Accessible Unlicensed Requires Authentication Published by De Gruyter July 31, 2020

Direct Interconnection and Investment Incentives for Content Quality

Soo Jin Kim

Abstract

This paper analyzes the effects of direct interconnection agreements in the Internet backbone on content quality investment for content providers (CPs). The model assumes that when the Internet service provider (ISP) has a vertical affiliation with one CP, the ISP directly interconnects the affiliated CP’s traffic to its network for free while collecting a direct interconnection fee from the unaffiliated CP. If the unaffiliated CP’s traffic is indirectly interconnected to the ISP’s network via a third party transit provider, its network quality is lower than that via a direct interconnection. For the CPs’ content quality investments, I find that the affiliated CP invests more in content when the rival indirectly interconnects, leading to a higher total level of content investment. Accordingly, there is a condition under which the ISP does not want to offer direct interconnection to the unaffiliated CP. However, consumers are not always worse off from this interconnection foreclosure. Thus, the regulation of a paid direct interconnection does not necessarily enhance welfare in terms of consumer surplus.

JEL Classification: L22; L42; D4

Corresponding author: Soo Jin Kim, School of Entrepreneurship and Management, ShanghaiTech University, Room 436, 393 Middle Huaxia Road, Shanghai, China, E-mail:

An earlier version of this paper was circulated under the title “Paid Peering and Investment Incentives for Network Capacity and Content Diversity.”


Appendix

A Further Discussion

A.1 Opportunity Cost of Free Direct Interconnection to CP1

When directly interconnecting the affiliated CP1’s content, the ISP faces an opportunity cost: if the cost of direct interconnection is high, the ISP might not want to offer free direct interconnection even for CP1. To guarantee that the ISP is incentivized to provide direct interconnection for free to CP1, I assume that the associated cost K is sufficiently low throughout the paper.

Specifically, I consider the case in which CP1’s content is indirectly interconnected even with its affiliation status. Given that CP1’s content can be either directly or indirectly interconnected, the equilibrium fee charged to CP2 for direct interconnection is different, as follows.

(13)fD,D=fIDα118[α7+2U(λ1,D,D*λ2,D,D*)];fID,D=fID+α118[α+52U(λ1,ID,D*λ2,ID,D*)],

where fD,D(fID,D) is the fee charged if CP1 is directly (indirectly) interconnected. Assuming that C(λj)=λj22, the set of equilibrium can be derived by the same logic in the main model. From the equilibrium profits for CP1, I first derive the profit difference with and without direct interconnection as follows.

(14)π1,ID,Dπ1,D,D=fID+K(α1)(U29)(9α+4U227)6(92U2)2.π1,ID,IDπ1,D,ID=fID+K(α1)(U29)[4U23(α+5)]6(92U2)2,

where π1,ID,Dπ1,D,D represents CP1’s profit difference with and without direct interconnection given that CP2 is directly interconnected. Similarly, π1,ID,IDπ1,D,ID is the profit difference given that CP2 is indirectly interconnected. From Equation (14), the threshold on K below which CP1’s profit from direct interconnection is greater than another for each case can be derived: π1,ID,D<π1,D,D if K<KD,DfID+(α1)(U29)[9(α3)+4U2]6(92U2)2, while π1,ID,ID<π1,D,ID if K<KD,IDfID+(α1)(U29)[3(α+5)+4U2]6(92U2)2. First, it can be easily shown that KD,IDKD,D=2(α1)2(U29)(92U2)2>0 by the concavity condition. Thus, the condition guaranteeing that the ISP offers free in-house direct interconnection to CP1 is given by K<KD,D, which is assumed to hold throughout the paper.

Additionally, I show that K¯ in Proposition 3 is always smaller than KD,D: KD,DK¯=(α1)[(α7)(8U6729)+(61272α)U4+243(α11)U2]162(92U2)2, which can be shown to be positive under the interior solution assumption. Per Proposition 3, if K<K¯, the ISP wants to offer direct interconnection to CP2 at the fee. Given that K¯<KDD, whenever CP2 is directly interconnected, so is CP1; thus, symmetric network quality specifications for both CPs emerge in equilibrium. Also, if K>K¯, which leads to indirect interconnection with CP2, an asymmetric network quality specification (direct interconnection for CP1 but the reverse for CP2) can prevail in equilibrium under K¯<K<KDD. That is, the implication in Proposition 3 still holds with the additional assumption on K.

