August 2006
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Answer:
Let the demand for the contract from A and B be a and b respectively. Negative values of a or b correspond to a desire to sell the contract short. Expected log wealth for A will is
p*log(X+a*(1-r))+(1-p)*log(X-a*r)
This has derivative with respect to a of
p*(1-r)/(X+a*(1-r)-(1-p)*r/(X-a*r)
Setting the derivative to 0 and solving for a we find
p*(1-r)*(X-a*r)=(1-p)*r*(X+a*(1-r))
( p*(1-r)-(1-p)*r)*X=((1-p)*r*(1-r)+p*r*(1-r))*a
(p-r)*X=r*(1-r)*a
a=(p-r)*X/(r*(1-r))
Similarly
b=(q-r)*Y/(r*(1-r))
Setting a+b=0 and solving for r we obtain
(p-r)*X/(r*(1-r)) + (q-r)*Y/(r*(1-r)) = 0
(p-r)*X + (q-r)*Y = 0
p*X+q*Y=(X+Y)*r
r=(p*X+q*Y)/(X+Y)
So the market clearing price is just the weighted average of the opinions of A and B as to the correct price.
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