Microeconomics I
c
Leopold S¨ogner
Department of Economics and Finance
Institute for Advanced Studies
Stumpergasse 56
1060 Wien
Tel: +43-1-59991 182
soegner@ihs.ac.at
http://www.ihs.ac.at/∼soegner
September, 2011
Consumer Theory 4
Revealed Preference Theory (1)
Micro I
• Weak Axiom of Revealed Preference.
• Strong Axiom of Revealed Preference.
• Revealed preferences and utility maximization.
MasColell, Chapter 1.C and 2.F., 3.J.
1
Consumer Theory 4
Revealed Preference Theory (2)
Micro I
• Samuelson’s idea: Cannot we start with observed behavior
instead of assumptions on preferences.
• Idea: if a consumer buys a bundle x0
instead of an other
affordable bundle x1
, then the first bundle is called revealed
preferred to x1
(see Consumer Theory 1).
• Definition - Weak Axiom on Revealed Preference: [D 2.F.1]
A Walrasian demand function x(p, w) satisfies the weak axiom of
revealed preference if for any two wealth price situations (p,w)
and (p’,w’) the following relationship holds: If p · x(p , w ) ≤ w
and x(p , w ) = x(p, w) then p · x(p, w) > w .
2
Consumer Theory 4
Revealed Preference Theory (3)
Micro I
• Interpret the weak axiom by means for Figure 2.F.1, page 30.
• We assume that x(p, w) is a function, which is homogeneous of
degree zero and Walras’ law holds.
3
Consumer Theory 4
Revealed Preference Theory (4)
Micro I
• From the former parts we already know:
Theorem - Weak Revealed Preference and Utility
maximization: If x(p, w) solves the utility maximization problem
with strictly increasing and strictly quasiconcave utility function,
then the weak axiom of revealed preference has to hold.
4
Consumer Theory 4
Revealed Preference Theory (5)
Micro I
Proof:
• Consider a pair x0
and x1
where x0
= x(p0
, w) solves the utility
maximization problem for p0
, x1
for p1
.
• Assume u(x0
) > u(x1
): w = p0
· x0
≥ p0
· x1
. Then
p1
· x0
> p1
· x1
= w. Otherwise a consumer would have chosen
x0
if it were affordable in the second maximization problem.
• I.e. p1
· x0
> p1
· x1
has to be fulfilled. Since x0
and x1
are
arbitrary pairs, the weak axiom of revealed preference has to hold.
5
Consumer Theory 4
Slutsky Compensation (1)
Micro I
• Definition - Slutsky compensation: Given a bundle
x0
= x(p, w) and income is compensated such that the consumer
can always buy the bundle x0
, i.e. w = p · x(p, w). Then
demand is called Slutsky compensated demand xS
(p, w(x0
)).
• Discuss this concept by means of Figure 2.F.2, page 31.
6
Consumer Theory 4
Slutsky Compensation (2)
Micro I
• Proposition: Suppose that the Walrasian demand function
x(p, w) is homogeneous of degree zero and satisfies Walras’ law.
Then x(p, w) satisfies the weak axiom if and only if the following
property holds:
For any compensated price change form the initial situation
(p, w) to a new pair (p , w ), where w = p · x(p, w), we have
(p − p) · [x(p , w ) − x(p, w)] ≤ 0
with strict inequality whenever x(p, w) = x(p , w ). [P 2.F.1]
• Remark: x(p , w ) = xS
(p , w(x0
)). In addition to the text book
we assume that x(p, w) solves the UMP with the assumptions of
the above theorem.
7
Consumer Theory 4
Slutsky Compensation (3)
Micro I
Proof:
• (i) The weak axiom implies (p − p) · [x(p , w ) − x(p, w)] ≤ 0
with strict inequality for different demands: If x(p , w ) = x(p, w)
then [x(p , w ) − x(p, w)] = 0.
• Suppose x(p , w ) = x(p, w) and expand
(p − p) · [x(p , w ) − x(p, w)] to
p · [x(p , w ) − x(p, w)] − p · [x(p , w ) − x(p, w)]. By Walras’ law
and the construction of compensated demand the first term is 0.
• By compensated demand we get p · x(p, w) = w . I.e.
x0
= x(p, w) can be bought with p , w . By the weak axiom
x(p , w ) /∈ Bp,w, such that p · x(p , w ) > w. Walras’ law implies
p · x(p, w) = w. This yields p · [x(p , w ) − x(p, w)] > 0, such
that ... holds. 8
Consumer Theory 4
Slutsky Compensation (4)
Micro I
Proof:
• (ii) (p − p) · [x(p , w ) − x(p, w)] ≤ 0 implies the weak law if
x(p , w ) = x(p, w):
• If we consider compensated demand, then the weak axiom has to
hold (replace Walrasian demand by compensated demand in the
Theorem - Weak Revealed Preference and Utility maximization.