A.2 Partial Market Coverage

By backward induction, CPs solve their profit maximization problem, considering the Internet market size, QI,D=1SDα or QI,ID=12SID1+α, as given. Using the same logic as in the main analysis, the set of content market equilibrium as a function of S and QI is derived as follows. Note that C(λj)=λj22 is assumed.

(15)P1,DPC=P2,DPC=1;Q1,DPC=Q2,DPC=QI,D2.P1,IDPC=13(α1)9+2U2QI,ID;Q1,IDPC=QI,ID[3(α+2)+2U2QI,ID]2(9+2U2)QI,ID.P2,IDPC=1+3(α1)9+2U2QI,ID;Q2,IDPC=QI,ID[3(α4)+2U2QI,ID]2(9+2U2)QI,ID.λ1,DPC=λ2,DPC=19(4α)UQI,D.λ1,IDPC=UQI,ID[3(α+2)+2U2QI,ID]3(9+2U2)QI,ID;λ2,IDPC=UQI,ID[3(α4)+2U2QI,ID]3(9+2U2)QI,ID.

Given the content market equilibrium, the ISP solves its joint profit maximization problem to set S as follows.

(16)maxSDπISP,D=SD(1SDα)+π1,DPC;maxSIDπISP,ID=SID(12SID1+α)+π1,IDPC.

The full set of equilibrium is numerically derived by assuming U=1,K=12, and fID=12. I show some important results graphically in Figures 3 and 4.

Figure 3: Internet subscription fee S and market share QI${Q}_{I}$ comparison.

Figure 3:

Internet subscription fee S and market share QI comparison.

Figure 4: The integrated firm’s profit from content (CP1$C{P}_{1}$) and from Internet comparisons.

Figure 4:

The integrated firm’s profit from content (CP1) and from Internet comparisons.

Both Figures plot how the equilibrium levels change as α increases. For example, the equilibrium Internet subscription fee is higher in the direct interconnection agreement, and the gap between two fees becomes wider as α increases. Similar patterns can be observed from other comparisons.

A.3 Bargaining Power

Denoting the ISP’s bargaining power β where β[0,1], the equilibrium fee charged to CP2 for direct interconnection is given by βfD*, where fD* is given in Equation (6). Although the set of equilibrium in the indirect interconnection agreement does not change, that in the direct interconnection agreement is different because the amount of content-specific investment is now a function of β. Assuming that C(λj)=λj22, the equilibria λ1,DB and λ2,DB, where the superscript B denotes the model with bargaining power, are derived as follows.

(17)λ1,DB=λ2,DB=U[3+β(1α)]9.

Comparing λ1,DB=λ2,DB to λ1,D*=λ2,D* as in Equation (7), it is immediately clear that λ1,DB=λ2,DB>λ1,D*=λ2,D* if β<1. Even if CPs’ investments in content quality increase, the equilibrium for price and market share is the same as in the main model; i.e., two CPs equally split the market by charging the content price of one to consumers because of the symmetry in content quality investment. Additionally, I find the following results.

(18)λ1,IDBλ1,DB=U(α1)9(β+992U2);λ2,IDBλ2,DB=2(α1)U2U29.
λ1,IDB+λ2,IDB(λ1,DB+λ2,DB)=2(α1)βU9.

From Equation (18), it is easy to see that Propositions 1 and 2 still hold in this extension. Given that the set of equilibria in the indirect interconnection agreement does not change, Corollary 1 remains the same. After some algebra, I can also show that Propositions 3 and 4 hold even if the quantitative threshold on K is different from K¯.[13]

For the welfare implications, consumer surplus in the indirect interconnection agreement is the same as before, whereas that in the direct interconnection agreement changes because of β: CSDB=α54+U29[3β(α1)]+U. From CSDB, it can be observed that, as β increases, i.e., as CP2 has less advantage in negotiation, consumer surplus from the direct interconnection agreement decreases. From the comparison of CSDB and CSID, I find a threshold on α, as in Proposition 5, below which CSDB is greater than CSID. The threshold on α, denoted α¯B, is given as follows.

(19)α¯B=16βU681+89(2β+1)U44(β+2)U2+19.

It can be verified that α¯Bβ=481U2(92U2)2<0, which suggests that, as CP2 gains more bargaining power, α¯B increases, implying that CSDB is more likely to be greater than CSID.

