Choice and revealed preference
Varian, Intermediate Microeconomics, chapter 5 and sections 7.1–7.7
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In this lecture, you will learn
• what the optimal choice is
• how to find it for different preferences
• whether Christmas is efficient
• how to find out from consumption choices whether
the consumer is rational and what are his preferences
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Optimal choice
The consumer chooses the most preferred bundle from her budget set.
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Optimal choice
The consumer chooses the most preferred bundle from her budget set.
If the consumer has monotonic and convex preferences and a smooth IC
and the problem has an inner solution, it always holds in optimum:
slope IC = MRS = −
p1
p2
= slope BL
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Optimal choice
The consumer chooses the most preferred bundle from her budget set.
If the consumer has monotonic and convex preferences and a smooth IC
and the problem has an inner solution, it always holds in optimum:
slope IC = MRS = −
p1
p2
= slope BL
Conversely, the slope of IC may not be equal to the slope of BL if:
• IC has a kink
• there is corner solution
• there are non-convex preferences
• there is a satiation point
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Convex, monotonic and smooth IC and inner solution
The optimal choice: slope IC = MRS = −p1
p2
= slope BL.
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Cobb-Douglas preferences (general solution)
The consumer chooses the bundle from her budget set in order to
maximize her utility:
max
x1,x2
u(x1, x2) = xc
1 xd
2
subject to p1x1 + p2x2 ≤ m
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Cobb-Douglas preferences (general solution)
The consumer chooses the bundle from her budget set in order to
maximize her utility:
max
x1,x2
u(x1, x2) = xc
1 xd
2
subject to p1x1 + p2x2 ≤ m
Cobb-Douglas preferences:
• monotonic preference =⇒ p1x1 + p2x2 = m
• monotonic, convex, and smooth IC, which do not touch the axes
=⇒ MRS = −p1/p2
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Cobb-Douglas preferences (general solution)
The consumer chooses the bundle from her budget set in order to
maximize her utility:
max
x1,x2
u(x1, x2) = xc
1 xd
2
subject to p1x1 + p2x2 ≤ m
Cobb-Douglas preferences:
• monotonic preference =⇒ p1x1 + p2x2 = m
• monotonic, convex, and smooth IC, which do not touch the axes
=⇒ MRS = −p1/p2
The optimal bundle (x∗
1 , x∗
2 ) is the solution of the following equations:
MRS = −
p1
p2
p1x∗
1 + p2x∗
2 = m
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Cobb-Douglas preferences (general solution)
The optimal bundle:
(x∗
1 , x∗
2 ) =
c
c + d
m
p1
,
d
c + d
m
p2
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Cobb-Douglas preferences (general solution)
The optimal bundle:
(x∗
1 , x∗
2 ) =
c
c + d
m
p1
,
d
c + d
m
p2
Property of Cobb-Douglas preferences: In optimum the consumer spends a
constant share of her income on each good:
p1x∗
1
m
=
p1
m
c
c + d
m
p1
=
c
c + d
p2x∗
2
m
=
p2
m
d
c + d
m
p2
=
d
c + d
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Cobb-Douglas preferences (general solution)
The optimal bundle:
(x∗
1 , x∗
2 ) =
c
c + d
m
p1
,
d
c + d
m
p2
Property of Cobb-Douglas preferences: In optimum the consumer spends a
constant share of her income on each good:
p1x∗
1
m
=
p1
m
c
c + d
m
p1
=
c
c + d
p2x∗
2
m
=
p2
m
d
c + d
m
p2
=
d
c + d
Convenient to have C-D function with exponents adding up to 1, e.g.
u(x1, x2) =
√
x1x2
Then the exponents are the shares of income spent on goods 1 and 2.
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Example – Cobb-Douglas preferences
Romana buys only apples A and bananas B.
Utility function: u(A, B) = A2B
Prices and income: pA = 5, pB = 10, m = 60
What is the optimum bundle?
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Example – Cobb-Douglas preferences
Romana buys only apples A and bananas B.
