Equilibrium with Complete Markets
Jesús Fernández-Villaverde
University of Pennsylvania
February 12, 2016
Jesús Fernández-Villaverde (PENN)
Equilibrium with Complete Markets
February 12, 2016
1 / 24
Arrow-Debreu versus Sequential Markets
In previous lecture, we discussed the preferences of agents in a
situation with uncertainty.
Now, we will discuss how markets operate in a simple endowment
economy.
Two approaches: Arrow-Debreu set-up and sequential markets.
Under some technical conditions both approaches are equivalent.
Jesús Fernández-Villaverde (PENN)
Equilibrium with Complete Markets
February 12, 2016
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Environment
We have I agents, i = 1, ..., I .
Endowment:
(e 1 , ..., e I ) = fet1 (s t ), ..., etI (s t )gt∞=0,s t 2S t
Tradition in macro of looking at endowment economies. Why?
Consumption, risk-sharing, asset pricing.
Advantages and shortcomings.
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Equilibrium with Complete Markets
February 12, 2016
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Allocation
De…nition
An allocation is a sequence of consumption in each period and event for
each individual:
(c 1 , ..., c I ) = fct1 (s t ), ..., ctI (s t )gt∞=0,s t 2S t
De…nition
Feasible allocation: an allocation such that:
cti (s t )
I
∑ cti (s t )
i =1
Jesús Fernández-Villaverde (PENN)
0 for all t, all s t 2 S t , for i = 1, 2
I
∑ eti (s t ) for all t,
i =1
all s t 2 S t
Equilibrium with Complete Markets
February 12, 2016
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Pareto E¢ ciency
De…nition
An allocation f(ct1 (s t ), ..., ctI (s t ))gt∞=0,s t 2S t is Pareto e¢ cient if it is
feasible and if there is no other feasible allocation
f(c̃t1 (s t ), ..., c̃tI (s t ))gt∞=0,s t 2S t
such that
u (c̃ i )
u (c i ) for all i
u (c̃ i ) > u (c i ) for at least one i
Ex ante versus ex post e¢ ciency.
Jesús Fernández-Villaverde (PENN)
Equilibrium with Complete Markets
February 12, 2016
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Arrow-Debreu Market Structure
Trade takes place at period 0, before any uncertainty has been
realized (in particular, before s0 has been realized).
As for allocation and endowment, Arrow-Debreu prices have to be
indexed by event histories in addition to time.
Let pt (s t ) denote the price of one unit of consumption, quoted at
period 0, delivered at period t if (and only if) event history s t has
been realized.
We need to normalize one price to 1 and use it as numeraire.
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Equilibrium with Complete Markets
February 12, 2016
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Arrow-Debreu Equilibrium
De…nition
An Arrow-Debreu equilibrium are prices fp̂t (s t )gt∞=0,s t 2S t and allocations
(fĉti (s t )gt∞=0,s t 2S t )i =1,..,I such that:
1
Given fp̂t (s t )gt∞=0,s t 2S t , for i = 1, .., I , fĉti (s t )gt∞=0,s t 2S t solves:
∞
∑ ∑
max
∑ ∑
s.t.
2
βt π (s t )u (cti (s t ))
fcti (s t )gt∞=0,s t 2S t t =0 s t 2S t
∞
∞
p
bt (s t )cti (s t )
t =0 s t 2S t
t =0 s t 2S t
i t
ct (s ) 0 for all t
∑ ∑
p
bt (s t )eti (s t )
Markets clear:
I
I
∑ ĉti (s t ) = ∑ eti (s t ) for all t,
i =1
Jesús Fernández-Villaverde (PENN)
i =1 with Complete Markets
Equilibrium
all s t 2 S t
February 12, 2016
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Welfare Theorems
Theorem
Let (fĉti (s t )gt∞=0,s t 2S t )i =1,..,I be a competitive equilibrium allocation.
Then, (fĉti (s t )gt∞=0,s t 2S t )i =1,..,I is Pareto e¢ cient.
Theorem
Let (fĉti (s t )gt∞=0,s t 2S t )i =1,..,I be Pareto e¢ cient. Then, there is a an A-D
equilibrium with price fp̂t (s t )gt∞=0,s t 2S t that decentralizes the allocation
(fĉti (s t )gt∞=0,s t 2S t )i =1,..,I .
Jesús Fernández-Villaverde (PENN)
Equilibrium with Complete Markets
February 12, 2016
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Sequential Markets Market Structure
Now we will let trade take place sequentially in spot markets in each
period, event-history pair.
One period contingent IOU’s: …nancial contracts bought in period t
that pay out one unit of the consumption good in t + 1 only for a
particular realization of st +1 = j tomorrow.
Qt (s t , st +1 ): price at period t of a contract that pays out one unit of
consumption in period t + 1 if and only if tomorrow’s event is
st +1 (zero-coupon bonds).
ati +1 (s t , st +1 ): quantities of these Arrow securities bought (or sold)
at period t by agent i.
