Analytical Exercise: Equilibrium in an Oligopoly Market with Capacity Constraints
Consider the economic environment we encountered in the online exercise on Monday, in
which a set of firms engage in Bertrand competition in four distinct markets. Each firm
produces a homogeneous good at a constant marginal cost � = 4, up to a capacity constraint �
which determines each firm’s maximum production. The firms compete in prices: each firm
posts a price in each market, where prices must be in whole numbers. Capacity constraints and
fixed costs of production vary by market according to the schedule below:
• Market 1: Capacity � = 1000 per firm, no fixed cost of production;
• Market 2: Capacity � = 1000 per firm, fixed cost � = 35,000 per firm;
• Market 3: Capacity � = 400 per firm, fixed cost � = 35,000 per firm;
• Market 4: Capacity � = 2000 per firm, fixed cost � = 35,000 per firm.
Demand is identical across markets and described by the schedule
on the right (also provided in more detail in the Excel file
attached). Consumers buy first from the firm with the lowest price
until its capacity is exhausted, then buy from the firm with the
second-lowest firm, and so on until no additional consumer wishes
to purchase at the lowest price charged among firms with
remaining free capacity. If two (or more) firms post the same
price, they split the remaining quantity demanded at that price,
after any firm charging a lower price has already sold the total
quantity it can produce.
You observe data on prices and quantities for a five-firm oligopoly
in these four markets, given below (generated from play in our inclass exercise on Monday). In this particular market, it appears
that four of the five firms are competing actively, while one firm
(Firm 3) is playing passively. Let us say that the passive Firm 3 is a
“Nonprofit,” while the remaining four firms are “For-profit.” Using
the models of oligopoly behavior we have discussed, your
assignment is to analyze the behavior of the four “For-profit” firms
in this market, taking the prices of the “Nonprofit” as given.
Assignment: In 1-2 pages, provide written answers to the following questions.
1. Analyze the Nash equilibrium of the pricing game between the four For-Profit players in
each of the four markets above, taking the price charged by the Nonprofit (Firm 3) in
each market as given. Based on this analysis, what theoretical predictions can you make
about the price in each market?
Market Demand
Schedule
Price Quantity
10 5120
20 4620
30 4190
40 3810
50 3470
60 3160
70 2880
80 2610
90 2360
100 2120
110 1890
120 1670
2. How do the observed prices charged by the four “For-profit” players compare with the
Nash equilibrium predictions in Part 1? Which Nash equilibrium predictions are borne
out in the data? Which predictions are not?
In your answer to Question 1, it may be helpful to consider the following:
• Should unavoidable fixed costs matter for Nash equilibrium prices? Why or why not?
(Note: By “unavoidable”, we mean that firms cannot leave the market this period.)
• If there were no capacity constraints, and ignoring the fact that price must be an
integer, what would be the Nash equilibrium price in each market?
• How do capacity constraints change our predictions about prices in each market? How
would we expect prices to be ranked with high, medium, and low capacity constraints?
• Consider any potential price � in the demand schedule below. Suppose that Firm 1’s
three for-profit rivals each charge this price � (holding the Nonprofit’s price fixed at its
observed value). Consider two strategies for Firm 1: “matching” the price charged by its
active rivals by setting �. = �, or “undercutting” all three active rivals by reducing price
by $1 to �. = � − 1. Bearing in mind Firm 1’s capacity constraints, what is the *highest*
price � on the demand schedule at which Firm 1 prefers to match than undercut? How
does this price vary with the capacity constraint �? And what does this tell you about
Nash equilibrium prices in each market?
[Hint: For a price � to be a symmetric Nash equilibrium, it must be that no for-profit
firm prefers to undercut when all for-profit rivals charge the given price. In this
particular game there may in general be many Nash equilibria, but from the firms’
perspective the best equilibrium will be the one with the highest price.]
Evaluation: This assignment will count for 10 percent of your final course grade. In your
answers, I will be looking for a clear discussion of both the key economic and strategic issues
governing competition in this market, and how predictions of the underlying theory fit (or
don’t!) the actual prices observed. Note that data *will not* line up with theory on every
dimension, although in some key dimensions it should. Long answers are not necessarily better;
the key is to identify and analyze the core questions involved.
Submissions are due via Canvas at midnight on Friday, April 10. Late submissions will receive a
one letter grade penalty for each day late.
Data on outcomes for each market is provided on the next page.
Market 1: Moderate capacity, no fixed costs
Firm Price Sales Capacity Revenues