STUDY QUESTIONS (WITH SUGGESTED ANSWERS) FOR FREE ENTRY COURNOT EQUILIBRIUM
CHAPTER 14 AND ENTRY AND STRATEGIC BEHAVIOR CHAPTER 15.
Entry costs and market structure (Chapter 14)
Differentiate this with respect to qi, set equal to zero to obtain the first order condition:
a − nq/S − q/S − c = 0
a c
Use the last line above to write (n) = 0 as S 
  F , and solve for n. This is the
 n 1 
number of firms, n*, in the free-entry equilibrium; n* is the largest integer at which profits
2
 a c 
S  *   F are not negative.
 n 1
n* =  a  c 
S
1 .
F
14.11 (From Cabral textbook) ---drop part (e)
Suggested Answer to 14.11
The following information is missing from the question: The cost of advertising is given by a i.
Example if Firm 1 makes “10 units of advertising” its advertising cost is equal to 10.
Then firm i’s profits (as a function of advertising levels) is
i = (ai/A)S − ai.
Remember : A = a1 + a2 + … … ai-1 + ai + ai+1 + … … + aN-1 + aN.
Firm i chooses ai to maximize i. We differentiate i with respect to ai, then set equal to 0:
This equation defines the optimal level of advertising for firm i. Of course, the optimal ai
depends on advertising levels of the other firms.
A  ai
S 1  0
A2
Remember the rule of differentiation d gf (( xx )) /dx = (fg − fg)/g2.
 
Remember that dA/dai = 1.
Let the advertising levels in equilibrium be denoted by (a*1, a*2, ... a*i, ... ... a*N).
In a symmetric equilibrium we have a*1 = a*2 = ... =...= a*N.
Then A = na* in equilibrium so, the above equation becomes
na *  a *
S 1  0
* 2
na
 
We can solve this equation for a*:
n 1
a*  2
n
Note that in equilibrium since all firms set ai = a*, all firms have equal market share: in
equilibrium we have si = 1/n. Then we can write the profits of firm i as
1
n 1
 = a *  S  2 S , which simplifies to S/n2.
n
n
Entry occurs as long as S/n2 ≥ F. Therefore, the number of firms in the free entry equilibrium
is the largest integer (= tam sayı) n for which we have S/n2 ≥ F.
S
n* =
.
F
One more question on the free entry Cournot equilibrium
The market demand is given by Q(P) = 200 – P, all firms have the cost function C(q) = q2.
There are 2 stages. In stage 1 firms decide whether to enter. If a firm decides to enter, it
has to pay an entry cost of F = 50. In stage 2 all firms that have decided to enter in stage 1,
compete in the style of Cournot.
How many firms will there be in the market in the Cournot equilibrium with free-entry?
To answer this question you need to compute the quantity produced by the individual firm,
q*, price, P*, and profits per firm, *, in the Cournot equilibrium. Note that all firms are
identical, so the Cournot equilibrium quantities will be the same.
Profit for firm 1 : (200 - Q)q1 – (q1)2. Differentiate this wrt q1 set equal to 0 : -q1 + (200 - Q)
-2q1 = 0 use the symmetry property of the Cournot-Nash equilibrium -q* + 200 – nq* - 2q* =
0  q* = 200/(n+3). Compute the Cournot-Nash equilibrium price and profit per firm.
Once you have the profit per firm as a function of n, find the maximum value of n for which
profits are greater than or equal to F = 50. This is the free entry equilibrium number of
firms.
Social Welfare and Free Entry Study Question (Chapter 14)
Consider and industry with inverse demand P = 21 − Q. There are N firms in the industry, all
firms have the cost function TC(q) = 6 + q. Firms compete in the style of Cournot.
a. The Cournot equilibrium for N = 4 is computed in the table below. Profit is computed as
(P−c)q, it is gross of F.
Complete the table.
