competitive equilibrium with complete markets: part ii · competitive equilibrium with complete...

Outline Motivation Model Setup Pareto (recap) ADE⇔ PO SME ADE ⇔ SME RCE

Competitive Equilibrium with CompleteMarkets: Part II

Timothy Kam

School of Economics & CAMAAustralian National University

ECON8022, This version April 8, 2008


ARC EDN Winter School on Money & Pricing

When: 04 Jul 2008 - 08 Jul 2008Where: Melbourne University

“Money and Pricing” will consist of three main components: lectures inmonetary theory (Randy Wright, UPenn) and pricing (Preston McAfee,CalTech/Yahoo!), and presentations by graduate students.

Participants may also be interested in attending the 3rd Annual Workshop onMacroeconomic Dynamics, which takes place at The University of Melbournedirectly preceding the Winterschool: July 2-3. This workshop includesProfessors Randall Wright (University of Pennsylvania) and Steven Turnovsky(University of Washington) as plenary speakers.

For more information on the EDN Winter School please visit:

http://pluto.ecom.unimelb.edu.au/winterschool2008/. Note: Limited

Travel Scholarships available! Apply early!

http://pluto.ecom.unimelb.edu.au/winterschool2008/


Outline

1 Model Setup

2 Pareto (recap)

3 ADE ⇔ PO

4 SMEA state variableSBC and Debt limitsTimingAgents’ problems

5 ADE ⇔ SME

6 RCERecursive formulationMarkovian Asset pricingArbitrage-free pricing and redundant assets


What Next?

Competitive/Decentralized equilibrium:Arrow-Debreu time-0-trading economy (ADE) with Arrow-Debreusecurities.Radner sequential-trading economy (SME) with Arrow securities.

Show ADE ⇔ SME ⇔ PO.


Motivation

Previously ...

Previously we look at a planned economy:

Single optimizing planner.

Characterized recursive optimal allocations as a DP problem.

In reality, we have decentralized or competitive economies.

Osamu Tezuka’s Metropolis


Motivation

And few lectures ahead ...

Want to work toward the stochastic growth model as also basic recursivecompetitive equilibrium model.

As a dynamic outcome where individuals and firms solve theirdecentralized optimal allocation problems independently.

No planner.

History (or State)-contingent, intertemporal relative prices, as theallocative mechanism.

Resulting versions of first- and second fundamental welfare theorems.(Why?)


Model Setup

Stochastic event st ∈ S = s1, ..., sn for t ∈ N.

Publicly observable history of events up to and including t:ht = (s0, s1, ..., st) ∈ St.Unconditional probability of ht given by probability measureπt(ht).

W.l.o.g., assume π (s0) = 1.

Probability of observing ht conditional on realization of hτ isπ(ht|hτ

), for any t ≥ τ .


Model Setup

I agents indexed by i = 1, ..., I.

Agent i’sEndowment: yit

`ht´

history-dependent consumption plan, ci =˘cit`ht´¯∞t=0

for each ht ∈ Stexpected utility criterion:

U`ci´

= E0

( ∞Xt=0

βtu`cit`ht´´)

=∞Xt=0

Xht

βtu`cit`ht´´πt`ht´

where

u′ (c) > 0, u′′ (c) < 0limc0 u

′ (c) = +∞to ensure ct > 0 for all t


Model Setup

A feasible allocation must satisfy

I∑i=1

cit(ht)≤

I∑i=1

yit(ht)

for all t and for all ht.


Remember?

A first-order neccesary condition for Pareto optimum is

βtu′(cit(ht))πt(ht)

=θt(ht)

λi

for all i = 1, ..., I, and for all t ≥ 0 and all ht.


Remember?

