Radner Equilibrium: De…nition and Equivalence
with Arrow-Debreu Equilibrium
Time and Uncertainty In Arrow-Debreu Economies
Econ 2100
Fall 2015
Lecture 24, November 30
Outline
1
Sequential Trade and Arrow Securities
2
Radner Equilibrium
3
Equivalence between Arrow-Debreu and Radner Equilibria
4
Time and Uncertainty
1
2
3
Information Structures
Date-Events
Measurability
Sequential Trading and Arrow Securities
Date 0
Only one physical good becomes a state-contingent commodity: money.
Amounts of money are traded for delivery if and only if a particular state occurs.
These are called Arrow securities (they are …nancial assets).
Date 1
After some state occurs, there are spot markets where physical goods trade at spot
prices (these prices can di¤er across states).
Individuals decide how much to trade of each good depending on (i) the spot
prices, and (ii) how much they have traded in the state-contingent commodity
that corresponds to the realized state.
Expectations Are Crucial
Date 0 trades re‡ect what consumers think will happen in the spot markets.
To decide how much to trade of the state-contingent commodity individuals
make consumption plans for each possible state; these plans depend on their
forecasts of the future spot prices.
Sequential Trading and Arrow Securities
Consider an exchange economy with Xi = RLS
+ .
Notation
zi = (z1i ; ::; zSi ) 2 RS denotes i’s trades in the state-contingent commodity;
these contracts specify amounts to be delivered, or received, of commodity 1 in
each of the s states.
note: these date zero trades can be negative or positive.
q = (q1 ; ::; qS ) 2 RS denotes the prices of the Arrow securities.
xi = (xi 1 ; ::; xiS ) 2 RLS denotes i’s consumption plans vector;
xsi = (x1si ; ::; xLsi ) 2 RL denotes i ’s expected consumption in state s;
x1i = (x11i ; ::; x1Si ) 2 RS represents expected trade in commodity 1, the only
commodity for which there are also date 0 markets.
p = (p1 ; ::; pS ) 2 RLS is the vector of expected prices;
ps 2 RL is the expected price vector for the L goods in state s.
Expected prices and expected consumption plans are elements of RLS .
Note: that many items are “expected”, they represent planned choices.
The consumer makes and plans choices given current and expected prices.
There is no date 0 consumption (for simplicity).
Optimization
Given current prices (q) and expected prices (p), each individual maximizes utility.
A plan includes current trades in the state-contingent commodity (zi ) and
expected spot market purchases (xi ).
Consumers’Choices at time 0
At date 0, consumer i solves the following maximization problem
max
zi 2RS ;x 2RLS
+
U (xi )
subject to
q zi 0
| {z }
budget constraint at time 0
and
p x
| s si
ps ! si + p1s zsi
{z
for each s
}
expected budget constraints at time 1
There will be S budget constraints at date 1.
These constraints are “in expectation” (from the point of view of date 0): xsi
is expected consumption; ps are expected prices.
Note that date 1 trading of good 1 has three components:
buy the desired amount for consumption,
sell the endowment, and
sell the realized state-contingent outcome of date 0 trades.
Date 0 Budget Constraint
Date 0
The budget constraint at date 0 is
S
X
qs zsi
0
s =1
The individual must engage in zero net trades of the state-contingent
commodity.
Since the price vector is non-negative, if she promises to buy good 1 in state s
(zsi > 0), she must also promise to sell it in some other state t (zti < 0).
This is because she has no income at date 0 and she cannot promise to spend
more than she makes
one could add a positive date 0 endowment (and consumption), without
a¤ecting this logic.
The date 0 budget constraint is homogeneous of degree zero in prices.
Therefore, we can normalize date zero prices (sum up to one).
Date 1 Budget Constraints
Date 1
The budget constraints at time 1 are
L
L
X
X
pls xlsi
pls ! lsi + p1s zsi
l =1
for each s = 1; :::; S
l =1
At date 1, consumer i trades in the spot markets corresponding to the realized
state; her wealth re‡ects the outcome of previous trades.
