### Walrasian Demand

```Walrasian Demand
Econ 2100
Fall 2015
Lecture 5, September 16
Outline
1
Walrasian Demand
2
Properties of Walrasian Demand
3
An Optimization Recipe
4
First and Second Order Conditions
Walrasian Demand
De…nition
Given a utility function u : Rn+ ! R, the Walrasian demand correspondence
x : Rn++ R+ ! Rn+ is de…ned by
x (p; w ) = arg max u(x)
x 2B p;w
where
B(p; w ) = fx 2 Rn+ : p x
w g:
Assumptions:
goods are perfectly divisible;
consumption is non negative;
income is non negative;
prices are strictly positive;
the total price of consumption cannot exceed income;
prices are linear in consumption.
Think of possible violations.
Exercise
Suppose w = \$100. There are two commodities, electricity and food. Each unit of
food costs \$1. The …rst 20Kwh electricity cost \$1 per Kwh, but the price of each
incremetal unit of electricity is \$1:50 per Kwh. Write the consumer’s budget set
formally and draw it.
Walrasian Demand Is Convex
Proposition
If u is quasiconcave, then x (p; w ) is convex.
If u is strictly quasiconcave, then x (p; w ) is unique.
Same as before (u (stricly) quasiconcave means % (strictly) convex).
Proof.
Suppose x; y 2 x (p; w ) and pick
2 [0; 1].
First convexity: need to show x + (1
)y 2 x (p; w ).
x % y by de…nition of x (p; w ).
u is quasiconcave, thus % is convex and x + (1
)y % y .
y % z for any z 2 B (p; w ) by de…nition of x (p; w ).
Transitivity implies x + (1
)y % z for any z 2 B (p; w ); thus
x + (1
)y 2 x (p; w ).
Now uniqueness.
x ; y 2 x (p; w ) and x 6= y imply x + (1
)y y for any 2 (0; 1) because
u is strictly quasiconcave (% is strictly convex).
Since B (p; w ) is convex, x + (1
)y 2 B (p; w ), contradicting
y 2 x (p; w ).
Walrasian Demand Is Non-Empty and Compact
Proposition
If u is continuous, then x (p; w ) is nonempty and compact.
We already proved this as well.
Proof.
De…ne A by
A = B(p; w ) = fx 2 Rn+ : p x
wg
This is a closed and bounded (i.e. compact, set) and
x (p; w ) = C% (A) = C% (B(p; w ))
where % are the preferences represented by u.
Then x (p; w ) is the set of maximizers of a continuous function over a
compact set.
Walrasian Demand: Examples
How do we …nd the Walrasian Demand?
Need to solve a constrained maximization problem, usually using calculus.
Question 2, Problem Set 3; due next Monday.
For each of the following utility functions, …nd the Walrasian demand
correspondence. (Hint: pictures may help)
n
Q
1
u(x) =
xi i with i > 0 (Cobb-Douglas).
i =1
2
3
4
u(x) = minf 1 x1 ; 2 x2 ; :::; n xn g with i > 0 (generalized Leontief).
Pn
u(x) = i =1 i xi for i > 0 (generalized linear).
1
Pn
u(x) =
(generalized CES).
i =1 i xi
Can we do the second one using calculus?
How about the third? Do we need calculus?
Always draw a picture.
Constant elasticity of substitution (CES) preferences are the most commonly
used homothetic preferences. Many preferences are a special case of CES.
An Optimization Recipe
How to solve
max f (x) subject to
1
Write the Langrange function L : Rn
2
Write the First Order Conditions:
gi (x)
0
with i = 1; ::; m
Rm ! R as
m
X
L (x; ) = f (x)
i gi (x)
i =1
n 1
z }| {
rL (x; ) = rf (x)
3
4
|
@f (x)
@xj
m
P
i =1
{z
@g i (x)
i @x =0
j
m
X
i =1
i rgi
(x) = 0
for all j =1;::;n
}
Write constraints, inequalities for , and complementary slackness conditions:
gi (x) 0 with i = 1; ::; m
0 with i = 1; ::; m
i
i gi (x) = 0 with i = 1; ::; m
Find the x and that satisfy all these and you are done...hopefully.
