Asymmetric Firms and their Willingness to Compete
Preliminary Version
Clemens Fiedler∗
Tilburg University, CentER, TILEC
February 18, 2016
Abstract
In this paper, we present a duopoly model of customer loyalty. Two firms compete in
research and development efforts. Efforts increase the firms’ number of customers while
incurring a cost that depends partially on the number of customers. We show how the
this cost structure can generate a inverted u-shaped reaction function of the firms’ efforts.
We derive conditions under which welfare is (not) raised by supporting the laggard and
show how this can help to understand the market for operating systems.
Keywords: Innovation, Competition, Loyalty
JEL Classification: D21, L11
∗
CentER, TILEC, Tilburg University,
[email protected]
P.O. Box 90153,
1
5000 LE Tilburg,
The Netherlands;
1
Introduction
Innovation and the incentives of firms to invest in research and development are of fundamental interest to society. Innovation is the engine that provides ever new improvements
to the well-being of customers and creates stepping stones for other industries to build on
and develop products never before considered possible. This is true today just as it was in
the last century when Willig et al. (1991) pondered the link of innovation and competition.
1 But how does market structure impact the firms’ incentive to create these innovations?
Might we be stalling innovation by regulating markets ignorant of the consequences of our
meddling?
The firms incentives to innovate are driven by two main aspects: Stealing market share
from competitors and obtaining higher profits from current customers. Ignoring dynamic
consideration, this shows that a monopolist tends to innovate less, as an innovation does
not help to steal market share from competitors. Competing firms tend to innovate more,
as gaining the edge over competitors can increase profits by generating additional revenue
from each customer who is already purchasing from the firm and stealing market share
from competitors.
These aspects are even more important in the modern consumer electronics market.
Take the market for smartphone operating systems. Here chiefly two firms compete with
each other Google with Android and Apple with iOS. Only Apple derives profits from
hardware sales, but both firms profit from app sales and advertisement. According to
documents leaked from an IP trial between Google and Oracle Inc, since 2008 Google
has obtained profits of 22 billion USD through both channels.2 On the other hand Apple
reported an Revenue of 32 billion USD in the third quarter of 2015 in hardware IPhone
sales. While it does not separately report the revenue from app sales total service profits
amount to 10 billion USD.3
According to Willig et al. (1991), an important paper on the US merger guidelines,
concentration is seen as highly suspicious. An important criterion to analyze the danger
imposed by a merger is the Herfindahl-Hirschman Index (HHI), which is the sum of the
squares of all market shares. The HHI is increasing in the degree of asymmetric market
shares.4 Furthermore, mergers between small firms, resulting in small changes of the HHI,
are treated more leniently.
Similarly, in duopoly models it is often the case that firms innovate more if they are
equal in strength and less if they are asymmetric. We tend to consider innovative efforts
as strategic substitutes - similar to output in Cournot competition. If one firm increases
its research efforts, it erodes the other firm’s incentives. Even worse this can lead to a
monopolization in the long run based on some initially small advantage of one firm.
1
”[It] is a commonplace in today’s economy that innovation is an important battleground for competition,
and it seems evident that market power and efficiencies based on asset holdings play significant roles in shaping
its contours and vigor.” Willig et al. (1991, p. 312)
2
Bloomberg http://bloom.bg/1PrmPTu, accessed 22/01/2016
3
Quartz P
http://qz.com/600416/,
accessed
23/01/2016
P
4
HHI = i∈N s2i = N1 + i∈N s2i − N12
2
This is often seen as evidence that a higher level of symmetry in the firms’ abilities
to engage and profit from research and development maximizes the incentives to innovate
and that regulators should reduce the possibility of strong asymmetries. Even more
importantly, competition authorities are well known to aim for symmetric market shares
and treat firms who compete with similar sized firms more leniently than if competing
with smaller firms.
Contrary to that Bloom et al. (2013) show empirically that while the technological
spillover benefits amount to twice the private benefits “[. . . ] smaller firms have significantly lower social returns because they tend to operate in technological niches [. . . ].”
Which grants a reason to focus on encouraging market leaders.
I will argue in this paper that the regulation of asymmetric markets might be even
more problematic than the literature leads us to believe. The efforts of firms can be
complements and substitutes on the same market depending on the relative strength.
I will argue that the existence of customer loyalty together with not-perfectly scaling
costs, requires a careful intervention policy. Supporting the weak firm can raise total
efforts exerted if the weak firm is slightly inferior to its stronger competitor, but might
also reduce total efforts if the laggard is strongly inferior. Consequently, even when only
considering the short run implications of a regulatory intervention, aiming for perfect
symmetry in such markets might reduce the incentives to innovate directly.
By creating a simple model that nests both strategic substitutes and complements I can
illustrate the difference between Bertrand and Cournot competition. A deeper mechanism
lies underneath that leads to the difference in firm behavior. I will elaborate how this can
enter the decision making of firms in a market that behaves more like a Bertrand or more
like a Cournot market. Do firms harm each other’s incentive to innovate or encourage
each other?
Furthermore, I will discuss how the underlying parameters effect different markets.
What markets are more prone to have firms raise their efforts or lower them in response
to their competitor’s expanses. Especial the different impact that market conditions have
on firms of different sizes is interesting as it can help to better understand the incentives
of leaders and laggards,
Finally, I will connect the model with the smartphone operating system market and
illustrate how the unique situation there gives raise to non-monotonic reactions of the
firms and how best to address this issue.
