Handout 13th Lecture, ECON 5200
Uncertainty and time
Kjell Arne Brekke
November 12, 2010
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1.1
Application of general equilibrium
Lindahl equilibrium
We consider an economy with a private and a public good
X
x2 = q = f (z) is the public good
X
xi1 + z =
! i market clearing in the private good
is i’s utility
Now, this does not correspond to a Walrasian equilibrium. Suppose however that the
…rm produces an public good for each consumer
q1 = q2 = ::: = qI = q = f (z)
and with market clearing in each market
xi2 = q2
with individualized prices
s.t. p1 xi1 + pi2 xi2
max ui (xi1 ; xi2 )
= wi
and pro…t maximizing production of the public good
X
max(
pi2 )f (z) p1 z
z
This is called a Lindahl equilibrium. In a sence it presumes that each consumer in fact
pays his willingness to pay (at the margin) for the public good. We note that if this was
the case then
X
(
pi2 )f 0 (z) = p1
That is the total willingness to pay for an additional unit of the public good should equal
the cost. This solution is of course Pareto e¢ cient.
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If individuals do in fact pay equally much for each unit, then increasing production at
a point where
X
(
pi2 )f 0 (z) < p1
would not be a Pareto improvement. Some will bene…t and some will lose.
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Uncertainty and Arrow Debreu equilibrium
We start with a straightforward extension of the general equilibrium model. Suppose that
there are a list of state of the world
f! 1 ; ::::; ! S g = fsnow, rain, sunnyg
as well as L commodities. We also have two periods, where all consumption happend
in period 1 but markets are only open in period 0. We now introduce a new set of
state.contingent commodities
xls
is a delivery of xls units of commodity l contingent on the state being ! s . The true state
is not known in period 0 when the market is open but will be revealed before consumption
in period 1.
Now, with a seperate market for each state dependent commodity, there will be equilibrium prices
pls
for each state dependent commodity.
The …rst and second welfare theorem applies as this formally is just an ordinary
Walrasian equilibrium.
2.1
Sequential trading - Radner
We do not today trade "appels tomorrow if it snows". We trade those apples tomorrow
after we have observed the state of the world. On the other hand: We have two kind of
decisions: How do we allocate our wealth within each state and how do we allocate our
entire wealth between di¤erent states. In some states we may want to have more resources
than in others. Suppose that there is only one commodity where the markets are open in
period 0, and that all other markets open …rst after the true state is revealed.
Now, when consumers trades in period 0, markets in period 1 are not yet open, hence
prices are not yet known. But we assume that consumers have perfect foresight and
correctly predicts all prices. The market in period 0 are called forward markets, and the
markets that opens in period 0 are sport markets. The resulting equilibrium is called an
Radner Equilibrium.
Proposition 19.D.1 The allocation emerging from an Arrow-Debreu equilibrium can
also be supported as a Radner equilibrium. As only relative prices in spot markets matter
we may have to transform prices such that
ps =
2
s ps
2.2
State prices
We need only forward trading in one commodity since what we need is to transfer wealth
between states. This is essentially what …nancial instruments does. As microeconomic
theory does not have a good theory of money we simply make a leap here, study the single
commodity traded between states and call it money, and ignore the other goods - those
markets are not yet open.
To gain some insight consider …rst the case where there is a contingent claim for each
state. The contingent claim 1s pays a $ in period 0 if state s emerges otherwise nothing.
And there is one asset for each state s = 1; :::; S. The price of each asset is denoted s .
Now consider a more standard asset with an uncertain return, it pays xs in state s.
Note that this is equivalent to buying
a portfolio of rs units of the contingent claim 1s for
P
each state. This will cost x0 =
x
s s : which should be the price of r in period 0. Note
that if we buy one unit of each contingent claim we get 1 for sure in period 1, which is
1
in period 0. But uisng the value formula above we see that
worth 1+r
F
X
1
1 X
=
qs
s =
1 + rF
1 + rF
qs = (1 + rF ) s
X
qs = 1 a kind of probabilities
Note that with these probabilities
x0 =
P
qs x s
Eq (x)
=
1 + rF
1 + rF
Thus expected return with these probabilities is equal to the risk free rate of interest.
THis is the basis for a technique used in …nanace. Adjust the expected return to assets
to the risk free rate, keep the volatility and value e.g. an option using the risk free rate
of return.
2.3
No arbitrage
If there are no such contingent claims 1s ? Well if there are two states s = 1; 2 and two
stocks with return (2,1) and (1,1) we get a (1,0) by owning one (2,1) and sitting short
(1,1). And (0,1) by owning 2 (1,1) and sitting short (2,1). Thus state prices should follow
from asset prices. In general let there be K assets with return
rks
and with prices in period 0 equal to
qk
If a portofolio is for free:
q
0
but pays something
rs
0 for all s, with strict inequality for at least one
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Then this is an arbitrage.
Proof. There is no arbitrage if and only if there exist state prices such that
X
qk =
s rks for all k
2.4
Incomplete markets
Note that if there are many states and few assets, we may not be able to generate a 1s
for all s. So some of the state prices are ambigous. We cannot trade on the asset market
to get rid of all the oil risk Norway is facing. That there are commodities for which there
is no separate market is thus a real phenomena here.
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Intertemporal equilibrium
As pointed out, it is trivial to extend general equilibrium theory to several time periods.
Disregaring uncertainty now, we simply label each commodity with its time period, and
open one market for each. Some issues arises about perfect forsight, but this is as discussed
for uncertainty.
There are some special issues involed however, with intertemporal consistency.
What would be the problem of assuming for three periods
U (c1 ; c2 ; c3 ; ::::) =
X1
t
u(ct )
There are two issues: Separability
Given separability we may consider the choice between c2 and c3 independently, but
the rankin changes over time. Plans are inconsistent.
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