Some Notes on Limit Pricing
The theory of limit pricing suggests that an incumbent firm may be able to make
it unprofitable for a potential entrant to enter the industry. The argument is that
the incumbent firm can produce a certain output before entry, and threaten to
continue producing that output even if entry occurs. If the potential entrant
believes the claim, he will decide it is unprofitable to enter.
Later economists were critical of “limit pricing” theory, except when the
incumbent firm can impose restrictions on itself to make its threat credible.
The example below explores limit pricing theory. (You will have to look at the
textbook (page 361) for the graph of this model).
Assume market demand is P = 100 – Q. There is one incumbent firm in the
industry (a monopoly), and its output is designated by qi. There is a potential
entrant to this industry and its output is designated by the symbol qe.
Both firms have the same costs of production: TC = 400 + 10q, and therefore AC
= (400/q) + 10.
The incumbent firm knows that there is a potential entrant, and believes that the
potential entrant believes that the incumbent will not change its output even if
the potential entrant decides to enter. The incumbent firm therefore wants to
choose qi so that entry will be unprofitable. In fact, the incumbent knows that
the potential entrant will not enter unless it earns a positive profit (∏e > 0), so the
incumbent will choose qi to make the entrant’s profit equal to zero. This will
happen if the residual demand curve of the potential entrant just touches (is
tangent to) its AC curve but does not rise above it anywhere.
To find the tangency point, take dAC/dq = -400q-2 and set this equal to the slope
of the residual demand curve dP/dq = -1. Therefore, -400q-2 = -1 or q = 20.
When q = 20, AC = (400/20) + 10 = $30. This means that the residual demand
curve must pass through the point q =20, P = $30 and have a slope of –1. The
general equation for this residual demand curve will be P = X - qe (where X is the
vertical intercept), and at the point of tangency, this equation will satisfy 30 = X –
20. Therefore, X = 50, and the residual demand curve which just touches the AC
curve will have the equation, P = 50 – qe.
The market demand curve is P = 100 – Q or P = 100 - qi – qe. To leave the
appropriate residual demand curve, qi must = 50. This is the entry-deterring
output for the incumbent firm. Given the beliefs of the potential entrant, it will
calculate its best output this way: P = 50 – qe, therefore MRe = 50 – 2qe. Setting
this equal to MC, we have 50 – 2qe = 10, or qe = 20. Therefore Pe = 50 – 20 = $30.
At this price and quantity, profit for the entrant is:
∏e = (30 x 20) – [400 + (10 x 20)] = $0. Given this calculation, the potential
entrant would decide not to enter (i.e., the limit pricing scheme works). The
price of $30 is called the “limit price” because the incumbent firm, by threatening
to produce 50 units of output after entry occurs, is threatening to drive the price
down to $30 after entry (a price which will limit the possibility of entry).
What would the price and profits be before entry occurred?
Before entry, the incumbent has to produce 50 units of output. To make 50 units
of output clear the market, the price must be P = 100 – 50 = $50. Therefore, the
profit earned before entry would be ∏i = (50 x 50) – (400 + [10 x 50]) = $1,600.
Is this the monopoly price and profit for the incumbent firm?
No, it’s not. The incumbent firm did not choose the output and price to
maximize profits, but to deter entry by the potential entrant. The monopoly
output would be: P = 100 – q; MR = 100 – 2q = MC = 10, so qi = 45 and P = $55.
Monopoly profit would be ∏i = (55 x 45) – (400 + [10 x 45]) = $1,625.
Would the incumbent firm make a profit if it produced 50 units of output after
the potential entrant entered the industry?
The price would be $30 and output 50 units, so ∏i = (30 x 50) – (400 + [10 x 50]) =
$600. The answer is yes…a reduced profit compared to before entry, but the
incumbent firm would profit while the entrant earned zero.
What’s wrong with the model?
The logic of the model is perfect, as long as you accept the assumption that the
potential entrant will believe that the incumbent will keep its output constant
after entry. But this assumption seems pretty shaky. Why would a potential
entrant believe this? For instance, let’s imagine that the potential entrant entered
the market and decided to produce 45 units of output. Would the incumbent
keep producing 50 units of output? Total output in the industry would be 95
units so price would be P = 100 – 95 = $5. At this price, the new entrant would be
earning: ∏e = (5 x 45) – (400 + [10 x 45]) = -$625 (a loss), but the incumbent would
be doing even worse: ∏i = (5 x 50) – (400 + [10 x 50]) = -$650 (a bigger loss).
Under these conditions, it’s unlikely the incumbent would keep producing 50
units of output.
If the two firms settled down into a Cournot duopoly result, the incumbent
would be much better off. As you can calculate on your own, the Cournot result
would be that each firm produces 30 units of output and the price is $40.
Therefore, profit of each firm would be: ∏ = (40 x 30) – (400 + [10 x 30]) = $500.
How could the incumbent firm make a credible threat?
The incumbent firm would have to be able to make a credible threat to produce
50 units of output if entry occurred. It could do this by precommitting to
produce 50 units of output. This would require adoption of an inflexible
production technology that only permitted this level of production (and not any
other level of output). Restricting its own options means that the potential
entrant can expect a protracted period of low prices and negative profits (losses)
if entry occurs. This makes entry of the potential entrant very unlikely.
In an extensive form game, you might look at the options and the payoffs this
way:
Enter
($600, $0)
Inflexible
Technology
Potential
Entrant (#2)
Don’t Enter
Incumbent
Firm (#1)
Enter
Flexible
Technology
($1,600, $0)
($500, $500)
Potential
Entrant (#2)
Don’t Enter
($1,625, $0)
The bottom result is the monopoly result (flexible technology, no entry).
However, if entry does occur and technology is flexible, we conclude that the
entrant would be able to force the incumbent to change its output and that the
Cournot result would occur ($500, $500). The adoption of inflexible technology
requires investment in changes to plant and equipment, so that could change the
costs of the incumbent. In these calculations, we assume the firm was able to sell
its old equipment and get this new plant and equipment for the same price, so its
costs do not change. If the potential entrant enters, it can no longer get the
incumbent to change its output by threatening to produce 45 units of output, so
the best the entrant could do would be to produce 20 units and earn a profit of
zero (which we assume is not enough to encourage entry). If the potential
entrant doesn’t enter, the incumbent does not earn the monopoly profit, because
the inflexible technology does not allow him to produce the monopoly output, so
the profit is $1,600.
Our analysis of this game tells us that the potential entrant will not enter if the
incumbent adopts the inflexible technology, but will enter if there is a flexible
technology (because the threat to produce 50 units of output is not credible).
Therefore, the best strategy for the incumbent firm is to adopt the inflexible
technology to deter entry.