Utility function: u(A, B) = A2B
Prices and income: pA = 5, pB = 10, m = 60
What is the optimum bundle?
Cobb-Douglas preference =⇒ in optimum it holds: MRS = −pA/pB
−2AB/A2
= −5/10
A = 4B
In optimum Romana will buy 4 times as many A than B.
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Example – Cobb-Douglas preferences
Romana buys only apples A and bananas B.
Utility function: u(A, B) = A2B
Prices and income: pA = 5, pB = 10, m = 60
What is the optimum bundle?
Cobb-Douglas preference =⇒ in optimum it holds: MRS = −pA/pB
−2AB/A2
= −5/10
A = 4B
In optimum Romana will buy 4 times as many A than B.
Substituting the ratio into the BL:
pA4B + pBB = m
20 × B + 10 × B = 60
B = 2 a A = 4B = 8
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Kink in the indifference curve
Graph: convex and monotonic IC – but not smooth
Optimal choice: the slope of IC is not defined.
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Example – perfect complements
u(x1, x2) = min{x1, x2} – goods consumed at a constant ratio 1:1
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Example – perfect complements
u(x1, x2) = min{x1, x2} – goods consumed at a constant ratio 1:1
If p1 > 0 a p2 > 0, it holds for the optimum that x∗
1 = x∗
2 .
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Example – perfect complements
John consumes only tee T and cookies C at a ratio 1 : 2.
Price of tee: pT = 5 CZK
Price of a cookie: pC = 2 CZK
John’s income: m = 90 CZK
Example of John’s utility function + optimal consumption bundle?
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Example – perfect complements
John consumes only tee T and cookies C at a ratio 1 : 2.
Price of tee: pT = 5 CZK
Price of a cookie: pC = 2 CZK
John’s income: m = 90 CZK
Example of John’s utility function + optimal consumption bundle?
John’s utility function is e.g.
u(T, C) = min{2T, C}
Positive prices =⇒ the optimal combination of goods (= kinks of ICs):
C = 2T. Substituting the ratio into the BL:
pT T + pC C = m
5 × T + 2 × 2T = 90
T = 10
C = 2T = 20
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Corner solution
Graph: convex, smooth and monotonic IC – but touches the axes
Optimal choice: slope of IC = slope of BL
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Example – perfect substitutes
u(x1, x2) = x1 + x2 – willingness to exchange goods 1 a 2 at a ratio 1:1
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Example – perfect substitutes
u(x1, x2) = x1 + x2 – willingness to exchange goods 1 a 2 at a ratio 1:1
If p1 < p2, the optimal choice is (x∗
1 , x∗
2 ) = (m/p1, 0).
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Example – perfect substitutes
Martha always willing to exchange 3 raspberries R for 1 blueberry B.
Price of raspberry: pR = 2 CZK
Price of blueberry: pB = 5 CZK
Martha’s income: m = 40 CZK
Example of Martha’s utility function + optimal consumption bundle?
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Example – perfect substitutes
Martha always willing to exchange 3 raspberries R for 1 blueberry B.
Price of raspberry: pR = 2 CZK
Price of blueberry: pB = 5 CZK
Martha’s income: m = 40 CZK
Example of Martha’s utility function + optimal consumption bundle?
Martha’s utility function: u(R, B) = R + 3B
MRS = −1/3 = slope BL = −2/5
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Example – perfect substitutes
Martha always willing to exchange 3 raspberries R for 1 blueberry B.
Price of raspberry: pR = 2 CZK
Price of blueberry: pB = 5 CZK
Martha’s income: m = 40 CZK
Example of Martha’s utility function + optimal consumption bundle?
Martha’s utility function: u(R, B) = R + 3B
MRS = −1/3 = slope BL = −2/5
Which corner solution has a higher utility u(R, B):
• u(20, 0) = 20
• u(0, 8) = 24 =⇒ Optimal choice: (R, B) = (0, 8).
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Nonconvex preferences
Graph: smooth and monotonic IC – but not convex
There may exist non-optimal bundles where slope of IC = slope of BL.
