Practice Problem Set 11. The XYZ Company expects to earn $2 million in profit per year, starting immediately

if tomorrow’s product test is successful, or starting one year from now if the product

test is not successful. In every case, XYZ has a 90% chance, each year, of surviving

until the following year. Calculate XYZ’s value if the test is successful and if it is

unsuccessful. If the test fails, the critical part could be purchased from a supplier, so

that profit can be earned immediately. What is the maximum additional cost XYZ

should incur if it sources the part externally in that case?

Solution: Since the firm has a 90% chance of surviving year-to-year, the discount

factor is 𝛿𝛿 = 0.9. The per-period profit is given as 𝜋𝜋 = 2,000,000. The present value,

for profits starting after 𝑡𝑡 periods, is 𝑃𝑃𝑃𝑃 𝑡𝑡 =

2,000,000

=

2,000,000

𝛿𝛿 𝑡𝑡 𝜋𝜋

1−𝛿𝛿

. This is equal to 𝑃𝑃𝑃𝑃 =

𝛿𝛿 0 𝜋𝜋

1−𝛿𝛿

=

𝜋𝜋

1−𝛿𝛿

=

= 20,000,000 if profits start immediately (after zero years). Or

1−0.9

0.1

𝛿𝛿 1 𝜋𝜋

𝛿𝛿𝛿𝛿

0.9∙2,000,000

1,800,000

1

𝑃𝑃𝑃𝑃 =

=

=

=

= 18,000,000 if profits start after 𝑡𝑡 = 1

1−𝛿𝛿

1−𝛿𝛿

1−0.9

0.1

year. The difference is $2,000,000, so that is the maximum investment worth

making.

2. For the XYZ Company in Practice Problem 1, suppose the test will be successful, so

that the firm will earn $2 million a year starting right away. What if, instead of a

90% chance of surviving each year, it has a 95% chance? And what if it has a 90%

chance, but annual profit is $4 million? Calculate the value of the firm in these two

scenarios. Which is better: 100% more profit or 5% better odds to last for another

year?

Solution: With a discount factor of 𝛿𝛿 = 0.95 and per-period profit 𝜋𝜋 = 2,000,000,

𝜋𝜋

2,000,000

2,000,000

present value is 𝑃𝑃𝑃𝑃 =

=

=

= 40,000,000. With a discount factor

1−𝛿𝛿

1−0.95

0.05

of 𝛿𝛿 = 0.9 and per-period profit 𝜋𝜋 = 4,000,000, present value is 𝑃𝑃𝑃𝑃 =

4,000,000

4,000,000

𝜋𝜋

1−𝛿𝛿

=

=

= 40,000,000. The value with 5% better survival odds is the same

0.1

as the value with double the profit.

1−0.9

Practice Quiz Questions 1

1. A start-up is about to conduct the final product test, but the Founder-CEO is already

thinking ahead. “If the test fails, I need to make a decision about buying the main

part from an external supplier, but that would be costly. It would allow us to roll out

the product right away and make $2 million in profit per year, except for the

payment to the supplier. Or, we could continue to work on the part during our

internal development work, and then I am sure we will be ready to start making a

profit beginning next year.” Assuming the start-up has a 90% chance of survival

each year, how much would it be worth to start earning profit immediately, rather

than next year?

o

o

o

o

Nothing

$1 million

$2 million CORRECT

$10 million

2. Delighted with a successful product test, which allows the company to earn $2

million per year in profit starting immediately, the Founder-CEO is thinking about

the next opportunities. “Currently, I believe we have a 90% chance of getting

through each of the next years. Part of the risk is software safety. We could invest to

reduce that risk, so that we have a 95% chance of survival each year. However, we

could also put that money into a marketing campaign that I expect to raise our

annual profits to $4 million.” Which is the better investment in terms of maximizing

the value of the firm?

o Neither would change the firm’s value, so the effect is the same.

o Reducing the risk: this increases the firm’s value by 50%, whereas increasing

profit only increases the firm’s value by 20%.

o Increasing the profit: this increases the firm’s value by 50%, whereas reducing

the risk only increases the firm’s value by 10%.

o Both would double the firm’s value, so the effect is the same. CORRECT

Problem Set 1

1. Mixtastic, a firm that sells a subscription for a mix-at-home cocktail kit, projects it

will earn $100K in profit each year, and each year it has an 80% chance of surviving

until the next year. Under these assumptions, what is the value of the firm today? In

case the firm only expects to start earning profit in three years, then approximately

by what percentage does its present value decrease?

2. Stuffitt, the “AirBnB for storage units,” expects to earn $5 million in profit per year,

once it is up and running. Currently, the expected start date is two years from now,

but with a $2 million additional investment, Stuffitt could go “live” in one year.

Whether or not Stuffitt is “live,” it always has a 60% chance of surviving until the

following year. Calculate the value if they open in two years and the value if they

open in one year. Is it worthwhile to make the investment (i.e. is the difference

greater than $2 million)?

3. For Mixtastic in Problem 1, suppose the firm will earn $100K a year starting

immediately. What if, instead of an 80% chance of surviving each year, it only has a

50% chance? And what if it has an 80% chance, but annual profit is only $50K?

Calculate the value of the firm in these two scenarios. Which is better: double the

profit or 30% better odds to last for another year?

4. For Stuffitt in Problem 2, suppose the firm will earn $5 million a year starting in two

years. What if, instead of a 60% chance of surviving each year, it has a 75% chance?

And what if it has a 60% chance, but annual profit is $10 million? Calculate the value

of the firm in these two scenarios. Which is better: double the profit or 15% better

odds to last for another year?

Practice Problem Set 2

1. Suppose a firm’s revenue depends on sales 𝑄𝑄 according to the function 𝑅𝑅 = 15𝑄𝑄 −

0.25𝑄𝑄2 . The firm’s cost depends on sales according to the function 𝐶𝐶 = 180 + 3𝑄𝑄 +

0.05𝑄𝑄2 .

(a) What is the firm’s marginal revenue function?

(b) What is the firm’s marginal cost function?

(c) What is “net” marginal revenue, the difference between marginal revenue and

marginal cost?

Solution: (a) Revenue is 𝑀𝑀𝑀𝑀 =

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

= 15 − 0.5𝑄𝑄. (b) Marginal cost is 𝑀𝑀𝑀𝑀 =

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

=3+

0.1𝑄𝑄. (c) The difference is 𝑀𝑀𝑀𝑀 − 𝑀𝑀𝑀𝑀 = 15 − 0.5𝑄𝑄 − (3 + 0.1𝑄𝑄) = 15 − 0.5𝑄𝑄 − 3 −

0.1𝑄𝑄 = 12 − 0.6𝑄𝑄.

To use the formulas for the derivatives, note that 𝑅𝑅 = 15𝑄𝑄 − 0.25𝑄𝑄2 is a function of

the form 𝑓𝑓(𝑥𝑥) = 𝛼𝛼 + 𝛽𝛽𝛽𝛽 + 𝛾𝛾𝑥𝑥 2 , where 𝛼𝛼 = 0, 𝛽𝛽 = 15, 𝛾𝛾 = −0.25, and 𝑥𝑥 = 𝑄𝑄.

𝜕𝜕𝜕𝜕(𝑥𝑥)

Therefore,

= 𝛽𝛽 + 2𝛾𝛾𝛾𝛾 = 15 − 2 ∙ (−0.25) ∙ 𝑄𝑄 = 15 − 0.5𝑄𝑄 = 𝑀𝑀𝑀𝑀. Also, 𝐶𝐶 =

𝜕𝜕𝜕𝜕

180 + 3𝑄𝑄 + 0.05𝑄𝑄2 is a function of the form 𝑓𝑓(𝑥𝑥) = 𝛼𝛼 + 𝛽𝛽𝛽𝛽 + 𝛾𝛾𝑥𝑥 2 , where 𝛼𝛼 = 180,

𝜕𝜕𝜕𝜕(𝑥𝑥)

𝛽𝛽 = 3, 𝛾𝛾 = 0.05, and 𝑥𝑥 = 𝑄𝑄. So,

= 𝛽𝛽 + 2𝛾𝛾𝛾𝛾 = 3 + 2 ∙ 0.05 ∙ 𝑄𝑄 = 3 + 0.1𝑄𝑄 = 𝑀𝑀𝑀𝑀.

𝜕𝜕𝜕𝜕

2. For the firm in Practice Problem 1:

(a) What is the marginal revenue when 𝑄𝑄 = 24?

(b) What is the marginal cost when 𝑄𝑄 = 24?

(c) Do sales of 𝑄𝑄 = 24 maximize the firm’s profit? If not, should sales be greater or

smaller, in order to increase profit?

Solution: (a) At 𝑄𝑄 = 24, marginal revenue is 𝑀𝑀𝑀𝑀 = 15 − 0.5 ∙ 24 = 3. (See Practice

Problem 1 for how to find the marginal revenue function.) (b) At 𝑄𝑄 = 24, marginal

cost is 𝑀𝑀𝑀𝑀 = 3 + 0.1 ∙ 24 = 5.4. (See Practice Problem 1 for how to find the marginal

cost function.) (c) Since 𝑀𝑀𝑀𝑀 ≠ 𝑀𝑀𝑀𝑀 at 𝑄𝑄 = 24, profit is not maximized. Given that

