Advanced Microeconomic Theory
Lecture Notes
Sérgio O. Parreiras
Economics Department, UNC at Chapel Hill
Fall, 2014
Announcements
Thursday, August 21st, 2014
▶
PS 01 posted (yesterday!)
▶
PPE reading groups
Dear All, If you have students who might be interested, I would
appreciate you letting them know about the Reading Groups being
offered this Fall by the Philosophy, Politics, and Economics Program.
We are offering three:
1. Ridley’s The Rational Optimist
2. Feminist Arguments For and Against the Market
3. Justice: Rawls and Nozick
The groups will meet over dinner (at Gourmet Kingdom in Carrboro).
The readings will be provided and the cost of dinner will be covered.
Students can find details at http://ppe.unc.edu/reading-groups/.
Thanks for your help in getting the word out, Geoff
Announcements
Thursday, August 21st, 2014
▶
PS 01 posted (yesterday!)
▶
PPE reading groups
Dear All, If you have students who might be interested, I would
appreciate you letting them know about the Reading Groups being
offered this Fall by the Philosophy, Politics, and Economics Program.
We are offering three:
1. Ridley’s The Rational Optimist
2. Feminist Arguments For and Against the Market
3. Justice: Rawls and Nozick
The groups will meet over dinner (at Gourmet Kingdom in Carrboro).
The readings will be provided and the cost of dinner will be covered.
Students can find details at http://ppe.unc.edu/reading-groups/.
Thanks for your help in getting the word out, Geoff
Decision Theory: Lotteries
A lottery is a pair of outcomes and their respective probabilities:
ℓ = ((x1, x2, . . . , xn ), (p1, p2, . . . , pn )) ,
where xk ∈ R and pk ≥ 0 for all k = 1, . . . , n and also
p1 + p2 + . . . + pn = 1.
x1
...
x2
xn
p2
p1
pn
ℓ
The Certain Lottery, Expectation and Variance
The lottery that gives outcome x with probability 1 (with
certainty) is denoted:
δx = ((x), (1)) .
The expected value of the ℓ1 = ((x1 , x2 , . . . , xn ), (p1 , p2 , . . . , pn ))
is:
n
∑
E[ℓ1 ] = p1 · x1 + p2 · x2 + . . . pn · xn =
pi · xi ;
i=1
and variance this lottery is
Var[ℓ1 ] =p1 · (x1 − E[ℓ1 ])2 + p2 · (x2 − E[ℓ1 ])2 + . . . pn · (xn − E[ℓ1 ])2 =
=
n
∑
i=1
pi · (xi − E[ℓ1 ])2 .
Composition of Lotteries
Given two lotteries, ℓ1 = ((x1 , x2 , . . . , xn ), (p1 , p2 , . . . , pn )) and
ℓ2 = (y1 , y2 , . . . , ym ), (q1 , q2 , . . . , qn )) and a number 0 < α < 1,
one can create a compound lottery by choosing ℓ1 with
probability α and ℓ2 with probability ℓ2 .
ℓ = αℓ1 ⊕ (1 − α)ℓ2 =
= ((x1 , x2 , y1 , y2 ), (αp, α(1 − p), (1 − α)q, (1 − α)(1 − q)).
x1
x2
ℓ
y1
y2
Composition of Lotteries
The compound lottery ℓ plays ℓ1 with probability α and ℓ2 with
probability ℓ2 :
ℓ = αℓ1 ⊕ (1 − α)ℓ2 =
= ((x1 , x2 , y1 , y2 ), (αp, α(1 − p), (1 − α)q, (1 − α)(1 − q)).
p
α
ℓ1
1−p
x1
x2
ℓ
1−α
q
ℓ2
1−q
y1
y2
Composition of Lotteries
The compound lottery ℓ plays ℓ1 with probability α and ℓ2 with
probability ℓ2 :
ℓ = αℓ1 ⊕ (1 − α)ℓ2 =
= ((x1 , x2 , y1 , y2 ), (αp, α(1 − p), (1 − α)q, (1 − α)(1 − q)).
x1
α·
p
α · (1 −
ℓ
x2
p)
(1 − α)
·q
(1 −
α)
· (1
−q
y1
)
y2
Preferences Over Lotteries
Given to lotteries ℓa and ℓb such that a decision maker (DM)
chooses ℓa over ℓb ,
the following statements are equivalent:
▶
The DM judges ℓa no worst than ℓb (everyday language);
▶
The DM prefers ℓa to ℓb (economics language);
▶
ℓa ⪰DM ℓb (mathematics language).
For simplicity we write:
▶
ℓa ≻ ℓb when ℓa ⪰ ℓb but ℓb ̸⪰ ℓa (strict preference)
▶
ℓa ∼ ℓb when ℓa ⪰ ℓb and ℓb ⪰ ℓa (indifference).
Preferences Over Lotteries
Given to lotteries ℓa and ℓb such that a decision maker (DM)
chooses ℓa over ℓb ,
the following statements are equivalent:
▶
The DM judges ℓa no worst than ℓb (everyday language);
▶
The DM prefers ℓa to ℓb (economics language);
▶
ℓa ⪰DM ℓb (mathematics language).
For simplicity we write:
▶
ℓa ≻ ℓb when ℓa ⪰ ℓb but ℓb ̸⪰ ℓa (strict preference)
▶
ℓa ∼ ℓb when ℓa ⪰ ℓb and ℓb ⪰ ℓa (indifference).
Preferences Over Lotteries
Given to lotteries ℓa and ℓb such that a decision maker (DM)
chooses ℓa over ℓb ,
the following statements are equivalent:
▶
The DM judges ℓa no worst than ℓb (everyday language);
▶
The DM prefers ℓa to ℓb (economics language);
▶
ℓa ⪰DM ℓb (mathematics language).
For simplicity we write:
▶
ℓa ≻ ℓb when ℓa ⪰ ℓb but ℓb ̸⪰ ℓa (strict preference)
▶
ℓa ∼ ℓb when ℓa ⪰ ℓb and ℓb ⪰ ℓa (indifference).
Preferences Over Lotteries
Given to lotteries ℓa and ℓb such that a decision maker (DM)
chooses ℓa over ℓb ,
the following statements are equivalent:
▶
The DM judges ℓa no worst than ℓb (everyday language);
▶
The DM prefers ℓa to ℓb (economics language);
▶
ℓa ⪰DM ℓb (mathematics language).
For simplicity we write:
▶
ℓa ≻ ℓb when ℓa ⪰ ℓb but ℓb ̸⪰ ℓa (strict preference)
▶
ℓa ∼ ℓb when ℓa ⪰ ℓb and ℓb ⪰ ℓa (indifference).
Preferences Over Lotteries
Given to lotteries ℓa and ℓb such that a decision maker (DM)
chooses ℓa over ℓb ,
the following statements are equivalent:
▶
The DM judges ℓa no worst than ℓb (everyday language);
▶
The DM prefers ℓa to ℓb (economics language);
▶
ℓa ⪰DM ℓb (mathematics language).
For simplicity we write:
▶
ℓa ≻ ℓb when ℓa ⪰ ℓb but ℓb ̸⪰ ℓa (strict preference)
▶
ℓa ∼ ℓb when ℓa ⪰ ℓb and ℓb ⪰ ℓa (indifference).
Preferences Over Lotteries
Given to lotteries ℓa and ℓb such that a decision maker (DM)
chooses ℓa over ℓb ,
the following statements are equivalent:
▶
The DM judges ℓa no worst than ℓb (everyday language);
▶
The DM prefers ℓa to ℓb (economics language);
▶
ℓa ⪰DM ℓb (mathematics language).
For simplicity we write:
▶
ℓa ≻ ℓb when ℓa ⪰ ℓb but ℓb ̸⪰ ℓa (strict preference)
▶
ℓa ∼ ℓb when ℓa ⪰ ℓb and ℓb ⪰ ℓa (indifference).
Preferences Over Lotteries
▶
A preference of the DM, ⪰DM , over the set of lotteries is
just the DM’s ranking of lotteries.
▶
We wish to have a numerical score that reflects the DM’s
ranking.
von Neuman & Morgenstern’s Assumptions:
Completeness For any two lotteries ℓ1 and ℓ2 ,
ℓ1 ⪰ ℓ2 and/or ℓ2 ⪰ ℓ1 .
Transitivity For any lotteries ℓ1 , ℓ2 and ℓ3 ,
if ℓ1 ⪰ ℓ2 and ℓ2 ⪰ ℓ3 then ℓ1 ⪰ ℓ3 .
Continuity If ℓ1 ⪰ ℓ2 ⪰ ℓ3 then exists p ∈ [0, 1] such that
ℓ2 ∼ pℓ1 ⊕ (1 − p)ℓ3 .
Independence If ℓ1 ≻ ℓ2 then for any ℓ3 and any 0 < p < 1,
pℓ1 ⊕ (1 − p)ℓ3 ≻ pℓ2 ⊕ (1 − p)ℓ3 .
von Neuman & Morgenstern’s Assumptions:
Completeness For any two lotteries ℓ1 and ℓ2 ,
ℓ1 ⪰ ℓ2 and/or ℓ2 ⪰ ℓ1 .
Transitivity For any lotteries ℓ1 , ℓ2 and ℓ3 ,
if ℓ1 ⪰ ℓ2 and ℓ2 ⪰ ℓ3 then ℓ1 ⪰ ℓ3 .
Continuity If ℓ1 ⪰ ℓ2 ⪰ ℓ3 then exists p ∈ [0, 1] such that
ℓ2 ∼ pℓ1 ⊕ (1 − p)ℓ3 .
Independence If ℓ1 ≻ ℓ2 then for any ℓ3 and any 0 < p < 1,
pℓ1 ⊕ (1 − p)ℓ3 ≻ pℓ2 ⊕ (1 − p)ℓ3 .
von Neuman & Morgenstern’s Assumptions:
Completeness For any two lotteries ℓ1 and ℓ2 ,
ℓ1 ⪰ ℓ2 and/or ℓ2 ⪰ ℓ1 .
Transitivity For any lotteries ℓ1 , ℓ2 and ℓ3 ,
if ℓ1 ⪰ ℓ2 and ℓ2 ⪰ ℓ3 then ℓ1 ⪰ ℓ3 .
Continuity If ℓ1 ⪰ ℓ2 ⪰ ℓ3 then exists p ∈ [0, 1] such that
ℓ2 ∼ pℓ1 ⊕ (1 − p)ℓ3 .
Independence If ℓ1 ≻ ℓ2 then for any ℓ3 and any 0 < p < 1,
pℓ1 ⊕ (1 − p)ℓ3 ≻ pℓ2 ⊕ (1 − p)ℓ3 .
von Neuman & Morgenstern’s Assumptions:
Completeness For any two lotteries ℓ1 and ℓ2 ,
ℓ1 ⪰ ℓ2 and/or ℓ2 ⪰ ℓ1 .
Transitivity For any lotteries ℓ1 , ℓ2 and ℓ3 ,
if ℓ1 ⪰ ℓ2 and ℓ2 ⪰ ℓ3 then ℓ1 ⪰ ℓ3 .
Continuity If ℓ1 ⪰ ℓ2 ⪰ ℓ3 then exists p ∈ [0, 1] such that
ℓ2 ∼ pℓ1 ⊕ (1 − p)ℓ3 .
Independence If ℓ1 ≻ ℓ2 then for any ℓ3 and any 0 < p < 1,
pℓ1 ⊕ (1 − p)ℓ3 ≻ pℓ2 ⊕ (1 − p)ℓ3 .
von Neuman & Morgenstern’s Assumptions:
Completeness For any two lotteries ℓ1 and ℓ2 ,
ℓ1 ⪰ ℓ2 and/or ℓ2 ⪰ ℓ1 .
Transitivity For any lotteries ℓ1 , ℓ2 and ℓ3 ,
if ℓ1 ⪰ ℓ2 and ℓ2 ⪰ ℓ3 then ℓ1 ⪰ ℓ3 .
Continuity If ℓ1 ⪰ ℓ2 ⪰ ℓ3 then exists p ∈ [0, 1] such that
ℓ2 ∼ pℓ1 ⊕ (1 − p)ℓ3 .
Independence If ℓ1 ≻ ℓ2 then for any ℓ3 and any 0 < p < 1,
pℓ1 ⊕ (1 − p)ℓ3 ≻ pℓ2 ⊕ (1 − p)ℓ3 .
von Neuman & Morgenstern’s Assumptions:
Completeness For any two lotteries ℓ1 and ℓ2 ,
ℓ1 ⪰ ℓ2 and/or ℓ2 ⪰ ℓ1 .
Transitivity For any lotteries ℓ1 , ℓ2 and ℓ3 ,
if ℓ1 ⪰ ℓ2 and ℓ2 ⪰ ℓ3 then ℓ1 ⪰ ℓ3 .
Continuity If ℓ1 ⪰ ℓ2 ⪰ ℓ3 then exists p ∈ [0, 1] such that
ℓ2 ∼ pℓ1 ⊕ (1 − p)ℓ3 .
Independence If ℓ1 ≻ ℓ2 then for any ℓ3 and any 0 < p < 1,
pℓ1 ⊕ (1 − p)ℓ3 ≻ pℓ2 ⊕ (1 − p)ℓ3 .
If ⪰ satisfy all of of the above, there exists u : R → R such that
((x1 , x2 , . . . , xn ), (p1 , p2 , . . . , pn )) ≻ (y1 , y2 , . . . , ym ), (q1 , q2 , . . . , qn ))
if and only if
n
∑
k=1
u(xk ) · pk >
m
∑
k=1
u(yk ) · qk .
Expected Utility
We write:
U (ℓ1 ) = u(x1 ) · p1 + . . . + u(xn ) · pn
and refer to U as the expected utility and to u as the:
1. utility for money
2. Bernoulli utility
3. von Neumann-Morgenstern utility (vNM)
Expected Utility
We write:
U (ℓ1 ) = u(x1 ) · p1 + . . . + u(xn ) · pn ,
and refer to U as the expected utility and to u as the:
1. utility for money
2. Bernoulli utility
3. von Neumann-Morgenstern utility (vNM)
Expected Utility
We write:
U (ℓ1 ) = u(x1 ) · p1 + . . . + u(xn ) · pn ,
and refer to U as the expected utility and to u as the:
1. utility for money
2. Bernoulli utility
3. von Neumann-Morgenstern utility (vNM)
Expected Utility
We made assumptions about the agents’ preferences over
lotteries so we can represent his/her preferences by an expected
utility.
The agent will choose the lottery that delivers the highest
expected utility.
Sometimes, it might be convenient to have the lottery
describing the flow (variation of wealth) and not the final
wealth.
