Markets and Uncertainty

Larry Blume

April 28, 2020

Contents

I Introduction

I The State-Preference

Model

I Quiz 1

I The 2-Period Exchange

Economy

I The Arrow-Debreu Model

I Quiz 2

I The Radner Model

I The No-Arbitrage Condition

I Quiz 3

I Arrow Securities

I Arrow’s (other) Theorem

I Quiz 4

I Equilibrium and The

Stochastic Discount Factor

I Quiz 5

I Welfare Economics

I Answers to Quizzes 1–4

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I Specialize the abstract Arrow-

Debreu model to interpret it as

trading under uncertainty.

I Interpret commodities,

preferences, endowments, budget

sets, etc., to apply to situations

with uncertainy.

I Examine alternative market

specifications.

What questions do we want to ask in these models?

I Comparative statics of preference parameters — risk,

ambiguity aversion.

I Welfare properties of markets.

I The structure of equilibrium — the composition of prices.

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Representation of Uncertainty

I Commodities are distributions on outcomes. Prices are linear

functionals of distributions — integration of a function.

I Commodities are random variables — measurable functions

of states. Prices are linear functionals of measurable

functions — measures on the state space.

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The State Preference Model

O The outcome space.

S The finite state space.

C The set of acts — functions

from states to outcomes.

< A preference relation on C .

Examples

U(f ) = minsu(f (s)

)U(f ) =

∑s

u(f (s)

)µ(s)

U(f ) = minµ∈P

∑s

u(f (s)

)µ(s)

U(f ) =1

1− γ

∫P

(∑s

u(f (s)

)µ(s)

)1−γdp(µ)

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The State Preference Model

O The outcome space.

S The finite state space.

C The set of acts — functions

from states to outcomes.

< A preference relation on C .

Examples

U(f ) = minsu(f (s)

)U(f ) =

∑s

u(f (s)

)µ(s)

U(f ) = minµ∈P

∑s

u(f (s)

)µ(s)

U(f ) =1

1− γ

∫P

(∑s

u(f (s)

)µ(s)

)1−γdp(µ)

3 / 31

The State Preference Model

O The outcome space.

S The finite state space.

C The set of acts — functions

from states to outcomes.

< A preference relation on C .

Examples

U(f ) = minsu(f (s)

)U(f ) =

∑s

u(f (s)

)µ(s)

U(f ) = minµ∈P

∑s

u(f (s)

)µ(s)

U(f ) =1

1− γ

∫P

(∑s

u(f (s)

)µ(s)

)1−γdp(µ)

3 / 31

The State Preference Model

O The outcome space.

S The finite state space.

C The set of acts — functions

from states to outcomes.

< A preference relation on C .

Examples

U(f ) = minsu(f (s)

)U(f ) =

∑s

u(f (s)

)µ(s)

U(f ) = minµ∈P

∑s

u(f (s)

)µ(s)

U(f ) =1

1− γ

∫P

(∑s

u(f (s)

)µ(s)

)1−γdp(µ)

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Quiz 1

I Suppose the state space is finite and that O = RL+. In each

caser, determine what conditions on u, µ, etc., guarantee the

existence of equilibrium in the private ownership economy.

I Minimax Regret Demand. On each budget set B(p, y) =

{x :∑s ps · x(s) ≤ y}, let v(s, p, y) = maxx∈B(p,y) u(x(s)).

State s regret of act x is then r(x , s, p, y) which is

v(s, p, y)− u(x(s)). Now define max regret

U(x , p, y) = maxs r(x , s, p, y). Demand on the budget set is:

d(p, y) = argminx∈B(p,y)

U(x , p, y).

Interpret this. Under what conditions on u will demand be

such that aggregate excess demand satisfies assumptions

1,2,3′, and 4 for the existence of a competitive equilibrium?

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The 2-Period Exchange Economy

I I traders live for two periods; t = 0, 1.

I State s ∈ S is revealed at the beginning of the second period.

I L physical commodities available at date 0 and states 1

through S . The consumption set for each trader is RL(S+1)+ .

I Each trader has a strictly positive endowment vector ωi � 0

in RL(S+1)+ .

