Download Preferences and Utility. X : set of alternatives (choice set or domain

Preferences and Utility.
X : set of alternatives (choice set or domain).
A preference relation % is a binary relation on X that
allows comparison of pairs of alternatives x; y 2 X:
x % y : alternative x is at least as good as alternative y:
From %, we can derive two other binary preference relations on X :
(i) Strict preference relation
de…ned by
y , x % y but not y % x
x
(ii) Indi¤erence relation
x
de…ned by
y , x % y and y % x
A basic assumption on % that underlies much of economics is that it is rational.
De…nition: The preference relation % on X is rational if
it satis…es the following properties:
(1) Completeness: for all x; y 2 X; either x % y or
y % x or both.
(2) Transitivity:for all x; y; z 2 X; if x % y and y % z ,
then x % z:
Sometimes a rational preference relation is also called an
"ordering".
Question:
If the weak preference relation % on X is complete, does
it imply that the strict preference relation
and the indi¤erence relation are complete?
If the weak preference relation % on X is transitive, does
it imply that the strict preference relation
and the indi¤erence relation are transitive?
Utility Function.
A utility function assigns a numerical value to each element in X in accordance with the preference.
More formally:
De…nition: A function u : X ! R is a utility function
representing preference relation % if, for all x; y 2 X;
x % y , u(x)
u(y ):
A utility function representing a preference relation % is
not unique.
Exercise: If u represents % on X , then for any strictly
increasing function f : R ! R; v (x) = f (u(x)) is also
a utility function that represents % :
Note: v is then said to be a strictly increasing transformation of u:
Properties of utility function that are invariant for all
strictly increasing transformations are known as ordinal
properties.
Otherwise, they are cardinal properties.
Exercise: If u represents %, then for all x; y 2 X;
x
y , u(x) > u(y )
x
y , u(x) = u(y ):
and
Proposition (1.B.2): A preference relation % can be
represented by a utility function only if it is rational.
Proof: Suppose there exists a utility function u that
represents % on X:
For any pair x; y 2 X; either
u(x)
u(y ) or u(y )
u(x) (or both)
and since u that represents %, this must imply:
x % y or y % x (or both)
which shows that % is complete.
Next, we show % is transitive. Suppose x % y and y % z:
We need to show that x % z:
As u that represents %, this must imply:
u(x)
u(y ) and u(y )
u(z )
so that
u(x)
u(z )
which implies (again because u that represents %)
x % z:
Thus, % is transitive. The proof is complete.
Not every rational preference relation can be represented
by a utility function.
Example: Lexicographic preferences (Example 3.C.1
in textbook)
X = R2+: For any pair of alternatives x; y 2 X; where
x = (x1; x2); y = (y1; y2);
x % y if either x1 > y1; or x1 = y1 and x2
y2:
Verify that this particular preference is rational.
Suppose that there is a utility function u that represents
this preference.
For every non-negative real number z we can look at the
real numbers u(z; 1) and u(z; 2).
Now, from the de…nition of lexicographic preferences, observe that (z; 2) % (z; 1) but not (z; 1) % (z; 2) which
implies (z; 2) (z; 1):
Hence (see exercise above):
u(z; 2) > u(z; 1):
Between any distinct real numbers there is a rational
number. So, for every non-negative real number z there
exists a rational number r(z ) such that
u(z; 2) > r(z ) > u(z; 1):
Further, from the de…nition of lexicographic ordering if
z > z 0, then
(z; 2)
(z; 1)
(z 0; 2)
(z 0; 1)
which implies
u(z; 2) > r(z ) > u(z; 1) > u(z 0; 2) > r(z 0) > u(z 0; 1):
Thus, for every non-negative real number z there exists
a distinct rational number r(z ):
But there are uncountable positive real numbers but only
countably many rational numbers. A contradiction.
Exercise: If X is …nite, then every rational preference relation can be represented by a utility function.
Consumer Behavior and Walrasian Demand.
L commodities, l = 1; 2; :::L:
L …nite.
A commodity vector (bundle)
2 3 lists amounts of the di¤erx
6 17
6 x2 7
6 7
7
ent commodities : x = 6
6 : 7
6 : 7
4 5
xL
xl : consumption level of good l
Consumption Set X RL: Set of all consumption bundles that the consumer can conceivably consume given
the physical constraints.
Assume: X = RL
+
(Any non-negative bundle may be consumed)
Consumer’s preference relation % de…ned on X = RL
+:
Notation:
2
3
2
3
y
x
6 17
6 17
6 y2 7
6 x2 7
6 7
6 7
7
6
7
For vectors x = 6 : 7,y = 6
6 : 7,
6 : 7
6 : 7
4 5
4 5
yL
xL
y
x () yl > xl ; for all l = 1; :::L;
y
x () yl
xl ; for all l = 1; :::L;
Note,
y
x; y 6= x () yl
xl ; for all l and yl > xl ; for some l:
v
u L
uX
L
Euclidean norm: For x 2 R ; kxk = t (xl )2
l=1
Euclidean distance: For x; y 2 RL; the (Euclidean) distance between vectors x and y is given by kx yk =
v
u L
uX
t
(x
l=1
l
yl )2
Some common assumptions on % used in classical demand theory:
Desirability assumptions:
De…nition: The preference relation % is monotone if
x 2 X and y
x implies y x:
De…nition: The preference relation % is strongly monotone
if x 2 X and y x; y 6= x implies y x:
(These imply goods are "good"; no "bads").
