Compatibility, network effects, and collusion

Economics Letters 151 (2017) 39–43

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Compatibility, network effects, and collusion✩ Alexander Rasch * Duesseldorf Institute for Competition Economics (DICE), University of Duesseldorf, Universitaetsstrasse 1, 40225 Duesseldorf, Germany

highlights • • • •

A market with network effects in which firms collude on prices is considered. Full collusion is easier to sustain under compatibility. Incentives to introduce compatibility under collusion may be higher or lower than under competition. Intertemporal preferences can have an ambiguous effect on firms’ compatibility decisions.

article

info

Article history: Received 7 September 2016 Received in revised form 17 November 2016 Accepted 23 November 2016 Available online 25 November 2016

a b s t r a c t I consider a market with network effects in which firms collude on prices. Depending on the fixed costs for achieving compatibility, there may be a non-monotone relationship between firms’ decisions to make their products compatible and their intertemporal preferences. © 2016 Elsevier B.V. All rights reserved.

JEL classification: L13 L14 L15 L41 Keywords: Collusion Compatibility Network effect Standard

1. Introduction Introducing a common standard to make products compatible is an important consideration in markets both with and without network effects. The compatibility issue has been extensively theoretically addressed in competitive market environments in which network effects are not present (see, e.g., Matutes and Regibeau, 1988, Economides, 1989, and Chou and Shy, 1990) and those in which they are (see, e.g., Katz and Shapiro, 1985 and Farrell and Saloner, 1986). However, the relationship between compatibility and collusion has received only limited attention. The sole exception is Lambertini et al. (1998), who consider a market without network ✩ I am grateful to Jesko Herre, Bernd Schauenberg, Jörg Schiller, and Achim Wambach for their helpful comments on an earlier version of the paper. I also thank an anonymous referee for his or her valuable suggestions. Fax: +49 0 211 81 15499. E-mail address: [email protected].

*

http://dx.doi.org/10.1016/j.econlet.2016.11.031 0165-1765/© 2016 Elsevier B.V. All rights reserved.

effects. The authors analyze the implications of an increasing discount factor on the incentives to introduce (costly) compatibility, showing that there is a non-monotone relationship. I complement this previous literature by investigating the relationship between collusion and firms’ incentives to introduce compatibility in a market with network effects. In industries with network effects, customers benefit from a larger network, i.e., from the larger number of customers who use the same network. In such cases, compatibility means that customers benefit not only from other customers opting for the same network but also from those who use a compatible network. One important industry in which network effects, compatibility, and collusion are prominently featured is the telecommunications sector. With regard to network effects, in their analysis of 1335 subscribers to one of the five South Korean mobile phone networks, Kim and Kwon (2003) find that consumers prefer larger networks; price discounts for on-net calls and quality signals are likely sources of this network effect. Similarly, based on quarterly data from Polish mobile phone providers between 1996 and

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A. Rasch / Economics Letters 151 (2017) 39–43

2001, Grajek (2007) reports that strong network effects exist in the Polish market for telecommunications.1 At first glance, compatibility in the telecommunications sector appears to be widespread, but this is not true for all countries. Various mobile operators use GSM (Global System for Mobiles) as a common standard in most European countries, but in the United States, there are two coexisting mobile phone technologies that are not compatible: GSM and CDMA (Code Division Multiple Access). With regard to collusive practices in the telecommunications sector, Chen and Lin (2002) describe a case of collusion among mobile phone service providers in Hong Kong in 2000. In their empirical study, Nunn and Sarvary (2004) analyze the mobile telecommunications industry, using price and quantity data from ten countries around the world; they conclude that ‘‘market power in different countries may originate from [...] collusive pricing among cellular operators’’ (p. 377). I use these observations as a starting point to investigate the interplay of collusion and compatibility in a market with network effects.

competition forever. The use of grim-trigger strategies leads to the critical discount factor, defined as