N
1
2
3
4
5
6
7
8
q
10,0
6,7
5,0
4,0
3,3
2,9
2,5
2,2
Q
10,0
13,3
15,0
16,0
16,7
17,1
17,5
17,8
P
11,0
7,7
6,0
5,0
4,3
3,9
3,5
3,2
50,0
100,0
100
150,0
88,9
44,4
88,9
177,8
112,5
25,0
75,0
187,5
128,0
16,0
64,0
192,0
138,9
11,1
55,6
194,4
146,9
8,2
49,0
195,9
153,1
6,3
43,8
196,9
158,0
4,9
39,5
197,5
Consumer Surplus
Profit per firm
total profits
total welfare
b. Argue that there will be 7 firms in the market in the free-entry Cournot equilibrium.
When making their entry decisions, firms consider how much profit they can make if they
enter. For example. If there are 3 firms in the industry already the new entrant looks at how
much profits each firm will make in the Cournot-Nash equilibrium with 4 firms. The table
show that per firm profit is bigger than F = 6 for N = 6 but less than that if N = 8, so there will
be 7 firms.
c. Argue that the socially optimal number of firms is N = 3. Much smaller than 7!! So there
is too much entry in the market equilibrium! Explain why this happens. In your answer also
describe the business stealing effect.
The additional entry means more competition and more competition lowers the equilibrium
price and increases the equilibrium quantity. This increases the sum of firm profits and
consumer surplus. Hence, for social optimality, the new entry is good if the increase in total
welfare (sum of firm profits and consumer surplus) that the new entry brings is larger than F
= 6. For example if N increases from 1 to 2, (from monopoly to duopoly) total welfare
increases by 27.8 which is larger than F = 6. Same is true as N increases from 2 to 3. But as
N increases from 3 to 4, total welfare inceases by 4.5 which is smaller than F = 6, hence the
entry by the 4th firms is inefficient from the social point of view.
Chapter 15.1
Entry Deterrence
Consider a market currently monopolized by an incumbent (I). A potential entrant (E) is
considering entry into the market. The (inverse) demand function is p = 100 − Q. The
marginal cost of both firms is 20. (The cost function is c(q) = 20q for both firms.)
The timeline:
The incumbent chooses its quantity qI first.
After observing the incumbent’s quantity choice, the entrant decides whether to enter.
To enter, the entrant has to incur a fixed cost, F = 225, which is sunk.
If the entrant enters, then he will choose its quantity qE.
If the entrant does not enter, the incumbent remains a monopolist, but it has to produce qI
set earlier.
We solve for the equilibrium by starting at the end: suppose q I is already set and entry has
already occurred, what is the best qE?
Given qI, suppose E enters. E Choose qE to maximize E = (100−qE−qI−20)qE.
Differentiate E with respect to qE, set equal to 0:
FOC: qE* = (80−qI)/2
E’s net profit is: E = (80−qI)2/4 − F (NOTE that entrant’s profit is a function of incumbent’s
capacity/quantity choice, qI. Higher qI lowers E’s profits. This is very important.)
E will NOT enter if:
(80−qI)2/4 − F ≤ 0, equivalently if: qI ≥ 80−2F1/2 = 50, otherwise E will enter.
We found that to deter entry, I must produce at least 50. Remember that I’s monopoly
quantity is 40, and his monopoly profit is 1600. So, to deter entry I must produce more than
what it would produce as a monopoly. If I chooses the monopoly quantity, E will enter.
The optimal quantity for I to deter entry is 50. Its corresponding profit is: 1500.
Now, the other option for I is to accommodate entry. If I allows for entry then the
competition becomes a Leader-Follower set up..
I chooses qI to maximize (100−qE*−qI−20)qI = (40 − qI/2)qI. Differentiate with respect to qI,
set equal to 0: qI*= 40 and qE* = 20. I’s profit is 800.
Therefore, it’s better for I to choose 50, and deter E’s entry.
Extensions:
Very large entry cost will lead to what is referred to as the “blockaded entry” case.
Now suppose that the entry cost increases to 625.
