Market equilibrium
Equilibrium of demand and supply yields
|
|
|
|
zj,t= |
|
|
⎛
⎜
⎜
⎝ |
|
zj,th |
⎞
⎟
⎟
⎠ |
= S,
(10) |
where
S is the total number of shares available in the market. Hence in equilibrium the total number of shares demanded by the agents should be equal to the total number of shares available. Equilibrium equation (6.10) can be rewritten as
where
Nth is the number of agents having belief
h at time
t. If both sides of equation (6.11) are divided by the total number of agents
N trading in the market, then
where
nth=
Nth/
N is the fraction of agents with belief
h and
s=
S/
N is the number of shares available per agent.
Further, BH assume that the conditional variance Vth(Pt+1+Dt+1)=σ2 is constant through time and equal for all beliefs. This assumption of homogeneous, constant beliefs on variance is made primarily for analytical tractability. Notice however that heterogeneity in conditional expectations in fact leads to heterogeneity in conditional variance as well, but this second-order effect will be ignored. Equilibrium equation (6.12) can be solved for Pt to yield the equilibrium price
|
Pt= |
|
|
|
|
{ |
nth Eth(Pt+1+Dt+1) |
} |
- |
|
a σ2 s.
(13) |
If the number of outside shares per trader is zero, i.e. if
s=0, then the equilibrium price at time
t is equal to the net present value of the average expected price plus dividends at time
t+1.
Evolutionary dynamics
The fraction of agents who choose to invest according to belief or forecasting rule
h are determined by a discrete choice model. Every agent chooses the belief with the highest
242