Asset Markets in General Equilibrium

Econ 3030 Fall 2020

Lecture 12, November 17

Outline

1 Asset Markets and Equilibrium2 Linear Asset Pricing3 Complete Markets4 Span5 No Arbitrage and Asset Pricing

Asset Markets in General Equilibrium

An asset is completely characterized by its return vector rk ∈ RS

rs is the dividend paid to the holder of a unit of rk if and only if state s occurs.

The return matrix R is an S ×K matrix whose kth column is the return vectorof asset k.

q ∈ RK+ are the asset prices.zi ∈ RK are consumer i’s holdings of each asset.Consumer i’s budget constraints areasset markets︷ ︸︸ ︷K∑k=1

qkzki ≤ 0︸ ︷︷ ︸time 0

and

spot markets︷ ︸︸ ︷L∑l=1

plsxlsi ≤L∑l=1

plsωlsi + p1sK∑k=1

zki rsk for each s︸ ︷︷ ︸time 1

or

q · zi ≤ 0 and p · xi ≤ p · ωi + Rziwhere we normalized p1s = 1.

Budget Set

What do budget sets look like?Normalizing p1s = 1, one can write i’s budget set as

Bi (p, q,R) =

xi ∈ RLS+ : there is zi ∈ RKsuch that q · zi ≤ 0andp · (xi − ωi ) ≤ Rzi

The second part of the budget set defines S inequalitiesp1 · (x1i − ω1i )

...ps · (xsi − ωsi )

...pS · (xSi − ωSi )

≤z1i r11 + ...+ zki rk1 + ...+ zKi rK 1

..z1i r1s + ...+ zki rks + ...+ zKi rKs

..z1i r1S + ...+ zki rkS + ...+ zKi rKS

.where each ps ∈ RL

Rzi represents the ‘financial income’attainable by a consumer who choosesportfolio zi .

For a fixed R, the set of all possible values for Rzi determines the possibleincomes available to the consumer. As R changes so does this set.

Radner Equilibrium with Asset MarketsDefinition

A Radner equilibrium with asset markets is given by assets prices q∗ ∈ RK , spotprices p∗s ∈ RL in each state s, portfolios z∗i ∈ RK , and spot market plans x∗si ∈ RLfor each s such that:

1 for each i , z∗i and x∗i solve

maxzi∈RK ,xi∈RSL+

Ui (xi )

subject toK∑k=1

q∗k zki ≤ 0 and p∗s · xsi ≤ p∗s · ωsi +K∑k=1

p∗1szki rsk

2 all markets clear; that is:I∑i=1

z∗ki ≤ 0 andI∑i=1

x∗si ≤I∑i=1

ωsi for all s and k

The definition is familiar, but there are two new objects: the optimal portfolioand the equilibrium asset prices.What can one say about them? What do we know about q∗?

Complete Markets

DefinitionAn asset structure R is complete if rank R = S .

ExampleThere are S different Arrow securities; then

R =

1 0 .. .. 00 1 0 .. 00 0 1 .. 0.. .. .. .. ..0 .. .. 0 1

which is the identity matrix and therefore has rank S .

Example

There are three states, so S = 3, and R =

1 0 01 1 01 1 1

which has full rank.

Completing Markets with Options

Spanning and Options

Consider the European Call Option on asset r = (r1, r2, r3, r4) at strike price c .

The return vector of this call option is

r (c) = (max {0, r1 − c} , ..,max {0, r4 − c})Let r = (4, 3, 2, 1), consider three options given by r (3.5), r (2.5), and r (1.5).

The return matrix given by these four assets

R =

4 0.5 1.5 2.53 0 0.5 1.52 0 0 0.51 0 0 0

which has full rank.

So even if there is only one asset and four states (so that K < S whichobviously would imply rank R < S), one can have complete markets byintroducing three call options: derivatives can complete markets.

SpanRemarkThe crucial feature of an asset structure is the wealth it can generate.

This is the set of all incomes that can be achived constructing some portfolio:Range R =

{w ∈ RS : w = Rz for some z ∈ RK

}When two return matrices have the same range (i.e. they ‘span’the samewealth levels) they yield the same equilibrium allocation.

Proposition

Suppose prices q∗ ∈ RK and p∗ ∈ RLS and plans x∗ ∈ RLSI and z∗ ∈ RKIconstitute a Radner equilibrium for an asset structure with S × K return matrix R.Let R̂ be an S × K̂ second asset returns structure. If Range R = Range R̂, then x∗is also an equilibrium allocation for the economy that has R̂ as its asset matrix.

The proof has two steps.First, show that the two budget sets corresponding to R and R̂ are the same.Second, find portfolios z such that∑

i

zi = 0 and mi = [p∗1 · (x∗1i − ω1i ) , .., p∗S · (x∗Si − ωSi )]T = R̂zi

This is question 1 in Problem Set 7

Budget Constraints

Remember, i’s budget set is:

Bi (p, q,R) =

xi ∈ RLS+ : there is zi ∈ RKsuch that q · zi ≤ 0andp · (xi − ωi ) ≤ Rzi

where we normalized p1s = 1.

