EC301 – Bob and Brian square off for the World Rock

EC301 – Bob and Brian square off for the World Rock

Subject: Economics / General Economics
Question
Chapter 12 Problems 1b, 4, 6. Chapter 19 Problems 1,2.

EC301 Problem Set 6

The homework totals 60 points: 30 points are awarded for completeness & 30 for accuracy. Chapter 12. Choices Involving Strategies
1. Bob and Brian square o? for the World Rock Paper Scissors Championship. The
winner will take home $10,000, the loser receives nothing, and they split the prize in
the event of a tie.
(a) Draw a table showing the players’ possible choices, and the payo?s that result
from each pair of actions.
(b) Solve for a mixed strategy Nash equilibrium.

2. Alice and Barbara are playing a one-stage guessing game. Each must choose a number
between 1 and 8 (inclusive). Alice’s target is to match Barbara’s number. Barbara’s
target is to name twice Alice’s number. Each receives $10 minus a dollar penalty that
is equal to the absolute di?erence between her guess and her target. Solve this game
by iteratively deleting dominating strategies. What will Alice and Barbara choose?

3. Explain why the pair of choices (squeal, squeal) is the only Nash equilibrium in the
Provost’s Nephew as illustrated below. Oskar Deny
Squeal Roger
Deny Squeal
0, ?2 ?6, ?1
?1, ?6 ?5, ?5

4. Explain why bidding a bidder’s own value weakly dominates any other strategy in a
second-price sealed-bid auction (a Vickrey auction).

5. Suppose Liz and Scott are writing a report together. They have 24 hours to do background research. The quality of their research will a?ect their prospects for raises and
promotions. The more time they spend on the research, the better, but each wants the
other to do most of the work. Let’s use the symbol X to indicate the number of hours
Scott spends on research and the symbol Y to indicate the number of hours Liz spends.
Both these numbers must be positive and neither can exceed 24. Suppose we can measure Scott and Liz’s costs and benefits on a utility scale. When Scott works for X hours
and Liz works for Y hours, each receives a total benefit of 60(X + 2Y ) ? (X + 2Y )2 .
The marginal benefit of Scott’s extra time is 60?2(X +2Y ), while the marginal benefit
1 of Liz’s extra time is 120 ? 4(X + 2Y ). The cost of their e?ort is X 2 for Scott and Y 2
for Liz, so the marginal cost is 2X for Scott and 2Y for Liz.
(a) Find the Nash equilibrium of this game.
(b) Is it a good outcome for Liz and Scott? Can they do better?

6. Dorothy and Henry are playing the one-stage game shown in the following figure.
Dorothy has three possible choices (left, middle, and right), as does Henry (up, middle,
down). Find all of the Nash equilibria in pure strategies and mixed strategies. Henry Left
3, 1
2, 4
1, 2 Up
Middle
Down Dorothy
Middle
6, 2
8, 7
4, 1 Right
3, 1
5, 5
4, 3

7. Dorothy and Henry are playing the one-stage game shown in the following figure.
Dorothy has three possible choices (left, middle, and right), as does Henry (up, middle,
down). Find all of the Nash equilibria in pure strategies and mixed strategies. Henry Left
5, 4
2, 3
1, 2 Up
Middle
Down Dorothy
Middle
3, 2
4, 5
0, 1 Right
2, 1
1, 0
0, 0

Chapter 19. Oligopoly
1. Joe and Rebecca are small-town ready-mix concrete duopolists. The market demand
function is Qd = 10000 ? 100P where P is the price of a cubic yard of concrete and
Qd is the number of cubic yards demanded per year. Marginal cost is $25 per cubic
yard. Competition in this market is described by the Cournot model. That is, the
firms simultaneously decide the quantity to produce and the price clears the market
given the quantity produced.
(a) What is Joe’s best response function?
(b) What is Rebecca’s best response function?
(c) Graph Joe and Rebecca’s best response functions on the same graph, with QJoe
on the horizontal axis and QRebecca on the vertical axis.
(d) What are Joe and Rebecca’s Nash equilibrium outputs? Label the Nash equilibrium in the same graph.
(e) What is the resulting price?
(f) How much profit do they each earn?
(g) What are Joe and Rebecca’s markups?
(h) How do the price and the two firms’ joint profit (the sum of their individual
profits) compare to the monopoly price and profit?

2. Joe and Rebecca are small-town ready-mix concrete duopolists. The market demand
function is Qd = 10000 ? 100P where P is the price of a cubic yard of concrete and Qd
is the number of cubic yards demanded per year. Joe’s marginal cost is $40 per cubic
yard and Rebecca’s marginal cost is $25 per cubic yard. Competition in this market is
described by the Cournot model. That is, the firms simultaneously decide the quantity
to produce and the price clears the market given the quantity produced.
(a) What is Joe’s best response function?
(b) What is Rebecca’s best response function?
(c) Graph Joe and Rebecca’s best response functions on the same graph, with QJoe
on the horizontal axis and QRebecca on the vertical axis.
(d) What are Joe and Rebecca’s Nash equilibrium outputs?
(e) What is the resulting price?
(f) What are the Joe and Rebecca’s markups, respectively?
(g) How much profit do they each earn? 3