De…nition: The preference relation % satis…es local nonsatiation if for every x 2 X and every > 0; there is
y 2 X such that k y x k< and y x:
(Implies that arbitrarily close to any consumption bundle,
there is a better bundle).
Exercise (Exercise 3.B.1) : Strong monotone ) Monotone
) Local Non-satiation
A digression:
For any x 2 X;the set fy 2 x : y % xg is called the
upper contour set of x; it is the set of all bundles that
are at least as good as x:
For any x 2 X;the set fy 2 x : x % yg is called the
lower contour set of x:
For any x 2 X;the set fy 2 x : x
indi¤erence set x:
yg is called the
Local non-satiation rules out "thick" indi¤erence sets.
Convexity:
A set S
RL is convex if for all x; y 2 S and
x + (1
2 [0; 1];
)y 2 S:
(the entire line segment connecting x and y is contained
in the set S ):
Beware: Convexity of a set is a very di¤erent concept
from convexity of a function.
Note: X = RL
+ is a convex set.
Convexity assumptions:
De…nition: The preference relation % is convex if for
every x 2 X; the upper contour set of x is convex i.e.,
for all y; z 2 X such that y % x and z % x; and for all
2 [0; 1];
y + (1
)z % x:
De…nition: The preference relation % is strictly convex
if for every x 2 X; and for all y; z 2 X such that y % x,
z % x; y 6= z; and for all 2 (0; 1);
y + (1
)z
x:
Convexity captures a taste for diversi…cation. Implies diminishing marginal rates of substitution between any pair
of goods.
For L = 2; implies that indi¤erence curves are convex to
the origin.
Continuity assumption:
De…nition: The preference relation % is continuous if for
any sequence of pairs of commodity bundles f(xn; y n)g1
n=1 ,
xn 2 X; y n 2 X for all n;with xn % y n for all n;
x = limn!1 xn; y = limn!1 y n; we have x % y:
(No reversal of preferences in the limit).
Digression: A set S RL is said to be closed if the limit
of every sequence of vectors in S is contained in S:
Equivalent de…nition of continuity: For all x 2 X; the
upper and lower contour sets are closed.
Example (Example 3.C.1): Lexicographic preferences are
not continuous.
y n = (0; 1) for all n
xn = ( n1 ; 0)
(0; 0); y n ! (0; 1) but (0; 1) (0; 0):
1, xn !
Proposition 3.C.1: Suppose that the rational preference
relation is % on X is continuous. Then there is a continuous utility function that represents % :
(Does not mean that all utility functions representing the
preference relation are continuous)
"Proof": Assume in addition that % is monotone.
Let e 2 RL
+ be the vector (1; 1; 1; :::1):
For every x 2 RL
+ ; monotone property implies
x%0
and for any positive number
such that
e
x;
e % x:
The sets A+ = f 2 R+ : e % xg and A = f
R+ : x % eg are nonempty and closed (why?).
2
Further, as % is complete, for every
2 R+; either
e % x or x % e so that 2 A+ [ A :
Thus, A+ [ A
= R+ :
One mathematical property of R+ is that it is a connected
set which means it cannot be the union of two disjoint,
nonempty, closed sets.
So, A+ \ A
6= :
So there exists
i.e., e x:
By monotonicity,
e % x and x %
0 such that
1e
2e
if
1>
e
2:
Therefore, there can be only one real number
that (x)e x:
(x) such
Set u(x) = (x) for every x 2 X:
Suppose u(x) u(y ):Then, (x)
(y ) so that using
monotonicity of preferences (x)e % (y )e: By construction, (x)e x; (y )e y so that using transitivity, x % y:
Conversely, suppose x % y: Then, by construction, (x)e
x; (y )e
y and so using transitivity (x)e % (y )e:
Using monotonicity of preferences, (x)
(y ) i.e.,
u(x) u(y ):
Thus, u(x)
u(y ) () x % y:
So, u represents % :
Continuity of u requires a bit more involved argument.
Assumptions on preferences imply certain properties of
ALL utility functions that represent them (they are ordinal
properties).
Monotone % ) u is increasing i.e., u(x) > u(y ) if
x
y:
Exercise: Strong monotone % ) ?
Let S
RL be a convex set.
A function f : S ! R is said to be quasiconcave if for
all x 2 S;the set fy 2 S : f (y )
f (x)g is a convex
set; or alternatively, for all x; y 2 S; 2 [0; 1];
f ( x + (1
)y )
minff (x); f (y )g:
A function f : S ! R is said to be strictly quasiconcave
if for all x; y 2 S; x 6= y; 2 (0; 1);
f ( x + (1
)y ) > minff (x); f (y )g:
% convex ) u is quasi-concave.
% strictly convex ) u is strictly quasi-concave.
Exercise: quasi-concavity is an ordinal property (invariant
to any strictly increasing transformation of u):
Let S
RL be a convex set.