𝑀𝑀𝑀𝑀 < 𝑀𝑀𝑀𝑀, sales should be smaller.
Practice Quiz Questions 2
1. The start-up you work for is in its fourth year and has been steadily expanding its
customer base. Recently, the firm hired a CFO to support its growth, and revenue
and cost data are now being systematically collected and analyzed. There are a few
concerns. Engineers say that the additional cost of serving one client is not constant,
but has been rising as the client base grew. Higher sales have been achieved through
price reductions, and management wonders whether the higher costs are being
recovered through the price the company charges. You were asked by the CFO to
investigate this issue and calculate the incremental revenue and cost per client at
different service levels. You are given estimated revenue and cost functions, which
are 𝑅𝑅 = 15𝑄𝑄 − 0.25𝑄𝑄2 and 𝐶𝐶 = 180 + 3𝑄𝑄 + 0.05𝑄𝑄2 (where 𝑅𝑅 is revenue, 𝐶𝐶 is cost,
and 𝑄𝑄 is number of clients served). Using your economics background, you
determine that you need to compute the marginal revenue and marginal cost
functions. You report that:
o Marginal revenue is 𝑀𝑀𝑀𝑀 = 10𝑄𝑄; marginal cost is 𝑀𝑀𝑀𝑀 = 2.5𝑄𝑄; each additional unit
brings in 7.5𝑄𝑄 in net revenue.
o Marginal revenue is 𝑀𝑀𝑀𝑀 = 15 − 0.5𝑄𝑄; marginal cost is 𝑀𝑀𝑀𝑀 = 3 + 0.1𝑄𝑄; each
additional unit brings in 12 − 0.6𝑄𝑄 in net revenue. CORRECT
o Marginal revenue is 𝑀𝑀𝑀𝑀 = 30𝑄𝑄 − 0.5𝑄𝑄2 ; marginal cost is 𝑀𝑀𝑀𝑀 = 6𝑄𝑄 + 0.1𝑄𝑄2 ;
each additional unit brings in 24𝑄𝑄 − 0.4𝑄𝑄2 in net revenue.
o Marginal revenue is 𝑀𝑀𝑀𝑀 = 150 − 2.5𝑄𝑄; marginal cost is 𝑀𝑀𝑀𝑀 = 30 + 0.5𝑄𝑄; each
additional unit brings in 120 − 2𝑄𝑄 in net revenue.
2. The CEO proudly announced at the all-hands meeting that, after winning an
important new contract, the business has expanded to 24 clients served. You were
previously analyzing the firm’s revenue and cost functions, which are 𝑅𝑅 = 15𝑄𝑄 −
0.25𝑄𝑄2 and 𝐶𝐶 = 180 + 3𝑄𝑄 + 0.05𝑄𝑄2 (where 𝑅𝑅 is revenue, 𝐶𝐶 is cost, and 𝑄𝑄 is number
of clients served). Looking to the future with visible excitement, the CEO goes
around the room, asking each employee for one suggestion that could further
increase the firm’s profitability. When it’s your turn, you say:
o Actually, we seem to be growing too fast. Reducing the number of clients we
serve would increase our profit, since marginal revenue is currently less than
marginal cost. CORRECT
o To be honest, we are in a great spot and should just continue like this. Revenue
from another client would be equal to the cost of serving a new client.
o Keep adding clients! Revenue will continue to grow a little faster than cost.
o Let’s double the number of clients we serve! Revenue will double, but our cost
will remain relatively low, so that we will earn nearly twice the profits!
Problem Set 2
1. Suppose a firm’s revenue depends on sales 𝑄𝑄 according to the function 𝑅𝑅 = 80𝑄𝑄.
The firm’s cost depends on sales according to the function 𝐶𝐶 = 1,100 + 16𝑄𝑄 + 𝑄𝑄2 .
(a) What is the firm’s marginal revenue function?
(b) What is the firm’s marginal cost function?
(c) What is “net” marginal revenue, the difference between marginal revenue and
marginal cost?
2. Suppose a firm’s revenue depends on sales 𝑄𝑄 according to the function 𝑅𝑅 = 40𝑄𝑄 −
5𝑄𝑄2 . The firm’s cost depends on sales according to the function 𝐶𝐶 = 7.5𝑄𝑄 + 0.25𝑄𝑄2 .
(a) What is the firm’s marginal revenue function?
(b) What is the firm’s marginal cost function?
(c) What is “net” marginal revenue, the difference between marginal revenue and
marginal cost?
3. For the firm in Problem 1:
(a) What is the marginal revenue when 𝑄𝑄 = 32?
(b) What is the marginal cost when 𝑄𝑄 = 32?
(c) Do sales of 𝑄𝑄 = 32 maximize the firm’s profit? If not, should sales be greater or
smaller, in order to increase profit?
4. For the firm in problem 2:
(a) What is the marginal revenue when 𝑄𝑄 = 3?
(b) What is the marginal cost when 𝑄𝑄 = 3?
(c) Do sales of 𝑄𝑄 = 3 maximize the firm’s profit? If not, should sales be greater or
smaller, in order to increase profit?
Practice Problem Set 3
1. Given demand function 𝑄𝑄 = 60 − 4𝑃𝑃:
o What are the sales at price 𝑃𝑃 = 10?
o What does the price need to be so that sales are 𝑄𝑄 = 40?
Solution: (a) Substitute 𝑃𝑃 = 10 into the demand function: 𝑄𝑄 = 60 − 4 ∙ 10 = 20. (b)
Rearrange the demand function for 𝑃𝑃 to obtain the inverse demand function: 4𝑃𝑃 =
60
1
60 − 𝑄𝑄 implies 𝑃𝑃 =
− 𝑄𝑄 = 15 − 0.25𝑄𝑄. Then substitute 𝑄𝑄 = 40: 𝑃𝑃 = 15 −
4
4
0.25 ∙ 40 = 5.
Or, for (b), we could recognize that 𝑄𝑄 = 60 − 4𝑃𝑃 has the form 𝑄𝑄 = 𝑎𝑎 − 𝑏𝑏𝑏𝑏, where
𝑎𝑎−𝑄𝑄
60−𝑄𝑄
𝑎𝑎 = 60 and 𝑏𝑏 = 4. Since the inverse demand function is 𝑃𝑃 =
, we have 𝑃𝑃 =
..
At 𝑄𝑄 = 40, it follows that 𝑃𝑃 =
60−40
4
=
20
4
= 5.
𝑏𝑏
4
2. Suppose the demand function for the ABC Company is 𝑄𝑄 = 100 − 4𝑃𝑃 + 2𝑃𝑃𝑋𝑋 −
0.05𝐼𝐼, where 𝑄𝑄 is ABC’s yearly sales, 𝑃𝑃 is ABC’s price, 𝑃𝑃𝑋𝑋 is the XYZ Company’s price,
and 𝐼𝐼 is the average annual consumer income after tax.
(a) Are ABC and XYZ complementary or substitute goods?
(b) Is ABC a normal or inferior good?
(c) What is the demand function for ABC if 𝑃𝑃𝑋𝑋 = 5 and 𝐼𝐼 = 1,000?
Solution: (a) Since 𝑃𝑃𝑋𝑋 has a positive coefficient, i.e. 2, the sales for ABC increase when
XYZ’s price increases (which implies XYZ’s sales fall). Hence, they are substitutes.
(b) Since 𝐼𝐼 has a negative coefficient, i.e. −0.05, ABC’s sales increase when consumer
incomes decrease. Thus, ABC is an inferior good. (c) Substitute 𝑃𝑃𝑋𝑋 = 5 and 𝐼𝐼 =
1,000 into the demand function: 𝑄𝑄 = 100 − 4𝑃𝑃 + 2 ∙ 5 − 0.05 ∙ 1,000 = 60 − 4𝑃𝑃.
Practice Quiz Questions 3
1. Your boss looks worried. “We need to get our sales up quickly. The company is
pursuing aggressive growth, and selling 30 of these machines a year isn’t enough.
We have some flexibility to give discounts. Where does our price need to be to get,
say, 10 more machines sold?” Fortunately, you have already analyzed past sales and
prices, and you feel pretty confident about the demand function you estimated:
𝑄𝑄 = 60 − 4𝑃𝑃, where 𝑄𝑄 is the company’s sales, and 𝑃𝑃 is the company’s price. Your
advice, to achieve sales of 40, is to set the price to …
o
o
o
o
5 CORRECT
15
30
40
2. After assembling a dataset that includes your employer’s, the ABC Company’s, sales
history and some relevant variables that might help predict sales, you ran a
regression and arrived at the following forecast model: 𝑄𝑄 = 100 − 4𝑃𝑃 + 2𝑃𝑃𝑋𝑋 −
0.05𝐼𝐼, where 𝑄𝑄 is ABC’s yearly sales, 𝑃𝑃 is ABC’s price, 𝑃𝑃𝑋𝑋 is the XYZ Company’s price,
and 𝐼𝐼 is the average annual consumer income after tax. Your boss is specifically
interested in how sales depend on ABC’s price. You explain that this information
would be captured by a standard demand function. A review of the current price of
the XYZ Company and current consumer incomes shows that 𝑃𝑃𝑋𝑋 = 5 and 𝐼𝐼 = 1,000.
You inform your boss that the demand function is:
o
o
o
o
𝑄𝑄 = 100 − 2𝑃𝑃
𝑄𝑄 = 100 + 2𝑃𝑃𝑋𝑋
𝑄𝑄 = 60 − 4𝑃𝑃 CORRECT
𝑄𝑄 = 60 + 4𝑃𝑃𝑋𝑋
Problem Set 3
1. Given demand function 𝑄𝑄 = 1,500 − 12.5𝑃𝑃:
(a) What are the sales at price 𝑃𝑃 = 40?
(b) What does the price need to be so that sales are 𝑄𝑄 = 1,000?
2. Given inverse demand function 𝑃𝑃 = 2,805 − 0.005𝑄𝑄:
(a) What does the price need to be so that sales are 𝑄𝑄 = 111,000?
(b) What are the sales at price 𝑃𝑃 = 1,695?
3. Suppose the demand function for United Soaps (US) is 𝑄𝑄 = 800 − 12.5𝑃𝑃 + 9𝑃𝑃𝑇𝑇 +
0.01𝐼𝐼, where 𝑄𝑄 is US yearly sales, 𝑃𝑃 is US price, 𝑃𝑃𝑇𝑇 is the price for Tupelo Honey
Eucalyptus Meds (THEM), and 𝐼𝐼 is the average annual consumer income after tax.
(a) Are US and THEM complementary or substitute goods?
(b) Is US a normal or inferior good?
(c) What is the demand function for US if 𝑃𝑃𝑇𝑇 = 50 and 𝐼𝐼 = 25,000?
4. Suppose the demand function for the Yay Co. is 𝑄𝑄 = 10,000 − 0.7𝑃𝑃 − 0.25𝑃𝑃𝑀𝑀 − 𝐼𝐼,
where 𝑄𝑄 is Yay’s monthly sales, 𝑃𝑃 is Yay’s price, 𝑃𝑃𝑀𝑀 is the price for the Meh Co., and 𝐼𝐼
is the average monthly consumer income after tax.
(a) Are Yay and Meh complementary or substitute goods?
(b) Is Yay a normal or inferior good?
(c) What is the demand function for Yay if 𝑃𝑃𝑀𝑀 = 400 and 𝐼𝐼 = 2,900?
Practice Problem Set 4
1. Given a cost function 𝐶𝐶 = 180 + 3𝑄𝑄 + 0.05𝑄𝑄2 :
(a) What is the average cost function? What is the average cost at 𝑄𝑄 = 20? At 𝑄𝑄 =
60?
(b) What is the average fixed cost function? What is the average fixed cost at 𝑄𝑄 =
20? At 𝑄𝑄 = 60?
(c) What is the average variable cost function? What is the average variable cost at
𝑄𝑄 = 20? At 𝑄𝑄 = 60?
(d) What is the marginal cost function? What is the marginal cost at 𝑄𝑄 = 20? At 𝑄𝑄 =
60?
Solution: (a) Average cost is 𝐴𝐴𝐴𝐴 =
180
