## Transcribed Text

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

Dorothy

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....