δ¯j :=

πjD − π¯ jK πjD − πjN

,

where πjN , πjD , and π¯ jK denote competitive (punishment) profits, deviation profits, and maximum collusive profits, respectively, and j ∈ {I , C } denotes the cases of incompatibility (subscript I) and compatibility (subscript C ). Note that collusion at maximum prices is stable if and only if δ ≥ δ¯ j .3 To avoid the possibility of a cornered market equilibrium, the transport-cost parameter must not be smaller than the network effect.4 In order to ensure that the market is always covered (i.e., that all customers along the line buy from one of the two firms), transport costs must not be too high. Hence, I make the following assumption: Assumption 1. Transport costs are such that ς ≤ τ ≤ 2/3 + 2ς/3 =: τ¯ .5

2. The model I use the model developed by Hotelling (1929) with added network effects similar to the setup in Farrell and Saloner (1992) (see also Doganoglu and Wright, 2006). There are two symmetric firms. Firm 1 is located at L1 = 0 on a linear city of unit length; firm 2 is located at L2 = 1. Firms’ marginal and fixed costs for providing the network product are normalized to zero. Firms compete in prices pi (with i ∈ {1, 2}) and may invest fixed costs of f (per firm) to make their products compatible. Firms have a common discount factor δ . Customers of mass one are uniformly distributed along the linear city. They derive an intrinsic utility of one from buying the network product. They also benefit from the participation of other customers who buy the same or a compatible product (network effect); the extent of this beneficial effect is measured by ς (with 0 ≤ ς ). Customers incur linear transport costs τ per unit of distance.2 Hence, a customer located at x derives the following net utility when buying from firm i: 1 − pi − τ |Li − x| + ς ni 1 − pi − τ |Li − x| + ς

{ ui =

under incompatibility under compatibility,

where ni denotes firm i’s customer base, which is equal to 1 under compatibility (see Assumption 1 below). The timing of the game is as follows: Period 1 Firms decide on the introduction of compatibility. Periods 2–∞ In each period, there are two stages: Stage 1 Firms simultaneously set prices. Stage 2 Customers observe prices and decide which firm to buy from. I look for subgame perfect equilibria, which implies that customers form rational expectations in determining the size of each network given the prices they observe. With regard to the price-setting decision, I assume that firms use grim-trigger strategies (Friedman, 1971), meaning that they collude as long as no firm has deviated from the collusive path in previous periods. Should such deviation occur, firms revert to 1 Again, lower on-net prices and other issues such as quality signals and conformist behavior are important factors. See Birke (2009) for a survey of the empirical contributions. 2 In light of the results in Rasch and Wambach (2009), assuming quadratic transport costs instead should not qualitatively change the results.

3. Analysis and results I first report competitive, collusive, and deviation profits in periods 2–∞. Subsequently, I analyze less-than-maximum collusive profits and firms’ compatibility decisions. 3.1. Prices and profits Competition I start by reviewing the competitive case.6 No compatibility. The symmetric competitive equilibrium price is given by pNI = τ − ς. The firms share the market equally, and the equilibrium profit per firm amounts to

πIN =

τ 2

−

ς 2

.

Compatibility. Suppose now that firms have decided to make their products compatible. In this case, competition is equal to the case without network effects, i.e., the equilibrium price is equal to pNC = τ . Given compatibility investments (which are sunk after period 1) and equal market shares, the per-period profit for each firm amounts to

πCN =

τ 2

.

3 Of course, one could also use optimal punishments (Abreu, 1986). But as Häckner (1996) observes when comparing his results to those of Chang (1991), who uses grim-trigger strategies, ‘‘the relationship between cartel stability and product differentiation is fairly robust to changes in the punishment mechanism for the class of models characterized by Bertrand competition and horizontal differentiation’’ (p. 613). 4 If this (standard) assumption is violated, a customer located at one of the extreme points may still want to buy from the distant firm at equal prices if he expects everyone else to do so. This raises the possibility of multiple consistent network sizes for given prices, which I ignore. 5 Note that this implies that ς ≤ τ¯ ⇔ ς ≤ 2.