𝐶𝐶
𝑄𝑄
=
180+3𝑄𝑄+0.05𝑄𝑄2
𝑄𝑄
=
180
𝑄𝑄
+ 3 + 0.05𝑄𝑄. At 𝑄𝑄 = 20,
180
average cost is 𝐴𝐴𝐴𝐴 =
+ 3 + 0.05 ∙ 20 = 13. At 𝑄𝑄 = 60, average cost is 𝐴𝐴𝐴𝐴 =
+
20
60
3 + 0.05 ∙ 60 = 9. (b) Fixed cost 𝐹𝐹𝐹𝐹 is the part of 𝐶𝐶 that does not depend on 𝑄𝑄, i.e.
𝐹𝐹𝐹𝐹
180
𝐹𝐹𝐹𝐹 = 180. Average fixed cost is therefore 𝐴𝐴𝐴𝐴𝐴𝐴 = =
. At 𝑄𝑄 = 20, average fixed
𝑄𝑄
180
𝑄𝑄
180
cost is 𝐴𝐴𝐴𝐴𝐴𝐴 =
= 9. At 𝑄𝑄 = 60, average fixed cost is 𝐴𝐴𝐴𝐴𝐴𝐴 =
= 3. (c) Variable
20
60
2
cost 𝑉𝑉𝑉𝑉 is the part of 𝐶𝐶 that depends on 𝑄𝑄, i.e. 𝑉𝑉𝑉𝑉 = 3𝑄𝑄 + 0.05𝑄𝑄 . Average variable
cost is therefore 𝐴𝐴𝐴𝐴𝐴𝐴 =
𝑉𝑉𝑉𝑉
𝑄𝑄
=
3𝑄𝑄+0.05𝑄𝑄2
𝑄𝑄
= 3 + 0.05𝑄𝑄. At 𝑄𝑄 = 20, average variable
cost is 𝐴𝐴𝐴𝐴𝐴𝐴 = 3 + 0.05 ∙ 20 = 4. At 𝑄𝑄 = 60, average variable cost is 𝐴𝐴𝐴𝐴𝐴𝐴 = 3 + 0.05 ∙
𝑑𝑑𝑑𝑑
60 = 6. (d) Marginal cost is 𝑀𝑀𝑀𝑀 =
= 3 + 0.1𝑄𝑄. At 𝑄𝑄 = 20, marginal cost is 𝑀𝑀𝑀𝑀 =
𝑑𝑑𝑑𝑑
3 + 0.1 ∙ 20 = 5. At 𝑄𝑄 = 60, marginal cost is 𝑀𝑀𝑀𝑀 = 3 + 0.1 ∙ 60 = 9.
Or we could recognize that 180 + 3𝑄𝑄 + 0.05𝑄𝑄2 has the form 𝐶𝐶 = 𝑓𝑓 + 𝑐𝑐𝑐𝑐 + 𝑑𝑑𝑑𝑑2 ,
𝑓𝑓
where 𝑓𝑓 = 180, 𝑐𝑐 = 3, and 𝑑𝑑 = 0.05. The average cost function is 𝐴𝐴𝐴𝐴 = + 𝑐𝑐 +
𝑑𝑑𝑑𝑑 =
180
𝑄𝑄
+ 3 + 0.05𝑄𝑄. Average fixed cost is 𝐴𝐴𝐴𝐴𝐴𝐴 =
𝑓𝑓
𝑄𝑄
=
180
𝑄𝑄
𝑄𝑄
. Average variable cost is
𝐴𝐴𝐴𝐴𝐴𝐴 = 𝑐𝑐 + 𝑑𝑑𝑑𝑑 = 3 + 0.05𝑄𝑄. Now we can substitute 𝑄𝑄 = 20 and 𝑄𝑄 = 60 in each
180
case. We find that, respectively, 𝐴𝐴𝐴𝐴 = 𝐴𝐴𝐴𝐴𝐴𝐴 + 𝐴𝐴𝐴𝐴𝐴𝐴 =
+ 3 + 0.05 ∙ 20 = 9 + 4 =
180
20
13 and 𝐴𝐴𝐴𝐴 = 𝐴𝐴𝐴𝐴𝐴𝐴 + 𝐴𝐴𝐴𝐴𝐴𝐴 =
+ 3 + 0.05 ∙ 60 = 3 + 6 = 9. Marginal cost is 𝑀𝑀𝑀𝑀 =
60
𝑐𝑐 + 2𝑑𝑑𝑑𝑑 = 3 + 2 ∙ 0.05𝑄𝑄 = 3 + 0.1𝑄𝑄. Therefore, respectively, 𝑀𝑀𝑀𝑀 = 3 + 0.1 ∙ 20 = 5
and 𝑀𝑀𝑀𝑀 = 3 + 0.1 ∙ 60 = 9.
2. Refer to the cost function in Practice Problem 1:
(a) A firm should operate in the short run only if the price it charges covers at least
the average variable cost. If the price is $5, is there a quantity where the average
variable cost is low enough for the firm to meet this condition?
(b) A firm breaks even if the average cost is equal to the price it charges. If the price
is $10, is there a quantity where the average cost is low enough for the firm to at
least break even?
(c) Which output level minimizes the average cost?
Solution: (a) Yes, there is. At 𝑄𝑄 = 20, we found that 𝐴𝐴𝐴𝐴𝐴𝐴 = 4, which is less than 5.
Therefore, the firm should operate at the profit-maximizing level. (b) Yes, there is.
At 𝑄𝑄 = 60, we found that 𝐴𝐴𝐴𝐴 = 9, which is less than 10. Therefore, the firm can
make a positive profit. (c) Average cost is minimized when 𝐴𝐴𝐴𝐴 = 𝑀𝑀𝑀𝑀. We can see
from our previous calculations in Practice Problem 1 that 𝐴𝐴𝐴𝐴 = 𝑀𝑀𝑀𝑀 when 𝑄𝑄 = 60.
Therefore, this must be the quantity where average cost is lowest. We could also
calculate this quantity directly by setting the 𝐴𝐴𝐴𝐴 and 𝑀𝑀𝑀𝑀 functions equal and solving
180
180
180
+ 3 + 0.05𝑄𝑄 = 3 + 0.1𝑄𝑄 implies
= 0.05𝑄𝑄, i.e. 𝑄𝑄2 =
= 3,600.
for 𝑄𝑄:
𝑄𝑄
Therefore 𝑄𝑄 = 60.
𝑄𝑄
0.05
For (c), we could also use our general formula for the minimum average cost, given
that the cost function 180 + 3𝑄𝑄 + 0.05𝑄𝑄2 has the form 𝐶𝐶 = 𝑓𝑓 + 𝑐𝑐𝑐𝑐 + 𝑑𝑑𝑑𝑑2 , where
𝑓𝑓 = 180, 𝑐𝑐 = 3, and 𝑑𝑑 = 0.05. The formula is 𝑄𝑄 = �𝑓𝑓/𝑑𝑑. Substituting 𝑓𝑓 = 180 and
𝑑𝑑 = 0.05, we find that 𝑄𝑄 = �𝑓𝑓/𝑑𝑑 = �180/0.05 = √3,600 = 60.
Practice Quiz Questions 4
1. The ABC Company is planning an aggressive new pricing strategy that offers its
product to wholesalers at $4 per unit. While the goal is to expand sales, rather than
necessarily maximize profits in the short term, it is critical that the company
maintain a positive operating profit, since it cannot raise any more debt in the
immediate future. Your manager would like to get a better sense of how realistic it is
to recover variable costs at a price of $4, given the cost function 𝐶𝐶 = 180 + 3𝑄𝑄 +
0.05𝑄𝑄2 . How many units must the firm sell so that the average variable cost exactly
equals the price of $4?
o
o
o
o
20 CORRECT
40
60
120
2. The XYZ Company’s management wants to increase its profit margin, but feels that
market conditions do not offer flexibility to raise the price. Therefore, the focus is
shifting to costs, and the CEO has announced a company-wide effort to cut unit cost.
When your boss mentions the need to identify potential savings, you make a
different suggestion: you believe that, if sales were increased, per unit cost would go
down. You explain that this phenomenon is called “economies of scale.” Your boss
asks whether you could come up with a precise sales target to take full advantage of
these economies. Fortunately, she has the cost function ready that the finance team
estimated: 𝐶𝐶 = 180 + 3𝑄𝑄 + 0.05𝑄𝑄2 , where 𝑄𝑄 is the quantity produced. In which
output range does the XYZ company have economies of scale?
o There are economies of scale up to an output level of 20, the point where the
firm breaks even.
o There are economies of scale up to an output level of 60, the point where
average cost is minimized. CORRECT
o There are economies of scale whenever the price is above $5, so that the firm
should operate instead of shutting down.
o There are economies of scale at every output level, since average cost falls
whenever output increases.
Problem Set 5
1. A competitive firm has cost function 𝐶𝐶 = 64 + 12𝑄𝑄 + 𝑄𝑄2 .
(a) What is the firm’s supply function (i.e. profit-maximizing output as a function of
price)?
(b) At a market price of $14, how much output should this firm produce to maximize
profit?
(c) At a market price of $28, how much output should this firm produce to maximize
profit?
2. A competitive firm has cost function 𝐶𝐶 = 360 + 20𝑄𝑄 + 2.5𝑄𝑄2 .
(a) What is the firm’s supply function (i.e. profit-maximizing output as a function of
price)?
(b) At a market price of $120, how much output should this firm produce to
maximize profit?
(c) At a market price of $80, how much output should this firm produce to maximize
profit?
3. For the competitive firm in Problem 1:
(a) What is the average cost when the firm produces the profit-maximizing quantity
at a price of $14? Is the firm making a profit or a loss at this price, and how
much? If it is making a loss, should it shut down?
(b) What is the average cost when the firm produces the profit-maximizing quantity
at a price of $28? Is the firm making a profit or a loss at this price, and how
much? If it is making a loss, should it shut down?
(c) At which quantity does the firm minimize its average cost?
(d) If this is a typical firm, and its cost function represents the long-run costs of each
firm in the industry, which price would you expect the industry to tend toward
in the long run?
(e) What is the expected long-run profit of a firm in this industry?
4. For the competitive firm in Problem 2:
(a) What is the average cost when the firm produces the profit-maximizing quantity
at a price of $120? Is the firm making a profit at this price, and how much?
(b) What is the average cost when the firm produces the profit-maximizing quantity
at a price of $80? Is the firm making a profit at this price, and how much?
(c) At which quantity does the firm minimize its average cost?
(d) If this is a typical firm, and its cost function represents the long-run costs of each
firm in the industry, which price would you expect the industry to tend toward
in the long run?
(e) What is the expected long-run profit of a firm in this industry?
Practice Problem Set 5
1. A competitive firm has cost function 𝐶𝐶 = 180 + 3𝑄𝑄 + 0.05𝑄𝑄2 .
(a) What is the firm’s supply function (i.e. profit-maximizing output as a function of
price)?
(b) At a market price of $6, how much output should this firm produce to maximize
profit?
(c) At a market price of $15, how much output should this firm produce to maximize
profit?
Solution: (a) The competitive firm will produce output 𝑄𝑄 such that 𝑀𝑀𝑀𝑀 = 𝑃𝑃. Since
𝜕𝜕𝜕𝜕
𝑀𝑀𝑀𝑀 =
= 3 + 0.1𝑄𝑄, optimal production implies that 3 + 0.1𝑄𝑄 = 𝑃𝑃. Rearranging for
𝜕𝜕𝜕𝜕
𝑄𝑄, we find, via 0.1𝑄𝑄 = 𝑃𝑃 − 3, that 𝑄𝑄 = 10𝑃𝑃 − 30. (b) At 𝑃𝑃 = 6, the supply function
implies that 𝑄𝑄 = 10 ∙ 6 − 30 = 30 is the profit-maximizing level of output. (c) At a
price of $15, the supply function implies that 𝑄𝑄 = 10 ∙ 15 − 30 = 120 is the profitmaximizing level of output.
2. For the competitive firm in Practice Problem 1:
(a) What is the average cost when the firm produces the profit-maximizing quantity
at a price of $6? Is the firm making a profit or a loss at this price, and how much?
If it is making a loss, should it shut down?
(b) What is the average cost when the firm produces the profit-maximizing quantity
at a price of $15? Is the firm making a profit or a loss at this price, and how
much? If it is making a loss, should it shut down?
