Microeconomics 3
at the Economics Programme, University of Copenhagen.
Spring semester 2006
Week 6
Today’s programme
Preference and demand. (1a) utility representations (summary from last
week)
Preference and demand. (1b) revealed preference: theory
Preference and demand. (1b) revealed preference: some experimental
evidence,
Preference and demand. (1c) aggregate demand
(last week)
We focused on a single consumer.
Three ways to model the consumer:
1. A preference relation
2. A utility function
3. A demand function
Question: Does it matter which model we choose to begin with?
Our aim: to formulate some basic results linking between 1.,2.,& 3.
(last week)
Consumption set X
x
Rk+:
is a binary relation on X .
y is interpreted as "x is at least as good as as y".
is complete (or "total") if for all x; y 2 X we have x
is transitive if x
y and y
z implies x
z:
y or y
x:
(last week)
A function u : X ! R represents
u(x)
if:
u( y ) , x
y,
for all x; y 2 X:
Theorem: If X is …nite, then the following is equivalent:
1)
is complete and transitive.
2)
can be represented by a utility function.
(last week)
Continuity: For all y 2 X the sets
fx j x
fx j y
yg
xg
are closed.
It is fair to say that continuity is only a "technical" assumption (agree? )
Strong monotonicity: If x
y and x 6= y then x
y.
Theorem: Suppose that
is complete, transitive, continuous, and strongly
monotonic. Then there exists a utility function u which represents
(last week)
Sometimes (a complete and transitive binary relation) represents preferences
over probability distributions (i.e. lotteries).
In this case, certain special assumptions ("axioms")on
have appeal, which
gives us not only existence of a utility function representing
; but - more
interestingly - a utility function which has a special form.
In fact, it allows us to write up the utility function as an expected utility.
(In the Problem set for week 5, you were asked to recall the details of this
expected utility model)
More on expected utility theory
We describe a lottery by a function p de…ned on the set of outcomes X where
p(x) > 0 for a …nite number of outcomes x; p(x) = 0 otherwise, and where
all probabilities sum to 1:
Let P denote the set of all such lotteries on X:
We now consider a complete and transitive binary relation on
represents preferences over lotteries.
on P which
Let supp(p) = fx 2 X j p(x) 6= 0g denote the support of a lottery p. Under
certain assumptions (see next slides), there is a utility function u on P and a
function q on X such that for any p; p0 2 P ,
p
p0 , u(p)
u(p0);
where
u(p) =
X
p(x)q (x);
x2supp(p)
Further, q is uniquely determined up to an a¢ ne transformation.(strictly increasing)
Additional remarks:
The function u represent preferences (over P ) in the usual - ordinal - sense.
The point here is that we are allowed to write u in a particularly simple way.
In the context of decision under uncertainty, the function q is typically also
referred to as a utility function. (such as, "the von Neumann Morgernstern
utility function").
Conditions on
resentatation:
(complete and transitive) that imply an expected utility rep-
(independence axiom) For all p; q; r 2 P and a 2 (0; 1]; p
ap + (1 a)r aq + (1 a)r:
q implies
(continuity axiom) For all p; q; r 2 P , if p
q
r then there exist a; b 2
(0; 1) such that ap + (1 a)r q bp + (1 b)r.
Revealed preferences
To sum up, we have considered the link between 1) binary relations and 2)
utility functions.
We now, in a similar spirit, investigate the link between 1) binary relations and
3) demand functions.
Consider a demand function x(p; m). Is there a utility function u from which
this demand function can be generated?
This classical problem is known as the integrability problem.
Interesting problem, but di¢ cult to handle.
Another problem is that who has even seen a demand function? We shall
(therefore) restrict our attention to the following variant:
Assume that we have a …nite number of observations of a consumer’s actual
demand (where we have varied prices and income).
We often refer to such data as revealed preference data (assuming thereby
implicitly that the consumer chooses what he/she prefers!)
Can we then …nd a utility function u consistent with these observations?
(We would then say that u "rationalizes" the observations)
Strictly speaking, the answer is trivially "yes!":
The utility function u(x) = c, for all x, rationalizes all imaginable revealed
preference data set.
But such function is not likely to represent the consumer’s actual preferences.
So let us reformulate the question:
Can we …nd a locally non-satiated utility function consistent with revealed
demand?
locally non-satiated: For any x 2 X and any " > 0, there is y 2 X with
jjx yjj < " such that u(y) > u(x):
or:
Can we …nd a strictly monotonic utility function consistent with revealed
demand?
We have data of the following form:
price vector income consumption bundle
obs 1
p1
m1
x1
...
...
...
...
obs T
pT
mT
xT
If the consumer does not always choose a bundle that lies on the budget line,
then it is clear that the consumer does not have a strictly monotonic utility
function.
We assume that the consumer always chooses a bundle that lies on the budget
line.
This is, in fact, quite convenient, because then we do not have to specify the
income (since we know that mt = ptxt)
Thus we consider a data set of the following form: (pt; xt); t = 1; :::; T .