A.4 Multiplicative Utility

From the utility specification, as in Equation (11), the indifferent consumer type xM is given as follows: xM=12[P1+P2+αλ1Uα2(λ2+1)U+αU+1]. From each CP’s profit maximization problem, the optimal price and market share are derived as follows.

(20)P1M=13[αλ1Uα2(λ2+1)U+αU+3];Q1M=16[αλ1Uα2(λ2+1)U+αU+3].P2M=13[αλ1U+α2(λ2+1)UαU+3];Q2M=16[αλ1U+α2(λ2+1)UαU+3].

By the same logic as in the main analysis, the equilibrium fee charged to CP2 for direct interconnection is derived by π2,DMπ2,IDM as follows:

(21)fDM=fID+118(α1)(λ2+1)U{U[1α+(1+α)λ22αλ1]+6}.

Assuming that C(λj)=λj22, the set of equilibria is given as follows.

(22)P1,DM=U2{α{(α1)U[α(2U+3)3]+9}+9}81U2{α2[2(α1)U2+9]+9}81.Q1,D*=U2{α{(α1)U[α(2U+3)3]+9}+9}812U2{α2[2(α1)U2+9]+9}162P2,DM=U2{α{18α+(α1)U[α(2U3)+3]9}+9}81U2{α2[2(α1)U2+9]+9}81.Q2,DM=U2{α{18α+(α1)U[α(2U3)+3]9}+9}812U2{α2[2(α1)U2+9]+9}162.P1,IDM=U(3α+2U+3)9(α2+1)U29;Q1,IDM=U(3α+2U+3)92[(α2+1)U29].P2,IDM=U[α(2αU+3)3]9(α2+1)U29;Q2,IDM=U[α(2αU+3)3]92[(α2+1)U29].fDM=fID81(7α)(1α)4U6+12(7α)U4108(4α)U21458.λ1,DM=αU[(α1)U3](2αU29)U2{α2[2(α1)U2+9]+9}81;λ2,DM=U((α1)U3)(2α2U29)U2(α2(2(α1)U2+9)+9)81.λ1,IDM=αU[U(3α+2U+3)9]3[(α2+1)U29];λ2,IDM=U{U[α(2αU+3)3]9}3[(α2+1)U29].

After some algebraic manipulation, I find the following results.

(23)λ1,IDMλ1,DM=2(α1)αU2(U29)(αU+3)[αU(2U+3)3(U+3)]3[(α2+1)U29](U2{α2[2(α1)U2+9]+9}81).λ2,IDMλ2,DM=2(α1)α2U4(αU+3)[αU(2U+3)3(U+3)]3[(α2+1)U29]{U2{α2[2(α1)U2+9]+9}81}.λ1,IDM+λ2,IDM(λ1,DM+λ2,DM)=2(α1)αU2(αU+3)[αU(2U+3)3(U+3)][(α+1)U29]3[(α2+1)U29]{U2{α2[2(α1)U2+9]+9}81}.Q1,IDMQ2,IDM=3(α1)αU2[(α1)U3]U2{α2[2(α1)U2+9]+9}81.

Under the interior solution condition, U<3(α+α(9α2)+11)4α2, and the concavity condition, Uα<3, Equation (23) shows that Propositions 1 and 2 still hold in this extension. Additionally, the comparison between Q1,IDM and Q2,IDM also shows that CP1 is dominant in terms of market share in the indirect interconnection agreement. From the profit comparison between π1,DM and π1,IDM, I derive a threshold on K, denoted as K¯M, below which CP1 wants to offer direct interconnection to CP2, as shown in Proposition 3. The welfare implication also qualitatively holds.[14]

B Proofs of the Propositions

Proof of Lemma 1

First, let ζD(λ1)U[4α+U(λ1λ2,D)]9C(λ1),ζD(λ2)U[4αU(λ1,Dλ2)]9C(λ2),ζID(λ1)U[2+α+U(λ1λ2,ID)]9C(λ1), and ζID(λ2)U[4αU(λ1,IDλ2)]9C(λ2). It can be shown that

(24)ζID(λ1)=ζD(λ1)+2U9(α1)Interconnection spillover effect (+)+U29(λ2,Dλ2,ID)Strategic effect (?).ζID(λ2)=ζD(λ2)+U29(λ1,Dλ1,ID)Strategic effect (?).