(c) At which quantity does the firm minimize its average cost?
(d) If this is a typical firm, and its cost function represents the long-run costs of each
firm in the industry, which price would you expect the industry to tend toward
in the long run?
(e) What is the expected long-run profit of a firm in this industry?
Solution: (a) Average cost is 𝐴𝐴𝐴𝐴 =
𝐶𝐶
𝑄𝑄
=
180
𝑄𝑄
+ 3 + 0.05𝑄𝑄. At 𝑃𝑃 = 6, we know from
Practice Problem 1 that the profit-maximizing quantity is 𝑄𝑄 = 30. Therefore, 𝐴𝐴𝐴𝐴 =
+ 3 + 0.05 ∙ 30 = 6 + 3 + 1.5 = 10.5. The firm is making a loss at this price,
30
since the average cost exceeds the price. Profit is 𝜋𝜋 = (𝑃𝑃 − 𝐴𝐴𝐴𝐴) ∙ 𝑄𝑄 = (6 − 10.5) ∙
30 = −135. It should not shut down, however, since the average variable cost
(𝐴𝐴𝐴𝐴𝐴𝐴 = 3 + 0.05𝑄𝑄 = 3 + 0.05 ∙ 30 = 4.5) is below the price. (b) At 𝑃𝑃 = 15, we know
from Problem 1 that the profit-maximizing quantity is 𝑄𝑄 = 120. Therefore, 𝐴𝐴𝐴𝐴 =
180
+ 3 + 0.05 ∙ 120 = 1.5 + 3 + 6 = 10.5. The firm is making a profit at this price,
180
120
since the average cost is less than the price. Profit is 𝜋𝜋 = (𝑃𝑃 − 𝐴𝐴𝐴𝐴) ∙ 𝑄𝑄 =
(15 − 10.5) ∙ 120 = 4.5 ∙ 120 = 540. (c) Average cost is minimized when 𝐴𝐴𝐴𝐴 = 𝑀𝑀𝑀𝑀.
180
Since marginal cost is 𝑀𝑀𝑀𝑀 = 3 + 0.1𝑄𝑄, the equation holds when
+ 3 + 0.05𝑄𝑄 =
180
2
3 + 0.1𝑄𝑄, i.e. 𝑄𝑄 =
0.05
= 3,600, and therefore 𝑄𝑄 = 60. (d) The industry price will
tend toward minimum average cost in the long run, i.e. 𝐴𝐴𝐴𝐴 =
180
𝑄𝑄
180
𝑄𝑄
+ 3 + 0.05𝑄𝑄 =
+ 3 + 0.05 ∙ 60 = 3 + 3 + 3 = 9. (e) In the long run, profit is zero: 𝜋𝜋 = (𝑃𝑃 − 𝐴𝐴𝐴𝐴) ∙
𝑄𝑄 = (9 − 9) ∙ 60 = 0.
60
Or, we could observe that the cost function has the form 𝐶𝐶 = 𝑓𝑓 + 𝑐𝑐𝑐𝑐 + 𝑑𝑑𝑄𝑄2 , where
𝑓𝑓
𝑓𝑓 = 180, 𝑐𝑐 = 3, and 𝑑𝑑 = 0.05. The average cost is, in general, 𝐴𝐴𝐴𝐴 = + 𝑐𝑐 + 𝑑𝑑𝑑𝑑 =
180
𝑄𝑄
𝑄𝑄
+ 3 + 0.05𝑄𝑄, and we can substitute the quantities implied by the supply function
𝑄𝑄 =
𝑃𝑃−𝑐𝑐
2𝑑𝑑
=
𝑃𝑃−3
2∙0.05
=
30 = 30 and 𝐴𝐴𝐴𝐴 =
and 𝐴𝐴𝐴𝐴 =
180
120
(𝑃𝑃−3)2
𝑃𝑃−3
0.1
180
30
= 10𝑃𝑃 − 30 at the different prices. I.e. at 𝑃𝑃 = 6, 𝑄𝑄 = 10 ∙ 6 −
+ 3 + 0.05 ∙ 30 = 10.5. At 𝑃𝑃 = 15, 𝑄𝑄 = 10 ∙ 15 − 30 = 120
+ 3 + 0.05 ∙ 120 = 10.5. Short-run profit is 𝜋𝜋 =
(𝑃𝑃−𝑐𝑐)2
4𝑑𝑑
− 𝑓𝑓 =
(𝑃𝑃−3)2
4∙0.05
−
180 =
− 180 = 5 ∙ (𝑃𝑃 − 3)2 − 180. I.e. at 𝑃𝑃 = 6, 𝜋𝜋 = 5 ∙ (6 − 3)2 − 180 =
0.2
−135. At 𝑃𝑃 = 15, 𝜋𝜋 = 5 ∙ (15 − 3)2 − 180 = 540. Average cost is minimized at 𝑄𝑄 =
𝑓𝑓
� =�
𝑑𝑑
180
0.05
= √3,600 = 60. Long-run price is 𝑃𝑃 = 𝑐𝑐 + 2�𝑑𝑑𝑑𝑑 = 3 + 2 ∙ √0.05 ∙ 180 =
3 + 2 ∙ √9 = 3 + 2 ∙ 3 = 9. Long-run profit is 𝜋𝜋 = 0.
Practice Quiz Questions 5
1. Engineers in your company have determined that the cost of production of good X is
accurately predicted by the function 𝐶𝐶 = 180 + 3𝑄𝑄 + 0.05𝑄𝑄2 , where 𝑄𝑄 is output of
X. In this industry, there are many competitors, making similar products, and each is
currently charging a price of $15. How much output do you recommend your
company should produce if it aims to maximize profit?
o
o
o
o
Q = 60
Q = 75
Q = 120 CORRECT
Q = 200
2. The COO of your company needs to make several decisions in the near future
regarding production facilitities and staffing. Factory space needs to be leased,
contracts with suppliers have to be negotiated, and workers must be hired and
trained. At the heart of it all is the question what sort of capacity is going to be
required – how much the firm should plan to produce and sell in the future. The
company is one of many similar players in an industry where costs are related to
output 𝑄𝑄 as follows: 𝐶𝐶 = 180 + 3𝑄𝑄 + 0.05𝑄𝑄2 . Can you tell the COO which output
level would yield the largest profit margin by minimizing the average cost of
production?
o
o
o
o
Q = 60 CORRECT
Q = 75
Q = 120
Q = 200
Problem Set 6
1. Given the demand function 𝑄𝑄 = 280 − 0.5𝑃𝑃:
(a) Find revenue as a function of 𝑄𝑄.
(b) What is revenue at 𝑄𝑄 = 100?
(c) Find the marginal revenue function.
(d) What is marginal revenue at 𝑄𝑄 = 100?
(e) At which quantity is revenue maximized?
(f) What is the price elasticity of demand at price 𝑄𝑄 = 100? Is demand elastic, unitelastic, or inelastic at this quantity?
2. Given the inverse demand function 𝑃𝑃 = 4,510 − 55𝑄𝑄:
(a) Find the revenue function.
(b) What is revenue at 𝑄𝑄 = 41?
(c) Find the marginal revenue function.
(d) What is marginal revenue at 𝑄𝑄 = 41?
(e) At which quantity is revenue maximized?
(f) What is the price elasticity of demand at 𝑄𝑄 = 41? Is demand elastic, unit-elastic,
or inelastic at this quantity?
3. You observe two data points. Initially, at price 𝑃𝑃1 = 8, sales were 𝑄𝑄1 = 320. Then the
price increased to 𝑃𝑃2 = 12, and sales fell to 𝑄𝑄2 = 80.
(a) Using the midpoint method, what is the approximate price elasticity of demand
in this price range?
(b) If the price increased by 9%, by approximately what percentage would sales
change, based on the elasticity you calculated?
(c) In order to increase sales by 9%, by approximately what percentage would the
price have to be cut, based on the elasticity you calculated?
4. In each of the following cases, you observe two data points. Initially, at price 𝑃𝑃1 ,
sales were 𝑄𝑄1 . Then the price changed to 𝑃𝑃2 , and sales changed to 𝑄𝑄2 . Using the
midpoint method, what is the approximate price elasticity of demand in each price
range?
(a) 𝑃𝑃1 = 110, 𝑄𝑄1 = 0; 𝑃𝑃2 = 90, 𝑄𝑄2 = 2
(b) 𝑃𝑃1 = 51, 𝑄𝑄1 = 50; 𝑃𝑃2 = 50, 𝑄𝑄2 = 51
(c) 𝑃𝑃1 = 2, 𝑄𝑄1 = 90; 𝑃𝑃2 = $0, 𝑄𝑄2 = 110
In addition, answer the following:
(d) If the price increased by 5%, in which scenario – (a), (b), or (c) – would sales fall
by the greatest percentage? Approximately what is that percentage, based on the
elasticity you calculated?
(e) In order to increase sales by 5%, in which scenario – (a), (b), or (c) – would the
price have to be cut by the greatest percentage? Approximately what is that
percentage, based on the elasticity you calculated?
Problem Set 7
1. Given the inverse demand function 𝑃𝑃 = 125 − 0.25𝑄𝑄 and cost function 𝐶𝐶 = 275 +
20𝑄𝑄 + 5𝑄𝑄2 :
(a) What is the profit-maximizing quantity 𝑄𝑄?
(b) What is the profit-maximizing price 𝑃𝑃?
(c) What is the elasticity at the profit-maximizing price and quantity?
(d) What is the maximum profit that can be earned?
2. Given the demand function 𝑄𝑄 = 280 − 0.5𝑃𝑃 and cost function 𝐶𝐶 = 1,200 + 20𝑄𝑄:
(a) What is the profit-maximizing quantity 𝑄𝑄?
(b) What is the profit-maximizing price 𝑃𝑃?
(c) What is the elasticity at the profit-maximizing price and quantity?
(d) What is the maximum profit that can be earned?
3. The Possum Run Railway takes tourists from Bear Lake to Eagle Lookout. The
railway’s non-senior customers demand 𝑄𝑄𝑁𝑁 = 800 − 10𝑃𝑃𝑁𝑁 tickets per year. Senior
customers demand another 𝑄𝑄𝑆𝑆 = 600 − 15𝑃𝑃𝑆𝑆 tickets. Since the trains operate
regardless of the number of passengers, marginal cost is zero. The fixed cost
associated with maintenance, fuel, and staff is reimbursed to the Possum Run
Railway from park fees, so it does not affect profit.
(a) Calculate the optimal prices to charge to non-seniors and seniors, respectively.
(b) How many tickets does Possum Run sell overall at the profit-maximizing prices?
(c) How much profit is Possum Run able to make?
(d) Now suppose Possum Run Railway was forced by state park regulations to
charge the same price to all customers. What price would maximize its profits,
and how many tickets would it sell?
(e) How much profit would the company now make? How much money (if any)
would Possum Run Railway lose from such a regulation?
4. Chessathlete.com has a monopoly on a unique “holistic health” game where players
compete in both chess and long-distance runs (which are tracked by their mobile
phones). They charge players a monthly flat fee to participate and improve their
mental and physical fitness. Past experience revealed that players registering
through iPhone and Android devices had a different willingness to pay for
membership. For iPhone users, the demand is 𝑄𝑄𝐼𝐼 = 798 − 19𝑃𝑃𝐼𝐼 . For Android users,
demand is 𝑄𝑄𝐴𝐴 = 1,470 − 35𝑃𝑃𝐴𝐴 . Chessathlete has a small marginal cost of $2 per user
(regardless of the device they are using), which has to do with the cost of server
capacity and managing accounts, which they contract out to an external service. The
fixed cost Chessathlete must cover each week is $20,000.
(a) Calculate the optimal prices to charge to iPhone and Android users, respectively.
(b) How many memberhips does Chessathlete sell overall at the profit-maximizing
prices?
(c) How much profit is Chessathlete able to make? (Hint: Be careful to subtract fixed
cost only once, since it applies to the whole company, rather than separately to
each market served.)