6 These results are taken from Doganoglu and Wright (2006).

A. Rasch / Economics Letters 151 (2017) 39–43

Full collusion Suppose that the discount factor is larger than the critical one (see below), which allows firms to (tacitly) set maximum collusive prices (full collusion). No compatibility. Firms share the market equally, and the (indifferent) customer at x = 1/2 is left with zero utility. The collusive price is thus given by

τ

ς

τ

ς

+ = 0 ⇔ p¯ KI = 1 − + . 2 2 2 2 The collusive per-period profit for each firm then amounts to 1 − p¯ KI −

π¯ = K I

1 2

−

τ 4

+

ς 4

.

Compatibility. Under compatibility, the collusive price is given by 1 − p¯ KC −

τ

+ ς = 0 ⇔ p¯ KC = 1 −

τ

+ ς.

2 2 This yields a collusive per-period profit for each firm of

π¯ CK =

1 2

−

τ 4

+

ς 2

.

Deviation When one firm deviates from the collusive path, whereas the other firm sets the maximum collusive price, two cases must be distinguished: either (i) the deviating firm covers the whole market (low transport costs), or (ii) the deviating firm covers only part of the market (high transport costs). No compatibility. Define τ˜I := 2/7 + ς .7 Then, the deviating firm sets an optimal price of

⎧ ⎪ ⎨1 − 3τ + 3ς D 2 2 p¯ I = 1 τ ς ⎪ ⎩ + −

if ς ≤ τ ≤ τ˜I

if τ˜I < τ ≤ τ¯ , 2 4 4 which gives a market share of 1 1

if ς ≤ τ ≤ τ˜I

{ n¯ DI =

8

+

1

if τ˜I < τ ≤ τ¯ .

4 (τ − ς )

As a result, its profit equals

π¯ ID =

⎧ 3τ 3ς ⎪ ⎨1 − +

if ς ≤ τ ≤ τ˜I

(2 + τ − ς ) ⎪ ⎩ 32 (τ − ς)

if τ˜I < τ ≤ τ¯ .

2

2

2

Compatibility. Analogously, define τ˜C := 2/7 + 2ς/7 ⋚ ς . The optimal deviation price is equal to

⎧ ⎪ ⎨1 − 3τ + ς D 2 p¯ C = 1 τ ς ⎪ ⎩ + +

if ς ≤ τ ≤ max{ς, τ˜C }

if max{ς, τ˜C } < τ ≤ τ¯ . 2 4 2 Given this price, a deviating firm has a market share of

{ nDC

¯ =

1 2 + τ + 2ς

8τ and makes a profit of

π¯ CD =

if ς ≤ τ ≤ max{ς, τ˜C } if max{ς, τ˜C } < τ ≤ τ¯

if ς ≤ τ ≤ max{ς, τ˜C }

2 ⎪ ⎩ (2 + τ + 2ς ) 32τ

if max{ς, τ˜C } < τ ≤ τ¯ .

2

3.2. Critical discount factors The critical discount factors are given by

⎧ 2 − 5(τ − ς ) ⎪ ⎨ 4(1 2(τ − ς )) δ¯I = 2 − − 3(τ − ς ) ⎪ ⎩ 2 + 5(τ − ς )

7 Note that at τ = τ˜ , there is a transition from full to partial market coverage. I Note further that τ˜I ≤ τ¯ ⇔ ς ≤ 8/7.

if ς ≤ τ ≤ τ˜I (1)

if τ˜I < τ ≤ τ¯

and

⎧ 2 − 5τ + 2ς ⎪ ⎨ 4(1 − 2τ + ς ) δ¯C = 2 − 3τ + 2ς ⎪ ⎩ 2 + 5τ + 2ς

if ς ≤ τ ≤ max{ς, τ˜C } (2)

if max{ς, τ˜C } < τ ≤ τ¯ .