A function f : S ! R is said to be concave if for all
x; y 2 S; 2 [0; 1];
f ( x + (1
)y )
f (x) + (1
)f (y ):
Concavity ) Quasi-concavity
A function f : S ! R is said to be strictly concave if for
all x; y 2 S; x 6= y; 2 (0; 1);
f ( x + (1
)y ) > f (x) + (1
Strict concavity ) Strict Quasi-concavity
)f (y ):
Concavity of the utility function is not an ordinal property.
Not invariant to strictly increasing transformations.
It is not based on any property of the underlying preference structure.
Law of diminishing marginal utility (implied by concavity,
but not necessarily by quasi-concavity) is not an ordinal
property.
The Utility Maximization Problem
Assume: % is a rational, continuous and locally nonsatiated preference relation represented by a continuous
utility function u(x) on X = RL
+:
We also assume:
(i) the L commodities are all traded in the market at
dollar prices that are publicly quoted (complete markets
assumption). In particular, price vector
2
3
p
6 17
6 p2 7
6 7
7
p=6
6 : 7
6 : 7
4 5
pL
0
(ii) consumers are price taking.
A¤ordability of a commodity bundle depends on price
vector p 2 RL
++ and wealth w 2 R++ :
De…nition: The Walrasian, or competitive budget set
Bp;w = fx 2 X : p x
wg is the set of all feasible consumption bundles for the consumer who faces
market prices p and has wealth w:
The upper boundary of the budget set fx 2 X : p x =
wg is called the budget hyperplane (or budget line for the
case L = 2).
Digression:
A set S RL is said to be bounded if there exists N > 0
such kxk < N for all x 2 S:
A set S
RL is compact if it is both closed and bounded.
Exercise: Bp;w is a compact and convex set.
Utility Maximization Problem (UMP):
Given p
0 and w > 0;
max u(x)
x2X
subject to p
x
w:
or equivalently,
max u(x):
x2Bp;w
Weirstrass’ Theorem: Let f : K ! R be a continuous function and K is a compact set: Then, there exists
x0; x00 2 K such that
f (x0)
f (x)
f (x00) for all x 2 K:
In other words, f attains a maximum and a minimum in
the set K:
Proposition 3.D.1: If p
has a solution.
0 and w > 0, then UMP
The proof follows from continuity of u and compactness
of Bp;w:
Note: Solution need not be unique.
Let x(p; w) = fx0 2 Bp;w : u(x0)
Bp;w g
u(x) for all x 2
In other words, x(p; w) is the set of all solutions to the
UMP.
We can view x(p; w) a set valued mapping or a correspondence that associates a set of optimal consumption
bundles with each (p; w): We call this the Walrasian demand correspondence.
If x(p; w) is single valued for each (p; w) i.e., there is a
unique solution to UMP for every (p; w); then x(p; w) is
a function often called the Walrasian demand function.
Proposition 3.D.2: Suppose that u(:) is a continuous
utility function representing a locally non-satiated preference relation % de…ned on the consumption set X = RL
+:
Then the Walrasian demand correspondence x(p; w) has
the following properties:
(i) Homogeneity of degree zero in (p; w) : x( p; w) =
x(p; w) for any p
0; w > 0 and scalar > 0:
(ii) Walras’law: p
x = w for all x 2 x(p; w):
(iii) Convexity/uniqueness: If % is convex so that u is
quasiconcave, then x(p; w) is a convex set for every (p; w)
0: If % is strictly convex so that u is strictly quasiconcave,
then x(p; w) consists of single point.
Proof.
(i) Homogeneity of degree zero in (p; w) : x( p; w) =
x(p; w) for any p
0; w > 0 and scalar > 0:
This follows from
fx 2 X : p
x
wg = fx 2 X : p
x
wg
and therefore, the UMP
max u(x)
x2X
subject to p
x
w:
is equivalent to
max u(x)
x2X
subject to
p
x
w:
so that the set of solutions to the two max problems are
identical.
(ii) Walras’law: p
x = w for all x 2 x(p; w)
For any bundle x such that p x < w; there exists > 0
small enough such that p x0 < w for all x0 2 X such
that x0 x < :
Using local non-satiation, there exists x0 such that x0 x <
and x0 x and as that x0 is in the budget set, it contradicts the optimality of x:
(iii)i) Convexity/uniqueness: If % is convex so that u
is quasiconcave, then x(p; w) is a convex set for every
(p; w)
0:
If x(p; w) has only one element, then it is trivially a convex set. So suppose that there are at least two distinct
elements x 6= x0 in x(p; w) i.e., both x; x0 solve UMP.
Let
u(x) = u(x0) = u :
For any
)x0:
2 [0; 1];consider the bundle x00 = x + (1
As Bp;w is a convex set, x00 2 Bp;w .
As u is quasiconcave,
u(x00) = u( x + (1
)x0)
minfu(x); u(x0)g = u
so that x00 must also be optimal i.e., x00 2 x(p; w): Thus,
x(p; w) is a convex set.
Next we show that if % is strictly convex so that u
is strictly quasiconcave, then x(p; w) consists of single
point.
Suppose to the contrary that there are at least two distinct solutions to UMP x; x0 2 x(p; w); x 6= x0:
Let
u(x) = u(x0) = u :
Consider the bundle x00 = 12 x + 21 x0:
As Bp;w is a convex set, x00 2 Bp;w .
