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Alex and Mitch are two farmers who grow vegetables on common land. Each farmer gets a benefit from the seeds that he plants, but the overall size and quality of the individual vegetables decreases as the number of seeds planted increases (since each vegetable has less room to grow and gets a smaller share of nutrients from the soil). If SA represents the total number of seeds planted by Alex and SM is the total number planted by Mitch, then the total benefit to Alex will be as follows: SA(60 - SA - SM), or (SA60 - SA² SMSA). The marginal benefit to Alex from planting additional seeds is, therefore, equal to (60 - 2SA - SM). Assuming the marginal cost of planting is zero, the best-response function that gives the optimal number of seeds Alex should plant, as a function of the number of seeds Mitch chooses to plant, is represented by

Alex and Mitch are two farmers who grow vegetables on common land. Each farmer gets a benefit from the seeds that he plants, but the overall size and quality of the individual vegetables decreases as the number of seeds planted increases (since each vegetable has less room to grow and gets a smaller share of nutrients from the soil). If SA represents the total number of seeds planted by Alex and SM is the total number planted by Mitch, then the total benefit to Alex will be as follows: SA(60 - SA - SM), or (SA60 - SA² SMSA). The marginal benefit to Alex from planting additional seeds is, therefore, equal to (60 - 2SA - SM). Assuming the marginal cost of planting is zero, the best-response function that gives the optimal number of seeds Alex should plant, as a function of the number of seeds Mitch chooses to plant, is represented by

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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SEEDS PLANTED BY ALEX 100 80 60 40 20 0 20 40 100 80 SEEDS PLANTED BY MITCH 60 Alex's BRF o- Mitch's BRF Nash Eq. Quantity Alex's BRF 2 Help Clear All When each farmer plants the equilibrium number of seeds, each farmer gets a benefit of each farmer plants 15 seeds, each farmer will get a benefit of . If, instead, Suppose Alex discovers a new method of fertilizing his seeds, such that his best-response function is now given by SA = 40 - 0.5(SM). Use the green line (triangle symbols) to plot Alex's new best-response function after this change on the original graph. As a result of this change, the optimal number of seeds for Alex to plant quantity for Mitch to plant and the optimal
Transcribed Image Text:SEEDS PLANTED BY ALEX 100 80 60 40 20 0 20 40 100 80 SEEDS PLANTED BY MITCH 60 Alex's BRF o- Mitch's BRF Nash Eq. Quantity Alex's BRF 2 Help Clear All When each farmer plants the equilibrium number of seeds, each farmer gets a benefit of each farmer plants 15 seeds, each farmer will get a benefit of . If, instead, Suppose Alex discovers a new method of fertilizing his seeds, such that his best-response function is now given by SA = 40 - 0.5(SM). Use the green line (triangle symbols) to plot Alex's new best-response function after this change on the original graph. As a result of this change, the optimal number of seeds for Alex to plant quantity for Mitch to plant and the optimal
7. Best responses and the tragedy of the commons Aa Aa Alex and Mitch are two farmers who grow vegetables on common land. Each farmer gets a benefit from the seeds that he plants, but the overall size and quality of the individual vegetables decreases as the number of seeds planted increases (since each vegetable has less room to grow and gets a smaller share of nutrients from the soil). If SA represents the total number of seeds planted by Alex and SM is the total number planted by Mitch, then the total benefit to Alex will be as follows: SA(60 - SA - SM), or (SA60 - SA² SMSA). The marginal benefit to Alex from planting additional seeds is, therefore, equal to (60 - 2SA - SM). Assuming the marginal cost of planting is zero, the best-response function that gives the optimal number of seeds Alex should plant, as a function of the number of seeds Mitch chooses to plant, is represented by Use the blue line (circle symbols) to plot Alex's best response function (BRF) on the following graph, with the total number of seeds he plants on the vertical axis as a function of the total number Mitch plants (horizontal axis). Then, use the orange line (square symbols) to plot Mitch's best-response function (assuming his total benefit is given by SM(60 - SM - SA), and thus he responds to Alex in the same way Alex responds to him), with the total number he plants (horizontal axis) as a function of the total number Alex plants (vertical axis). Finally, place the black point (X symbol) on the graph to indicate the Nash equilibrium number each plants, given the other's equilibrium choice. Dashed drop lines will automatically extend to both axes.
Transcribed Image Text:7. Best responses and the tragedy of the commons Aa Aa Alex and Mitch are two farmers who grow vegetables on common land. Each farmer gets a benefit from the seeds that he plants, but the overall size and quality of the individual vegetables decreases as the number of seeds planted increases (since each vegetable has less room to grow and gets a smaller share of nutrients from the soil). If SA represents the total number of seeds planted by Alex and SM is the total number planted by Mitch, then the total benefit to Alex will be as follows: SA(60 - SA - SM), or (SA60 - SA² SMSA). The marginal benefit to Alex from planting additional seeds is, therefore, equal to (60 - 2SA - SM). Assuming the marginal cost of planting is zero, the best-response function that gives the optimal number of seeds Alex should plant, as a function of the number of seeds Mitch chooses to plant, is represented by Use the blue line (circle symbols) to plot Alex's best response function (BRF) on the following graph, with the total number of seeds he plants on the vertical axis as a function of the total number Mitch plants (horizontal axis). Then, use the orange line (square symbols) to plot Mitch's best-response function (assuming his total benefit is given by SM(60 - SM - SA), and thus he responds to Alex in the same way Alex responds to him), with the total number he plants (horizontal axis) as a function of the total number Alex plants (vertical axis). Finally, place the black point (X symbol) on the graph to indicate the Nash equilibrium number each plants, given the other's equilibrium choice. Dashed drop lines will automatically extend to both axes.
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