(d) Now suppose Chessathlete.com cannot tell iPhone and Android users apart, due
to a freely available masking technology. Therefore, it has to charge the same
price to all customers. What price would maximize its profits, and how many
memberships would it sell?
(e) How much profit would it make? How much money (if any) does Chessathlete
lose due to the existence of the masking technology?
Practice Problem Set 7
1. Given the demand function 𝑄𝑄 = 60 − 4𝑃𝑃 and cost function 𝐶𝐶 = 180 + 3𝑄𝑄 + 0.05𝑄𝑄2 :
(a) What is the profit-maximizing quantity 𝑄𝑄?
(b) What is the profit-maximizing price 𝑃𝑃?
(c) What is the elasticity at the profit-maximizing price and quantity?
(d) What is the maximum profit that can be earned?
Solution: (a) To maximize profit, we need to set 𝑀𝑀𝑀𝑀 = 𝑀𝑀𝑀𝑀. Inverse demand is 𝑃𝑃 =
15 − 0.25𝑄𝑄 (from 𝑄𝑄 = 60 − 4𝑃𝑃). Therefore, 𝑅𝑅 = 𝑃𝑃 ∙ 𝑄𝑄 = 15𝑄𝑄 − 0.25𝑄𝑄2 , and 𝑀𝑀𝑀𝑀 =
𝑑𝑑𝑑𝑑
= 15 − 0.5𝑄𝑄. We know from before that 𝑀𝑀𝑀𝑀 = 15 − 0.5𝑄𝑄. Marginal cost is 𝑀𝑀𝑀𝑀 =
𝑑𝑑𝑑𝑑
𝑑𝑑𝐶𝐶
= 3 + 0.1𝑄𝑄. The profit-maximizing 𝑄𝑄 will satisfy 15 − 0.5𝑄𝑄 = 3 + 0.1𝑄𝑄, which
𝑑𝑑𝑄𝑄
implies via 0.6𝑄𝑄 = 12 that 𝑄𝑄 = 20. (b) The profit-maximizing price is the price at
which the profit-maximizing quantity 𝑄𝑄 = 20 is sold, i.e. 𝑃𝑃 = 15 − 0.25 ∙ 20 = 10.
𝜕𝜕𝜕𝜕 𝑃𝑃
10
(c) Elasticity is 𝜀𝜀 = ∙ = −4 ∙ = −2. (d) Profit at the optimal price and quantity
𝜕𝜕𝜕𝜕
𝑄𝑄
20
is 𝜋𝜋 = 𝑅𝑅 − 𝐶𝐶 = 20 ∙ 10 − (180 + 3 ∙ 20 + 0.05 ∙ 202 ) = 200 − 260 = −60.
(Remember that maximum profit can be negative, since the firm may still be better
off producing in the short run while it’s making a loss. A loss of 60 is better than a
loss of 180, which is what would happen if the firm did not produce, but still
incurred the fixed cost.)
Or, we could recognize that the demand function 𝑄𝑄 = 60 − 4𝑃𝑃 is of the form 𝑄𝑄 =
𝑎𝑎 − 𝑏𝑏𝑏𝑏, where 𝑎𝑎 = 60 and 𝑏𝑏 = 4; the cost function 𝐶𝐶 = 180 + 3𝑄𝑄 + 0.05𝑄𝑄2 is of the
𝑎𝑎−𝑏𝑏𝑏𝑏
=
form 𝐶𝐶 = 𝑓𝑓 + 𝑐𝑐𝑐𝑐 + 𝑑𝑑𝑄𝑄2 , where 𝑓𝑓 = 180, 𝑐𝑐 = 3, and 𝑑𝑑 = 0.05. Therefore, 𝑋𝑋 =
60−4∙3
24
2
1.2
2
= 24, and 𝑌𝑌 = 1 + 𝑏𝑏𝑏𝑏 = 1 + 4 ∙ 0.05 = 1.2. The profit maximizers are 𝑄𝑄 =
= 20 and 𝑃𝑃 =
180 =
576
4.8
𝑎𝑎𝑎𝑎−𝑋𝑋
𝑏𝑏𝑏𝑏
=
60∙1.2−24
4∙1.2
=
48
4.8
− 180 = 120 − 180 = −60.
= 10. Maximum profit is 𝜋𝜋 =
𝑋𝑋 2
𝑏𝑏𝑏𝑏
− 𝑓𝑓 =
242
𝑋𝑋
𝑌𝑌
4∙1.2
2. A monopolist has demand of 𝑄𝑄𝐴𝐴 = 60 − 2𝑃𝑃𝐴𝐴 in market 𝐴𝐴 and 𝑄𝑄𝐵𝐵 = 60 − 6𝑃𝑃𝐵𝐵 in
market 𝐵𝐵. The monopolist is able to charge different prices 𝑃𝑃𝐴𝐴 and 𝑃𝑃𝐵𝐵 in these
markets and has a constant marginal cost of 3.
(a) Calculate the optimal prices to charge to market A and market B customers,
respectively.
(b) How many units does the monopolist sell overall at the profit-maximizing
prices?
(c) How much profit is the monopolist able to make?
=
−
(d) Now suppose the monopolist must charge a uniform price to each of the
consumer types. What price would maximize its profits, and how many units
would it sell?
(e) How much profit will the monopolist make? How much money (if any) does the
monopolist lose if it cannot price discriminate?
Solution: (a) In order to maximize profit in each market, we need to set 𝑀𝑀𝑀𝑀 = 𝑀𝑀𝑀𝑀 =
3 in each case. For market A customers, with demand 𝑄𝑄𝐴𝐴 = 60 − 2𝑃𝑃𝐴𝐴 , inverse
demand is 𝑃𝑃𝐴𝐴 = 30 − 𝑄𝑄𝐴𝐴 /2 = 30 − 0.5𝑄𝑄𝐴𝐴 . Therefore, 𝑅𝑅𝐴𝐴 = 𝑃𝑃𝐴𝐴 ∙ 𝑄𝑄𝐴𝐴 = 30𝑄𝑄𝐴𝐴 − 0.5𝑄𝑄𝐴𝐴2
𝑑𝑑𝑅𝑅
and 𝑀𝑀𝑀𝑀𝐴𝐴 = 𝐴𝐴 = 30 − 𝑄𝑄𝐴𝐴 . Setting 𝑀𝑀𝑀𝑀𝐴𝐴 = 𝑀𝑀𝑀𝑀, in order to maximize profit, implies
𝑑𝑑𝑄𝑄𝐴𝐴
30 − 𝑄𝑄𝐴𝐴 = 3, i.e. 𝑄𝑄𝐴𝐴 = 27 and 𝑃𝑃𝐴𝐴 = 30 − 0.5 ∙ 27 = 16.5. For market B customers,
with demand 𝑄𝑄𝐵𝐵 = 60 − 6𝑃𝑃𝐵𝐵 , inverse demand is 𝑃𝑃𝐵𝐵 = 10 − 𝑄𝑄𝐵𝐵 /6. Therefore, 𝑅𝑅𝐵𝐵 =
𝑑𝑑𝑅𝑅
𝑃𝑃𝐵𝐵 ∙ 𝑄𝑄𝐵𝐵 = 10𝑄𝑄𝐵𝐵 − 𝑄𝑄𝐵𝐵2 /6 and 𝑀𝑀𝑀𝑀𝐵𝐵 = 𝐵𝐵 = 10 − 𝑄𝑄𝐵𝐵 /3. Setting 𝑀𝑀𝑀𝑀𝐵𝐵 = 𝑀𝑀𝑀𝑀, in order to
𝑑𝑑𝑄𝑄𝐵𝐵
maximize profit, implies 10 − 𝑄𝑄𝐵𝐵 /3 = 3, i.e. 𝑄𝑄𝐵𝐵 = 21, and 𝑃𝑃𝐵𝐵 = 10 − 21/6 = 6.5. (b)
Since 𝑄𝑄𝐴𝐴 = 27 and 𝑄𝑄𝐵𝐵 = 21, the total quantity sold is 𝑄𝑄 = 𝑄𝑄𝐴𝐴 + 𝑄𝑄𝐵𝐵 = 48. (c) The
company has revenue 𝑅𝑅𝐴𝐴 = 𝑃𝑃𝐴𝐴 ∙ 𝑄𝑄𝐴𝐴 = 16.5 ∙ 27 = 445.5 in market A and revenue
𝑅𝑅𝐵𝐵 = 𝑃𝑃𝐵𝐵 ∙ 𝑄𝑄𝐵𝐵 = 6.5 ∙ 21 = 136.5 from market B. Profit is the total revenue 𝑅𝑅 = 𝑅𝑅𝐴𝐴 +
𝑅𝑅𝐵𝐵 = 582, less the total cost 𝐶𝐶 = 3 ∙ 48 = 144, i.e. 438. (d) We need to add up the
demands from the two types of customers, and then maximize profit with respect to
the total demand. At a uniform price 𝑃𝑃, with 𝑄𝑄𝐴𝐴 = 60 − 2𝑃𝑃 and 𝑄𝑄𝐵𝐵 = 60 − 6𝑃𝑃, total
demand is 𝑄𝑄 = 𝑄𝑄𝐴𝐴 + 𝑄𝑄𝐵𝐵 = 120 − 8𝑃𝑃. Then, inverse demand is 𝑃𝑃 = 15 − 𝑄𝑄/8.
𝑑𝑑𝑅𝑅
Therefore, 𝑅𝑅 = 𝑃𝑃 ∙ 𝑄𝑄 = 15𝑄𝑄 − 𝑄𝑄2 /8 and 𝑀𝑀𝑀𝑀 =
= 15 − 𝑄𝑄/4 = 15 − 0.25𝑄𝑄. Setting
𝑑𝑑𝑄𝑄
15−3
this equal to the marginal cost, 15 − 0.25𝑄𝑄 = 3, yields 𝑄𝑄 =
= 48. Using the
0.25
inverse demand function, the price is 𝑃𝑃 = 15 − 48/8 = 9. (e) Profit is 𝑃𝑃 ∙ 𝑄𝑄 − 𝐶𝐶 = 9 ∙
48 − 3 ∙ 48 = 288, which is less by 150 than the profit of 438 with price
discrimination. Not being able to price discriminate costs the company about half of
its profit in this case.
Or, if we recognize that the demand functions have the linear form, 𝑄𝑄 = 𝑎𝑎 − 𝑏𝑏𝑏𝑏, and
the cost function has the quadratic form, 𝐶𝐶 = 𝑓𝑓 + 𝑐𝑐𝑐𝑐 + 𝑑𝑑𝑄𝑄2 , we can apply the
general formulas. For Market A, 𝑎𝑎𝐴𝐴 = 60 and 𝑏𝑏𝐴𝐴 = 2. For Market B, 𝑎𝑎𝐵𝐵 = 60 and
𝑏𝑏𝐵𝐵 = 6. For both, 𝑓𝑓 = 0, 𝑐𝑐 = 3, and 𝑑𝑑 = 0. (Since marginal cost is 3.) Therefore, 𝑋𝑋𝐴𝐴 =
𝑎𝑎𝐴𝐴 −𝑏𝑏𝐴𝐴 𝑐𝑐
60−2∙3
𝑎𝑎 −𝑏𝑏 𝑐𝑐
=
= 27 and 𝑌𝑌𝐴𝐴 = 1 + 𝑏𝑏𝐴𝐴 𝑑𝑑 = 1 + 2 ∙ 0 = 1. Similarly, 𝑋𝑋𝐵𝐵 = 𝐵𝐵 𝐵𝐵 =
2
60−6∙3
2
𝑃𝑃𝐴𝐴 =
2
= 21 and 𝑌𝑌𝐵𝐵 = 1 + 𝑏𝑏𝐵𝐵 𝑑𝑑 = 1 + 6 ∙ 0 = 1. For Market A, 𝑄𝑄𝐴𝐴 =
𝑎𝑎𝐴𝐴 𝑌𝑌𝐴𝐴 −𝑋𝑋𝐴𝐴
𝑏𝑏𝐴𝐴 𝑌𝑌𝐴𝐴
=
Market B, 𝑄𝑄𝐵𝐵 =
2
𝑋𝑋𝐵𝐵
𝑏𝑏𝐵𝐵 𝑌𝑌𝐵𝐵
− 𝑓𝑓 =
212
6∙1
60∙1−27
=
2∙1
𝑋𝑋𝐵𝐵
21
𝑌𝑌𝐵𝐵
=
1
33
2
= 16.5, and 𝜋𝜋𝐴𝐴 =
= 21 and 𝑃𝑃𝐵𝐵 =
2
𝑋𝑋𝐴𝐴
− 𝑓𝑓 =
272
=
6∙1
=
𝑌𝑌𝐴𝐴
=
27
1
2
= 27,
− 0 = 364.5. For
𝑏𝑏𝐴𝐴 𝑌𝑌𝐴𝐴
2∙1
𝑎𝑎𝐵𝐵 𝑌𝑌𝐵𝐵 −𝑋𝑋𝐵𝐵
60∙1−21
39
𝑏𝑏𝐵𝐵 𝑌𝑌𝐵𝐵
𝑋𝑋𝐴𝐴
6
= 6.5, and 𝜋𝜋𝐵𝐵 =
− 0 = 73.5. Total output is 𝑄𝑄𝐴𝐴 + 𝑄𝑄𝐵𝐵 = 27 + 21 = 48, and total profit
is 𝜋𝜋𝐴𝐴 + 𝜋𝜋𝐵𝐵 = 364.5 + 73.5 = 438. In (d), the demand parameters change to 𝑎𝑎 = 120
and 𝑏𝑏 = 8. Now, 𝑋𝑋 =
𝑋𝑋
𝑌𝑌
=
48
1
= 48, 𝑃𝑃 =
𝑎𝑎−𝑏𝑏𝑏𝑏
𝑎𝑎𝑎𝑎−𝑋𝑋
𝑏𝑏𝑏𝑏
2
=
=
120−8∙3
2
120∙1−48
8∙1
= 48 and 𝑌𝑌 = 1 + 𝑏𝑏𝑏𝑏 = 1 + 8 ∙ 0 = 1. So 𝑄𝑄 =
= 9, and 𝜋𝜋 =
𝑋𝑋 2
𝑏𝑏𝑏𝑏
− 𝑓𝑓 =
482
8∙1
− 0 = 288.
Practice Quiz Questions 7
1. The XYZ Company is evaluating several strategic options. Half of the board is
focused on maximizing profit. Given this goal, and the analysts’ estimate of demand,
𝑄𝑄 = 60 − 4𝑃𝑃, and cost, 𝐶𝐶 = 180 + 3𝑄𝑄 + 0.05𝑄𝑄2 , which price would the company
have to charge? The other half of the board wants to pursue an aggressive growth
strategy, and is therefore focused on maximizing revenue, regardless of costs. If the
company wants to maximize revenue, should it charge more or less than if it wants
to maximize profit?
o
o
o
o
$10 to maximize profit, less to maximize revenue. CORRECT
$90 to maximize profit, less to maximize revenue.
$8 to maximize profit, more to maximize revenue.
$50 to maximize profit, more to maximize revenue.
2. Your employer, the XYZ Company, is serving customers through its car wash
facilities in the towns of Assyria and Babylonia, which are similar in population size.
Because of this, and since the cost of providing an automated car wash is $3 in both
locations, the company has been charging the same price of $9 for a car wash in
these towns. However, customers in Assyria tend to be older and wealthier. Their
daily demand function has been estimated as 𝑄𝑄𝐴𝐴 = 60 − 2𝑃𝑃𝐴𝐴 , where 𝑄𝑄𝐴𝐴 is average
daily sales in Assyria and 𝑃𝑃𝐴𝐴 is the price charged at the Assyria facility. On the other
hand, customers in Babylonia are mostly younger and less affluent. Here, daily
demand has been estimated as 𝑄𝑄𝐵𝐵 = 60 − 6𝑃𝑃𝐵𝐵 , where 𝑄𝑄𝐵𝐵 is average daily sales in
Babylonia and 𝑃𝑃𝐵𝐵 is the price charged at the Babylonia facility. When Xenia Yolanda
Zalazar, the CEO of XYZ, stopped by your cubicle during her weekly walk-through,
you brought up the possibility to charge different prices in Assyria and Babylonia.
“Interesting idea,” she muses. “How much more profit do you think we could make
on a typical day?” Your best forecast is:
o
o
o
o
$120
$150 CORRECT
$360
$480
Practice Problem Set 8
1. A Cournot duopoly operates in an industry with the following inverse demand
function: 𝑃𝑃 = 15 − 0.25𝑄𝑄1 − 0.25𝑄𝑄2 , where 𝑄𝑄1 and 𝑄𝑄2 are quantities produced by
firms 1 and 2, respectively. The firms' marginal costs are identical at 3 + 0.1𝑄𝑄𝑖𝑖 ,
where 𝑖𝑖 is either firm 1 or firm 2.
(a) What are firm 1 and 2's marginal revenue functions?
(b) What are firm 1 and 2’s reaction functions?
Solution: (a) Revenue is 𝑅𝑅𝑖𝑖 = 𝑃𝑃 ∙ 𝑄𝑄𝑖𝑖 = �15 − 0.25𝑄𝑄𝑖𝑖 − 0.25𝑄𝑄𝑗𝑗 � ∙ 𝑄𝑄𝑖𝑖 = 15𝑄𝑄𝑖𝑖 −
0.25𝑄𝑄𝑖𝑖2 − 0.25𝑄𝑄𝑖𝑖 𝑄𝑄𝑗𝑗 . Therefore, marginal revenue is 𝑀𝑀𝑀𝑀𝑖𝑖 =
𝑑𝑑𝑅𝑅𝑖𝑖
𝑑𝑑𝑄𝑄𝑖𝑖
= 15 − 0.5𝑄𝑄𝑖𝑖 −