In this case, how do we compute the expected utility of a
lottery (X , p), where X = (x1 , . . . , xn ) and p = (p1 , . . . , pn ) ?
U (X ) =
n
∑
u(xs + ω) · ps
s=1
Remark 1: U (X ) is the expected utility of lottery (X , p).
Expected Utility
We made assumptions about the agents’ preferences over
lotteries so we can represent his/her preferences by an expected
utility.
The agent will choose the lottery that delivers the highest
expected utility.
Sometimes, it might be convenient to have the lottery
describing the flow (variation of wealth) and not the final
wealth.
In this case, how do we compute the expected utility of a
lottery (X , p), where X = (x1 , . . . , xn ) and p = (p1 , . . . , pn ) ?
U (X ) =
n
∑
u(xs + ω) · ps
s=1
Remark 1: U (X ) is the expected utility of lottery (X , p).
Expected Utility
We made assumptions about the agents’ preferences over
lotteries so we can represent his/her preferences by an expected
utility.
The agent will choose the lottery that delivers the highest
expected utility.
Sometimes, it might be convenient to have the lottery
describing the flow (variation of wealth) and not the final
wealth.
In this case, how do we compute the expected utility of a
lottery (X , p), where X = (x1 , . . . , xn ) and p = (p1 , . . . , pn ) ?
U (X ) =
n
∑
u(xs + ω) · ps
s=1
Remark 1: U (X ) is the expected utility of lottery (X , p).
Expected Utility
We made assumptions about the agents’ preferences over
lotteries so we can represent his/her preferences by an expected
utility.
The agent will choose the lottery that delivers the highest
expected utility.
Sometimes, it might be convenient to have the lottery
describing the flow (variation of wealth) and not the final
wealth.
In this case, how do we compute the expected utility of a
lottery (X , p), where X = (x1 , . . . , xn ) and p = (p1 , . . . , pn ) ?
U (X ) =
n
∑
u(xs + ω) · ps
s=1
Remark 1: U (X ) is the expected utility of lottery (X , p).
Expected Utility
We made assumptions about the agents’ preferences over
lotteries so we can represent his/her preferences by an expected
utility.
The agent will choose the lottery that delivers the highest
expected utility.
Sometimes, it might be convenient to have the lottery
describing the flow (variation of wealth) and not the final
wealth.
In this case, how do we compute the expected utility of a
lottery (X , p), where X = (x1 , . . . , xn ) and p = (p1 , . . . , pn ) ?
U (X ) =
n
∑
u(xs + ω) · ps
s=1
Remark 1: U (X ) is the expected utility of lottery (X , p).
Expected Utility
We made assumptions about the agents’ preferences over
lotteries so we can represent his/her preferences by an expected
utility.
The agent will choose the lottery that delivers the highest
expected utility.
Sometimes, it might be convenient to have the lottery
describing the flow (variation of wealth) and not the final
wealth.
In this case, how do we compute the expected utility of a
lottery (X , p), where X = (x1 , . . . , xn ) and p = (p1 , . . . , pn ) ?
U (X ) =
n
∑
u(xs + ω) · ps
s=1
Remark 1: U (X ) is the expected utility of lottery (X , p).
Computing Expected Utility
Examples with zero initial wealth (or lottery already gives final wealth).
1. Lotteries: A = ((1000, 0), (1/2, 1/2)) and
√
B = ((500, 0), (1, 0)). vN-M utility: u(x) = x. Then
√
√
√
U (A) = 21 1000 + 12 0 = 5 10 (see the prob. tree) and
√
√
√
U (B) = 1 · 500 = 500 = 10 5 ( prob. tree).
2. Lotteries: C = ((4, 0), (2/3, 1/3)) and D = ((2.66, 0), (1, 0)).
vN-M utility: u(x) = x 2 . Then
U (C ) = 23 16 + 13 0 = 32
3 = 10.66 and U (D) = 7.11.
3. Lotteries A = ((1000, 0), (1/2, 1/2)) and
B = ((500, 0), (1, 0)) and vN-M utility: u(x) = 2x . Then
U (A) = 12 21000 + 12 20 = 2999 + 12 and U (B) = 2500 .
4. Lotteries C = ((4, 0), (2/3, 1/3)) and D = ((2.66, 0), (1, 0)).
vN-M utility: u(x) = ln(x). Then
U (C ) = 23 ln(4) + 13 ln(0) = −∞ and U (D) = ln(2.66).
Computing Expected Utility
Examples with zero initial wealth (or lottery already gives final wealth).
1. Lotteries: A = ((1000, 0), (1/2, 1/2)) and
√
B = ((500, 0), (1, 0)). vN-M utility: u(x) = x. Then
√
√
√
U (A) = 21 1000 + 12 0 = 5 10 (see the prob. tree) and
√
√
√
U (B) = 1 · 500 = 500 = 10 5 ( prob. tree).
2. Lotteries: C = ((4, 0), (2/3, 1/3)) and D = ((2.66, 0), (1, 0)).
vN-M utility: u(x) = x 2 . Then
U (C ) = 23 16 + 13 0 = 32
3 = 10.66 and U (D) = 7.11.
3. Lotteries A = ((1000, 0), (1/2, 1/2)) and
B = ((500, 0), (1, 0)) and vN-M utility: u(x) = 2x . Then
U (A) = 12 21000 + 12 20 = 2999 + 12 and U (B) = 2500 .
4. Lotteries C = ((4, 0), (2/3, 1/3)) and D = ((2.66, 0), (1, 0)).
vN-M utility: u(x) = ln(x). Then
U (C ) = 23 ln(4) + 13 ln(0) = −∞ and U (D) = ln(2.66).
Computing Expected Utility
Examples with zero initial wealth (or lottery already gives final wealth).
1. Lotteries: A = ((1000, 0), (1/2, 1/2)) and
√
B = ((500, 0), (1, 0)). vN-M utility: u(x) = x. Then
√
√
√
U (A) = 21 1000 + 12 0 = 5 10 (see the prob. tree) and
√
√
√
U (B) = 1 · 500 = 500 = 10 5 ( prob. tree).
2. Lotteries: C = ((4, 0), (2/3, 1/3)) and D = ((2.66, 0), (1, 0)).
vN-M utility: u(x) = x 2 . Then
U (C ) = 23 16 + 13 0 = 32
3 = 10.66 and U (D) = 7.11.
3. Lotteries A = ((1000, 0), (1/2, 1/2)) and
B = ((500, 0), (1, 0)) and vN-M utility: u(x) = 2x . Then
U (A) = 12 21000 + 12 20 = 2999 + 12 and U (B) = 2500 .
4. Lotteries C = ((4, 0), (2/3, 1/3)) and D = ((2.66, 0), (1, 0)).
vN-M utility: u(x) = ln(x). Then
U (C ) = 23 ln(4) + 13 ln(0) = −∞ and U (D) = ln(2.66).
Computing Expected Utility
Examples with zero initial wealth (or lottery already gives final wealth).
1. Lotteries: A = ((1000, 0), (1/2, 1/2)) and
√
B = ((500, 0), (1, 0)). vN-M utility: u(x) = x. Then
√
√
√
U (A) = 21 1000 + 12 0 = 5 10 (see the prob. tree) and
√
√
√
U (B) = 1 · 500 = 500 = 10 5 ( prob. tree).
2. Lotteries: C = ((4, 0), (2/3, 1/3)) and D = ((2.66, 0), (1, 0)).
vN-M utility: u(x) = x 2 . Then
U (C ) = 23 16 + 13 0 = 32
3 = 10.66 and U (D) = 7.11.
3. Lotteries A = ((1000, 0), (1/2, 1/2)) and
B = ((500, 0), (1, 0)) and vN-M utility: u(x) = 2x . Then
U (A) = 12 21000 + 12 20 = 2999 + 12 and U (B) = 2500 .
4. Lotteries C = ((4, 0), (2/3, 1/3)) and D = ((2.66, 0), (1, 0)).
vN-M utility: u(x) = ln(x). Then
U (C ) = 23 ln(4) + 13 ln(0) = −∞ and U (D) = ln(2.66).
Computing Expected Utility
Examples with initial wealth equal to ωs = 8
1. Lottery A = ((1000, 0), (1/2, 1/2)), lottery
√
B = ((500, 0), (1, 0)), and u(x) = x. Then
√
√
√
U (A) = 12 1000 + 12 0 = 5 10 (see the prob. tree) and
√
√
√
U (B) = 1 · 500 = 500 = 10 5 ( prob. tree).
2. The vN-M utility is u(x) = − exp(−x) and a coin if flipped
twice, the lottery pays $ 10 if HH, $ 5 if HT, $0 if TH and
-$3 if TT. The agent must pay 1 for the lottery and her
initial wealth is $ 8. Please see the prob. tree for how to
compute her expected utility.
Computing Expected Utility
Examples with initial wealth equal to ωs = 8
1. Lottery A = ((1000, 0), (1/2, 1/2)), lottery
√
B = ((500, 0), (1, 0)), and u(x) = x. Then
√
√
√
U (A) = 12 1000 + 12 0 = 5 10 (see the prob. tree) and
√
√
√
U (B) = 1 · 500 = 500 = 10 5 ( prob. tree).
2. The vN-M utility is u(x) = − exp(−x) and a coin if flipped
twice, the lottery pays $ 10 if HH, $ 5 if HT, $0 if TH and
-$3 if TT. The agent must pay 1 for the lottery and her
initial wealth is $ 8. Please see the prob. tree for how to
compute her expected utility.
Decision/Probability Tree
Lottery A, initial wealth ω = 0 and u(x) =
$1000
√
1000
1
2
√
1000
2
t=0
1
2
$0
√
x.
√
0
+
√
0
2
Decision/Probability Tree
Lottery B, initial wealth ω = 0 and u(x) =
$500
√
x
√
500
1
2
√
500
t=0
1
2
$500
√
500
Decision/Probability Tree
Lottery A, initial wealth ω = 8
$1000 + 8
√
1008
1
2
√
1008
2
t=0
1
2
$0 + 8
√
8
+
√
8
2
Decision/Probability Tree
Lottery B, initial wealth ω = 8
$508
√
508
1
2
√
508
t=0
1
2
$508
√
508
Decision/Probability Tree
Two coins example, u(x) = − exp(x).
E[U (X )] =
8 − 1 + 10
1
2
t=1
1
2
1
2
t=0
+
− exp(−17)
4
8−1+5
− exp(−12)
+
4
8−1−0
+
− exp(−7)
4
1
2
1
2
− exp(−4)
4
t=1
1
2
8−1−3
Extracting u from ⪰
Extracting u from ⪰
≻
Extracting u from ⪰
≺
≻
Extracting u from ⪰
≺
≻
≺
A Behavioral Look at Choice
▶
Anchoring
▶
Availability
▶
Representativeness
▶
Optimism and over confidence
▶
Gains and losses
▶
Status Quo Bias
▶
Framming
Risk Aversion
Let’s go back to expected utility theory, consider the two
lotteries:
1 1
ℓ1 = ((100, 200), ( , )
2 2
and
δ150 = ((150), (1))
We have
U (ℓ1 ) = u(150 − 50) ·
1
1
+ u(150 + 50) ·
2
2
U (δ150 ) = u(150) · 1.
and
Risk Aversion
Let’s go back to expected utility theory, consider the two
lotteries:
1 1
ℓ1 = ((100, 200), ( , )
2 2
and
δ150 = ((150), (1))
We have
U (ℓ1 ) = u(150 − 50) ·
1
1
+ u(150 + 50) ·
2
2
U (δ150 ) = u(150) · 1.
and
Risk Aversion
Let’s go back to expected utility theory, consider the two
lotteries:
1 1
ℓ1 = ((100, 200), ( , )
2 2
and
δ150 = ((150), (1))
We have
U (ℓ1 ) = u(150 − 50) ·
1
1
+ u(150 + 50) ·
2
2
U (δ150 ) = u(150) · 1.
and
Risk Aversion
Let’s go back to expected utility theory, consider the two
lotteries:
1 1
ℓ1 = ((100, 200), ( , )
2 2
and
δ150 = ((150), (1))
We have
U (ℓ1 ) = u(150 − 50) ·
1
1
+ u(150 + 50) ·
2
2
U (δ150 ) = u(150) · 1.
and
Risk Aversion
[
]
u(200) − u(150) u(150) − u(100) 50
U (ℓ1 ) − U (δ150 ) =
−
·
50
50
2
u
u(200)
u(100)
100
150
E[ℓ1 ]
200
x
Risk Aversion