Consider two agents, i 6= j. The ratio of their marginal utilities ateach period, for all possible histories, is

u′(cit(ht))

u′(cjt (ht)

) =λjλi

This implies:

cit(ht)

= u′−1

[λjλiu′(cjt(ht))]


Theorem

A Pareto optimal allocation is a function of the realized aggregateendowment and does not depend on

1 the particular history ht leading up to that outcome, nor

2 the realization of individual endowments,

so that if ht 6= hτ are such that∑

j yjt

(ht)

=∑

j yjτ (hτ ) then

cit(ht)

= ciτ (hτ ).


Corollary (First fundamental welfare theorem)

The competitive equilibrium is a particular Pareto optimalallocation, where µi = λ−1

i for all i = 1, ..., I, is unique (up to amultiplication by a positive scalar). Furthermore, the shadow pricesfor the planner θt

(ht)

are equal to Arrow-Debreu equilibriumprices q0t

(ht).


What Next?

We have already charactered Pareto-optimal allocation (PO).

We have studied one assumption for a decentralized marketeconomy: ADE

Next we study alternative SME. Note along the way,connections btw SME and ADE via asset pricing relationships.

W.t.s. FWT: Allocative “equivalence” between ADE andSME and PO.


Sequential Markets Economy

Market trading structure – assumptions:

Trade occurs at each t ∈ N.

Trade in one-period complete Arrow securities.

At each t reached with history ht, traders meet to trade forhistory ht+1-contingent goods deliverable in t+ 1.


Example

b

t = 0

rt = 1

r@@

t = 2

r 1|(0, 1, 1)

r 0|(0, 1, 1)

t = 3

S = 0, 1. At t = 2, trades occurs for only t = 3 goods at statesthat can be reached from the realized t = 2 history, h2 = (0, 1, 1).


Preliminaries: Two new items ...

Now markets exist sequentially.

At start of each period t, after each ht, traders need to keep trackof what is feasibly tradable.

This depend on:

1 Wealth as a state variable.

2 A restriction that prevents forever-borrowing schemes byagents.

Remark. These two things not needed in ADE. Why?


Relevant state variable

Need to find an appropriate individual state variable.

This state variable tracks available opportunity set;

to provide the right choices of consumption so that there willbe enough resources left for future trades on contingentclaims.

State variable is the present value (in terms of history ht anddate t) of expected current and future net claims – i.e.current wealth of the consumer.


ADE again! Agent i’s wealth at time-t is just the time-t expectedvalue of all her current and future net claims conditional on time-t,history ht:

Ωit

(ht)

=∞∑τ=t

∑hτ |ht

qtτ (hτ ) dτ (hτ )

=∞∑τ=t

∑hτ |ht

qtτ (hτ )[ciτ (hτ )− yiτ (hτ )

]But,

qtτ (hτ ) = βτ−tu′(ciτ (hτ )

)u′(cit (ht)

) πτ (hτ |ht)Note Ωi

t

(ht)

has the same expression as the value of a tail asset!


Since, aggregate endowment must equal aggregate consumption(following any ht), then

I∑i=1

Ωit

(ht)

= 0.

for all t and all ht.

Remarks:

The cross-sectional distribution of tail wealth across all agents i sums tozero, since all contingent debt sellers are balanced out by buyers.

When we move from this ADE to the SME, we can relate this tail wealthof each i, Ωit

`ht´, to the time t history ht individual asset of the

sequential markets world.


SBC and Debt limits

In ADE, households face a single intertemporal budgetconstraint that ensures intertemporal solvency.

In the sequential markets setting, there will be a sequence ofbudget constraints, indexed by t and ht.

Need to ensure sequential asset trades are not open to “Ponzischemes” – i.e. consumers cannot forever be consuming morethan their endowments.

We will consider the weakest possible restrictions called“natural debt limits” .


Definition (Natural debt limit)

Let the Arrow-Debreu price in terms of the time t history ht

numeraire good be qtτ (hτ ) for τ ≥ t. The value of the tail of i’sendowment sequence at time t given history ht,

Ait(ht)

=∞∑τ=t

∑sτ |ht

qtτ (hτ ) yiτ (sτ ) ,

is the natural debt limit at time t and history ht.