I Each trader has beliefs πi and concave and increasing date t

payoff functions uit : RL+ → R.

U i(x i) = ui0(xi0) +

S∑s=1

πisui1(xis)

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The Arrow-Debreu Model

At date 0 traders trade current consumption bundles and

contracts promising the delivery of xsl units of good l if state s

occurs in date 1.

I Prices are φ = (φ0, . . . , φS) ∈ RL+/0.

I The Arrow-Debreu budget set is

BAD(φ, ωi) =

{y ∈ RL(S+1)

+ :

S∑s=0

φs(ys − ωis) = 0

}.

Equilibrium in the AD model is a price φ and allocation x such

that

a) Each trader i is maximixing U i(x i) on BAD(φ, ωi);

b) For all s and l ,∑i xisl − ωisl = 0.

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The Arrow-Debreu Model

We learn from the general analysis:

I Equilibrium exists.

I The First and Second Welfare Theorems apply

What is the meaning of Pareto Optimality here?

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Quiz 2

I Draw an Edgeworth box for an economy with two expected

utility maximizers, one commodity, and two states of the

world. Neither preferences nor endowments need be the

same, but the aggregate endowment of the economy is the

same in both states of nature. Suppose first that they have

identical beliefs. Second, suppose they have different beliefs.

I With expected utility preferences and one good in each state,

what is the MRS for any diagonal consumption bundle?

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The Radner Model

I Traders can trade physical commodities in date 0 and state s

spot markets, and J assets.

I Asset j is a promise to pay to its holder ajs units of the

numeraire good 1 in state s. The asset return matrix is A,

with rows As .

I The vector of state-contingent returns from portfolio z ∈ RJis Az .

I The set of feasible portfolios is unbounded. z j < 0 is a

sale/short position on the numeraire payouts; z j > 0 is a

purchase/long position on the numeraire payouts.

Prices in the Radner model are spot prices (p0, . . . , pS) ∈ RL(S+1)+

and asset prices q ∈ RJ .

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The Radner ModelThe Date-Event Tree

H T

H

H T

T

t = 0

t = 1

t = 2

Each node is a spot market. Each edge is a state. The red node

is the spot market after 2 H states are realized. People trade

goods at spot markets. THey trade assets between spot markets.

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The Radner Model

The Radner budget set is

BR(p, q, ωi) =

(y , z) ∈ RL(S+1)

+ × RJ s.t.

p0y0 + qz = p0ωi0

for s ∈ S , psys = psωis + ps1Asz

.

Equilibrium in the Radner model is a vector of spot prices p, asset

prices q, a commodity allocation x and an asset allocation z such

that

a) Each trader i is maximizing U i(x i) on BR(p, q, ωi);

b) For all s and l ,∑i xisl − ωisl = 0;

c)∑j zj = 0.

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The No-Arbitrage Condition

If there is a portfolio with a semi-positive

return vector, then some spot-market

budget sets are unbounded. Equilibrium

requires is that no such portfolio exists.

This condition may restrict asset prices.

Let

M =

[−qA

]with column

space < M >.

The no arbitrage condition is satisfied if there is no portfolio z

such that Mz > 0. That is, < M > ∩RS+1+ = {0}.

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The No-Arbitrage Theorem

Theorem. The no-arbitrage condition is satisfied iff there is a

π̃ � 0 such that π̃M = 0.

Let P(q) ={π ∈ RS++ : (1, π) ·M = 0

}. π ∈ P(q) iff

qj = π1aj1 + · · ·+ πSajS .

Any member of P(q) is called a state price vector.

dimP(q) = S − rankA. So π will be unique iff rankA = S . In this

case, markets are said to be complete.

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The No-Arbitrage Theorem

Theorem. The no-arbitrage condition is satisfied iff there is a

π̃ � 0 such that π̃M = 0.

Let P(q) ={π ∈ RS++ : (1, π) ·M = 0

}. π ∈ P(q) iff

qj = π1aj1 + · · ·+ πSajS .