As u is strictly quasiconcave,
1
1
u(x00) = u( x + x0) > minfu(x); u(x0)g = u
2
2
which contradicts the fact that x solves UMP and u is
the maximum utility attainable on Bp;w:
Applications:
Suppose x(p; w) is a di¤erentiable function.
Homogeneity of degree zero of the function xl (p; w) implies:
L
X
@xl
k=1
@pk
pk +
@xl
w = 0 for all l = 1; :::L:
@w
If xl (p; w) > 0;dividing through by xl :
L
X
@xl pk
@xl w
+
=0
@p
x
@w
x
k l
l
k=1
so that:
L
X
"lk (p; w) + "lw (p; w) = 0
k=1
where "lk (p; w) is the (cross) price elasticity of demand
for good l with respect to price of good k; "lw (p; w) is
the wealth (or income) elasticity of demand for good l:
An equi-proportionate change in all prices and wealth has
no net e¤ect on demand.
Walras Law:
p x(p; w) = w for all p
0; w > 0
gives us an identity in p; w:Di¤erentiating through with
respect to pk we have
L
X
@xl (p; w)
l=1
@pk
If x(p; w)
pl + xk (p; w) = 0 for all k = 1; :::L:
0; multiplying through by pk and dividing
through by w
L
X
@xl pk pl xl
p x
+ k k = 0 for all k = 1; :::L:
@pk xl w
w
l=1
so that
L
X
"lk (p; w)bl (p; w) + bk (p; w) = 0 for all k = 1; :::L:
l=1
l xl is the share of expenditure (or
where bl (p; w) = pw
budget share) on good l:
p x(p; w) = w for all p
0; w > 0
Di¤erentiating through with respect to w we have
L
X
@xl (p; w)
l=1
If x(p; w)
@w
pl = 1 :
0;
L
X
@xl w pl xl
l=1
@w xl w
=1
so that
L
X
l=1
"lw (p; w)bl (p; w) = 1:
Continuity of Demand:
In general, there may not be any continuous demand
function that one can select from the correspondence
x(p; w): Even though utility is continuous.
Example:L = 2:
u(x1; x2) = x1 + x2
w
; 0); if p1 < p2
p1
= fx : px = wg if p1 = p2
w
= (0; ); if p1 > p2:
p2
x(p; w) = (
Maximum Theorem ) If there is a unique solution to
UMP for every (p; w), then x(p; w) is continuous at
every (p; w)
0.
Thus, strict quasi-concavity of u ensures continuity of
demand function.
Assume: u is continuously di¤erentiable on X
Kuhn-Tucker necessary condition: If x 2 x(p; w); then
there exists a (scalar) multiplier
0 such that for all
l = 1; :::L;
@u(x )
@xl
pl
and
@u(x )
= pl if xl > 0:
@xl
These are the …rst order conditions.
If
ru(x) = [
@u(x)
@u(x)
]
; ::::;
@x1
@xL
then the above conditions can be written as
ru(x )
and
x
[ru(x )
p
p] = 0 :
If ru(x )
0; ru(x ) 6= 0 and x
(why?) and for any two goods l; k
@u(x )
@xl
@u(x )
@xk
0; then
>0
pl
= :
pk
The left hand side the marginal rate of substitution (MRS)
between goods l and k:
Note MRS need not equal price ratio if we have corner
solution.
Interpretation of multiplier
:
Suppose x(p; w) is a di¤erentiable function and x(p; w)
0:
Then, the maximum utility is u(x(p; w)):
L
X
@u(x(p; w))
@u(x(p; w)) @xl
=
@w
@xl
@w
l=1
L
X
@xl
=
pl
@w
l=1
= ;
using an implication of Walras’Law. Thus, is the marginal value of wealth. This result holds much more generally (do not need x(p; w) to be a di¤erentiable or even
continuous function); all one needs is that the maximum
utility u(x(p; w)) should be di¤erentiable in wealth.
Su¢ ciency of …rst order conditions.
Suppose x
0 satis…es the …rst order conditions
@u(x )
@xl
pl
and
@u(x )
= pl if xl > 0:
@xl
for some
0: Further,
px = w:
Under what conditions is x optimal (i.e., x 2 x(p; w))?
Answer: if u is quasi-concave, monotone and ru(x) 6= 0
for all x 2 RL
+:
Concept of Open Set
A set S
RL is open if its complement is closed.
A set S
RL is open if for every x 2 S there exists
> 0 such that fy : kx yk < g S:
L
RL and RL
++ are open sets (in R ):
A set may be neither open nor closed.
A set may be both open and closed - for instance,
RL :
and
Verifying quasi-concavity: a useful result.
An interesting characterization of twice continuously differentiable quasi-concave functions can be given in terms
of the “bordered” Hessian matrix associated with the
functions.
Let A be an open subset of Rn, and f : A ! R be a
twice continuously di¤erentiable function on A.
The bordered Hessian matrix of f at x 2 A is denoted
by Gf (x) and is de…ned as the following (n +1) (n +1)
matrix
Gf (x) =
"
0
rf (x)
rf (x)
Hf (x)
#
where Hf (x) is the Hessian matrix of second order partial
derivatives whose (i; j )-th element is
@ 2f
@xi @xj
We denote the (k +1)th leading principal minor of Gf (x)
by Gf (x; k) , where k = 1; :::; n. The (k +1)th leading
principal minor is the determinant of the matrix obtained
after deleting all but the …rst k +1 rows and k +1 columns
of the matrix.