By the conditions in (4) and (5), ζD(λj,D*)=0 and ζID(λj,ID*)=0. Then, ζID(λ1,D*) =  if either strategic effect is positive, or it is negative but with a small effect compared to the interconnection spillover effect. Similarly, if the strategic effect is negative. If the inequality holds, by the concavity of profit functions, ζ(λ) is non-increasing; therefore, λ1,D*<λ1,ID* and λ2,D*>λ2,ID*. □

Proof of Proposition 1

From the proof of Lemma 1, it can be shown that ζ(λj)λj<0 where jj, i.e.,λ1 and λ2 are strategic substitutes. Here, the positive interconnection spillover effect for CP1 leads to λ2,ID*<λ2,D*. This outcome suggests that the strategic effect for CP1 is positive, implying thatλ1,D*<λ1,ID*. □

Proof of Proposition 2

It can be shown by the proof of Lemma 1. Also, under the parametric example ofC(λj)=λj22, it is readily shown that λ1,ID*+λ2,ID*(λ1,D*+λ2,D*)=2U(α1)9, which is positive for α>1.

Proof of Corollary 1

It is obvious from Equation (6) and by Proposition 1 that Q1,ID*>12.

Proof of Proposition 3

The profit comparison for CP1 with and without the direct interconnection agreement is shown as follows.

(25)π1,Dπ1,ID=fIDKC(λ1,D*)+C(λ1,ID*)+[3+U(λ1,D*λ2,D*)]2[2+α+U(λ1,ID*λ2,ID*)]218.

The threshold K¯ is the solution to π1,Dπ1,ID=0. Under the parametric example of C(λj)=λj22, the profit comparison for CP1 with and without the direct interconnection agreement is shown as follows.

(26)π1,Dπ1,ID=fIDK(α1)[729(α1)4(α7)(9U2)U4486U2]81(92U2)2.

From Equation (26), I find the threshold, denoted K¯, as follows.

(27)K¯=fID(α1)[729(α1)4(α7)(9U2)U4486U2]81(92U2)2.

Given K¯, it is easy to show that π1,D<π1,ID if K¯<K and vice versa. □

Proof of Proposition 4

Given K¯ derived from Equation (25), the comparative statics with respect to α can be simplified by applying the implicit function theorem to ζD(λj)andζID(λj), as defined in the proof of Lemma 1, which is given as follows.

(28)K¯α=2UU29C{CU9[2+α+U(λ1,ID*λ2,ID*)},

where U2<9C holds by the concavity condition. Thus, as long as α is sufficiently large, such that α>9CU2U(λ1,ID*λ2,ID*), K¯α is negative. For instance, under the parametric example, the threshold on K is given as in Equation (27). The comparative statics with respect to α is given as follows.

(29)K¯α=1458(α1)8(α4)(U29)U4+486U281(92U2)2.

After some algebraic manipulation, it can be shown that Equation (29) is negative as long as α is larger than 243(U29)4U636U4+729+4. The interior solution condition, i.e., U<6(4α)2, guarantees that 243(U29)4U636U4+729+4<α. □

Proof of Proposition 5

Equation (9) can be further simplified as follows.

(30)CSIDCSD=U[λ1,ID*xID*+λ2,ID*(1xID*)λ1,D*+λ2,D*2]+(1α)(1xID*)+xID*(1xID*)14+(1P1,ID*)(2xID*1).

Under the parametric example of C(λj)=λj22, consumer surplus levels with and without the direct interconnection agreement are derived as follows.

(31)CSD=α54(α4)U29+U;
CSID=27α2+4U(U+3)[U(4U345U+27)+81]+24α(U49U2+18)70212(92U2)2.

From Equation (31), the difference between the two levels is obtained as follows.

(32)CSIDCSD=(α1)[81(α19)+16U6216U4+972U2]36(92U2)2.

It can be immediately shown that CSID<CSD if α<16U681+8U4312U2+19α¯. Under the interior solution condition, α¯ is larger than one, indicating that CSID<CSD can prevail in equilibrium if 1<α<α¯.[15]

Acknowledgments

I would like to thank Jay Pil Choi, Thomas Jeitschko, John Wilson, and Aleksandr Yankelevich; and the participants at the Red Cedar Conference at Michigan State University, the Fall 2016 Midwest Economic Theory Conference, and the Eastern Economic Association 2017 Conference. I am also grateful to the editor and two anonymous referees for their constructive comments, which greatly improved this paper.

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Received: 2020-02-06
Accepted: 2020-05-21
Published Online: 2020-07-31
Published in Print: 2020-09-25

© 2019 Walter de Gruyter GmbH, Berlin/Boston