If (x1; x2) is chosen when prices are p1,p2; and p1x1 + p2x2
(x1; x2) is directly revealed preferred to (y1; y2).
p2y1 + p2y2,
Notation: (x1; x2)RD (y1; y2):
If the consumer maximizes utility u, then we must have u(x)
u(y ):
This is a very elementary observation, but notice the general point: If the
consumer maximizes utility, then the data allows us to say something about
the utility function.
x2
(x1,x2)
slope: -p1/p2
(y1,y2)
x1
If y is not on the budget line (but still in the budget set) when x is choosen,
we say that x is strictly directly revealed preferred to y .
Notation: (x1; x2)P D (y1; y2):
If u exists, and is strictly monotonic, we must have u(x) > u(y):(why ?)
x2
violates WARP
xt
xs
x1
To rule out this situation, we can de…ne:
Weak Axiom of Revealed Preferences (WARP): If xtRD xs; xt 6= xs; then
it is not the case that xsRD xt.
x2
satisfies WARP
xt
xs
x1
we have the following:
Theorem (Rose, 1958): When k = 2; the following is equivalent:
1) The data satis…es WARP.
2) There exists a strictly monotonic utility function that rationalizes the data.
3) There exists a strictly monotonic, continuous, strictly concave, utility function that rationalizes the data.
Nice results, but there are two limitations:
First, the results does not hold.when k > 2: In this case WARP is necessary
but not su¢ cient for utility max.
In this case WARP is (unfortunately) not enough to ensure utility max.
We need a condition that is stronger that WARP.
Suppose that we have a sequence xt; xj ; xk ; :::; xn; xs where
xtRD xj
xj RD xk
...
xnRD xs
Then we say that xt is revealed preferred to xs:
Notation: xtRxs.
If a strictly concave utility function exists (rationalizing data), then we must
have u(xt) > u(xs).
The following condition rules out that we at the same time can conclude that
u(xt) < u(xs) if u is strictly concave (which is clearly nonsense and would
therefor rule out the utility maximizing hypothesis).
Strong Axiom of Revealed Preferences (SARP): If xtRxs; xt 6= xs; then
it is not the case that xsRxt.
x2
xt
xj
xs
x1
Since Houthakker (1950), it has been known that SARP is closely related to
the utility maximizing hypothesis.
During the years, the connection between SARP and utility max has been
extensively studied, culminating with the following remarkable result by Matzkin
and Richter:
Theorem (Matzkin and Richter, 1991): For an arbitrary number k of
commodities; the following is equivalent:
1) The data satis…es SARP.
2) There exists a strictly monotonic utility function that rationalizes the data.
3) There exists a strictly monotonic, continuous, strictly concave utility function
that rationalizes the data.
NB: If k = 2; WARP,SARP.
We can think of situations, where SARP is violated not because of inconsistence
with utility max, but simply due to non-uniqueness of the most preferred bundle
(for example when indi¤erence curves are linear!)
x2
budget line
x1
perfect substitutes
Consider the following weaker version of SARP
Generalized Axiom of Revealed Preferences (GARP): If xtRxs; then
xsRxt implies psxs psxt.
"if xt is revealed preferred to xs; then when you choose xs (at prices ps) xt is
not strictly within the budget set"
or using Varian’s terminology: If xtRxs then xs is not strictly directly revealed preferred to xt.
Theorem (Afriat 1967, Varian 1982): For an arbitrary number k of commodities;
the following is equivalent:
1) The data satis…es GARP.
2) There exists a locally nonsatiated utility function that rationalizes the data.
3) There exists a locally nonsatiated, continuous, concave, monotonic, utility
function that rationalizes the data.
Note the implication of the result: If some data set can be rationalized by
a non-satiated utility function, then it can also be rationalized by a locally
nonsatiated, continuous, concave, monotonic, utility function!
Hence given utility maximization holds, the following assumption on u can
never be rejected from a …nite data set: continuity, concavity, monotonicity.
Similar conclusion from the WARP/SARP results:
Given utility maximization holds, the following assumption on u can never be
rejected from a …nite data set: continuity, strict concavity, strong monotonicity
(and you are not likely to have an in…nite number of observations, aren’t you? )
Provides us with a strong "theoretical justi…cation" for making such assumptions.
Illustration:
If upper contour sets are not convex, we can never "see" it from choices made
under linear budget constraints.
Hence we can just as well assume that the utility functions is quasi-concave
(in fact, we are allowed to assume that u is not only quasi-concave but also
concave, as the theorem shows).
x2
actual indifference
curves
x1
x2
convexified
indifference curves
x1
Some experimental evidence
Harbaugh et al, AER, 2001:
Harbaugh et al, AER, 2001 (continued):
Andreoni and Miller, Econometrica, 2002:
Andreoni and Miller, Econometrica, 2002: (continued).
Aggregate demand
Standard consumer theory is developed in the context of an individual consumer.
But it is often easier to obtain consumption data at some aggregate level, for
example at a household level.