Note that it is a priori unclear under which compatibility regime full collusion is easier to sustain. Whereas punishment is harsher under incompatibility (i.e., collusion at maximum prices tends to be more stable), collusive profits are lower (i.e., collusion tends to be less stable). Moreover, deviation profits in one regime may be larger or smaller than in the other regime. The following proposition clarifies this issue: Proposition 1. It holds that δ¯ I ≥ δ¯ C , i.e., full collusion is easier to sustain under compatibility than under incompatibility. Proof. Follows from a comparison of (1) and (2). □ Under incompatibility, the relative gains from deviation are greater than those under compatibility. This stems from the fact that a deviating firm’s offer under incompatibility is attractive for customers for two reasons: a lower price and a larger network. Under compatibility, only a lower price can be offered by the deviator. This leads to a relative increase in the profitability of deviation under incompatibility. As the network effect becomes stronger, the difference between the critical discount factors increases (see Fig. 1). 3.3. Partial collusion As pointed out by Chang (1991), collusion at less-thanmaximum prices can be sustained by horizontally differentiated firms even though the discount factor is below the critical threshold. Again, I differentiate between the cases in which a deviating firm covers (i) the whole market or (ii) only part of it. No compatibility When the competitor sets a price pKI and the deviating firm covers the whole market, the optimal price equals pKI − τ + ς . In this case, collusion can be sustained for

δ=

pKI − τ + ς − pKI − τ + ς −

τ 2

pK I 2

+

ς

⇔ pKI =

2

(2 − 3δ )(τ − ς ) 2(1 − 2δ )

.

For the case in which the deviator only covers part of the market, the optimal price is given by pKI /2 + τ /2 − ς /2, resulting in a profit of (pKI + τ − ς )2 /8(τ − ς ). Hence, (pK +τ −ς )2 I

⎧ 3τ ⎪ ⎨1 − +ς

41

δ=

8(τ −ς )

(pK +τ −ς )2 I 8(τ −ς )

−

−

τ 2

pK I 2

+

ς

⇔ pKI =

(1 + 3δ )(τ − ς ) 1−δ

.

2

A comparison of the resulting deviation profits shows that full (partial) market coverage is optimal if and only if δ ≥ (<)1/3.8 8 Note that δ¯ = 1/3 for τ = τ˜ . I I

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A. Rasch / Economics Letters 151 (2017) 39–43

Combining these findings with the previous results, I can express collusive prices and profits as follows:

⎧ (1 + 3δ) (τ − ς) ⎪ ⎪ ⎪ ⎪ ⎪ 1−δ ⎨ pKI (δ ) = (2 − 3δ )(τ − ς ) ⎪ ⎪ 2(1 − 2δ ) ⎪ ⎪ ⎪ ⎩1 − τ + ς 2

if 0 ≤ δ < min

{ if min

1 3

, δ¯I

{

1

, δ¯I

}

} 3 ≤ δ < δ¯I

if δ¯ I ≤ δ ≤ 1

2

and

⎧ (1 + 3δ) (τ − ς ) ⎪ ⎪ ⎪ ⎪ 2(1 − δ ) ⎪ ⎨ (2 − 3δ )(τ − ς ) K πI ( δ ) = ⎪ 4(1 − 2δ ) ⎪ ⎪ ⎪ ⎪ ⎩1 − τ + ς 2

4

if 0 ≤ δ < min

{

1

if min

3

{

1

, δ¯I

}

} 3 , δ¯I ≤ δ < δ¯I

(a) ς = 0.25.

Fig. 1. Collusive per-period profits under incompatibility and compatibility. Note: There are three cases: (i) full collusion, (ii) partial collusion under incompatibility and full collusion under compatibility, and (iii) partial collusion.

if δ¯ I ≤ δ ≤ 1.

4

Compatibility Proceeding as above, a deviating firm that covers the whole market sets pKC − τ . In this case, collusion can be sustained for

δ=

pKC pKC

−τ − −τ −

pK C 2

⇔ pKC =

τ

2

(2 − 3δ )τ 2(1 − 2δ )

.

For the case in which the deviator only covers part of the market, the optimal price is given by pKC /2 + τ /2, resulting in a profit of (pKC + τ )2 /8τ . Hence, (pK +τ )2 C

δ=

8τ

(pK +τ )2 C 8τ

− −

pK C 2

⇔ pKC =

τ

(1 + 3δ )τ 1−δ

.

