capacity commitment and licensing - fep.up.pt iv/iv.d... · this was challenged by dixit (1980),...
TRANSCRIPT
Commitment and excess capacity: a new look with licensing
Arijit Mukherjee*
University of Nottingham and The Leverhulme Centre for Research in Globalisation and Economic Policy, UK
February 2005
Abstract: This paper provides a new rationale for holding excess capacity. We show
that incumbent firms may hold excess capacity not to deter entry but to get more
benefit from technology licensing. Our results are robust with respect to capacity
commitment by the entrant. We also show that if the entrant commits to a capacity
level, the incumbent is better off by licensing its technology after rather than before
the entrant’s capacity choice.
Key Words: Capacity commitment, Entry, Excess capacity, Incumbent, Licensing
JEL Classification: D43, L13, O33
Correspondence to: Arijit Mukherjee, School of Economics, University of
Nottingham, University Park, Nottingham, NG7 2RD, U.K.
E-mail: [email protected]
Fax: +44-115-951 4159
___________________________________ * I would like to thank David Greenaway for his comments and suggestions on the earlier version of this paper. I am very much indebted to Arnab Bhattacharjee for providing me with the empirical findings. Discussion with Achintya Ray was also rewarding. Financial support from the Netherlands Technology Foundation (STW) is gratefully acknowledged. The usual disclaimer applies.
Commitment and excess capacity: a new look with licensing
1. Introduction
The existence of excess capacity in firms has attracted considerable amount of
attention in the industrial organization literature. Previous work has posited entry
deterrence, collusion and demand uncertainty as the motives of incumbent firms for
holding excess capacity. In this paper, we provide a new rationale for holding excess
capacity. We show that the benefit from technology licensing can induce an
incumbent firm to install more capacity than it needs.
In earlier work, Spence (1977) has shown that an incumbent firm may keep
excess capacity to deter entry. This was challenged by Dixit (1980), who argued that
excess capacity cannot occur in a subgame perfect equilibrium.1 In turn that paper
stimulated a large amount of theoretical and empirical research. Many researchers
have shown that the Dixit (1980) result is very much dependent on its assumptions.
Bulow et al. (1985) show that excess capacity may be the equilibrium outcome if we
do not consider each firm’s marginal revenue is decreasing in other firms’ outputs,
which is assumed in Dixit (1980). Spulber (1981) and Basu and Singh (1990) have
found that if the post-entry game is Stackelberg instead of Cournot, excess capacity
might be the outcome in a perfect equilibrium.2
These theoretical works have inspired empirical works, which test entry
deterrence as the motive for holding excess capacity. Though Masson and Shaanan
1 Ware (1984) criticizes the structure of the game considered in Dixit (1980) and extends Dixit’s two-stage game to a three-stage game. Though Ware (1984) shows that capacity commitment by the entrant reduces incumbent’s first-mover advantage, excess capacity does not occur in equilibrium. 2 Ware (1985) shows the role of inventory in order to deter entry.
1
(1986) and Lieberman (1987) confirm the presence of excess capacity, they provide
very little evidence for entry deterrence as the motive.
In this paper we show that the benefit from technology licensing may induce
an incumbent to install capacity that it ends up not using. Significant investment in
capacity reduces the reservation payoff of the entrant and helps the incumbent to
extract a higher price for its technology under licensing. We show that this benefit
outweighs the cost of the excess capacity and provides a rationale for holding excess
capacity in equilibrium. We find that the co-existence of licensing and excess
capacity is more likely to occur in industries with relatively low costs of capacity,
which provides a testable hypothesis. When the cost of capacity is relatively low, it
neither gives the incumbent a significant strategic advantage nor creates large waste.
But, capacity commitment helps the incumbent to extract higher surplus under
technology licensing.
We further show that our result holds only if the industry marginal revenue
curve is downward sloping. This condition is consistent with Dixit (1980). Thus, we
confirm that an incumbent may also hold excess capacity under the demand
conditions considered in Dixit (1980) if it licenses its technology to the
technologically inferior entrant.
As in Ware (1984), we extend our basic model to incorporate capacity
installation by the entrant. We consider two possibilities: first, where the entrant
installs capacity after licensing and second, where the entrant installs capacity before
licensing. We show that the co-existence of excess capacity and licensing may occur
in equilibrium even for these extended games. We further show that the incumbent
earns higher profit if it licenses its technology after the entrant’s capacity installation.
2
This paper falls in the area of theoretical works, which shows that firms may
hold excess capacity even if that does not deter entry. Benoit and Krishna (1987),
Davidson and Deneckere (1990) and Fershtman and Gandal (1994) have shown that
collusion in an oligopolistic industry creates the incentive for holding excess capacity.
Kim (1996), Poddar (1998) and Robles (2001) have shown that excess capacity can
be the equilibrium outcome if market demand is variable.3 In contrast, we show that,
even in a world with certainty and no collusion, excess capacity can be the
equilibrium outcome when it provides benefit to the other non-productive activities
such as technology licensing.