The budget constraint is homogeneous of degree zero in spot prices.
Here, we normalize by assuming that ps 1 = 1 for each s.
Thus, an Arrow security promises to deliver one unit of money (good 1).
Since there are no restrictions on z, we may have zt <
! 1ti for some t:
i sells money ‘short’(she promises to deliver more than she will have); if so, she
must buy some money on the spot market.
the ability to sell short is limited: consumption cannot be negative.
Arrow securities allow wealth transfers across states: at date 0, a consumer
can buy a state s dollar and pay for it with a state t dollar.
If state s occurs, she uses the extra dollar to buy other goods;
if state t occurs, she has one less dollar to buy other goods.
Radner Equilibrium
In equilibirum, expected prices must equal realized prices, and expected trades must
equal actual trades. Everything happens exactly as planned.
De…nition
A Radner equilibrium is composed by state-contingent commodities prices q 2 RS ,
trades zi 2 RS , spot prices ps 2 RL for each s, and consumption xi 2 RLS for
each i such that:
1
for each individual, zi and xi solve
(i) :
max
zi 2RS ;xi 2RLS
+
2
U (xi ) subject to
PS
s =1
qs zsi
(ii) : ps xsi
all markets clear:
I
X
zsi
i =1
0
and
I
X
i =1
xsi
0
ps ! si + p1s zsi
I
X
! si
for each s
for all s
i =1
Spot and forward markets must clear at consumption plans that maximize
individuals’utility, given current and expected prices; the expected prices equal
the realized prices that clear the spot markets.
Radner and Arrow-Debreu Are Equivalent
Under the “rational expectations hypothesis” implicit in Radner equilibirum,
planned behavior equals actual behavior and the timing of decisions is
unimportant.
Proposition (Equivalence of Arrow-Debreu and Radner equilibria)
1
2
1
2
Suppose the allocation x 2 RLSI and the prices p 2 RLS
++ constitute an
Arrow-Debreu equilibrium. Then, there are prices q 2 RS++ and trades
z = (z1 ; ::; zI ) 2 RSI for the state-contingent commodity such that: z , q ,
x , and spot prices ps for each s form a Radner equilibrium.
Suppose consumption plans x 2 RLSI and z 2 RSI and prices q 2 RS++ and
p 2 RLS
++ constitute a Radner equilibrium. Then, there are S strictly positive
numbers 1 ; ::; S such that the allocation x and the state-contingent
commodities price vector ( 1 p1 ; ::; S pS ) 2 RLS
++ form an Arrow-Debreu
equilibrium.
An Arrow-Debreu equilibrium becomes a Radner equilibrium if one
appropriately chooses trades and prices of the state contingent commodity.
A Radner equilibrium becomes an Arrow Debreu equilibrium if one
appropriately modi…es spot prices to make them state-contingent prices.
The proof only needs to show that the two budget sets are the same.
From Arrow-Debreu to Radner
First, choose p1s = qs for each s (we can do this because...).
Write the Arrow-Debreu budget set as
(
S
X
AD
Bi = xi 2 RLS
ps (xsi
+ :
)
! si )
0
s =1
Write the Radner budget set as
8
>
>
>
>
<
R
S
Bi = xi 2 RLS
+ : there is zi 2 R s.t.
>
>
>
>
:
PS
qs zsi
s =1
ps (xs
We need to show these two sets are the same.
0
and
! si )
p1s zsi for all s
9
>
>
>
>
=
>
>
>
>
;
From Arrow-Debreu to Radner
n
PS
Suppose x 2 BiAD = xi 2 RLS
+ :
s =1 ps (xsi
For each s let
zsi =
Then
S
X
s =1
qs zsi =
S
X
1
ps (xs
p1s
p1s zsi =
s =1
S
X
o
0 .
