Dynamic Playing-The-Field Contest without Fundamental Risk

If there is no fundamental risk, i.e.

, then the population dynamic for the playing-the-field contest reduces to the following:

Solving ˙x₂(t) = 0 yields the steady states of the population dynamic in (30). The states x₂ = 0 and x₂ = 1 are always stationary in the dynamic. If noise traders on average have negative sentiment, i.e., P^¤ < 0, then the expected return differential is negative for all x(t). This means that starting from any nonstationary states

, the population share of the noise traders will decline to zero in the long run. On the other hand, if noise traders on average have positive sentiment, i.e., P^¤ > 0, then for some parameter values the expected return differential is also negative. In this case, the population share of the noise traders will also decline to zero in the long run. In addition, there may be one interior steady state, x₂ = µ,
where µ is given by

. Interestingly, it turns out
that the interior steady state is unstable and the long run equilibrium depends on the current population state relative to the interior steady state. To see this, note that if the current noise trader share is below the interior steady state, i.e., x₂(t) < µ, then the expected return differential is always negative and, as a result, the population share of the noise traders will decline to zero in the long run. On the other hand, if
the current noise trader share is above the interior steady state, i.e., x₂(t) > µ, then the expected return differential is always positive and the population share of the noise traders will increase to one. Such a dichotomy of long-run equilibria implies that the unique interior steady state itself is asymptotically unstable. Theorem 2.1 summarizes the dynamic equilibrium result

Prof. F. Albert Wang

Next: Theorem 2

Summary: Index