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Example – concave preferences
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Example – concave preferences
Optimal choice (corner solution Z) – slope of IC = slope of BL
Non-optimal choice (inner solution X) – slope of IC = slope of BL
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Example – concave preferences
Libor buys mushrooms M and cuckoos C
Libor’s utility function: u(M, C) = M2 + C
Price of mushrooms: pM = 20 CZK
Price of cuckoos: pC = 10 CZK
Libor’s income: m = 100 CZK
What is Libor’s optimal consumption bundle?
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Example – concave preferences
Libor buys mushrooms M and cuckoos C
Libor’s utility function: u(M, C) = M2 + C
Price of mushrooms: pM = 20 CZK
Price of cuckoos: pC = 10 CZK
Libor’s income: m = 100 CZK
What is Libor’s optimal consumption bundle?
We derive the IC for the utility u = 10:
C = 10 − M2
Second derivative IC: C = −2 =⇒ Libor has concave preferences.
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Example – concave preferences
Libor buys mushrooms M and cuckoos C
Libor’s utility function: u(M, C) = M2 + C
Price of mushrooms: pM = 20 CZK
Price of cuckoos: pC = 10 CZK
Libor’s income: m = 100 CZK
What is Libor’s optimal consumption bundle?
We derive the IC for the utility u = 10:
C = 10 − M2
Second derivative IC: C = −2 =⇒ Libor has concave preferences.
Finding out, which corner solution brings higher utility u(M, C):
• u(0, 10) = 10
• u(5, 0) = 25 =⇒ Optimal choice: (M, C) = (5, 0)
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Example – concave preferences (non-optimal bundle)
Libor buys mushrooms M and cuckoos C
Libor’s utility function: u(M, C) = M2 + C
Price of mushrooms: pM = 20 CZK
Price of cuckoos: pC = 10 CZK
Libor’s income: m = 100 CZK
What is Libor’s optimal consumption bundle?
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Example – concave preferences (non-optimal bundle)
Libor buys mushrooms M and cuckoos C
Libor’s utility function: u(M, C) = M2 + C
Price of mushrooms: pM = 20 CZK
Price of cuckoos: pC = 10 CZK
Libor’s income: m = 100 CZK
What is Libor’s optimal consumption bundle?
Assuming that in optimum the slope of IC = the slope of BL:
MRS = −p1/p2
−2M = −20/10
M = 1
C = m/pC − (pM/pC )M = 8
Non-optimal choice: X = (M, C) = (1, 8)
X has the lowest utility of all bundles on BL: u(1, 8) = 9.
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Satiation point
Graph: convex and smooth IC – but not monotonic
Optimal choice: the slope of IC in the satiation point not defined
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Example – satiation point
Milena consumes only marmalade M and croissants C.
The bundle that maximizes her utility is (M∗, C∗) = (10, 5).
Her ICs are concentric circles – the farther from (M∗, C∗), the worse.
Price of marmalade: pM = 10 CZK
Price of tee: pC = 5 CZK
Milena’s income: m = 140 CZK
What is her optimal consumption bundle?
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Example – satiation point
Milena consumes only marmalade M and croissants C.
The bundle that maximizes her utility is (M∗, C∗) = (10, 5).
Her ICs are concentric circles – the farther from (M∗, C∗), the worse.
Price of marmalade: pM = 10 CZK
Price of tee: pC = 5 CZK
Milena’s income: m = 140 CZK
What is her optimal consumption bundle?
Her optimal bundle is the satiation point (M∗, C∗) = (10, 5).
Can she afford the bundle?
pMM∗
+ pC C∗
= 10 × 10 + 5 × 5 = 125.
The optimal bundle is available (125 < 140).
Milena buys the bundle (M∗, C∗) = (10, 5).
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APPLICATION: Estimating a utility function
What utility function corresponds to this consumption data?
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APPLICATION: Estimating a utility function
What utility function corresponds to this consumption data?
Consumption shares (s1, s2) are roughly constant.