Mukherjee (2001) also considers licensing in an incumbent-entrant
framework. However, the present paper differs in two important ways from
Mukherjee (2001). Firstly, unlike Mukherjee (2001), in this paper the incumbent firm
does not have an advantage after technology licensing. So, Mukherjee (2001) may be
applicable to situations where firms need time-to-build capacity, whereas this paper
looks at the situation where the incumbency advantage comes mainly from earlier
acquisition of technology. Therefore, to exploit the incumbency advantage, the
incumbent firm needs to install capacity before entry and so, before technology
licensing. Secondly, this paper allows the entrant to choose its capacity after the
incumbent, whereas Mukherjee (2001) ignores capacity choice by the entrant.
Though our analysis is theoretical, there is evidence of the co-existence of
excess capacity and technology licensing in many industries. For example, Ericsson
has licensed its technology to other manufacturers and holds excess capacity (see,
e.g., Ericsson’s Annual report, 2001 and First quarter report, 2002). The Draft
3 Ungern-Sternberg (1988) and Marchionatti and Usai (1997) show rationale for holding excess capacity in a vertical structure and in the prospect of voluntary export restraint respectively.
3
Commission Decision of EEC (1994) also provides evidence of the co-existence of
excess capacity and technology licensing in the float glass industry. It shows that
Pilkington licenses its technology to competitors whilst holding excess capacity.
Unfortunately, we know of no empirical study that attempts to sort out the strategic
effects of licensing on capacity installation. Our results provide testable hypothesis
for this topic.
The remainder of the paper is organized as follows. Section 2.1 provides the
basic argument with a geometric representation and proves the main result. This
section also considers lower cost of capacity. Section 2.2 shows the implications of
higher costs of capacity. Section 3 considers capacity installation by the entrant.
Section 4 concludes.
2. Model
2.1 Geometric representation
Assume there is an incumbent firm, firm 1, and a potential entrant, firm 2. Production
requires investment in capacity. We assume the per-unit cost of capacity is and
is the same for both firms. The firms also need to incur a variable cost of production,
which depends on technology. Assume that firm 1 has better technology
0>z
1c
4 compared to
firm 2 and produces its product with the per-unit variable cost of production ,
while firm 2 produces with c , where c .
0>
2 12 c> 5 Think of firm 1 having a patent for its
technology, which corresponds to the variable cost of production , whilst its patent 1c
4 We consider that lower variable cost of production implies better technology. 5 One may refer to Allen et al. (2000) for a distinction between the cost of capacity and the variable cost of production.
4
for the previous technology has expired and creates potential threat of entry.6 So, total
per-unit cost of production or marginal cost of production, which aggregates the cost
of capacity and the variable cost of production, is 1cz + for firm 1 and for firm
2. We further assume that the per-unit costs are such that both always produce
positive outputs in equilibrium. Since, our purpose is not to address the issue of entry-
deterrence, we abstract from other costs such as fixed costs and entry costs. We
assume the post-entry game is characterized by Cournot-Nash competition.
2cz +
q1
q11
1cz +
Assume that the firms produce a homogeneous product. Inverse market
demand function is with P P q q= +( 1 2 ) ′ <P 0 , where P is price of the product
and, and are the outputs of firms 1 and 2 respectively. We assume that market
demand is such that it ensures unique and stable equilibrium output.
q1 q2
We consider the following game. At stage 1, firm 1 invests up to capacity
level x . At stage 2, firm 2 enters the market.7 At stage 3, firm 1 decides whether or
not to license its technology to firm 2. At stage 4, production takes place and the
profits are realized. We solve the game through backward induction. So, in this
section we do not allow firm 2 to install capacity before production. We will relax this
in section 3.
If firm 1’s output is less than its installed capacity level (i.e., ), its
marginal cost of production at the output stage is c . But, if it produces , its
marginal cost of production at the output stage is
x≤
x>
.
6 There may be other justifications for the differences in the variable costs of production. The technologies may require different types of inputs and the competitive prices for the inputs show the variable costs of production. Or, the technologies may require same inputs but with different combinations and therefore, creates difference in the variable cost of production. 7 Since in this analysis we do not consider the possibility of entry-deterrence, entry always occurs even with different marginal costs of production.
5
Figure 1 shows equilibria of the above game. For expositional convenience,
we draw the reaction functions as linear. The reaction functions are downward
sloping when the industry marginal revenue is downward sloping, which we will
show as a necessary condition for our results.
Figure 1
Let’s first consider the situation under no licensing. In Figure 1, the lines AB
and CD show the reaction functions for firms 1 and 2 respectively when the firms
produce with the marginal costs 1cz + and 2cz + respectively. Point S shows the
equilibrium outputs if firm 1 is a Stackelberg leader. It is well known that firm 1’s
capacity commitment will shift its reaction function to the right and it will try to
commit its output corresponding to point S through its capacity installation. However,
this amount of capacity commitment is subgame perfect provided firm 1’s
Stackelberg leader’s output with its marginal cost 1cz + is less than its Cournot-Nash
output with its marginal cost . Otherwise, subgame perfect capacity commitment is
equal to firm 1’s Cournot-Nash output with its marginal cost c . To prove our result
in the simplest way, we assume that firm 1’s Stackelberg leader’s output with its
marginal cost is more than its Cournot-Nash output with marginal cost . It
follows from Ware (1984) that this will be the outcome for sufficiently low values of
.