3 Questions:
1. When can aggregate demand be expressed as a function of aggregate income?
2. If WARP holds for individual demand, will WARP the hold for aggregate
demand?
3. Does it make sense to talk about a "representative" consumer?
Notation (following DDL note):
k commodities. n consumers.
Consumer i demand:
xi(p; mi) = (x1i (p; mi); :::; xki (p; mi))
Aggregate demand:
X (p; m1; :::; mn) =
n
X
i=1
xi(p; mi):
Question: Under what conditions do we have:
X (p; m1; :::; mn) = X (p;
n
X
mi)?
i=1
This holds if and only if for any two income distributions
m0 = (m01; :::; m0n)
and
m00 = (m001 ; :::; m00n)
where
P
m0i =
P
m00i ; we have:
X (p; m01; :::; m0n) = X (p; m001 ; :::; m00n)
Or formulated di¤erently (looking only at small displacemants in income):
From any initial distribution (m1; :::; mn) then for a small income restribution
Pn
(dm1; :::; dmn) where i=1 dmi = 0, we must have:
n @xj (p; m )
X
i
i
dmi = 0;
@m
i
i=1
for j = 1; :::; k:
This is true for all small redistributions if and only if:
j
j
@xh(p; mh)
@xi (p; mi)
=
;
@mi
@mh
for any j , and any pair i ; h; all (m1; :::; mn).
In other words, the income e¤ect must be the same for all consumers at all
income levels.
Income´
ns
o
C
ri
e
um
er
m
n su
o
C
h
Good j
Engel curves
Good l´
th
a
p
n
o
r i ansi
e
um exp
s
n e
Co o m
th
c
a
n
p
I
n
o
i
h
s
r an
e
p
sume ex
n
Co om
Inc
Good j
Engel curves /
Income expansion paths
Only a restrictive class of utility functions is consistent with this property.
Proposition: A group of consumers (i = 1; :::; n) have straight, parallel Engel
curves if and only if preferences admit indirect utility functions of the Gorman
form:
vi(p; mi) = ai(p) + b(p)mi;
for all i:
First: let us recall de…nition of indirect utility vi:
For a consumer i and utility function ui,
vi(p; mi) = max ui(x1; :::; xk )
s.t. xp = mi:
Restrictive class of (indirect) utility functions, but satis…ed for two important
classes:
a) Symmetric, homothetic utility functions. Implies linear Engel curves, and
vi(p; mi) = v (p)mi.
Examples: Cobb-Douglas, perfect substitutes, perfect complements.
a) Quasi-linear utility, that is: ui(x; mi) = mi + ui(x); where mi is consumption of composite good (money). Implies vi(p; mi) = v (pi) + mi (at least
when mi is su¢ ciently large).
Does aggregate demand satisfy WARP?
No! Even if WARP is satis…ed at individual level this might not be the case.
We look at example with 2 goods, and 2 consumers who share aggregate income
M equally ( 12 M for each consumer).
1 x (p; M ) + 1 x (p; M ):
)
=
We consider average demand: 12 X (p; M
2
2 1
2
2 2
2
(aggregate demand is just twice the average demand).
good 2
good 1
Why focus on WARP (and not SARP or GARP)?
WARP is the simplest condition. Further, WARP is necessary (but not su¢ cient, unless k = 2) for SARP.
Problems with "representative consumer"
There are many examples from economic theory of use of a "representative
consumer".
Very convenient, since it is much easier to handle a single ("representative")
consumer in a model, than a population of heterogeneous consumers.
Two main Questions:
1) Is aggregate demand rationalizable from utility function of a single representative consumer?
2) If "Yes" to 1), would it then be a reasonable social objective to maximize
the utility of the "representative" consumer?
Ad 1.: Let X (p; m) denote aggregate demand. A positive representative
consumer exists if there is a utility function (preference relation) such that
X (p; m) is the demand function generated by this utility function.
This is (again) the so-called integrability problem, which we shall not discuss.
However, if we have only a …nite number of aggregate demand observations, it
is the revealed preference problem studied earlier. We have just seen that for
the average consumer such a utility function may not exist (because of violation
of WARP at aggregate level), even if WARP is satis…ed at individual level.
Ad 2. The question is whether society’s welfare max. corresponds to utility
max. for the representative consumer.
Problem: When we look at the representative consumer, only aggregate consumption counts (by construction).
Hence it is only a meaningful objective to maximize the representative consumers’s utility if the underlying the consumer’s utility functions and society’s
social welfare function are such that distribution of income is does not matter
for overall welfare- a very special case!
Main example:
1) Individual utility of the Gorman form: vi(p; mi) = ai(p) + b(p)mi; i =
1; :::; n:
2) Utilitarian social welfare function: Social welfare =
P
vi .
v (p; M ) =
X
ai(p) +
X
b(p)mi
= A(p) + B (p)M:
Then v is the social welfare fuction (by de…nition),
v also has the Gorman form. In particular, it is the utility function of a
representative consumer (why? )