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Advanced Microeconomic TheoryLecture Notes

Sérgio O. Parreiras

Economics Department, UNC at Chapel Hill

Spring, 2015

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 alsop1 + p2 + . . .+ pn = 1.

x1 x2 . . . xn

The Certain Lottery, Expectation and Variance

The lottery that gives outcome x with probability 1 (withcertainty) is denoted:

δx = ((x), (1)) .

The expected value of the ℓ1 = ((x1, x2, . . . , xn), (p1, p2, . . . , pn))

E[ℓ1] = p1 · x1 + p2 · x2 + . . . pn · xn =n∑

pi · xi ;

and variance this lottery is

Var[ℓ1] =p1 · (x1 − E[ℓ1])2 + p2 · (x2 − E[ℓ1])

2 + . . . pn · (xn − E[ℓ1])2 =

pi · (xi − E[ℓ1])2.

Composition of Lotteries

Given two lotteries, ℓ1 = ((x1, x2, . . . , xn), (p1, p2, . . . , pn)) andℓ2 = (y1, y2, . . . , ym), (q1, q2, . . . , qn)) and a number 0 < α < 1,one can create a compound lottery by choosing ℓ1 withprobability 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)).

The compound lottery ℓ plays ℓ1 with probability α and ℓ2 withprobability ℓ2:

1− α

1− p

1− q

ℓ = αℓ1 ⊕ (1− α)ℓ2 =

= ((x1, x2, y1, y2), (αp, α(1− p), (1− α)q, (1− α)(1− q)).

The compound lottery ℓ plays ℓ1 with probability α and ℓ2 withprobability ℓ2:

α · p

α · (1− p)

(1− α) · q(1− α) · (1− q)

ℓ = αℓ1 ⊕ (1− α)ℓ2 =

= ((x1, x2, y1, y2), (αp, α(1− p), (1− α)q, (1− α)(1− q)).

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).

A preference of the DM, ⪰DM, over the set of lotteries isjust the DM’s ranking of lotteries.

We wish (for convenience) a numerical score that reflectsthe 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 thatℓ2 ∼ 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 ifn∑

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 ,

1 utility for money

2 Bernoulli utility

Expected Utility

We write:

U (ℓ1) = u(x1) · p1 + . . .+ u(xn) · pn ,

1 utility for money

2 Bernoulli utility

Expected Utility

Recapitulating so far:1. We made assumptions about the agents’ preferences overlotteries so we can represent his/her preferences by an expectedutility.2. The agent will choose the lottery that delivers the highestexpected utility.

Sometimes, in real life applications, the word "lottery" describesthe flow (variation of wealth) and not the final wealth.

To compute the expected utility of a "lottery" (X , p), whereX = (x1, . . . , xn) and p = (p1, . . . , pn) and xk denotes thewealth flow (income/loss) if the event k happens – we need toknow the value of the initial wealth ω0. In this case theexpected utility is:

U (X , p, ω0) =n∑

u(ω0 + xs) · ps

Remark 1: U (X , p) is the expected utility of lottery (X , p).Remark 2: u(·) is the vN-M utility for the prizes.Remark 3: ω0 is the agent’s initial wealth

Expected Utility

U (X , p, ω0) =n∑

u(ω0 + xs) · ps

Expected Utility

U (X , p, ω0) =n∑

u(ω0 + xs) · ps

Expected Utility

U (X , p, ω0) =n∑

u(ω0 + xs) · ps

Expected Utility

U (X , p, ω0) =n∑

u(ω0 + xs) · ps

Expected Utility

U (X , p, ω0) =n∑

u(ω0 + xs) · ps

Computing Expected UtilityExamples with zero initial wealth (or lottery already gives final wealth).

1 Lotteries: A = ((1000, 0), (1/2, 1/2)) andB = ((500, 0), (1, 0)). vN-M utility: u(x) =

√x. Then

U (A) = 12

√1000 + 1

√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) = x2. ThenU (C ) = 2

316 +130 = 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 . ThenU (A) = 1

221000 + 1

220 = 2999 + 1

2 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). ThenU (C ) = 2

3 ln(4) + 13 ln(0) = −∞ and U (D) = ln(2.66).

√x. Then

U (A) = 12

√1000 + 1

√0 = 5

U (B) = 1 ·√500 =

√500 = 10

√5 ( prob. tree).

316 +130 = 32

221000 + 1

220 = 2999 + 1

2 and U (B) = 2500.

3 ln(4) + 13 ln(0) = −∞ and U (D) = ln(2.66).

√x. Then

U (A) = 12

√1000 + 1

√0 = 5

U (B) = 1 ·√500 =

√500 = 10

√5 ( prob. tree).

316 +130 = 32

221000 + 1

220 = 2999 + 1

2 and U (B) = 2500.

3 ln(4) + 13 ln(0) = −∞ and U (D) = ln(2.66).

√x. Then

U (A) = 12

√1000 + 1

√0 = 5

U (B) = 1 ·√500 =

√500 = 10

√5 ( prob. tree).

316 +130 = 32

221000 + 1

220 = 2999 + 1

2 and U (B) = 2500.

3 ln(4) + 13 ln(0) = −∞ and U (D) = ln(2.66).

Computing Expected UtilityExamples with initial wealth equal to ωs = 8

1 Lottery A = ((1000, 0), (1/2, 1/2)), lotteryB = ((500, 0), (1, 0)), and u(x) =

√x. Then

U (A) = 12

√1000 + 1

√0 = 5

U (B) = 1 ·√500 =

√500 = 10

√5 ( prob. tree).

2 The vN-M utility is u(x) = − exp(−x) and a coin if flippedtwice, 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 herinitial wealth is $ 8. Please see the prob. tree for how tocompute her expected utility.

Computing Expected UtilityExamples with initial wealth equal to ωs = 8

1 Lottery A = ((1000, 0), (1/2, 1/2)), lotteryB = ((500, 0), (1, 0)), and u(x) =

√x. Then

U (A) = 12

√1000 + 1

√0 = 5

U (B) = 1 ·√500 =

√500 = 10

√5 ( prob. tree).

2 The vN-M utility is u(x) = − exp(−x) and a coin if flippedtwice, 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 herinitial wealth is $ 8. Please see the prob. tree for how tocompute her expected utility.

Decision/Probability Tree

Lottery A, initial wealth ω = 0 and u(x) =√

√1000

√10002 +

Lottery B, initial wealth ω = 0 and u(x) =√

√500

Lottery A, initial wealth ω = 8

$1000 + 8

$0 + 8

√1008

√10082 +

Lottery B, initial wealth ω = 8

√508

Decision/Probability TreeTwo coins example, u(x) = − exp(−x).

8− 1 + 10

8− 1 + 5

8− 1− 3

8− 1− 0

− exp(−17)4 +

− exp(−12)4 +

− exp(−7)4 +

− exp(−4)4

E [U (X)] =

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 twolotteries:

ℓ1 = ((100, 200), (1

andδ150 = ((150), (1))

We have

U (ℓ1) = u(150− 50) · 12+ u(150 + 50) · 1

U (δ150) = u(150) · 1.