0.25𝑄𝑄𝑗𝑗 . The marginal functions are 𝑀𝑀𝑀𝑀1 = 15 − 0.5𝑄𝑄1 − 0.25𝑄𝑄2 and 𝑀𝑀𝑀𝑀2 = 15 −
0.5𝑄𝑄2 − 0.25𝑄𝑄1 . (b) The reaction functions maximize profit for each firm, given the
other firm’s output. To maximize profit, we must set 𝑀𝑀𝑀𝑀𝑖𝑖 = 𝑀𝑀𝑀𝑀. Thus, 15 − 0.5𝑄𝑄𝑖𝑖 −
0.25𝑄𝑄𝑗𝑗 = 3 + 0.1𝑄𝑄𝑖𝑖 . Expressing 𝑄𝑄𝑖𝑖 as a function of 𝑄𝑄𝑗𝑗 , we have, via 0.6𝑄𝑄𝑖𝑖 = 12 −
0.25𝑄𝑄𝑗𝑗 , 𝑄𝑄𝑖𝑖 =
20 −
5
12
12
0.6
−
0.25
0.6
𝑄𝑄𝑗𝑗 = 20 −
𝑄𝑄2 and 𝑄𝑄2 = 20 −
5
12
𝑄𝑄1 .
5
12
𝑄𝑄𝑗𝑗 . Therefore, the reaction functions are 𝑄𝑄1 =
Or, we could apply the general formulas. Since the firms’ outputs add up to 𝑄𝑄1 +
𝑄𝑄2 = 𝑄𝑄 (the overall quantity sold in the industry), we can write the inverse demand
function as 𝑃𝑃 = 15 − 0.25𝑄𝑄1 − 0.25𝑄𝑄2 = 15 − 0.25(𝑄𝑄1 + 𝑄𝑄2 ) = 15 − 0.25𝑄𝑄. This
implies, via 0.25𝑄𝑄 = 15 − 𝑃𝑃, that 𝑄𝑄 = 60 − 4𝑃𝑃. The demand function has the linear
form, 𝑄𝑄 = 𝑎𝑎 − 𝑏𝑏𝑏𝑏, where 𝑎𝑎 = 60 and 𝑏𝑏 = 4. Moreover, the cost function has the
quadratic form, 𝐶𝐶 = 𝑓𝑓 + 𝑐𝑐𝑐𝑐 + 𝑑𝑑𝑄𝑄2 , where 𝑐𝑐 = 3, and 𝑑𝑑 = 0.05. (This is implied by
the marginal cost, which is 3 + 0.1𝑄𝑄𝑖𝑖 = 𝑐𝑐 + 2𝑑𝑑𝑄𝑄𝑖𝑖 .) The general formula for marginal
𝑎𝑎−2𝑄𝑄𝑖𝑖 −𝑄𝑄𝑗𝑗
60−2𝑄𝑄𝑖𝑖 −𝑄𝑄𝑗𝑗
revenue in the Cournot model is: 𝑀𝑀𝑀𝑀𝑖𝑖 =
=
= 15 − 0.5𝑄𝑄𝑖𝑖 − 0.25𝑄𝑄𝑗𝑗 .
𝑏𝑏
X
4
The general formula for the reaction function is 𝑄𝑄𝑖𝑖 = −
60−4∙3
1
2
2.4
Y
1 𝑄𝑄𝑗𝑗
2 Y
. Since X =
= 24 and Y = 1 + 𝑏𝑏𝑏𝑏 = 1 + 4 ∙ 0.05 = 1.2, we have 𝑄𝑄𝑖𝑖 =
𝑄𝑄𝑗𝑗 = 20 −
5
12
𝑄𝑄𝑗𝑗 .
2. For the Cournot duopolists in Practice Problem 1:
24
1.2
−
1 𝑄𝑄𝑗𝑗
2 1.2
𝑎𝑎−𝑏𝑏𝑏𝑏
2
=
= 20 −
(a) What are the equilibrium quantities produced by each firm?
(b) What is the overall quantity produced by both firms?
(c) What is the market price in equilibrium?
(d) What is the profit each firm makes in equilibrium? (Assume that there is no fixed
cost. )
Solution: (a) Substituting the reaction functions, 𝑄𝑄1 = 20 −
5
𝑄𝑄 , into each other, we find that 𝑄𝑄1 = 20 −
12 1
140
25
5
12
∙ �20 −
5
12
5
12
𝑄𝑄2 and 𝑄𝑄2 = 20 −
𝑄𝑄1 � = 20 −
100
12
+
17
=
25
144
𝑄𝑄1 =
35
25
25
35
119
35
𝑄𝑄 = +
𝑄𝑄 . Therefore, �1 − � 𝑄𝑄1 = , i.e.
𝑄𝑄 = , which implies
144 1
3
144 1
144
3
144 1
3
144 35
5,040
240
5 240
100
340
100
∙ =
=
≈ 14 and then 𝑄𝑄2 = 20 − ∙
= 20 −
=
−
=
𝑄𝑄1 =
119 3
357
17
12 17
17
17
17
240
480
. (b) The overall quantity produced is 𝑄𝑄 = 𝑄𝑄1 + 𝑄𝑄2 =
. (c) From the inverse
17
17
120
255
12
+
demand function 𝑃𝑃 = 15 − 0.25𝑄𝑄, we find the market price 𝑃𝑃 = 15 −
−
17
=
≈ 8. (d) The cost function is 𝐶𝐶 = 𝑓𝑓 + 𝑐𝑐𝑄𝑄𝑖𝑖 + 𝑑𝑑𝑄𝑄𝑖𝑖2 = 0 + 3𝑄𝑄𝑖𝑖 + 0.05𝑄𝑄𝑖𝑖2 . Thus,
17
17
135 240
240
240 2
∙
−3∙
− 0.05 ∙ � � =
profit for either firm is 𝑃𝑃 ∙ 𝑄𝑄𝑖𝑖 − 3𝑄𝑄𝑖𝑖 − 0.05𝑄𝑄𝑖𝑖2 =
17
17
17
17
32,400
12,240
2,880
17,280
120
135
289
−
289
−
289
=
289
≈ 60.
Or, we could recognize that the demand function has the linear form, 𝑄𝑄 = 𝑎𝑎 − 𝑏𝑏𝑏𝑏,
where 𝑎𝑎 = 60 and 𝑏𝑏 = 4, and the cost function has the quadratic form, 𝐶𝐶 = 𝑓𝑓 + 𝑐𝑐𝑐𝑐 +
𝑑𝑑𝑄𝑄2 , where 𝑓𝑓 = 0, 𝑐𝑐 = 3, and 𝑑𝑑 = 0.05. Therefore, X = 24 and Y = 1.2 (see Practice
Problem 1), and we can apply the general formula for equilibrium output in the
2𝑋𝑋
2∙24
48
240
Cournot model: 𝑄𝑄𝑖𝑖 =
=
=
=
. Market price and profit follow from
2𝑌𝑌+1
2∙1.2+1
3.4
17
the inverse demand function and profit function.
Practice Quiz Questions 8
1. In a strategy meeting, the Chief Operating Officer (COO) discusses production
planning for the next year. She observes that, in recent years, the price we could
charge depended not only on our own, but also on the main competitor’s, decisions.
When they supplied a lot of product, the price would drop. In fact, the behavior of
the price has been estimated as follows: 𝑃𝑃 = 15 − 0.25𝑄𝑄1 − 0.25𝑄𝑄2 , where 𝑃𝑃 is the
market price, 𝑄𝑄1 is the quantity sold by your company, and 𝑄𝑄2 is the quantity sold by
your competitor. Since the price impacts the profitability of our choices, the COO
says we need to plan flexibly until we get some sense of how the competitor is
planning. You are being tasked with designing a strategy that responds to each
possible output level 𝑄𝑄2 of the competitor with our best response 𝑄𝑄1 . In order to
solve this problem, you decide to model the industry as a Cournot duopoly and find
your company’s reaction function. You shoot a quick e-mail to the COO to find out
about costs of production. It turns out that the cost increases with every additional
unit that is produced by $3, plus an additional amount that increases in the overall
production volume (10 cents for each unit). The marginal cost can be expressed as
𝑀𝑀𝑀𝑀1 = 3 + 0.1𝑄𝑄1 . The competitor uses the same technology and therefore has the
same cost structure (i.e. 𝑀𝑀𝑀𝑀2 = 3 + 0.1𝑄𝑄2 ). Based on these data, you derive the
reaction function for your firm as:
o 𝑄𝑄1 = 12 − 3𝑄𝑄2
1
o 𝑄𝑄1 = 15 − 𝑄𝑄2
4
5
o 𝑄𝑄1 = 20 − 𝑄𝑄2 CORRECT
12
o 𝑞𝑞1 = 25 − 𝑄𝑄2
2. In a recent strategy meeting, the Chief Operating Officer (COO) had asked you to
report on the best way to respond to your main competitor’s production decisions.
You modeled the industry as a Cournot duopoly and calculated profit-maximizing
production levels for your firm, depending on what we learn about the competitor’s
output. The market price 𝑃𝑃 depends on the quantity sold by your company (𝑄𝑄1 ), and
the quantity sold by your competitor (𝑄𝑄2 ) as follows: 𝑃𝑃 = 15 − 0.25𝑄𝑄1 − 0.25𝑄𝑄2 .
Costs per unit depend on the production level: they are 𝑀𝑀𝑀𝑀1 = 3 + 0.1𝑄𝑄1 for us, and
similarly 𝑀𝑀𝑀𝑀2 = 3 + 0.1𝑄𝑄2 for our competitor. Impressed with your work on
production strategy, the COO looks at you musingly. “You seem to have a good grasp
of this. Is it somehow possible to predict the price?” “Actually, yes,” you say. “I can
use game theory to do that.” It doesn’t take you long to work out the Nash
equilibrium quantities, and an hour later you e-mail your price prediction to the
COO. The equilibrium price is:
o
o
o
o
$5
Approximately $8 CORRECT
Approximately $59
$125
Practice Problem Set 9
1. The ABC Company and the XYZ Company are the only two competitors in a market
that has a completely inelastic demand of 3,000 units for a good that they both
make. Because customers regard ABC’s and XYZ’s product as excellent substitutes,
they buy from whoever is cheaper. If the prices are the same, customers will split
equally between ABC and XYZ, so that each sells 1,500 units. Both companies have
constant marginal costs of $10 per unit. Consider three claims, and determine
whether they are true and why / why not.
(a) There can be a Nash equilibrium where both firms charge $10.
(b) There can be a Nash equilibrium where both firms charge $20.
(c) There can be a Nash equilibrium where one company charges $15, and the other
charges $20.
Solution: (a) True. In a Nash equilibrium, each player responds optimally to the
other player. Consider ABC. If XYZ charges $10, then ABC would make a loss by
charging less than $10 (it would sell $3,000 units, but each below the cost of $10). It
would not make a profit by charging more than $10 (since all customers would buy
from XYZ, as it is cheaper). Since ABC cannot do better than break even, charging
$10 is a best response. Now consider XYZ. If ABC charges $10, then XYZ cannot make
a positive profit for the same reasons; therefore, charging $10 is a best response.
Because $10 is a best response for both, it constitutes a Nash equilibrium. (b) False.
If both firms charge $20, then either firm can do better by slightly reducing its price.
For example, ABC currently makes a profit of (20 − 10) ∙ 1,500 = 15,000 (they are
splitting the market with equal pricing). If ABC reduced its price, say, to $19, then it
would make a profit of (19 − 10) ∙ 3,000 = 27,000 (now, ABC is cheaper and
captures the entire market). Therefore, it is not a best response for ABC to charge
$20 when XYZ charges $20. So it cannot be a Nash equilibrium. (c) False. Let’s say
ABC’s price is $15, and XYZ’s price is $20. Both firms can do better. For ABC,
increasing its price to $16 would not change the fact that it captures the whole
market, but it would increase the profit margin by one dollar. Profit would increase
from (15 − 10) ∙ 3,000 = 15,000 to (16 − 10) ∙ 3,000 = 18,000. So a price of $16 is
not a best response for ABC to XYZ’s price of $20. For XYZ, too, $20 is not optimal:
currently, it has no sales because ABC is cheaper. By reducing its price to $14, XYZ
could grab all of ABC’s customers and make a profit of (14 − 10) ∙ 3,000 = 12,000
instead of zero. Clearly, ABC and XYZ are not playing best responses; therefore, it
cannot be a Nash equilibrium.
2. Suppose now that the ABC Company’s engineers found a way to reduce the marginal
