Micro Solutions 1-2-2010

8/14/2019 Micro Solutions 1-2-2010

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Economics Honors Exam 2010 Solutions: Microeconomics

Questions 1-2

Question 1 (Microeconomics, 30 minutes). Consider a small exchange economywith two consumers,AandB, and two commodities,xandy.

ConsumersAandBhave preferences

uA x

A,y

A( ) =xAyA , uB xB ,yB( ) =xByB

The initial endowments of the goods are thatAhas 12 units ofxand 2 units ofy

whileBhas 8 units ofxand 18 units ofy.

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 onexandyrequired, 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.


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 consumerAgiven the utility of consumerB. Is the initialendowment 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 pointE, theMRSs are not equal.

At pointE:


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We can see thatAs indifference curve is a lot flatter thanBs indifference at pointE.


2 points for the correct answer (not Pareto efficient)

4 points for the justification

d. (6 points)Now assumeAgets to choose a new allocation to maximize utility,

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

Answer:


Consumption point must be onBs indifference curve that cuts throughE(3 points)

Must have tangency betweenAs indifference curve andBs indifference curve (3

points)


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e. (8 points)Solve formally for the Pareto efficient allocations ofxandyunder theassumptions in part (d).

Answer:

max uA x

A,y

A( ) =xAyA subject to

and

and

since uBat the endowment pointEis 144.

This can be rewritten:

max xAyA

subject to

We must have tangency between the twoMRSs 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 amountofxfor each individual must be equal to the amount ofyfor that individual. Proof:

The tangency condition is:

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

" yA

xA

=

20# yA

20# xA

" 20# yA

yA

=

20# xA

xA

" yA =xA and yB =xB

We can therefore rearrange our constraint above:

" 20# xA=12

" xA=8


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Therefore,Ahas an equilibrium consumption bundle of (8, 8).AndBhas an equilibrium consumption bundle of (12, 12).

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

need to work through the proof to show thatx=yfor each individual. If they write orinfer thatx=yfor 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 onBs 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 otheras 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 ADon'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 DriverBwas to swerve, the best response ofAis not to swerve.

If DriverBwas not to swerve, the best response ofAis to swerve.


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, dont swerve} and {dont swerve,swerve}


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You can find these by finding the best responses of each party to each of the other

partys actions.


3 points for the correct answer3 points for eachcorrect 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 DriverBplaysswervewith probabilityp, then DriverAhas the following payoffs:

Payoff (A playsswerve) = 1

Payoff (A plays dont swerve) = 4p

In equilibrium the payoffs must be equal (otherwise DriverAwould choose theaction with the highest payoff)

4p= 1

Therefore, p =1

4

This implies that in equilibrium:

DriverBplaysswervewith probability 14

DriverBplays dont swerve with probability 34

DriverAplaysswervewith probability 14

DriverAplays dont swerve with probability 34


Partial credit is to be awarded for the correct set-up but the wrong answer

Documents

Micro Solutions 1-2-2010