 u(200) − u(150) u(150) − u(100)  50
·
−
U (ℓ1 ) − U (δ150 ) = 
 2

50
50
|
{z
} |
{z
}
≃ Mu(150)
≃ Mu(100)
u
u(200)
U (ℓ1 )
u(100)
100
150
E[ℓ1 ]
200
x
Risk Aversion


 u(200) − u(150) u(150) − u(100)  50
·
−
U (ℓ1 ) − U (δ150 ) = 
 2

50
50
|
{z
} |
{z
}
≃ Mu(150)
≃ Mu(100)
u
u(200)
U (ℓ1 )
u(100)
100
150
E[ℓ1 ]
200
x
Risk Aversion


 u(200) − u(150) u(150) − u(100)  50
·
−
U (ℓ1 ) − U (δ150 ) = 
 2

50
50
|
{z
} |
{z
}
≃ Mu(150)
≃ Mu(100)
u
u(200)
U (ℓ1 )
u(100)
100
150
E[ℓ1 ]
200
x
Risk Aversion


 u(200) − u(150) u(150) − u(100)  50
·
−
U (ℓ1 ) − U (δ150 ) = 
 2

50
50
|
{z
} |
{z
}
≃ Mu(150)
≃ Mu(100)
u
u(200)
U (ℓ1 )
u(100)
100
150
E[ℓ1 ]
200
x
Expected Utility Theory
Attitudes Towards Risk
1. Diminishing marginal utility, u is concave, u ′′ < 0 ⇒, the
consumer is risk-averse.
U (X ) < u(E[X ]) for all X
2. Increasing marginal utility, u is convex, u ′′ > 0 ⇒, the
consumer is risk-loving.
U (X ) > u(E[X ]) for all X
3. Constant marginal utility, u is affine (linear plus a
constant),u ′′ = 0 ⇒, the consumer is risk-neutral,
U (X ) = u(E[X ]) for all X
Expected Utility Theory
Attitudes Towards Risk
1. Diminishing marginal utility, u is concave, u ′′ < 0 ⇒, the
consumer is risk-averse.
U (X ) < u(E[X ]) for all X
2. Increasing marginal utility, u is convex, u ′′ > 0 ⇒, the
consumer is risk-loving.
U (X ) > u(E[X ]) for all X
3. Constant marginal utility, u is affine (linear plus a
constant),u ′′ = 0 ⇒, the consumer is risk-neutral,
U (X ) = u(E[X ]) for all X
Expected Utility Theory
Attitudes Towards Risk
1. Diminishing marginal utility, u is concave, u ′′ < 0 ⇒, the
consumer is risk-averse.
U (X ) < u(E[X ]) for all X
2. Increasing marginal utility, u is convex, u ′′ > 0 ⇒, the
consumer is risk-loving.
U (X ) > u(E[X ]) for all X
3. Constant marginal utility, u is affine (linear plus a
constant),u ′′ = 0 ⇒, the consumer is risk-neutral,
U (X ) = u(E[X ]) for all X
Measuring the Degree of Risk-Aversion
The Arrow-Pratt or Absolute Measure of Risk Aversion
Definition
The Arrow-Pratt absolute measure of risk-aversion of an agent
with VN-M utility u at wealth level w is:
ρu (w) =
−u ′′ (w)
.
u ′ (w)
If for two individual with VN-M utilities u and u
e we have that
ρu (w) > ρue (w) for all wealth levels w then we say that the agent
with utility u is more risk-averse than the agent with utility u
e.
Measuring the Degree of Risk-Aversion
The Arrow-Pratt or Absolute Measure of Risk Aversion
Definition
The Arrow-Pratt absolute measure of risk-aversion of an agent
with VN-M utility u at wealth level w is:
ρu (w) =
−u ′′ (w)
.
u ′ (w)
If for two individual with VN-M utilities u and u
e we have that
ρu (w) > ρue (w) for all wealth levels w then we say that the agent
with utility u is more risk-averse than the agent with utility u
e.
Measuring the Degree of Risk-Aversion
The Relative Measure of Risk-Aversion
We are not covering this material, please skip this slide...
Definition
The relative absolute measure of risk-aversion of an agent with
VN-M utility u at wealth level w is:
ru (w) =
−u ′′ (w) w
.
u ′ (w)
Mathematics Review
Taylor’s Approximation
Consider a function of one variable defined on the real line,
f : R → R. If f is differentiable, we write the first order Taylor
approximation:
f (x + h) − f (x) ≃ f ′ (x) · h
The approximation works well only if |h| is "small".
For a function of two variables and h = (h1 , h2 ) we have a
similar expression:
f (x + h1 , y + h2 ) − f (x, y) ≃
∂
∂
f (x, y) · h1 +
f (x, y) · h2
∂x
∂y
Mathematics Review
Taylor’s Approximation
Consider a function of one variable defined on the real line,
f : R → R. If f is differentiable, we write the first order Taylor
approximation:
f (x + h) − f (x) ≃ f ′ (x) · h
The approximation works well only if |h| is "small".
For a function of two variables and h = (h1 , h2 ) we have a
similar expression:
f (x + h1 , y + h2 ) − f (x, y) ≃
∂
∂
f (x, y) · h1 +
f (x, y) · h2
∂x
∂y
Mathematics Review
Taylor’s Approximation
Consider a function of one variable defined on the real line,
f : R → R. If f is differentiable, we write the first order Taylor
approximation:
f (x + h) − f (x) ≃ f ′ (x) · h
The approximation works well only if |h| is "small".
For a function of two variables and h = (h1 , h2 ) we have a
similar expression:
f (x + h1 , y + h2 ) − f (x, y) ≃
∂
∂
f (x, y) · h1 +
f (x, y) · h2
∂x
∂y
Math. Review
Marginal Utility & Taylor’s Approximation
U(x + ∆x, y + ∆y) − U(x, y) =
MUx · ∆x + MUy · ∆y
Mathematical Review
Interior Solutions
Maximizing a function of one variable defined on the real line,
f : R → R.
Maximization Problem
max f (x)
(P)
First order condition
f ′ (x) = 0
(FOC)
Second order condition
x∈R
′′
f (x) ≤ 0
(SOC)
Any point x satisfying FOC and SOC is a candidate for an
interior solution.
Mathematical Review
Interior and corner Solutions
Maximizing a function of one variable defined on an interval,
f : [a, b] → R. As before,
Maximization Problem
max f (x)
(P)
First order condition
f ′ (x) = 0
(FOC)
Second order condition
f ′′ (x) ≤ 0
(SOC)
b≥x≥a
Any point x satisfying FOC and SOC is a candidate for an
interior solution and now,
▶
x = a is a candidate for a corner solution if f ′ (a) ≤ 0.
▶
x = b is a candidate for a corner solution if f ′ (b) ≥ 0.
Mathematical Review
Concavity and convexity
Consider any function f : Rk → R.
Definition: f is concave if and only if, for all α ∈ [0, 1], and
any two points x, y ∈ Rk , we have
f (α x + (1 − α) y) ≥ α f (x) + (1 − α) f (y).
Another definition: We say that f is convex if −f is concave.
Math. Review
Global Maxima
Proposition. Assume f is concave and also assume that x
satisfy the FOC then x is a solution to the maximization
problem (i.e. x is a global maximum).
Math. Review
The implicit Function Theorem
Let f (x, y) be a real-valued function of two variables and let
g(x) be a real-valued function of one-variable. Moreover,
assume that g has the following special property:
f (x, g(x)) = c
for all values of x and c is a constant. Then,
∂
f (x, g(x))
g ′ (x) = − ∂x
∂
f (x, g(x)).
∂y
.
Intertemporal Consumption
Key concepts:
1. Present Value
2. Arbitrage
3. Intertemporal Marginal Rate of Substitution - MRIS
Learning Goals:
1. Be able to compute PV .
2. Solve for the optimal consumption bundle.
3. Be able to justify the PV by arbitrage arguments.
Intertemporal Consumption
Key concepts:
1. Present Value
2. Arbitrage
3. Intertemporal Marginal Rate of Substitution - MRIS
Learning Goals:
1. Be able to compute PV .
2. Solve for the optimal consumption bundle.
3. Be able to justify the PV by arbitrage arguments.
Intertemporal Consumption
Key concepts:
1. Present Value
2. Arbitrage
3. Intertemporal Marginal Rate of Substitution - MRIS
Learning Goals:
1. Be able to compute PV .
2. Solve for the optimal consumption bundle.
3. Be able to justify the PV by arbitrage arguments.
Intertemporal Consumption
Key concepts:
1. Present Value
2. Arbitrage
3. Intertemporal Marginal Rate of Substitution - MRIS
Learning Goals:
1. Be able to compute PV .
2. Solve for the optimal consumption bundle.
3. Be able to justify the PV by arbitrage arguments.
Intertemporal Consumption
Key concepts:
1. Present Value
2. Arbitrage
3. Intertemporal Marginal Rate of Substitution - MRIS
Learning Goals:
1. Be able to compute PV .
2. Solve for the optimal consumption bundle.
3. Be able to justify the PV by arbitrage arguments.
Intertemporal Consumption
Key concepts:
1. Present Value
2. Arbitrage
3. Intertemporal Marginal Rate of Substitution - MRIS
Learning Goals:
1. Be able to compute PV .
2. Solve for the optimal consumption bundle.
3. Be able to justify the PV by arbitrage arguments.
Intertemporal Consumption
Key concepts:
1. Present Value
2. Arbitrage
3. Intertemporal Marginal Rate of Substitution - MRIS
Learning Goals:
1. Be able to compute PV .
2. Solve for the optimal consumption bundle.
3. Be able to justify the PV by arbitrage arguments.
Intertemporal Model (no uncertainty)
▶
t = 0, 1, . . . , T periods.
▶
one good at each period, ct consumption at period t
▶
πt = 1 is the spot price for all t (pay at the "spot")
▶
pt is the forward price (contingent price) (pay today)
t=0
p0 = π0
p1
p2
..
.
pT
t=1
π1
t=2
π2
t
πt
t=T
πT
Definition: A forward contract is a non-standardized contract
between two parties to buy or to sell an asset at a specified
future time at a price agreed upon today.
Intertemporal Model (no uncertainty)
▶
t = 0, 1, . . . , T periods.
▶
one good at each period, ct consumption at period t
▶
πt = 1 is the spot price for all t (pay at the "spot")
▶
pt is the forward price (contingent price) (pay today)
t=0
p0 = π0
p1
p2
..
.
pT
t=1
π1
t=2
π2
t
πt
t=T
πT
Definition: A forward contract is a non-standardized contract
between two parties to buy or to sell an asset at a specified
future time at a price agreed upon today.