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APPLICATION: Estimating a utility function
What utility function corresponds to this consumption data?
Consumption shares (s1, s2) are roughly constant.
=⇒ Cobb-Douglas utility function u(x1, x2) = x
1/4
1 x
3/4
2 .
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APPLICATION: Estimating a utility function (cont’d)
What is this estimation for?
For example, we can evaluate political choices.
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APPLICATION: Estimating a utility function (cont’d)
What is this estimation for?
For example, we can evaluate political choices.
Example:
New tax system leads to (p1, p2) = (2, 3) and m = 200.
Is it a good or a bad result?
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APPLICATION: Estimating a utility function (cont’d)
What is this estimation for?
For example, we can evaluate political choices.
Example:
New tax system leads to (p1, p2) = (2, 3) and m = 200.
Is it a good or a bad result?
Demanded quantities of goods at these prices and income are:
x1 =
1
4
200
2
= 25
x2 =
3
4
200
3
= 50
Estimated utility of the bundle is u(x1, x2) = 251/4503/4 ≈ 42.
We can compare the results with the past – higher than in year 2 but
lower than in year 3 (see the table in the previous slide).
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APPLICATION: The cost of Christmas
Joel Waldfogel, “The Deadweight Loss of Christmas” (AER, 1993):
•
”
In the standard microeconomic framework of consumer choice, the
best a gift-giver can do with, say, $10 is to duplicate the choice that
the recipient would have made.“ (p. 1328) In most cases, the
recipient is worse of.
• Gift-giving destroys 10 – 33 % of the value of gifts: loss ≈ $4 billion.
(10 % of the estimated dead-weight loss of the income tax).
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Revealed preference
In previous slides we derived choices from preferences.
In reality we do not observe preferences of people directly.
Conversely, revealed preferences derive preferences from choices.
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Revealed preference
In previous slides we derived choices from preferences.
In reality we do not observe preferences of people directly.
Conversely, revealed preferences derive preferences from choices.
We assume that consumer’s preferences are stable
= they do not change in the time we observe consumer’s choices.
For simplicity we assume that the derived preferences are
• strictly convex =⇒ exactly one bundle is demanded.
• monotonic =⇒ consumer spends the entire income.
These two assumptions are not necessary for the theory of RP!
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Concept of revealed preference
If the consumer chooses bundle X even though bundle Y is available, then
she reveals that she prefers X to Y .
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Concept of revealed preference
If the consumer chooses bundle X even though bundle Y is available, then
she reveals that she prefers X to Y .
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Concept of revealed preference
If the consumer chooses bundle X even though bundle Y is available, then
she reveals that she prefers X to Y .
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Revealed preference and preference
If X is revealed preferred to Y , does it mean that X is preferred to Y ?
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Revealed preference and preference
If X is revealed preferred to Y , does it mean that X is preferred to Y ?
No.
”
X revealed preferred to Y“ only means that X is chosen when Y is
available.
If the consumer maximizes utility (chooses best bundle available) then
”
revealed preference“ implies
”
preference“.
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Revealed preference and preference
If X is revealed preferred to Y , does it mean that X is preferred to Y ?
No.
”
X revealed preferred to Y“ only means that X is chosen when Y is
available.
If the consumer maximizes utility (chooses best bundle available) then
”
revealed preference“ implies
”
preference“.
The following exposition in two steps:
1 Assuming utility maximization (and other assumptions) we use
revealed preference to derive preferences from consumer choices.
2 We show how to test whether the consumer behaves in line with
utility maximization.
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Directly revealed preference
The chosen bundle (x1, x2) is directly revealed preferred to bundle
(y1, y2) if the bundle (y1, y2) is available, that is if
p1x1 + p2x2 ≥ p1y1 + p2y2.
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Indirectly revealed preference
It follows from transitivity that if
• bundle X is directly revealed preferred to bundle Y and
• bundle Y is directly revealed preferred to bundle Z
then bundleX is indirectly revealed preferred to bundle Z.
