Suppose that in the absence of human intervention the stock of tuna (z) would change according to the following equation: dz/dt = 10z −0.02z2. Furthermore, suppose E units of fishing effort will produce a harvest of tuna (x) given by x = 0.02zE. Finally, suppose the cost of Eunits of effort is $4E, where $4 is the wage rate, and that the price of tuna is constant at $1.5 dollar per unit of x.A.Find the profit-maximizing sustainable equilibrium for this model when the tuna are not common property (that is, when the level of effort can be controlled). Solve for the levels of Z, E, x and profit. (8points)2.This question explores a simple exampleof resources where the cost of mining increases with the the total amount of resource extracted. Suppose that the profit on a copper mine at time t is given by πt=pqt −cqt2 −kst , where p, c and k are exogenous constants, qt is the amount of copperextracted in period t, and st is the total amount of silver extracted in all periods before t. The evolution of s is given by st+1 −st = qt . The value of s is initially zero. Also, the resource is infinite supply in the sense that there is no upper limit on s. The firm will own and operate the mine in periods 0 (now) through T−1; at T the firm expects to be nationalized and to have no subsequent revenues or costs. The firm’s goal is to maximize the sum of the profits it earns in periods 0 to T−1. You may assume the interest rate is zero.A. Set up the firm’s optimization problem and take first order conditions. Briefly explain what each equation shows (6points)B.Solve for an expression giving the firm’s output (qt) at each point in time from 0 to T -1. Discuss your result and give an intuitive explanation

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Suppose that in the absence of human intervention the stock of tuna (z) would change according to the following equation: dz/dt = 10z −0.02z2. Furthermore, suppose E units of fishing effort will produce a harvest of tuna (x) given by x = 0.02zE. Finally, suppose the cost of Eunits of effort is $4E, where $4 is the wage rate, and that the price of tuna is constant at $1.5 dollar per unit of x.A.Find the profit-maximizing sustainable equilibrium for this model when the tuna are not common property (that is, when the level of effort can be controlled). Solve for the levels of Z, E, x and profit. (8points)2.This question explores a simple exampleof resources where the cost of mining increases with the the total amount of resource extracted. Suppose that the profit on a copper mine at time t is given by πt=pqt −cqt2 −kst , where p, c and k are exogenous constants, qt is the amount of copperextracted in period t, and st is the total amount of silver extracted in all periods before t. The evolution of s is given by st+1 −st = qt . The value of s is initially zero. Also, the resource is infinite supply in the sense that there is no upper limit on s. The firm will own and operate the mine in periods 0 (now) through T−1; at T the firm expects to be nationalized and to have no subsequent revenues or costs. The firm’s goal is to maximize the sum of the profits it earns in periods 0 to T−1. You may assume the interest rate is zero.A. Set up the firm’s optimization problem and take first order conditions. Briefly explain what each equation shows (6points)B.Solve for an expression giving the firm’s output (qt) at each point in time from 0 to T -1. Discuss your result and give an intuitive explanation

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