2005・20

Industry profits and free entry in input markets
Noriaki Matsushima

Industry proﬁts and free entry in input markets∗

Noriaki Matsushima†
Graduate School of Business Administration, Kobe UniversityAbstract
When upstream ﬁrms compete in quantity and freely enter the input market, competition among downstream ﬁrms reduces the input price (the marginal cost of downstream ﬁrms). The industry proﬁts of downstream ﬁrms competing in quantity canincrease with the number of downstream ﬁrms.

JEL Classiﬁcation Codes: D43, L11, L13
Key Words: Cournot competition, input markets, free entry, proﬁts∗

I thank Ikuo Ishibashi, Toshihiro Matsumura, Naoko Nishimura, and Hikaru Ogawa for helpful com-ments and suggestions. I also thank the seminar participants at Institute of Social Science in Universityof Tokyo. The author gratefully acknowledges ﬁnancial supports from Grant-in-Aid for Encouragement ofYoung Scientists from the Japanese Ministry of Education, Science and Culture. Needless to say, I am
responsible for any remaining errors.
†Correspondence author: Noriaki Matsushima, Graduate School of Business Administration, KobeUniversity, 2-1, Rokkodai, Nada-Ku, Kobe-City, Hyogo, 657-8501, Japan. Phone: (81)-78-803-6981. Fax:(81)-78-803-6977. E-mail: nmatsush@kobe-u.ac.jp1

1Introduction

In the standard Cournot model of oligopoly, as the number of ﬁrms competing in themarket increases, industry proﬁts (the sum of the ﬁrms’ proﬁts) decrease through increased
product market competition. The nature of the relationship between the number of ﬁrms
and industry proﬁts inﬂuences the incentives of ﬁrms. For instance, when ﬁrms collude in a
product market, they control their own quantities supplied like a monopolist, because theirmonopolistic behavior maximizes the industry proﬁts.
In this paper, we show that, under free entry in input markets, the relationship between
industry proﬁts and the number of downstream ﬁrms depends on ﬁxed costs (the easinessconcerning entries) in the input markets. When the ﬁxed costs are large, industry proﬁts
can increase with the number of downstream ﬁrms. We also show that, as the number ofdownstream ﬁrms increases, the input price decreases.
Naylor (2002) also shows a situation in which industry proﬁts increase as the numberof downstream ﬁrms increases. However, the structure of his model is quite diﬀerent fromthat of ours. Naylor (2002) considers a quantity-setting model in which input prices are
not exogenous but are determined by bargaining in bilateral oligopoly. A pair made up ofa labor union (an upstream ﬁrm) and a downstream ﬁrm bargains the wages of the labors(the wholesale price). The bargaining structure is similar to that in Horn and Wolinsky(1988), who discuss Nash bargaining.
On the contrary, in our model, upstream ﬁrms compete in quantity and freely enter
into input markets. Each upstream ﬁrm faces the derived demand of downstream ﬁrms
for input. The wholesale price is determined by the quantities supplied by upstream ﬁrms.
Negotiations between upstream and downstream ﬁrms do not exist.
The driving force of our result is as follows. The number of downstream ﬁrms aﬀects thederived demand for input. The increase in the number of downstream ﬁrms enhances thederived demand and attracts potential entrants to the input market. The additional entries
reduce the input price and are beneﬁcial to the downstream ﬁrms. Under some conditions,2the reduction of the input price has a signiﬁcant eﬀect for industry proﬁts.
The property of our result is related to that in Lahiri and Ono (1995). They introduce
Cournot oligopoly to the Heckscher-Ohlin model and show that free trade reduces theoligopoly price and increases welfare under free entry. Free trade enlarges the market sizefor the oligopolists and induces additional entries. The entries reduce the oligopoly price.
The mechanism of their result is similar to that of ours, but they do not consider industry
proﬁts and a vertical relationship between upstream and downstream ﬁrms.
The rest of the paper is organized as follows. Section 2 contains the basic model. Section
3 has the main results.