! si )
! si ) :
ps (xs
! si )
0
s =1
and
ps (xs
! si ) = p1s zsi
Thus
8
<
S
x 2 BiR = xi 2 RLS
+ : 9zi 2 R s.t.
:
for all s
9
=
qs zsi 0
.
and
;
! si ) p1s zsi for all s
PS
s =1
ps (xs
From Arrow-Debreu to Radner
Let x 2 BiR =
8
<
:
S
xi 2 RLS
+ : 9zi 2 R s:t:
Then, for some zi 2 RS we have
S
X
qs zsi 0 and ps (xs
9
=
qs zsi 0
.
and
;
! si ) p1s zsi for all s
PS
s =1
ps (xs
! si )
p1s zsi
for all s
s =1
Summing over s yields
S
X
ps (xs
s =1
! si )
S
X
s =1
p1s zsi =
n
PS
Therefore x 2 BiAD = xi 2 RLS
+ :
s =1 ps (xsi
S
X
qs zsi
s =1
! si )
0
o
0 .
From Arrow-Debreu to Radner
We have shown that the budget set for Radner and Arrow-Debreu are the same.
An Arrow-Debreu equilibrium yields a Radner equilibrium
If x ; p is an Arrow-Debreu equilibrium, then
x ; z ; q = (p11 ; :::; p1S ) ;
and zsi =
1
p (xsi
p1s s
! si )
is a Radner equilibrium.
Why?
If the budget sets are the same, maximizing utility in BiAD implies maximizing
utility in BiR .
The spot markets clear because the Arrow-Debreu markets clear.
The state contingent markets clear since
!
I
I
I
X
X
X
1
1
zsi =
ps (xsi ! si ) =
ps
(xsi ! si )
0
p1s
p1s
i =1
i =1
i =1
From Radner to Arrow-Debreu
Choose s such that
the same.
s p1s
= qs for each s. Next, show the budget sets are
We have s p1s zsi = qs zsi for each s by construction.
Write the Radner budget set as
8
9
PS
<
=
0
s =1 qs zsi
S
BiR = xi 2 RLS
and
+ : 9zi 2 R s.t.
:
;
! si )
s ps (xs
s p1s zsi = qs zsi for all s
Summing over all s,
S
S
X
X
qs zsi =
! si )
s ps (xs
s =1
Thus
Therefore, any x
S
X
qs zsi
s =1
2 BiR
BiAD
=
s =1
0)
belongs to
(
xi 2
RLS
+
S
X
s ps
(xs
! si )
0
s =1
:
S
X
s ps
s =1
and the budget sets are the same as desired.
(xs
! si )
)
0
From Radner to Arrow-Debreu
We have shown that the budget set for Radner and Arrow-Debreu are the same
A Radner equilibrium yields an Arrow-Debreu equilibrium
If x ; z ; q ; p form a Radner equilibrium, one can get an Arrow-Debreu
equilibrium x ; p by letting
p =(
1 p1 ; :::;
S pS )
Why?
If the budget sets are the same, maximizing utility in BiR implies maximizing
utility in BiAD , and
all markets clear by construction.
This concludes the proof of the proposition.
A Model with Time and Uncertainty
A State-contingent commodity promises to deliver units of physical goods if
and only if a state occurs.
These contracts are traded now; consumers exchange binding promises to
obtain/deliver certain quantities of goods in given states.
Consumer i’s preferences are de…ned over Xi
contingent commodities.
RL
S
, the space of state
How to extended this setup to more time periods
A State-contingent commodity promises to deliver units of physical goods if
and only if a state occurs at some particular point in time.
Suppose there are T time periods in which consumption is possible, one is
tempted to de…ne Xi RL S T .
State contingent commodity l at time t in state s could be written as xlts .
Draw a tree and think about whether or not this can work for all t and s.
