2. [25%] There are two citizens, A and B, living in a neighborhood. They both consume a good 3: which can produce garbage during consumption. There is a cost to clean the garbage, which is determined by the total consummion of both citizens. Citizen i's utility function is: 1wort) = x.- - 091,963) = 994 - 0-5094 + x3)2; i = A, B, where x,- is i's consummion, and C(xA, x3) is the cleaning cost borne by each citizen. In this question, we assume that both citizens have plenty of endowments so you don't have to consider them. a. b. [3%] Dene the social optimum as the allocation (xi, x;) that maximizes the m utility of the two citizens, uA + 113. Write down the maximization problem. Find the social optimum. [6%] Now let the citizens choose their own consumption simultaneously to maximize their own utility. Write down the maximization problem for each citizen. Find the best response function. What is the set of Nash equilibria, denoted by (x3, xg)? Draw a gure to show the best response functions and indicate the set of Nash equilibria in the gure. [4%] Compare the to_tal consumption levels determined in the social optimum and Nash equilibria. Explain the difference. [5%] Now focus on the symmetric case where xA = x3. To solve the garbage problem, government decides to impose a per unit tax $t on the consumption. Citizen i's utility function becomes: ui(xi) = xi 0.5(xA + x3)2 t - xi. The citizens still choose their own consumption simultaneously to maximize their own utility. In order to achieve the social optimum, how much should t be? [7%] Consider another situation. Now Citizen A values xA more because his habit changes. His utility function becomes: uA(xA) = axA 0-5001 + 953?; where a > 1 . Other things remain unchanged. What are the new social optimum and Nash equilibrium? How will it affect the optimal tax rate described in d.? Explain your nding. (1m: In this case, the social optimal and Nash equilibrium consumption levels are not symmetric.)