The set of income levels that can be achieved using some portfolio is:

Range R ={w ∈ RS : w = Rz for some z ∈ RK

}This is a linear space.

By the previous proposition, if two return matrices have the same range they‘span’the same wealth levels and yield the same equilibrium allocation.

Complete Markets and SpanPropositionAssume the asset markets are complete. Then

1 Suppose consumption plans x∗ ∈ RLSI and prices p∗ ∈ RLS++ constitute anArrow-Debreu equilibrium. Then, there are asset prices q∗ ∈ RK++ andportfolio choices z∗ = (z∗1 , .., z

∗I ) ∈ RKI such that: z∗, q∗, x∗, and spot prices

p∗s ∈ RS++ for each s form a Radner equilibrium.2 Suppose consumption plans x∗ ∈ RLSI , portfolio plans z∗ ∈ RKI and pricesq∗ ∈ RS++ and spot prices p∗s ∈ RS++ for each s constitute a Radnerequilibrium. Then, there are S strictly positive numbers (µ1, .., µS ) ∈ RS++such that the allocation x∗ and the state-contingent commodities price vector(µ1p

∗1 , .., µSp

∗S ) ∈ RLS form an Arrow Debreu Equilibrium.

When asset markets are complete, agents are effectively unrestricted in theamounts of trades they can make across states. Then, the Radner equilibriumwith assets is equivalent to an Arrow-Debreu equilibrium.

Because we can always set p1s = 1, each multiplier is interpreted as the valueat time 0 of a dollar at time 1 in state s.

Prove this as an exercise.

ArbitrageArbitrageAn abitrage opportunity is the possibility to make strictly positive profits at no risk.

Under what conditions on asset prices such opportunities do not exist?

Definition

Asset prices q̂ ∈ RK satisfy absence of arbitrage if there does not exists a z ∈ RKsuch that

q̂ · z ≤ 0, Rz ≥ 0 and Rz 6= 0.

q̂ are sometimes called no-arbitrage prices.

This can be interpreted as

q̂ · z ≤ 0︸ ︷︷ ︸affordable portfolio

, Rz ≥ 0 and Rz 6= 0︸ ︷︷ ︸strictly positive profits for sure

No arbitrage means there does not exists an affordable portfolio that yieldsnon-negative returns in all states and strictly positive returns in some state.

Preferences are nowhere to be seen in this definition.

Arbitrage and Asset Prices

No arbitrage imposes restrictions on prices.

No arbitrage and pricesLet

R =

1 1 11 1 01 0 0

so that Rz = z1 + z2 + z3z1 + z2

z1

Here, no arbitrage implies that if z1 > 0 then q1 > 0. Why?

Can always pick z2 and z3 equal to zero.

This result generalizes beyond the example.

Arbitrage and Asset Prices

No arbitrage and prices

Fact

If rk ≥ 0 and rk 6= 0 for all k (all assets have non-negative, non-zero, returns), thenno arbitrage implies qk > 0 for all k.

Think of the portfolio z defined as

zk > 0 and zk ′ = 0 for all k ′ 6= k.The portfolio returns are

Rz = rskzk for s = 1, ..,S

this must be strictly positive in some state.

Therefore, the only way to avoid arbitrage is for the price of asset k to bestrictly positive.

Thus no arbitrage imposes

q · z = qkzk > 0since we already have Rz ≥ 0 and Rz 6= 0.

No Arbitrage and Equilibrium Asset Prices

Theorem

Assume rk ≥ 0 and rk 6= 0 for all k (return vectors are nonnegative and non zero),and suppose preferences are strictly monotone.Then, any q∗ ∈ RK that is part of a Radner equilibrium satisfies no-arbitrage.

Under simple restrictions on assets and preferences, equilibrium impliesabsence of arbitrage.

Intuitively, arbitrage implies some money is left on the table.

Given the effi ciency properties of equilibirum, this should not happen.

This is similar in spirit to the First Welfare Theorem.

No Arbitrage and Equilibrium Asset Prices

We want to show that there cannot be arbitrage in equilibrium.

Proof.Suppose not; then, q∗ is an equilibrium and there exists a portfolio z such that

q∗ · z ≤ 0, Rz ≥ 0, and Rz 6= 0

Suppose an individual buys some amount of this portfolio:

it doesn’t cost her anything, since q∗ · z ≤ 0;given Rz ≥ 0 and Rz 6= 0, this action strictly increases her ‘financial’income(all columns of R are positive and no entry is negative).

Hence, she can relax her budget constraint.

Because preferences are strictly monotone, she will increase her utility by usingthis additional income to buy more of some goods on the spot markets.

This contradicts the assumption that q∗ is an equilibrium.

Equilibrium and Linear Asset PricesTheorem (Martingale Pricing)

Assume rk ≥ 0 and rk 6= 0 for all k (return vectors are nonnegative and non zero),and suppose preferences are strictly monotone. Then, for any q∗ ∈ RK that is partof a Radner equilibrium, we can find non-negative numbers µs ≥ 0 which satisfy

[q∗]T = µ · Ror q∗k =

∑Ss=1 µs rsk for all k.