The Recipe In Action: Cobb-Dougals Utility
Compute Walrasian demand when the utility function is u(x1 ; x2 ) = x1 x21
Here x (p; w ) is the solution to
max
x1 x21
(p1 x1 + p2 x2
w)
x1 ;x2 2fp 1 x1 +p 2 x2 w ; x1 0, x2 0g
1
The Langrangian is
L (x; ) = x1 x21
w
The First Order Conditions for x is:
0
1 0
x1 1 x21
w p1 + 1
A =B
rL (x; ) = @
@
|
{z
}
(1
)
x
x
p
+
(1
w
2
2
1
2
2 1
(
1 x1 )
(
2 x2 )
2
u(x1 ;x2 )
x1
) u(xx12;x2 )
w p1
+
1
w p2
+
3
The constraints, inequalities for , and complementary slackness are:
p 1 x1 + p 2 x2 w 0
x1 0, and
x2 0
0,
0,
and
0
w
1
2
(p
x
+
p
x
w
)
=
0,
x
=
0,
and
x
=
0
w
1 1
2 2
1 1
2 2
4
Find a solution to the above (easy for me to say).
1
2
C
A=0
The Recipe In Action: Cobb-Dougals Utility
Compute Walrasian demand when the utility function is u(x1 ; x2 ) = x1 x21
We must solve:
u(x1 ;x2 )
x1
w
w p1
+ 1 = 0 and
p1 x1 + p2 x2 w 0
and
x1 0;
x2 0
(p1 x1 + p2 x2 w ) = 0 and
) u(xx12;x2 )
(1
0;
w
1 x1
w p2
1
= 0;
x (p; w ) must be strictly positive (why?), hence
1
The budget constraint must bind (why?), hence
w
=
+
2
0;
2
2 x2
=0
2
=0
0
= 0.
0.
Therefore the top two equalities become
u(x1 ; x2 ) =
w p1 x1
Dividing one by the other
1
and
= pp12 xx12
(1
) u(x1 ; x2 ) =
w p 2 x2
and use Full Expenditure.
Summing both sides and using Full Expenditure we get
u(x1 ; x2 ) =
w (p1 x1
+ p 2 x2 ) =
ww
Some algebra then yields
x1 (p; w ) =
w
(1
)w
; x2 (p; w ) =
; and
p1
p2
w
=
1
1
p1
p2
Marginal Rate of Substitution
The First Order Conditions for utility maximization take the form:
X
utility
budget
non negativity
r(
)
)
)=0
non negativity r(
budget r(
function
constraint
constraints
constraints
constraint
If the non-negativity constraints hold, the correspoding multipliers equal 0.
Then, at a solution x , the expression above is:
@u (x )
for all j = 1; ::; n
w pj = 0
@xj
One way to rearrange this expression gives
@u(x )
@xj
@u(x )
@xk
=
pj
pk
for any j; k
This is the familiar condition about equalizing marginal rates of substituitons
across goods.
Marginal Utility Per Dollar Spent
Again, the First Order Conditions for utility maximization take the form
@u (x )
for all j = 1; ::; n
w pj
j =0
@xj
Another way to rearrange it gives
@u(x )
@xj
pj
=
w
+
j
pj
for all j = 1; ::; n
In words, if any two goods are consumed in strictly positive amounts, the
marginal utility per dollar spent must be equal across them.
If not, there are goods j and k for which
@u (x
@xj
)
@u (x
)
k
< @x
pj
pk
Then DM can spend " less on j and " more on k so that the budget constraint
still holds (why?).
By Taylor’s theorem, the utility at this new choice is
0
1
u (x )+
@u (x )
@xj
"
@u (x )
+
pj
@xk
@u(x )
"
+o (") = u (x )+" @ @xk
pk
pk
@u(x )
@xj
pj
which implies that x is not an optimum.
Think about the case in which some goods are consumed in zero amount.
A+
Luca’s Rough Guide to Convex Optimization
Where does the recipe come from? Roee told you, but here is my informal
summary.