2
Literature
The model outlined here is similar in spirit to Caselli et al. (2015). The authors discuss
conflicts between two countries and the effect an asymmetric allocation of resources has
on the escalation of conflicts. They come to the conclusion that the existence of natural
resources and their geographical position has a drastic impact on the escalation likelihood.
If only one country possesses natural resources, conflict becomes more likely, the closer the
resources are to the border. If both countries possess natural resources conflict becomes
more likely if the positioning relative from the border is asymmetric. I.e. war is more
3
likely if one country has a resources in its safe hinterland, while the other country has its
resources close to the border.
Their model is similar to ours, but instead of natural resources and countries’ borders,
we look at the number of customers and the separation of market shares. The main
difference is that in their model the outcome of the conflict is uncertain, while in the
model presented here, it is determined result of the efforts of both parties.
Ishida et al. (2011) show how and increase in competition might have surprising effects
on firm profits and outcomes, when asymmetries are at play. They analyze a oligopolistic
market with high and low costs firms and shows how the entry of additional high costs
firms might stimulate R&D by low cost firms and reduce it by the high cost firms. Further
more it raises the profits of the low cost firms.
Closely related is Salant and Shaffer (1999, AER). The authors show two interesting
facts about the Cournot competition using a two-stage model with a R&D stage followed
by a standard quantity competition: First, a change to the constant marginal costs of
firms leaves aggregate output unchanged as long as the sum of all marginal costs remains
constant and all firms remain in the market. Second, an increase in the variance of the
costs (with the mean remaining constant) leads to a shift of production from high cost
to low cost firms and decreases aggregate production costs. Thus an increase in the
cost spread that leaves the mean cost unchanged leads to higher industry profits. If all
firms remain in the market the consumer outcome remains unchanged and total welfare
increases with asymmetry.
Bloom et al. (2013) provide an empiric analysis of firms innovative efforts, their private
benefits and spill-over effects. They find that the externalities of innovation are about
twice the private benefits and underinvestment is likely.
Another possibility how asymmetry can impact the competition intensity in a market
is the waterbed effect. Inderst and Valletti (2011) show that a powerful buyer might
negotiate a better price with its supplier. The latter is then forced to charge a higher
price from other firms. This leads to a snowballing where initial asymmetries are amplified.
3
Model
Consider the market for smartphones. Two firms i ∈ {1, 2} compete over customers with
heterogeneous preferences. We are interested in competition in R&D efforts, thus any
price competition is abstracted away from. Devices are sold at an exogenously given price
such that firm i derives a profit of γi from each customer.5
Demand is given by a continuum of customers with unit mass and a heterogeneous
level of loyalty towards one of the brands. Customer k preference is given by θk ∈ Θ.
θk 0 implies that customer k has a strong preference for the device sold by firm 2,
θk 0 that she has a strong reference for product 1 and θk = 0 that she is indifferent
between the firms. Preferences are exogenously determined and distributed according to
the cdf G(·) and the pdf g(·).
5
Think of prices being determined by the retail market.
4
Assumption 1 G(·) is twice continuously differentiable on Θ such that g(·) is continuously differentiable and the distribution lacks atoms.
Preferences are exogenously. They are non-marketable characteristic of the products such
as past experiences of the customer with the firms. This can be used as the starting point
for a dynamic model.
Firms compete in R&D efforts to increase the attractiveness (or quality) of their
own products which steals customers from their competitors. The profits derived from
each customer (γi0 (xi )) are weakly increasing in the efforts such that: γi0 (xi ) ≥ 0. R&D
can lowers the manufacturing and distribution costs or generates a higher return from
improved third party services. Furthermore, profits are concave γi00 (xi ) < 0. Consider
a the development of a chip with higher transistor density. The chip is faster, enabling
better uses, but it also has lower variable costs.
Customer k compares the utility presented by both choices and purchases product
2 if and only if θk ≥ x1 − x2 . By assumption 1 ties occur with measure 0. We define
the indifferent customer as θ∗ ≡ x1 − x2 . If firm 1 provides a product superior to its
competitor’s, only customers highly loyal to firm 2 purchase from firm 2. The demand for
firm 1 is q1 = G(x1 − x2 ).
Example 1 Consider the sector for (smartphone) operating systems and other platform
services. Here the suppliers of operating systems - chiefly Apple, Google and Microsoft provide a platform use by other firms to supply end users. Improving the operating system
not only increases their market share but also generates revenue from third party sales.
As third parties develop better applications, the share in app sales that the OS suppliers
receive, grow.
The unique characteristic of this model are the costs of R&D:
Ci (x1 , x2 ) = ci qiα (x1 , x2 )x2i /2
∂Ci (x1 , x2 )
= ci qiα
∂xi
∂qi
x2 ∂x
α i i + xi
2 qi
!
The costs depend on the efforts and the market share of firm i. α ∈ [0, 1] measures how
directly costs scale with the number of customers. If α = 0, costs are independent of the
number of customers - e.g. an improvement to the code can easily be implemented on
every device. Conversely, for α = 1 R&D directly raises the unit costs - e.g. licensing
a new chip from a third party for a fixed unit price. In between, α ∈ (0, 1) and efforts
partially scale with quantity.6
6
Think of Apple adding a piece of functionality to iOS. The initial development is quite expensive, and
adjusting the code for different hardware requires further investment. As the number of devices compatible
increase adjustment becomes cheaper.