cost per unit to $9. The XYZ Company does not have access to this technology, so its
marginal cost remains $10 per unit. What is the Nash equilibrium? (Assume that the
firms can only charge prices in dollars and whole cents.)
Solution: In a Nash equilibrium, ABC charges $9.99 and XYZ charges $10. There is no
deviation that benefits ABC. Lowering the price does not increase sales (since ABC
already charges less than XYZ and captures the full market demand); increasing the
price to $10 would reduce ABC’s sales and profit: instead of (9.99 − 9) ∙ 3,000 =
2,970, it would make (10 − 9) ∙ 1,500 = 1,500. Raising the price even further, to
more than $10, would mean that ABC loses all its sales and therefore makes no
profit. XYZ can also not benefit from a price change. Lowering the price, say to $9.99,
would generate sales for XYZ, but every sale would generate a loss, and profit would
be negative: (9.99 − 10) ∙ 1,500 = −15. Raising the price above $10 would still leave
XYZ without sales and therefore no profit. There exists a second Nash equilibrium,
where ABC charges $10 and XYZ charges $10.01. The reasoning is similar: ABC could
not gain sales by lowering price and would lose too many sales if it increased the
price, even by one cent. XYZ can only make sales by lowering the price to $10 or less,
where its profit margin is zero or negative. Since neither has a deviation (an
opportunity to change its price) that increases its profit, they are each optimizing
and therefore it’s a Nash equilibrium.
Practice Quiz Questions 9
1. You are working as an analyst for an investment firm that is potentially interested in
a stake in the ABC Company or its competitor, the XYZ Company. Both firms have
developed a drug to treat a disease which affects 3,000 people. The drugs sold overthe-counter by the ABC Company and the XYZ Company, work equally well and have
very similar risks of side effects. Therefore, you conclude that patients will buy
whichever drug is cheaper. If the prices will be the same, you expect that customers
will split equally between ABC and XYZ, so that each sells to 1,500 patients. From
newspaper reports, you gathered that both firms can produce the drugs at a cost of
$10 per patient. Which of the following is the most plausible result of your gametheoretic analysis of this market?
o Since the ABC Company and the XYZ Company are the only two firms that can
make this drug, and the cost is low, their profits should be high.
o The ABC Company and the XYZ Company will have strong profit margins per
patient, but since they need to share the market, profit for both firms will be
moderate.
o Either of the two companies has an opportunity to undercut the other with an
aggressive pricing strategy and control the market. One will be very profitable,
and one cannot yet predict which. The other will not be profitable.
o These companies cannot be very profitable because the price will be close to
their cost of $10, so the profit margin is essentially zero. CORRECT
2. Your boss has become interested in the analysis of strategic competition and
recently read a book about game theory. He remembers from it that “Nash
equilibrium” describes a kind of stable arrangement that we might expect to play
out between two rivaling firms. As it happens, the investment firm you both work
for is currently trying to assess whether to buy a stake in the ABC Company. It sells
an over-the-counter drug that has 3,000 potential customers. A rival, the XYZ
Company, markets a very similar drug. It is clear that patients will choose between
the drugs based on price. Since the companies use the same distribution channels, it
can be assumed that each would capture half the market, i.e. 1,500 patients, if they
set the same price. Your boss thinks that, if the ABC Company plays it smart, it can
undercut the XYZ Company’s price and sell to all 3,000 patients. He wonders if he
could say in an upcoming meeting that this view is supported by the concept of Nash
equilibrium. Do you agree?
o No, it is never possible that one firm charges less than the other in Nash
equilibrium, regardless of whether the firms have identical or different costs.
o It would happen in a Nash equilibrium only if ABC has lower marginal cost than
XYZ, and is therefore able to charge a lower price that XYZ cannot match. CORRECT
o It would happen in a Nash equilibrium only if ABC has higher marginal cost than
XYZ, and is therefore at a disadvantage, so that it has to charge a lower price.
o Yes, in a Nash equilibrium one firm will always charge a lower price than the
other, regardless of whether the firms have identical or different costs.
Problem Set 10
1. Two stores on the same plaza independently consider whether to run a big sale.
Payoffs in each scenario are as follows.
Store 1
No Sale
Sale
Store 2
No Sale
50, 50
100, 10
Sale
10, 100
20, 20
(a) Does Store 1 have a dominant strategy?
(b) Does Store 2 have a dominant strategy?
(c) Find the Nash equilibria of the game.
(d) Is this game a Prisoner’s Dilemma (i.e. does each player have a dominant
strategy, and is there is an outcome that would be better for both players than
the Nash equilibrium)?
2. Salubric, a pharmaceutical company, currently sells a profitable drug that treats a
rare condition. Both Salubric and a competitor, Bewell, have been developing an
improved drug. They will decide independently whether or not to put this drug on
the market. The market is, however, not large enough to sustain two players: each
will make a loss if both enter. Payoffs in each scenario (measured in millions) are as
follows:
Bewell
Enter
Stay Out
Salubric
Enter
-5, -5
0, 30
(a) Does one of the firms have a dominant strategy?
(b) Is the game a Prisoner’s Dilemma?
(c) Find any Nash equilibria of the game.
Stay Out
20, 5
0, 10
3. Consider a sequential-move version of the game in Problem 1, which can be
represented by the following game tree.
No Sale
No Sale
Store 2
Store 1
(50, 50)
Sale
(10, 100)
No Sale
(100, 10)
Sale
Sale
(20, 20)
Derive the Nash equilibrium through backward induction.
4. Consider a sequential-move version of the game in Problem 2, which can be
represented by the following game tree.
Enter
Enter
Salubric
Bewell
Stay Out
(-5, -5)
Stay Out
(20, 5)
Enter
(0, 30)
Stay Out
(0, 10)
Derive the Nash equilibrium through backward induction.
Practice Problem Set 10
1. Two players in a game each choose simultaneously whether to “be nice” or “be
tough.” Payoffs in each scenario are as follows.
Player 1
Nice
Tough
Player 2
Nice
2, 2
3, 0
Tough
0, 3
1, 1
(a) Does Player 1 have a dominant strategy?
(b) Does Player 2 have a dominant strategy?
(c) Find the Nash equilibria of the game.
(d) Is this game a Prisoner’s Dilemma (i.e. does each player have a dominant
strategy, and is there is an outcome that would be better for both players than
the Nash equilibrium)?
Solution:
(a) Tough is a dominant strategy for Player 1, since it yields a higher payoff than
Nice against both of Player 2’s strategies: 3 vs. 2 against Nice, and 1 vs. 0 against
Tough.
Player 1
Player 1
Nice
Tough
Nice
Tough
Player 2
Nice
2, 2
3, 0
Tough
0, 3
1, 1
Nice
2, 2
3, 0
Player 2
Tough
0, 3
1, 1
(b) Similarly, Tough is also a dominant strategy for Player 2, since it yields a higher
payoff than Nice against both of Player 1’s strategies: 3 vs. 2 against Nice, and 1
vs. 0 against Tough.
Player 1
Player 1
Nice
Tough
Player 2
Nice
2, 2
3, 0
Tough
0, 3
1, 1
Nice
Tough
Player 2
Nice
2, 2
3, 0
Tough
0, 3
1, 1
(c) A Nash equilibrium is a strategy for each player such that everyone is
maximizing their payoff, given what the other player does. The only Nash
equilibrium here is (Tough, Tough), which leads to the outcome (1, 1). In fact,
dominant strategies always form a Nash equilibrium, since a dominant strategy
is a best response against any of the other player’s strategies. When both players
have a dominant strategy, it is for each the best response against the other
player’s strategy.
Player 1
Nice
Tough
Player 2
Nice
2, 2
3, 0
Tough
0, 3
1, 1
(d) This game is a Prisoner’s Dilemma, since the Nash equilibrium in dominant
strategies is worse for both than (Nice, Nice). (Nice, Nice) yields (2, 2): more for
each player than the equilibrium (Tough, Tough), which yields (1, 1).
Alternatively, we could use the conditions for dominant strategies and Nash
equilibria in a 2-by-2 game:
1
1
𝑟𝑟1 = 𝜋𝜋𝑈𝑈𝑈𝑈
− 𝜋𝜋𝐷𝐷𝐷𝐷
1
1
𝑟𝑟2 = 𝜋𝜋𝑈𝑈𝑈𝑈 − 𝜋𝜋𝐷𝐷𝐷𝐷
2
2
𝑐𝑐1 = 𝜋𝜋𝑈𝑈𝑈𝑈
− 𝜋𝜋𝑈𝑈𝑈𝑈
2
2
𝑐𝑐2 = 𝜋𝜋𝐷𝐷𝐷𝐷 − 𝜋𝜋𝐷𝐷𝐷𝐷
Player 1
Up
𝑟𝑟1 > 0, 𝑟𝑟2 > 0