Remarks

Ait(ht)

=∞∑τ=t

∑sτ |ht

qtτ (hτ ) yiτ (sτ )

The maximal amount that i can repay his debt starting fromtime t is thus the tail value of his endowment starting outfrom time t given history ht.

Alternatively, this says the worst i can do is to consume zeroforever from time t to repay existing debt at time t history ht.

At each time t, i will face one such borrowing constraint foreach possible realization ht+1 the next period.


Sequential trades: Timing and Actions

Markets are open for trade in one period-ahead state-contingentclaims every period.

-Start t Start t + 1

?

1. ht

realized

?

2. Relevant claim ait(ht)

realized

62. Endowment yit(h

t)

realized

?

3. Agent i chooses

cit(ht)

63. Agent i choosesn

ait+1(st+1, ht)

o∈ Rn

?

ht+1 = (ht, st+1)

realized

6ait+1(ht+1)

realized


Suppose the pricing kernel Qt(st+1|ht

)exists.

Qt(st+1|ht

): price of one unit of time t+ 1 consumption,

contingent on realization of st+1 in t+ 1, given t-history ht.

Agent i’s sequence of budget constraints:

cit(ht)

+∑st+1

ait+1

(st+1, h

t)Qt(st+1|ht

)≤ yit

(ht)

+ ait(ht)

for t ≥ 0, at each ht.


Note at time t, given ht, i chooses

Current consumption: cit(ht), and

Quantities of all possible n number of state-contingent claimsnext period:(

ait+1

(st+1, h

t))∈ Rn


No-Ponzi borrowing constraint:

−ait+1 (st+1) ≤ Ait+1

(ht+1

)=

∞∑τ=t+1

∑hτ |ht+1

qt+1τ (hτ ) yiτ (hτ ) .

Huh? ...

Amount of debt i brings into all possible st+1 ∈ S,

Must be repayable, in the worst case,

by expected discounted (real) value of tail endowments, remaining fromt+ 1 onward.

Worst case: consume nothing forever from t+ 1 on ...! Merd!


Agent i chooses cit(ht), ait+1(st+1, ht)∞t=0 to:

max∞∑τ=0

∑ht

βtu

(cit(ht))πt(ht)

+ ηit(ht) [yit(ht)

+ ait(ht)

− cit(ht)−∑st+1

ait+1

(st+1, h

t)Qt(st+1|ht

) ]

+ νit(ht; st+1

) [ait+1

(st+1, h

t)

+Ait+1

(ht+1

)]

for given initial wealth ai0(h0).


Remarks: Following each ht,

There are n no-Ponzi constraints to consider. Why?

So there are n Lagrange multipliers νit(ht; st+1

), one for each

possible st+1 ∈ S.

For each st+1 need to calculate upper bound on negativeassets:

Ait+1

(ht+1

)=

∞∑τ=t+1

∑hτ |ht+1

qt+1τ (hτ ) yiτ (hτ ) .


Optimal decision by agents i:No-Ponzi constraints not binding. Why? So νit

(ht; st+1

)= 0 for

all t, all ht.

Then necessary (and sufficient) condition for optimalconsumption-asset-accumulation strategy is

Qt(st+1|ht

)= β

u′(cit+1

(ht+1

))u′(cit (ht)

) πt(ht+1|ht

)for all st+1, t ≥ 0 and ht.

Crickey! This is a familiar looking one-period pricing kernel weencountered in the Arrow-Debreu economy!


Definition

A distribution of wealth is a vector−→ea t `ht´ =

˘eait `ht´¯Ii=1satisfyingP

i eait `ht´ = 0.

