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exam2002 - Final Exam ARE 261(2002 Answer all parts of all...

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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 0-isoclines, the directional arrows, and the stable saddle path (the optimal control rule).
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