• It is necessary that the weak axiom holds for all compensated
demand changes: Assume u(x0
) > u(x1
): w = p0
· x0
≥ p0
· x1
.
Suppose that p1
· x0
≤ p1
· x1
= w. Then x0
cannot be an
optimum by local non-satiation.
• By these arguments the weak law holds if p · x(p , w ) > w
whenever p · x(p, w) = w and x(p , w ) = x(p, w). 9
Consumer Theory 4
Slutsky Compensation (5)
Micro I
Proof:
• By this argument we can test for the weak axiom by looking at
compensated price changes. (We show that ¬H ⇒ ¬C.) If the
weak law does not hold, there is a compensated price change
from p , w ti some p, w such that p · x(p, w) = w and
p x(p, w ) ≤ w . By Walras law we get
p · [x(p , w ) − x(p, w)] = 0
and
p · [x(p , w ) − x(p, w)] = 0.
• This results in (p − p) · [x(p , w ) − x(p, w)] ≥ 0 and
x(p , w ) = x(p, w). This contradicts the
(p − p) · [x(p , w ) − x(p, w)] ≤ 0 holds. 10
Consumer Theory 4
Revealed Preference Theory (6)
Micro I
• From the above sections we know that the substitution matrix
S(p, w) is symmetric and negative semi-definite when we start
with utility maximization and non-satiated preferences.
• For a general Walrasian demand function satisfying Walras’ law
and the weak axiom, the substitution matrix has to be negative
semi-definite.
11
Consumer Theory 4
Revealed Preference Theory (7)
Micro I
• Theorem - Revealed Preference and Demand (I): If the
weak axiom of revealed preference and budget balancedness hold,
then xC
(p, w(x0
)) is homogeneous of degree zero and the
Slutsky matrix is negative semidefinite.
• The existence of the Slutsky matrix requires that x(p, w) is
differentiable.
12
Consumer Theory 4
Revealed Preference Theory (8)
Micro I
Proof:
• Homogeneous of degree zero: xC
(p, w(x0
)) = xC
(νp, w(x0
)) for
ν > 0. With p1
= νp0
and w1 = νw0 and the assumption that all
income is spent we get w1 = p1
· x1
= νp0
x0
, such that x1
= x0
since p1
= νp0
.
• In addition w1 = νw = p1
· x1
= νp0
· x1
, also results in x0
= x1
.
Therefore, xC
(p, w(x0
)) is homogeneous of degree zero.
• The weak axiom of revealed preference implies: p0
· x0
≤ p0
· x1
.
Equality only holds for x0
= x1
. For x0
= x1
we get (i)
p0
· x0
< p0
· x1
.
13
Consumer Theory 4
Revealed Preference Theory (9)
Micro I
Proof:
• With a Slutsky compensated choice function we get:
x0
= xC
(p0
, w(x0
)) and x1
= xC
(p1
, w(x0
)) (here w1 = p1
· x0
and w0 = p0
· x0
, i.e. with w1 and w0 we can buy x0
given p1
or
p0
).
• With Slutsky compensated choice and Walras’ law, we get (ii)
p1
· x0
= p1
· xC
(p1
, w(x0
)) .
• The difference (ii)-(i) now yields for arbitrary prices p1
:
(p1
− p0
) · x0
≥ (p1
− p0
) · xC
(p1
, w(x0
)) .
14
Consumer Theory 4
Revealed Preference Theory (10)
Micro I
Proof:
• Choose an arbitrary z. Let p1
:= p0
+ νz is an arbitrary vector in
Rn
, choose ν such that p1
remains non-negative. With ν = 0, we
get p1
= p0
.
• In terms of p1
= p0
+ νz, wν(x0) = (p0
+ νz) · x0
we get (iii)
z · x0
≥ z · xC
(p0
+ νz, w(x0
)) .
• Define the function
g(ν) = z · xC
(p0
+ νz, w(x0
)) .
15
Consumer Theory 4
Revealed Preference Theory (11)
Micro I
Proof:
• By relationship (iii) this function is maximized with ν = 0;
w0 = p0
· x0
.
• Due to ≤ in (iii), the first derivative has to be negative (≤ at
ν = 0):
dg(ν)
dν
=
n
i=1
n
j=1
zi
∂xC
i (p0
, w0)
∂pj
+ x
C
j (p
0
, w0)
∂xC
i (p0
, w0)
∂w
zj ≤ 0 .
