The total wealth of all N agents at time t is equal to
|
|
| Wt= |
|
| |
| R |
|
Wj,t-1 + (Pt+Dt-R Pt-1) |
|
zj,t-1= |
|
| |
R Wt-1 + (Pt+Dt-R Pt-1) s = |
| |
R (Wt-1-s Pt-1) + s Dt + s Pt, |
|
(29) |
where
s is the total number of shares available to trade.
R (
Wt-1-
s Pt-1) +
s Dt is the total amount of money invested in the risk free asset and
s Pt is the total amount of money invested in the risky asset by all agents at time
t. If the total initial market wealth is equal to
W0, then the total wealth at time
t is equal to
|
|
| Wt= |
| Rt (W0-s P0) + s |
|
(Ri Dt-i) + s Pt = |
|
| |
| (W0-s P0) + |
|
|
( |
Ri (rF (W0-s P0)+s Dt-i) |
) |
+ s Pt . |
|
|
(30) |
Naturally
Dt≥ 0 at each date. Suppose that
W0=
M+
s P0, then the total amount of money invested in the risk free asset,
Wt-
s Pt, at time
t is greater than or equal to zero for
t=0,...,
T, if the initial amount of money invested in the risk free asset,
M, is positive. We assume throughout this chapter that
M ≥ 0.
We assume that at time t each of the N agents hands over his demand function (6.27) for the risky asset to a market auctioneer. The auctioneer collects the demand functions and computes the final equilibrium price Pt so that the market clears. Equilibrium of demand and supply so that the market clears yields
|
|
|
|
zj,t =
|
|
|
⎛
⎜
⎜
⎝ |
|
zj,th |
⎞
⎟
⎟
⎠ |
=s.
(31) |
By substituting (6.27) in (6.31) the equilibrium equation (6.31) can be rewritten as
|
|
|
|
⎛
⎜
⎜
⎝ |
|
|
|
⎞
⎟
⎟
⎠ |
=
|
|
|
⎛
⎜
⎜
⎝ |
|
|
|
Wj,t |
⎞
⎟
⎟
⎠ |
=
|
|
|
⎛
⎜
⎜
⎝ |
|
⎞
⎟
⎟
⎠ |
=s, |
|
|
|
(32) |
Here
Wth is the total wealth of all agents who use forecasting rule
h at time
t. Recall that the demand
yth is a function of
Pt. To solve the equilibrium equation (6.32) for
Pt
250