2nd Lecture, ECON 5200, Classical Demand Theory
Kjell Arne Brekke
August 31, 2010
1
Preferences
Preferences are rational.
We usually also invoke some additional properties. The …rst one comes in three variesties.
1. Preferences are monotone if y >> x implies y
2. Preferences are strongly monotone if y
x
x and x 6= y implies y
x
3. Preferences are locally nonsatiated if for every x and every " > 0, there is an y such
that jjx
yjj
" and y
x:
Proposition 1 These are properties are related as follows
2 =) 1 =) 3
Another important assumption is:
De…nition 2 Preferences are convex if for every x the upper contour set is fy 2 X : y
xg is convex.
De…nition 3 Preferences are strictly convex if for every x the y
y + (1
x for
)z
1
2 (0; 1)
x and z
x implies
The text also states two properties about preferences that are often stated in terms of
utility functions: Homothetic and quasi-linear. It is useful to note that these are about
preferences. Utility functions are only means to express preferences. I’ll only mention the
quasi-linear case:
De…nition 4 Preferences are homothetic i¤
x
y () x
y for all
0
De…nition 5 Preferences are quasi-linear with respect to commodity 1 if
x
1.1
y () x + e1
y + e1 for all
0
Utility
Not all preferences can be represented by a utility function, one exampel are lexicogra…c
preferences.
When we construct the utility function the idea we will use is the following:
For any x choose the intersection between the indi¤erence curve thorugh x and the
45 degree line. That intersection is u(x)e; for some number x and where e = (1; 1; 1:::; 1).
u(x) represent the preferences if property 1 hold.
De…nition 6 Preferences are continuous i¤ for any sequence of pair (xn ; y n )1
n=1 with
xn
y n and with x = limn!1 xn and y = limn!1 y n then x
y: (Equivalently, the upper
contour set is closed.)
Problem 7 Show that lexiographic preferences are not continuous. Hint, for the case
where the …rst component of x = (x1 ; x2 ) has lexicographic priority, sonsider the sequences:
xn = (1 +
2
1
1
1
; 1 + ) and y n = (1 + ; 2 + )
n
n
n
n
Proposition 8 Suppose rational preference relation
a utility function representing it.
2
on X is continuous, then there is
2
Utility maximization
The utility maximization problem
max u(x)
x 0
s.t. p x
w
Proposition 9 If p >> 0 and u is continuous, the utility maximization problem (UMP)
has a solution
Proof: The budget set is compact (compactness will be discussed in class).
The solution, the Walrasian demand correspondence x(p; w) is in general a correspondence, that is there can be more than one solution to the UMP. The correspondence
satis…es the assumptions we invoked last lecure:
Proposition 10 If u is continuous representing locally nosatiated preferences, then x(p; w)
satis…es
1. Homogeneity of degree zero
2. Walras law p x(p; w) = w
3. If
is convext then x(p; w) is a covex set, and if
is strictly convex then x(p; w)
is a singleton, that is x(p; w) is a function.
The UMP is a standard constrained optimization problem. The Kuhn-Tucker condition are then
ru(x )
x [ru(x )
p
p] = 0
or in single components
@u(x )
@xl
L
X
l=1
xl
@u(x )
@xl
pl
3
pl
= 0
Note that both
@u(x )
@xi
0 and xl
pi
0, thus no term can become positive. For
the sum to be zero, every therm has to be zero, thus:
@u(x )
= pl
@xl
if xl > 0 then
Problem 11 Solve the UMP with utility
u(x) = ax1 + bx2 with a; b > 0
Using Kuhn-Tucker. (Note that except for
2.1
p1
p2
=
a
b
the solution is a corner solution.)
Indirect utility
The indirect utility is
v(p; w) = max u(x)
px w
Proposition 12 Suppose u represent preference satisfying local nonsatiation. Then the
indirect utility has the following properties
1. Homogenous of degree zero
2. Strictly increasing in w and non-increasing in pl for any l
3. Quasiconvex: f(p; w) : v(p; w)
vg is convex
4. Continuous in p and w.
3
Expenditure minimization
The expenditure minimization problem (EMP) is
e(p; u) = min px
u(x) u
There is a close connection between EMP and UMP:
Proposition 13 Suppose that
satis…es local nonsatiation and p
4
0
1. If x solves the UMP then x is optimal in the EMP with required utility u(x ).
2. If x solves the EMP, then x solves the UMP with w = px .
We will approach this duality between the two problem further in the following. Note
that one implication is
e(p; u(x )) = w
or in other words
e(p; v(p; w)) = w
and similarly
v(p; e(p; u)) = u
We denote the solution to the EMP the Hicksian demand correspondence, h(p; u)
Proposition 14 Under standard assumption h(p; u) satis…es
1. Homogenous of degree zero in p: h(p; u) = h( p; u)
2. No excess utility: For any x 2 h(p; u): u(x) = u.
3. If
is convexm then h(p; u) is a convex set, if
is strictly convex then there is a
unique element
4
Duality
The economic discussion of duality is part of a more general matematical structure:
A convex set K is characterized by the support function
K (p)
L
= inffpx : x 2 Kg for all p 2 RL (Note, not just p 2 R+
)
and we get back K
K = fx 2 RL : px
K (p)
5
for all p 2 RL g
Theorem 15 Duality theorem:Let K be a nonempty closed set and let
K (p)
port function. Given p, there is a unique x such that px =
is di¤erentiable.
