Final Exam ARE 261 (2001)
Answer all parts of all three questions.
The value of the question is
given in parentheses next to the question number.
Question 1)
(45 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).
(iii) Write the equations that determine the optimal steady state.
(iv) Explain how to
fi
nd the comparative statics of the steady state with
respect to
α
. You should set up the comparative statics expression and ex
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 Fall '06
 KARP
 Optimization, optimal control rule, unknown threshold

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