Result: Let A be an open convex set in Rn, and f : A !
R be a twice continuously di¤erentiable function on A.
If
( 1)k Gf (x; k) > 0
for x 2 A, and k = 1; :::; n; then f is strictly quasiconcave on A
n and quasiResult: If h : Rn
!
R
is
continuous
on
R
+
+
n
concave on R++; then it is quasi-concave on Rn
+:
Indirect Utility Function: value of the utility maximization
problem i.e., the maximum utility as a function of prices
and wealth.
v (p; w) =
max u(x)
x2Bp;w
= u(x ) where x 2 x(p; w):
Proposition 3.D.3: Suppose that u(:) is a continuous
utility function representing a locally non-satiated preference relation % de…ned on the consumption set X = RL
+:
Then the indirect utility function v (p; w) has the following properties:
(i) Homogeneity of degree zero in (p; w):
(ii) Strictly increasing in w and nonincreasing in pl for
any l:
(iii) Quasiconvex: the set f(p; w) : v (p; w)
convex for any v:
vg is
(iv) Continuous in p; w:
Note: indirect utility function depends on the speci…c
utility function chosen to represent the preferences.
The Expenditure Minimization Problem (EMP)
For p
0 and u > u(0);
min p x
x2RL
+
s:t: u(x)
u:
Here we continue to assume u(:) is a continuous utility function representing a locally nonsatiated preference
relation on RL
+:
Proposition 3.E.1: Suppose that u(:) is a continuous utility function representing a locally nonsatiated preference
relation on X = RL
0:
+ and that the price vector p
Then
(i) If x solves UMP when wealth is w > 0; then x solves
EMP when the required utility level is u(x ): Moreover,
the minimum expenditure in the latter EMP is exactly w:
(ii) If x solves EMP when the required utility level is
u > u(0); then x solves UMP when wealth is p x :
Moreover, the maximum utility in the latter UMP is exactly u:
Proof. (i) To show: If x solves UMP when wealth is
w > 0; then x solves EMP when the required utility
level is u(x ):
Suppose to the contrary that x does not solve EMP
when the required utility level is u(x ):
Then there exists x0 such that p x0 < p x and u(x0)
u(x ):
As x solves UMP, p x = w:
Thus, p x0 < w:
By local nonsatiation, there exists x00 2 Bp;w such that
u(x00) > u(x ) which contradicts the optimality of x in
the UMP.
Thus, x solves EMP when the required utility level is
u(x ) and the minimum expenditure is p x = w.
(ii) To show: If x solves EMP when the required utility
level is u > u(0); then x solves UMP when wealth is
p x . Moreover, the maximum utility in the latter UMP
is exactly u:
Since x solves EMP given u > u(0); x 6= 0:
Hence, p x > 0:
Suppose x does not solve UMP when wealth is p x .
Then, there exists x0 such p x0
u(x ):
p x and u(x0) >
By continuity, there exists x00 = x0 where 2 (0; 1)
such that p x00 < p x and u(x00) > u(x ):
This contradicts optimality of x in EMP.
Thus, x solves UMP when wealth is p x and the maximized utility level is therefore u(x ):
A later proposition shows that u(x ) = u i.e., the inequality constraint holds with equality at optimal solution
to EMP.
Existence of solution to EMP: all we need is to ensure
that there exists x such that u(x) u:
Why?
The Expenditure Function: Given prices p
0 and required utility level u > u(0); the expenditure function is
given by
e(p; u) =
s:t: u(x)
min p x
x2RL
+
u:
Proposition 3.E.2: Suppose that u(:) is a continuous utility function representing a locally nonsatiated preference
relation on X = RL
+ : The expenditure function e(p; u)
is:
(i) Homogenous of degree one in p:
(ii) Strictly increasing in u and nondecreasing in pl for
any l
(iii) Concave in p
(iv) Continuos in p and u:
Implication of Proposition 3.E.1: Two identities: for any
p > 0; w > 0; u > u(0)
e(p; v (p; w)) = w
v (p; e(p; u)) = u
For a …xed price vector p = p;
e(p; u) = v 1(p; u)
v (p; w) = e 1(p; w)
Set of optimal solutions to EMP: h(p; u)
Called the Hicksian or compensated demand correspondence (or function if h(p; u) is single valued).
Proposition 3.E.3: Suppose that u(:) is a continuous utility function representing a locally nonsatiated preference
relation on X = RL
0; u > u(0);
+ : Then for any p
the Hicksian demand correspondence h(p; u) satis…es:
(i) Homogenous of degree zero in p : h( p; u) = h(p; u)
for any > 0
(ii) For any x 2 h(p; u); u(x) = u:
(iii) Convexity/uniqueness: If % is convex so that u is
quasiconcave, then h(p; u) is a convex set: If % is strictly
convex so that u is strictly quasiconcave, then h(p; u)
consists of single point.
Suppose that u(:) is continuously di¤erentiable.