1c
1
1cz +
1c
1c
z 8 Therefore, firm 1 installs capacity up to its Cournot-Nash output level with
marginal cost , i.e., up to B′ in Figure 1. A′EB′ shows the reaction function for firm
1 when it installs capacity up to B′ and A′EB′ corresponds to the marginal cost of
8 If the inverse market demand is qaP −= , firm 1’s Stackelberg leader’s output with its marginal
cost is more than its Cournot-Nash output with the marginal cost provided 1cz + 1c
5)21 c
z+
<2( ca −
.
6
production . So, equilibrium output is at E where firm 1 produces 0B′ and firm 2
produces B′E. This equilibrium shows outputs when the firms produce with their own
technologies and provides the benchmark for our licensing game.
1c
Now, suppose that firm 1 installs capacity up to B′ and decides to license its
technology to firm 2. Following Katz and Shapiro (1985), Marjit (1990), Mukherjee
(2001) and others, we focus on a fixed-fee licensing contract. The possibility of
imitation by the licensee or lack of information needed for a provision of royalty in
the licensing contract could be the reason for licensing with an up-front fixed-fee only
(see, e.g., Katz and Shapiro, 1985 and Rockett, 1990). We assume that, in case of
licensing, firm 1 extracts the entire surplus generated from its technology through a
licensing fee.9
If firm 1 licenses its technology to firm 2, firm 2’s per-unit variable cost of
production is . Hence, firm 2’s marginal cost of production under licensing is
. Suppose the new reaction function of firm 2 is C′D′ in Figure 1. Therefore, if
firm 1 licenses its technology, equilibrium in the product market is at L. So, under
licensing, firms 1 and 2 produce 0L′ and LL′ respectively. Hence, ( shows the
amount of excess capacity.
1c
1cz +
)LB ′−′
The above has assumed that firm 1 installs capacity up to B′ and also licenses
its technology. Since equilibrium in the product market is at L, another possibility
may be to install capacity up to L′. In that case, there exists no excess capacity and
firm 1 can save the waste of zLB )( ′−′ . However, if capacity installation is up to L,
equilibrium under no licensing is at E′ and firm 2’s profit at E′ is greater than at E.
Hence, the reservation payoff (i.e., the profit under no licensing) of firm 2 is higher
7
under firm 1’s capacity commitment up to L′ than under firm 1’s capacity
commitment up to B′. Therefore, in the case of capacity commitment up to L′, the
licensing fee is lower compared to the situation where firm 1 installs capacity up to
B′. Thus, capacity installation up to B′ helps firm 1 to extract a higher licensing fee.
However, it creates waste from excess capacity under licensing. If the benefit from
the higher licensing fee outweighs the loss from wastage, it is better for firm 1 to
install more capacity than L′.
Note that due to the trade-off between a higher licensing fee and waste from
excess capacity, it may be that it is better for firm 1 to install capacity between L′ and
B′. Capacity installation below B′ reduces the licensing fee but saves waste from
excess capacity. However, for any capacity installation more than L′, there exists
excess capacity if firm 1 licenses after capacity installation.
2.1.1 Analytical result
Consider the structure as specified in the previous subsection. The per-unit cost of
capacity is sufficiently small that credible commitment by firm 1 without licensing is
up to its Cournot output level with the marginal costs and 1c 2cz + for firms 1 and 2
respectively. Since we solve the game through backward induction, let us first we see
when licensing is profitable, given the capacity commitment up to B′. Then we will
consider the incumbent’s decision on capacity installation.
Licensing is profitable if it increases industry profit. Given the capacity
installation, the decision on licensing should not consider firm 1’s cost of capacity
installation since that investment is sunk at the time of licensing. Therefore, with
9 Our qualitative results will hold for other types of pricing for the technology (e.g., pricing through
8
capacity installation up to B′, Cournot equilibrium without licensing corresponds to
marginal costs and for firms 1 and 2 respectively but, under licensing, it
corresponds to and for firms 1 and 2 respectively. So, the joint profit ex-post
firm 1’s capacity installation is
1c
1c
2cz +
1cz +
2112121 ))(( cqqcqqqqP −−++ (1)
where c without licensing and 2cz += 1czc += under licensing.
Proposition 1: Suppose firm 1 cannot credibly commit to its Stackelberg leader’s
output when both firms produce with their own technologies.
(i) Licensing is profitable only if the industry marginal revenue is downward sloping.
(ii) Given that the industry marginal revenue is downward sloping, licensing is
profitable if the own technologies of the firms are sufficiently close and the per-unit
cost of capacity installation is sufficiently small.
Proof: (i) First, consider the effect of on industry revenue, i.e., 2c
))(( 2121 qqqqPR ++= . We find after rearranging that
22
)(cqPqP
cR
∂∂′+=
∂∂ . (2)
Next, we consider the effect of c on total cost at the stage of production, i.e.,
. We find after rearranging that
2
2211 )( qczqcC ++=
2
12
2122
2
)(cqc
cqcczq
cC
∂∂
+∂∂
−++=∂∂ . (3)
Therefore, the effect of on industry profit (excluding the cost of capacity of firm 1)
is
2c
Nash bargaining).