Any member of P(q) is called a state price vector.

dimP(q) = S − rankA. So π will be unique iff rankA = S . In this

case, markets are said to be complete.

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Proof of the No-Arbitrage Theorem

Let ∆ denote the set of y ∈ RS+1+ such that

∑n yn = 1. Mz > 0

has a solution iff Mz ∈ ∆ has a solution.

Lemma. If A,B ⊂ Rm are convex, A closed and B compact, then

there is a π that separates them, that is, supa∈A πa < infb∈B πb.

(Sufficiency) If the condition of the theorem is satisfied, then

πMz = 0 for all z . Since π � 0, Mz 6> 0.

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Proof of the No-Arbitrage Theorem

(Necessity) Let D = {y : y = Mz , z ∈ RJ}. The no-arbitrage

condition implies that ∆ ∩D = ∅. Thus there is a π that

separates them.

To apply the lemma, take A = D and B = ∆. If for some s

πs ≤ 0 then supd∈D πd ≥ 0 > infδ∈∆ πδ, a contradiction. Thus

π � 0.

If πd 6= 0 for some d ∈ D, then πMz 6= 0 for some z ∈ RJ . If so,

supd∈D πd = supz∈RJ πMz = +∞, contradicting the separation

property. Thus πMz = 0 for all z ∈ RJ , so πM = 0.

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Implications of No Arbitrage

Redundant Assets

Suppose asset Aj =∑Kk 6=j αkA

k . Each qk = πAk , so

qj = πAj = π∑k 6=j

αkAk =

∑k 6=j

αkπAk =

∑k 6=j

αkqk .

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Quiz 3

1. Suppose there are two states and three assets, and

A =

(1 0 2

2 1 0

)Find all state prices, and asset prices such that the

no-arbitrage condition is satisfied.

2. Suppose there are three state and two assets.

A =

1 0

0 2

1 1

Find all state prices, and asset prices such that the

no-arbitrage condition is satisfied.

3. Under what circumstances do asset prices uniquely define the

state price vector?

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Arrow Securities

Suppose A is the S × S identity matrix. No arbitrage implies

q � 0. Then π = q.

πs is the price of a unit of numeraire in state s.

The securities that pay off 1 in some state s and 0 otherwise are

called Arrow securities.

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Arrow’s (other) Theorem

Theorem. Suppose that rankA = S .

a) Suppose Radner prices are (p, q), with state prices π, and

define φ0 = p0; for s ≥ 1, φsl = πspsl/ps1.

(x , z) ∈ BR(p, q, ωi) iff x ∈ BAD(φ, ωi).

b) Suppose A-D prices are φ, and define p0 = φ0; πs = φs1;

psl = φsl/φs1, and q = πA. x ∈ BAD(φ, ωi) iff there is a z

such that (x , z) ∈ BR(p, q, ωi).

Implications:

I When markets are complete, Radner and Arrow-Debreu

equilibrium commodity allocations are identical.

I Prices of any one equilibrium can be derived from the other.

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Quiz 4

Suppose in an economy with two physical goods that a consumer

has endowment (1, 1) at time 0, (1, 0) in state 1 and (0, 2) in

state 2. Arrow-Debreu prices at time 0 are (1, 1), (1, 2) in state 1

and (1, 1) in state 2.

I Check that the consumption bundle x which contains (1, 1)at time 0, (1/2, 1/2) in state 1 and (3/4, 3/4) in state 2 is

affordable.

I Suppose now a Radner economy with the same endowments

and the equivalent Radner prices. Suppose asset 1 is a bond

that pays off 1 unit of numeraire in each state, and asset 2

pays off 1 unit in state 1 and 0 otherwise. Find the asset

prices, and the portfolio z that makes the consumption

bundle (x , z) budget-feasible.

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Equilibrium FOC

λis is the multiplier for constraint s, s = 0, . . . ,S .

for all s and l , DslUi(x i)− λspsl = 0

for all j , − λi0qj +∑j

λisajs = 0

budget feasibility

x � 0 λ� 0

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The Stochastic Discount Factor

Suppose expected utility: There is a probability distribution ρ and

a function u such that U(x) = u(x0) +∑s πsu(xs).