Risk Aversion

ℓ1 = ((100, 200), (1

andδ150 = ((150), (1))

We have

U (ℓ1) = u(150− 50) · 12+ u(150 + 50) · 1

U (δ150) = u(150) · 1.

Risk Aversion

ℓ1 = ((100, 200), (1

andδ150 = ((150), (1))

We have

U (ℓ1) = u(150− 50) · 12+ u(150 + 50) · 1

U (δ150) = u(150) · 1.

Risk Aversion

ℓ1 = ((100, 200), (1

andδ150 = ((150), (1))

We have

U (ℓ1) = u(150− 50) · 12+ u(150 + 50) · 1

U (δ150) = u(150) · 1.

Risk Aversion

U (ℓ1)− U (δ150) =

[u(200)− u(150)

50− u(150)− u(100)

]· 502

100 150 200

u(100)

u(200)

E[ℓ1]

Risk Aversion

U (ℓ1)− U (δ150) =

u(200)− u(150)50︸︷︷︸

≃Mu(150)

− u(150)− u(100)50︸︷︷︸

≃Mu(100)

· 502

100 150 200

u(100)

u(200)

U (ℓ1)

E[ℓ1]

Risk Aversion

U (ℓ1)− U (δ150) =

u(200)− u(150)50︸︷︷︸

≃Mu(150)

− u(150)− u(100)50︸︷︷︸

≃Mu(100)

· 502

100 150 200

u(100)

u(200)

U (ℓ1)

E[ℓ1]

Risk Aversion

U (ℓ1)− U (δ150) =

u(200)− u(150)50︸︷︷︸

≃Mu(150)

− u(150)− u(100)50︸︷︷︸

≃Mu(100)

· 502

100 150 200

u(100)

u(200)

U (ℓ1)

E[ℓ1]

Risk Aversion

U (ℓ1)− U (δ150) =

u(200)− u(150)50︸︷︷︸

≃Mu(150)

− u(150)− u(100)50︸︷︷︸

≃Mu(100)

· 502

100 150 200

u(100)

u(200)

U (ℓ1)

E[ℓ1]

Expected Utility TheoryAttitudes Towards Risk

1 Diminishing marginal utility, u is concave, u′′ < 0 ⇒, theconsumer is risk-averse.

U (X) < u(E[X ]) for all X

2 Increasing marginal utility, u is convex, u′′ > 0 ⇒, theconsumer is risk-loving.

U (X) > u(E[X ]) for all X

3 Constant marginal utility, u is affine (linear plus aconstant),u′′ = 0 ⇒, the consumer is risk-neutral,

U (X) = u(E[X ]) for all X

Measuring the Degree of Risk-AversionThe Arrow-Pratt or Absolute Measure of Risk Aversion

Definition

The Arrow-Pratt absolute measure of risk-aversion of an agentwith 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 we have thatρu(w) > ρu(w) for all wealth levels w then we say that the agentwith utility u is more risk-averse than the agent with utility u.

Measuring the Degree of Risk-AversionThe Arrow-Pratt or Absolute Measure of Risk Aversion

Definition

The Arrow-Pratt absolute measure of risk-aversion of an agentwith 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 we have thatρu(w) > ρu(w) for all wealth levels w then we say that the agentwith utility u is more risk-averse than the agent with utility u.

Measuring the Degree of Risk-AversionThe 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 withVN-M utility u at wealth level w is:

ru(w) =−u′′(w)w

u′(w).

Mathematics ReviewPartial Derivatives

Often we wish to evaluate the marginal impact of ONE givenvariable on some function of several variables.

MY f =∂

∂yf (x, y, z) = lim

∆→0

f (x , y +∆, z), z − f (x, y, z)∆

We call MY :

1 The marginal change f with respect to y.

2 The partial derivative of f wrt y.

3 The slope of f wrt y.

To compute partial derivatives, we use the exact same rules of derivation ofcalculus with one variable and we treat all the other variables that are notof interest (in the example above, x and z) as constants.

If h(x) = f(g(x)) then

h′(x) = f′(g(x)) · g′(x)

Math. Review The Chain Rule

The chain rule is one of the most useful calculus tricks

we have.

Chain RuleLearning By Doing Exercises

For each of the composite functions below tell us, what are thecorresponding f and g and compute h′.

1 h(x) =√2x.

2 h(x) = − exp(−ρ · x)

3 h(x) = (4 + xσ)1σ .

If k(x) = f (g(h(x))) is a composition of three functions, insteadof two, apply the chain rule twice to compute k ′(x).

Mathematics ReviewTaylor’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 Taylorapproximation:

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 asimilar expression:

f (x + h1, y + h2)− f (x, y) ≃ ∂

∂xf (x, y) · h1 +

∂yf (x, y) · h2

f (x + h)− f (x) ≃ f ′(x) · h

f (x + h1, y + h2)− f (x, y) ≃ ∂

∂xf (x, y) · h1 +

∂yf (x, y) · h2

f (x + h)− f (x) ≃ f ′(x) · h

f (x + h1, y + h2)− f (x, y) ≃ ∂

∂xf (x, y) · h1 +

∂yf (x, y) · h2

∆U = U(x +∆x,y +∆y)−U(x,y)

= MUx ·∆x +MUy ·∆y

Math. Review

Marginal Utility & Taylor’s Approximation

Mathematical ReviewInterior Solutions

Maximizing a function of one variable defined on the real line,f : R → R.

Maximization Problem maxx∈R

f (x) (P)

First order condition f ′(x) = 0 (FOC)

Second order condition f ′′(x) ≤ 0 (SOC)

Any point x satisfying FOC and SOC is a candidate for aninterior solution.

Mathematical ReviewInterior and corner Solutions

Maximizing a function of one variable defined on an interval,f : [a, b] → R. As before,

Maximization Problem maxb≥x≥a

f (x) (P)

First order condition f ′(x) = 0 (FOC)

Second order condition f ′′(x) ≤ 0 (SOC)

Any point x satisfying FOC and SOC is a candidate for aninterior 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 ReviewConcavity and convexity

Consider any function f : Rk → R.

Definition: f is concave if and only if, for all α ∈ [0, 1], andany 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. ReviewGlobal Maxima

Proposition. Assume f is concave and also assume that xsatisfy the FOC then x is a solution to the maximizationproblem (i.e. x is a global maximum).

Math. ReviewThe implicit Function Theorem

Let f (x , y) be a real-valued function of two variables and letg(x) be a real-valued function of one-variable such that if we sety = g(x) we have that f remains a constant when we change x.That is, we have

f (x, g(x)) = c for all x, where c is a constant.

g′(x) = −

∂xf (x , g(x))

∂yf (x, g(x)).

Math. ReviewThe implicit Function Theorem

Proof: Taking the total derivative of f with respect to x,

f (x, g(x)) = ∂

∂xf (x, g(x)) · ∂

∂xx +

∂yf (x, g(x)) · ∂

∂xg(x)

∂xf (x, g(x)) + ∂

∂yf (x, g(x)) · g′(x) ⇒

g′(x) = −

∂xf (x , g(x))

∂yf (x, g(x))

As f is constant along the curve g(x), we call g an iso-curve off . Familiar iso-curves examples are: indifference curves andiso-cost curves.

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.