9
22
212
22
2
)( qcqccz
cqPq
c−
∂∂
−+−∂∂′=
∂∂π . (4)
The reduction in increases industry profit (excluding the cost of capacity of firm
1), i.e.,
2c
02
<∂∂cπ if and only if
2
212
22 )()1(
cqccz
cqPq
∂∂
−+−>∂∂′− . (5)
Differentiating the first order condition of profit maximization for firm 2 with respect
to c , we get that 2 02
2 <∂∂
cq . So, condition (5) holds only if
2cqP
∂1 ∂′> .
Adding the first order conditions of profit maximization for both firms, ex-
post firm 1’s capacity installation, we get
02 21 =−−−′+ czcPqP . (6)
Differentiating (6) with respect to c and after rearranging we find 2
0)2()1(22
=∂∂′′+′+−
∂∂′
cqPqP
cqP . (7)
Since, the absolute slope of the reaction function of firm 1 ex-post capacity
installation and up to the installed capacity level is less than 1,10 we get
0)1(2
1
2
2
2
<∂∂
+∂∂
=∂∂
cq
cq , since 0
2
2 <∂∂
cq . Therefore, it follows from (7) that
2cqP
∂∂′>1
only if 02 <′′+′ PqP , i.e., industry marginal revenue is downward sloping.
10 Ex-post capacity installation by firm 1, the reaction function of firm 1 up to the installed capacity level is given by the first order condition of profit maximization 011 =−′+ cPqP . Differentiating
this first order condition by and after rearranging, we find that 2q 1<1 1
2
1
′′′
+=∂∂
−PPq
.
10
(ii) When , (5) holds if 02 <′′+′ PqP )( 12 ccz −+ is sufficiently small, i.e., when
and c is sufficiently close, and is sufficiently small.1c 2 z 11 Q.E.D.
Proposition 1(i) shows the necessary condition for profitable licensing when
firm 1 invests up to B′. If the necessary condition holds, i.e., if the industry marginal
revenue is downward sloping, licensing occurs when condition (5) holds. Licensing
has two effects on industry profit. On the one hand, licensing increases cost efficiency
of firm 2 and increases its profit excluding licensing fee. This benefit of cost
efficiency is transferred to firm 1 through the licensing fee. On the other hand,
licensing reduces firm 1’s profit in the product market (i.e., profit excluding licensing
fee) as it makes firm 2 more competitive. When the marginal cost of firm 2 at the
output stage (i.e., ) is very close to the variable cost of firm 1, which is c , the
effect of competition due to licensing is sufficiently small. In this situation, the cost
efficiency effect in firm 2 dominates the competition effect and makes licensing
profitable. As is getting larger compared to , the effect of competition is
becoming stronger compared to the cost efficiency effect in firm 2 and makes
licensing unlikely.
2cz +
2c+
1
z 1c
The conditions for profitable licensing shown in the above proposition are
similar to the conditions shown in Katz and Shapiro (1985) with an additional
restriction on the cost of capacity. Unlike Katz and Shapiro (1985), we consider a
situation where the licenser (the incumbent) commits to a capacity level before
production and saves the cost of capacity installation, i.e., , at the stage of
production. Thus, capacity commitment makes the licenser more cost efficient at the
z
11 Note that we have already assumed that the per-unit cost of capacity, , is sufficiently small to z
11
stage of production and the cost of capacity becomes important for profitable
licensing. If the per-unit cost of capacity is sufficiently small, capacity commitment
does not provide much benefit to firm 1 and the conditions for profitable licensing are
more like the conditions shown in Katz and Shapiro (1985). But, given the difference
between and c , as increases, the benefit from capacity commitment increases
and makes licensing less likely to be profitable. It is clear from (5) that as
increases, the right hand side of (5) gets bigger and makes it less likely to satisfy
condition (5).
1c 2 z
z
z
Given that the industry marginal revenue is downward sloping, the per-unit
variable costs are sufficiently close and the per-unit cost of capacity is sufficiently
small, subgame perfect capacity installation is up to B′ and firm 1 licenses its
technology to firm 2. Hence, it creates excess capacity of ( )LB ′−′ in equilibrium.
Though, investment up to B′ increases the licensing fee, it creates waste of
. So, it remains to check whether firm 1 is willing to invest more than L′.LB )( ′−′ 12
Proposition 1 shows that ‘investment up to B′ and licensing’ dominates
‘investment up to B′ and no licensing’. So, given the capacity installation up to B′,
firm 1 is better off at L than at E. It is also well known that profit increases from its
Cournot-Nash equilibrium up to its Stackelberg leader equilibrium. Hence, firm 1 is
better off at E than at E′. Therefore, ex-post capacity installation, firm 1 is better off at
L with capacity installation up to B′ than at E′ with capacity installation up to L′. So,
ensure equilibrium under no licensing at E. 12 It is important to note that capacity commitment is helpful with the possibility of licensing if L′ is to the right of K. Otherwise, commitment up to L′ does not reduce reservation payoff of firm 2 compared to the Cournot-Nash equilibrium G. If the difference in variable costs is sufficiently small, it satisfies that L is to the right of G.
