EEP101/ECON125 Spring 00Prof.: D. ZilbermanGSIs: Malick/McGregor/St-PierrePROBLEM SET 5 Solutions1.Suppose r=0.1, price (P) of timber is $10 per boardfoot, and the volume of trees in a stand obeys thefunction Q(T) = 12 T2- 1/3 T3.a)Consider a single rotation. Set up the profit maximization problem and derive the equilibriumcondition. Solve for the optimal rotation length (T*). Be sure to check to see if your answer makessense.The problem assumed no harvesting costs. The grower’s objective is to choose the rotation length tomaximize profits (or revenues in this case). Of course, since this profit is realized T years from now, weneed to discount this future profit to the present. This objective is expressed asrTTeTPQVPMAX-=Π)(..Maximizing this objective with respect to T gives one first order condition which implicitly defines T*:*****)(*)('0*)(*)('0:rTrTrTrTeTrPQeTPQeTrPQeTPQdTdFOC----=?=-?=ΠSince e-rtand P cancel, this leaves us withor,( 29rTQTQ=)('(1)This says that the profit maximizing rotation length is such that the growth of the tree volume is equal tothe interest rate. Notice that Q’(T) refers to change in growth from one period to the next and representsthe change in volume of trees. The growth rate of trees multiplied by 100 is the percentage growth oftrees from one period to the next. Using the information given in the description of the problem, andQ’(T) = 24T-T2, equation 1 is rewritten as1.0T1/3-T1224322=-TT(2)To find T*, we need to solve equation (2) for T. After factoring out a T and rearranging terms, equation(2) can be rewritten as*)(*)('TrQTQ=
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