What does this equality imply?Prices are a linear combination of dividends.

Suppose there is an asset which pays a constant amount in all states: that isrsk = c for all s.

Normalize the price of this asset to be c , so that

c =S∑s=1

µsc ⇒ 1 =S∑s=1

µs

Thus, equilibrium asset prices must equal an expected value of their returns.

The ‘probability’used to compute the expected value is the same for all assets.

No-Arbitrage Implies Linear Asset Prices

Since we have already shown that equilibrium implies no-arbitrage, to provethe Martingale Pricing Theorem we only need to prove the following lemma.

Lemma

Suppose rk ≥ 0 and rk 6= 0 for all k (return vectors are nonnegative and non zero).If asset prices q ∈ RK satisfy the no-arbitrage condition, then there exists a vectorµ ∈ RS+ such that

qT = µ · R

The proof follows from convexity of the set of no arbitrage portfolios.

The main step uses the separating hyperplane theorem, and should remind youof the proof of the welfare theorems.

Separate the positive orthant from the set of incomes that can be obtainedgiven a no-arbitrage (equilibirum) price vector.

Proof of Lemma: Convex SetsAssume no row of R is zero (there is no state where all assets yield zero).

This is without loss of generality because if such a row existed, one can pick anarbitrary µs for that row.

Pick a no arbitrage price vector q̂ ∈ RK

Given q̂, consider the following set

V ={v ∈ RS : v = Rz for some z ∈ RK with q̂ · z = 0

}This is the set of ‘financial’incomes attainable with ‘zero cost’portfolios.V is a convex set (prove it as an exercise).Note that if v ∈ V , then −v ∈ V

one can always ‘reverse’all trades in z and q̂ · (−z) = − (q̂ · z) = 0.

Furthermore, 0 ∈ V .

Since q̂ · z = 0, no-arbitrage implies the financial income from z cannot bestrictly positive (Rz ≤ 0).Thus, V can intersect the S-dimensional non-negative orthant only at 0:

V ∩{RS+\ {0}

}= ∅.

Graph these sets.

Proof of Lemma: Separation

Since V and{RS+\ {0}

}are convex, and the origin ({0}) belongs to V , we can

apply the separating hyperplane theorem at 0.

By separating hyperplane theorem, there exists a nonzero µ′ ∈ RS such thatµ′ · v ≤ 0 for each v ∈ V

andµ′ · v ≥ 0 for each v ∈ RS+.

Notice that µ′ ≥ 0;if not, µ′ < 0 and for some v ∈ RS++ one would have µ′ · v < 0 contradictingseparation.

Because v and −v are in V we haveµ′ · v ≤ 0 and µ′ · (−v) = −(µ′ · v) ≤ 0

Since µ′ ≥ 0, these two inequalities imply µ′ · v = 0.

Proof of Lemma: Linear Pricing

Consider the row vector µ′ · R ∈ RK .

We need to show that q̂T is proportional to µ′ · R.

Because R has all nonnegative entries (and no zero row) and µ′ ≥ 0, we haveµ′ · R ≥ 0T and µ′ · R 6= 0T .

Suppose q̂T is not proportional to µ′ · R (there exists no real number α > 0such that q̂T = α(µ′ · R)) and find a contradiction.

If this is the case, there exists a z̄ such that q̂ · z̄ = 0 (z̄ is ortogonal to q̂) andµ′ · Rz̄ > 0.Let v̄ = Rz̄ ; then v̄ ∈ V and µ′ · v̄ > 0 contradicting separation (µ′ · v = 0 foreach v ∈ V ).

Thus, q̂T is proportional to µ′ · R and q̂T = αµ′ · R for some real numberα > 0.

If we let µ = αµ′ we get the result q̂T = µ · R concluding the proof of theLemma.

No-Arbitrage Implies Linear Asset Prices

We have proved the following

LemmaSuppose rk ≥ 0 and rk 6= 0 for all k (return vectors are nonnegative and non zero).If asset prices q ∈ RK satisfy the no-arbitrage condition, then there exists a vectorµ ∈ RS+ such that

qT = µ · R

Notice that all we need for linear pricing is no-arbitrage.

There are no restrictions on preferences for the Lemma!

The connection with equilibrium is made by adding strictly monotonepreferences.

Linear Pricing at Work

Implications of Linear PricingSuppose there is an asset that delivers one unit of good 1 in all states:

r1 = (1, .., 1)

Let q be a no-arbitrage price vector and normalize q1 = 1.

If µ satisfies qT = µ · R, we have

µ = (µ1, .., µS ) ≥ 0 andS∑s=1

µs = µ · r1 = q1 = 1

Hence, for all assets different from asset 1,

qk =S∑s=1

µs rsk ≥ mins rsk and qk =S∑s=1

µs rsk ≤ maxs rsk

Intuitively, prices must be between the lowest and highest dividend.

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