1. (Mixed Strategies.) Dorothy and Henry are playing one-stage game...

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1. (Mixed Strategies.) Dorothy and Henry are playing one-stage game shown in the table below. Dorothy has three possible choices (Left, Middle, and Right), as does Henry (Up, Middle, and Down). Dorothy Left Middle Right Up (A,B) (6,2) (3,1) Henry Middle (2,4) (8,7) (5,5) Down (1, (4,1) (C,D) Find all of the Nash equilibrium (equilibria) in pure strategies and mixed strategies if (a) A = 1, B = 0. C = 6. and D = 3. (b) A = 3, B = 1. C = 4, and D = 0. 2. Suppose that the demand and supply function for ice cream are Qf = 85-4P,+6Pp and Q; = 5P, - 5. respectively. Suppose the demand and supply function for pie are of = 110 - 5Pp -2P, and QP = 3Pp - 10. respectively. (a) Are pie and ice cream substitutes or complements? (b) Solve for and graph the market-clearing curves for pie and ice cream. (c) Find the general equilibrium prices and levels of consumption of both goods. 3. Suppose Helen starts out with 4 kilograms of food and 7 litres of water, while Lily starts out with 8 kilograms of food and 5 litres of water. (a) Draw an Edgeworth box that shows all possible allocations of these goods, and plot the endowment points. Now suppose that both Helen and Lily's preference can be described by the utility function U(F,W = min{F,W} (where F refers to kilograms of food and W refers to litres of water). (b) Add Helen and Lily's indifference curves to the Edgeworth box, then draw the contract curve. (c) To which points on the contract curve is each consumer willing to trade? 4. Amy (A) and Brian (B) have the following utility functions: UA = = where their endowments of consumption goods x and Y are as follows: XA = 50,YA = 20, XB = = 100. (a) At the endowment point, what are the marginal rates of substitution of x for Y for Amy? For Brian? (b) If they could engage in voluntary exchange, would they? (c) Find the competitive equilibrium price and allocation. (d) Is the allocation of resources achieved in (c) Pareto-Optimal? Explain. 5. In an exchange economy with two consumers and two goods, is it possible for one of the markets to clear but not the other? Explain your answer conceptually and graphically using an Edgeworth box.

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Question 1:
(a) When A=1, B=0; C=6 and D=3
                                        Left           Middle           Right
Henry          Up                (1,0)           (6,2)              (3,1)
                   Middle          (2,4)           (8,7)              (5,5)
                   Down           (1,2)           (4,1)                (6,3)

The pure strategy Nash equilibria when the values of A,B,C,D are as above, are (middle,middle) and (down, right). For finding the mixed strategy Nash equilibrium, we have to see whether there is a dominated strategy for the players. For Henry, Up is dominated by Middle and for Dorothy, Left is dominated by Right. As dominated strategies are not used by the players in an equilibrium, Henry would randomize between Middle and Down and Dorothy would randomize between Middle and Right....

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