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
linear-quadratic 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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