Definition

A sequential trading competitive equilibrium is an initial distribution of

wealth−→ea 0 (s0), an allocation (sequence of allocations for all agents)

˘eci¯Ii=1

and pricing kernels eQt `st+1|ht´

such that

1 for all i, given eai0 (s0) and eQt `st+1|ht´, the consumption allocationeci =

˘ecit¯∞t=0solves agent i’s optimization problem;

2 for all realizations of˘ht¯∞t=0

the agent’s consumption allocation and

implied asset portfoliosnecit `ht´ ,˘eait+1

`st+1, h

t´¯st+1

ot∈N

satisfyPi ecit `ht´ =

Pi yit

`ht´

andPi eait+1

`st+1, h

t´

= 0 for all st+1.


Theorem

The time-0 trading arrangement in the Arrow-Debreu equilibriumwith complete markets has the same allocations as the sequentialtrading arrangement with one-period complete Arrow securities,

ciIi=1

=ciIi=1

,

for an appropriate initial distribution of wealth in the sequential

markets equilibrium,ai0 (s0)

Ii=1

.


Proof.

First we show “ADE ⇒ SME”.

Take Arrow-Debreu equilibrium q0t`ht´

as given.

Suppose ∃ eQt `st+1|ht´

satisfying recursion

q0t+1

`ht+1´ = eQt `st+1|ht

´q0t`ht´

⇔ eQt `st+1|ht´

=q0t+1

`ht+1

´q0t (ht)

= qtt+1

`ht+1´ .

To show guess is true, take Arrow-Debreu equilibrium first-orderconditions from two succesive periods and write:

βu′`cit+1

`ht+1

´´u′ (cit (ht))

πt`ht+1|ht

´=q0t+1

`ht+1

´q0t (ht)


Proof (cont’d).

But then if guess is true, it must be that

βu′`cit+1

`ht+1

´ú′ (cit (ht))

πt`ht+1|ht

´= eQt `st+1|ht

´= β

u′ècit+1

`ht+1

´ú′ (ecit (ht))

πt`ht+1|ht

´.

So then, Arrow-Debreu equilibrium is equivalent to the sequential marketsequilibrium in terms of allocations,n

cioIi=1

=necioI

i=1.


Proof (cont’d).

Next we show “ADE ⇐ SME”:

Pick˘eai0 (s0)

¯Ii=1

s.t. SBCs in SME consistent with IBC for ADE. Guess

that˘eai0 (s0)

¯Ii=1

= 0I×1.

Why? In Arrow-Debreu equilibrium, at time 0, agents bring in only theirendowments, y0 (s0).

At t ≥ 0 and history ht, i chooses asset portfolio,eait+1

`st+1, h

t´

= Ωit+1

`ht+1

´for all st+1.

The expected value in date t terms isXst+1

eait+1

`st+1, h

t´ eQt `st+1|ht´

=Pst+1

Ωit+1

`ht+1

´qtt+1

`ht+1

´=P∞τ=t+1

Phτ |ht q

tτ (hτ )

ˆciτ (hτ )− yiτ (hτ )

˜just the tail value of wealth!


Proof (cont’d).

Show that i can afford this portfolio strategy. Use SME SBCs. At time 0given eai0 (s0) = 0,

eci0 (s0) +∞Xt=1

Xht

q0t (st)hcit`ht´− yit

`ht´i

= yit (s0) + 0

But this is the same as IBC in the ADE.

So eci0 (s0) = ci0 (s0).


Proof.

For all t > 0, we can write eait `ht´ = Ωit`ht´, and the time t, ht-BC is

ecit `ht´+Xst+1

eait+1

`st+1, h

t´ eQt `st+1|ht´

= yit`ht´

+ eait `ht´⇒Xst+1

eait+1

`st+1, h

t´ eQt `st+1|ht´

= Ωit`ht´−hecit `ht´− yit `htí

⇒Xst+1

Ωit+1

`ht+1´ qtt+1

`ht+1´ = Ωit

`ht´−hecit `ht´− yit `htí

⇒∞X

τ=t+1

Xhτ |ht

qtτ (hτ )hciτ (hτ )− yiτ (hτ )

i= Ωit

`ht´−hecit `ht´− yit `htí

It then follows that ecit `ht´ = cit`ht´

for all t and ht.


