Consider a village of two farmers, 1 and 2. Each summer they graze their goats on...

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Consider a village of two farmers, 1 and 2. Each summer they graze their goats on the village green. The cost to each farmer for buying and caring for a goat is c20. The value to each farmer of grazing a goat on the green when a total of G goats are grazing is v(G)-a-G2 per goat, where a>c. During the spring, the farmers simultaneously choose how many goats to own. Denoting by g1 and g2 the choices by farmers 1 and 2, the net benefit (payoff) to farmer 1 is (a-G^2)g1-cg1. where G=g1+g2. (a) Write the corresponding payoff function for farmer 2. Next assume that g1 and g2 can take any value equal to or greater than zero (so that, for example, it makes sense for a farmer to choose "half a goat"). (b) Let (g1*, g2*) be a Nash equilibrium of the normal-form game representing this situation. What system of simultaneous equations in g1 and g2 should (g1*.g2*) satisfy? Explain. (c) Let g* denote the common value of g1* and g2*. Compute g*. A benevolent social planner leading the village would choose G to maximize (a-G^2)G-CG, which is the net value extracted from the village goats. (d) Compute the social planner's optimal choice of G, call it G**, and show that it is smaller than G* =g1* + g2*. Thus, self-interested villagers overuse the village green. The problem of overuse of common resources is known as the tragedy of the commons and permeates many aspects of social life. Consider a village of two farmers, 1 and 2. Each summer they graze their goats on the village green. The cost to each farmer for buying and caring for a goat is c20. The value to each farmer of grazing a goat on the green when a total of G goats are grazing is v(G)-a-G2 per goat, where a>c. During the spring, the farmers simultaneously choose how many goats to own. Denoting by g1 and g2 the choices by farmers 1 and 2, the net benefit (payoff) to farmer 1 is (a-G^2)g1-cg1. where G=g1+g2. (a) Write the corresponding payoff function for farmer 2. Next assume that g1 and g2 can take any value equal to or greater than zero (so that, for example, it makes sense for a farmer to choose "half a goat"). (b) Let (g1*, g2*) be a Nash equilibrium of the normal-form game representing this situation. What system of simultaneous equations in g1 and g2 should (g1*.g2*) satisfy? Explain. (c) Let g* denote the common value of g1* and g2*. Compute g*. A benevolent social planner leading the village would choose G to maximize (a-G^2)G-CG, which is the net value extracted from the village goats. (d) Compute the social planner's optimal choice of G, call it G**, and show that it is smaller than G* =g1* + g2*. Thus, self-interested villagers overuse the village green. The problem of overuse of common resources is known as the tragedy of the commons and permeates many aspects of social life.

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A a The payoff function for Farmer 2 is aG2g2cg2 b A Nash equilibrium g1 g2 must satisfy the conditi... View the full answer