These contracts are often called Arrow securities, contingent claims or
one-period insurance contracts.
Jesús Fernández-Villaverde (PENN)
Equilibrium with Complete Markets
February 12, 2016
9 / 24
Period-by-period Budget Constraint
The period t, event history s t budget constraint of agent i is given by
cti (s t ) +
∑
s t +1
Qt (s t , st +1 )ati +1 (s t , st +1 )
eti (s t ) + ati (s t )
js t
Note: we only have prices and quantities.
Many economists use expectations in the budget constraint. We will
later see why. However, this is bad practice.
Jesús Fernández-Villaverde (PENN)
Equilibrium with Complete Markets
February 12, 2016
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Natural Debt Limit
We need to rule out Ponzi schemes.
Tail of endowment distribution:
Ait (s t ) =
∞
∑ ∑
τ =t s τ js t
pτ (s τ ) i τ
e (s )
pt ( s t ) τ
Ait (s t ) is known as the natural debt limit.
Then:
ati +1 (s t +1 )
Jesús Fernández-Villaverde (PENN)
Ait +1 (s t +1 )
Equilibrium with Complete Markets
February 12, 2016
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Sequential Markets Equilibrium
De…nition
b t (s t , st +1 )g∞ t t
A SME is prices for Arrow securities fQ
t =0,s 2S ,st +1 2S and
∞
allocations
1
ĉti (s t ),
âti +1 (s t , st +1 ) st +1 2S
such that:
i =1,..,I
t =0,s t 2S t
b t (s t , st +1 )g∞ t t
For i = 1, .., I , given fQ
t =0,s 2S ,st +1 2S , for all i,
∞
i
t
i
t
fĉt (s ), ât +1 (s , st +1 ) st +1 2S gt =0,s t 2S t solves:
∞
max
fcti (s t ),fati +1 (s t ,st +1 )gs 2S gt∞=0,s t 2S t
t +1
s.t. cti (s t ) +
∑
s t +1 j s t
βt π (s t )u (cti (s t ))
t =0 s t 2S t
b t (s t , st +1 )ati +1 (s t , st +1 )
Q
cti (s t )
ati +1 (s t , st +1 )
Jesús Fernández-Villaverde (PENN)
∑ ∑
eti (s t ) + ati (s t )
0 for all t, s t 2 S t
Ait +1 (s t +1 ) for all t, s t 2 S t
Equilibrium with Complete Markets
February 12, 2016
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De…nition (cont.)
2. For all t
0
I
∑ ĉti (s t ) =
i =1
I
∑ âti +1 (s t , st +1 )
i =1
Jesús Fernández-Villaverde (PENN)
I
∑ eti (s t ) for all t, s t 2 S t
i =1
= 0 for all t, s t 2 S t and all st +1 2 S
Equilibrium with Complete Markets
February 12, 2016
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Equivalence of Arrow-Debreu and Sequential Markets
Equilibria
A full set of one-period Arrow securities is su¢ cient to make markets
“sequentially complete.”
Any (nonnegative) consumption allocation is attainable with an
appropriate sequence of Arrow security holdings fat +1 (s t , st +1 )g
satisfying all sequential markets budget constraints.
Later, when we talk about asset pricing, we will discuss how to use
Qt (s t , st +1 = j ) to price any other security.
Jesús Fernández-Villaverde (PENN)
Equilibrium with Complete Markets
February 12, 2016
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Pareto Problem
We will extensively exploit the two welfare theorems.
Negishi’s (1960) method to compute competitive equilibria:
1
We …x some Pareto weights.
2
We solve the Pareto problem associated to those weights.
3
We decentralize the resulting allocation using the second welfare
theorem.
All competitive equilibria correspond to some Pareto weights.
Jesús Fernández-Villaverde (PENN)
Equilibrium with Complete Markets
February 12, 2016
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Social Planner’s Problem
We solve the social planners problem:
I
max
∑ αi
I
I
(fcti (s t )gt∞=0,s t 2S t )i =1,..,I i =1
s.t.
∞
∑ ∑
βt π (s t )u (cti (s t ))
t =0 s t 2S t
∑ cti (s t ) = ∑ eti (s t ) for all t, s t 2 S t
i =1
i =1
cti (s t )
0 for all t, s t 2 S t
where αi are the Pareto weights.
De…nition
An allocation (fcti (s t )gt∞=0,s t 2S t )i =1,..,I is Pareto e¢ cient if and only if it
solves the social planners problem for some (αi )i =1,...,I 2 [0, 1].
Jesús Fernández-Villaverde (PENN)
Equilibrium with Complete Markets
February 12, 2016
16 / 24
Perfect Insurance
We write the lagrangian for the problem:
(
t
∞
∑Ii =1 αi β π (s t )u (cti (s t ))
max
∑ ∑ +λt (s t ) ∑Ii =1 eti (s t ) cti (s t )
(fcti (s t )gt∞=0,s t 2S t )i =1,..,I t =0 s t 2S t
)
where λt (s t ) is the state-dependent lagrangian multiplier.
