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exam2001

exam2001 - Final Exam ARE 261(2001 Answer all parts of all...

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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 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). (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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