5
Combining everything gives the profit functions:
πi (x1 , x2 ) = qi (x1 , x2 )γi (xi ) − qiα (x1 , x2 )ci x2i /2
(1)
q1 (x1 , x2 ) = G(x1 − x2 )
q2 (x1 , x2 ) = 1 − G(x1 − x2 )
Without loss of generality we normalize the costs to ci = 1 for i ∈ {1, 2} by setting
γi (xi ) ≡ γi (xi )/ci and ci ≡ 1
3.1
Optimal Firm Behavior
C
Let xC
i be the optimal effort level under competition for firm i and let θ be the indifferent
customer under the optimal effort levels. The first and second derivatives are given as:
∂qi (x1 , x2 ) ∂πi (x1 , x2 )
(α−1) 2
=
γi (xi ) − αqi
xi /2 + qi γi0 (xi ) − qiα xi
∂xi
∂xi
2
2
∂ πi (x1 , x2 )
∂ qi (x1 , x2 ) (α−1) 2
=
γ
(x
)
−
αq
x
/2
i
i
i
i
∂ 2 xi
∂ 2 xi
∂qi (x1 , x2 )
α(1 − α) (α−2) 2 ∂qi (x1 , x2 )
(α−1)
0
2γi (xi ) +
qi
xi
− 2αqi
xi
+
∂xi
2
∂xi
− qiα (x1 , x2 ) + qi (x1 , x2 )γi00 (xi )
For firm 1 this expression becomes:
∂π1 (x1 , x2 )
= g(x1 − x2 ) γ1 − αG(x1 − x2 )(α−1) x21 /2
∂x1
− G(x1 − x2 )α x1 + G(x1 − x2 )γ10 (x1 )
extensive margin
intensive margin
An increase in firm i’s efforts helps it twice: It attracts customers from firm j (extensive
margin) and it raises the return form customers already purchasing from it (intensive
margin).
The first order condition gives an implicit solution for the firms problem. For simplicity
we assume that firms efforts are such that both firms remain in the market: qi > 0 for
i ∈ {1, 2}.
Lemma 1 (Shared Market) Let Θ = [−a2 , a1 ] be the support of G(·) with ai > 0 for
i ∈ {1, 2}. If
γi0 (ai ) − ai ≤ 0
a2
lim g(a) γ1 (a1 ) − 1 + γ10 (a1 ) − a1 ≤ 0
2
a→a−
1
a22
lim g(a) γ2 (a2 ) −
+ γ20 (a2 ) − a2 ≤ 0
+
2
a→−a2
both firms command a positive market share in equilibrium.
6
Proof: First, for ai = ∞ it is trivial to show that this is fulfilled. If the inequality is
satisfied for ai it is satisfied for all a0i > ai as g(a0i ) = 0 and γi00 (·) < 0. If firm j provides
zero efforts and firm i provides enough effort that qi = 1 the latter finds it profitable to
lower its efforts. An increase in xj has the same affect as an increase in a1 , which lowers
the RHS of the inequality by
!
0
00
lim g(a) γ1 (a1 ) − a1 + γ1 (a1 ) − 1 dxj
a→a−
1
Thus, the inequality is enough to guarantee xi < 1 for all xj .
Lemma 2 (Quasiconcavity) For g 0 (·) = 0 and γ 000 (xi ) < 0 the profit function is quasi
concave.
Proof : Ci (x, 1, x2 ) is convex in xi as it is the product of a convex function with an
increasing function. Thus, the profit function can only be convex if qi (xi , xj )γi (xi ) is
convex which requires:
2g(xi − xj )γi0 (xi ) > −qi (xi , xj )γi00 (xi )
|
{z
} |
{z
}
>0
<0
As γi0 (xi ) is decreasing in xi so is the LHS. The RHS is increasing in xi . Thus, if
the inequality is violated for one xi it is violated for any x0i > xi . Furthermore as the
inequality is strict it also holds for γ 000 (·) = 0 + ε for some small ε.
In-fact lemma 3.1 can be relaxed, as shown by figure 1, showing the profit function for
different values of γi (xi ). 7
1 ,x2 )
Theorem 1 (Existence) If ∂qi (x
> 0, γi00 (·) < 0 and if the requirements of Lemma
∂xi
1 are satisfied, a pair (x1 , x2 ) exists in which both firms set their efforts to maximize their
profits given their competitors choice with xi , qi > 0 for i ∈ {1, 2}.
Proof: By design the efforts x1 , x2 ≥ 0. Furthermore as γi00 (·) < 0 and q1 , q2 ∈ [0, 1] a
x exists such that ∀xi > x : γi0 (xi ) < xi and γi (xi ) < x2i /2. Thus the best response for
the players is bound between [0, x]. The profit functions are continuously differentiable
for interior solutions and the best response functions are continuous. Thus, by Brouwer’s
Fixed Point Theorem a Nash Equilibrium exists.
By Lemma 1 market shares are qi ∈ (0, 1). Thus for xi = 0, profits are increasing in
xi and for xi = x profits are decreasing in xi .
The optimal efforts are given by:
∂qi 1 (α−1) 2
1−α
0
xC
=
q
γ
−
αq
x
/2
+
γ
(x
)
(2)
i
i
i
i i
i
i
∂xi qi
7
The parameter values are x2 = 0 and α = 0.