Intertemporal Model (no uncertainty)
▶
t = 0, 1, . . . , T periods.
▶
one good at each period, ct consumption at period t
▶
πt = 1 is the spot price for all t (pay at the "spot")
▶
pt is the forward price (contingent price) (pay today)
t=0
p0 = π0
p1
p2
..
.
pT
t=1
π1
t=2
π2
t
πt
t=T
πT
Definition: A forward contract is a non-standardized contract
between two parties to buy or to sell an asset at a specified
future time at a price agreed upon today.
Intertemporal Model (no uncertainty)
▶
t = 0, 1, . . . , T periods.
▶
one good at each period, ct consumption at period t
▶
πt = 1 is the spot price for all t (pay at the "spot")
▶
pt is the forward price (contingent price) (pay today)
t=0
p0 = π0
p1
p2
..
.
pT
t=1
π1
t=2
π2
t
πt
t=T
πT
Definition: A forward contract is a non-standardized contract
between two parties to buy or to sell an asset at a specified
future time at a price agreed upon today.
Intertemporal Model (no uncertainty)
▶
t = 0, 1, . . . , T periods.
▶
one good at each period, ct consumption at period t
▶
πt = 1 is the spot price for all t (pay at the "spot")
▶
pt is the forward price (contingent price) (pay today)
t=0
p0 = π0
p1
p2
..
.
pT
t=1
π1
t=2
π2
t
πt
t=T
πT
Definition: A forward contract is a non-standardized contract
between two parties to buy or to sell an asset at a specified
future time at a price agreed upon today.
Intertemporal Model (no uncertainty)
▶
t = 0, 1, . . . , T periods.
▶
one good at each period, ct consumption at period t
▶
πt = 1 is the spot price for all t (pay at the "spot")
▶
pt is the forward price (contingent price) (pay today)
t=0
p0 = π0
p1
p2
..
.
pT
t=1
π1
t=2
π2
t
πt
t=T
πT
Definition: A forward contract is a non-standardized contract
between two parties to buy or to sell an asset at a specified
future time at a price agreed upon today.
Intertemporal Model (no uncertainty)
OTC = over the counter
▶
t = 0, 1, . . . , T periods.
▶
one good at each period, ct consumption at period t
▶
πt = 1 is the spot price for all t (pay at the "spot")
▶
pt is the forward price (contingent price) (pay today)
t=0
p0 = π0
p1
p2
..
.
pT
t=1
π1
t=2
π2
t
πt
t=T
πT
Definition: A forward contract is a non-standardized contract
between two parties to buy or to sell an asset at a specified
future time at a price agreed upon today.
The Relationship between Forward and Spot Prices
In practice, forward contracts are over-the-counter (OTC) bilateral contracts between two parties that are customized as
opposed to standard contracts that are traded in markets.
Here, however, we assume forward contracts are traded in a
competitive market. As a result, by arbitrage, we mud have:
pt =
Can you explain why?
πt
.
(1 + ı)t
Present Value
▶
It cash-flow in period t
▶
ı interest rate period t to t + 1 (constant)
Present value formula:
PV (I0 , I1 , I2 , . . . , IT ) =I0 +
=
I1
I2
IT
+
+ ... +
1 + ı (1 + ı)2
(1 + ı)T
T
∑
t=0
It
(1 + ı)t
(PV)
Present Value
▶
It cash-flow in period t
▶
ı interest rate period t to t + 1 (constant)
Present value formula:
PV (I0 , I1 , I2 , . . . , IT ) =I0 +
=
I1
I2
IT
+
+ ... +
1 + ı (1 + ı)2
(1 + ı)T
T
∑
t=0
It
(1 + ı)t
(PV)
Inter-temporal Consumption
2-Period (T = 2) Consumer Problem
c1
max U (c0 , c1 )
c1 , c2
subject to
c0 +
1
≤ Y0 + (1+ı)
Y1
c0 ≥ 0 and c1 ≥ 0
1
(1+ı) c1
(1 + ı)Y0 + Y1
Y1
0
Y0
Y1
Y0 +
1+ı
c0
Inter-temporal Consumption
2-Period (T = 2) Consumer Problem
c1
max U (c0 , c1 )
c1 , c2
subject to
c0 +
(1 + bı)Y0 + Y1
1
≤ Y0 + (1+ı)
Y1
c0 ≥ 0 and c1 ≥ 0
1
(1+ı) c1
ı ↗ bı
(1 + ı)Y0 + Y1
Y1
0
Y0
Y0 +
Y1
1 + bı
Y1
Y0 +
1+ı
c0
Inter-temporal Consumption
2-Period (T = 2) Consumer Problem
c1
max U (c0 , c1 )
c1 , c2
subject to
c0 +
1
≤ Y0 + (1+ı)
Y1
c0 ≥ 0 and c1 ≥ 0
1
(1+ı) c1
ı ↘ eı
(1 + ı)Y0 + Y1
Y1
(1 + eı)Y0 + Y1
0
Y0
c0
Y1
Y0 +
1+ı
Y0 +
Y1
1 + eı
The idea of arbitrage
The A’s front office realized right away, of course, that they
couldn’t replace Jason Giambi with another first baseman just
like him. There wasn’t another first baseman just like him and
if there were they couldn’t have afforded him and in any case
that’s not how they thought about the holes they had to fill.
"The important thing is not to recreate the individual," Billy
Beane would later say. "The important thing is to recreate the
aggregate." He couldn’t and wouldn’t find another Jason
Giambi; but he could find the pieces of Giambi he could least
afford to be without, and buy them for a tiny fraction of the
cost of Giambi himself. – Moneyball by Micheal Lewis, p. 103
The idea of arbitrage
continuation
The A’s front office had broken down Giambi into his obvious
offensive statistics: walks, singles, doubles, home runs along
with his less obvious ones: pitches seen per plate appearance,
walk to strikeout ratio and asked: which can we afford to
replace? And they realized that they could afford, in a
roundabout way, to replace his most critical offensive trait, his
on-base percentage, along with several less obvious ones. The
previous season Giambi’s on-base percentage had been .477, the
highest in the American League by 50 points. (Seattle’s Edgar
Martinez had been second at .423; the average American
League on-base percentage was .334.) There was no one player
who got on base half the time he came to bat that the A’s could
afford; – Moneyball by Micheal Lewis, p. 103
The idea of arbitrage
continuation
on the other hand, Jason Giambi wasn’t the only player in the
Oakland A’s lineup who needed replacing. Johnny Damon
(onbase percentage .324) was gone from center field, and the
designated hitter Olmedo Saenz (.291) was headed for the
bench. The average on-base percentage of those three players
(.364) was what Billy and Paul had set out to replace. They
went looking for three players who could play, between them,
first base, outfield, and DH, and who shared an ability to get on
base at a rate thirty points higher than the average big league
player. – Moneyball by Micheal Lewis, p. 103
Understanding Present Value
Arbitrage
What is the value today of Y1 dollars tomorrow?
1. What should we do if someone else thinks that x dollars
x
tomorrow are worth less than
today?
1+ı
2. Or alternatively, believes x dollars tomorrow are worth
x
more than
dollars today?
1+ı
Understanding The Solution to the Consumer Problem
MU0
=1+ı
MU1
1
c0 +
c1 = PV(Y0 , Y1 )
(1 + ı)
MRIS ≡
To understand the first equation: MRIS is how many units of
consumption tomorrow are equal to one unit of consumption
today for the consumer and 1 + ı is how many units of
consumption tomorrow are equal to one unit of consumption
today for the market. In equilibrium they ought to be equal.
The second equation just says the consumer expends her
income during her lifetime.
Financial Instruments:
Options
A stock option is an option (not an obligation) to buy (call
option) or to sell (put option) some specified number of shares
of the stock at price (per-share) K (the strike price) at expire
date T (European option) or, alternatively at any point in time
before T (American option).
Example: Two periods: at t = 0 the price of the stock is 27 and
at t = 1 the it is 28 with prob. 32 or 26 with prob. 13 . The
option is an European (you can use it only at the expire date
T = 1) call (gives you the right to buy 1 stock) option with
strike price K = 26.5.
Note: the strike price is not the price of the option (in practice,
the price of the option is called premium).
A Call Option Example
continuation
The payoffs associated to this call option are:
$ − 26.5 + 28 = 1.5
2
3
t=0
1
3
$0
A Call Option Example
continuation
If a DM with utility for money u and initial wealth $36 buys
the option paying P at t = 0, his/her expected utility is:
$36 − P − 26.5 + 28 = 37.5 − P
t=1
2
3
2u(37.5−P)
3
t=0
1
3
36 − P
t=1
u(37.5 − P)
+
u(36−P)
3
u(36 − P)
A Call Option Example
continuation
The expected utility of not buying the call is
U (not buy) = u(36).
The expected utility of the call is
U (buy) =
2u(37.5 − P) u(36 − P)
+
.
3
3
If P = 0 then U (not buy) = u(36) < U (buy) = 2u(37.5)
+ u(36)
3
3 .
2u(36)
u(35.5)
If P = 1.5 then U (not buy) = u(36) > U (buy) = 3 + 3 .
As P ↗ we have U (buy) ↘ and U (not buy) =cte.
There exists Pmax such that U (not buy) = U (buy).
A Call Option Example
continuation
1. u(x) = x =⇒ Pmax = 1.
2. u(x) = x 2 =⇒ Pmax = 1.0069. Making the DM indifferent,
2
we get 73.5
√ − 74P + P = 0 so
74−
742 −4(73.5)
2
≃ 1.0069.
√
3. u(x) = √
x =⇒ Pmax = 0.9965. Making the DM indifferent,
√
we get 2 36 − P + 32 + 36 − P = 3 · 6. The "trick" is to
√
call
√ y = 36 − P and remove the square root in
2 y 2 − 32 + y = 18 to get a quadratic equation in y. We
solve it for y and set P = 36 − y 2 .
P=
A Call Option Example
continuation
Now there is a bond that costs $1 at t = 0 and pays 1 + ı at
t = 1. If we buy/sell s shares of the stock and b bonds such
that:
 

