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Example text

In econometric practice, two approximate solutions have been proposed, one taking a feedback form and the other an open-loop form. To obtain an approximately optimal feedback rule, one can work with a certainty equivalent model by omitting the random term 8 t in (18), linearize the resulting model, and find the optimal feedback rule for the linearized model. Specifically, after dropping 8 t from (18) and given the initial state Yo and the vector At for t, t = 1, ... , T, one assumes a tentative policy path x~ and solves the certainty-equivalent model Yt = for Yt.

XT , which is an open-loop policy, and applies only Xl to the original stochastic control problem. The latter yields a set of optimal feedback control equations XI = GIYI~1 + gl (t = I, ... , T) which are valid for all T periods. Thus the solution is in a closed-loop feedback form. When these feedback control equations are combined with the original model (2), the combined equations determine the dynamic behavior of all XI and YI through time, and the dynamic properties of the system under optimal control can be conveniently studied.

Gandolfo and A. Isidori for helpful comments. 43 2. Point controllability is also known as output controllability or pointwise reproducibility, while path controllability is also known as perfect controllability or as functional reproducibility. 3. Buiter [5] presents an economic model which is not statically controllable (or control1able across steady states) while it is path control1able. 4. We follow here Aoki's definition of path control1ability, refering therefore to the case in which either (a) the target function y*(t) can be any function in the target space and the admissible control space includes impulse control functions, or (b) the admissible target space is restricted only to sufficiently smooth target functions and the admissible control space does not contain impulse functions.

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