Key concepts:

1 Present Value

2 Arbitrage

Learning Goals:

Key concepts:

1 Present Value

2 Arbitrage

Learning Goals:

Key concepts:

1 Present Value

2 Arbitrage

Learning Goals:

Key concepts:

1 Present Value

2 Arbitrage

Learning Goals:

Key concepts:

1 Present Value

2 Arbitrage

Learning Goals:

Key concepts:

1 Present Value

2 Arbitrage

Learning Goals:

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=0p0 = π0

p1p2...

t=1π1

t=2π2

t=TπT

Definition: A forward contract is a non-standardized contractbetween two parties to buy or to sell an asset at a specifiedfuture time at a price agreed upon today.

t = 0, 1, . . . ,T periods.

t=0p0 = π0

p1p2...

t=1π1

t=2π2

t=TπT

t = 0, 1, . . . ,T periods.

t=0p0 = π0

p1p2...

t=1π1

t=2π2

t=TπT

t = 0, 1, . . . ,T periods.

t=0p0 = π0

p1p2...

t=1π1

t=2π2

t=TπT

t = 0, 1, . . . ,T periods.

t=0p0 = π0

p1p2...

t=1π1

t=2π2

t=TπT

t = 0, 1, . . . ,T periods.

t=0p0 = π0

p1p2...

t=1π1

t=2π2

t=TπT

t = 0, 1, . . . ,T periods.

t=0p0 = π0

p1p2...

t=1π1

t=2π2

t=TπT

OTC = over the counter

The Relationship between Forward and Spot Prices

In practice, forward contracts are over-the-counter (OTC) -bilateral contracts between two parties that are customized asopposed to standard contracts that are traded in markets.Here, however, we assume forward contracts are traded in acompetitive market. As a result, by arbitrage, we mud have:

pt =πt

(1 + ı)t .

Can you explain why?

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

1 + ı+

I2(1 + ı)2

+ . . .+IT

(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

1 + ı+

I2(1 + ı)2

+ . . .+IT

(1 + ı)T

T∑t=0

It(1 + ı)t (PV)

Inter-temporal Consumption2-Period (T = 2) Consumer Problem

max U (c0, c1)c1, c2subject to

c0 + 1(1+ı)c1 ≤ Y0 +

1(1+ı)Y1

c0 ≥ 0 and c1 ≥ 0

0 Y0 +Y1

1 + ı

(1 + ı)Y0 + Y1

c0 + 1(1+ı)c1 ≤ Y0 +

1(1+ı)Y1

c0 ≥ 0 and c1 ≥ 0

0 Y0 +Y1

1 + ıY0 +Y1

1 + ı

(1 + ı)Y0 + Y1

ı ↗ ı

c0 + 1(1+ı)c1 ≤ Y0 +

1(1+ı)Y1

c0 ≥ 0 and c1 ≥ 0

0 Y0 +Y1

1 + ı

Y0 +Y1

1 + ı

(1 + ı)Y0 + Y1

ı ↘ ı

The idea of arbitrage

The A’s front office realized right away, of course, that theycouldn’t replace Jason Giambi with another first baseman justlike him. There wasn’t another first baseman just like him andif there were they couldn’t have afforded him and in any casethat’s not how they thought about the holes they had to fill."The important thing is not to recreate the individual," BillyBeane would later say. "The important thing is to recreate theaggregate." He couldn’t and wouldn’t find another JasonGiambi; but he could find the pieces of Giambi he could leastafford to be without, and buy them for a tiny fraction of thecost of Giambi himself. – Moneyball by Micheal Lewis, p. 103

The idea of arbitragecontinuation

The A’s front office had broken down Giambi into his obviousoffensive statistics: walks, singles, doubles, home runs alongwith his less obvious ones: pitches seen per plate appearance,walk to strikeout ratio and asked: which can we afford toreplace? And they realized that they could afford, in aroundabout way, to replace his most critical offensive trait, hison-base percentage, along with several less obvious ones. Theprevious season Giambi’s on-base percentage had been .477, thehighest in the American League by 50 points. (Seattle’s EdgarMartinez had been second at .423; the average AmericanLeague on-base percentage was .334.) There was no one playerwho got on base half the time he came to bat that the A’s couldafford; – Moneyball by Micheal Lewis, p. 103

The idea of arbitragecontinuation

on the other hand, Jason Giambi wasn’t the only player in theOakland A’s lineup who needed replacing. Johnny Damon(onbase percentage .324) was gone from center field, and thedesignated hitter Olmedo Saenz (.291) was headed for thebench. The average on-base percentage of those three players(.364) was what Billy and Paul had set out to replace. Theywent looking for three players who could play, between them,first base, outfield, and DH, and who shared an ability to get onbase at a rate thirty points higher than the average big leagueplayer. – Moneyball by Micheal Lewis, p. 103

Understanding Present ValueArbitrage

What is the value today of Y1 dollars tomorrow?

1 What should we do if someone else thinks that x dollarstomorrow are worth less than x

1 + ıtoday?

2 Or alternatively, believes x dollars tomorrow are worthmore than x

1 + ıdollars today?

Understanding The Solution to the Consumer Problem

MRIS ≡ MU0

MU1= 1 + ı

(1 + ı)c1 = PV(Y0,Y1)

To understand the first equation: MRIS is how many units ofconsumption tomorrow are equal to one unit of consumptiontoday for the consumer and 1 + ı is how many units ofconsumption tomorrow are equal to one unit of consumptiontoday for the market. In equilibrium they ought to be equal.

The second equation just says the consumer expends herincome during her lifetime.

Financial Instruments:Options

A stock option is an option (not an obligation) to buy (calloption) or to sell (put option) some specified number of sharesof the stock at price (per-share) K (the strike price) at expiredate T (European option) or, alternatively at any point in timebefore T (American option).

Example: Two periods: at t = 0 the price of the stock is 27 andat t = 1 the it is 28 with prob. 2

3 or 26 with prob. 13 . The

option is an European (you can use it only at the expire dateT = 1) call (gives you the right to buy 1 stock) option withstrike 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 Examplecontinuation

The payoffs associated to this call option are:

$− 26.5 + 28 = 1.5

If a DM with utility for money u and initial wealth $36 buysthe option paying P at t = 0, his/her expected utility is:

$36− P − 26.5 + 28 = 37.5− Pt = 1

36− Pt = 1

u(37.5− P)

u(36− P)

2u(37.5−P)3 + u(36−P)

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)

If P = 0 then U (not buy) = u(36) < U (buy) = 2u(37.5)3 + u(36)

3 .If P = 1.5 then U (not buy) = u(36) > U (buy) = 2u(36)

3 + u(35.5)3 .

As P ↗ we have U (buy) ↘ and U (not buy) =cte.

There exists Pmax such that U (not buy) = U (buy).

1 u(x) = x =⇒ Pmax = 1.

2 u(x) = x2 =⇒ Pmax = 1.0069. Making the DM indifferent,we get 73.5− 74P + P2 = 0 soP =

74−√

742−4(73.5)

2 ≃ 1.0069.

