Capacity choice with upstream investment
Qing Hu
Tomomichi Mizuno
February 2022
Discussion Paper No. 2202

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANCapacity choice with upstream investment
Qing Hu∗
Faculty of Economics, Kushiro Public University of Economics,
Graduate School of Economics, Kobe University
Tomomichi Mizuno†
Graduate School of Economics, Kobe University
February 15, 2022

Abstract
We consider a vertically related market with an upstream ﬁrm engaging in costreducing investment and n downstream ﬁrms competing on quantity. We analyze
the capacity choice by downstream ﬁrms and ﬁnd that over-capacity occurs inequilibrium if the number of downstream ﬁrms is large or the upstream investment
is eﬃcient.

JEL codes: D43, L13, M21.
Keywords: capacity, investment, vertical relationship.

∗

Corresponding author: Qing Hu, Faculty of Economics, Kushiro Public University of Economics, Ashino 4-1-1, Kushiro-shi, Hokkaido, 085-8585, Japan. Phone: (81)-154-37-3211. E-mail:huqing549@gmail.com†
Graduate School of Economics, Kobe University, 2-1 Rokkodai, Nada, Kobe, Hyogo, 657-8501,
Japan. Phone: (81)-78-803-6802. E-mail: mizuno@econ.kobe-u.ac.jp1

1

Introduction

In a “capacity-then-quantity” game in a pure oligopoly market, it is anticipated that
increasing production capacity will also increase the market share; thus, ﬁrms choose
over-capacity as a strategic device (Dixit, 1980; Brander and Spencer, 1983; Horiba andTsutsui, 2000). Additionally, in a vertical market, an upstream ﬁrm has an incentive
to extract rent from a capacity activity; hence, downstream ﬁrms choose under-capacity(Choi and Lee, 2020). This study challenges these results by introducing an upstream
investment and downstream competition.
Particularly, we consider a three-stage game with an upstream ﬁrm and n downstreamﬁrms. In the ﬁrst stage, each downstream ﬁrm decides its production capacity. In thesecond stage, the upstream ﬁrm chooses the input price and investment level for marginal
cost reduction. In the third stage, the downstream ﬁrms compete in terms of quantity.
We ﬁnd that input price decreases with production capacity if many downstreamﬁrms exist or the investment is eﬃcient. Meanwhile, the investment level always increases with the aggregate production capacity. Additionally, downstream ﬁrms choose
over-capacity if the number of downstream ﬁrms is large or the upstream investment
is eﬃcient; otherwise, they choose under-capacity. In our analysis, capacity choice isdependent on investment eﬃciency, and this assumption diﬀers from that of previous
research.
The following is the explanation of the intuition for our results. On expanding its production capacity, a downstream ﬁrm obtains a larger market share. This eﬀect becomesstrong if downstream competition is tough; that is, many downstream ﬁrms exist. Hence,
downstream ﬁrms choose over-capacity. Additionally, the total output increases with theaggregate production capacity. For the upstream ﬁrm, an increase of the downstream
output strengthens its incentive to invest; thus, the investment level always increases
with the production capacity. However, the investment level reduces the upstream ﬁrm’s
marginal cost. Hence, the upstream ﬁrm has an incentive to reduce the input price. The
2

input price reduction causes the ﬁrms to choose over-capacity. However, this reductioneﬀect plays a limited role when the investment cost is large. Fontana and Guerzoni(2008), using data for European Union countries, show that process R&D becomes lessimportant when market size decreases. Therefore, our result can be inferred indirectly
by empirical studies.
Our study is related to previous research that reexamined whether the well-known
result of over-capacity always stands in various contexts (Barcena-Ruiz and Garzon,
2010; Chen et al., 2021; Fanti and Meccheri, 2017). Choi and Lee (2020) allow capacity
choice to be considered in a vertical market. All these studies consider models without
upstream investment.1
We ﬁrst analyze the capacity choice with upstream cost-reducing investment. The
assumption that downstream ﬁrms decide the production capacities before the upstreamﬁrm is crucial for our main result.
The rest of the paper is organized as follows. Section 2 describes the model, Section
3 presents the analysis, and ﬁnally, Section 4 concludes the paper.