Question
Let f : Rn+ ! R be a continuous, increasing, and quasi-concave function, and let
C Rn be a convex set. We want to …nd a solution to the following problem:
max f (x)
x 2C
where
C = fx 2 Rn : gi (x)
0 with i = 1; :::; Ng
This is the most general way to state a maximization problem.
Examples
1
If C = Rn , we have an unconstrained problem.
2
If you have constraints like h (x)
b, de…ne gj (x) = h (x)
3
If you have constraints like h (x)
b, de…ne gj (x) =
4
If you have constraints like h (x) = b, de…ne gj (x) = h (x) b and
gk (x) = [h (x) b] and rewrite as fgj (x) 0 and gk (x) 0g.
b.
[h (x)
b].
Simple Geometry Observations
A level curve for some function f : Rn+ ! R is given by f (x) = c for some c 2 R.
The ‘better than’set is
% (x) = fy 2 Rn : f (y )
f (x)g :
Draw C and some level curves (when are better than sets convex?).
The tangent plane to C at a point is a plane through the point that does not
intersect the interior of C .
The tangent to the level curves at a point is a plane through the point that
does not intersect the interior of % (x).
At a maximum, level curves and constraint set are tangent to the same plane.
Geometry In Action
De…nition
An hyperplane is H = fx 2 Rn :
(x
y ) = 0g.
Fix x on the boundary of C , the tangent to C at x is a supporting hyperplane.
De…nition
An hyperplane H supports C at a point x if
C is a strict subset of H
= fx 2 Rn :
(x
y)
0g
The tangent to the better than set at x is also a supporting hyperplane.
De…nition
An hyperplane H supports the better than set at x if
% (x) is a strict subset of H + = fx 2 Rn :
(x
y)
0g
Supporting Hyperplane Theorem
A non-empty convex set has at least one supporting hyperplane.
An optimum is a point x where the same hyperlane supports C and % (x).
Details To Remember
If the better than set or the constraint sets are not convex: big trouble.
If functions are not di¤erentiable: small trouble.
If the geometry still works we can …nd a more general theorem (see convex
analysis).
When does the recipe fail?
If the constraint quali…cation condition fails.
If the objective function is not quasi concave.
This means you must check the second order conditions when in doubt.
these have to do with the matrix of second derivatives;
look at Section M.K in MWG for details (boring and mechanical, but one needs
to know them);
Varian also has a good quick and dirty guide.
Summary
Convex Optimization Summary
gi (x)
0
max f (x) subject to
When x is a solution of the problem
i = 1; ::; m
then
rL (x ; ) = rf (x )
m
X
i =1
i rgi
(x ) = 0
with
i
0
and
i gi
(x ) = 0
and
Dx2 L (x ; ) = D 2 f (x )
for each i = 1; ::; m
m
X
iD
2
gi (x )
i =1
is negative semide…nite on fz 2 Rn : rgi (x ) z = 0 for each i = 1; ::; mg...
...provided constraint quali…cation holds.
Demand and Indirect Utility Function
The Walrasian demand correspondence x : Rn++ R+ ! Rn+ is de…ned by
x (p; w ) = arg max u(x)
where
B(p; w ) = fx 2 Rn+ : p x w g:
x 2B (p;w )
De…nition
Given a continuous utility function u : Rn+ ! R, the indirect utility function
v : Rn++ R+ ! R is de…ned by
v (p; w ) = u(x (p; w ))
where
x (p; w ) 2 arg max u(x):
x 2B (p;w )
This indirect utility function measures changes in the ‘optimized’value of the
objective function as the parameters (prices and wages) change and the
consumer adjusts her optimal consumption accordingly.
Results
The Walrasian demand correspondence is upper hemi continuous
To prove this we need properties that characterize continuity for
correspondences.
The indirect utility function is continuous.
To prove this we need properties that characterize continuity for
correspondences.
Berge’s Theorem of the Maximum
The theorem of the maximum lets us establish the previous two results.