• This corresponds to the definition of negative semidefiniteness.
The matrix implied by the terms
∂xC
i (p0
,x0)
∂pj
+ xC
j (p0
, x0)
∂xC
i (p0
,x0)
∂w
is the Slutsky matrix for the compensated demand xC
(p, x0
).
16
Consumer Theory 4
Revealed Preference Theory (12)
Micro I
• If we can show that the Slutsky matrix for xC
(p, w(x0
)) is
symmetric, then the integrability theorem tells us the
xC
(p, w(x0
)) is a demand function arising from an UMP.
• Answer: yes for the two good case, no for L > 2.
• Problem: intransitive circles.
17
Consumer Theory 5
Integrability (1)
Micro I
• We already know that a Walrasian demand function satisfies
homogeneity of degree zero, Walras’ law, symmetry and negative
semidefiniteness and Cournot and Engel aggregation.
• Cournot and Engel aggregation: follow directly from Walras’ law.
(see Chapter 2)
• Walras’ law and a symmetric Slutsky substitution matrix imply
homogeneity of degree zero (see Chapter 1).
18
Consumer Theory 5
Integrability (2)
Micro I
• Currently we know that we get a Walrasian demand system
fulfilling budget balancedness, symmetry and negative
semidefiniteness of the Slutsky matrix from utility maximization.
• Is this list complete or are there any further properties? ⇒ yes
• Theorem - Integrability Theorem: A continuously
differentiable function x(p, w) is a demand function generated by
some increasing, quasiconcave utility function if it satisfies
budget balancedness and symmetry and negative semidefiniteness
of the Slutsky matrix.
If and only if holds when utility is continuous, strictly increasing
and strictly quasiconcave.
19
Consumer Theory 4
Revealed Preference Theory (13)
Micro I
• Definition - Strong Axiom of Revealed Preference: [3.J.1]
The market demand satisfies the strong axiom of revealed
preference if for any list
(p1
, w1
), . . . , (pN
, wN
)
with x(pn+1
, wn+1
) = x(pn+1
, wn+1
) for all n ≤ N − 1, we have
pN
x(p1
, w1
) > wN
whenever pn
· x(pn+1
, wn+1
) ≤ wn
for all
n ≤ N − 1.
• I.e. if x(p1
, w1
) is directly or indirectly revealed preferred to
x(pN
, wN
), then x(pN
, wN
) cannot be directly or indirectly be
revealed preferred to x(p1
, w1
). Or for different bundles
x1
, x2
, . . . , xk
: If xq
is revealed preferred to x2
and x2
is preferred
to x3
, then x1
is revealed preferred to x3
. 20
Consumer Theory 4
Revealed Preference Theory (14)
Micro I
• Theorem - Revealed Preference and Demand (II): If the
Walrasian demand function x(p, w) satisfies the strong axiom of
revealed preference then there is a rational preference relation
that rationalizes x(p, w). I.e. for all (p, w), x(p, w) y for every
y = x(p, w) with y ∈ Bp,w. [P 3.J.1].
• Proof - see page 92.
21
Consumer Theory 5
Welfare Analysis (1)
Micro I
• Measurement of Welfare
• Concept of the Equivalent Variation, the Compensating Variation
and the Consumer Surplus.
• Pareto improvement and Pareto efficient
Literature: Mas-Colell, Chapter 3.I, page 80-90.
22
Consumer Theory 5
Welfare Analysis (2)
Micro I
• From a social point of view - can we judge that some market
outcomes are better or worse?
• Positive question: How will a proposed policy affect the welfare
of an individual?
• Normative question: How should we weight different effects on
different individuals?
23
Consumer Theory 5
Welfare Analysis (3)
Micro I
• Definition - Pareto Improvement: When we can make
someone better off and no one worse off, then a Pareto
improvement can be made.
• Definition - Pareto Efficient: A situation where there is no way
to make somebody better off without making someone else worth
off is called Pareto efficient. I.e. there is no way for Pareto
improvements.
• Strong criterion.
24
Consumer Theory 5
Consumer Welfare Analysis (1)
Micro I
• Preference based consumer theory investigates demand from a
behavioral perspective.
• Welfare Analysis provides a normative analysis of consumer
theory.
• E.g. how do changes of prices or income affect the well being of
a consumer.
25
Consumer Theory 5
Consumer Welfare Analysis (2)
Micro I
• Given a preference relation and Walrasian demand x(p, w).