3 u(x) =√

x =⇒ Pmax = 0.9965. Making the DM indifferent,we get 2

√36− P + 3

2 +√36− P = 3 · 6. The "trick" is to

call y =√36− P and remove the square root in

y2 − 32 + y = 18 to get a quadratic equation in y. We

solve it for y and set P = 36− y2.

Now there is a bond that costs $1 at t = 0 and pays 1 + ı att = 1. If we buy/sell s shares of the stock and b bonds suchthat:

1 + ı

It must be that 2s = 1.5 so s = 0.75 and b = −0.75·26

1+ı s As aresult we can figure out the price of the call:

P = 0.75 · 27− 0.75 · 261 + 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 selectioncontinuation...

The problem of the investor is to choose α ∈ [0, 1] to maximizeher expected utility:

U (α) = p · u ((α(1 + ı) + (1− α)RH )w0)+

+(1− p) · u ((α(1 + ı) + (1− α)RL)w0) .

U (α) = p · u ((α(1 + ı) + (1− α)RH )w0)+

+(1− p) · u ((α(1 + ı) + (1− α)RL)w0) .

wealth whenstate H happens

U (α) = p · u ((α(1 + ı) + (1− α)RH )w0)+

+(1− p) · u ((α(1 + ı) + (1− α)RL)w0) .

wealth whenstate L happens

The corresponding first-order condition for a maximum is:

U ′(α) = 0 or equivalently,

p · u′ ((α(1 + ı) + (1− α)RH )w0)︸︷︷︸expected MU

·(1 + ı− RH ) · w0

︸︷︷︸MC of riskless asset

(1− p) · u′ ((α(1 + ı) + (1− α)RL)w0)︸︷︷︸expected MU

) · (1 + ı− RL)w0

︸︷︷︸MB of riskless asset

Intertemporal Choice (recap.)

max u(c0) + δu(c1)c1, c2subject to

c0 + 1(1+ı)c1 ≤ Y0 +

1(1+ı)Y1

c0 ≥ 0 and c1 ≥ 0

Indiference curve with utility uc1(c0) = u−1 (u − u(c0)/δ)

c′1(c0) = −MRIS

0 Y0 +Y1

1 + ı

(1 + ı)Y0 + Y1

U (c0, c1) = 4

c0 + 1(1+ı)c1 ≤ Y0 +

1(1+ı)Y1

c0 ≥ 0 and c1 ≥ 0

c′1(c0) = −MRIS

0 Y0 +Y1

1 + ı

(1 + ı)Y0 + Y1

U (c0, c1) = 5

c0 + 1(1+ı)c1 ≤ Y0 +

1(1+ı)Y1

c0 ≥ 0 and c1 ≥ 0

c′1(c0) = −MRIS

0 Y0 +Y1

1 + ı

(1 + ı)Y0 + Y1

U (c0, c1) = 6

c0 + 1(1+ı)c1 ≤ Y0 +

1(1+ı)Y1

c0 ≥ 0 and c1 ≥ 0

c′1(c0) = −MRIS

0 Y0 +Y1

1 + ı

(1 + ı)Y0 + Y1

U (c0, c1) = 5.9

General Equilibrium

Consider a 2× 2 pure trade economy (2 consumers, A & B; 2goods c1 & c2, no firms).Endowment The initial basket that a consumer owns:

eA = (eA1 , eA

2 ) and eB = (eB1 , eB

2 ).Feasible Allocation Division of the total endowment between

the consumers: (cA, cB) such thatcA1 + cB

1 = eA1 + eB

1 and cA2 + cB

2 = eA2 + eB

2 .Efficient Allocation An allocation where trade can never

benefit all consumers: MRSA = MRSB in the casethat cA > 0 and cB > 0 and MRSs are decreasing.

Individual Rational Allocations Given consumers higher orequal utility than if they just consume theirendowments.

Contract Curve Set of allocations that are individuallyrational and efficient.

General Equilibrium

eA = (eA1 , eA

1 = eA1 + eB

1 and cA2 + cB

2 = eA2 + eB

General Equilibrium

eA = (eA1 , eA

1 = eA1 + eB

1 and cA2 + cB

2 = eA2 + eB

General Equilibrium

eA = (eA1 , eA

1 = eA1 + eB

1 and cA2 + cB

2 = eA2 + eB

General Equilibrium

eA = (eA1 , eA

1 = eA1 + eB

1 and cA2 + cB

2 = eA2 + eB

General Equilibrium

Efficient Allocation An allocation where trade can neverbenefit all consumers: MRSA = MRSB in the casethat cA > 0 and cB > 0 and MRSs are decreasing.

General Equilibrium

General EquilibriumcA1

endowment

A’s endowment of c0

endowment

B’s endowment of c0

endowment

utility of Abetter thanendowment

utility of Bbetter thanendowment

better

MRSA = MRSB

efficient

alloca-

MRSA = MRSB

efficient

alloca-

efficiency:

indiff.

curves are

tangent

MRSA = MRSB

efficient

alloca-

efficiency:

indiff.

curves are

tangent

MRSA = MRSB

efficient

alloca-

efficiency:

indiff.

curves are

tangent

efficient

alloca-

tionsMRSA = MRSB

efficient

alloca-

MRSA = MRSB

efficient

alloca-

MRSA = MRSB

efficient

alloca-

MRSA = MRSB

efficient

alloca-

MRSA = MRSB

efficient

alloca-

MRSA = MRSB

efficient

alloca-

MRSA = MRSB

efficient

alloca-

MRSA = MRSB

efficient

alloca-

MRSA = MRSB

General Equilibrium: Time and UncertaintyArrow-Debreu goods

Definition: An Arrow-Debreu good is defined by (at least) 4dimensions:

1 Its physical properties and qualities.2 The geographic location where it is available.3 The time when it is available for consumption.4 The state of the world where it is available for

consumption.

Say there are K distinct physical attributes, L locations, Ttime periods and S states.

The the total number of goods is n = K · L · T · S . That is, wehave n prices and n markets.

A bundle or basket of goods is a vector with n entries.

UncertaintyArrow-Debreu goods, an example

Assume that:K ∈ {umbrella,parasol},K ∈ {Hillsborough,Chicago},T ∈ {today,tomorrow}, andS ∈ {sun,rain}.

Then we have 16 goods! 16 markets! 16 prices!For instance, the first good is an umbrella in Hillsboroughavailable today provided today is a sunny day, the second goodis an umbrella in Hillsborough available today if it is rainy, thethird good is is an umbrella in Hillsborough available tomorrowif tomorrow is sunny, ..., the last good is a parasol availabletomorrow in Chicago if it rains.

Remark: the order in which one may label the goods is arbitrary, but wemust select some order and fix it.

Complete Markets

When all the markets for Arrow-Debreu goods exists we call itthe case of complete markets.

The assumption of complete markets may appear too extremeand unrealistic, however there are more markets out there thanmeets the eye:

1 Weather Markets

2 A weather contract

3 Events Markets

Complete Markets

1 Weather Markets

3 Events Markets

Complete Markets

1 Weather Markets

3 Events Markets

Complete Markets

1 Weather Markets

3 Events Markets

General Equilibrium

1 General v. Partial Equilibrium.Assume the demand for a good produce in a competitivemarket increases. What are the effects of that?

