2. (11 pts) I propose the following game: I flip a one dollar coin, and while it is in the air, you call either heads or tails. If you call the coin correctly, you get to keep the coin. (a) (2 pts) Suppose that you know that the coin is not fair and always comes up heads. What is the best strategy for you to pursue? (b) (2 pts) Suppose that the coin is unbalanced and comes up heads 80% of the time and tails 20% of the time. Now what is yvour best strategy? (c) (2 pts) What if the coin is fair, i.e. comes up heads 50% of the time and tails 50% of the time? What is your best strategy? (d) (2 pts) Now, suppose that I am able to choose the type of coin that I will toss (where a coin's type is the probability that it comes up heads), and that you will know my choice. What type of coin should I choose to minimize my losses? (e) (3 pts) Given that T choose a type of coin and you choose calling strategy, what is the mixed strategy Nash equilibrium for us in this game? Explain. (It may help to recognize that a lot of symmetry exists in the game.) Y o ' 5. (15 pts) Dean Interface and Professor Nightsoil exchange bromides (B) and platitudes (P) in com- petitive markets. Dean Interface's utility function is Ur(By, Pr) = B + 2+/P;. Professor Nightsoil's utility function is Un(By, Py) = By + 4,/Pp. Dean Interface's initial endowment is 8 bromides and 12 platitudes. Professor Nightsoil's initial endowment is 8 bromides and 4 platitudes. (a) (4 pts) Find an equation that states the condition for the contract curve. How does this condition depends on their consumption of platitudes and bromides? Explain. (b) (3 pts) Solve for the Pareto optimal allocations of platitudes, P; and Py. () (5 pts) What is the shape of the contract curve? Draw a plot to display several indifference curves and the contract curve in the Edgeworth box. Put Dean Interface at the left bottom corner and Professor Nightsoil at the right top corner. (d) (3 pts) Let pp and pp be the market prices for bromides and platitudes. Determine an equation of pp and pp that clear markets in competitive equilibrium