7
1.5
p
π1 (x1 ,x2 )
1.0
γ(x) =0.1 x +1.0
0.5
p
γ(x) =0.1 x
γ(x) =1.0
0.0
0.5
0.0
0.5
1.0
x1
1.5
2.0
Figure 1: Profit function for different γ(x)
Depending on the shape of G(·) the game might feature multiple equilibria. To avoid
having to assume uniqueness the following discussion focuses only on local changes.
Efforts depend
on
three effects. First, efforts are the higher, the more customers are
∂qi 1
under threat ∂xi qi . This effect is more powerful for the laggard than for the leader
who has less customers to gain relative to their share.
Second, the higher γi , the greater are the efforts of firm i. A firm that derives a
higher profit from each customer has a higher incentive to exert effort. As γi0 (xi ) > 0 this
augments asymmetry. The efforts also depend on γ 0 (xi ). If the profits per customer react
strongly to efforts, firms will use higher efforts to raise them.
Third, if the costs per customer increase strongly with the number of customers firms
will provide less efforts. For low market shares this decreases in α while it increases
for high market shares. Consequently, a low α can boost an initial asymmetry in value
extraction.
Figure 2 shows the equilibrium efforts for a two different levels of asymmetry in the
profits per customers, such that γ2 (·) = l2 γ2 (·) for l2 > 1. In general efforts are increasing
in α and the leader exerts more efforts than the laggards. However, for high values of α
the market leader exerts less efforts against a weak opponent than against a only slightly
weaker opponent. The cost structure works against asymmetry. For low αs this is the
other way around.
The driving mechanism is the change to the costs as a function of the marketshare.
For a low α the costs of the leader change little with changes to quantity.
8
0.16
0.14
0.12
Leader - high asymmetry
Leader - low asymmetry
x1C
0.10
0.08
Laggard - low asymmetry
0.06
Laggard - high asymmetry
0.04
0.02
0.0
0.2
0.4
α
0.6
0.8
1.0
Figure 2: Efforts for high and low levels of asymmetries
4
4.1
Comparative Analysis
Baseline: Uniform Distribution
First, consider the most basic case. Customers are distributed uniformly on [−k, k] with
∂qi (x1 ,x2 )
1
= g(θ) = 2k
≡ g̃ and g 0 (·) = 0. Secondly, the quality does not effect the value
∂xi
extracted per customers (γi0 (xi ) = 0) and costs scale one-to-one with quantity (α = 1).
For simplicity we define γi (xi )) = li γ(xi ) with l1 = 1. The optimal efforts are:
s
2
qi
qi
C
xi = 2li γ +
−
(3)
g̃
g̃
The implications are standard. An increase in qg̃i leads to more customers at the margin
and intensifies competition. An increase in the efforts of firm j reduces firm i’s market
share, making its efforts cheaper and raising xi . Efforts are strategic complements, similar
to Bertrand competition.8
The other extreme is α = 0 and li γ 0 (xi ) > 0, which gives a market that behaves like a
Cournot market. The optimal efforts of firm i become:
g̃
γ0
C
+
(4)
xi = qi li γ
qi
γ
8
In Bertrand competition a decrease of the competitors price leads to a reduction in the number of customers
makes price cuts less expensive and causes the firm to lower its price.
9
0.12
Leader - α = 0
0.10
Leader - α = 1
x1C
0.08
0.06
Laggard - α = 1
0.04
Laggard - α = 0
0.02
0.00
0.0
0.2
0.4
0.6
0.8
1.0
l2
Figure 3: Effect of Asymmetry
Again the efforts are increasing in both, the share of customers under threat and the
return generated from each customer. Efforts are also increasing in the market share as
they raise the return per customer. The market behaves like a Cournot market as efforts
are strategic substitutes.9
To sum up: γ 0 (xi ) adds substitutionary (submission) pressure, while α adds complementary (escalation) pressure. If a firm can increase the return from its product by
increasing its quality, any reduction of its market share leads to a decline of efforts.
If a firm has less to lose as it falls behind its competitor, efforts act like strategic
substitutes. α determines if a firm becomes more aggressive (α = 1) as it loses market
share or if the firm becomes more complacent (α = 0).
For completeness, if α = 0 and γ 0 (xi ) = 0 the efforts are xC
i = g̃γi . Firms still compete
but their efforts are independent.10
Figure 3 illustrates how different levels of α affect the outcome on an asymmetric
market. The γi (·) was scaled so that the symmetric outcome is the same. In both cases
the efforts of the laggard are increasing in l2 and are lower for α = 0. More interestingly
for α = 0 the efforts of the leader are decreasing in l2 , for α = 1 they are increasing.
The two extreme cases help to understand the mechanics, but more interesting are
intermediate cases with α ∈ (0, 1) and γi0 (xi ) > 0 in which case both pressures exist. The
optimal efforts are:
9
In a Cournot market if the competing firm raises its output it lowers the price for the other firm which
leads to it reducing its output.
10
Conditional on the parameters of the model.
