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Econ 251a
John Geanakoplos
CAPM with Quadratic Utilities
1 ArrowDebreu Equilibrium
Given uncertainty about the future, Ken Arrow and Gerard Debreu (both members
of the Cowles Foundation, both future Nobel Prize winners, though Debreu did not
get promoted to tenure at Yale) suggested that the economy could be understood
just like any trading economy, by interpreting the same commodity in two di
f
erent
states of nature as two di
f
erent commodities. Instead of trading apples for oranges,
one could trade apples today for the Arrow security promising apples in state of
nature s tomorrow. The utility
W
h
(
x
0
,x
1
2
3
)
to agent h of the consumption
bundle
(
x
0
1
2
3
)
can be taken to be the expected utility of consumption
u
h
0
(
x
0
)+
P
S
s
=1
u
h
(
x
s
)
γ
h
s
,
where
γ
h
s
is the probability
h
assigns to state s, and
u
h
(
x
)
is his von
Neumann Morgenstern utility of consumption.
The price of an Arrow security
s
is determined just as the price of any commodity
in an exchange economy, by its marginal utility of consumption. The marginal utility
of
W
h
is the probability
γ
s
multiplied by the vNM marginal utility of consumption
in state s. The latter rises as consumption declines. Thus we get the conclusion
that Arrow prices are determined by probablities, but states where consumption is
small get weighted by more than their probability, while states where consumption
is relatively large get weighted by less than their probabilities.
By no arbitrage, the price of an asset is just its payo
f
s times the corresponding
Arrowprices.Itfollowsfromwhatwejustsaidthatassetswhichpayo
f
when people
are poor will be worth more than assets that have the same expected payo
f
, but tend
to pay in states when people are rich.
2 Quadratic Utility and CAPM
Tjalling Koopmans, the director of the Cowles Foundation, suggested that his PhD
student Harold Markowitz work on the problem of
f
nance by assuming that all the
vNM utilities are taken to be quadratic and that everyone agree on the probablities.
3 Example with no aggregate risk
Consider a world with 3 states of nature next period, three
f
rms and objective prob
abilities of each state. Two consumers,
A
and
B
have endowments in period zero
of
e
α
0
= 133
.
5
e
β
0
=6
6
.
5
1
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α
owns
f
rm
A
and
β
owns
f
rms
B
and
C
.
Period 0 Period 1
ABC
Totals
.
25
50
150 300
500
•
%
−→
&
.
25
100 180 220
500
.
5
75
365
60
500
Firm
A
has a payo
f
of 50 if state 1 is realized, 100 if state 2 occurs and 75 if state 3
is realized. The probabilities of the three states are .25,.25,.5, respectively.
We wish to answer the question: How much is each
f
rm worth? For example,
what should each
f
rm’s stock market valuation be?
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 '09
 GEANAKOPLOS,JOHN

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