12
if firm 1 was to install capacity up to L′ in equilibrium, it would have to license its
technology ex-post capacity installation. So, investment up to L′ saves investment
costs of . But, investment up to L′ instead of B′ increases firm 2’s
reservation payoff from to
, where is the optimal output of firm 2 when
firm 1 produces at and
zLB )( ′−′
())(*2 zL −′
L
)())())((( *22
*2 BqczBqBP ′+−′+′
(.)*2)())(( *
22 LqcqLP ′++′
′
q
B′ respectively.
Proposition 2: Suppose condition (5) holds. Excess capacity occurs if and only if the
per-unit profit in firm 2 is greater than the per-unit cost of capacity, i.e.,
. zzcLqLP >−−′+′ ]))(([ 2*2
Proof: If firm 1 invests up to L′ instead of B′, its net gain is
. )]()))((()()))(([()( *22
*2
*22
*2 BqczBqBPLqczLqLPzLBH ′−−′+′−′−−′+′−′−′=
(8)
Under Cournot conjectures, the marginal gain to firm 1 by increasing capacity from L′
is
)())(( *2
*2 LqLqLPz
LH ′′+′′−−=′∂
∂ . (9)
Due to the profit maximization of firm 2 and after rearranging, condition (9) reduces
to
zzcLqLPLH
−−−′+′=′∂
∂ ]))(([ 2*2 . So, 0>
′∂∂LH if and only if
zzcLqLP >−−′+′ ]))(([ 2*2 . (9’)
Since [ , (9’) holds for sufficiently small . Q.E.D. 0]))(( 2*2 >−−′+′ zcLqLP z
13
The above result is very intuitive. If the cost of capacity is sufficiently small,
excess capacity is not costly to firm 1 but it helps firm 1 to extract a higher price for
its technology. Since, price must exceed the marginal cost of firm 2, i.e.,
, firm 1 invests up to B′ than up to L′ when the per-unit cost of
capacity is sufficiently small.
zcLqLP +>′+′ 2*2 ))((
Hence, combining Propositions 1 and 2, we get the conditions required for the
co-existence of licensing and excess capacity.
Proposition 3: Suppose firm 1 cannot credibly commit to its Stackelberg leader’s
output when both firms produce with their own technologies.
(i) Licensing and excess capacity occurs only if the industry marginal revenue is
downward sloping.
(ii) Licensing and excess capacity occurs if and only if (5) and (9’) hold.
The above results show the necessary and sufficient conditions for the co-
existence of licensing and excess capacity. If these conditions are violated, we can get
results similar to previous works. If the industry marginal revenue is not downward
sloping, it follows from Proposition 1 that licensing does not occur. However, when
the industry marginal revenue is not downward sloping, excess capacity occurs only if
it deters entry, as shown in Bulow et al. (1985).
Excess capacity does not occur if either condition (5) or condition (9’) does
not hold.13 In the event of the former, ex-post capacity installation up to B′, firm 1
does not license its technology to firm 2. Hence, firm 1’s profit at E is greater than its
profit both at L with licensing and at E′. Further, it is easy to understand that,
14
following capacity installation, firm 1’s profit under licensing is higher for capacity
installation up to B′ than for capacity installation up to L′, since the licensing fee will
be lower in the latter situation while the profit in the product market remains the
same. Therefore, in this situation, firm 1’s optimal capacity installation is up to B′ and
there will be no excess capacity in equilibrium, which is similar to Dixit (1980).
If licensing after capacity commitment is profitable but condition (9’) is not
satisfied, we again do not observe the co-existence of licensing and excess capacity. If
(9’) is not satisfied then firm 1’s marginal gain from increasing its capacity above L′
is negative and therefore, it has no incentive to install capacity above L′, given that
licensing is profitable after capacity installation. In this situation, firm 1’s optimal
capacity installation is up to L′. So, as the per-unit cost of capacity increases, it
reduces the possibility of excess capacity simultaneously. Though, here excess
capacity does not occur in equilibrium, the possibility of licensing after capacity
installation induces firm 1 to reduce its capacity installation compared to the situation
without licensing, which is again similar to Dixit (1980).
Hence, the following corollary is immediate from the above discussion.
Corollary 1: Even if the industry marginal revenue is downward sloping, we are less
likely to observe excess capacity for relatively larger cost of capacity (i.e., ). z
We have considered licensing with up-front fixed-fee only, which is the
optimal licensing contract when the licensee (firm 2) can imitate the licensed
technology costlessly (see, Rockett, 1990). On the other hand, if imitation is not a
credible threat, following Rockett (1990) and Mukherjee and Balasubramanian
13 As the cost of capacity increases, it is less likely to satisfy (5) or (9’).
15
(2001), we find that it is optimal for the licenser (firm 1) to charge a per-unit output
royalty only and the optimal rate of royalty is . Hence, the reaction
function of firm 2 is similar under licensing and no licensing. Therefore, excess
capacity does not occur in equilibrium when imitation is not a credible threat. But, if
the threat of imitation is credible and imitation costly, it follows from the above-
mentioned two papers that the optimal licensing contract consists of royalty and up-
front fixed-fee, where the optimal royalty rate is less than . In this
situation, licensing shifts the reaction function of firm 2 rightwards and excess
capacity occurs in equilibrium. Though, the assumption of costless imitation helps us
to prove the results in the simplest way, the above argument implies that our
qualitative results hold as long as imitation is a credible threat.