2The model

The setting of the model is somewhat similar to those in Salinger (1988) and Lin (2004).
There are m upstream ﬁrms and n downstream ﬁrms. As discussed later, m is endogenously
determined, and n is exogenously given. All downstream ﬁrms buy an input from theupstream ﬁrms and then transform it into the ﬁnal product. One unit of the ﬁnal productrequires exactly one unit of input. The unit cost of producing the input is c. For simplicity,c and the cost of transforming the input into the ﬁnal product are normalized to zero. The
demand for the ﬁnal product is given by p = a−bQ, where Q is the quantity supplied by thedownstream ﬁrms. Upstream ﬁrms freely enter the input market. When an upstream ﬁrmenters the market, it incurs a ﬁxed cost, F . Free entry in input markets is not consideredby Salinger (1988) and Lin (2004).
The input price is determined by Cournot competition at both levels of the industry:
The downstream ﬁrms choose their output levels given the input price, leading to the derived
demand for input; the upstream ﬁrms then compete in a Cournot fashion with respect tothe derived demand.

3

3

Results

Given the input price w, downstream ﬁrm i (i = 1, 2, . . . , n) maximizes the following function:
πi = (a − b(Q−i + qi ) − w)qi ,where Q−i is the sum of the quantity supplied by the other ﬁrms. The ﬁrst-order conditions
lead toa−w
a + nw1
qi =, p=, πi =b(n + 1)
n+1b(a−wn+1)2, Q=n(a − w).
b(n + 1)

Using the last equation, we have the derived demand for input:
w =a−b(n + 1)
Qd .
n(1)

Upstream ﬁrm j (j = 1, 2, . . . , m) maximizes the following function:(πj = a −)b(n + 1)(Q−j + qj ) qj .
nThe ﬁrst-order conditions lead toqj =naana2mna, w=, πj =, Qd =. (2)2(m + 1)(n + 1)bm+1(m + 1) (n + 1)b(m + 1)(n + 1)b

Upstream ﬁrms freely enter the input market. Their actions lead the proﬁt of each upstreamﬁrm to zero. Therefore, πj − F = 0, that is,√na2
πj == F, → m + 1 = a(m + 1)2 (n + 1)bn, w=b(n + 1)F√b(n + 1)F.
n(3)To assure that the number of upstream ﬁrms is larger than 1, we assume that F < a2 /(8b).
From (3), we have the following lemma:
Proposition 1 As the number of downstream ﬁrms increases, the input price w decreases.
Under free entry in the input market, the gross proﬁt of each upstream ﬁrm must be equal
to F . As the number of downstream ﬁrms increases, the slope of the demand function for4the input becomes gentler (see w in (1)). The average total cost curve and the slope of thedemand function for the input must come in contact with each other (see ﬁg 1). We ﬁndthat the gentler the slope of the demand function for the input is, the lower the input price
is. Therefore, Proposition 1 holds.
In this paper, we assume a linear demand function. If we assume a general inverse
demand function P (Q) which satisﬁes P ′′ (Q)Q + P ′ (Q) < 0 and P ′ (Q) < 0, we can derive
Proposition 1 (see Appendix).
Note that, in the analysis, we ignore the integer problem in the number of upstreamﬁrms. When we take the problem into account, the number of entrants may not change with
the number of downstream ﬁrms. Even though we take the integer problem into account,the qualitative property of our result holds, if the rise in the number of downstream ﬁrms
increases the number of upstream ﬁrms. (Obviously, the rise in the number of downstreamﬁrms never decreases the number of upstream ﬁrms.)
Substituting the last equation in (3) into the proﬁt of each downstream ﬁrm, we have:1
πi =b(a−wn+1)21=b(a−n+1√bFn(n + 1))2.

