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Unformatted text preview: Problem Set 3 Solutions Numerical Questions Renewable Resources and Trade 1. (35 Points) Suppose the Tru ula trees can grow. Each period they seed and produce new trees with a growth rate described by the equation g ( S ) = 4 S . 004 S 2 , where S is the stock (#) of Tru ula trees. (a) (5 Points) Graph the growth rate as a function of Tru ula tree stock. (b) (4 Points) Calculate the maximum sustainable stock level if there is no harvesting of trees. Max. sustainable stock: S CC = max { S : g ( S ) = 0 } = { S : 4 . 004 S = 0 } = 1000 . (c) (8 Points) Figure out the maximum sustainable harvest level and the stock level that supports it. Max. sustainable harvest stock: S MSY = { S : g ( S ) = 0 } = { S : 4 . 008 S = 0 } = 500 . Max. sustainable harvest level: H MSY = g ( S MSY ) = 4(500) . 004(500) 2 = 1000 . (d) (6 Points) If each Tru ula tree that is harvested can produce one Thneed, what's the production possibilities frontier (PPF) of Thneed and Tru ula trees each period (in steady state)? (e) (3 Points) If the Onceler cares only about selling Thneed (and the costs of harvesting do not depend on the stock of Tru ula trees), what sort of indi erence curves would represent the 1 Onceler's preferences on this graph? If the Onceler truly did not care about the amount of trees in the ground, but only for how they could produce Thneed, then his indi erence curves would be straight lines, perpendicular to the Thneed axis (so horizontal in the graph above). (f) (3 Points) What steadystate level of Thneed and Tru ula trees would he desire each period? The Onceler would want the most Thneed per period, which would be the maximum sustain able yield: (Thneed, Trees) = (1000, 500) . (g) (3 Points) Since the Lorax cares only about the Tru ula trees, what sort of indi erence curves would represent the Lorax's preferences on the graph? If the Lorax cares only about the amount of trees in the ground, his indi erence curves would be straight lines as well, perpendicular to the Trees axis (so vertical in the graph above). (h) (3 Points) What steadystate level of Thneed and Tru ula trees would the Lorax desire each period? The Lorax would want the most trees in the ground, which would be the carrying capacity stock: (Thneed, Trees) = (0, 1000) . 2. (10 Points) Suppose to the dismay of the Lorax, word of the Onceler's creation spreads far and wide and the going price of a Thneed on the global market is $3.98. As people learn about the Thneed, so too do they learn about the trees from whence it comes. Many people come to visit the forest; those that don't hope to see it some day, or just like its existence. The Lorax, in seeing simultaneously the threat to the trees as well as the increased appreciation, creates an adoptatree program through which people can pay to save the glorious Tru ula trees....
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 Spring '09
 ZILBERMAN

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