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Unformatted text preview: 4.4. APPLICATIONS TO OPTIMAL BEHAVIOR 535 One possible explanation for this discrepancy is that the model doesnt account for the energetic costs of traveling. Cowie adjusted the model to account for these energetic costs and the resulting prediction is plotted as a solid curve in the figure above. In the problem set you are asked to account for these energetic costs. a50 The marginal value theorem has a simple graphical interpretation, but also some limitations; both of which are explored in the next example. Example 5. Optimal time to harvest Over a sixty year period a forestry company has collected data on the profit P ( t ) of stands harvested at various ages of t years. Initially, P ( t ) is negative because the costs required to bring in the heavy equipment needed to harvest the trees exceeds the value of the harvest itself. Once the trees reach a certain size a profit is possible and it steadily increases as the stand of trees ages. The company found that the function that best fit their data has the following graph: 0.5 1 1.5 2 t 20 40 60 P where the profit P is measured in thousands of US dollars and t is measured in years. a. The company wants to maximize the profits it makes per year not taking into account the costs needed to clear the land during the harvesting cycle (i.e. the so-called clear cutting part of the operation). Write down the function A ( t ) that they want to maximize and illustrate the solution graphically. b. The company wants to maximize the profits it makes per year, but now taking into account the costs needed to clear the land during the harvesting (i.e. clear cutting operation) Write down the function B ( t ) that they want to maximize and illustrate the solution graphically. c. Discuss the differences between the two solutions and the role of the Marginal Value Theorem....
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- Fall '10