, we first have to determine how much wealth is assigned to each belief, Wth, by all agents. In Appendix B we show that under the assumption that at date t=0 wealth is equally divided among agents and under the assumption that each agent has zero market power at each date, it is true that the fraction of total wealth invested according to belief h at time t converges in probability to the probability that an agent chooses belief h, that is
Notice that we use slightly different choice probabilities as in (6.15) by introducing a lower bound on the probabilities as motivated by Westerhoff (2002); see Appendix B for details.
The heterogeneous agents model equilibrium equation
Now that we have shown that the fraction of total market wealth invested according to a certain belief converges in probability to the probability that the belief is chosen, we can solve equation (6.32) for
Pt to get the equilibrium price. Equilibrium equation (6.32) can be rewritten as
|
|
⎛
⎜
⎜
⎝ |
|
|
⎞
⎟
⎟
⎠ |
|
|
|
|
⎛
⎜
⎜
⎝ |
|
⎞
⎟
⎟
⎠ |
=s |
|
, |
or equivalently
The left hand side of equation (6.34) is the demand for the risky asset as a fraction of total wealth, while the right hand side is the worth of the supply of shares as a fraction of the total wealth. Using (6.29) the right hand side can be rewritten to
| S(Pt)= |
| Pt |
|
| R |
⎛
⎜
⎜
⎝ |
|
-Pt-1 |
⎞
⎟
⎟
⎠ |
+Dt+Pt |
|
|
. |
The first and second derivative of the supply function
S(
Pt) are equal to
where
c=
R (
Wt-1/
s-
Pt-1)+
Dt is the amount of money invested in the risk free asset per risky share. We assume that
c>0 (see also equation (6.30)). Thus for
Pt≥ 0 the supply function is a continuous monotonically increasing and concave function of
Pt which starts
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