ISSN 1791-3144
University of Macedonia
Department of Economics
Discussion Paper Series
Equilibrium downstream mark-up and upstream free
entry
Ioannis N. Pinopoulos
Discussion Paper No. 2/2014
Department of Economics, University of Macedonia, 156 Egnatia str, 540 06 Thessaloniki,
Greece, Fax: + 30 (0) 2310 891292
http://www.uom.gr/index.php?newlang=eng&tmima=3&categorymenu=2
Equilibrium downstream mark-up and upstream free entry
Ioannis N. Pinopoulos∗
Department of Economics, University of Macedonia, 156 Egnatia Street,
Thessaloniki, Greece
This version
August, 2014
Abstract
We consider a successive Cournot oligopoly model where firms freely enter into the
upstream market. We show that, under specific conditions, a higher number of downstream
firms can lead to a higher mark-up in the downstream market. Although downstream market
power may increase, consumer prices still decrease with the number of downstream firms
implying that higher market power does not necessarily imply lower consumer surplus.
Keywords: Vertical relations; Cournot competition; Free entry; Market Power
JEL Classification Codes: D43; L11; L13
∗
Corresponding address: Ioannis N. Pinopoulos, Department of Economics, University of Macedonia, 156
Egnatia Street, Thessaloniki, Greece. Phone: (+30) 2310-891-663. E-mail address: [email protected]
1. Introduction
It is well-known that in standard Cournot oligopoly models, the equilibrium mark-up, i.e., the
difference between price and marginal cost, decreases with the number of firms. In other
words, market power is diminished when more firms are present in the market.
In this paper, we consider a successive Cournot oligopoly model where firms freely enter
into the upstream market. We show that, under specific conditions, a higher number of
downstream firms can lead to a higher mark-up in the downstream market. Therefore, the
commonly held view that market power decreases with the number of firms does not always
hold. Moreover, although downstream market power may increase, consumer prices still
decrease with the number of downstream firms implying that higher market power does not
necessarily imply lower consumer surplus.
The intuition is as follows. For any given input price (marginal cost of downstream firms),
a higher number of downstream firms imply lower final good prices and thus lower mark-up
in the market. However, when there is free entry in the upstream market, the equilibrium
input price decreases with the number of downstream firms for any degree of product
differentiation, causing a further decrease in final good prices. When products are sufficiently
differentiated and fixed costs in the upstream market are large enough, the decrease in input
price is more pronounced than the decrease in final goods price that market power is
enhanced with more downstream firms.
Downstream firms produce differentiated final goods using a homogeneous input procured
from upstream firms. Upstream firms face the derived demand of downstream firms for the
input. The linear and uniform input price is determined in an open market by the quantities
supplied by the upstream firms. A standard reference for models with linear and uniform
input prices is the seminal work of Salinger (1988). Following Matsushima (2006), we
introduce free entry in the upstream market and we extend his model by assuming
differentiated final goods.
Although the assumption of a linear pricing scheme may be somewhat restrictive, uniform
linear prices are often observed in practise (see Smith and Thanassoulis, 2009). Moreover, the
assumption of uniform linear prices seems suitable when a high number of up- and
downstream firms interact with each other in spot markets and there is little scope for price
discrimination due to arbitrage opportunities. The petroleum industry is likely to be a case in
point.
The rest of the paper is organized as follows. Section 2 describes the model while Section
3 provides the main results. Section 4 concludes the paper.
2. The Model
We consider goods requiring two production stages and assume Cournot oligopolists at each
stage, as in the seminal work of Salinger (1988). Following Matsushima (2006), we introduce
free entry in the upstream market, however, we assume differentiated final goods.
At the downstream stage, N ≥ 2 firms serve final consumers. Downstream firms transform
one unit of a homogeneous input into one unit of a differentiated final good. We assume that
the consumer demand for the final good of a downstream firm i is given by the following
inverse demand function,
N −1
pi = 1 − qi − θ ∑ q j ,
i≠ j
where pi is the price of downstream firm i’s final good and qi , q j are the final good outputs
of downstream firms i and j respectively ( i, j = 1, 2,....N , i ≠ j ). The parameter θ ∈ [0,1] shows
the degree of differentiation. As θ approaches 0, the final goods of downstream rivals
become independent, while as θ approaches 1, final goods become closer substitutes. Each
downstream firm has a constant marginal cost which for simplicity we normalize to zero.
At the upstream stage, there are M firms that produce a homogeneous input. The input
quantity supplied by an upstream firm i is denoted by xi , i = 1, 2,......M . We assume that the
number of upstream firms is endogenously determined by a zero profit condition, i.e., there is
upstream free entry. We require that at least two firms enter the market at equilibrium. Each
upstream firm has a constant marginal cost which for simplicity we normalize to zero.
The timing of the game is as follows. First, each upstream firm decides whether to enter
the market or not. If an upstream firm enters, it must incur a fixed set-up cost f. Second, the
upstream firms that enter compete by choosing quantities of the input. Finally, taking the
price of the input, w, as given, downstream firms compete by choosing quantities of the final
good. We solve the model by using backwards induction. We first solve for the equilibrium
on the downstream market and obtain a derived demand function. We then use this to
characterize the equilibrium in the upstream market. Thus, we model the upstream market by
using a “market interface”, implying a linear and uniform input price.
3. Results
Each downstream firm maximizes its profits Π iD = ( pi − w)qi , i = 1,......N . Using the first order
conditions we obtain the following symmetric quantities and final good price as a function of
the input price:
q = ( p − w) =
p=
1− w
,
2 + θ ( N − 1)
1 + w(1 + θ ( N − 1))
.
