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Question 3. In this question, we consider some variations of the production externality example discussed in Lecture 21. To begin, suppose Farm 1 has cost function Ci(q1,z)=7-4log(1 + 2)+ 5q, + WDz, where q is Farm I's quantity of soybeans produced, z is the quantity of herbicide used, and wp is the cost of the herbicide. Suppose the price of Farm 1's soybeans is p. (a) Prove that Farm 1 cost function is a convex function of q, and z. [Hint: Since 22C, /2q1 0 z = 0, it is enough to verify that C, is a convex function of q, and of z separately.] (b) What are the profit maximizing choices of q and z if p, =4 and wa = 1? What are the profit maximizing choices of q and z if p, =4 and wa = 5? Farm 2 has cost function C2 (92 , 2 ) = = 12 + =43. 15 where q2 is Farm 2's quantity of soybeans produced and z is the quantity of herbicide used by Farm 1. Suppose the price of Farm 2's soybeans is pz. 2 (c) Suppose p1 = 4, p2 =5.2, and wp = 1. What is Farm 2's profit maximizing choice of q2? How much would Farm 2 be willing to pay Farm 1 in exchange for Farm 1 choosing z = 0? (d) This will not be enough to induce Farm 1 to choose z = 0. What is the minimum payment necessary to induce Farm 1 to choose z = 0? (e) Explain why there is a tax rate that the government can impose on the use of herbicides that increases total profits above the no tax level. Do not attempt to calculate it! Suppose, unexpectedly, the government were to return the tax revenue raised to Farm 1 (the point of this is to ensure that Farm I's choice of z is determined by the tax rate). Does the final outcome Pareto dominate the no tax allocation