Kuhn-Tucker …rst order necessary conditions: : If x 2
h(p; u); then there exists a (scalar) multiplier
0 such
that for all l = 1; :::L;
pl
@u(x )
@xl
and
@u(x )
pl =
if xl > 0:
@xl
Also,
u(x ) = u:
The …rst set of conditions can be written as
p
ru(x )
and
x
[p
ru(x )] = 0:
If h(p; u) is single valued everywhere, then it is a continuous function.
First order conditions are su¢ cient for optimality in the
EMP if u is quasi-concave.
Implication of Proposition 3.E.1: Two more identities:
for any p > 0; w > 0; u > u(0)
h(p; u) = x(p; e(p; u))
x(p; w) = h(p; v (p; w))
Note x(p; e(p; u)) is the Walrasian demand if for every
price vector, wealth is adjusted to a level that allows the
agent to just attain utility u:
This is the Hicksian compensation use to decompose effect of price change into substitution and income e¤ect.
The …rst identity therefore explains why h(p; u) is called
the compensated demand.
Compensated Law of Demand
Modi…ed Proposition 3.E.4: Suppose that u(:) is a continuous utility function representing a locally nonsatiated
0 00
preference relation on X = RL
+ : Then, for all p ; p
0; u > u(0); x0 2 h(p0; u); x00 2 h(p00; u), the following
holds
(p0
p00)(x0
x00)
0:
Proof: As x0 2 h(p0; u); x00 2 h(p00; u)
p 0 x0
p0x00
p00x00
p00x0
and subtracting the second from the …rst inequality yields
the result.
Implication of the proposition: If p0k = p00k for all k 6= l;
(p0
p00)(x0
x00)
= (p0l
p00l )(x0l
x00l )
so that we have
p0l > p00l ) x0l
p0l < p00l ) x0l
x00l
x00l
i.e., other things being equal, compensated demand for
any commodity is non-increasing in own price.
Restated as: "substitution e¤ect of price increase is negative"
Note: Law of demand does not apply to Walrasian demand. Income e¤ect can overtake substitution e¤ect.
Some Useful Relationships between Demand, Indirect Utility and Expenditure Functions
Continue to assume: u(:) is a continuous utility function
representing a locally nonsatiated preference relation on
X = RL
+:
Also, assume u is strictly quasi-concave.
Thus, x(p; w) and h(p; u) are single valued at each p
0; w > 0; u > u(0):
x(p; w) and h(p; u) are functions.
The maximum theorem can be used to show that the
functions v (p; w); x(p; w); e(p; u) and h(p; u) are continuos at every p
0; w > 0; u > u(0).
Further, it can be shown that e(p; u) is di¤erentiable at
every p
0; u > u(0):
Proposition 3.G.1: At every p
0; u > u(0);
@e(p; u)
; l = 1; ::; L:
hl (p; u) =
@pl
"Proof": Easy to show under additional restriction that
the utility function u is di¤erentiable, h(p; u) is di¤erentiable in p and h(p; u)
0:
First order necessary condition for EMP: for some
(check is in fact > 0)
pk =
0
@u(h(p; u))
; k = 1; :::L:
@xk
Now,
u(h(p; u)) = u
As the latter is identity that holds for all p
0; u >
u(0); we have by di¤erentiating with respect to pl :
L
X
@u(h(p; u)) @hk (p; u)
k=1
@xk
@pl
=0
and using the …rst order condition:
L
X
@u(h(p; u)) @hk (p; u)
=
k=1
L
X
@xk
@pl
pk @hk (p; u)
@pl
k=1
so that
L
X
@hk (p; u)
pk
= 0:
@p
l
k=1
Further, identity:
e(p; u) = p h(p; u); for all p
0; u > u(0)
Di¤erentiating with respect to pl :
L
X
@e(p; u)
@hk (p; u)
= hl (p; u) +
pk
@pl
@pl
k=1
= hl (p; u):
Can also be shown to be a direct consequence of the
"envelope theorem".
Implication.
Suppose that h(p; u) is continuously di¤erentiable at (p; u):
Then, e(p; u) is twice continuously di¤erentiable in p and
@hl (p; u)
@ 2e(p; u)
=
:
@pk @pl
@pk
Young’s theorem:
@ 2e(p; u)
@ 2e(p; u)
=
@pk @pl
@pl @pk
(for a twice continuously di¤erentiable function, the second order partial derivatives are independent of the order
of di¤erentiation).
For a twice di¤erentiable concave function, the matrix
of second order cross-partial derivatives (i.e., the Hessian
matrix) is negative semi-de…nite matrix:Thus,
2
@ 2e
6 @p21
6
6 @ 2e
6 @p @p
6 2 1
6
6
6
6
6
4 @ 2e
@pL @p1
@ 2e
@p1 @p2
@ 2e
@p22
3
@ 2e
@p1 @pL 7
7
@ 2e 7
@p2 @pL 7
7
7
7
7
7
7
2
@ e 5
@p2L
is a symmetric negative semi-de…nite matrix. This implies, the (Jacobian) matrix of …rst order derivatives of
h with respect to prices (i.e., the matrix of substitution
e¤ects)
2
@h1
6 @p1
6 @h1
6
6 @p2
6
6
6
6
4
@h1
@pL
@h2
@p1
@h2
@p2
3
@hL
@p1 7
@hL 7
7
@p2 7
7
7
7
7
5
@hL
@pL
is a symmetric negative semi-de…nite matrix. This matrix
is sometimes called the substitution matrix.