Theorem (Theorem of the Maximum)
If f : X ! R is a continuous function and ' : Q ! X is a continuous
correspondence with nonempty and compact values, then
the mapping x : Q ! X de…ned by
x (q) = arg max f (x)
x 2'(q)
is an upper hemicontinuous correspondence and
the mapping v : Q ! R de…ned by
v (q) = max f (x)
is a continuous function.
x 2'(q)
Berge’s Theorem is useful when exogenous parameters enter the optimization
problem only through the constraints, and do not directly enter the objective
function.
This happens in the consumer’s problem: prices and income do not enter the
utility function, they only a¤ect the budget set.
Properties of Walrasian Demand
Proposition
If u(x) is continuous, then x (p; w ) is upper hemicontinuous and v (p; w ) is
continuous.
Proof.
Apply Berge’s Theorem:
If u : Rn+ ! R a continuous function and B (p; w ) : Rn++ R+ ! Rn+ is a
continuous correspondence with nonempty and compact values. Then:
(i): x : Rn++ R+ ! Rn+ de…ned by x (p; w ) = arg maxx 2B (p;w ) u(x) is an upper
hemicontinuous correspondence and
(ii): v : Rn++ R+ ! R de…ned by v (p; w ) = maxx 2B (p;w ) u(x) is a continuous
function.
We need continuity of the correspondence from price-wage pairs to budget
sets.
We must show that B : Rn++ R+ ! Rn+ de…ned by
B (p; w ) = fx 2 Rn+ : p x w g
is continuous and we are done.
Continuity for Correspondences
Reminder from math camp.
De…nition
A correspondence ' : X ! Y is
upper hemicontinuous at x 2 X if for any neighborhood V Y containing
'(x), there exists a neighborhood U X of x such that '(x 0 ) V for all
x 0 2 U.
lower hemicontinuous at x 2 X if for any neighborhood V Y such that
'(x) \ V 6= ;, there exists a neighborhood U X of x such that
'(x 0 ) \ V 6= ; for all x 0 2 U.
A correspondence is upper (lower) hemicontinuous if it is upper (lower)
hemicontinuous for all x 2 X .
A correspondence is continuous if it is both upper and lower hemicontinuous.
Conditions for Continuity (from Math Camp)
The following su¢ cient conditions are sometimes easier to use.
Proposition (A)
Suppose X Rm and Y
Rn . A compact-valued correspondence ' : X ! Y is
upper hemicontinuous if, and only if, for any domain sequence xj ! x and
corresponding range sequence yj such that yj 2 '(xj ), there exists a convergent
subsequence fyjk g such that lim yjk 2 '(x).
Note that compactness of the image is required.
Proposition (B)
Suppose A R m , B R n , and f : A ! B. Then ' is lower hemicontinuous if, and
only if, for all fxm g 2 A such that xm ! x 2 A and y 2 '(x), there exist
ym 2 '(xm ) such that ym ! y .
Continuity for Correspondences: Examples
Exercise
Suppose ' : R ! R is de…ned by:
f1g if x < 1
:
[0; 2] if x 1
Prove that ' is upper hemicontinuous, but not lower hemicontinuous.
'(x) =
Exercise
Suppose ' : R ! R is de…ned by:
f1g if x 1
:
[0; 2] if x > 1
Prove that ' is lower hemicontinuous, but not upper hemicontinuous.
'(x) =
Continuity of the Budget Set Correspondence
Question 4, Problem Set 3; due next Monday.
Show that the correspondence from price-wage pairs to budget sets,
B : Rn++ R+ ! Rn+ de…ned by
is continuous.
B(p; w ) = fx 2 Rn+ : p x
wg
First show that B(p; w ) is upper hemi continuous.
Then show that B(p; w ) is lower hemi continuous.
In both cases, use the propositions in the previous slide (and
Bolzano-Weierstrass).
There was a very very similar problem in math camp (PS 6, #9).
Econ 2100
Fall 2015
Problem Set 3
Due 21 September, Monday, at the beginning of class
1. Prove that if u represents %, then:
(a) % is weakly monotone if and only if u is nondecreasing;
% is strictly monotone if and only if u is strictly increasing.
(b) % is convex if and only if u is quasiconcave;
% is strictly convex if and only if u is strictly quasiconcave.