• A price change from p0
to p1
increases the well-being of a
consumer if indirect utility increases. I.e. v(p1
, w) > v(p0
, w).
• Here we are interested in so called money metric indirect
utility functions.
26
Consumer Theory 5
Consumer Welfare Analysis (3)
Micro I
• Suppose u1 > u0, u1 arises from p1
, w1
and u0 from p0
, w0
.
• With p fixed at ¯p, the property of the expenditure function the
e(p, u) is increasing in u yields:
e(¯p, u1)) = e(¯p, v(¯p, ˜w1
)) = ˜w1
> e(¯p, v(¯p, ˜w0
)) = e(¯p, u0) = ˜w0
- i.e. it is an indirect utility function which measures the degree
of well-being in money terms.
27
Consumer Theory 5
Consumer Welfare Analysis (4)
Micro I
• Based on these considerations we set ¯p = p0
or p1
and
w = e(p0
, u0
) = e(p1
, u1
); we define:
– Definition - Equivalent Variation:
EV (p0
, p1
, w) = e(p0
, u1
) − e(p1
, u1
) = e(p0
, u1
) − w
– Definition - Compensating Variation:
CV (p0
, p1
, w) = e(p0
, u0
) − e(p1
, u0
) = w − e(p1
, u0
)
• EV measures the money amount that a consumer is indifferent
between accepting this amount and the status after the price
change.
• CV measures the money amount that is required to compensate
a consumer to remain on the same level of well-being.
28
Consumer Theory 5
Consumer Welfare Analysis (5)
Micro I
• Discuss Figure 3.1.2, page 82; if p1 falls then the consumer is
prepared to pay the amount CV , i.e. CV > 0.
• Both measures are associated with Hicksian demand.
• Suppose the only p1 changes, then p0
1 = p1
1 and p0
l = p1
l for
l ≥ 2. With w = e(p0
, u0
) = e(p1
, u1
) and
h1(p, u) = ∂e(p, u)/∂p1 we get
EV (p0
, p1
, w) =
p0
1
p1
1
h1((p1, p−), u1
)dp1
CV (p0
, p1
, w) =
p0
1
p1
1
h1((p1, p−), u0
)dp1
29
Consumer Theory 5
Consumer Welfare Analysis (6)
Micro I
• Discuss these integrals - Mas-Colell, Figure 3.1.3, page 83.
• EV, CV increase if utility increases and vice versa.
• If x1 is a normal good than the slope if the Walrasian demand
function is smaller (in absolute terms).
• We get EV (p0
, p1
, w) > CV (p0
, p1
, w) if the good is normal (in
absolute value), the converse is true for inferior goods.
• EV (p0
, p1
, w) = CV (p0
, p1
, w) with zero income effect for good
1. This is caused with quasilinear preferences for good one (see
[D 3.B.7])
30
Consumer Theory 5
Consumer Welfare Analysis (7)
Micro I
• EV (p0
, p1
, w) = CV (p0
, p1
, w) with zero income effect for good
1. In this case EV (p0
, p1
, w) = CV (p0
, p1
, w) is also equal to the
change in Marshallian Consumer Surplus.
• Definition - Marshallian Consumer Surplus:
MCSl(p, w) =
∞
p
xl((pl, p−), w)dpl
• Definition - Area Variation:
AV (p0
, p1
, w) =
p0
l
p1
l
x(pl, p−, w)dpl.
31
Consumer Theory 5
Area Variation Measure (1)
Micro I
• Definition - Area Variation:
AV (p0
, p1
, w) =
p0
1
p1
1
x(p1, p−, w)dp1.
• It measures the change in Marshallian consumer surplus.
• If the income effect is zero this measure corresponds to EV and
CV . (see Marshallian Consumer Surplus)
• The argument that AV provides are good approximation of EV
or CV can but need not hold. See Figure 3.1.8, page 90.
Jehle/Reny, 1st edition, Theorem 6.3.2, page 278: Willing’s
upper and lower bounds on the difference between CS and CV.
32
Consumer Theory 5
Dead Weight Loss (1)
Micro I
• Consider a price change in x1 or an alternative change in w by a
lump-sum tax TL.
• E.g. we put a commodity tax of t on x1 or levy a lump-sum TL
tax such that income is decreased.
• Restriction: TL has to be equal to
TC = x1(p1
, w)t = h1(p1
, u1
)t, u1
= v(p1
, w).
• Then the change in welfare with the commodity tax is:
−EV (p0
, p1
, w) = e(p1
, u1
) − e(p0
, u1
) = −e(p0
, u1
) + w. (Note
that EV < 0, a consumer is willing to give up the money
amount EV instead of being taxed with t per unit of x1.)