2

A comparison of the resulting deviation profits shows that full (partial) market coverage is again optimal if and only if δ ≥ (<)1/3.9 I summarize my results as follows:

⎧ (1 + 3δ) τ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1−δ pKC (δ ) = (2 − 3δ )τ ⎪ ⎪ 2(1 − 2δ ) ⎪ ⎪ ⎪ ⎩1 − τ 2

if 0 ≤ δ < min

{ if min

1 3

, δ¯C

{

1

, δ¯C

(b) ς = 0.5.

}

} 3 ≤ δ < δ¯C

if δ¯ C ≤ δ ≤ 1

Analogously, if firms collude, compatibility is profitable for f ≤ f˜ K (δ ) :=

δ 1−δ

(

) πCK (δ ) − πIK (δ ) =

⎧ δς (1 + 3δ) ⎪ ⎪ ⎪ ⎪ 2(1 − δ)2 ⎪ { } ⎪ ⎪ 1 ⎪ ⎪ ¯C if 0 ≤ δ < min , δ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ δ (2 (1 − δ) − τ (3 + 5δ) + 4ς (1 + δ)) ⎪ ⎪ ⎪ ⎪ ⎪ 4}(1 − δ)2 { ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ if min , δ¯C ≤ δ < ⎪ ⎪ 3 3 ⎪ ⎨ δς (2 − 3δ) ⎪ 2 (1 − δ) (1 − 2δ) { ⎪ } ⎪ ⎪ 1 1 ⎪ ⎪ ¯ , δC ⎪ if ≤ δ < max ⎪ ⎪ 3 3 ⎪ ⎪ δ (2 (1 − 2δ) − τ (5 − 8δ) + 2ς (3 − 5δ)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ { 4 (1 }− δ) (1 − 2δ) ⎪ ⎪ ⎪ ⎪ if max 1 , δ¯ ≤ δ < δ¯ ⎪ I C ⎪ ⎪ 3 ⎪ ⎪ ⎪ δς ⎪ ⎩ if δ¯ I ≤ δ ≤ 1. 4 (1 − δ)

(4)

A comparison of the two thresholds yields:

and

⎧ (1 + 3δ) τ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 2(1 − δ ) (2 − 3δ )τ K πC ( δ ) = ⎪ 4(1 − 2δ ) ⎪ ⎪ ⎪ ⎪ ⎩1 − τ 2

4

if 0 ≤ δ < min

{ if min

1 3

{

1

, δ¯C

Proposition 2. Competition may lead to more or less compatibility than collusion, as

}

} 3 , δ¯C ≤ δ < δ¯C

⎧ K ⎨< f˜ (δ ) ˜f N (δ ) ⋚ f˜ K (δ ) ⎩ ˜K > f (δ )

if δ¯ C ≤ δ ≤ 1.

Fig. 1 illustrates the regions of collusive per-period profits for two examples. 3.4. Firms’ incentives to introduce compatibility Under competition, firms opt for compatibility as long as f ≤ f˜ N (δ ) :=

δ 1−δ

(

) πCN − πIN =

δς 2 (1 − δ)

.

(3)

9 Note that δ¯ = 1/3 for τ = τ˜ , and that for large values of ς , full market C C coverage may never be relevant when δ < δ¯ C (see Fig. 1(b)). This is due to the ¯ fact that at τ = ς , δI = 1/2, whereas δ¯ C ⋚ 1/3.

if if if

0 ≤ δ < δ¯ C

δ¯C ≤ δ < δ¯I δ¯I ≤ δ ≤ 1.

Proof. Follows from a comparison of (3) and (4). □ Hence, firms’ incentives to choose compatibility are weaker in a situation in which full collusion can be sustained, i.e., when the discount factor is relatively large. This is due to the fact that the price-increasing effect under competition following the introduction of compatibility is stronger than that of the additional mark-up under collusion. When the critical discount factor is low (such that collusion under incompatibility is only sustainable at lower-thanmaximum prices), full collusion or at least partial collusion at much higher prices is sustainable under compatibility. As a result, firms’ incentives to introduce compatibility are greater under collusion than under competition (see Fig. 2).

