Problems with solutions, Intermediate microeconomics, part 3
Niklas Jakobsson,
Problem 1. Game theory
Find the solutions to the following games using backward induction.
a)
b)
c)
Problem 2. Game theory
In the game below the payoffs have the following relation: T>R>P>S.
Player A / Player BCooperation / Conflict
Cooperation / R, R / S, T
Conflict / T, S / P, P
a) What is the equilibrium in a one shot game?
b) Is the equilibrium Pareto efficient?
c) What is this type of game called?
d) We now assume that the game is repeated an infinite number of times. We also assume that Player Bs patience is given by δ. If δ=1 the next round of the game is as important as this round of the game, if δ=0 the next round of the game is not important at all. Player A will always play the grim-trigger-strategy: cooperate in the first round, and cooperate as long as Player B cooperates. If B plays conflict, A plays conflict in all following games. How large must δ at least be for Player B to cooperate?
e) What is the tit-for-tat-strategy?
Problem 3. Game theory
Using the game matrix below, explain why an equilibrium in dominant strategies is also a pure Nash equilibrium.
Player A / Player BLeft / Right
Top / A, B / C, D
Bottom / E, F / G, H
Problem 4. Game theory
The questions in this problem refer to the following game:
L / M / RU / 1, 2 / 3, 5 / 2, 1
M / 0, 4 / 2, 1 / 3, 0
D / -1, 1 / 4, 3 / 0, 2
Player 2
Player 1
a. Determine if either player has any dominated strategies. If so, identify them.
b. Does either player have a dominant strategy? Why or why not?
c. Use iterated elimination of dominated strategies to solve this game. Be clear about the order in which you are eliminating strategies. Also specify whether you are eliminating strictly or weakly dominated strategies. (2p)
Problem 1. Behavioral economics
a) Explain the law of small numbers.
b) What is loss aversion?
c) How does loss aversion differ from risk aversion? Describe using pictures of utility functions.
d) What is time inconsistency? Give an example.
Problem 2. Behavioral economics
Lisa Late has to finish a project in three days. The total amount of time that she has to spend finishing the project is 12 hours. Lisa’s preferences over working time over the next three days can be described by the following utility function: , where xt is the number of hours she spends on the project day t. As you can see, working day t has a higher cost than working day t+1 and t+2. (5 p)
a) On the first day (Monday) Lisa makes a plan for finishing the project that maximizes her Monday utility function: , subject to the constraint of 12 working hours in total xM+xT+xW=12. How many hours does Lisa plan to work each day?
b) Lisa spent 12/7 hours working on the project on Monday. On Tuesday morning her utility function is: . Since she must finish the project before Thursday she will set xTh=0and her utility function to be maximized is thus: , subject to the budget constraint xT+xW=12-(12/7). How many hours do Lisa plan to work each day?
c) Given your answers in question a and b, does Lisa have time-consistent preferences? Explain.
Problem 1. Exchange
In a pure exchange economy with two goods, and two individuals that act competitively, we know that at some Pareto efficient allocation, the MRS between the two goods for one individual is -2.
a) What is the MRS for the other individual?
b) What is the relative price?
c) Show this situation in an Edgeworth box.
d) Explain why the allocation is Pareto efficient.
Problem 1. Welfare economics
A parent has two children named A and B and she loves both of the equally. She has a total of 1000 SEK to give them.
a) The parent has the following utility function: , where a is the amount of money that the parent gives to A, and b is the amount of money that the parent gives to b. How will the parent divide the money?
b) How will the parent divide the money if the utility function is: .
c) How will the parent divide the money if the utility function is: .
d) How will the parent divide the money if the utility function is: .
Problem 2. Welfare economics
A social planner has 1000 units of utility to distribute in a society with only two individuals, she can therefore enforce any allocation such that 1000=U1+U2.
a) If the social welfare function (SWF) is utilitarian, how will the planner distribute the money between the two individuals?
b) If the social welfare function (SWF) is Rawlsian, how will the planner distribute the money between the two individuals?
c) Explain the median voter theorem.
Problem 1. Externalities
The government issues permits to trap lobsters and is trying to determine how many permits to issue. It costs 2000 a month to operate a lobster boat.
If there are x boats in operation, the total revenue from the lobster catch per month will be f(x)=1000(10x-).
a) If the permits are free of charge, how many boats will trap lobsters?
b) What number of boats maximises total profits?
c) If the government wants to restrict the number of boats to the number that maximises total profits, how much should it charge per month for a lobster permit?
d) What is the name of the well known problem this example is an illustration of? Describe briefly another situation where this problem exists.
Problem 2. Externalities
Major Steelbone consumes two goods: bones (b) and steel (s). Her utility function is: U=74b-6b2+s. Her budget is 200 and the price of bones (pb) is 2, and the price of steel (ps) is 1.
a) How much of each good will she consume?
b) Suppose that her friends do not like her consumption of bones. Their disutility can be described by the function: b2. What quantity of bones should she consume to maximize social utility?
c) What price of bones would be needed for her to consume the social optimal amount of bones?
d) Explain two ways in which she could be induced to consume the social optimal amount of bones. What are the potential problems with these methods?
e) What kind of externality is this?
Problem 3. Externalities
Two farmers has the opportunity to put zero, one or two cows each on a common field. Each cows daily milk production depends on how many cows there is on the common field.