The state s intertemporal MRS (IMRSis) is D1ui(x is1)/D1u

i(x i01).

From trader i ’s FOC:

qj =∑s

ρsIMRSisajs ≡ Eρ IMRS

iaj .

That is, (ρsIMRSis)s∈S ∈ P(q).

In finance, the function s → IMRS is is called a stochastic discount

factor.

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Quiz 5

I Prove the relationship between Radner and Arrow-Debreu

budget sets.

I Derive the asset price equation on slide 22.

I Consider an economy with 2 states and a single consumption

good. U i(x) = log x0 + πi1 log x1 + πi2 log x2. For all i and

s ≥ 0, ωis = 1.

I Derive the first-order conditions for the case of a single asset

which is a bond, paying off 1 in each state s ≥ 1.

I What is the equilibrium consumption and portfolio allocation?

I Suppose that π11 6= π21. Argue intuitively why equilibrium is

not Pareto optimal.

I Prove it is not optimal from the first order conditions.

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Welfare Economics

Arrow’s other Theorem implies that if markets are complete, then

the first and second welfare theorems hold for Radner equilibria.

What can go wrong when markets are incomplete?

The FOC with Arrow securities:

DslUi(x i)− λispsl = 0,

−λi0qs + λis = 0,

etc.

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Welfare Economics

DslUi(x i)

DsmU i(x i)=pslpsm

Marginal rates of substitution

between goods in the same state

are equal.

DslUi(x i)

DtmU i(x i)=λisλit

pslptm

λisλit

=qsqt

Marginal rates of substitution

between goods in different states

equal a constant times the

marginal rate of substitution

between the numeraires in the

different states. But they are

also equal.

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Welfare Economics

Suppose only a bond that pays off 1 in each state:

DslUi(x i)− λispsl = 0,

−λi0q1 +∑s

λis = 0,

etc.

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Welfare Economics

The vector

(λi1λi0, . . . ,

λiSλi0

)is a state price vector.

The first order conditions do not uniquely determine state prices!

Marginal rates of substitution between goods in different states

are not necessarily equal.

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Quiz 1 Answers

1. (Maxmin): u is continuous, strictly increasing. (EU): u is

continuous, concave, increasing. Convince yourself that the

sum of two quasi-concave functions need not be

quasi-concave.) Maxmin EU): u is continuous, concave,

increasing. P is closed. (Smooth Ambiguity): u is

continuous, concave, increasing. γ ≥ 0, 6= 1.

2. v is the maximal utility achievable in any state Regret in

state s is the loss from the maximal achievable utility at s. U

is maximum regret. Sufficient conditions are that u is

continuous, strictly increasing and strictly quasi-concave, and

indifference curves do not intersect the axes. Does minimax

regret demand satisfy WARP?

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Quiz 2 Answers

I Boxes

Same Beliefs Different Beliefs

I MRS is the ratio of probabilities.

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Quiz 3 Answers

I For any (π1, π2) > 0, take q1 = π1 + 2π2, q2 = π2 and

q3 = 2π1. Check that (1, π1, π2) ·M = 0. The no-arbitrage

condition implies that q3 = 2q1 − 4q2. Any arbitrage-free

asset price vector uniquely defines state prices.

I q1 = π1 + 2π3, q2 = 2π2 + π3. Asset prices do not uniquely

define state prices. For any π3 ≤ min{q1/2, q2},π1 = q1 − 2π3 and π2 = (q2 − π3)/2.

I When the rank of A is S and there are no arbitrage

possibilities.

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Quiz 4 Answers

Wealth in state 1 is 1, and in state 2 is 2. The states 1 and 2

consumption bundles each cost 3/2. So 1/2 unit of wealth has to

be transfered from state 2 to state 1. The state prices (Arrow

security prices) are (1, 1). The prices of the two assets are q1 = 2

and q2 = 1. The portfolio which moves 1 unit of wealth from

state 2 to state 1 is z = (−1/2, 1); sell a half-unit of the bond,

and buy one unit of the asset that pays off only in state 1. The

cost of this portfolio is, no surprise, 0.

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