Final Exam ARE 261 (2002)
Answer all parts of all three questions.
The value of the question is
given in parentheses next to the question number. Don’t spend more than 3
hours on the exam.
If you feel pressed for time, outline your answer without
providing details.
Question 1)
(40 points)
S
is an index of environmental quality and
x
is
the
fl
ow of pollution. The evolution of
S
is given by
˙
S
=
x
−
g
(
S
;
α
)
, where
g is increasing and concave in
S
, and
α
is a parameter of the function g. The
instantaneous (
fl
ow) payo
ff
is
x
−
x
2
2

S
2
2
and the instantaneous discount rate
is
r
.
You want to maximize the present discounted integral of the
fl
ow of
welfare from time 0 to in
fi
nity. The initial condition
S
0
is given.
(i) Write the current value Hamiltonian and necessary conditions.
(ii) Sketch the phase portrait in
(
S, x
)
, space. The portrait should include
the 0isoclines, the directional arrows, and the stable saddle path (the optimal
control rule).
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 Fall '06
 KARP
 Steady State, Convex function, steady state stock

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