As the diagonal terms of any negative semi-de…nite matrix is non-positive,
@hl
@pl
0; l = 1; :::L:
which the own price e¤ect in the law of compensated
demand.
@hl
@pk
0 : goods l; k are substitutes
@hl
@pk
0 :goods l; k are complements
As h(p; u) is homogenous of degree zero in p; Euler’s
theorem:
L
X
@hl
k=1
@hl
l
As @h
0
;
@pl
@pk
one substitute).
@pk
pk = 0; l = 1; :::; L:
0 for some k (every good has at least
Proposition 3.G.3: (Slutsky equation) For all p; w
and u = v (p; w)
0
@hl (p; u)
@xl (p; w) @xl (p; w)
=
+
xk (p; w) for all l; k
@pk
@pk
@w
Proof. Choose any p; w
0:Fix Let u = v (p; w): Then,
e(p; u) = w: Identity in p:
hl (p; u) = xl (p; e(p; u))
Di¤erentiating through with respect to pk and evaluating
at p; u :
@hl (p; u)
@xl (p; e(p; u)) @xl (p; e(p; u)) @e(p; u)
=
+
@pk
@pk
@w
@pk
@xl (p; w) @xl (p; w)
+
hk (p; u)
=
@pk
@w
@xl (p; w) @xl (p; w)
=
+
xk (p; e(p; u))
@pk
@w
@xl (p; w) @xl (p; w)
=
+
xk (p; w):
@pk
@w
Implication: E¤ect of change of own price on Walrasian
demand
@xl (p; w)
@pl
@hl (p; u) @xl (p; w)
xl (p; w)
=
@pl
@w
= substitution e¤ect + income e¤ect
@x (p;w)
l
For a normal good,
0; so that income and
@w
substitution e¤ects work in same direction (negative).
You can recover Walrasian demand function from the indirect utility function.
Proposition 3.G.4. Suppose that the indirect utility function v (p; w) is di¤erentiable at p; w
0:
xl (p; w) =
@v (p; w)=@pl
:
@v (p; w)=@w
0:Fix Let u = v (p; w): Then,
Proof. Choose any p; w
e(p; u) = w:Identity in p:
v (p; e(p; u)) = u
Di¤erentiating through with respect to pl and evaluating
at p = p :
@v (p; e(p; u)) @v (p; e(p; u)) @e(p; u)
+
=0
@pl
@w
@pl
so that
@v (p; w) @v (p; w)
+
hl (p; u) = 0
@pl
@w
and as hl (p; u) = hl (p; v (p; w)) = xl (p; w); we have
@v (p; w) @v (p; w)
+
xl (p; w) = 0
@pl
@w
and this yields the result.
Welfare Evaluation of Economic Changes
Fix wealth w > 0:
Suppose that price vector changes from p0 to p1:
Consumer is better o¤ as a result of this change if
v = v ( p1 ; w )
v (p0; w) > 0
and worse o¤ if the opposite is true. The speci…c amount
of this change in welfare depends on the choice of utility
function (indirect utility function depends on u):
How then do we measure this welfare change?
Use money metric (indirect) utility.
Choose any utility function u, derive the indirect utility
0:
function v: Now, choose any arbitrary price vector p
Consider the function
e(p; v (p; w)):
It is the (minimum) amount of money you need to spend
at price vector p to attain same utility level as the maximum utility you can attain where price vector is p and
wealth is w: As e(p; v (p; w)) is strictly increasing in the
second argument, it is nothing but a strictly increasing
transformation of the indirect utility function. So, there
is a utility function representing the same preference ordering for which e(p; v (p; w)) is in fact the indirect utility
(why?).
Also note that e(p; v (p; w)) does not depend on which
indirect utility v is chosen.
A dollar measure of the welfare change:
e(p; v (p1; w))
e(p; v (p0; w))
This measure is independent of which indirect utility v is
chosen.
If we choose p = p0; we obtain a measure of welfare
change called Equivalent Variation (EV):
EV (p0; p1; w) = e(p0; v (p1; w))
= e(p0; v (p1; w))
e(p0; v (p0; w))
w
It is the dollar amount that the consume would be indifferent about accepting in lieu of the price change. Let
u0 = v (p0; w); u0 = v (p1; w);
Then,
EV (p0; p1; w) = e(p0; u1)
w
If we choose p = p1; we obtain a measure of welfare
change called Compensating Variation (CV):
CV (p0; p1; w) = e(p1; v (p1; w))
= w
e(p1; u0)
e(p1; v (p0; w))
It is the amount by which the agent must be compensated
after a price change to make him as well o¤ as before the
price change.
Both CV and EV can be interpreted as area to the left of
the Hicksian demand curves.
