Economics work

Assignment 3: Economics of Business Decisions
Directions: Answer every question below – each question is worth 20 points. Submit your
file (pdf) with answers to the Canvas website using the Assignment Tab. This assignment is
due Sunday, January 10 by 7pm. You should 1) submit a single PDF file before the submission
deadline, 2) include clear, neatly organized, and well-explained answers, and 3) include your
name and page number on each page.
You may or may not need to round any numerical answers, but our general rule on this will
be to two decimal points. Also, in reporting answers, you do not need to report units of
measurement.
This is an open-book/notes assignment, but remember you are on your honor to work on
your own. This is effectively an exam, so only questions of clarification will be answered.
(Notes: Answers need not be long! We do give partial credit so show your work.)
Questions:
1. Consider a duopoly in which two firms produce and sell a homogenous product. Demand
is characterized by the function: P=48,000-2Q, where P is the market-clearing price, and
Q is the market quantity (i.e., Q=q1+q2). Each firm incurs total cost c(q)=8q2 – that is,
c1(q1)=8q12 and c2(q2)=8q22.
Part (i): Suppose that the two firms form a cartel. Identify the quantity of output that each
firm should produce and sell (q1 and q2) under the cartel agreement.
Part (ii): Suppose that the two firms produce and sell the quantities specified by the cartel
agreement, which you identified in part (i). What is the resulting market price, and how
much profit does each firm earn?
Part (iii): Suppose the Firm 2 produces the amount of output specified by the cartel
agreement, which you identified in part (i). What is the profit-maximizing quantity that
Firm 1 would produce and sell if it chose to cheat on the cartel agreement?
Part (iv): Suppose the Firm 2 produces the amount of output specified by the cartel
agreement, which you identified in part (i), and Firm 1 cheats on the agreement and
produces the quantity that you identified in part (iii). Confirm that Firm 1’s profits
increase by cheating on the cartel agreement.
Part (v): Suppose the Firm 2 produces the amount of output specified by the cartel
agreement, which you identified in part (i), and Firm 1 cheats on the agreement and
produces the quantity that you identified in part (iii). Does the market price fall or rise
when Firm 1 cheats on the cartel agreement?
2. Consider a duopoly in which two firms produce and sell a homogenous product. The
firms form a cartel agreement with the sole objective of maximizing their combined
profits. They identify that the cartel optimum involves jointly producing 60 units of
output, but they are unsure how many of those 60 units each firm should produce. You
will help them solve this problem.
The table below provides the marginal cost for several different quantities that could be
produced by each firm. For example, the marginal cost of the 27th unit of output produced
by Firm 1 is $49,000, and the marginal cost of the 33rd unit of output produced by Firm 2
is $51,000.
Unit of Output MC1 MC2
27th $49,000 $42,000
28th $49,500 $43,500
29th $50,000 $45,000
30th $50,500 $46,500
31st $51,000 $48,000
32nd $51,500 $49,500
33rd $52,000 $51,000
Part (i): Identify and carefully explain why the combined profits of the cartel cannot be
maximized when each firm produces 30 units of output.
Part (ii): Identify the cartel optimum – meaning, the individual firms’ quantities of output
that maximize the cartel’s combined profits. Explain why this must be the combinedprofits-
maximizing way of allocating production of the 60 units.
3. Consider a duopoly in which two firms choose to advertise or not advertise. This
interaction and the resulting profits are depicted by the game below.
Part (i): Identify the Nash Equilibrium of this game.
Part (ii): Suppose that the firms form a cartel agreement with the sole objective of
maximizing their combined profits. The firms are not colluding on the typical choice
variables of price or quantity, but on their advertising strategies. Identify the cartel
agreement strategies.
Part (iii): Suppose that the firms adopt the cartel agreement that you identified in part
(ii), and that each adopts the Grim Trigger Strategy. Identify the range of 𝛿 over which
the firms would uphold that cartel agreement. Explain this result.
Part (iv): Suppose that the firms adopt the cartel agreement that you identified in part
(ii), and that each adopts the Tit-for-Tat Trigger Strategy. Identify the range of 𝛿 over
which the firms would uphold that cartel agreement. Explain this result.
Firm 1
Firm 2
Advertise Don’t Advertise
Don’t
Advertise Advertise
€50 billion €0 billion
€100 billion €65 billion
€50 billion €100 billion
€0 billion €65 billion
4. Consider a duopoly in which two firms produce and sell a homogenous product. The
firms compete by choosing their respective quantities of output, but they are unable to
establish a cartel (or, at the very least, choose not to establish one). The firms compete à
la Cournot. Demand is characterized by the function: P=12,600-3Q, where P is the
market-clearing price, and Q is the market quantity (i.e., Q=q1+q2). Each firm incurs total
cost c(q)=6q2 – that is, c1(q1)=6q12 and c2(q2)=6q22.
Part (i): Identify the quantity of output that each firm should produce and sell (q1 and q2)
and the resulting market-clearing price (P).
Part (ii): The diagram below depicts the duopolists’ best reply functions. Each best reply
function identifies that firm’s profit-maximizing quantity of output when the other firm
produces a specific quantity of output. What is Firm 1’s best reply when Firm 2 produces
2,100 units of output?
Part (iii): Refer to the figure above. Explain why there is not a Nash Equilibrium in
which Firm 2 produces 2,100 units of output.
Part (iv): Refer to the figure above. What is Firm 1’s profit-maximizing quantity of output
when Firm 2 produces 0 units of output? How does this relate to the single-price (i.e.,
assuming no price discrimination) monopoly solution in which a monopolist with total
cost c(Q)=6Q2 serves this market?
Quantity Produced by Firm 2 (q!)
(in hundreds)
Quantity Produced by Firm 1 (q”)
(in hundreds)
7 14 21 28 35 42
7
14
21
28
35
42
49
49
Firm 2’s B.R.
Firm 1’s B.R.
5. Consider a duopoly in which two firms produce and sell a differentiated product. The
firms compete by choosing their respective prices of output, but they are unable to
establish a cartel (or, at the very least, choose not to establish one). The firms compete à
la Bertrand.
Demands for the two firms’ products are characterized by the functions:
q1=4,800-12p1+bp2 —AND— q2=4,800-12p2+bp1,
where b is some number between 0 and 12 that measures the substitutability between
the two products. When b is greater, the two products are more substitutable. For
simplicity, assume that the firms incur no costs of production.
Part (i): Suppose that b=8. Identify the price that each firm should charge (p1 and p2) and
the resulting quantities that each firm will produce and sell. Show your work; do not rely
on the answer provided in the next part of the question.
Part (ii): In this setting, each firm’s equilibrium price is 𝑝 = !,#$$
%!&’ . Does this price increase
or decrease when the two differentiated products are more substitutable? Does this
result make intuitive sense? Explain.
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