2 The consumer problem revisited.What is another name for the optimal basket/bundle ofgoods chosen by a given consumer?What is income? How do we calculate it?

General Equilibrium

General EquilibriumConsumers’ Budget Constraint

Income is the value of the consumer endowment. It mayinclude:

1 Value of number of hours of labor sold (wages = wage rate× hours).

2 Value of the amount of capital lend (interest rate ×capital).

3 Share of the profits of firms.

General Equilibrium

An equilibrium is a vector of prices p such that:

1 Consumers maximize their utility ⇒ individual demands.

2 Firms maximize their profits ⇒ supply output and inputdemand.

3 Markets clear: for any good j, either Dj(p) = Sj(p) orSj(p) > Dj(p) and pj = 0.

General Equilibrium

GE: Example 1One consumer (Chuck), two goods (X and Y) and no firms.

Chuck’s preferences are: u(x, y) = ln(x) + 2 ln(y); hisendowment is: e = (3, 4); and so his budget is:pX · x + pY · y ≤ pX · 3 + pY · 4 .

Solving for his optimal basket, we find his demand functions:

X(px , py) =pX · 3 + pY · 4

3pxand Y (px , py) =

3· pX · 3 + pY · 4

Market demand of good X is DX (pX , pY ) =pX · 3 + pY · 4

which coincides with Chuck’s demand, and the market supplyof good X, which coincides with Chuck’s endowment of X, isSX (pX , pY ) = 3 (perfectly inelastic). Equilibrium prices mustclear the market:

DX (pX , pY ) = SX (pX , pY ) ⇔ pX · 3 + pY · 43pX

= 3 ⇒ pXpY

GE: Example 2Two consumers (Chuck & Bob), two goods (X and Y) and no firms.

Chuck’s preference and endowments are as in example 1, so hisdemand remains the same. Bob’s preferences areuB(x, y) = 2 ln(x) + ln(y); his endowment is eB = (1, 1); so hisbudget is: pX · x + pY · y ≤ pX · 1 + pY · 1 . Solving for Bob’soptimal basket, we find his demand functions:

XB(px , py) =2

pX + pYpx

and Y B(px , py) =pX + pY3pY

The market demand of good X is the sum of Chuck’s and Bob’sdemand for X:

DX (pX , pY ) =3pX + 4pY

pX + pYpX

=5pX + 6pY

GE: Example 2continuation

We found the market (aggregated) demand for X was:

DX (pX , pY ) =3pX + 4pY

pX + pYpX

=5pX + 6pY

The total supply of X is the amount of X available in theeconomy (sum of the amounts of X consumers were endowedwith):

SX (pX , pY ) = eCX + eB

X = 3 + 1 = 4 .

For prices to clear the market:

5pX + 6pY3pX

= 4 ⇒ pXpY

GE: comparing the economies of examples 1 & 2

1 What happened with the supply of good X when withmoved to example 2?

2 What would have been the market demand in example 2 atthe equilibrium prices of example 1?

3 If the quantity demanded was as in item 2 above, in theeconomy of example 2, the market for good X would haveexcess demand or excess supply?

4 What happened with the equilibrium price (ratio) whenyou moved from example 1 to example 2? Increased?Decreased? Remained constant? In economy 2 is Xcheaper? or more expensive?

5 Can you explain (please try): Why the prices moved in thisdirection?

GE: Example 3, inter-temporal consumption

Assume two consumers (Telma & Louise) with utilities forpresent and future consumption give by:

uT (c0, c1) = ln(c0)+9

10ln(c1) and uL(c0, c1) = ln(c0)+

10ln(c1)

Assume their endowments are

eT = (6, 6) and eL = (3, 3).

1 Write down T and L’s respective budget constraints.2 Find the demand function of T and of L for present

consumption (c0).3 Find the market demand for present consumption (c0).4 Find the total supply for present consumption.5 Find the equilibrium price (ratio): p0

GE: Example 4Production

Consider an economy with 3 periods t = 0, 1 and 2. There isone consumer whose (ex-post) utility is

U (c) = s ·√

c1 + (1− s)√

where c1 is the amount of consumption in period 1 and c2 is theamount of consumption in period 2 (he does not valueconsumption in period zero).

At t = 0, the value of the state s variable is not known but theprobabilities Pr[s = 0] = p and Pr[s = 1] = 1− p arecommon-knowledge.At t = 1, the realization (value) of s is publicly observed.

t = 1s = 0

t = 1s = 1

t = 2s = 0

1− p

The consumer is endowed with: one unit of consumption att = 0; nothing at t = 1, 2; and ownership of firms S and L.

Firms operate in competitive markets (price takers).

Firm S uses qS units of consumption good as input att = 0 to produce qS units of consumption good at t = 1.

Firm L uses qL units of consumption good at t = 0 toproduce R · q units of output at t = 2.

ES [UD(c)] = (1− p) · √c1 + p · √c2The are 5 goods in this economy:

1 consumption contigent on t = 0, c02 consumption contingent on t = 1 and s = 0, c1,0.

3 consumption contingent on t = 1 and s = 1, c1,1.

With this notation, expected utility becomes

(1− p) · √c1,1 + p · √c2,0.

The budget constraint is:

p0c0 + p1,0c1,0 + p1,1c1,1 + p2,0c2,0 + p2,1c2,1 ≤

≤ p0 · 1 + πS + πL

πS = p1,0qS + p1,1qS − p0qS

πL = p2,0RqL + p2,1RqL − p0qL

where πS and πL are the profits of the firms S and L.

MRS(1,1),(2,0) =

(1− p)2√c1,1

p2√c2,0

=p1,1p2,0

π′S = p1,0qS + p1,1qS − p0qS ⇒ p1,0 + p1,1 − p0 = 0.

π′L = p2,0RqL + p2,1RqL − p0qL ⇒ R(p2,0 + p2,1)− p0 = 0.

Market clearing conditions:

c0 + qS + qL = 1,

c1,0 = qS ,

c1,1 = qS ,

c2,0 = RqL,

c2,1 = RqL.

The Consumer Problem under Complete MarketsTwo States and One Good

maxcL,cH

st.pL cL+pH cH≤pL YL+pH YH

πL u(cL) + πH u(cH ).

L(cL, cH , λ) = πL u(cL) + πH u(cH )− λ (pL YL + pH YH − pL cL − pH cH )

∂cLL(cL, cH , λ) = πL u′(cL) + λ pL = 0 (FOCcL)

∂cHL(cL, cH , λ) = πH u′(cU ) + λ pH = 0 (FOCcH )

∂λL(cL, cH , λ) = pL YL + pH YH − pL cL − pH cH = 0 (FOCλ)

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 MarketsTwo States, Two Consumers and One Good

• Consumers A and B with expected utilities:πL uA(cL) + πH uA(cH ) and πL uB(cL) + πH uB(cH ).• Their endowments are given, Y A = (Y A

H ,Y AL ) and Y B = (Y B

H ,Y BL ).