(c) u is quasi-linear if and only
1. (x; m) % (x; m0 ) if and only if m m0 , for all x 2 Rn 1 and all m; m0 2 R;
2. (x; m) % (x0 ; m0 ) if and only if (x; m+m00 ) % (x0 ; m0 +m00 ), for all x 2 Rn 1 and m; m0 ; m00 2
R;
3. for all x; x0 2 Rn 1 , there exist m; m0 2 R such that (x; m) (x0 ; m0 ).
(d) a continuous % is homothetic if and only if it is represented by a utility function that is homogeneous of degree 1.
2. Consider a setting with 2 goods. For each of the following utility functions, …nd the Walrasian
demand correspondence. (Hint: draw pictures)
(a) u(x) =
n
Q
i=1
xi i with
i
> 0. (Cobb-Douglas)
(b) u(x) = minf 1 x1 ; 2 x2 ; :::; n xn g with i > 0. (generalized Leontief)
P
(c) u(x) = ni=1 i xi with i > 0. (linear)
1
Pn
(CES)
(d) u(x) =
i=1 i xi
From now on consider CES preferences when n = 2.
1. Show that CES preferences are homothetic.
2. Show that these preferences become linear when = 1, and Leontie¤ as ! 1.
3. Assume strictly positive consumption and show that these preferences become Cobb-Douglas
as ! 1. [Hint: use L’Hopital Rule]
4. Compute the Walrasian Demand when < 1.
5. Verify that when < 1 the Walrasian Demand converges to the Walrasian Demands for
Leontie¤ and Cobb-Douglas utility functions as ! 1 and ! 0 repectively.
6. The elasticity of substitution between x1 and x2 is
h
i
x (p;w)
p1
@ x1 (p;w)
p2
2
=
1;2
p1
x
(p;w)
1
@ p2
x2 (p;w)
Prove that for CES preferences
Douglas preferences?
1;2
=
1
1
1
. What is
1;2
for linear, Leontie¤, and Cobb-
3. (Properties of Di¤erentiable Walrasian Demand) Assume that the Walrasian demand x (p; w) is a
di¤erentiable function.
(a) Show that for each (p; w),
n
X
i=1
pi
@xk (p; w)
@x (p; w)
+w k
=0
@pi
@w
for k = 1; : : : ; n
Hint: x is homogeneous of degree 0 in (p; w). What does this imply?
(This is a special case of Euler’s formula for functions that are homogeneous of degree r. See
MWG M.B for more.)
(b) Suppose in addition that % is locally nonsatiated. Show that for each (p; w),
n
X
k=1
pk
@xk (p; w)
+ xi (p; w) = 0
@pi
for i = 1; : : : ; n
and
n
X
pk
k=1
@xk (p; w)
= 1:
@w
Give a simple intuitive (short) version of these two results (sometimes called Cournot and Engel
aggregation, respectively) in words.
4. Prove that the correspondence from price-wage pairs to budget sets, B : Rn++
B(p; w) = fx 2 Rn+ : p x
R+ ! Rn+ de…ned by
wg
is continuous.
5. Suppose a consumer can choose to move to either Philadelphia or Queens. The same n goods are
available in both locations, and he must make all of his purchases in the location where he chooses to
move. The prices may di¤er, however; let p denote the price vector in Philadelphia, and let q denote
the price vector in Queens. Notice that he can decide where to move, but he cannot purchase some
goods in Philadelphia and other goods in Queens.
(a) Describe the consumer’s budget set B(p; q; w) for …xed p; q 2 Rn++ and wealth w 2 R+ .
(b) Suppose the consumer’s utility function u : Rn+ ! R is continuous. Prove the following:
The Walrasian demand correspondence, which is now de…ned on two prices and wealth, x :
Rn++ Rn++ R+
Rn+ , de…ned by
x (p; q; w) = arg
max
x2B(p;q;w)
u(x)
is upper hemicontinuous; and the indirect utility function, v : Rn++
by
v(p; q; w) = max u(x)
Rn++
R+ ! R, de…ned
x2B(p;q;w)
is continuous.
(c) Prove or disprove the following: If the consumer’s utility function is strictly quasiconcave, then
jx (p; q; w)j 1.
2
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