  
27
1
P
 

  
s 28 + b 1 + ı = 1.5
26
1+ı
0
It must be that 2s = 1.5 so s = 0.75 and b = − 0.75·26
1+ı s As a
result we can figure out the price of the call:
P = 0.75 · 27 −
0.75 · 26
.
1+ı
If 0 ≤ ı < +∞ then 0.75 ≤ P < 20.25.
Portfolio selection
A simple model of portfolio choice, there is one investor and:
▶
Two periods t = 0, 1 and two states at t = 1 (H or L).
▶
Investor has wealth only at t = 0, w0 > 1 and w1 = 0.
▶
Investor uses assets to transfer wealth across periods.
▶
There are two assets: the riskless and the risky one.
▶
The riskless asset’s rate of return is (1 + ı) in both states.
▶
The risky asset returns RH in state H and RL in state L.
▶
No asset is dominated that is, RH > 1 + ı > RL .
▶
The fraction (of w0 ) invested in the riskless asset is α.
▶
The probability of state H is p and the prob. of L is 1 − p.
Portfolio selection
continuation...
The problem of the investor is to choose α ∈ [0, 1] to maximize
her expected utility:
U (α) = p · u ((α(1 + ı) + (1 − α)RH )w0 ) +
+(1 − p) · u ((α(1 + ı) + (1 − α)RL )w0 ) .
Portfolio selection
continuation...
The problem of the investor is to choose α ∈ [0, 1] to maximize
her expected utility:
U (α) = p · u ((α(1 + ı) + (1 − α)RH )w0 ) +
+(1 − p) · u ((α(1 + ı) + (1 − α)RL )w0 ) .
wealth when
state H happens
Portfolio selection
continuation...
The problem of the investor is to choose α ∈ [0, 1] to maximize
her expected utility:
U (α) = p · u ((α(1 + ı) + (1 − α)RH )w0 ) +
+(1 − p) · u ((α(1 + ı) + (1 − α)RL )w0 ) .
wealth when
state L happens
Portfolio selection
continuation...
The corresponding first-order condition for a maximum is:
U ′ (α) = 0 or equivalently,
p · u ′ ((α(1 + ı) + (1 − α)RH )w0 ) ·(1 + ı − RH ) · w0 +
|
{z
}
|
expected MU
{z
}
MC of riskless asset
(1 − p) · u ′ ((α(1 + ı) + (1 − α)RL )w0 )) · (1 + ı − RL )w0 = 0
|
{z
}
|
expected MU
{z
MB of riskless asset
}
Intertemporal Choice
max u(c0 ) + δu(c1 )
c1 , c2
recap.
subject to
c1
c0 +
1
≤ Y0 + (1+ı)
Y1
c0 ≥ 0 and c1 ≥ 0
1
(1+ı) c1
Indiference curve with utility ū
(1 + ı)Y0 + Y1
c1 (c0 ) = u −1 (ū − u(c0 )/δ)
c1′ (c0 ) = −MRIS
Y1
U (c0 , c1 ) = 4
0
Y0
Y1
Y0 +
1+ı
c0
Intertemporal Choice
max u(c0 ) + δu(c1 )
c1 , c2
recap.
subject to
c1
c0 +
1
≤ Y0 + (1+ı)
Y1
c0 ≥ 0 and c1 ≥ 0
1
(1+ı) c1
Indiference curve with utility ū
(1 + ı)Y0 + Y1
c1 (c0 ) = u −1 (ū − u(c0 )/δ)
c1′ (c0 ) = −MRIS
Y1
U (c0 , c1 ) = 5
0
Y0
Y1
Y0 +
1+ı
c0
Intertemporal Choice
max u(c0 ) + δu(c1 )
c1 , c2
recap.
subject to
c1
c0 +
1
≤ Y0 + (1+ı)
Y1
c0 ≥ 0 and c1 ≥ 0
1
(1+ı) c1
Indiference curve with utility ū
(1 + ı)Y0 + Y1
c1 (c0 ) = u −1 (ū − u(c0 )/δ)
c1′ (c0 ) = −MRIS
Y1
U (c0 , c1 ) = 6
0
Y0
Y1
Y0 +
1+ı
c0
Intertemporal Choice
max u(c0 ) + δu(c1 )
c1 , c2
recap.
subject to
c1
c0 +
1
≤ Y0 + (1+ı)
Y1
c0 ≥ 0 and c1 ≥ 0
1
(1+ı) c1
Indiference curve with utility ū
(1 + ı)Y0 + Y1
c1 (c0 ) = u −1 (ū − u(c0 )/δ)
c1′ (c0 ) = −MRIS
Y1
U (c0 , c1 ) = 5.9
0
Y0
Y1
Y0 +
1+ı
c0
c1A
General Equilibrium
0
c0B
endowment
c0A
0
c1B
c1A
General Equilibrium
0
c0B
endowment
c0A
0
A’s endowment of c0
c1B
c1A
General Equilibrium
0
c0B
endowment
c0A
0
B’s endowment of c0
c1B
c1A
c0B
General Equilibrium
B’s endowment of c0
0
endowment
c0A
0
A’s endowment of c0
c1B
c1A
c0B
General Equilibrium
B’s endowment of c0
0
B’s
endowment
of c1
A’s
endowment
of c1
endowment
c0A
0
A’s endowment of c0
c1B
c1A
c0B
General Equilibrium
B’s endowment of c0
0
B’s
endowment
of c1
A’s
endowment
of c1
c0A
0
A’s endowment of c0
c1B
c1A
c0B
General Equilibrium
B’s endowment of c0
0
B’s
endowment
of c1
A’s
endowment
of c1
c0A
0
A’s endowment of c0
c1B
c1A
General Equilibrium
0
c0B
c0A
0
c1B
c1A
General Equilibrium
0
c0B
util
ity
of
bet
A
ter
tha
end
n
ow
me
nt
c0A
0
c1B
c1A
General Equilibrium
0
c0B
c0A
0
c1B
c1A
General Equilibrium
0
c0B
util
ity
of
bet
B
ter
tha
end
n
ow
me
nt
c0A
0
c1B
c1A
General Equilibrium
0
c0B
both
are
r
bette
off
c0A
0
c1B
c1A
General Equilibrium
0
c0B
efficient
allocations
both
are
r
bette
off
MRISA
SB
= MRI
c0A
0
c1B
c1A
General Equilibrium
0
c0B
efficient
allocations
MRISA
SB
= MRI
c0A
0
c1B
c1A
General Equilibrium
0
c0B
efficient
allocations
MRISA
SB
= MRI
c0A
0
c1B
c1A
General Equilibrium
0
c0B
efficient
allocations
MRISA
SB
= MRI
c0A
0
c1B
Uncertainty
Arrow-Debreu goods
Definition: An Arrow-Debreu good is defined by 4 dimensions:
1.
2.
3.
4.
Its physical properties.
The geographic location where it is available.
The time when it is available for consumption.
The state where it is available for consumption.
Say there are K distinct physical attributes, L locations, T
time periods and S states.
The the total number of goods is n = K · L · T · S. That is, we
have n prices and n markets.
A bundle or basket of goods is a vector with n entries.
Uncertainty
Arrow-Debreu goods, an example
Assume that:
▶
▶
▶
▶
K ∈ {umbrella,parasol},
K ∈ {Hillsborough,Chicago},
T ∈ {today,tomorrow}, and
S ∈ {sun,rain}.
Then we have 16 goods! 16 markets! 16 prices!
For instance, the first good is an umbrella in Hillsborough
available today provided today is a sunny day, the second good
is an umbrella in Hillsborough available today if it is rainy, the
third good is is an umbrella in Hillsborough available tomorrow
if tomorrow is sunny, ..., the last good is a parasol available
tomorrow in Chicago if it rains.
Remark: the order in which one may label the goods is
Complete Markets
When all the markets for Arrow-Debreu goods exists we call it
the case of complete markets.
The assumption of complete markets may appear too extreme
and or unrealistic, however there are more markets out there
than meets the eye:
1. Weather Markets
2. A weather contract
3. Events Markets
Complete Markets
When all the markets for Arrow-Debreu goods exists we call it
the case of complete markets.
The assumption of complete markets may appear too extreme
and or unrealistic, however there are more markets out there
than meets the eye:
1. Weather Markets
2. A weather contract
3. Events Markets
Complete Markets
When all the markets for Arrow-Debreu goods exists we call it
the case of complete markets.
The assumption of complete markets may appear too extreme
and or unrealistic, however there are more markets out there
than meets the eye:
1. Weather Markets
2. A weather contract
3. Events Markets
Complete Markets
When all the markets for Arrow-Debreu goods exists we call it
the case of complete markets.
The assumption of complete markets may appear too extreme
and or unrealistic, however there are more markets out there
than meets the eye:
1. Weather Markets
2. A weather contract
3. Events Markets
The Consumer Problem under Complete Markets
Two States and One Good
max
cL ,cH
πL u(cL ) + πH u(cH ).
st.
pL cL +pH cH ≤pL YL +pH YH
L(cL , cH , λ) = πL u(cL ) + πH u(cH ) − λ (pL YL + pH YH − pL cL − pH
∂
L(cL , cH , λ) = πL u ′ (cL ) + λ pL = 0
(FOCcL )
∂cL
∂
L(cL , cH , λ) = πH u ′ (cU ) + λ pH = 0
(FOCcH )
∂cH
∂
L(cL , cH , λ) = pL YL + pH YH − pL cL − pH cH = 0 (FOCλ )
∂λ
Two states: H with probability of πH and L with prob. πL .
Endowment: Y = (YH , YL ).
Income (value of endowment): I = pH YH + pL YL .
Price of unit of good delivered if H (L) happens: pH (pL ).
Equilibrium Prices under Complete Markets
Two States, Two Consumers and One Good
• Consumers A and B with expected utilities:
πL u A (cL ) + πH u A (cH ) and πL u B (cL ) + πH u B (cH ).
• Their endowments are given, Y A = (YHA , YLA ) and Y B = (YHB , YLB ).
• Solve the consumer problem to find their individual demands (see
A
previous page). Notice cLA and cH
depend only the probabilities, the
B
prices and A’s endowment and likewise, cLB and cH
depend only the
probabilities, the prices and B’s endowment.
• To find the price ratio:
⇒ Equate total supply with total demand,
A
A
B
YH + YHB = cH
(pH , pL ) + cH
(pH , pL ).
| {z } |
{z
}
supply
demand
we solved for it previously
• Important: we can always normalize one price to one. If we set
pH = 1 and then solve the above equation for pL then we actually get
the value of ppHL .
• Important: See Mathematica file on General Equilibrium.
Uncertainty
Arrow Securities
But even if markets are not complete we can “complete” the