10
xC
i =
g̃ 1−α
α qi li γ(xi ) − x2i + qi1−α li γ 0 (xi )
qi
2
(5)
Multiple equilibria can exist. If firm i exerts high efforts it raises the costs for firm j
who is then limited to exert less efforts. Thus, even for symmetric conditions asymmetric
C
equilibria can exists. For xC
i → ∞ the RHS is negative, for xi = 0 it is positive thus it
C
must intersect an odd number of times with xi . The RHS can be increasing in xi but
not around the optimal choice.
∂ 2 πi (x1 , x2 )
= li 2γ 0 (xi ) + q α
∂ 2 xi
α(1 − α)
2
xi
qi
2
For α = 0 and α = 1 the optimal efforts become:
g̃
γ0
C
xi = qi li γ
+
qi
γ
g̃
1 2
C
xi =
li γ − xi + li γ 0
qi
2
xi
− 2α − 1
qi
!
+ qi li γ 00 (xi )
α=0
(6)
α=1
(7)
The reaction function of firm i is:
li γ 0 (xi ) C
αg̃xC
C
i
dxC
dxi
i − dxj +
qi
qiα
(−2)
(−1)
C
+ (1 − α)αqi x2i /2g̃ dxC
xi dxC
i − dxj − αqi
i
00 (x )
l
γ
li γ 0 (xi )g̃
i
i
C
dxC
+
dxC
i − dxj +
i
qiα
qiα−1
dxC
i =−
B
dxC
A+B j
xi
li γ 0 (xi ) li γ 00 (xi )
+ α−1
A=1+α −
qi
qiα
qi
!
2
xi (1 − α)α xi
li γ 0 (xi )
B = g̃ α −
−
qi
2
qi
qiα
dxC
i =
(8)
dxC
−li γ 0 (xi )
i
=
g̃
1 − 2li γ 0 (xi ) − qi li γ 00 (xi )
dxC
j
α=0
(9)
xi − li γ 0 (xi )
dxC
i
=
g̃
qi + 2xi − 2li γ 0 (xi ) − qi li γ 00 (xi )
dxC
j
α=1
(10)
11
0.30
0.25
Reaction dxxji
0.20
0.15
0.10
Leader
Laggard
0.05
0.00
0.05
0.10
0.0
different signed reaction
0.2
0.4
α
0.6
0.8
1.0
Figure 4: Reaction of firms
For α = 0 and γ 0 (·) > 0 this expression is negative and for α = 1 it is positive.11
Increasing γ 0 (·) raises xi and qi . Ignoring this effect, an increase of γ 0 (·) lowers the slope
C
of the reaction function. Furthermore, for any γ(·) and (xC
1 , x2 ) an α̃ exists such that the
reaction of firm 1 in equilibrium is negative for smaller α.
(
>0
dxi dxj xC ,xC < 0
i
j
∀α > α̃
∀α < α̃
(11)
Now imagine an exogenous shock reducing the efficiency of firm 1 in extracting value,
such that γ(·) is increased by a factor li such that γi (xi ) = γ(xi )li and γi0 (xi ) = γ 0 (xi )li .
This directly leads to an increase in the efforts of firm 1 and its demand, which in turn
lowers the costs of firm 2 but reduces the incentive to increase the value extraction from
customers.
If α > α̃ the first effect dominates and firm 2 exerts more efforts. If firm 2 loses market
share it - paradoxically - becomes more competitive as the costs are reduced encouraging
it to exert more efforts.
Such a market is characterized by unit costs that directly depend on R&D and do not
depend on the quantity provided.
In the other case firms suffer from high start-up costs of their research and development
programs. Increasing its efforts is quite expensive if only a small number of customers are
served, but gets increasingly cheaper as the customer base is expanded. Then the reaction
of the firms becomes negative and losing market shares raises the costs of research.
Figure 4 illustrates the reaction of the firms to an increase of its competitors efforts
for different levels of α. For small α the reactions of both firms are negative, for large
11
If xi − γi0 (xi ) was negative a reduction in xi would raise the profit per customer and thus the total profit.
12
values they are positive. However, for a small range of values the reaction is positive for
the laggard, but negative for the leader.
The reason for this is that the reaction depends on the market shares. For a large
market share losing customers only effects the costs of R&D by a small amount, while the
other channels do not impact firms differently. Thus for a small α the market leader uses
efforts substitutionary while the laggard acts complementary.
The sign of the reaction function depends on B as A + B is proportional to the second
order condition and thus has a positive sign.
From equation 5 we have that:
α
2
2
1 xi
li γ(xi )
li γ 0 (xi )
=−
+
+
g̃ qi
qiα
q 1−α
i 0
1 xi
li γ(xi )
li γ (xi )
<
+
1−α
g̃ qi
qiα
qi
xi
qi
Which substituted into the reaction function 9 gives:
li γ 0 (xi )
xi
1
li γ(xi )
−
(2
−
α)
B1 = α + (1 − α)
− (1 − α)
g̃
qi
qiα
qi1−α
2
α xi
γ 0 (xi )
xi
−
− li α
B1 = α
qi
2 qi
qi
For xi = 0 this expression is negative, for xi → ∞ it is negative. The market shares
impact the result for any α ∈ [0, 1] and a larger market share makes the expression smaller
and more likely to be negative.
Keeping efforts, quantity and the base return per customer (γ(·)) fixed, the sign depends on li γ 0 as it determines the importance of the intensives margin for efforts. Any
loss of customers erodes the effort incentives the stronger, the greater li is.