2 + θ ( N − 1)
(1)
(2)
The equilibrium output of the downstream firms equals their demand for the input, and since
one unit of the input is required for each unit of the final good, we have
X = Q = Nq =
N (1 − w)
.
2 + θ ( N − 1)
(3)
Solving Eq. (3) for w, we obtain the inverse derived demand for the upstream producers’
product:
w = 1−
2 + θ ( N − 1)
X.
N
(4)
Based on this inverse derived demand, each upstream firm maximizes its profits
ΠUi = wxi − f , i = 1,......M . Imposing symmetry, we obtain the final equilibrium input quantity
and price,
x=
N
,
(2 + θ ( N − 1))( M + 1)
(5)
w=
1
,
M +1
(6)
The net profits of each active upstream firm are
ΠU = wx − f =
N
−f .
(2 + θ ( N − 1))( M + 1) 2
Free entry in the upstream sector implies ΠU = 0 ; solving the latter for M we obtain the free
entry equilibrium number of upstream firms
Mf =
N
f
2 + θ ( N − 1)
−1.
(7)
where the subscript f is used to denote free entry equilibrium outcomes.
We require that at least two firms enter the upstream market, i.e., M f ≥ 2 , which implies that,
f ≤
N
.
9(2 + θ ( N − 1))
(8)
The final free-entry equilibrium outcomes are readily shown to be the following:
xf =
f
N
2 + θ ( N − 1)
q f = ( p f − wf ) =
,
f
wf =
1− wf
2 + θ ( N − 1)
2 + θ ( N − 1)
N
,
pf =
,
1 + w f (1 + θ ( N − 1))
2 + θ ( N − 1)
.
(9)
It is well-known that, in the case without entry, the equilibrium input price does not
depend on the number of competing downstream firms or/and the degree of downstream
product differentiation (see (6) and Inderst and Valletti, 2007; 2009). However, in the case of
upstream entry, the equilibrium input price depends positively on θ and negatively on N.
Proposition 1. In a successive Cournot oligopoly model with linear demand and upstream
free entry, the equilibrium input price increases with the degree of product differentiation
and decreases with the number of downstream firms.
Proof. From (8), it is easy to verify that ∂w f ∂θ > 0 and ∂w f ∂N < 0 .
■
The result that the equilibrium input price depends negatively on N has already been
pointed out by Matsushima (2006) in his Proposition 1 for completely homogeneous goods.
The average total cost curve and the derived demand for the input must be tangent under a
free entry situation. The flatter (steeper) the slope of the derived demand is, the lower
(higher) the input price is. From (4), as θ increases, the slope of the derived demand for the
input becomes steeper, while as N increases the slope of the derived demand becomes flatter.
Taking the derivative of ( p f − w f ) from (9) with respect to N we obtain our main result.
Proposition 2. In a successive Cournot oligopoly model with linear demand and upstream
free entry, a higher number of downstream firms imply a higher mark-up in the downstream
market, if the following conditions hold:
θ ≤θ* =
2
1 + 4N
Proof. First we show that
and
4 N 3θ 2
N
< f ≤
.
2
(2 + θ ( N − 1))(2 + θ (2 N − 1))
9(2 + θ ( N − 1))
4 N 3θ 2
N
< f ≤
.
2
9(2 + θ ( N − 1))
(2 + θ ( N − 1))(2 + θ (2 N − 1))
stems from condition (8). From (9), we obtain
The second inequality
∂ ( p f − w f ) ∂N =
f 2N
[ f (2 + θ ( N − 1))(2 + θ (2 N − 1) − 2 Nθ Nf (2 + θ ( N − 1))] .
2( Nf (2 + θ ( N − 1))5 2
The sign of
the above expression is determined by the sign of the term in brackets, which is positive
whenever
4 N 3θ 2
< f
(2 + θ ( N − 1))(2 + θ (2 N − 1)) 2
be shown that
holds. After some straightforward calculations, it can
2
4 N 3θ 2
N
holds when θ ≤
≤
2
1 + 4N
9(2 + θ ( N − 1))
(2 + θ ( N − 1))(2 + θ (2 N − 1))
■
For any given input price, a higher number of downstream firms imply lower final good
prices and thus lower mark-up in the market (see (1) and (2)). However, when there is free
entry in the upstream market, the equilibrium input price decreases with the number of
downstream firms for any degree of product differentiation, causing a further decrease in
final goods prices (see (8)). When products are sufficiently differentiated and fixed costs in
the upstream market are large enough, the decrease in input price is more pronounced than
the decrease in final goods price that market power is enhanced with more downstream firms.
It is clear from the above analysis that, although downstream market power may increase,
consumer prices still decrease with the number of downstream firms implying that higher
market power does not necessarily imply lower consumer surplus.
4. Conclusions
In this paper, we consider a successive Cournot oligopoly model where firms freely enter into
the upstream market. We show that the equilibrium downstream mark-up may increase with
the number of downstream firms; the commonly held view that market power decreases with
the number of firms does not always hold. Moreover, we find that, although downstream
market power may increase, consumer prices still decrease with the number of downstream
firms, implying that higher market power does not necessarily imply lower consumer surplus.
Acknowledgements
I am much indebted to Christos Constantatos and Dimitrios Bakas for valuable comments and
expository improvements. Financial support from the Greek State Scholarships Foundation
(I.K.Y.) is also gratefully acknowledged. All remaining errors are solely mine.
References
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integration. Scandinavian Journal of Economics 111, 527–546.
Matsushima, N., 2006. Industry profits and free entry in input markets. Economics Letters
93(3), 329-336.
Salinger, M.A., 1988. Vertical mergers and market foreclosure. Quarterly Journal of
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Smith, H., Thanassoulis, J., 2009. Upstream competition and downstream buyer power.
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