)( 12* ccr −=
r )( 12* cc −=
2.2 Higher values of z
Now consider the implications of the situation where firm 1 can credibly commit to
its Stackelberg leader’s output when both firms produce with their own technologies.
This is shown in Figure 2.
Figure 2
AB shows firm 1’s reaction function before its capacity installation and CD firm 2’s
reaction function without licensing. If S" is firm 1’s Stackelberg leader’s output
without licensing, firm 1’s optimal capacity installation is up to B" without licensing.
If licensing is not profitable after capacity installation, the equilibrium is at S". But, if
licensing is profitable after capacity installation, it shifts firm 2’s reaction function to
C"D".14 Following Proposition 1, we can find the condition required for profitable
14 We consider that licensing occurs with up-front fixed-fee.
16
licensing with the exception that now the output of firm 1 does not change with c .
To avoid repetition, we are omitting the details here.
2
15 However, it is clear from
Figure 2 that excess capacity does not occur in equilibrium since firm 2’s reaction
function after licensing intersects the vertical segment of firm 1’s reaction function
A"B". So, even if licensing is profitable, we find that here excess capacity does not
occur in equilibrium.
If S"', rather than S", is firm 1’s Stackelberg leader’s output without licensing,
firm 1’s optimal capacity installation is up to B"' without licensing. If licensing occurs
after capacity installation, it shifts firm 2’s reaction function to C"D". In this situation,
excess capacity of L"'B"' occurs in equilibrium.16 We need the per-unit cost of
capacity to be sufficiently small for the co-existence of licensing and excess capacity.
However, now we are considering the situation for relatively higher per-unit costs of
capacity. So, it is less likely to satisfy the conditions required for the co-existence of
licensing and excess capacity, and it may be more likely to observe capacity
installation up to L"'.
So, there are two factors that make excess capacity less likely when firm 1 can
credibly commit to its Stackelberg leader’s output when the firms produce with their
own technologies. As the cost of capacity increases, it increases the value of waste
due to excess capacity and makes it less likely to occur. Further, higher cost of
capacity installation gives firm 1 a large strategic advantage in the product market due
to its pre-commitment to the capacity installation. In this situation, even if firm 2 gets
a license and becomes more cost efficient in the product market, it does not affect
15 Here licensing is profitable provided
2
212 c
qPqq
∂∂′> .
16 Since the procedures of Propositions 1 and 2 can be used to find the required conditions for the co-existence of licensing and excess capacity, we are omitting details to avoid repetition.
17
firm 1’s pre-committed output level and does not generate excess capacity in
equilibrium.
So, the implication of higher cost of capacity installation for which firm 1 can
credibly commit to its Stackelberg leader’s output without licensing is immediate and
is given in the following proposition.
Proposition 4: If the per-unit cost of capacity installation is such that firm 1 can
credibly commit to its Stackelberg leader’s output when both firms produce with their
own technologies, the co-existence of licensing and excess capacity is less likely to
occur.
Thus, the analysis of this section shows that the co-existence of licensing and
excess capacity is more likely to hold in industries with relatively lower cost of
capacity. This finding may provide a testable hypothesis.
3. Capacity commitment by the entrant
We have so far assumed that the entrant (firm 2) cannot install capacity prior to
production. However, as pointed out by Ware (1984), capacity installation by the
entrant may affect the incumbent’s optimal capacity installation and may change
industry profits. In this subsection we extend the model of the subsections 2.1 and
2.1.1 to incorporate capacity installation by firm 2.17 We will consider two scenarios:
(i) where firm 2 installs capacity after the decision on licensing, and (ii) where firm 2
17 We have shown that licensing and excess capacity is more likely to observe in the situations considered in these subsections.
18
installs capacity before the decision on licensing. So, our analysis in this section is a
useful check on the robustness of our results.
3.1 Firm 2’s capacity installation after licensing
We consider the following game. At stage 1, firm 1 invests up to capacity level x . At
stage 2, firm 2 enters the market. At stage 3, firm 1 decides whether or not to license
technology to firm 2. At stage 4, firm 2 installs capacity. At stage 5, production takes
place and profits are realized. We again solve the game through backward induction.
Since firm 1 cannot credibly commit to its Stackelberg leader’s output when
the firms produce with their own technologies, it follows from Ware (1984) that, if
firm 2 commits to the capacity level before production, equilibrium under no
licensing is not at E of Figure 1 but it is somewhere between G and E of Figure 1.18
Let us consider Figure 3 where we allow firm 2 to install capacity before
production.
