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Questions 1-2

Question 1 (Microeconomics, 30 minutes). Consider a small exchange economy with two consumers, A and B, and two commodities, x and y.

Consumers A and B have preferences

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uA xA ,yA( ) = xA yA , uB xB ,yB( ) = xB yB The initial endowments of the goods are that A has 12 units of x and 2 units of y

while B has 8 units of x and 18 units of y.

a. (5 points) Draw an Edgeworth box for this economy. Be sure to label everything clearly (including the endowment point).

Answer:

This is worth 5 points To be awarded 5 points, graphs must be fully labeled (including axis (only one x and

y required, the two origins (which parties?), endowment point)

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b. (5 points) Draw in the efficient lens that corresponds to the initial endowment. Explain this lens.

Answer:

The efficient lens is the set of all bundles where at least one party is better off than

the endowment point however neither party is worse off.

This is worth 5 points 3 points for the graph 2 points for the explanation

c. (6 points) For an allocation in this economy to be Pareto efficient, it must maximize the utility of consumer A given the utility of consumer B. Is the initial endowment Pareto efficient? Why or why not?

Answer:

The initial endowment point is not Pareto efficient. A Pareto efficient point equates the marginal rates of substitution of the two parties.

At point E, the MRSs are not equal.

At point E:

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We can see that A’s indifference curve is a lot flatter than B’s indifference at point E.

This is worth 6 points 2 points for the correct answer (not Pareto efficient) 4 points for the justification d. (6 points) Now assume A gets to choose a new allocation to maximize utility,

subject to the constraint that B's utility be no lower than at the endowment point. Illustrate this situation on a separate graph.

Answer:

This is worth 6 points Consumption point must be on B’s indifference curve that cuts through E (3 points) Must have tangency between A’s indifference curve and B’s indifference curve (3

points)

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e. (8 points) Solve formally for the Pareto efficient allocations of x and y under the

assumptions in part (d).

Answer:

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max uA xA ,yA( ) = xA yA subject to and and

since uB at the endowment point E is 144.

This can be rewritten:

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max xA yA subject to

We must have tangency between the two MRSs along any point on the contract

curve. It turns out that when we have these particular symmetrical Cobb-Douglas utility functions and we have a square Edgeworth box that along the contract curve, the amount of x for each individual must be equal to the amount of y for that individual. Proof:

The tangency condition is:

But this can be rewritten as follows if the constraint is taken into account:

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⇒yAxA

=20 − yA20 − xA

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⇒20 − yAyA

=20 − xAxA

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⇒ yA = xA and

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yB = xB

We can therefore rearrange our constraint above:

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⇒ 20 − xA =12

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⇒ xA = 8

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Therefore, A has an equilibrium consumption bundle of (8, 8). And B has an equilibrium consumption bundle of (12, 12).

This is worth 8 points Students would only need to get the correct answer to earn 8 points. They do not

need to work through the proof to show that x = y for each individual. If they write or infer that x = y for each individual in equilibrium, this is sufficient.

4 points are to be awarded for setting up the correct maximization problem with the correct constraint (i.e., the constraint on B’s utility)

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Question 2 (Microeconomics, 30 minutes). Consider the following game, known as Chicken. It represents a situation where two players drive their cars directly at each other as fast as possible. If one swerves away at the last minute, that person is a “chicken,” which is not good, but the person avoids serious injury. The other person is the “hero” and also avoids injury. If both swerve they are both chickens but avoid injury. If no one swerves, they both experience serious injury.

The payoff matrix for this game is as follows:

Driver B

Swerve Don't Swerve

Swerve 1,1 1,4 Driver A Don't Swerve 4,1 0,0

a. (9 points) Does the game have a dominant strategy for Driver A or Driver B? If

so, state the dominant strategies. Answer:

Neither party has a dominant strategy here. Therefore, there can be no dominant

strategy equilibrium. You can see from the best responses that the best response of a player depends on the

action taken by the other player.

If Driver B was to swerve, the best response of A is not to swerve. If Driver B was not to swerve, the best response of A is to swerve.

This is worth 9 points 9 points for the correct answer

b. (9 points) Does the game have any pure strategy Nash Equilibria? If so, what are they?

Answer: Yes, there are two PSE. They are:

{swerve, don’t swerve} and {don’t swerve, swerve}

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You can find these by finding the best responses of each party to each of the other party’s actions.

This is worth 9 points 3 points for the correct answer 3 points for each correct equilibria

c. (12 points) Does the game have any mixed strategy Nash Equilibria? If so, what are they?

Answer: Yes, there is a mixed strategy equilibrium.

If Driver B plays swerve with probability p, then Driver A has the following payoffs:

Payoff (A plays swerve) = 1 Payoff (A plays don’t swerve) = 4p

In equilibrium the payoffs must be equal (otherwise Driver A would choose the action with the highest payoff)

4p = 1

Therefore,

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p =14

This implies that in equilibrium:

Driver B plays swerve with probability

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14

Driver B plays don’t swerve with probability

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34

Driver A plays swerve with probability

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14

Driver A plays don’t swerve with probability

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34

This is worth 12 points Partial credit is to be awarded for the correct set-up but the wrong answer

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