Answer the questions.

For questions (d), (e) and (f) assume that B = (1,2,3,4,5}, v, = 3 and v, =5 (d) Suppose that both players are selfish and uncaring. Find all the pure-strategy Nash equilibria. (e) Suppose that both players are selfish and benevolent. Find all the pure-strategy Nash equilibria. (f) Suppose that both players are selfish and spiteful. Find all the pure-strategy Nash equilibria. Now suppose that we have a situation of incomplete information. As a matter of fact, both players are selfish and benevolent (B), but this is not common knowledge. The following is common knowledge: 1. Each player knows her own preferences. 2. Player 1 is either selfish and benevolent (B) or selfish and uncaring (U). If Player 1 is B then she attaches probability 1 to Player 2 being B, while if Player 1 is ( then she attaches probability + to Player 2 being B and probability $ to Player 2 being $ (= selfish and spiteful). 3. Player 2 is either B or S. If Player 2 is B then she attaches probability + to Player 1 being B and probability * to Player 1 being U, while if Player 2 is $ then she attaches probability 1 to Player 1 being U. (g) Use an interactive knowledge-belief structure to represent this situation. (h) For this question assume that B = (P. P,}, V =V, = p, and the auction is not simultaneous: Player 1 bids first and Player 2 bids second after having been informed of Player 1's bid. (h.1) Apply the Harsanyi transformation to the incomplete-information situation of part (g), without writing payoffs but writing the outcomes. (h.2) Suppose that you know that the players have von Neumann-Morgenstern preferences, but all you know is that their von Neumann-Morgenstern preferences induce the ordinal rankings given above. Is it possible to find a pure-strategy weak sequential equilibrium of the game of part (h.1)? If you claim that it is not possible because not enough information is given, state what extra information you would need, otherwise findQuestion 3 3. Consider an economy with two agents (agents a and b), two goods - labor/leisure and a con- sumption good - and two firms. Both firms transform labor into the consumption good. Firm 1, with production function y1 = f(Li), belongs to agent a, and firm 2, with production function 12 = g(L2), belongs to agent b. (L denotes a quantity of labor and y a quantity of consumption good produced). Each agent has 3 units of time which can be used as leisure or sold as labor. Firm 2 uses a technique of production which creates a negative externality z, a chemical which has an adverse effect on the consumers and whose quantity is proportional to the level of production: = = 12. The utility and production functions are as follows wi(li, ri, 2) = Viri -2, i=a,b, f(LI) = L1, g(12) = 2VL2 where { denotes leisure and r the quantity of consumption good. (a) Derive the competitive equilibrium of this economy, i.e. equilibrium prices, production and consumption plans. Restrict attention to allocations where both firms produce a positive amount. (b) To show that the equilibrium is not Pareto optimal consider the following marginal change from the equilibrium allocation: add z/2 of leisure to each agent and decrease the labor used by firm 2 by E. Decrease equally the consumption of the consumption good of the two agents. Do not change anything to firm 1's plan. Compute the marginal change in utility for each agent and show that both agents are better off. Explain. It may be easier to keep the partial derivative notation for estimating the change in utilities and then evaluate the values of these derivatives at the equilibrium.] (c) Assume that the government fixes a limit 2* on the amount of pollution emitted by firm 2. Solve for the 'regulated competitive equilibrium' as a function of 2* . [Assume that 2* is smaller than the value found in (a)]. (d) To find the optimal 2" come back to your calculations in (b) and find the value of a* such that the change considered in that question (or the reverse when