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Indirectly revealed preference
It follows from transitivity that if
• bundle X is directly revealed preferred to bundle Y and
• bundle Y is directly revealed preferred to bundle Z
then bundleX is indirectly revealed preferred to bundle Z.
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Example – revealed preferrence
Derivation of an IC for strictly convex a monotonic preferences.
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Example – revealed preferrence
Derivation of an IC for strictly convex a monotonic preferences.
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Example – revealed preferrence
Derivation of an IC for strictly convex a monotonic preferences.
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Example – revealed preferrence
Derivation of an IC for strictly convex a monotonic preferences.
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Weak axiom of revealed preference
Weak axion of revealed preference (WARP)
If bundle X is directly revealed preferred to bundle Y , then Y cannot be
directly revealed preferred to X.
More formally: For each bundle (x1, x2) bought at prices (p1, p2) and a
different bundle (y1, y2) bought at prices (q1, q2) holds that if
p1x1 + p2x2 ≥ p1y1 + p2y2,
then it must not be true that
q1y1 + q2y2 ≥ q1x1 + q2x2.
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Weak axiom of revealed preference
Weak axion of revealed preference (WARP)
If bundle X is directly revealed preferred to bundle Y , then Y cannot be
directly revealed preferred to X.
More formally: For each bundle (x1, x2) bought at prices (p1, p2) and a
different bundle (y1, y2) bought at prices (q1, q2) holds that if
p1x1 + p2x2 ≥ p1y1 + p2y2,
then it must not be true that
q1y1 + q2y2 ≥ q1x1 + q2x2.
A necessary condition for consistency with utility maximization.
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Weak axiom of revealed preference (cont’d)
Consumer choices that are not consistent with WARP:
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Weak axiom of revealed preference (cont’d)
Consumer choices that are not consistent with WARP:
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Weak axiom of revealed preference (cont’d)
Consumer choices that are consistent with WARP:
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Weak axiom of revealed preference (cont’d)
Consumer choices that are consistent with WARP:
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How to test WARP?
Consider the following consumer data:
Costs of bundles 1, 2, and 3 at different prices:
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How to test WARP?
Consider the following consumer data:
Costs of bundles 1, 2, and 3 at different prices:
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How to test WARP?
Consider the following consumer data:
Costs of bundles 1, 2, and 3 at different prices:
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How to test WARP?
Consider the following consumer data:
Costs of bundles 1, 2, and 3 at different prices:
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How to test WARP?
Consider the following consumer data:
Costs of bundles 1, 2, and 3 at different prices:
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How to test WARP?
Consider the following consumer data:
Costs of bundles 1, 2, and 3 at different prices:
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How to test WARP?
Consider the following consumer data:
Costs of bundles 1, 2, and 3 at different prices:
The chosen bundles are directly revealed preferred to bundles with *
in the same line (e.g. at prices 1 is bundle 1 preferred to bundle 2).
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How to test WARP?
Consider the following consumer data:
Costs of bundles 1, 2, and 3 at different prices:
The chosen bundles are directly revealed preferred to bundles with *
in the same line (e.g. at prices 1 is bundle 1 preferred to bundle 2).
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How to test WARP?
Consider the following consumer data:
Costs of bundles 1, 2, and 3 at different prices:
The chosen bundles are directly revealed preferred to bundles with *
in the same line (e.g. at prices 1 is bundle 1 preferred to bundle 2).
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How to test WARP?
Consider the following consumer data:
Costs of bundles 1, 2, and 3 at different prices:
The chosen bundles are directly revealed preferred to bundles with *
in the same line (e.g. at prices 1 is bundle 1 preferred to bundle 2).
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How to test WARP?
Consider the following consumer data:
Costs of bundles 1, 2, and 3 at different prices:
The chosen bundles are directly revealed preferred to bundles with *
in the same line (e.g. at prices 1 is bundle 1 preferred to bundle 2).
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How to test WARP?
Consider the following consumer data:
Costs of bundles 1, 2, and 3 at different prices:
The chosen bundles are directly revealed preferred to bundles with *
in the same line (e.g. at prices 1 is bundle 1 preferred to bundle 2).