0
An increase in α has a more unpredictable consequence. li γ q(xα i ) is increasing in α,
2
while α xqii − α2 xqii
is increasing if positive and decreasing if negative. Consequently,
for a negative reaction an increase in α makes it more negative, for a positive reaction the
change can be negative or positive.
For small efforts an increase in xi makes the reaction more positive, for large efforts
an increase in xi makes the reaction more negative. A high effort level means steep effort
costs, thus firms are more willing to cut costs if possible.
Figure 5 shows efforts and reactions for different values of γ, l2 and g̃ for l1 = 1. Not
surprisingly the efforts of firm 2 are increasing in l2 (x-axis) and in g̃ (y-axis). However,
the contour lines of the efforts are similar for the values of α.
The efforts of firm 1 are increasing in g̃ and either increasing (α = 0) or decreasing
(α = 1) in l2 . This is seen one to one in the reaction of firm 1, that is negative for small
values of α and positive for large values.
13
0.559
0.528
0.528
0.497
0.45
0.497
0.40
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.40
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
(a) Efforts 1
(b) Efforts 2
l2
0.3
0.50
16
0.0
0
4
0.0
01
0.45
2
0.28
7
0.24
.213
079
0.10.076
51
0.14
0.11
0.247
0.042
-0.032
0.213
2
-0.004
-0.033
α =1.0
0.007
-0.031
4
α =0.08
0.55
-0.024
-0.018
-0.011
-0.005
-0.03
α =0.0
0.010
0.282
-0.0
36
0.60
0.316
37
α =1.0
-0.00
0.45
α =0.08
-0.007
0.50
-0.0
0.002 0
-0.00
0.55
α =0.0
l2
g̃
0.60
g̃
0.50
0.800
0.590
0.55
α =1.0
0.700
0.600
0.500
0.400
0.300
0.200
0.100
0.590
α =0.08
0.600
0.621
α =0.0
0.500
0.400
0.300
0.200
0.100
g̃
0.60
0.600
0.652
0.621
0.559
0.45
α =1.0
0.683
0.500
0.400
0.300
0.200
0.100
0.50
α =0.08
0.838.807
0
0.776 .745
0
0.714
0.68352
0.6 0.621
0.55
α =0.0
0.683
0.652
g̃
0.60
-0.0
0.40
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
03
0.40
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
(c) Reaction 1
(d) Reaction 2
-0.030
l2
l2
Figure 5: Comparison
14
The reaction of firm one is negative for small a α and positive for a large α. In between
the pattern is more complicated, interestingly for intermediate values of α the reaction
function changes non-monotone in l2 and reaches its maximum for an interior value of l2 .
For α = 0.08 the reaction function of firm 2 is dominated by the ratio of xq22 , which
leads to a inverse-U shaped reaction. If the efforts of firm 2 are high an increase in x1
reduces the market share.
4.2
Loyalty: Double Peaked Distribution
As discussed the market for smartphones exhibits a high decree of customer loyalty. According to a 2013 survey by WDS 76% of all IPhone customers continue to buy a new
IPhone. This percentage is 58% for Samsung - the manufacturer with the greatest sales
among Android phones.12 This shows that both customers staying with a firm and switching are relevant in numbers.
0.30
0.25
g( θ )
0.20
0.15
0.10
0.05
0.00
3
2
1
0
1
2
3
θ
Figure 6
Loyalty is modeled with a double peaked distribution of g(·). Half of the customers
exhibit a preference for each of the brands. Customers loyal to firm 1 are drawn from a
2
2q
i
> 0. Figure 6
distribution with E(θ) < 0. More generally this implies that ∂∂xq2i = − ∂x∂i ∂x
j
j
illustrates one example for such a g(θ). The peaks represent two groups of loyal customers.
The first order condition remain unchanged and the second derivative with respect to
xj is:
12
WDS http://www.wds.co/apple-samsung-vs-rest, Accessed 24/01/2016
15
∂πi (x1 , x2 )
∂qi (x1 , x2 ) (α−1) 2
=
γi (xi ) − αqi
xi /2 − qiα xi + qi γi0 (xi )
∂xi
∂xi
2
2
∂ qi (x1 , x2 ) ∂πi (x1 , x2 )
(α−1) 2
γi (xi ) − αqi
xi /2
=
∂xi ∂xj
∂xi ∂xj
∂qi (x1 , x2 ) (α−2) 2
α(1 − α)qi
xi /2
+
∂xi
∂qi (x1 , x2 ) (α−1)
−
αqi
xi − γi0 (xi )
∂xj
2
2
∂πi (x1 , x2 )
∂ qi (x1 , x2 ) (α−1) 2
=
γ
(x
)
−
αq
x
/2
i i
i
i
∂x2i
∂x2i
∂qi (x1 , x2 ) (α−2) 2
+
α(1 − α)qi
xi /2
∂xi
∂qi (x1 , x2 ) (α−1)
−
αqi
xi − γi0 (xi )
∂xi
∂qi (x1 , x2 ) 0
(α−1)
+
γi (xi ) − αqi
xi − qiα + qi γi00 (xi )
∂xi
The reaction function is given as:
−1
∂πi2 (x1 , x2 )
∂πi2 (x1 , x2 )
dxi
= −
dxj
∂xi ∂xj
∂x2i
|
{z
}
>0
Using that
∂qi (x1 ,x2 )
∂xi
1 ,x2 )
= − ∂qi (x
and
∂xj
∂ 2 qi (x1 ,x2 )
∂xi ∂xj
= −∂
2 q (x ,x )
i 1 2
∂x2i
the equations simplify to:
∂qi (x1 , x2 ) ∂ 2 πi (x1 , x2 )
(α−2) 2
(α−1)
=
α(1 − α)qi
xi /2 + αqi
xi − γi0 (xi )
∂xi ∂xj
∂xi
2
∂ qi (x1 , x2 ) (α−1) 2
γ
(x
)
−
αq
−
x
/2
i
i
i
i
∂x2i
∂ 2 πi (x1 , x2 )
∂ 2 πi (x1 , x2 )
−
=
2
∂xi ∂xj
∂xi
∂qi (x1 , x2 ) 0
(α−1)
γi (xi ) + αqi
xi + qiα + qi γi00 (xi )
−
∂xi
The only new aspect of the reaction function is
∂π
∂ 2 qi (x1 , x2 ) (α−1) 2
i
γ
(x
)
−
αq
x
/2
=
i i
i
i
∂qi
∂x2i
which gives the change to revenue per customer in response to a change in the number of
customers, ignoring firm reactions.
