).
2. Suppose a government has to decide on a budget. Its tax revenue in 2012 is Y and its tax revenue in
2013 will be kY , where k > 1 is the growth rate of the economy. The government has to decide how
much to spend this year, x1 , and how much to spend next year, x2 . Because of the poor economy this
year, the government will certainly run a de…cit x1 Y > 0. For each dollar of de…cit, the government
will have to pay bond holders 1 + r dollars next year, where r > 0 is the interest rate. Next year,
the government can spend its 2013 tax revenue kY , minus what it will owe bond holders to …nance
the 2012 de…cit. (The government cannot continue to run a de…cit in 2013). The government’s
objective function is U (x1 ; x2 ) = v(x1 ) + v(x2 ), where v refers to the annual social bene…t generated
by government spending. Assume v is C 2 , v 0 (c) > 0 and v 00 (c) < 0 for each c. Assume a strictly
positive solution (i.e., x
0).
(a) Write the government’s budget constraint over 2012 and 2013 spending.
(b) Using the Implicit Function Theorem, determine the comparative statics of 2012 and 2013
@x
@x
@x
spending with respect to the interest rate and the growth rate. That is, …nd @r1 , @k1 , @r2 ,
@x2
. as explicitly as possible, and determine the signs of these expressions as fully as you can.
@k
Interpret brie‡y.
3. Prove that if u is continuous and locally nonsatiated, then the expenditure function is homogeneous
of degree 1 and concave in p.
4. Using this fact and h (p; v) = x (p; e(p; v)), provide su¢ cient conditions for di¤erentiability with
respect to p of the indirect utility function and extend the su¢ cient conditions for continuity and
di¤erentiability in p of Walrasian demand to Hicksian demand.
5. Suppose u : Rn+ ! R is continuous, locally nonsatiated, strictly quasiconcave (equivalently, you need
h to be single-valued) and h is continuously di¤erentiable at (p; v). Let Dp h (p; v) denote the n n
derivative matrix of the Hicksian demand function h with respect to price (its …rst n dimensions),
which we call the Slutsky matrix. Prove the following.
2
(a) Dp h (p; v) = Dpp
e(p; v) (the Slutsky matrix is the Hessian of the expenditure function);
(b) Dp h (p; v) is symmetric and negative semide…nite (M is symmetric if Mij = Mji for all i; j =
1; : : : ; n, and it is negative semide…nite if z > M z 0 for all z 2 Rn . );
(c) Dp h (p; v)p = 0n .
1