We now compare the industry proﬁts in monopoly and duopoly cases. The proﬁts are:ΠM
ΠD1 a≡ π1 =−b 22∑√

2bF ,22 a≡−πi =b 3i=1√

2bF .
6

From the equations, we have
Proposition 2 If the following inequality holds, the industry proﬁts in a duopoly case arelarger than those in a monopoly case:√√√(85 − 60 2 − 48 3 + 34 6)a2
F >≅ 0.0486a2 /b.
6bThe result provides a market structure in which industry proﬁts increase with the number ofdownstream ﬁrms. In this paper, as mentioned in Proposition 1, the input price decreases
5with the number of downstream ﬁrms. This feature crucially depends on the free entry
condition in the input market. When the number of upstream ﬁrms is exogenously given,
the increase in the number of downstream ﬁrms does not aﬀect the input price but aﬀects
the quantity supplied by each upstream ﬁrm (see w and qj in (2)). Under free entry, theincrease in the derived demand for input leads to a decrease in input price. When the ﬁxed
cost is large, the reduction of the input price is eﬀective because the number of upstreamﬁrms is not large, and the input price is then set at a high level (see w in (3)). Therefore,
Proposition 2 holds.

[2005.7.7 710]

6

w

w
w′AT C
q0Fig. 1: The input price(s) under free entry.
(The arrow indicates the increase in the number of downstream ﬁrms)

7Appendix
We now suppose that the demand for the ﬁnal product is given by P (Q), where Q is thequantity supplied by the downstream ﬁrms. Given the input price w, downstream ﬁrm i(i = 1, 2, . . . , n) maximizes the following function:
πi = (P (Q−i + qi ) − w)qi ,where Q−i is the sum of the quantity supplied by the other ﬁrms. The ﬁrst-order conditions
lead toP (Q) − w + P ′ (Q)qi = 0, ⇒ nP (Q) − nw + P ′ (Q)Q = 0.
Using the last equation, we have the derived demand for input:w(Q, n) = P (Q) +P ′ (Q)Q.
n(4)Upstream ﬁrm j (j = 1, 2, . . . , m) maximizes the following function:
πj = w(Q−j + qj , n)qj .
The ﬁrst-order conditions lead to∂w(Q, n)
qj + w(Q, n) = 0, ⇒∂Q∂w(Q, n)Q + mw(Q, n) = 0.
∂Q

(5)

Upstream ﬁrms freely enter the input market. Their actions lead the proﬁt of each upstreamﬁrm to zero. Therefore, πj − F = 0, that is,w(Q, n)qi = F, → w(Q, n)Q = mF.

(6)

Diﬀerentiating w(Q, n) in (4) with respect to Q, we have:∂w(Q, n)P ′′ (Q)Q + P ′ (Q)= P ′ (Q) +.
∂Qn(7)

Substituting m in (6) and (7) into (5), we have:2[w(Q, n)]()∂w(Q, n)P ′′ (Q)Q + P ′ (Q)= −F = − P ′ (Q) +F,∂Qn(8)

If P ′′ (Q)Q + P ′ (Q) < 0, then [w(Q, n)]2 is decreasing with n, that is, the input price isdecreasing with the number of downstream ﬁrms.
8

References
Horn, H. and Wolinsky, A. “Bilateral monopolies and incentives for merger.” Rand Journalof Economics, Vol. 19 (1988), pp. 408–419.
Lin, P. “Strategic spin-oﬀs of input divisions.” mimeo, 2004, forthcoming European Economic Review.
Lahiri, S. and Ono, Y. “The role of free entry in an oligopolistic Heckscher-Ohlin model.”International Economic Review, Vol. 36 (1995), pp. 609–626.
Naylor, R. A. “Industry proﬁts and competition under bilateral oligopoly.” Economics
Letters, Vol. 77 (2002), pp. 169–175.
Salinger, M. A. “Vertical mergers and market foreclosure.” Quarterly Journal of Economics, Vol. 103 (1988), pp. 345–356.

9