• Solve the consumer problem to find their individual demands (seeprevious page). Notice cA

L and cAH depend only the probabilities, the

prices and A’s endowment and likewise, cBL and cB

H depend only theprobabilities, the prices and B’s endowment.• To find the price ratio:⇒ Equate total supply with total demand,

Y AH + Y B

H︸︷︷︸supply

= cAH (pH , pL) + cB

H (pH , pL)︸︷︷︸demand

we solved for it previously

• Important: we can always normalize one price to one. If we setpH = 1 and then solve the above equation for pL then we actually getthe value of pL

• Important: See Mathematica file on General Equilibrium.

UncertaintyArrow Securities

But even if markets are not complete we can “complete” themissing markets if we have Arrow securities.

An Arrow-security is a financial instrument that pays $1 unit ofaccounting in a given location, date t, and state and it is tradedin a market at date t − 1. Thus at each point in time we needonly L · S markets.

Also even if we do not have Arrow securities we can completethe markets if we have enough financial instruments (as we didin 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

maxcL,cH

st.pL cL+pH cH≤pL YL+pH YH

U (cL, cH ).

and with Arrow-securities is

maxcL,cH ,zL,zH

st.qL zL+qH zH≤0

pL cL≤pL YL+zL

pH cH≤pH YH+zH

U (cL, cH ).

Financial Market Eq. with Arrow Securities

Only two states, s ∈ {L,H}, and consumption takes place only atdate T = 1. But consumer makes decisions and markets operate atdate T = 0. The consumer problem with with Arrow-securities is:

maxcL,cH ,zL,zH

st.qL zL+qH zH≤0

pL cL≤pL YL+zL

pH cH≤pH YH+zH

U (cL, cH ).

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 under uncertaintywith Arrow securities

maxcL,cHst.

qL zL+qH zH≤0pL cL≤pL YL+zL

pH cH≤pH 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 (pL cL − pL YL − zL)− λ3 (pH cH − pH YH − zH )

∂cLL(cL, cH , λ) = πL u′(cL) + λ2 pL = 0 (FOCcL)

∂cHL(cL, cH , λ) = πH u′(cU ) + λ3 pH = 0 (FOCcH )

∂λ1L(cL, cH , λ) = qL zL + qH zH = 0 (FOCλ1)

∂λ2L(cL, cH , λ) = pL cL − pL YL − zL = 0 (FOCλ2)

∂λ3L(cL, cH , λ) = pH cH − pH YH − zH = 0 (FOCλ3)

Risk-Sharing

Let’s assume:two consumers (A and B)complete markets (with Arrow-securities we will obtainidentical results).total endowment constant across states,

Y = Y AL + Y B

L and Y = Y AH + Y B

From the first-order condition, we have that:

u′A(cA

u′B(cB

H )⇒ cA

L > cAH ⇔ cB

L > cBH

u′A(cA

u′B(Y − cA

H )⇒ cA

L > cAH ⇔ Y − cA

L > Y − cAH

⇔ cAL < cA

H But this is a contradiction !

Portfolio Choice

maxθ,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 − θRH W0 − (1− θ)W0)

∂θL(θ, cL, cH , λ1, λ2) = λ1 (RL − 1)W0 + λ2 (RH1)W0 = 0

(FOCθ)∂

∂cLL(θ, cL, cH , λ1, λ2) = πL u′(cL)− λ1 = 0 (FOCcL)

∂cHL(θ, cL, cH , λ1, λ2) = πH u′(cH ) + λ2 = 0 (FOCcH )

∂λ1L(θ, cL, cH , λ1, λ2) = cL − θ RL W0 − (1− θ)W0 = 0

(FOCλ1)∂

∂λ2L(θ, cL, cH , λ1, λ2) = cH − θRH W0 − (1− θ)W0 = 0

(FOCλ2)

Asset Pricing

If we have complete-markets and we know the equilibrium pricevector, p = (ps). We can price ANY financial asset/security. Afinancial 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 PricingExample

Say we have one state today and two tomorrow,S = {s0, s1H , s2L} and the price of delivery of one unit ofconsumption if the state s happens is p0, p1H and p2H for therespective states.

1 A financial security that always pays 1 in period 1 (arisk-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 andnothing 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 atdate 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.

maxβst.

0≤β≤1

π1u (β) + π2u (β + (1− β)R) (2.1)

Liquidity ShocksAn 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. 1

2 consumption takes place only at date 1.The fraction of wealth in short asset is β.

maxβst.

0≤β≤1

2log (β 10) +

2log (β 10 + (1− β)40)

Liquidity ShocksAn Example

maxβst.

0≤β≤1

2log (β 10) +

2log (β 10 + (1− β)40)

2 (β 10)+

10− 40

2 (β 10 + (1− β)40)= 0 (FOC)

Liquiditywith risk-pooling

Two agents, i = 1, 2.

Initial individual wealth, W i0 = 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 aliquidity shock if the other agent did not suffer a liquidityshock.

Liquiditywith risk-pooling (continuation)

If agents do not pool their resources, each agent choose its ownβ to maximize:

maxβst.

0≤β≤1

πu (r β ω) + (1− π)u(r2 β ω + (1− β)R ω

)(INDIVIDUAL)

If agents pool their resources, each agent choose its own β tomaximize:

maxβst.

0≤β≤10≤d≤2r βω

π2u(r β ω)+π(1−π) u(d)++(1−π)π u(r(r β 2ω−d)+(1−β)R 2ω)+

+(1−π)2u(r2 β ω+(1−β)R ω)(POOL)

To understand the payoffs in the previous page, let’s look at thefollowing tables:

Without pooling:State Probability Agent 1’s consumption

All suffer the shock π2 r β ω

Only 1 suffers π(1− π) r β ω

Only 2 suffers (1− π)π r2 β ω + R (1− β)ω

None suffers (1− π)2 r2 β ω + R (1− β)ω

With pooling:State Probability Agent 1’s consumption

All suffer the shock π2 β ω

Only 1 suffers π(1− π) dOnly 2 suffers (1− π)π r (r β 2ω − d) + R (1− β)2ω

None suffers (1− π)2 r2 β ω + R (1− β)ω

Remark 1: The agents are always better-off by pooling theirresources. The optimal (β∗, d∗) that solves the maximizationproblem (POOL) always delivers a higher utility than the β∗∗ thatsolves the maximization problem (INDIVIDUAL).

Remark 2: For risk-pooling to occur is crucial that not all agentsreceive liquidity shocks at the same time. If they face aggregateuncertainty, 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 andsame 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 whosuffer 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 arefeasible if π1c1 = β and π2c2 = (1− π1)c2 = (1− β)R.

maxβst.

0≤β≤1π1c1=β

π2c2=(1−β)R

π1u (c1) + (1− π1)u (c2) (2.2)

Law of Large Numberseliminating uncertainty thru averages, side comment

We saw before that under risk-pooling, the company promisesc1 = 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 agentswho do suffer a liquidity shock, N , is less or equal than theaverage number of consumers who suffer a liquidity shock π1 N .

• However, if N > π1 N , the company can not honor itspromises.

Assume that instead of promising the maximum possible,c1 = β/π1 for some small ε > 0, the company promises to payc1 = (1− ε)c1.

What is the probability that the company can honor itspromises?