16
The sign of
∂πi
∂qi
must be positive. Although it can be locally optimal to exert efforts
∂πi
∂qi
C
< 0 if the γi0 is sufficiently large, this would imply that πi (xC
such that
i , xj ) < 0 <
πi (0, xC
j ) and violate the participation constraint.
Theorem 2 Let xC
i be the optimal efforts of the original game. Increasing the split between both customer groups such that more customers become loyal to their respective
1 ,x2 ) remains unbrands while the market shares remain unchanged such that ∂qi (x
∂xi
xi =xC
i
∂q 2 (x ,x ) increases, leaves the equilibrium unchanged but reduces the
changed and i ∂x12 2 C
i
xi =xi
reaction functions of firm i.
Proof: The second derivative of the demand does not enter the FOCs of the firms.
i
Thus the optimal efforts remain optimal. Additionally, the sign of ∂π
∂qi must be positive.
C
C
Otherwise, πi (xC
i , xj ) < 0 < πi (0, xj ) and the firm would prefer to exert no efforts.
Consequently, a positive term is subtracted from the numerator and denominator. Thus,
the reaction depends on the size of the original reaction.
5
Discussion
Based on the previous analysis we can now discuss three typical markets.
First, a hardware market exhibits innovation costs that scale nearly perfect with market shares (α = 1). Firms add more hardware into their devices which raises manufacturing costs. The implication is simple. A lower market share lowers the cost of competition
and encourages the firm.
Second, a software market exhibits innovation costs that are independent from the
market share (α = 0). Improving the underlying code has a fixed costs irrespective of the
number of users. Again the implication is simple. A lower market share lowers the return
to innovation and discourages the firm.
Third, a platform market features firms competing with their platforms for developers
to use. Costs only partially scale with market share. Developing a platform for a small
number of customer is expensive as the costs need to be split up on fewer people. On
the other hand a higher market share also means that more efforts need to be exerted to
guarantee compatibility and debugging services to platform third party companies.
Consider Apple adding an additional feature to iOS that allows applications to better
connect with the operating system. The initial development is costly and barely depends
on the number of customer using iOS. However, with a large user base also come more
complaints about compatibility with specific programs, thus raising the costs for debugging.
In this case a very small market share blocks the firm from competing efficiently as
it faces very steep start up costs. However if the firms are equal in size losing a few
customers lowers the costs of the firm, which helps it to compete more fiercely. Thus
efforts exerted are maximal for an intermediate value of asymmetry.
17
5.1
Third Party
Now we consider a third party that can increase the profits li of one firm, while extracting
a lump sum payment of li γ(xC
i ) from the market participants.
If costs are independent of the quantity sold, intervention on the side of one firm
will crowed out efforts of the other firm. If the costs scale perfectly with the demand,
encouraging one firm raises both efforts.
However, for intermediate levels of α, effort can be strategic complements for the leader
and substitutes for the laggard. In which case encouraging the laggard raises the efforts
of both firms.
The driver of this is the different change caused to the costs of R&D to the leader and
the laggard. A change in the quantity sold, barely affects the leader, while it strongly
affects the laggards. Thus, the costs channel that adds complementary pressure to the
firms’ reaction is stronger for the laggard than the leader.
The second consequence is that small changes to the underlying parameters can have
big consequences to the symmetry within the market. A positive shock to the market
leader lowers their costs of development and raises the costs for the laggard, which leads
to an even larger gap in research.
The implications are very useful for the consumer electronics market: Here improving
the quality of the product has a high fixed proportion and high economies of scale. A
plant that produces 100,000 chips is not much more expensive than a plant for 10,000.
5.2
Welfare
Consumer welfare in this model is given as:
Z ∞
W (x1 , x2 ) =
θg(θ)dθ + x1 G(θ∗ ) + x2 (1 − G(θ∗ ))
∗
Zθ ∞
=
θg(θ)dθ + (x1 − x2 )G(θ∗ ) + x2
(x1 −x2 )
An increase in xi raises the welfare of the customers of firm i and attracts customers from
firm j to i, who are indifferent between firm i and j. Additionally, firm j reacts.
dxj
dW (x1 , x2 )
= qi (x1 , x2 ) + (1 − qi (x1 , x2 ))
dxi
dxi
If efforts are strategic complements a small intervention always increase customer welfare.