2

Model

We assume a market with an upstream ﬁrm and n downstream ﬁrms. The upstreamﬁrm produces input and sells it to downstream ﬁrms at the input price w. To produce
one unit of a homogenous ﬁnal product, downstream ﬁrms use one unit of the input.
We denote the output of the downstream ﬁrm i (= 1, . . . , n) as qi . The inverse demandPin the ﬁnal good is given by p = 1 − Q, where p is the price and Q = nj=1 qj is theaggregate output.
We assume that all downstream ﬁrms use the same technology. Following the established literature (Nishimori and Ogawa, 2004; Lu and Poddar, 2005; Fanti and Meccheri,1

For cost-reducing upstream investment, Hu et al. (2020) analyze the relationship of cross-holdings
and upstream R&D.

3

2017), the cost function of the downstream ﬁrm i is denoted by Ci = wqi + (xi − qi )2 ,where xi is the production capacity of ﬁrm i. This cost function shows that the average
cost is minimized when quantity equals production capacity. Thus, both over-capacity(xi − qi > 0) and under-capacity (xi − qi < 0) are ineﬃcient. Then, the proﬁt of thedownstream ﬁrm i is πi = (p − w)qi − (xi − qi )2 .
The upstream ﬁrm can make investments to reduce its own marginal cost. To reducethe marginal cost by k, it incurs investment cost γk 2 /2, where γ relates to investmenteﬃciency and c (< 1) is the marginal cost without investment. Hence, the marginal
cost of the upstream ﬁrm is c − k. Then, the proﬁt of the upstream ﬁrm is πU =[w − (c − k)]Q − γk 2 /2. We deﬁne
1 + 2n
if n = 1, 2,

15
γ=

 1
otherwise.
2
With small γ or small c, the marginal cost of the upstream ﬁrm may take a non-positive
value. Therefore, to guarantee positive equilibrium outcomes, we assume that γ > γ andc>3n2 [(2γ − 1)n + 6γ].
2γ[(6γ − 3)n3 + (28γ − 5)n2 + (38γ − 4)n + 12γ]

The following is the timing of the game. In the ﬁrst stage, each downstream ﬁrm
decides its production capacity. In the second stage, the upstream ﬁrm chooses theinvestment level k and the input price w. In the third stage, each downstream ﬁrmcompetes in terms of quantity. We solve this game using backward induction.

3

Results

3.1

Calculating equilibrium

In the third stage, summing up the ﬁrst-order conditions, ∂πi /∂qi = 1−3qi −Q−w+2xi =0, and solving it for Q, we obtain an aggregate output.
Q (w, X) =n(1 − w) + 2X,
3+n4

where X =Pn

j=1 xj the aggregate capacity.

Substituting the aggregate output into theﬁrst-order condition for the downstream ﬁrm i, the output of the downstream ﬁrm i isqi (w, xi , X) =3(1 − w) + 2xi (3 + n) − 2X.
3(3 + n)In the second stage, the upstream ﬁrm chooses the input price and investment level.
Substituting the aggregate output into the proﬁt of the upstream ﬁrm and solving theﬁrst-order conditions for w and k, we obtain the outcomes in this stage.
w(X) =(3 + n)(n + cn + 2X)γ − n(n + 2X),
n[2(3 + n)γ − n]k(X) =n(1 − c) + 2X.
2(3 + n)γ − n