A. Rasch / Economics Letters 151 (2017) 39–43

43

δ¯C < δ < δ¯I , firms collude at maximum prices under compatibility, and an increase in the discount factor does not change maximum collusive profits. At the same time, however, such an increase leads to higher partial-collusion profits under incompatibility. Overall, this leads to lower incentives for firms to introduce compatibility. Fig. 2 illustrates this finding. When fixed costs lie in between the two dotted horizontal lines, the pattern described above can be observed. References

Fig. 2. Incentives to introduce compatibility under competition and collusion (for τ = 0.5 and ς = 0.25). Note: There are four scenarios: (i) incompatibility, (ii) incompatibility under competition and compatibility under collusion, (iii) compatibility under competition and incompatibility under collusion, and (iv) compatibility. For those values of the fixed costs that lie in between the two dotted horizontal lines, there is a non-monotone relationship between the discount factor and firms’ incentives to introduce compatibility.

Whereas f˜ N (δ ) strictly increases in δ , the following proposition highlights that this is not the case under collusion: Proposition 3. There is a non-monotone relationship between f˜ K (δ ) and δ :

⎧ K ˜ ∂ f (δ ) ⎨ > 0 ∂δ

⋚0 ⎩ >0

if 0 ≤ δ < δ¯ C if δ¯ C ≤ δ < δ¯ I if δ¯ I ≤ δ ≤ 1.

Proof. Follows from differentiating (4). □ For some levels of the fixed costs, firms’ compatibility decisions depend on their intertemporal preferences in a non-monotone fashion. More precisely, firms oppose compatibility for low levels of the discount factor. As this factor increases, firms start to introduce compatibility before they opt against it again. For large values of the discount factor, making products compatible is once again preferred. This is due to the fact that full collusion is sustainable for a wider range of the discount factor (see Proposition 1). For

Abreu, D., 1986. Extremal equilibria of oligopolistic supergames. J. Econ. Theory 39, 191–225. Birke, D., 2009. The economics of networks: A survey of the empirical literature. J. Econ. Surveys 23, 762–793. Chang, M.-H., 1991. The effects of product differentiation on collusive pricing. Int. J. Ind. Org. 9, 453–469. Chen, E.K.Y., Lin, P., 2002. Competition policy under laissez-faireism: Market power and its treatment in Hong Kong. Rev. Ind. Org. 21, 145–166. Chou, C., Shy, O., 1990. Network effects without network externalities. Int. J. Ind. Org. 8, 259–270. Doganoglu, T., Wright, J., 2006. Multihoming and compatibility. Int. J. Ind. Org. 24, 45–67. Economides, N., 1989. Desirability of compatibility in the absence of network externalities. Amer. Econ. Rev. 79, 1165–1181. Farrell, J., Saloner, G., 1986. Standardization and variety. Econ. Lett. 20, 71–74. Farrell, J., Saloner, G., 1992. Converters, compatibility, and the control of interfaces. J. Ind. Econ. 40, 9–35. Friedman, J., 1971. A non-cooperative equilibrium for supergames. Rev. Econ. Stud. 38, 1–12. Grajek, M. (2007) Estimating network effects and compatibility in mobile telecommunications. ESMT Working Paper No. 07-001. Häckner, J., 1996. Optimal symmetric punishments in a Bertrand differentiated products duopoly. Int. J. Ind. Org. 14, 611–630. Hotelling, H., 1929. Stability in competition. Econ. J. 39, 41–57. Katz, M., Shapiro, C., 1985. Network externalities, competition, and compatibility. Amer. Econ. Rev. 75, 424–440. Kim, H.-S., Kwon, N., 2003. The advantage of network size in acquiring new subscribers: A conditional logit analysis of the Korean mobile telephony market. Inf. Econ. Policy 15, 17–33. Lambertini, L., Poddar, S., Sasaki, D., 1998. Standardization and the stability of collusion. Econ. Lett. 58, 303–310. Matutes, C., Regibeau, P., 1988. Mix and match: Product compatibility without network externalities. RAND J. Econ. 19, 221–234. Nunn, D., Sarvary, M., 2004. Pricing practices and firms? Market power in international cellular markets, an empirical study. Int. J. Res. Market. 21, 377–395. Rasch, A., Wambach, A., 2009. Internal decision-making rules and collusion. J. Econ. Behav. Organ. 72, 703–715.