If the efforts are strategic substitutes an intervention raises welfare if:
dxj
qi (x1 , x2 )
>−
qj (x1 , x2 )
dxi
(12)
If firm 1 is the market leader with x1 > x2 and q1 > q2 an intervention in the leader
tends to be more likely to increase total welfare.
18
θ=
xC
1
−
xC
2
1 (α−1) 2
0
γ(x1 ) − αq1
x1 /2 + γ (x1 )
=
g(θ)
q1
1 (α−1) 2
1−α
0
l2 γ(x2 ) − αq2
x2 /2 + l2 γ (x2 )
− q2
g(θ)
q2
q11−α
For α = 0, g(θ) = g̃ and qi (0, 0) = 1/2 we have:
1
1
0
0
θ = g̃ (γ(x1 ) − l2 γ(x2 )) +
+ g̃θ γ (x1 ) −
− g̃θ l2 γ (x2 )
2
2
g̃ (γ(x1 ) − l2 γ(x2 )) + 21 (γ 0 (x1 ) − l2 γ 0 (x2 ))
g̃θ =
1/g̃ − (γ 0 (x1 ) + l2 γ 0 (x2 ))
By using a tailor expansion at θ = 0, we get:
qi (x1 , x2 )
≈ 1 − g̃θ
qj (x1 , x2 )
Comparing this with equation 9 allows us to make some inferences. Starting form
dx
parameter values such that qq21 = dxji and increase in g̃ makes an intervention welfare
enhancing.
Consider the case of strategic substitutes with 1 being the leader with x1 > x2 . An
intervention on x1 directly, raises welfare of the larger group of people, while the crowding
out effects only a relative small number. As the reaction is less than unity. Intervening
on the side of the leader will always raise welfare.
For the laggard the situation is slightly different. Intervening on its efforts raises
welfare, however if the reaction of firm 1 is sufficient strong and its market share sufficiently
large total welfare will decrease.
This implication changes slightly, once we consider the costs of efforts as social costs.
Total costs are given as:
C(x1 , x2 ) = q1 (x1 , x2 )α x21 /2 + (1 − q1 (x1 , x2 ))α x22 /2
dC(x1 , x2 )
(q α−1 x21 − (1 − q1 )α−1 x22 ) dq1
dx2
=α 1
+ q1α x1 + (1 − q1 )α x2
dx1
2
dx1
dx1
If development costs do not depend on the demand (α = 0) the total costs are equal
to the sum of both efforts. If both efforts increase cost go up. However, as the costs are
higher for the leader an increase of the laggard that leads to a reduction of the costs of
the leader can reduce total costs.
6
Conclusion
I presented a model to illustrate the difference between markets that can behave more like
a Cournot, where efforts are strategic substitutes, or like a Bertrand market, with efforts
being strategic complements.
19
First, if the costs of efforts closely depend on the market share of the firms (α → 1)
efforts for both firms are strategic complements. Consequently, any intervention encouraging any party raises efforts. A competition authority need not pay much attention to
the identity of the firms it encourages. In this “hardware” case the third party should
could red tape, offer subsidies and relax legal barriers for all firms.
Second, if the costs of efforts do not depend on the market shares (α → 0) efforts are
strategic substitutes and every positive intervention leads to crowding out. In general it
will still lead to a positive gain, but with two caveats. If the regulator decides to encourage
the laggard the loss of efforts of the leader might lead to a total decrease of welfare,
especially if the market is split unequal. This implies that intervention in markets where
a higher quality does not raise unit costs should target a leader or should be avoided.
This is the case for software systems, where adding a new feature has fixed costs but
hardly variable costs. On the other hand to avoid “snowballing” (the continuous increase
in asymmetry) it can support the laggard, by having laws that treat smaller business more
lenient or by putting stricter requirements on monopolists.
In between lies the platform market, the special case occupied by the market for
operating systems. Two firms compete here, both with a low but positive α and high
start up costs of innovation. Encouraging the leader in terms of market share, raises total
efforts in the market, while encouraging the laggard does not do so.
This intuition can be applied to the smartphone market. We can infer that an increase
in the efforts of Google would motivate Apple to exert more efforts, whereas an increase
in the efforts of Apple would reduce the efforts by Google.
The model shows that regulators should be more reluctant to support (directly or
indirectly) firms on markets with variable costs. Within platform markets they should
treat the leader more lenient than the laggards.
The model used can also be applied to incumbent-entrant markets. A heavily entrenched incumbent has a higher market shares and any shock to its market share leads
to an increase in its efforts if costs are variable. If costs are mainly fixed, the entry and
increase in the efforts of a competitor reduces the incentives to innovate by the leader.
This also impacts the decision of an entrant who has to decide if he wants to enter with a
large market share or with a small market share. Because of the strategic substitutes the
latter is more attractive in case R&D costs are fixed.
Of course a policy supporting the market leader or incumbent can have severe problems
in the long run as it creates a natural monopoly. However, the impact that supporting
firms in an asymmetric situation has should be better taken into account by regulators to
perform a more differentiated policy.
This paper illustrated the underlying mechanisms behind these effects and helps regulators to understand when encouraging the incumbent might be advantageous for competition. By doing so it also emphasized the importance of asymmetric markets in which
firms compete on different level with each other.
20
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21