K (p)
i¤
K
be the sup-
Morover, in this case
r
K (p)
=x
No proof. This illustrates that what follows is part of a more general mathematical
structure, but without a proof it does not add much more. Intuition of the equation. let
(p) = px
K (p)
then since x 2 K
px
K (p)
but
(p) = px
K (p)
0 = (p)
Hence since it is a minimum
@
(p) = 0
@pi
thus
@ k
@
(p) = xi +
(p) = xi
@pi
@pi
4.1
Relationship between Demand, Indirect utility and Expenditure functions
Lemma 16 Shepards lemma
h(p; u) = rp e(p; u)
The most clarifying proof is the envelop theorem argument:
L
X
@
@
@xk
e(p; u) =
( min px) = xl +
pk
@pl
@pl u(x)=u
@pl
k=1
but the last term must be zero by the …rst order condition, asd we will se in class (and in
the textbook).
Proposition 17 Under standard assumptions
6
1. Dp h(p; u) = Dp2 e(p; u)
2. Dp h(p; u) is negative semide…nite
3. Dp h(p; u) is symmetric
4. Dp h(p; u)p = 0
4.2
Slutsky
Di¤erentiating the identy hl (p; u) = xl (p; e(p:u)) yields:
Proposition 18 Under standard assumption
@hl
@xl
@xl
=
+ xk
@pk
@pk
@w
Note here that the slutsky matrix
2
3 2
6
6
6
6
6
6
4
@h1
@p1
@h1
@p2
@h1
@pL
@h2
@p1
@hL
@pL
7 6
7 6
7 6
7=6
7 6
7 6
5 4
@x1
@p1
+
1
x1 @x
@w
@x1
@pL
+
@xL
@p1
L
+ x1 @x
@w
@xL
@pL
L
+ xL @x
@w
1
xL @x
@w
must be symmetric and negative semide…nte.
3
7
7
7
7
7
7
5
The meaning of negative semide…nite:
f (x)
f (x0 ) + rf (x0 )(x
x0 )T D2 f (x0 )(x
x0 ) + (x
x0 )
Now if
z T D2 f (x0 )z
0
the second order term will always be negative, as with a concave function.
Proposition 19 Roy
xi (p; w) =
The Slutsky matrix Negative semide…nite
7
@v
@pi
@v
@w
5
Integrability
Proposition 20 If e(p; u) is strictly increasing in u and is continuous, increasing, homogenous of degree one concave and di¤erentiable in p:Then e is the expenditure function
associated with the at least as good as set
Vu = fx : px
e(p; u); 8p
0g
That is e(p; u) = minfpx : x 2 Vu g
What about the other way around, to start with the demand and …nd the preferences,
since we get the preferences from e we can start with …nding the expenditure function:
We know that
@e
= x1 (p; e(p; u))
@p1
@e
= x2 (p; e(p; u))
@p2
:::
@e
= x2 (p; e(p; u))
@pL
Now the solution to be an expenditure function we know that it must be concave,
hence the Slutsky matrix must be symmetric and negative semide…nite.
6
Welfare changes
As with utility function, any monoton transform of the indirect utility function makes it
still a utility function. One particular transform is
e(p; v(p; w))
where we keep p …xed. This is called the money-metric utility as utility is measured in
terms of the income required to reach the utility level v(p; w) under prices p. We can use
this to assess the e¤ect of a price change
e(p; v(p1 ; w))
e(p; v(p0 ; w))
8
and in particular two choices of p seems natural p = p1 or p0 . The changes are then
EV (p0 ; p1 ; w) = e(p0 ; v(p1 ; w))
= e(p0 ; u1 )
e(p0 ; v(p0 ; w))
e(p0 ; u0 ) = e(p0 ; u1 )
CV (p0 ; p1 ; w) = e(p1 ; v(p1 ; w))
= e(p1 ; u1 )
w
e(p1 ; v(p0 ; w))
e(p1 ; u0 ) = w
e(p1 ; u0 )
To interpret the …rst, note that at prices p0 if the consumer get e(p0 ; u1 ) in income (that
is an additional EV is given to him) he get to the utility level u1 that same utility level as
he will get with no compenastion if prices changes to p1 . That is There are two eqivalent
ways to change the consumers utility from u0 to u1 either change prices or equivalently
change his income with EV.
Now suppose that commodity 1 is a normal good, so we want to consume more of
it when we are better o¤. If the price changes from a high p01 to a lower level p11 then
u0 < u1 and hence
EV =
Z
p01
1
h1 (p1 ; p 1 ; u )dp1 >
Z
p01
h1 (p1 ; p 1 ; u0 )dp1 = CV
p11
p11
Similarly if the price increase, then u1 < u0 and
EV =
Z
p11
1
h1 (p1 ; p 1 ; u )dp1 >
Z
p11
h1 (p1 ; p 1 ; u0 )dp1 = CV
p01
p01
Problem 21 Some of you may be familiar with the use of consumer surplus as an individual welfare measure in the case of quasi-linear preferences:
u(x1 ; x2 ) = v(x1 ) + x2
In this case EV=CV. Explain intuitively why. (Hint: How do the hicksian demand for
commodity 1 respond to changes in utility level?)
9