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How to test WARP? (cont’d)
The WARP is violated if there is * in line t and column s and line s and
column t
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How to test WARP? (cont’d)
The WARP is violated if there is * in line t and column s and line s and
column t (e.g. bundle 1 + price 2 and bundle 2 + price 1).
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How to test WARP? (cont’d)
The WARP is violated if there is * in line t and column s and line s and
column t (e.g. bundle 1 + price 2 and bundle 2 + price 1).
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How to test WARP? (cont’d)
The WARP is violated if there is * in line t and column s and line s and
column t (e.g. bundle 1 + price 2 and bundle 2 + price 1).
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How to test WARP? (cont’d)
The WARP is violated if there is * in line t and column s and line s and
column t (e.g. bundle 1 + price 2 and bundle 2 + price 1).
Data in the table violate WARP.
What does it mean that data violate WARP? Two options:
• The consumer does not choose the best available bundle.
• The consumer does not have stable or strictly convex preferences.
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Strong axiom of revealed preference
WARP = necessary condition for consistency with utility maxim.
Does not test, though, whether the preferences are transitive.
Strong axiom of revealed preference (SARP)
If X is directly or indirectly revealed preferred to Y ,
then Y cannot be directly or indirectly revealed preferred to X.
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Strong axiom of revealed preference
WARP = necessary condition for consistency with utility maxim.
Does not test, though, whether the preferences are transitive.
Strong axiom of revealed preference (SARP)
If X is directly or indirectly revealed preferred to Y ,
then Y cannot be directly or indirectly revealed preferred to X.
Necessary and sufficient condition for consistency with utility maxim.
If SARP holds we can find such preferences for which consumer behavior
will be consistent with utility maximization.
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How to test SARP?
The table below shows expenditures on bundles 1, 2, and 3 at different
prices:
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How to test SARP?
The table below shows expenditures on bundles 1, 2, and 3 at different
prices:
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How to test SARP?
The table below shows expenditures on bundles 1, 2, and 3 at different
prices:
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How to test SARP?
The table below shows expenditures on bundles 1, 2, and 3 at different
prices:
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How to test SARP?
The table below shows expenditures on bundles 1, 2, and 3 at different
prices:
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How to test SARP?
The table below shows expenditures on bundles 1, 2, and 3 at different
prices:
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How to test SARP?
The table below shows expenditures on bundles 1, 2, and 3 at different
prices:
At prices 1 (line 1) bundle 1 is indirectly revealed preferred to bundle 3
with (∗).
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How to test SARP?
The table below shows expenditures on bundles 1, 2, and 3 at different
prices:
At prices 1 (line 1) bundle 1 is indirectly revealed preferred to bundle 3
with (∗).
SARP is violated if each pair of diagonal fields with has ∗ or (∗).
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How to test SARP?
The table below shows expenditures on bundles 1, 2, and 3 at different
prices:
At prices 1 (line 1) bundle 1 is indirectly revealed preferred to bundle 3
with (∗).
SARP is violated if each pair of diagonal fields with has ∗ or (∗).
SARP is not violated.
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What should you know?
• The consumer chooses the most preferred
bundle from her budget set.
• If we have monotonic, convex and smooth
IC and an inner solution, MRS equals to
the slope of BL in optimum.
• This equality does not hold for perfect
substitutes (usually) and for prefect
complements.
• It always holds for Cobb-Douglas
preferences. The consumer also spends
fixed shares of her income on each good.
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What should you know? (cont’d)
• If a consumer buys bundle A when bundle B is
available, she reveals that she prefers A to B.
• For a rational consumer, it also means that
she prefers A to B.
• We can use WARP and SARP to test whether
the consumer is rational.
• WARP: if the consumer prefers A to B, with
a different BL she cannot prefer B to A.
• SARP = WARP + indirectly revealed
preferences (transitivity).
• If SARP holds, we can use consumer choices
to estimate preferences.
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