PREDATORY PRICING IN A NETWORK INDUSTRY WITH MULTIPLE MARKETS Mark A. Jamison* Director of Telecommunications Studies Public Utility Research Center University of Florida Draft Date: October 25, 2001
Abstract
I examine conditions under which a multimarket firm in a network industry would price below marginal cost. Network providers choose quality for exchanging information within the provider’s own network, quality for exchanging information between the provider’s network and other networks, and output. I show that a firm serving markets A and B would price below marginal cost in A when its price elasticity of demand times its externality elasticity of demand is greater than the ratio of its revenue in A and B. If the reverse is true, then its prices could be considered predatory. Despite the firm’s below-cost prices in A, a single-market rival in A can receive non-negative profits, but its output in A must be greater than the multimarket firm’s total output. Under similar conditions, the multimarket firm prefers a duopoly market over a monopoly market in A.
*
This is a work in progress. Please do not quote or cite without permission of the author. Comments should be sent to
[email protected].
1
1. Introduction
US v. Microsoft has stimulated new interest in predatory pricing, the situation where an aggressor firm sacrifices short-run profits to cause rivals to exit the market. Microsoft’s software products are generally network goods, goods whose value for an individual customer increases with the number of users. This property of network goods calls into question traditional views of predatory pricing. Economists have generally questioned whether predatory pricing was rational because market entry by future rivals would prohibit the aggressor firm from recovering its short-term losses. In US v. Microsoft, there was general consensus that Microsoft’s prices did not maximize short-run profits (Fudenberg and Tirole, 2000; Gilbert and Katz, 2001). The US believed that network externalities and tipping effects (Evans and Schmalensee, 1996; Farrell and Katz, 2001) enabled Microsoft to profit in the long run from driving rivals out of its markets. (Gilbert and Katz, 2001) Microsoft’s consultants argued that its prices were not predatory because it was using below-cost prices to stimulate demand for complementary products and to stimulate demand in future periods. (Evans and Schmalensee, 1996) Fudenberg and Tirole (2000) and Farrell and Katz (2001) analyze this issue by considering two-period games in which a firm in a network industry would lower prices in the initial period to deter entry in the subsequent period. Farrell and Katz (2000) also consider this issue by examining the incentives of a monopolist of one component of a network system to weaken or destroy incentives for independent innovation in another component of the system. This literature does not examine how positive network externalities between markets affect incentives to price below marginal cost. This issue may provide insights into Microsoft’s pricing strategies as well as pricing by incumbent telecommunications providers, mobile
2
communications providers, and Internet providers that operate in multiple markets. Gabel (1994) found evidence of below-cost pricing by AT&T in selected markets during the early years of the telephone industry. To analyze this issue, I extend the Katz and Shaprio (1985) model for a network industry to examine prices of firm serving multiple markets in a network industry. I examine an oligopoly model in which firms choose output levels, quality within a network (internal quality), and quality between networks (external quality). There are two markets and positive network externalities between the markets. When a single firm serves multiple markets with network externalities, the firm internalizes network externalities and may choose higher levels of output than separate firms would choose. (Jamison, 2001) When a firm internalizes network externalities, its extra revenue from an increase in output in market A reflects not only the effects on price and quantity sold in market A, but also the effects that higher market demands have on prices in other markets in which the firm operates. The higher prices in these other markets provide extra revenue from market A's output. I call this additional revenue from market A's output the marginal extra-market revenue. Positive marginal extra-market revenue may cause the firm to have prices that are below marginal cost in market A. These facts imply that belowcost pricing may increase welfare. I find results that revise conclusions in the current literature. I find that a firm would be willing to price below marginal cost in market A even if the firm had no prospects of recovering these short-term losses from that market. This result holds because the firm benefits from the higher demand in market B, which can make below-cost pricing optimal even in a single-period game. Furthermore, a single-market rival in market A can receive non-negative profits despite the multimarket firm’s below-cost price, but the rival’s output in market A must be greater than
3
the multimarket firm’s combined output in both markets A and B. Under similar circumstances, the multimarket firm would prefer that its rival choose to serve market A. Lastly, a traditional measure of market power, the Lerner Index, is an unreliable indicator of market power for the multimarket firm because the index reflects the effects of marginal extra-market revenue. The firm’s Lerner Index for market A may be zero or negative even when the firm is a monopoly. The analysis proceeds as follows. Section 2 describes the model, which is similar to the model developed in Jamison (2001). Section 3 provides the primary results regarding below-cost prices. Section 4 describes the effects on the Lerner Index. Section 5 is the conclusion. All proofs are in the Appendix.
2. The Model I consider a game in which consumers form expectations about each firm's output and quality choices in the first stage. Firms perfectly observe these expectations and choose qualities simultaneously in the second stage. Firms set quantities simultaneously in the third stage and these quantities generate market-clearing prices. Lastly, customers choose their preferred providers.