Notes

The equivalence between Arrow-Debreu equilibrium and Arrow’s sequentialmarkets equilibrium follows from two key factors:

Agents are v.N-M expected utility maximizers – their once-and-for-alltime 0 choices are time consistent. Past actions affect future payoffs butfuture actions do not affect past payoffs.

Under complete markets, the budget sets defined by the two formulationsare equivalent, and thus Arrow-Debreu equilibrium prices of contingentclaims are equal to Arrow’s spot prices weighted by the price in period 1of the appropriate Arrow security.


Recursive (Markov) Competitive Equilibrium

The assumptions about the state variables so far in theArrow-Debreu equilibrium and sequential markets equilibriumeconomies are too general to be useful – at each t, statevariables are made up of the entire history leading up to t, i.e.ht := (s0, s1, ..., st).

For practical purposes, we need to discipline the evolution ofthe state further – e.g. to be Markovian – so only a few statevariables suffice to describe the position of the economy ateach time period.

We’ll look at a recursive competitive equilibrium formulationof the sequential markets equilibrium and Arrow-Debreuequilibrium.


Endowments with Markov property

Consider the state space, S. So exogenous event is s ∈ S. Let sbe governed by a Markov chain:

π0 (s) = Pr (s0 = s).

π (s′|s) = Pr (st+1 = h′|st = s) .The Markov chain induces a sequence of probability measures onhistories ht. The probability of realizing history ht is

πt(ht)

= π (st|st−1)π (st−1|st−2) · · ·π (s1|s0)π0 (s0) .

where it is assumed π0 (s0) = 1.


The Markov property says that

πt

(ht|hk

)= π (st|st−1)π (st−1|st−2) · · ·π (sk+1|sk)

where πt(ht|hk

)depends only on state sk at k < t and the history

prior to k is redundant.

Example

π3

(h3|h2

)= π3 ((s0, s1, s2, s3) | (s0, s1, s2)) = π (s3|s2) .

Then, for each i = 1, ..., I, we can write endowments as

yit(ht)

= yi (st)

Since st is a Markov process, yit (st) will also be a Markov process.


Equilibrium inherits Markov property

Theorem

Given yit(st)

a Markov process, the Arrow-Debreu equilibriumprice of date-τ history hτ consumption goods in terms of date t,0 ≤ t ≤ τ , history ht goods is not history dependent:

qtτ (hτ ) = qkj

(hk)

for j, k ≥ 0 such that τ − t = k − j and

(st, st+1, ..., sτ ) = (sj , sj+1, ..., sk).


Remark. Natural debt limits and household wealth are also historyindependent: Ait

(st)

= Ai (st) and Ωit

(st)

= Ωi (st).

Each agent enters every period with wealth independent ofpast endowment realizations.

Past trades have fully insured away all idiosyncraticendowment risks.

So an agent enters the current period with current-statecontingent wealth just sufficient to fund a trading schemethat insures against future idiosyncratic risks.

The pricing kernel Q (st|st−1) thus provides the correct signal,along with market clearing, to coordinate trade in time t− 1such that all idiosyncratic risks are eliminated.

However, if there are aggregate risks, they would still have tobe borne by all agents.


Now yit(st) Markov.

Agent i’s competitive equilibrium sequence problem, given Q(s′|s),is now recursive:

vi (a, s) = maxc,ba(s′)

u (c) + β

∑s′

vi(a(s′), s′)π(s′|s)

subject to

c+∑s′

a(s′)Q(s′|s)≤ yi (s) + a

c ≥ 0

−a(s′)≤ Ai

(s′)

for all s′ ∈ S.


Let the optimal decision rules associated with the fixed-pointsolution of the Bellman equation be

c = hi (a, s)

a(s′)

= gi(a, s, s′

)for each i = 1, ..., I.


We can show that this optimal solution depends on the price kernelQ (s′|s).