Substituting w(X) into Q(w, X), the aggregate output is expressed as follows:
Q(X) =[n(1 − c) + 2X]γ.
2(3 + n)γ − nFrom γ > γ, the denominators of w(X), k(X), and Q(X) are positive. Hence, k(X) andQ(X) increase with X. Diﬀerentiating w(X) with respect to X, we obtainw′ (X) =2[(3 + n)γ − n].
n[2(3 + n)γ − n]Hence, we obtain the following result:Lemma 1 The input price decreases with the aggregate capacity if γ < γ < n/(3 +n), while the aggregate output and investment level always increase with the aggregate
capacity.
From γ < n/(3 + n), the input price decreases with the aggregate capacity if the
upstream investment technology is eﬃcient or the number of downstream ﬁrms is large.
The following is an intuition behind this result. Given that, by choosing a larger capacity, downstream ﬁrms can make a commitment to expand their production, an upstreamﬁrm faces large input demand. Then, the upstream ﬁrm decides to increase its investment. Hence, large capacity by downstream ﬁrms can have diﬀerent eﬀects on the input
5price. Meanwhile, the demand increasing eﬀect raises the input price, but the investment
increasing eﬀect reduces it. If the upstream investment technology is eﬃcient, the upstream ﬁrm will increase its investment with the aggregate capacity. Additionally, if thenumber of downstream ﬁrms is large, the marginal proﬁtability of investment becomeslarge. In these cases, the investment increasing eﬀect dominates the demand increasing
eﬀect. Therefore, with small γ or large n, the input price decreases with the aggregate
capacity.
Finally, we obtain the outcomes in the ﬁrst stage. Substituting the outcomes in thesecond and third stages into the proﬁt of the downstream ﬁrm i and solving the ﬁrstorder condition for xi , we obtain equilibrium capacity and output of the downstreamﬁrms as follows:4(1 − c)γ [(2γ − 1)n2 + 4γn + n − 3γ],
3(1 − 2γ)2 n3 + 28γ(2γ − 1)n2 + 4γ(19γ − 2)n + 24γ 23(1 − c)nγ[(2γ − 1)n + 6γ]qi∗ =.
3(1 − 2γ)2 n3 + 28γ(2γ − 1)n2 + 4γ(19γ − 2)n + 24γ 2x∗i =Note that the denominators of x∗i and qi∗ are positive because we assume that γ < γ.

3.2

Condition for over-capacity investment

Comparing x∗i with qi∗ , we obtain the main result as follows.
Proposition 1 The condition in which downstream ﬁrms choose over-capacity investment, x∗i > qi∗ , is (i) n ≥ 3 or (ii) n ∈ {1, 2} and γ < γ < (n − 4)n/[2(n − 3)(n + 2)].
Proof Comparing x∗i with qi∗ yieldsx∗i − qi∗ =(1 − c)γ [2(n − 3)(n + 2)γ − (n − 4)n].
3(1 − 2γ)2 n3 + 28γ(2γ − 1)n2 + 4γ(19γ − 2)n + 24γ 2First, we consider the case with n ≥ 3. As the denominator is positive, the sign ofx∗i − qi∗ only depends on the terms in square brackets. Rearranging the terms, we obtain6

2(n − 3)(n + 2)γ − (n − 4)n = (2γ − 1)n2 + 2n(2 − γ) − 12γ. Solving (2γ − 1)n2 + 2n(2 −γ) − 12γ > 0, we obtain n < n0 or n > n1 , whereppγ − 2 + 4 − 16γ + 25γ 2
γ − 2 − 4 − 16γ + 25γ 2, n1 =.
n0 =2γ − 12γ − 1Given that n1 converges to 3 as γ → ∞, by showing that n1 is an increasing function,
we can conclude that the least upper bound of n1 is 3. Diﬀerentiating n1 with respectto γ leads top
2 − 3γ
4
−
16γ
+
25γ
∂n1
p
.
=
∂γ
(2γ − 1)2 4 − 16γ + 25γ 2
3