6 1:Let
Suppose p01 6= p11 and p0l = p1l = pl for all l =
p 1 = (p2; :::pL):Then (assuming appropriate di¤erentiability), as
@e
h1(p; u) =
;
@p1
EV (p0; p1; w) = e(p0; u1)
w
= e(p0; u1) e(p1; u1)
Z p0
1
1 @e(p1 ; p 1 ; u )
=
dp1
@p1
p11
=
Z p0
1
p11
h1(p1; p 1; u1)dp1:
Similarly,
CV (p0; p1; w) =
Z p0
1
p11
h1(p1; p 1; u0)dp1
Suppose good 1 is a normal good (income e¤ect is strictly
positive). If p11 < p01 . In that case, for all p1 2 (p11; p01 );
h1(p1; p 1; u1) > x1(p1; p 1; w) > h1(p1; p 1; u0)
and
h1(p11; p 1; u1) = x1(p11; p 1; w) > h1(p11; p 1; u0)
h1(p01; p 1; u1) > x1(p01; p 1; w) = h1(p01; p 1; u0):
To see these inequalities, suppose that the agent is initially facing prices vector p1 and enjoying utility u1:
If price of good 1 now increases to p1 > p11, then the
quantity bought will decline to h1(p1; p 1; u1) due to
substitution e¤ect and then decline further to x1(p1; p 1; w)
due to income e¤ect (as the good is a normal good) so
that h1(p1; p 1; u1) > x1(p1; p 1; w) .
Next, suppose that the agent is initially facing prices vector p0 and enjoying utility u0:
If price of good 1 now decreases to p1 < p01, then the
quantity bought will increase to h1(p1; p 1; u0) due to
substitution e¤ect, and then increase further to x1(p1; p 1; w)
due to income e¤ect (as the good is a normal good), so
that h1(p1; p 1; u0) < x1(p1; p 1; w):
It follows therefore, that
EV (p0; p1; w)
=
Z p0
1
h1(p1; p 1; u1)dp1
Z p0
1
h1(p1; p 1; u0)dp1:
p11
> CV (p0; p1; w)
=
p11
It is easy to check that the same inequality holds if p11 >
p01 (keep in mind that the integrals equal to EV and CV
are now negative numbers).
If good 1 is an inferior good, EV < CV:
If there is no income e¤ect, EV = CV ; in particular, the
Hicksian and Walrasian demands coincide (as a function
of prices) and
EV (p0; p1; w) =
Z p0
1
p11
x1(p1; p 1; w)dp1 = CV (p0; p1; w)
i.e., as area to the left of the Walrasian demand curve for
good 1 (Marshallian consumer surplus).
This is what happens when the utility function is quasilinear.
If the utility function is one where income e¤ect is not
zero then using the area to the left of the Walrasian demand curve can still be an approximation of CV or EV
provided price changes are very small.
Revealed Preference Approach to Law of Demand
(Samuelson)
Preferences are not observable, only choices are.
If choices made by individuals always satisfy some basic
consistency axioms, then we can obtain certain patterns
of economic behavior including the law of demand.
Instead of imposing structures on unobservable preferences, we need to think about what consistency requirements on choices can generate.
Assume that for each (p; w), the consumer chooses a
unique bundle x(p; w):[We say nothing about why the
consumer makes that choice.]
We call x(p; w) a demand function.
Further, assume x(p; w) is homogenous of degree zero
and satis…es Walras’law.
Weak Axiom of Revealed Preference (WARP): For any
two price-wealth situations (p; w) and (p0; w0); the following holds: if p x(p0; w0)
w and x(p0; w0) 6=
x(p; w), then p0 x(p; w) > w0:
Reasoning behind the axiom:
p x(p0; w0)
w and x(p0; w0) 6= x(p; w)
) the bundle x(p0; w0) was a¤ordable in the situation
where consumer chose a di¤erent bundle x(p; w)
) consumer revealed a preference for x(p; w) over x(p0; w0)
) consumer should choose x(p; w) over x(p0; w0) whenever both are a¤ordable
) since consumer chooses x(p0; w0) in situation (p0; w0);
the bundle x(p; w) must not be a¤ordable in this situation
) p0 x(p; w) > w0:
Now, …x w and consider a price change from p to p0.
To remove income e¤ect, adjust wealth to w0 so that at
price p0, the consumer can just a¤ord the bundle x(p; w)
chosen prior to price change.:
(Slutsky compensation criterion)
w0 = p0 x(p; w)
Then, x(p0; w0) is the (Slutsky) compensated demand at
price p0:
(Proposition 2.F.1) Compensated Law of Demand.
If x(:; :) satis…es homogeneity of degree zero, Walras’
Law and WARP, then the following property holds:
for any compensated price change from a initial pricewealth situation (p; w) to a new situation (p0; w0) =
(p0; p0 x(p; w)); we have
(p0
p)[x(p0; w0)
x(p; w)]
0
with strict inequality whenever x(p; w) 6= x(p0; w0) i.e.,
Proof: The inequality is immediate if x(p; w) = x(p0; w0):
So, suppose x(p; w) 6= x(p0; w0):
Then,
(p0
p)[x(p0; w0)
= p0 x(p0; w0)
= w0
x(p; w)]
p x(p0; w0)
p x(p0; w0)
p0 x(p; w) + p x(p; w)
w0 + w;
using Walras Law and Slutsky compensation criterion,
and the latter expression is
=w
p x(p0; w0):
If w p x(p0; w0) 0, then WARP implies p0 x(p; w) >
w0 which violates the fact that w0 = p0 x(p; w). Thus,
(p0
= w
p)[x(p0; w0)
x(p; w)]
p x(p0; w0) < 0:
This concludes the proof.