We forget about the non-negativity constraints and take FOCs:
αi βt π (s t )u 0 (cti (s t )) = λt s t
for all i, t, s t 2 S t
Then, by dividing the condition for two di¤erent agents:
u 0 (cti (s t ))
u 0 (ctj (s t ))
=
αj
αi
De…nition
An allocation (fcti (s t )gt∞=0,s t 2S t )i =1,..,I has perfect consumption insurance
if the ratio of marginal utilities between two agents is constant across time
and
states.
Jesús Fernández-Villaverde (PENN)
Equilibrium with Complete Markets
February 12, 2016
17 / 24
Irrelevance of History
From previous equation, and making j = 1:
cti (s t ) = u 0
1
α1 0 1 t
u (ct (s ))
αi
Summing over individuals and using aggregate resource constraint:
I
∑ eti (s t ) =
i =1
I
∑ u0
i =1
1
α1 0 1 t
u (ct (s ))
αi
which is one equation on one unknown, ct1 (s t ).
Then, the pareto-e¢ cient allocation (fcti (s t )gt∞=0,s t 2S t )i =1,..,I only
depends on aggregate endowment and not on s t .
Jesús Fernández-Villaverde (PENN)
Equilibrium with Complete Markets
February 12, 2016
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Theory Confronts Data
Perfect insurance implies proportional changes in marginal utilities as
a response to aggregate shocks.
Do we see perfect risk-sharing in the data?
Surprasingly more di¢ cult to answer than you would think.
Let us suppose we have CRRA utility function. Then, perfect
insurance implies:
1
cti (s t )
αi γ
=
αj
ctj (s t )
Jesús Fernández-Villaverde (PENN)
Equilibrium with Complete Markets
February 12, 2016
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Individual Consumption
Now, cti (s t ) =
ctj (s t )
1
γ
αj
I
∑
1
αiγ and we sum over i
cti (s t )
I
=
∑
eti (s t )
=
ctj (s t )
i =1
i =1
Then:
1
γ
αj
I
1
γ
∑ αi
i =1
1
ctj (s t )
=
I
αjγ
1
γ
∑Ii =1 αi
∑ eti (s t ) = θ j yt (s t )
i =1
i.e., each agent consumes a constant fraction of the aggregate
endowment.
Jesús Fernández-Villaverde (PENN)
Equilibrium with Complete Markets
February 12, 2016
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Individual Level Regressions
Take logs:
log ctj (s t ) = log θ j + log yt (s t )
If we take …rst di¤erences,
∆ log ctj (s t ) = ∆ log yt (s t )
Equation we can estimate:
∆ log ctj (s t ) = α1 ∆ log yt (s t ) + α2 ∆ log eti (s t ) + εit
Jesús Fernández-Villaverde (PENN)
Equilibrium with Complete Markets
February 12, 2016
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Estimating the Equation
How do we estimate?
∆ log ctj (s t ) = α1 ∆ log yt (s t ) + α2 ∆ log eti (s t ) + εit
CEX data.
We get α2 is di¤erent from zero (despite measurement error).
Excess sensitivity of consumption by another name!
Possible explanation?
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Equilibrium with Complete Markets
February 12, 2016
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Permanent Income Hypothesis
Build the Lagrangian of the problem of the household i:
∞
∑ ∑
βt π (s t )u (cti (s t ))
t =0 s t 2S t
∞
µi
∑ ∑
pt (s t ) eti (s t )
cti (s t )
t =0 s t 2S t
Note: we have only one multiplier µi .
Then, …rst order conditions are
βt π (s t )u 0 (cti (s t )) = µi pt (s t )for all t, s t 2 S t
Substituting into the budget constraint:
∞
∑ ∑
t =0 s t 2S t
Jesús Fernández-Villaverde (PENN)
1 t
β π (s t )u 0 (cti (s t )) eti (s t )
µi
Equilibrium with Complete Markets
cti (s t ) = 0
February 12, 2016
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No Aggregate Shocks
Assume that ∑Ii =1 eti (s t ) is constant over time. From perfect
insurance, we know then that cti (s t ) is also constant. Let’s call it b
ci .
Then (cancelling constants)
∞
∑ ∑
t =0
βt π (s t ) eti (s t )
s t 2S t
∞
b
c i = (1
β)
∑ ∑
b
ci = 0 )
βt π (s t )eti (s t )
t =0 s t 2S t
Later, we will see that with no aggregate shocks, β
Then,
b
ci =
Jesús Fernández-Villaverde (PENN)
r
1+r
∞
∑ ∑
t =0
s t 2S t
1
1+r
Equilibrium with Complete Markets
1
= 1 + r.
t
π (s t )eti (s t )
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