Two consumers, 1 and 2, constitute a small part of the overall economy. The consumers have preferences over the consumption of two traded goods and also over some action taken by consumer 1. Let xi (resp. yi) represent the quantity of the first (resp. second) good consumed by agent i, and let a ≥ 0 be the action taken by consumer 1. Let ui(xi , yi , a) denote consumer i’s utility function, and suppose that u1(x1, y1, a) = x1+ln y1+2a−a 2 and u2(x2, y2, a) = x2+ln y2− a 2 2 + a . The prices of good 1 and 2 are px = 1 and py = 1 2 respectively, and consumer i’s income is wi = 10. (a) Solve for consumer 1’s utility maximizing action and commodity bundle. Solve for consumer 2’s utility maximizing commodity bundle. (b) Show that the allocation found in (a) is inefficient, i. e., that there are alternative allocations that improve at least one consumer’s well being without making anyone else worse off. (c) Find the optimal level of a. (d) Suppose that consumer 1 is made to pay a tax of t per unit of a. Is there a value of t that implements the optimal level of the externality a? (e) Suppose that the government subsidizes the reduction of a as follows: it gives consumer 1 a subsidy of $s for every one unit decrease in a. Is there a value of s that implements the optimal level of the externality a? (f) Suppose that the government assigns the right to an externality-free environment to consumer 2. That is, consumer 1 cannot engage in