Equilibrium

25.4.1 Decision making of private brokers

The conditional expectation and the conditional variance that the representative private brokers have are

*

(P2

— Pi) Bi + (F~

B,

= (E, [P2]-P1)B1 +

(F-E, [

P2]) B2

E-,

(A

-P1)B1 + (F-

p2;

Bi

= (P2-P1)S1 + (P-

-Pi)B*.

(

V'arj

B,

= <rliB\ + V«r^ [(p

~ Pi) ^2

, and

Var^

(P,

- Pi) B, +

P2;

Bz

= *%B\.

We consider a linear equilibrium in the following. Since a linear combination of variables that follow normal distributions also follows normal distributions, the equilibrium prices follow normal distributions.

The utility of the representative private broker at ¿=1,2 is

Et [Up] = - exP (~pEt [(P2 - P,) Bt + (F- Pi) B2] - ^Vwrt [(P, - Pl)Bl + (F - Pi) B,

The first-order condition is

25.4.2 Decision making of nations

By the sellout constraint of each nation i Si;2 = Wi;2 and by the market clearing condition Si;2 +MB2+n=0,

The variance conditional on the events by the time 1 is equal to

Take the expectation of both sides conditional on the events by the time 1, and obtain

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