• With the lump sum tax the decrease in welfare is −TL.
33
Consumer Theory 5
Dead Weight Loss (2)
Micro I
• Definition - Dead Weight Loss of Commodity Taxation: Loss in
welfare due to tax t, where TC = th1(p0
1 + t, p−, u1
) = TL.
• The difference is: −TL − EV (p0
, p1
, w) =
−TL + e(p1
, u1
) − e(p0
, u1
) = −TL + w − e(p0
, u1
) ≥ 0.
• Why? With TL = TC = T:
−T − EV (p
0
, p
1
, w) = e(p
1
, u
1
) − e(p
0
, u
1
) − T
= e(p
1
, u
1
) − e(p
0
, u
1
) − th1(p
0
1 + t, p−, u
1
)
=
p0
1+t
p0
1
h1(p1, p−, u
1
) − h1(p
0
1 + t, p−, u
1
) dp1
34
Consumer Theory 5
Dead Weight Loss (3)
Micro I
• Since h1(p, u) is non-increasing in prices the above integrand is
non-negative ⇒ dead weight loss.
• Figures 3.I.4 and 3.I.5, page 85
• Equivalently: −CV (p0
, p1, w) − th1(p1
, u) ≥ 0, since h1 is
non-increasing in prices.
• Mas-Colell, Figure 3.I.6, page 87.
35
Consumer Theory 5
Partial Information (1)
Micro I
• Often a complete Walrasian demand function cannot be
observed, however:
• Theorem - Welfare and Partial Information I: Consider a
consumer with complete, transitive, continuous, and locally
non-satiated preferences. If (p1
− p0
) · x0
< 0, then the consumer
is strictly better of with (p1
, w) compared to (p0
, w). [P 3.I.1]
36
Consumer Theory 5
Partial Information (2)
Micro I
Proof:
• With non-satiation the consumer chooses a set on the boarder of
the budget set, such that p0
· x = w. Then p1
· x < w.
• ⇒ x is affordable within the budget set under p1
. By the
assumption of local non-satiation there must be an open
neighborhood including a better bundle which remains within the
budget set. Then the consumer is strictly better off with p1
.
37
Consumer Theory 5
Partial Information (3)
Micro I
• (p1
− p0
) · x0
> 0 ?
• Theorem - Welfare and Partial Information II: Consider a
consumer with a twice differentiable expenditure function. If
(p1
− p0
) · x0
> 0, then there exists an ¯α ∈ (0, 1) such that for
all 0 ≤ α ≤ ¯α, we have e((1 − α)p0
+ αp1
), w) > w the
consumer is strictly better off under p0
, w than under
(1 − α)p0
+ αp1
, w. [P 3.I.2]
38
Consumer Theory 5
Partial Information (4)
Micro I
Proof:
• We want to show that CV decreases we move from p0
to p1
. I.e.
CV = e(p0
, u0
) − e(p1
, u0
) < 0 ⇒ e(p1
, u0
) − e(p0
, u0
) > 0.
• Taylor expand e(p, u) at p0
, u0
:
e(p1
, u0
) = e(p0
, u0
) + (p1
− p0
) pe(p0
, u0
) + R(p0
, p1
)
where R(p0
, p1
)/||p1
− p0
|| → 0 if p1
→ p0
. e(., .) has to be at
least C1
. (fulfilled since second derivatives are assumed to exist).
39
Consumer Theory 5
Partial Information (5)
Micro I
Proof:
• By the properties of this approximation, there has to exist an ¯α,
where the Lagrange residual can be neglected. Then
sgn(e(p1
, u0
) − e(p0
, u0
)) = sgn(p1
− p0
) pe(p0
, u0
) for all
α ∈ [0, ¯α].
• This results in e(p1
, u0
) − e(p0
, u0
) > 0 by the assumption that
(p1
− p0
) pe(p0
, u0
) > 0 and the fact that
pe(p0
, u0
) = h(p0
, u0
) = x(p0
, e(p0
, u0
)).
40
Consumer Theory 5
Partial Information (6)
Micro I
Proof:
• Remark: Note that with a differentiable expenditure function the
second order term is non-positive, since the expenditure function
is concave.
• Remark: We can show the former theorem also in this way
(differentiability assumptions have to hold in addition). There the
non-positive second order term does not cause a problem, since
there we wanted to show that e(p1
, u0
) − e(p0
, u0
) < 0 if
(p1
− p0
) · x0
< 0.
41