missing markets if we have Arrow securities.
An Arrow-security is a financial instrument that pays $1 unit of
accounting in a given location, date t, and state and it is traded
in a market at date t − 1. Thus at each point in time we need
only L · S markets.
Also even if we do not have Arrow securities we can complete
the markets if we have enough financial instruments (as we did
in class).
Financial Market Eq. with Arrow Securities
Only two states, s ∈ {L, H }, and consumption takes place only at
date T = 1. But consumer makes decisions and markets operate at
date T = 0. The consumer problem with complete markets is
max
c ,c
L
U (cL , cH ).
H
st.
pL cL +pH cH ≤pL YL +pH YH
and with Arrow-securities is
max
cL ,cH ,zL ,zH
st.
qL zL +qH zH ≤0
p̂L cL ≤p̂L YL +zL
p̂H cH ≤p̂H YH +zH
U (cL , cH ).
Financial Market Eq. with Arrow Securities
Only two states, s ∈ {L, H }, and consumption takes place only at
date T = 1. But consumer makes decisions and markets operate at
date T = 0. The consumer problem with with Arrow-securities is:
max
cL ,cH ,zL ,zH
U (cL , cH ).
st.
qL zL +qH zH ≤0
p̂L cL ≤p̂L YL +zL
p̂H cH ≤p̂H YH +zH
where:
▶
qs is the price of one unit of the security s.
▶
zs is the amount of securities s the consumer buys
(negative if he or she sells).
The consumer problem with expected utility (Arrow
securities)
max
cL ,cH
st.
qL zL +qH zH ≤0
p̂L cL ≤p̂L YL +zL
p̂H cH ≤p̂H YH +zH
πL u(cL ) + πH u(cH ). (CP - Arrow securities)
L(cL , cH , λ) = πL u(cL ) + πH u(cH ) − λ1 (qL zL + qH zH ) +
− λ2 (p̂L cL − p̂L YL − zL ) − λ3 (p̂H cH − p̂H YH − zH )
∂
L(cL , cH , λ) = πL u ′ (cL ) + λ2 p̂L = 0
∂cL
∂
L(cL , cH , λ) = πH u ′ (cU ) + λ3 p̂H = 0
∂cH
∂
L(cL , cH , λ) = qL zL + qH zH = 0
∂λ1
∂
L(cL , cH , λ) = p̂L cL − p̂L YL − zL = 0
∂λ2
∂
L(cL , cH , λ) = p̂H cH − p̂H YH − zH = 0
(FOCcL )
(FOCcH )
(FOCλ1 )
(FOCλ2 )
(FOCλ3 )
Risk-Sharing
Let’s assume:
▶
two consumers (A and B)
▶
complete markets (with Arrow-securities we will obtain
identical results).
▶
total endowment constant across states,
Y = YLA + YLB
and
Y = YHA + YHB .
From the first-order condition, we have that:
′ (c A )
′ (c B )
uB
uA
L
L
=
′ (c A )
′ (c B )
uA
u
H
B H
′ (c A )
′ (Y − c A )
uA
uB
L
L
=
′ (c A )
′ (Y − c A )
uA
u
B
H
H
A
B
⇒ cLA > cH
⇔ cLB > cH
A
A
⇒ cLA > cH
⇔ Y − cLA > Y − cH
A
⇔ cLA < cH
But this is a contradiction !
Portfolio Choice
max
θ,cL ,cH
st.
0≤θ≤1
cL =θ RL W0 +(1−θ)W0
cH =θ RH W0 +(1−θ)W0
πL u(cL ) + πH u(cH ).
(CP - portfolio choice)
L(θ, cL , cH , λ1 , λ2 ) = πL u(cL ) + πH u(cH )+
− λ1 (cL − θ RL W0 − (1 − θ)W0 ) − λ2 (cH − θ RH W0 − (1 − θ)W0 )
∂
L(θ, cL , cH , λ1 , λ2 ) = λ1 (RL − 1)W0 + λ2 (RH 1)W0 = 0
∂θ
(FOCθ )
∂
L(θ, cL , cH , λ1 , λ2 ) = πL u ′ (cL ) − λ1 = 0
(FOCcL )
∂cL
∂
L(θ, cL , cH , λ1 , λ2 ) = πH u ′ (cH ) + λ2 = 0
(FOCcH )
∂cH
∂
L(θ, cL , cH , λ1 , λ2 ) = cL − θ RL W0 − (1 − θ)W0 = 0
∂λ1
Asset Pricing
If we have complete-markets and we know the equilibrium price
vector, p = (ps ). We can price ANY financial asset/security. A
∑
financial asset that pays fs at state s must value s∈S ps · fs
where S is the set of all states (including time periods).
Asset Pricing
Example
Say we have one state today and two tomorrow,
S = {s0 , s1H , s2L } and the price of delivery of one unit of
consumption if the state s happens is p0 , p1H and p2H for the
respective states.
1. A financial security that always pays 1 in period 1 (a
risk-less bond) and zero today must be worth (today)
0 · p0 + 1 · p1H + 1 · p2H .
2. A bet that pays 1 if H happens and −1 if L happens and
nothing today is worth p1H − p1L today.
Liquidity
Let’s assume:
▶
Three dates (0, 1 and 2) and one consumer.
▶
Investment occurs at dates 0 and 1.
▶
Consumption occurs at dates 1 or 2.
▶
With prob. π1 consumption takes place only at date 1.
▶
With prob. π2 = 1 − π1 consumption takes place at date 2.
▶
Safe (short asset) investment of x yields x at next date.
▶
Risky (long asset) investment of x at date 0 yields R x at
date 2 where R > 1. The long asset is illiquid at date 1.
▶
Initial wealth: W0 = 1.
▶
The fraction of wealth in short asset is β.
max π1 u (β) + π2 u (β + (1 − β)R)
β
st.
0≤β≤1
(2.1)
Liquidity Shocks
An Example
▶
Dates: = t = 0, 1, 2.
▶
Consumer with utility u(c) = log(c).
▶
Safe (short asset): investment of x yields x at next date.
▶
Risky (long asset): invest. x at t = 0 yields 4 x at t = 2.
▶
Long asset is illiquid at t = 1.
▶
Initial wealth: W0 = 10.
▶
Investment at t = 0, 1.
▶
Consumption at t = 1, 2 but not both.
▶
With prob.
▶
The fraction of wealth in short asset is β.
max
β
st.
0≤β≤1
1
2
consumption takes place only at date 1.
1
1
log (β 10) + log (β 10 + (1 − β)40)
2
2
Liquidity Shocks
An Example
max
β
st.
0≤β≤1
1
1
log (β 10) + log (β 10 + (1 − β)40)
2
2
10
10 − 40
+
=0
2 (β 10) 2 (β 10 + (1 − β)40)
β=
2
3
(FOC)
Liquidity
with risk-pooling
▶
Two agents, i = 1, 2.
▶
Initial individual wealth, W0i = ω.
▶
Short asset with rate of return r, 1 ≤ r < R.
▶
Long asset with rate of return R.
▶
β fraction of wealth invest in short-asset.
▶
π prob. of liquidity shock (independent across agents).
▶
d amount of short-asset promised to an agent who had a
liquidity shock if the other agent did not suffer a liquidity
shock.
Liquidity
with risk-pooling (continuation)
If agents do not pool their resources, each agent choose its own
β to maximize:
(
)
max πu (r β ω) + (1 − π)u r 2 β ω + (1 − β)R ω
β
st.
0≤β≤1
(INDIVIDUAL)
If agents pool their resources, each agent choose its own β to
maximize:
max
β
st.
0≤β≤1
0≤d≤2r βω
π 2 u(r β ω)+π(1−π) u(d)+
+(1−π)π u(r(r β 2 ω−d)+(1−β)R 2 ω)+
+(1−π)2 u (r 2 β ω+(1−β)R ω )
(POOL)
Liquidity
with risk-pooling (continuation)
To understand the payoffs in the previous page, let’s look at the
following tables:
Without pooling:
State
Probability
Agent 1’s consumption
All suffer the shock
π2
rβω
Only 1 suffers
π(1 − π)
rβω
Only 2 suffers
(1 − π)π
r 2 β ω + R (1 − β) ω
None suffers
(1 − π)2
r 2 β ω + R (1 − β) ω
With pooling:
State
Probability
Agent 1’s consumption
All suffer the shock
π2
βω
Only 1 suffers
π(1 − π)
d
Only 2 suffers
(1 − π)π
r (r β 2 ω − d) + R (1 − β)2 ω
None suffers
(1 −
π)2
r 2 β ω + R (1 − β)ω
Liquidity
with risk-pooling (continuation)
Remark 1: The agents are always better-off by pooling their
resources. The optimal (β ∗ , d ∗ ) that solves the maximization
problem (POOL) always delivers a higher utility than the β ∗∗ that
solves the maximization problem (INDIVIDUAL).
Remark 2: For risk-pooling to occur is crucial that not all agents
receive liquidity shocks at the same time. If they face aggregate
uncertainty, they can not (fully) diversify their risks. Individual
(idiosyncratic uncertainty) can be diversified.
Risk Pooling
Let’s assume:
▶
▶
▶
▶
▶
▶
▶
Three dates (0, 1 and 2).
Infinitely many consumers i ∈ [0, 1], each one with Wi = 1 and
same preferences ui = u.
Investment opportunities and consumption are as before.
Probabilities of liquidity shocks are independent.
π1 prob. of a ‘bad’ liquidity shock or fraction of consumers who
suffer a ‘bad’ shock.
Company decides on investment decision for the pool of consumers,
it promises c1 to early consumers and c2 to late consumers.
Company faces no risk (Law of Large Numbers), its plans are
feasible if π1 c1 = β and π2 c2 = (1 − π1 )c2 = (1 − β)R.
max
β
st.
0≤β≤1
π1 c1 =β
π2 c2 =(1−β)R
π1 u (c1 ) + (1 − π1 )u (c2 )
(2.2)
Law of Large Numbers
eliminating uncertainty thru averages, side comment
We saw before that under risk-pooling, the company promises
c1 = β/π1 to each depositor who needs to consume in period 1.
Let’s assume we have a finite number of consumers, N .
• Clearly this promise can be carried out if the number of agents
e , is less or equal than the
who do suffer a liquidity shock, N
average number of consumers who suffer a liquidity shock π1 N .
e > π1 N , the company can not honor its
• However, if N
promises.
Law of Large Numbers
eliminating uncertainty thru averages, side comment
Assume that instead of promising the maximum possible,
c1 = β/π1 for some small ε > 0, the company promises to pay
ĉ1 = (1 − ε)c1 .
What is the probability that the company can honor its
promises?