As 4 − 16γ + 25γ 2 − (3γ)2 = 4(2γ − 1)2 > 0, n1 increases with γ. Hence, for any γ,
n1 < 3, indicating that, for any n ≥ 3, we have x∗i − qi∗ > 0.
The remaining cases are for n = 1, 2. Solving x∗i − qi∗ > 0 for γ, we obtain γ <(n − 4)n/[2(n − 3)(n + 2)]. Additionally, the threshold value (n − 4)n/[2(n − 3)(n + 2)
is larger than γ at n = 1, 2. Therefore, we obtain this proposition.2

From this proposition, downstream ﬁrms choose over-capacity if the upstream investment is eﬃcient or the number of downstream ﬁrms is large. The following is an intuition
behind this result. Capacity expansion is a commitment to increase production. With
the presence of many downstream ﬁrms, each downstream ﬁrm’s share is small. Hence,the eﬀect of one unit of capacity expansion becomes signiﬁcant. Further, over-capacity
investment occurs if the number of downstream ﬁrms is large. When downstream ﬁrms
are few, this eﬀect becomes weak. However, each downstream ﬁrm has an incentive tochoose over-capacity investment if the upstream investment technology is eﬃcient. From
Lemma 1, the input price decreases with the aggregate capacity if the upstream ﬁrm’s
investment is eﬃcient. Given that the lower input price is proﬁtable for the downstream
ﬁrms, they have incentives to increase capacity, resulting in over-capacity investment.

7

3.3

Comparative statics of over-capacity investment

By performing comparative statics of over-capacity investment, we can identify a situation wherein overcapacity becomes serious. Diﬀerentiating x∗i − qi∗ with respect to γ, we
obtain the following:∂(x∗i − qi∗ )3(1 − c)n2 [−4γ 2 (n + 2)(n2 − 3) + 4γ(n − 3)(n + 2)n − (n − 4)n2 ].
=∂γ[24γ 2 + 3(1 − 2γ)2 n3 + 28γ(2γ − 1)n2 + 4γ(19γ − 2)n]2
The sign of ∂(x∗i − qi∗ )/∂γ is the same as that of the terms in square brackets in the
numerator. Then, we obtain the following result.
Proposition 2 For n ≥ 2, the degree of over-capacity investment, x∗i − qi∗ , decreases
with γ. With n = 1, the degree of over-capacity investment decreases with γ if γ <√(2 + 2)/4 (≈ 0.854).
Proof The sign of ∂(x∗i − qi∗ )/∂γ only depends on Ψ = −4γ 2 (n + 2)(n2 − 3) + 4γ(n −3)(n + 2)n − (n − 4)n2 .
First, we consider the case with n ≥ 4. As the coeﬃcient of γ 2 in Ψ is negative,
solving ∂(x∗i − qi∗ )/∂γ < 0 for γ, we obtain γ < γ0 or γ > γ1 , where γ0 = n(n2 − n − 6 −pp6(2 + n))/[2(2 + n)(n2 − 3)] and γ1 = n(n2 − n − 6 + 6(2 + n))/[2(2 + n)(n2 − 3)].
Here, we show γ1 < 1/2. Diﬀerentiating γ1 with respect to n, we obtain
√
√
√
√
√
√
√
∂γ1
6 n + 2n3 + 36 n + 2 − 3 6n3 − 4 6n2 − 6 n + 2n − 3 6n − 12 6=.
∂n4(n + 2)3/2 (n2 − 3)2√√√√
As n ≥ 4, we have 6 n + 2n3 > 3n3 n + 2 + 4n2 n + 2 + 32n n + 2. Then, thenumerator of ∂γ1 /∂n is positive and γ1 increases with n. Additionally, γ1 converges
1/2 as n → ∞. Hence, for any n ≥ 4, we obtain γ1 < γ = 1/2, which becomes∂(x∗i − qi∗ )/∂γ < 0.
Subsequently, we consider the cases with n = 1, 2, and 3. Substituting n = 1, 2, and8