N · c1 ≤ N · β︸︷︷︸total amount of short-asset

N · (1− ε)β/π1 ≤ N · β︸︷︷︸total amount of short-asset

[N ≤ π1N

(1− ε π1)

[π1N(1−ε)

]∑k=0

k!(N − k)!πk(1− π1)

where [x] is the floor function, it gives the largest integer below x.

What is the probability that the company can honor itspromises?

Pr[N ≤ π1N

(1− ε)

[π1N(1−ε)

]∑k=0

k!(N − k)!πk(1− π1)

Assume π = 13 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-assetpages 60–64 in the textbook

The model is the same as before in the risk-pooling case withthe exception that consumers investment decisions are againindividual 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.

maxx,yx+y≤1

λu(y + Px) + (1− λ)u((x +

In eq. we have thatP = 1 ⇒ c1 = 1& c2 = R ⇒ u∗ = λu(1) + (1− λ)u (R)

Bank Runspages 72-76

A deposit contract is a pair of consumption promises(c1, c2) such that if a consumer deposits his entire wealthW0 = 1 at the bank at t = 1, the consumer has the right towithdraw c1 at date 1 and c2 at date 2.

The optimal deposit contract is the one that solves therisk-pooling problem.

Liquidation technology: long asset is worth r ≤ 1 units ofthe good at date t = 1

If c1 > rx + y the bank will no be able to honor the depositcontract if all consumers ‘run’.

If c2 > c1 > rx + y we may or may not have a bank-run (itdepends on the consumers/investors) expectations.

Bank Runsas an equilibrium, pages 76–82

In the previous analysis, we assumed that the deposit contractwas the same regardless if the Bank expected a bank run or not.But if the Bank expects a bank-run with certainty, the bank willoffer a deposit contract (c1, c2) = (1, 1) and so late consumersare indifferent between withdrawing at t = 1 or at t = 2.

Bank Runscont.

To simplify the model, we shall assume that the liquidationtechnology is perfect: that is r = 1 so there is no penalty inliquidating the long-asset at t = 1. The bank can transform thelong-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 isno reason for the bank to hold the short-asset. If the bankneeds cash to pay depositors at t = 1 the bank can justliquidate the long-asset.

Bank Runscont.

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 thewithdraws which amount: λ · N · c1 where c1 is the amountpromised by the deposit contract and λ is the fraction ofearly consumers.At t = 2 , the total wealth of the bank is R ·N · (W0 − λ c1)which must cover the withdraws by late consumers whichamount to (1− λ) · N · c2.Assume for simplicity that W0 = 1 and so the budgetconstraint for any feasible deposit contract must be:

0 ≤ c1 ≤1

λand 0 ≤ c2 ≤

R(1− λ c1)1− λ

Notice that N cancels out so this is the budget constraint percapita.

Avoiding a run

Important remark: If c1 ≤ 1 the bank will be solvent to payany late consumers at t = 2 even if a bank-run takes place. Soif c1 ≤ 1 and c1 < c2 no late consumer will ever want to join abank-run.

Thus if the bank wants to avoid a run, it can choose c1 ≤ 1. Inthis case, the bank maximizes:

max0≤c1≤1

0≤c2≤R(1−λ c1)1−λ

λu(c1) + (1− λ)u(c2) (PNR)

To solve this we substitute R(1−λ c1)1−λ for c2 and take derivative

with respect to c1, equate to zero, and solve for c1. If thesolution satisfies c1 ≤ 1 we are done but if the ‘solution’ hasc1 > 1 then we are violating the constraint c1 ≤ 1. In this casethe true solution must be c1 = 1 and soc2 = R(1−λ c1)

1−λ = R(1−λ)1−λ = R.

Not avoiding a run

Assume that in the previous problem PNR the optimal depositcontract that avoids a run is (1,R) that is, ideally the bankwould like to have c1 > 1 but that may cause a run so the banksets c1 = 1. In this event, what will be the optimal depositcontract if the bank does not avoid a run. In the case the bankexpects a run with probability π, the bank maximizes:

max0≤c1

0≤c2≤R(1−λ c1)1−λ

πu(1) + (1− π) [λu(c1) + (1− λ)u(c2)] (PN )

To solve this we again substitute R(1−λ c1)1−λ for c2 and take

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 optimaldeposit contract when the bak wants to avoid a run and when itdoes not. The overall optimal deposit contract is the one thatyields the greatest utility. The bank will avoid a run if and onlyif:

λu(1) + (1− λ)u(R) = max0≤c1≤1

0≤c2≤R(1−λ c1)1−λ

λu(c1) + (1− λ)u(c2) >

> max0≤c1

0≤c2≤R(1−λ c1)1−λ

πu(1) + (1− π) [λu(c1) + (1− λ)u(c2)] =

= πu(1) + (1− π) max0≤c1

0≤c2≤R(1−λ c1)1−λ

λu(c1) + (1− λ)u(c2)

that is, only if π is sufficiently large.

Asset Markets and LiquidityModel Set-Up

The model is similar the the ones we saw before. But now thereis uncertainty regarding the fraction of the population whosuffers a liquidity shock: it could be high or low, λH (in state Hthat happens with prob. π) or λL (in state L that happens withprob. 1− π) .

Dates: t = 0, 1, 2. States: s = H ,L.At date 0: consumers make deposit decisions and banksoffer deposit contracts and then make portfolio decisions(how much to invest in the short-asset and how much toinvest in the long-asset).At the start of date 1: all learn what is the current stateand mkts. open for trade.At t = 1 there are two markets: the good market (where c2is exchanged for c1) and the asset market (where thelong-asset is exchanged for the short-asset).At date 2, the long-asset yields a payoff (rate of returnR > 1).

Asset Markets and LiquidityModel Set-Up (cont.)

Let ps be the price of c2 (in terms of c1) and Ps the price ofthe long-asset (in terms of the short-asset). (Notice we canalways convert c1 into the short-asset and vice-versa).Banks are profit maximizing and they compete to attractdepositors.At t = 1, consumers can not trade in the asset market norin the forward good market.At t = 1, banks can trade in both markets.Consumers place all their wealth in one bank.

Asset Markets and LiquidityEQUILIBRIUM

Banks are profit maximizing and compete to attractdepositors.

Consumers can not trade in the asset market nor in theforward (good) market.

Ps = R · ps

ps ≤ 1

Asset Markets and Liquidity Simplified ModelNOT COVERED Chapter 4- No Banks,

Let’s assume for now that consumers do their own investment: y in theshort-asset and x in the long asset, x + y = 1.

Behavior in the consumption good marketObviously their consumption will be lower since there is no risk-pooling:

c1s = y where s = H ,L and

c2s = R ·(

+ (1− y))

since in equilibrium R/Ps ≤ 1.

The consumer takes P = (PH ,PL) as given and chooses y to maximizethe 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 ModelNOT COVERED Chapter 4- No Banks

Behavior in the asset marketIn 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 solvefor the asset prices that equate supply and demand.

Ss(Ps) = (1− y)λs, supply of long-asset (inelastic, early consumerssell).

Ds(Ps) =

(1− λs)y

Psif Ps < R

[0,(1− λs)y

Rif Ps = R

0 if Ps > R

, demand of long-asset (late

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 ModelChapter 5 - Banks,

With financial itermediaries (banks) consumers are able to consumemore:

d if incentive constraint holds

y + (1− y)Ps otherwise.

y + (1− y)Ps − λs d

(1− λs) · psif incentive constraint holds

y + (1− y)Ps otherwise.