Figure 3
Assume that equilibrium under no licensing is at E". So, firm 1 and firm 2 produce B"
and B"E" respectively. Industry profit under no licensing is
)()())())((( *221
*2
*2 BqczBcBqBBqBP ′′+−′′−′′+′′′′+′′ . (10)
Now, assume that firm 1 licenses its technology ex-post capacity installation
up to B". So, firm 2’s marginal cost of production after licensing is . Since, firm
2 can install capacity before production, it will try to commit its capacity level to its
Stackelberg leader’s output with its marginal cost of production and firm 1’s
marginal cost of production is c . In other words, firm 2 will choose its capacity level
1cz +
1cz +
1
18 Refer to Ware (1984) for the details.
19
to maximize its profit given the reaction function A'OB". However, firm 2’s
Stackelberg leader’s output with its marginal cost of production and firm 1’s
marginal cost of production c is greater than firm 2’s Cournot-Nash output when
both firms’ marginal cost of production is . Hence, firm 2’s credible capacity
commitment cannot exceed the value corresponding to point W, i.e., J. So, ex-post
licensing, capacity installation by firm 2 generates the equilibrium in the product
market at W. Hence, under licensing, firm 1 produces
1cz +
1
1c
I and firm 2 produces
corresponding to its marginal cost of production , i.e., .
)(*2 Iq
1c
)I
IW
(*2q)) −
z
)) − (*2q I
2c
z
Therefore, industry profit under licensing is
)(())((( 11*2
*2 czIcIqIIqIP +−++ . (11)
Now, we are in a position to examine whether licensing is profitable to the
firms and capacity installation up to B" is better for firm 1 compared to its capacity
installment up to I.
Proposition 5: If the own technologies of the firms are sufficiently close, licensing is
profitable when 2
2
cqP∂∂′>1 and is sufficiently small.
Proof: The industry profit at W is
(12) )(())((( 21*2
*2 czIcIqIIqIP +−++ )
with . Differentiating (12) with respect to and following the procedure used
in Proposition 1, we can show that, if c and c is sufficiently close and is
sufficiently small, industry profit over WO in Figure 3 reduces with higher if and
only if
12 cc =
2 1
2c
20
)(12
1
2
2
cq
cqP
∂∂
+∂∂′> . (13)
Since 02
1 >∂∂cq , (13) holds if
2
2
cqP∂∂′>1 .19
Now consider the effect of on the industry profit over B"E". If is very
small and, and is sufficiently close then q
2c z
1c 2c 21 q≈ . In this situation, we find that
industry profit reduces with higher if and only if 2c
2
21cqP∂∂′> . (14)
Q.E.D.
Proposition 6: Capacity installation more than I makes firm 1 better off when the
per-unit cost of capacity is sufficiently small.
Proof: If firm 1 invests up to I instead of B", its net gain is
. )]()))((()()))(([()( *22
*2
*22
*2 BqczBqBPIqczIqIPzIBH ′′−−′′+′′−−−+−−′′=′′
(15)
Following the proof of Proposition 2, it is easy to show that 0>∂′′∂
IH , when is
sufficiently small. Q.E.D.
z
Combining Propositions 5 and 6, we get the following result immediately.
19 It follows from Proposition 1 that (13) holds if the industry marginal revenue is downward sloping.
21
Proposition 7: Assume that firm 2 installs capacity after licensing and prior to
production. Licensing and excess capacity co-exist when 2
2
cqP∂∂′>1 , the own
technologies of the firms are sufficiently close and is sufficiently small. z
Proposition 7 shows that our qualitative result of subsections 2.1 and 2.1.1
showing the co-existence of licensing and excess capacity is robust even if firm 2 (the
entrant) installs capacity after licensing and before production.
3.2 Firm 2’s capacity installation before licensing
Now, we consider another possibility where firm 2 installs capacity before licensing.
So, the game is as follows. At stage 1, firm 1 invests up to capacity level x . At stage
2, firm 2 enters the market. At stage 3, firm 2 installs capacity. At stage 4, firm 1
decides whether or not to license the technology to firm 2. At stage 5, production
takes place and the profits are realized.
It is easy to understand that if there is no licensing after capacity installation
of firm 2, both firms’ capacity installation is like subsection 3.1. Given the capacity
installation by firm 1 up to B", firm 2 has no incentive to change its capacity
installation from Y, since any deviation reduces firm 2’s payoff under no licensing.
Since, under licensing, firm 2 receives its reservation payoff, firm 2 installs capacity
to maximize its reservation payoff. Hence, given that firm 1’s capacity installation is
up to B", firm 2’s optimal capacity installation is up to Y. Further, since, under no
licensing, firm 2’s reservation payoff is minimized when firm 1 installs capacity up to
B", firm 1 also has no incentive to install capacity different from B". So, firm 1
22
installs capacity up to B", firm 2 installs capacity up to Y and the equilibrium is at E"
(see Figure 4).
