Your uncle plans to invest in the tuna fishery for the coming 15 years. The natural...

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Your uncle plans to invest in the tuna fishery for the coming 15 years. The natural growth function of the tuna population (X₁) is rXln(K/X). The amount of tuna caught during a season (Yt) is a function of effort (E) and X, according to Y-qEX. The season market price (p) depends on the quantity harvested: p-a/(1+Y₁). The seasonal fishing cost per unit effort is c-b ln(1+E). The total cost of all effort is the number of effort units multiplied by this per unit cost. Finally, no matter how many tunas your uncle catches, he has to leave some fish behind when he withdraws from the fishery after 15 years. Note, however, that he does not value any of the remaining stock. Assume that the discount rate, 8, is equal to 10% and the following parameter values: a = 200, b = 10, q= 0.2, Xo =40, r= 1.8 and K = 50. 1. Write the problem of your uncle if he wants to maximize profit over the 15 year period. Take a picture of the setup and place it in sheet 7 or save it in a separate pdf file. 2. Also in sheet 7, setup the problem in excel using an initial guess of 1 for all Et. How many boats should he use each season? To answer this, copy everything into sheet 8 and use solver to derive the optimal number of boats. In the same sheet, plot Xt and Et as a function of time on separate graphs. Entitle your graphs, stock over time and effort over time respectively. 3. Setup sheet 9 as an infinite time horizon problem with time periods 0 to 14 as a transition period using an initial guess of 1 for all Et. Time period 15 will be the steady state period. Copy everything from sheet 9 to sheet 10 and use solver to solve the problem. Hint: In steady state: YT+1=F(XT+1). Also, you can use, ET+1=YT+1/qXT+1=F(XT+1)/qXT+1. Your uncle plans to invest in the tuna fishery for the coming 15 years. The natural growth function of the tuna population (X₁) is rXln(K/X). The amount of tuna caught during a season (Yt) is a function of effort (E) and X, according to Y-qEX. The season market price (p) depends on the quantity harvested: p-a/(1+Y₁). The seasonal fishing cost per unit effort is c-b ln(1+E). The total cost of all effort is the number of effort units multiplied by this per unit cost. Finally, no matter how many tunas your uncle catches, he has to leave some fish behind when he withdraws from the fishery after 15 years. Note, however, that he does not value any of the remaining stock. Assume that the discount rate, 8, is equal to 10% and the following parameter values: a = 200, b = 10, q= 0.2, Xo =40, r= 1.8 and K = 50. 1. Write the problem of your uncle if he wants to maximize profit over the 15 year period. Take a picture of the setup and place it in sheet 7 or save it in a separate pdf file. 2. Also in sheet 7, setup the problem in excel using an initial guess of 1 for all Et. How many boats should he use each season? To answer this, copy everything into sheet 8 and use solver to derive the optimal number of boats. In the same sheet, plot Xt and Et as a function of time on separate graphs. Entitle your graphs, stock over time and effort over time respectively. 3. Setup sheet 9 as an infinite time horizon problem with time periods 0 to 14 as a transition period using an initial guess of 1 for all Et. Time period 15 will be the steady state period. Copy everything from sheet 9 to sheet 10 and use solver to solve the problem. Hint: In steady state: YT+1=F(XT+1). Also, you can use, ET+1=YT+1/qXT+1=F(XT+1)/qXT+1.

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1 Problem formulation for maximizing profit over the 15year period Variables Xt Tuna population in year t Et Effort in year t ... View the full answer