e · ĉ1 ≤ N · β
Pr N
| {z }
e · (1 − ε)β/π1 ≤ N · β
 = Pr N
| {z }
total amount of short-asset
[
e ≤
= Pr N


total amount of short-asset
π1 N
[ (1−ε)
]
]
∑
N!
π1 N
=
π k (1 − π1 )N −k
(1 − ε π1 )
k!(N − k)!
k=0
where [x] is the floor function, it gives the largest integer below x.
Law of Large Numbers
eliminating uncertainty thru averages, side comment
What is the probability that the company can honor its
promises?
[
]
e ≤ π1 N
Pr N
=
(1 − ε)
Assume π =
1
3
π N
1
[ (1−ε)
]
∑
k=0
N!
π k (1 − π1 )N −k
k!(N − k)!
then:
ε\N
2
10
100
1,000
9,000
.1
0.444444
0.559264
0.812311
0.993344
1
.01
0.444444
0.559264
0.518803
0.585493
0.75259
.005
0.444444
0.559264
0.518803
0.559216
0.63596
0
0.444444
0.559264
0.518803
0.505947
0.504956
Please see the Mathematica file largenumbers.nb
Liquidity Shocks with a spot market for the long-asset
pages 60–64 in the textbook
The model is the same as before in the risk-pooling case with
the exception that consumers investment decisions are again
individual and there is a market at t = 1 for the long-asset.
▶
0 ≤ x fraction of wealth invested in the long-asset,
▶
0 ≤ y fraction of wealth invested in the short-asset.
▶
P is the price of the long-asset at date t = 1.
▶
It must be that P ≤ R in eq.
▶
λ prob. investor wants to consume only at t = 1.
(
y )
max
λu(y
+
Px)
+
(1
−
λ)u
(x
+
)R
x,y
P
x+y≤1
In eq. we have that
P = 1 ⇒ c1 = 1 & c2 = R ⇒ u ∗ = λu(1) + (1 − λ)u (R)
Bank Runs
pages 72-76
▶
A deposit contract is a pair of consumption promises
(c1 , c2 ) such that if a consumer deposits his entire wealth
W0 = 1 at the bank at t = 1, the consumer has the right to
withdraw c1 at date 1 and c2 at date 2.
▶
The optimal deposit contract is the one that solves the
risk-pooling problem.
▶
Liquidation technology: long asset is worth r ≤ 1 units of
the good at date t = 1
▶
If c1 > rx + y the bank will no be able to honor the deposit
contract if all consumers ‘run’.
▶
If c2 > c1 > rx + y we may or may not have a bank-run (it
depends on the consumers/investors) expectations.
Bank Runs
as an equilibrium, pages 76–82
In the previous analysis, we assumed that the deposit contract
was the same regardless if the Bank expected a bank run or not.
But if the Bank expects a bank-run with certainty, the bank will
offer a deposit contract (c1 , c2 ) = (1, 1) and so late consumers
are indifferent between withdrawing at t = 1 or at t = 2.
Bank Runs
cont.
To simplify the model, we shall assume that the liquidation
technology is perfect: that is r = 1 so there is no penalty in
liquidating the long-asset at t = 1. The bank can transform the
long-asset into the good or cash or short-asset in a 1-to-1 ratio.
In this case the long-asset dominates the short-asset as there is
no reason for the bank to hold the short-asset. If the bank
needs cash to pay depositors at t = 1 the bank can just
liquidate the long-asset.
Bank Runs
cont.
▶
▶
▶
▶
At t = 0, the total wealth of the bank is N · W0 where W0
is the initial wealth and N is the number of depositors.
At t = 1, the total wealth of the bank is reduced by the
withdraws which amount: λ · N · c1 where c1 is the amount
promised by the deposit contract and λ is the fraction of
early consumers.
At t = 2 , the total wealth of the bank is R · N · (W0 − λ c1 )
which must cover the withdraws by late consumers which
amount to (1 − λ) · N · c2 .
Assume for simplicity that W0 = 1 and so the budget
constraint for any feasible deposit contract must be:
R(1 − λ c1 )
1
and 0 ≤ c2 ≤
λ
1−λ
Notice that N cancels out so this is the budget constraint per
0 ≤ c1 ≤
Avoiding a run
Important remark: If c1 ≤ 1 the bank will be solvent to pay
any late consumers at t = 2 even if a bank-run takes place. So
if c1 ≤ 1 and c1 < c2 no late consumer will ever want to join a
bank-run.
Thus if the bank wants to avoid a run, it can choose c1 ≤ 1. In
this case, the bank maximizes:
max
0≤c1 ≤1
R(1−λ c )
0≤c2 ≤ 1−λ 1
λu(c1 ) + (1 − λ)u(c2 )
(PNR )
c1 )
To solve this we substitute R(1−λ
for c2 and take derivative
1−λ
with respect to c1 , equate to zero, and solve for c1 . If the
solution satisfies c1 ≤ 1 we are done but if the ‘solution’ has
c1 > 1 then we are violating the constraint c1 ≤ 1. In this case
the true solution must be c1 = 1 and so
c1 )
c2 = R(1−λ
= R(1−λ)
1−λ
1−λ = R.
Not avoiding a run
Assume that in the previous problem PNR the optimal deposit
contract that avoids a run is (1, R) that is, ideally the bank
would like to have c1 > 1 but that may cause a run so the bank
sets c1 = 1. In this event, what will be the optimal deposit
contract if the bank does not avoid a run. In the case the bank
expects a run with probability π, the bank maximizes:
max
0≤c1
R(1−λ c )
0≤c2 ≤ 1−λ 1
πu(1) + (1 − π) [λu(c1 ) + (1 − λ)u(c2 )]
(PN )
c1 )
for c2 and take
To solve this we again substitute R(1−λ
1−λ
derivative with respect to c1 , equate to zero, and solve for c1 .
The optimal solution does not depend on the value of π!
Optimal Deposit Contract
Which is the optimal deposit contract? We saw the optimal
deposit contract when the bak wants to avoid a run and when it
does not. The overall optimal deposit contract is the one that
yields the greatest utility. The bank will avoid a run if and only
if:
λu(1) + (1 − λ)u(R) =
>
max
0≤c1
R(1−λ c )
0≤c2 ≤ 1−λ 1
max
0≤c1 ≤1
R(1−λ c )
0≤c2 ≤ 1−λ 1
λu(c1 ) + (1 − λ)u(c2 ) >
πu(1) + (1 − π) [λu(c1 ) + (1 − λ)u(c2 )] =
= πu(1) + (1 − π)
max
0≤c1
R(1−λ c )
0≤c2 ≤ 1−λ 1
λu(c1 ) + (1 − λ)u(c2 )
that is, only if π is sufficiently large.
Asset Markets and Liquidity
Model Set-Up
The model is similar the the ones we saw before. But now there
is uncertainty regarding the fraction of the population who
suffers a liquidity shock: it could be high or low, λH (in state H
that happens with prob. π) or λL (in state L that happens with
prob. 1 − π) .
▶
▶
▶
▶
Dates: t = 0, 1, 2. States: s = H , L.
At date 0: consumers make deposit decisions and banks
offer deposit contracts and then make portfolio decisions
(how much to invest in the short-asset and how much to
invest in the long-asset).
At the start of date 1: all learn what is the current state
and mkts. open for trade.
At t = 1 there are two markets: the good market (where c2
is exchanged for c1 ) and the asset market (where the
Asset Markets and Liquidity
Model Set-Up (cont.)
▶
▶
▶
▶
▶
Let ps be the price of c2 (in terms of c1 ) and Ps the price of
the long-asset (in terms of the short-asset). (Notice we can
always convert c1 into the short-asset and vice-versa).
Banks are profit maximizing and they compete to attract
depositors.
At t = 1, consumers can not trade in the asset market nor
in the forward good market.
At t = 1, banks can trade in both markets.
Consumers place all their wealth in one bank.
Asset Markets and Liquidity
EQUILIBRIUM
▶
Banks are profit maximizing and compete to attract
depositors.
▶
Consumers can not trade in the asset market nor in the
forward (good) market.
▶
Ps = R · p s
▶
ps ≤ 1
Asset Markets and Liquidity Simplified Model
NOT COVERED Chapter 4- No Banks,
Let’s assume for now that consumers do their own investment: y in the
short-asset and x in the long asset, x + y = 1.
Behavior in the consumption good market
Obviously their consumption will be lower since there is no risk-pooling:
▶
▶
▶
c1s = y where
s = H,L )
and
(
y
c2s = R ·
+ (1 − y) since in equilibrium R/Ps ≤ 1.
Ps
The consumer takes P = (PH , PL ) as given and chooses y to maximize
the expected utility U (c1H , c2H , c1L , c2L ) =
π · λH · u(c1H ) + π · (1 − λH ) · u(c2H )+
(1 − π) · λL · u(c1L ) + (1 − π) · (1 − λL ) · u(c2L )
Asset Markets and Liquidity Simplified Model
NOT COVERED Chapter 4- No Banks
Behavior in the asset market
In the good market we took as given the price of the assets P = (PH , PL ).
In the asset market, we take as given the consumption decisions and solve
for the asset prices that equate supply and demand.
▶
▶
Ss (Ps ) = (1 − y)λs , supply of long-asset (inelastic, early consumers
sell).