Total number of cows: 1 2 3 4
Each cows production: 8 5 3 2
That is, the first cow produces 8 litres of milk per day, the second 5 litres and so on. Each farmer likes to maximize her total production, and they make their decisions simultaneously.
(a) Illustrate this game in a normal form game matrix.
(b) Is there a dominant strategy equilibrium? If not, is there a pure strategy Nash equilibrium? Explain.
(c) Which is the pareto optimal number of cows on the field?
(d) What is the actual problem here?
(e) What is this situation called?
(f) Give an example of another situation with this problem.
Problem 1. Public goods
Ten people have dinner together at the restaurant “Glada Ankan”. They agree that the bill will be divided equally among them.
a) What is the additional cost to any one of them of ordering an appetizer that costs 200SEK)
b) Why may this be an inefficient system?
Answers to the problems
Problem 1. Game theory
a)
b)
c)
Problem 2. Game theory
a) (Conflict, Conflict)
b) No, in (Cooperation, Cooperation) both players are better off.
c) The prisoners dilemma
d) If B choose not to cooperate she will get: T+δP+δ2P+δ3P+…=T+Pδ(1+δ+δ2+ δ3+…)=T+Pδ/(1- δ). If B choose to cooperate she will get: R+δR+δ2R+δ3R+…=R(1+δ+δ2+ δ3+…)=R/(1- δ). If we set these equal to each other we find the lowest value for δ that makes cooperation beneficial for B: δ=(T-R)/(T-P).
e) Play Cooperation in the first round, and then play whatever the other player did the round before.
Problem 3. Game theory
(1) For a strategy to be a dominant strategy it must always be the best choice of a player. In an equilibrium in dominant strategies, the strategy chosen must be a dominant strategy for each player. (2) In a pure Nash equilibrium, the strategy of a player is the best choice given the other players choice. (3) Thus, (2) is always true if (1) is true.
In the figure: (1) Assume that (Top, Right) is an equilibrium in dominant strategies. Then a>e, c>g, and d>b, h>f. (2) Assume the (Top, Right) is a pure Nash equilibrium. Then c>g and d>b. (3) If (1) is true, (2) will also be true.
Problem 4. Game theory
a) R for player 2 is dominated by M. For each player 1 strategy, M gives player 2 a higher payoff than does R.
b) No. For either player to have a dominant strategy, 2 of her 3 strategies would need to be dominated.
c) Eliminate R as above (Strict). In the 3 x 2 game, U strictly dominates M. In the
2 x 2 game, M strictly dominates L. In the 2 x 1 game, D strictly dominates U.
IEDS Solution = (D,M).
Problem 1. Behavioral economics
a) Page 571-573.
b) Page 573-574.
c) Page 573-574 + lecture notes.
d) Page 574-575.
Problem 2. Behavioral economics
a) Differentiate the utility function with respect to hours each day and set equal to zero:.
.
Set: -xt=-2xM → xT=2xM and -0,5xW-2xM → xW=4xM, substitute into the budget constraint and solve.
Answer: xM=12/7, xT=24/7, xW=48/7
b) xT=24/7, xW=48/7
c) Yes.
Problem 1. Exchange
a) -2
b) -2
c) You should be able to do this yourself…
d) The points where one individual is better off are disjoint from the points where the other individual gets better off.
Problem 1. Welfare economics
The utility possibilities frontier will be a+b=1000.
a) a=500 and b=500
b) a=500 and b=500
c) All money to either A or B
d) Everything to one of the kids. The “maximum” you find with the standard methods is actually a minimum. By trial and error you will clearly see that giving everything to one of the kids maximizes utility.
Problem 2. Welfare economics
a) Any distribution of the money maximizes the SWF, SWF=U1+U2.
b) An equal division (U1=500, U2=500), SWF=min{U1,U2}.
Problem 1. Externalities
a) The total profit if x boats trap are π=1000(10x-x2)-2000x. At what value of x is this equal to zero? This is the equilibrium since boats will enter until the profit is equal to zero: 1000(10x-x2)-2000x=0, x=8000/1000=8 boats.
b) Profit maximum requires that MR = MC. 10000-2000x-2000=0, x=4 boats
c) With a license fee of F thousand dollars per month, the marginal cost of operating a boat for a month would be 2000+F. Boats (or firms) enter until the profit is equal to zero: π=1000(10x-x2)-2000x-Fx=0 (we set x = 4), F=4000 per month and boat.
d) The tragedy of the commons, see p. 659-663.
Problem 2. Externalities
a) b=6, s=188
b) b≈5,143
c) 12,286
d) A tax t=12,286-10=10,286, or a quantity restriction on her consumption b≈5,143. The government (or any planner) may not have correct information, e.g. the government does not know the optimal quantity.
e) A negative consumption externality
Problem 3. Externalities
a)
Farmer 20 / 1 / 2
0 / 0,0 / 0,8 / 0,10
Farmer 1 / 1 / 8,0 / 5,5 / 3,6
2 / 10,0 / 6,3 / 4,4
b) (2,2) is an equilibrium in dominant strategies
c) 2
d) Each farmer only consider her own payoff, not the total payoff
e) The tragedy of the commons
f) Over-fishing, over-extraction of ground water, animal extinction etc
Problem 1. Public goods
a) 20SEK
b) Each individual pays less than the full cost of her own meal, so everyone buy too much
2