Then to deter entry qI ≥ 80 − 2F1/2 = 30.
Even if I chooses the monopoly quantity 40, the entrant has no incentive to enter.
We refer to this case as the blockaded entry. The threat of entry is irrelevant.
With lower entry cost, entry deterrence is not optimal for I.
Now suppose that the entry cost decreases to 25.
Then to deter entry qI ≥ 80−2F1/2 = 70.
If I want to deter entry, the best choice is qI = 70
I’s profit under deterred entry is 700, which is lower than its profit if entry is accommodated.
I will choose qI = 40 and entry is accommodated. We call this case entry accommodation.
SUMMARY
This is the intuition: as the entry cost increases, to deter entry becomes less costly for the
incumbent, so it becomes more likely that entry will be deterred. As the entry cost becomes
very big, the entrant will not enter, and entry is blocked.
Note that to deter entry, the incumbent’s quantity choice must be irreversible.
Otherwise, Incumbent can adjust its quantity after the Entrant chooses its quantity, since E’s
quantity choice is not a best response to E’s ex post, it would be like Cournot model, and
entry deterrence is impossible.
So it’s better to think that I is choosing capacity, instead of quantity.
Only irreversible choices are credible.
We say that irreversible choices have commitment value.
Incumbent gains by restricting its own flexibility.
Chapter 15 Strategic behavior entry and exit
15.6
(From Cabral textbook Entry Deterrence)
P = 100 − 2Q, both the incumbent and the entrant have variable costs C = 10q. Entry cost is
F. Incumbent chooses its output (this choice is irreversible) the entrant observes this and
then decides to enter. If he enters he pays F and chooses its output to maximize profits.
a. Find the incumbent;s profits maximizing q in the absence of entry.
We solve for monopoly output
MR = 100 − 4Q, MC = 10
MR = MC  QM = 22.5, PM = 55.
b. Compute the limit output (that is the lowest output the incumbent needs to produce to
deter entry)
E
= [P − MC]qE
= [(100 − 2(qE + qI) − 10]qE
Differentiate and set equal to zero and solve for qE.
qE = (90−2qI)/4
This is the optimal quantity for the incumbent, given the previously set qI.
E
= [P − MC]qE
= [(100 − 2( 9042qI + qI) − 10] 9042qI
This simplifies to
E
= ½[45 − qI]2
E will enter if ½[45 − qI]2 ≥ F, this can be rewritten as qI ≥ 45 − 2F . For example F = 112.5,
then to deter entry, I must produce at least 30 units, which is greater than the monopoly
quantity. Doing so may or may not be profitable. In contrast, if F = 312.5, then to deter
entry, I must produce at least 20 units, since as a monopoly I will produce 22.5 units,
deterring entry occurs naturally (without effort).
With a little bit of algebra we can show that the Incumbent’s profits with the minimum entry
deterrence quantity (also called “the limit output”!) is
Limit output is 45 − 2F , the limit price is 100−2(45 − 2F ). Profits are [100−2(45 − 2F ) −
10](45 − 2F ), this simplifies to 90 2F −4F.
Lerner’s index with entry deterrence:
L = (P−MC)/P
P = 100 − 2(45 − 2F ), MC = 10
So, P − MC = 2 2F , and L 
2 2F
.
45  2F
Entry accommodation
Incumbent’s profits with entry:
E = [(100 − 2( 9042qI + qI) − 10]qI
This simplifies to
E = (45 − qI)qI
Differentiate and set equal to zero:
q*I = 22.5
then q*E = 11.25
The equilibrium price is 100 − 2(22.5 + 11.25) = 32.5
The incumbent’s profits with entry accommodation are (32.5 − 10)22.5 = 22.52 = 506.25
Entry deterrence is profitable if 90 2F −4F ≥ 506.25, equivalently, if F ≥ 21.7. [this can be
solved numerically or by trial]. If the sunk entry cost is less than 21.7 then entry
accommodation is more profitable than entry deterrence.