Perfect foresightΒΆ

We consider an fg model, that is a model with in the form:

\(s_t = g\left(s_{t-1}, x_{t-1}, \epsilon_t \right)\)

\(0 = E_t \left( f\left(s_{t}, x_{t}, s_{t+1}, x_{t+1}\right) \right) \ \perp \ \underline{u} <= x_t <= \overline{u}\)

We assume that \(\underline{u}\) and \(\overline{u}\) are constants. This is not a big restriction since the model can always be reformulated in order to meet that constraint, by adding more equations.

Given a realization of the shocks \((\epsilon_i)_{i>=1}\) and an initial state \(s_0\), the perfect foresight problem consists in finding the path of optimal controls \((x_t)_{t>=0}\) and the corresponding evolution of states \((s_t)_{t>=0}\).

In practice, we find a solution over a finite horizon \(T>0\) by assuming that the last state is constant forever. The stacked system of equations satisfied by the solution is:

\(s_0 = s_0\) \(f(s_0, x_0, s_1, x_1) \perp \underline{u} <= x_0 <= \overline{u}\)
\(s_1 = g(s_0, x_0, \epsilon_1)\) \(f(s_1, x_1, s_2, x_2) \perp \underline{u} <= x_1 <= \overline{u}\)
\(s_T = g(s_{T-1}, x_{T-1}, \epsilon_T)\) \(f(s_T, x_T, s_T, x_T) \perp \underline{u} <= x_T <= \overline{u}\)