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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
f
ow of pollution. The evolution of
S
is given by
˙
S
=
x
−
g
(
S
;
α
)
,where
gisincreasingandconcavein
S
,and
α
is a parameter of the function g. The
instantaneous (
f
ow) payo
f
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
f
ow of
welfare from time 0 to in
F
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
the0isoclines,thedirectionalarrows,andthestablesaddlepath(theoptimal
control rule).
(iii) Write the equations that determine the optimal steady state.
(iv) Explain how to
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This note was uploaded on 08/01/2008 for the course ARE 263 taught by Professor Karp during the Fall '06 term at University of California, Berkeley.
 Fall '06
 KARP

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