The bank takes P = (PH ,PL) as given and chooses (d, y) tomaximize 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 ModelChapter 5 - Banks,

In state s banks will be able to avoid a bank-run if and only if lateconsumers lack the incentive to withdraw earlier (at t = 1), this happensonly when

y + (1− y)Ps ≥ λs d + ps(1− λs)d, or equivalently

y + (1− y)Ps ≥ λs d +PsR

(1− λs)d

When we consider the equilibrium, there are essentially two cases: theincentive condition always holds in both states (“no default or crises”) andthe incentive condition fails to hold in the “bad” state (s = H ).

Asset Markets and Liquidity, Chapter 5No Default Scenario

If no bank ever defaults in period 1, then given (PH ,PL), all banks aremaximizing the same function (the expected utility of a depositor). Animportant consequence of this fact is that:

All banks will choose the same deposit contract (d, y).

Asset Markets and Liquidity, Chapter 5No Default Scenario

Behavior in the consumption good market:This is as before, the banks take PH ,PL as given and choose d and y tomaximize 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 ofthe long-asset. If they have more cash than they need, they must be willingto hold cash until t = 2. As the supply of cash is Ss(Ps) = y and thedemand is Ds(Ps) = λs · d, we have:

y > λs · d ⇒ Ps = R

y ≤ λs · d ⇒ Ps ≤ R

Behavior in the asset marketcontinuation

Because (in the no default case) banks are choose the samestrategy (d, y). No banks can ever be short of cash at state sbecause if one bank has to liquidate (forced to sell) some of thelong asset, then all banks will be selling (and none will bebuying) so the price would fall to Ps = 0. But if the price of thelong-asset is zero in any state, then in period t = 0, any bankshould invest all in the short-asset and buy an infinite amountof the long-asset in period t = 1 in the state where Ps = 0. Butall banks doing this cannot be part of an equilibrium...

Behavior in the asset marketno 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 haveexcess liquidity in both states but then then bank would be ableto reduce its position in the short-asset without compromisingits ability to pay depositors. Alternatively if y < λH · d then thebank would need to (partially) divest from the long-asset butbecause all banks are using the same strategy this can nothappen as we pointed before.Now remember that when we solved for y and d in bankproblem we take PH and PL as given so both y and d arefunctions of (PH ,PL) = (PH ,R). We can use this to find thevalue of PH by solving:

y(PH ,R) = λH · d(PH ,R).

The Optimal Deposit Contractno default

The expected utility of the representative consumer/depositoris:

U (d, y) =(π · λH + (1− π) · λL) · u (d)+

+ π · (1− λH ) · u(

y + (1− y)PH − λH d(1− λH ) · pH

+ (1− π) · (1− λL) · u(

y + (1− y)R − λL d(1− λL) · 1

The Optimal Deposit Contractno default

We know that d = y/λH and PL = R and there is no default, sowe can write the expected utility of the depositor as a functionof y only (of course it also depends of PH but the bank takes itas given).

U (yλH

, y) =(π · λH + (1− π) · λL) · u(

+ π · (1− λH ) · u

(y + (1− y)PH − λH

(1− λH ) · pH

+ (1− π) · (1− λL) · u

(y + (1− y)R − λL

(1− λL) · 1

The Optimal Deposit Contractno default, continuation

Remember that Ps = R · ps so we can simplify:

U (yλH

, y) =(π · λH + (1− π) · λL) · u(

+ π · (1− λH ) · u((1− y)R1− λH

+ (1− π) · (1− λL) · u

((1− λL

λH)y + (1− y)R

(1− λL)

We now solve ddy U =

∂dU · ∂

∂yyλH

∂yU = 0 for y ...

Asset Markets again!no default, continuation

Notice that we solved for y and d without using the value ofPH ! It cancelled because we assumed d = y

λH...

How should we obtain the equilibrium value of PH then? Theprice PH must be such that the bank chooses y = λH · d ...

U (d, y) =(π · λH + (1− π) · λL) · u (d)+

+ π · (1− λH ) · u(

y + (1− y)PH − λH · d(1− λH )PH/R

+ (1− π) · (1− λL) · u(

y + (1− y)R − λL · d(1− λL)

Asset Markets - no default, continuation

The price PH must be such that the bank chooses y = λH · d.

∂yU (d, y) = 0+

π · (1− λH ) · u′(

y + (1− y)PH − λH · d(1− λH )PH/R

)· 1− PH(1− λH )PH/R

(1− π) · (1− λL) · u′(

y + (1− y)R − λL · d(1− λL)

)· (1− R(1− λL)

Simplifying and using d = yλH

∂yU (d, y) = π · u′

((1− y)R(1− λH )

)· (1− PH ) · R

(1− π) · u′

((1− λL

λH)y + (1− y)R

(1− λL)

)· (1− R) = 0

Notice this is a linear equation in PH ! We solve it for PH . Notice that inthe solution we must have PH < 1! This implies the short-asset is notdominated at t = 0.At t = 1 and s = H , because PH < 1 < R, banks lose money if they sell thelong-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 5Default Scenario

Remember that under no default, no banks ever sold amounts of thelong-asset. They carried enough cash to pay depositors. But if defaultoccurs (say in the H state), banks need to sell the long-asset. But inthis case we can not have the all banks in default otherwise PH = 0 (allwant to sell the long-asset and none wants to buy) which is notcompatible with equilibrium. The only way that some banks escapedefault is if they use a more conservative strategy (they hold more cashand promise lower payments at date 1):

safe banks’ choice (yB, dA)

risky banks’ choice (yR, dA)

where dB > yR and dB < dA

ρ is the fraction of risky banks (A=active)1− ρ is the fraction of safe banks (B=boring)

We are going to construct examples of equilibrium where:

The risky banks always sell long-asset to the safe banks att = 1.

In the state H , there will be no excess liquidity and theprice 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, therisky banks are insolvent. They go bankrupt.

The banks’ objectives

U B(dB, yB) =(π · λH + (1− π) · λL) · u(

+ π · (1− λH ) · u(

yB + (1− yB)PH − λH d(1− λH ) · PH/R

+ (1− π) · (1− λL) · u(

yB + (1− yB)PH − λL d(1− λL) · PL/R

U A(dA, yA) =(1− π) · λL · u(

+ π · u(

yA + (1− yA)PH

+ (1− π) · (1− λL) · u(

yA + (1− yA)PL − λL dA

(1− λL) · PL/R

Market Clearing Conditions

ρc1(s, dA, yA)+(1−ρ)c1(s, dB, yB)

= ρyA + (1− ρ)yB if ps < 1,

≤ ρyA + (1− ρ)yB if ps = 1

ρC (s, dA, yA) + (1− ρ)C (s, d, yB) =ρ(yA + R(1− yA))+

(1− ρ)(yB + R(1− yB)),

where C = c1 + c2.If ps < 1 banks are not willing to carry cash from date 1 intodate 2.

Other Eq. ConditionsConsumers utility of putting money on A or B bank is thesame.

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