(1 − λs )y


if Ps < R



Ps
Ds (Ps ) = [0, (1 − λs )y if Ps = R , demand of long-asset (late


R


0 if P > R
s
consumers buy as long as price is not too high, Ps ≤ R)
We solve supply equal demand to find Ps for s = H , L.
Asset Markets and Liquidity Model
Chapter 5 - Banks,
With financial itermediaries (banks) consumers are able to consume
more:

d if incentive constraint holds
▶ c1s =
.
y + (1 − y)P otherwise
s


 y + (1 − y)Ps − λs d if incentive constraint holds
(1 − λs ) · ps
▶ c2s =
.

y + (1 − y)Ps otherwise
▶
The bank takes P = (PH , PL ) as given and chooses (d, y) to
maximize the expected utility U (c1H , c2H , c1L , c2L ) =
π · λH · u(c1H ) + π · (1 − λH ) · u(c2H )+
(1 − π) · λL · u(c1L ) + (1 − π) · (1 − λL ) · u(c2L )
Asset Markets and Liquidity Model
Chapter 5 - Banks,
In state s banks will be able to avoid a bank-run if and only if late
consumers lack the incentive to withdraw earlier (at t = 1), this happens
only when
y + (1 − y)Ps ≥ λs d + ps (1 − λs )d, or equivalently
y + (1 − y)Ps ≥ λs d +
Ps
(1 − λs )d
R
When we consider the equilibrium, there are essentially two cases: the
incentive condition always holds in both states (“no default or crises”) and
the incentive condition fails to hold in the “bad” state (s = H ).
Asset Markets and Liquidity, Chapter 5
No Default Scenario
If no bank ever defaults in period 1, then given (PH , PL ), all banks are
maximizing the same function (the expected utility of a depositor). An
important consequence of this fact is that:
All banks will choose the same deposit contract (d, y).
Asset Markets and Liquidity, Chapter 5
No Default Scenario
Behavior in the consumption good market:
This is as before, the banks take PH , PL as given and choose d and y to
maximize the consumer’s expected utility.
Behavior in the asset market:
In state s, banks have to pay d to early consumers and they hold y as cash.
So they have to cover the difference (if positive) by selling some amount of
the long-asset. If they have more cash than they need, they must be willing
to hold cash until t = 2. As the supply of cash is Ss (Ps ) = y and the
demand is Ds (Ps ) = λs · d, we have:
y > λ s · d ⇒ Ps = R
y ≤ λ s · d ⇒ Ps ≤ R
Behavior in the asset market
continuation
Because (in the no default case) banks are choose the same
strategy (d, y). No banks can ever be short of cash at state s
because if one bank has to liquidate (forced to sell) some of the
long asset, then all banks will be selling (and none will be
buying) so the price would fall to Ps = 0. But if the price of the
long-asset is zero in any state, then in period t = 0, any bank
should invest all in the short-asset and buy an infinite amount
of the long-asset in period t = 1 in the state where Ps = 0. But
all banks doing this cannot be part of an equilibrium...
Behavior in the asset market
no default, continuation
So we have that y > λL · d ⇒ PL = R and y = λH · d.
Why y = λH · d? Because if y > λH · d then we would have
excess liquidity in both states but then then bank would be able
to reduce its position in the short-asset without compromising
its ability to pay depositors. Alternatively if y < λH · d then the
bank would need to (partially) divest from the long-asset but
because all banks are using the same strategy this can not
happen as we pointed before.
Now remember that when we solved for y and d in bank
problem we take PH and PL as given so both y and d are
functions of (PH , PL ) = (PH , R). We can use this to find the
value of PH by solving:
y(PH , R) = λH · d(PH , R).
The Optimal Deposit Contract
no default
The expected utility of the representative consumer/depositor
is:
U (d, y) =(π · λH + (1 − π) · λL ) · u (d) +
(
)
y + (1 − y)PH − λH d
+ π · (1 − λH ) · u
+
(1 − λH ) · pH
(
)
y + (1 − y)R − λL d
+ (1 − π) · (1 − λL ) · u
(1 − λL ) · 1
The Optimal Deposit Contract
no default
We know that d = y/λH and PL = R and there is no default, so
we can write the expected utility of the depositor as a function
of y only (of course it also depends of PH but the bank takes it
as given).
(
)
y
y
U(
, y) =(π · λH + (1 − π) · λL ) · u
+
λH
λH
(
)
y + (1 − y)PH − λH λyH
+ π · (1 − λH ) · u
+
(1 − λH ) · pH
(
)
y + (1 − y)R − λL λyH
+ (1 − π) · (1 − λL ) · u
(1 − λL ) · 1
The Optimal Deposit Contract
no default, continuation
Remember that Ps = R · ps so we can simplify:
(
)
y
y
U(
, y) =(π · λH + (1 − π) · λL ) · u
+
λH
λH
)
(
(1 − y) R
+
+ π · (1 − λH ) · u
1 − λH
(
)
(1 − λλHL )y + (1 − y)R
+ (1 − π) · (1 − λL ) · u
(1 − λL )
We now solve
d
dy U
=
∂
∂ y
∂
U·
+
U = 0 for y ...
λ
H
∂d
∂y
∂y
Asset Markets again!
no default, continuation
Notice that we solved for y and d without using the value of
PH ! It cancelled because we assumed d = λyH ...
How should we obtain the equilibrium value of PH then? The
price PH must be such that the bank chooses y = λH · d ...
U (d, y) =(π · λH + (1 − π) · λL ) · u (d) +
(
)
y + (1 − y)PH − λH · d
+ π · (1 − λH ) · u
+
(1 − λH )PH /R
(
)
y + (1 − y)R − λL · d
+ (1 − π) · (1 − λL ) · u
(1 − λL )
Asset Markets - no default, continuation
The price PH must be such that the bank chooses y = λH · d.
∂
U (d, y) = 0+
∂y
(
)
y + (1 − y)PH − λH · d
1 − PH
π · (1 − λH ) · u ′
·
+
(1 − λH )PH /R
(1 − λH )PH /R
(
)
y + (1 − y)R − λL · d
(1 − R
(1 − π) · (1 − λL ) · u ′
·
(1 − λL )
(1 − λL )
Simplifying and using d =
y
λH :
(
)
∂
(1 − y)R
(1 − PH ) · R
+
U (d, y) = π · u ′
·
∂y
(1 − λH )
PH
(
)
(1 − λλHL )y + (1 − y)R
(1 − π) · u ′
· (1 − R) = 0
(1 − λL )
Notice this is a linear equation in PH ! We solve it for PH . Notice that in
the solution we must have PH < 1! This implies the short-asset is not
dominated at t = 0.
At t = 1 and s = H , because PH < 1 < R, banks lose money if they sell the
long-asset so no wants to sell it and also no bank has cash to buy it.
An Example
Let u(x) = ln(x) then solving U ′ (y) = 0 for y gives us:
Asset Markets and Liquidity, Chapter 5
Default Scenario
Remember that under no default, no banks ever sold amounts of the
long-asset. They carried enough cash to pay depositors. But if default
occurs (say in the H state), banks need to sell the long-asset. But in
this case we can not have the all banks in default otherwise PH = 0 (all
want to sell the long-asset and none wants to buy) which is not
compatible with equilibrium. The only way that some banks escape
default is if they use a more conservative strategy (they hold more cash
and promise lower payments at date 1):
▶
▶
▶
▶
▶
safe banks’ choice (y B , d A )
risky banks’ choice (y R , d A )
where d B > y R and d B < d A
ρ is the fraction of risky banks (A=active)
1 − ρ is the fraction of safe banks (B=boring)
Asset Markets and Liquidity, Chapter 5
Default Scenario
We are going to construct examples of equilibrium where:
▶
The risky banks always sell long-asset to the safe banks at
t = 1.
▶
In the state H , there will be no excess liquidity and the
price of the long-asset will be less than one, PH < 1.
▶
In the state H , as the price of the long-asset is too low, the
risky banks are insolvent. They go bankrupt.
The banks’ objectives
( )
U B (d B , y B ) =(π · λH + (1 − π) · λL ) · u d B +
( B
)
y + (1 − y B )PH − λH d
+
+ π · (1 − λH ) · u
(1 − λH ) · PH /R
( B
)
y + (1 − y B )PH − λL d
+ (1 − π) · (1 − λL ) · u
(1 − λL ) · PL /R
( )
U A (d A , y A ) =(1 − π) · λL · u d A +
(
)
+ π · u y A + (1 − y A )PH +
( A
)
y + (1 − y A )PL − λL d A
+ (1 − π) · (1 − λL ) · u
(1 − λL ) · PL /R
Asset Markets and Liquidity, Chapter 5
Default Scenario
Market Clearing Conditions

= ρy A + (1 − ρ)y B if p < 1,
s
A A
B B
ρc1 (s, d , y )+(1−ρ)c1 (s, d , y )
≤ ρy A + (1 − ρ)y B if p = 1
s
ρC (s, d A , y A ) + (1 − ρ)C (s, d, y B ) =ρ(y A + R(1 − y A ))+
(1 − ρ)(y B + R(1 − y B )),
where C = c1 + c2 .
If ps < 1 banks are not willing to carry cash from date 1 into
date 2.
Asset Markets and Liquidity, Chapter 5
Default Scenario
Other Eq. Conditions
▶
Consumers utility of putting money on A or B bank is the
same.