Information Structures
There are T + 1 dates, and S states unfold over time.
This is a tree that starts at 0 and ends at T when some state is realized.
Subsets of S are called events.
De…nition
A collection of events S is an information structure if it is a partition; that is
(i): for any s there is an E 2 S with s 2 E
and
(ii) for any two E ; E 0 2 S, with E 6= E 0 , we have E \ E 0 = ;
As time goes by, there is a family of information structures (S0 ; S1 ; :::; St ; :::; ST )
where the partitions are becoming increasingly …ne.
States that belong to the same event cannot be distinguished.
Since partitions become …ner as time passes, if you learn that two events are
di¤erent you cannot forget it.
Example
If t = 0; 1; 2 and at each t > 0 there are two states:
S0 = (f1; 2; 3; 4g)
S1 = (f1; 2g ; f3; 4g)
ST = (f1g ; f2g ; f3g ; f4g)
Date-Events
There are T + 1 dates, and S states unfold over time.
Subsets of S are called events.
A collection of events S is an information structure if it is a partition; that is
(i): for any s there is an E 2 S with s 2 E
and
(ii) for any two E ; E 0 2 S, with E 6= E 0 , we have E \ E 0 = ;
De…nition
A date-event is a pair (t; E ) where t is a date and E is an event in St .
Date-events “force” particular partitions (and family of partitions) to be
formed according to the unfolding of time along the tree.
Each date-event (except the one at t = 0) has a unique predecessor.
Each date-event (except the ones at t = T ) has a multiple successors.
Measurability
De…nition
A vector v 2 RL (T +1) S is (Lebesgue) measurable with respect to the family of
information partitions (S0 ; S1 ; :::; St ; :::; ST ) if for any lts and lts 0 , we have
vlts = vlts 0 whenever s and s 0 belong to the same element of the partition St .
In words, whenever s and s 0 cannot be distinguished at time t, the values any
measurable vector assigns to the two states cannot be di¤erent.
Example
Suppose we have
S0 = (f1; 2; 3; 4g)
S1 = (f1; 2g ; f3; 4g)
ST = (f1g ; f2g ; f3g ; f4g)
then
vl 01 = vl 02 = vl 03 = vl 04 for all l
and
vl 11 = vl 12 , vl 13 = vl 14 for all l
State Contingent Goods With Many Periods
What happens in an Arrow-Debreu model?
We de…ne state contingent commodities using date-events.
A state contingent commodity is de…ned according to the date-event at which
it is available: lts indicates physical commodity l at time t along the path
that gets to s.
Everything must be measurable with respect to the same family of information
partitions.
Let the consumption space and production sets be subsets of RL (T +1) S .
Thus, for any phisical commodity l, endowment, consumption, and production
must satis…es
! lts = ! lts 0 whenever s and s 0 belong to the same element of the partition St .
xlts = xlts 0 whenever s and s 0 belong to the same element of the partition St .
ylts = ylts 0 whenever s and s 0 belong to the same element of the partition St .
If at time t one cannot distinguish among two states then consumption,
endowments, and production must be the same in those two states.
Given these restrictions, all de…nitions are the same as in Arrow-Debreu.
Problem Set 13 (Incomplete)
Due Monday 7 December
1
Consider the following three consumers (no production), one good economy
with two periods (0 and 1) and two states (s and t). Each consumers utility is
the sum of utility in period 0 and expected utility in period 1; the utility
function is ln(x), where x is the amount consumed of the good, in all periods
and states. Consumers all agre that both future states are equally likely. The
initial endowments of the three consumers are
! 1 = (0; 4; 0) ! 2 = (0; 0; 4) ! 3 = (4; 0; 0) where ! i = (! 0i ; ! 1si ; ! 1ti ).
De…ne an Arrow-Debreu equilibrium and a Radner equilibirum for this two
period economy, and …nd both normalizing the price of the good in period 0 to
be 1.