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Problem Set 7
The purpose of this problem set is to be sure that you can solve a linear
quadratic control problem, and to be sure that you know what the Principle
of Certainty Equivalence says, and when it holds.
1. (Hoel and Karp 2001, Taxes versus Quotas for a Stock Pollutant) Equa
tion 16 in Appendix A is the Dynamic Programming equation for a
linearquadratic problem with additive errors.
Using the sketch below
this equation, solve the dynamic programming equation. The solution
requires that you …nd the control rule and the unknown parameters
½
0
;
½
1
; ½
2
.
The paper contains the expressions for
½
0
and
½
2
so you need
only con…rm these equations and …nd the expression for
½
1
, and …g
ure out the relation between these coe¢cients and the optimal control
rule.
(The optimal control rule is a linear function of the state.
The
coe¢cients of this linear function depend on
½
i
.)
Note that the control rule is independent of the variance of the stochas
tic term; in other words, we would obtain the same control rule if we
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 Fall '06
 KARP

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