Evaluate the first-order condition for the RHS of each Bellmanequation and apply the Benveniste-Scheinkman formula to get

Q (st+1|st) = βu′(cit+1

)u′(cit) π (st+1|st)

where c = hi (a, s) and a (s′) = gi (a, s, s′).


Definition

A recursive competitive equililibrium is an initial distribution of

wealthai0Ii=1

a pricing kernel Q(s′|s), sets of value functionsvi (a, s)

Ii=1

and decision ruleshi (a, s) , gi (a, s, s′)

Ii=1

suchthat

1 for all i, given ai0 and the pricing kernel, the decision rulessolve the household i’s problem;

2 for all histories st∞t=0, the consumption and assetscit, ait+1 (s′)s′i∞t=0 implied by the decision rules satisfy∑

i cit =

∑i yi (st) and

∑i ait+1 (s′) = 0 for all t and s′.


j-step ahead pricing kernel

Since a complete set of markets exists for all j periods aheadcontingent claims,

a consumer i, at the end of period t, can always buy zit,j(st+j)units of contingent consumption claims, for j ≥ 1.

Recall that agent i’s sequential budget constraint is

cit +∑st+1

Q1(st+1|st)ait+1(st+1) ≤ yi(st) + ait.


The agent’s next period wealth depends on next period statest+1 and the composition of the asset portfolio:

ait+1(st+1) = zit,1(st+1)+∞∑j=2

∑st+j

Qj−1(st+j |st+1)zit,j(st+j).

So the outcome st+1 will determine which element of then-dimensional vector zit,1(st+1 = s1)

...zit,1(st+1 = sn)

pays off at time t+ 1.

But this is only one component of time t+ 1 wealth.


ait+1(st+1) = zit,1(st+1) +∞∑j=2

∑st+j

Qj−1(st+j |st+1)zit,j(st+j).

The second term on the RHS, conditional of outcome st+1

when time t+ 1 arrives, is the expected (capital gains orlosses) from holding longer term claims on t+ 1 + j, j ≥ 1,consumption.

Together they make up next-period – i.e. t+ 1 – wealth ifst+1 ∈ S is to be realized.

Using this fact, the sequence of budget constraints becomes

cit +∞∑j=1

∑st+j

Qj(st+j |st)zit,j(st+j) ≤ yi(st) + ait.


Note that the first-order condition for a optimal plan by agent iwill imply that

Qj(st+j |st) = β∑

st+1∈S

u′[cit+1(st+1)]u′[cit(st)]

π(st+1|st)Qj−1(st+j |st+1).

Since agent’s optimal RCE strategy also requires satisfying,

Q1 (st+1|st) := Q (st+1|st) = βu′(cit+1

)u′(cit) π (st+1|st)

we then have

Qj(st+j |st) =∑

st+1∈SQ1(st+1|st)Qj−1(st+j |st+1),

a recursive formula for computing j-step ahead pricing kernels forj = 2, 3, ....


Arbitrage-free pricing and redundant assets

Suppose, apart from purchasing zt,j(st+j) units of j-stepahead complete Arrow securities, Sam also trades anex-dividend stock called a Lucas tree.

A unit of this stock allows Sam to have the right to a unit offruit or dividend d(st+1) from this Lucas tree, if state st+1

occurs.

Sam can buy Nt units of this stock. The ex-divident price isp(st).

So Sam can obtain Nt[p(st+1) + d(st+1)] units ofconsumption in t+ 1 if st+1 occurs then.


See notes and LS, Ch.8 for details....

In equilibrium, two arbitrage-free pricing conditions

p(st) =∑st+1

Q1(st+1|st)[p(st+1) + d(st+1)],

Qj(st+j |st) =∑st+1

Qj−1(st+j |st+1)Q1(st+1|st), j = 2, 3, ....

for all t ∈ N.

Meaning?

competitive equilibrium with complete markets: part ii · competitive equilibrium with complete...

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