Down

𝑟𝑟1 < 0, 𝑟𝑟2 < 0
Player 2
Left
𝑐𝑐1 > 0, 𝑐𝑐2 > 0

𝑟𝑟1 > 0,

𝑐𝑐1 > 0

𝑟𝑟1 < 0,
𝑐𝑐2 > 0

Right

𝑐𝑐1 < 0, 𝑐𝑐2 < 0
𝑟𝑟2 > 0,

𝑐𝑐1 < 0
𝑟𝑟2 < 0,
𝑐𝑐2 < 0
Since 𝑟𝑟1 = 2 − 3 < 0, 𝑟𝑟2 = 0 − 1 < 0, 𝑐𝑐1 = 2 − 3 < 0, and 𝑐𝑐2 = 0 − 1 < 0, Player 1’s
dominant strategy is “Down,” i.e. Tough, and Player 2’s dominant strategy is “Right,”
i.e. Tough. The Nash equilibrium is the bottom right cell, i.e. (Tough, Tough).
2. Consider a sequential-move version of the game in Problem 1, which can be
represented by the following game tree.
Nice
Nice
Player 2
Player 1
Tough
(2, 2)
Tough
(0, 3)
Nice
(3, 0)
Tough
(1, 1)
Derive the Nash equilibrium through backward induction.
Solution: The backward induction solution is as follows. Player 2, who moves
second, will respond to Nice with Tough (for a payoff of 3 vs. 2) and to Tough with
Tough as well (for a payoff of 1 vs. 0).
Nice
Nice
Player 2
Player 1
Tough
(2, 2)
Tough
(0, 3)
Nice
(3, 0)
Tough
(1, 1)
Anticipating this, Player 1 faces a choice between a payoff of 0 when playing Nice
and a payoff of 1 when playing Tough, so Player 1 prefers Tough.
Nice
Nice
Player 2
Player 1
Tough
(2, 2)
Tough
(0, 3)
Nice
(3, 0)
Tough
(1, 1)
Since both play Tough, the outcome is (1, 1).
Nice
Nice
Player 2
Player 1
Tough
(2, 2)
Tough
(0, 3)
Nice
(3, 0)
Tough
(1, 1)
Note that a sequential (tree) game does not always have the same outcome as the
corresponding simultaneous-move (table) game. The difference is that, in the
sequential game, there is an order of moves and, as a result, Player 2 can make her
move dependent on what Player 1 has done. This is not possible in the
simultaneous-move game, where Player 2 must select her move before observing
Player 1’s move.
Practice Quiz Questions 10
1. The annual collective bargaining meetings with the union are coming up.
Management isn’t looking forward to them. Fearing there will be a strike, your boss
sighs and says: “Why can’t we just all be reasonable and agree?” You point out that
this situation can be understood through game theory. You draw the following table
on a sheet of paper.
Player 1
Nice
Tough
Player 2
Nice
2, 2
3, 0
Tough
0, 3
1, 1
“Say Player 1 is management, and Player 2 is the union,” you say. “Each of them can
be a nice negotiator, who wants to agree, or a tough negotiator, whose priority is to
get the best outcome for their side. “Tough” means demanding small wage increases
from management’s point of view, and large wage increases from where the union
sits.” Your boss is intrigued. “So what do you think is going to happen?” Based on the
hypothetical payoffs in your table, you respond:
o Both players have a dominant strategy, which is to play tough. In Nash
equilibrium, these will be difficult negotiations. CORRECT
o Both players have a dominant strategy, which is to play nice. In Nash
equilibrium, they will quickly agree to a fair solution.
o In a Nash equilibrium, one of the players will play nice and the other will play
tough. Negotiations should be over soon, and it won’t come to a strike.
o None of the players have a dominant strategy. That makes it difficult to predict,
since there is no Nash equilibrium.
2. You are discussing the upcoming collective bargaining meetings between
management and the union with your boss. The CEO just announced in an e-mail
that the company definitely cannot afford a general salary increase. So Management
is taking a tough stance in the negotiations. Your colleague wonders whether the
union will give in. You draw the following game tree on the lunch room whiteboard:
Nice
Nice
Player 2
Player 1
Tough
(2, 2)
Tough
(0, 3)
Nice
(3, 0)
Tough
(1, 1)
“Each side can take a friendly or tough approach. If they adopt different strategies,
the tough side will win. If they adopt the same strategies, they will probably
compromise in the end, but if they are both playing tough, it’s going to be costly to
get there, with a possible strike on the cards.” Next, you explain your prediction
based on the backward-induction Nash equilibrium:
o Management is making a mistake by playing tough, because the union’s best
likely strategy is to play tough against a tough management and nice against a
nice management. The union will prevail.
o Management is playing tough, because they expect the union to be intimidated
and play nice. This will be a quick win for management.
o Management is playing tough, because they anticipate that the union will play
tough regardless of what Management does. As a result, we might have a strike.
CORRECT
o Management is just bluffing and will change its strategy to playing nice, which
will convince the union to respond by playing nice, too. They will soon agree
amicably on a moderate wage increase.