3 into ∂(x∗i − qi∗ )/∂γ yields∂(x∗i − qi∗ )(1 − c)(8γ 2 − 8γ + 1)=,∂γ
(1 − 16γ + 56γ 2 )2n=1∂(x∗i − qi∗ )3(1 − c)(2γ 2 + 4γ − 1)
=−
< 0,
∂γ
2(3 − 28γ + 62γ 2 )2n=2∂(x∗i − qi∗ )9(1 − c)(40γ − 3)
=−
< 0.
∂γ
(27 − 200γ + 360γ 2 )2n=3√
At n = 1, solving ∂(x∗i − qi∗ )/∂γ < 0, we obtain (2 − 2)/4 (≈ 0.146) < γ < (2 +√2)/4 (≈ 0.854). From the assumption γ = 1/5 at n = 1, x∗i − qi∗ decreases with γ if√
γ < (2 + 2)/4.2

The following is an intuition behind this result. As explained in the intuition for
Proposition 1, capacity expansion is a commitment to increase production. Hence, capacity expansion promotes investment by an upstream ﬁrm and allows downstream ﬁrms
to capture their rivals’ market share. If the upstream investment is ineﬃcient, the investment enhancing eﬀect becomes weak, leading to small over-capacity investment. However, when n = 1, the market share capturing eﬀect disappears. Therefore, downstreamﬁrms determine their capacity by comparing the input price reduction eﬀect for the enhanced upstream investment and the input price increasing eﬀect due to the increasing
demand. If the upstream ﬁrm’s investment is eﬃcient, the input price reduction eﬀect
due to investment dominates. However, if it is ineﬃcient, the input price increasing
eﬀect dominates. In our model, the input price reduction eﬀect is the strongest when
the investment eﬃciency is intermediate. Therefore, x∗i − qi∗ takes the minimum value.

3.4

Alternative timing

In the previous analysis, we assume that each downstream ﬁrm decides its production
capacity before the upstream ﬁrm. In this subsection, we consider the reversed timingof the game, which is changed to the following. In the ﬁrst stage, the upstream ﬁrmchooses its investment level and input price; in the second stage, each downstream ﬁrm
9decides its production capacity; and in the third stage, downstream ﬁrms choose their
outputs.
The outcomes in the third stage are the same as in the previous section: Q(w, X) andqi (w, xi , X). In the second stage, a downstream ﬁrm i maximizes its proﬁt with respect
to production capacity xi yields
xi (w) =4(2 + n)(1 − w).
11 + 10n + 3n2

Substituting xi = xi (w) and X = nxi (w) into qi (w, xi , X), we obtain the downstream
output asqi (w) =3(3 + n)(1 − w).
11 + 10n + 3n2

Comparing xi (w) with qi (w) yields
xi (w) − qi (w) =(n − 1)(1 − w)
> 0.
11 + 10n + 3n2

Hence, we ﬁnd that, in the reversed timing, a downstream ﬁrm always chooses overcapacity. The following is a simple intuition behind this result. When downstreamﬁrms choose their capacity, the input price and upstream investment level are ﬁxed.
Hence, input price reduction and investment promotion eﬀects disappear. Given thatthe capacity expansion of each downstream ﬁrm will only have the eﬀect of capturing amarket share, all downstream ﬁrms will choose over-capacity investments.

4

Conclusions

We considered a vertically related market with an upstream ﬁrm engaging in costreducing investment and n downstream ﬁrms. We found that the input price decreases
with the production capacity if the number of downstream ﬁrm is large or the investment is eﬃcient, while the investment level always increases with the aggregate capacity.
Additionally, downstream ﬁrms choose over-capacity if the number of downstream ﬁrm
is large or the upstream investment is eﬃcient; otherwise, they choose under-capacity.
10

We also analyzed the model wherein the upstream ﬁrm makes the input price contract
with the downstream ﬁrms. It is worthwhile to consider the case where the upstreamﬁrm proposes a two-part tariﬀ contract to the downstream ﬁrms. Moreover, an extension
is possible by analyzing product diﬀerentiation. We leave these for future research.

11

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