Figure 4
Now, assume that firm 1 licenses its technology ex-post capacity installation
by both firms. If licensing ex-post capacity installation shifts the reaction function of
firm 2 in a way to create equilibrium between W and O then firm 1 has excess
capacity in equilibrium. This is shown in Figure 4 with the reaction function
D"UM"N" of firm 2 and the equilibrium is at W". But, if equilibrium after license is
between E" and O, say at W"', as shown in Figure 4 with the reaction function
D"UM"'N"' of firm 2, firm 1’s capacity is fully utilized and there is no excess
capacity.20 So, if firm 2 installs capacity before licensing, it reduces its flexibility
about capacity installation and production and may help firm 1 to utilize its full
capacity.
The above argument however assumed that licensing is profitable after
capacity installation and firm 1 has no incentive to install its capacity in a way to
eliminate excess capacity in equilibrium, if ex-post licensing, equilibrium is between
W and O. It can be shown, following the arguments of subsections 2.1.1 and 3.1, that
both assumptions are satisfied when the per-unit cost of capacity installation is
sufficiently small and the own technologies of the firms sufficiently close. To avoid
repetition, we are omitting the details here.
Hence:
20 Difference between and is important in determining whether the equilibrium will be either on the segment OW or on the segment B"O.
z )( 12 cc −
23
Proposition 8: Licensing and excess capacity may co-exist even if firm 2 installs
capacity before licensing.
3.3 Timing of licensing
The previous two subsections have considered different timing of capacity installation
by firm 2. If firm 1 licenses its technology prior to firm 2’s capacity installation,
excess capacity occurs in equilibrium. But, if firm 1 licenses its technology after firm
2’s capacity installation, equilibrium is different from the previous situation and firm
1 may not have excess capacity in equilibrium. If firm 1 can choose the timing of
licensing, the natural question is to ask whether it prefers to license its technology
before or after firm 2’s capacity installation.
Proposition 9: Suppose firm 2 installs capacity before production and the condition
for profitable licensing holds. If the cost of capacity (i.e., ) is sufficiently small, firm
1 prefers to license its technology after firm 2’s capacity installation.
z
Proof: The profits of the firms under no licensing are same irrespective of whether
firm 1 licenses its technology before or after capacity installation of firm 2. So, firm 1
prefers that situation which gives it higher profit under licensing.
Assume that firm 1 experiences excess capacity irrespective of firm 2’s timing
of capacity installation. Hence, we find from Figures 3 and 4 that firm 1’s profit under
licensing is higher if it licenses after firm 2’s capacity installation than if it licenses
before firm 2’s capacity installation provided21
21 Note that if firm 1 licenses its technology after firm 2’s capacity installation then firm 2’s cost of capacity installation is sunk at the time of licensing and hence, is not included in firm 2’s reservation payoff.
24
,0)(
))(()())((
1
111
>+++++−′′′′+−′′−′′′′+′′′′′′+′′
IWczIcIWIIWIPWIczIcWIIWIIP
(16)
or equivalently
0])())(([
2
21112121 <∂
+−−++∂q
qczqcqqqqP . (17)
Given that is sufficiently small, which also helps to make licensing profitable,
condition (17) always holds. This implies that firm 1 prefers to license its technology
after rather than before firm 2’s capacity installation.
z
Following similar logic, we can show that even if firm 1 does not have excess
capacity in equilibrium when it licenses after firm 2’s capacity installation, it prefers
to license its technology after firm 2’s capacity installation. Q.E.D.
The reason for the above proposition is as follows. If firm 1 licenses before
firm 2’s capacity installation then firm 2 has more flexibility about its output level
compared to the situation where firm 1 licenses after firm 2’s capacity installation.
So, firm 1 can induce firm 2 to produce relatively lower amount of output if it licenses
after firm 2’s capacity installation. This, in turn, increases firm 1’s output and profit
in the product market. Further, licensing after firm 2’s capacity installation increases
firm 1’s capacity utilization and reduces the cost of excess capacity, however small it
is. So, higher strategic advantage in the product market makes firm 1 better off if it
licenses after firm 2’s capacity installation compared to the situation where it licenses
before firm 2’s capacity installation.
25
4. Conclusions
Researchers have paid a considerable amount of attention to the existence of excess
capacity by dominant firms. Though earlier theoretical contributions have argued that
entry deterrence is the motive for holding excess capacity, empirical analysis does not
provide much support for this. More recent work shows that the possibility of variable
demand or price competition in the product market may be the reasons for holding
excess capacity even if it does not deter entry.
This paper provides a new rationale for holding excess capacity. We show that
the dominant firms may hold excess capacity not to deter entry but to get more benefit
from technology licensing. In a model with an incumbent and an entrant we show that
the incumbent firm invests in capacity that it ends up not using completely and the
rationale for this type of investment may be to extract a higher price for its superior
technology when licensing it to the entrant. We show that this result is robust with
respect to the possibility of capacity installation by the entrant. Further, we find that
when the entrant has the option for capacity installation, the incumbent’s profit is
higher when it licenses its technology after the entrant’s capacity installation than
when it licenses before its capacity installation.
26
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29
Figure 1: When firm 1 cannot credibly commit to its Stackelberg leader’s output and
firm 2 does capacity installation and production at the same time
Figure 2: When firm 1 can credibly commit to its Stackelberg leader’s output and
firm 2 does capacity installation and production at the same time
30