2.1. Demand and Supply There are two markets for the network service, which I designate as markets A and B. A market is a customer group, such as customers located in a specific geographic region or that purchase a particular product with network externalities, such as database software. Customers cannot migrate between markets to buy the service. Each customer buys at most a single unit of output. Each market m has a finite number of customers Q m .
4
As many as two firms may operate in the model. The firms are labeled 1 and 2. Firm 1 serves both markets and firm 2 serves only market A. qi,m ≥ 0 will denote the number of customers that firm i serves in market m. q will denote the vector of all outputs of all firms in all markets. I assume that firms “interconnect” their networks. In the setting of physical communications networks, this interconnection would be the lines and technical arrangements that allow customers to communicate. In the setting of virtual networks, such as computer software, this interconnection could be interpreted as features that allow customers to benefit from other customers. For example, software providers may create features that allow spreadsheet users to exchange data with database users. Let θ i ,i ∈ [0,θ ] represent firm i’s quality choice for communications between its customers, for all i = 1, 2, and let θ i , j ∈ [0,θ ] represent firm i’s quality choice for external interconnection between its network and j’s network, for j ≠ i. For simplicity, I assume that each firm chooses a single internal quality and a single external quality for interconnecting with each other firm. Let θi represent the vector of i’s quality choices. Quality includes such things as capacity for customers of physical networks to exchange messages, and features, such as instant messaging. A choice of zero represents a refusal to interconnect. Network quality is perfectly observable to firms and customers alike. Furthermore, only one quality choice can prevail for each network interconnection. This is because only a single physical capacity can exist at a single point of interconnection, and if a technical feature cannot be used by customers of network i to communicate with customers of network j, then neither can customers of network j use the feature to communicate with customers of network i. It is possible that firms can either bargain
5
to establish external quality or let the firm with the lower external quality preference choose it (Cr-mer et al., 2000). As will become evident in the next section, this paper’s analysis is simpler if I assume that the firm with the lower external quality preference chooses it, so I adopt this
{
}
* * assumption. Let θˆ1, 2 min θ 1, 2 ,θ 2,1 represent the external quality that prevails.
(
)
The inverse demand curve for firm i in market m can be expressed as p i ,m q, θ i . I consider a market-clearing price at equilibrium, given output and qualities. I assume that price decreases continuously with quantity sold in the market. This will be the case, for example, when there is a continuum of customer types, higher-type customers place more value on communications than lower type customers, and the rate of decrease in value by successively lower customer types is greater in absolute value than the value that customers place on network externalities, i.e., p qi ,jm, m < 0 for i, j = 1, 2, where the subscript represents the first partial derivative. Marginal willingness to pay in one market increases with quantity sold in other markets because of positive network externalities, i.e., p qi ,jm, mˆ > 0 for i, j = 1, 2, and m ≠ mˆ . Price also increases with quality because customers value higher quality network service more than lower quality network service, all other things being equal, i.e., pθi ,im, j > 0 for i, j = 1, 2. I assume that consumers have decreasing marginal utility for the network service. As a result, the marginal value of quality decreases as quality increases and the marginal value of positive network externalities decreases with the quantity sold in other markets. Quality and network externalities interact in that the marginal value of the size of the system increases with quality, i.e., p qi ,jm, mˆ ,θ i , j > 0 for all i, j = 1, 2, and m, mˆ = A, B. To ensure that an internal solution exists for output choices, I assume that each firm’s marginal revenue in a market declines as its rivals
6
increase their output in the market and that each firm’s residual demand curve intersects its marginal cost curve from above (Dixit, 1986). Customers make purchasing decisions after firms have made their quality and output choices. Each customer chooses to purchase from the firm that provides the greatest net consumer surplus. The marginal customer is the customer that is indifferent between buying and not buying the network service in equilibrium when quantity sold is strictly positive (Katz and Shapiro, 1985). At equilibrium, the marginal customer will receive zero net surplus and, for each firm that sells a positive amount, price is equal to the value of the network service to the marginal customer. Prices vary among firms if quality and network size vary among firms. Demand elasticities may also vary between firms in the same market if the firms choose different internal qualities. To illustrate, consider the effects of a marginal increase in the output for firms 1 and 2 in market A on each firm’s price in A. Because of network externalities, the increase in output increases the value of each firm’s network service. The increase in the value of firm 1’s network service depends on its internal quality and the quality of its interconnection with firm 2. The increase in the value of firm 2’s network service likewise depends on its internal quality and the quality of its interconnection with firm 1. Unless these firms’ internal qualities are equal, the increases in value of their individual network services will be unequal. The firm with the higher internal quality experiences the higher increase in value. Lemmas 1 and 2 summarize this result. Lemma 1: If firms 1 and 2 change their external quality from a given level to a level equal to firm 2’s internal quality, and if firm 1 changes its internal quality from a given level to a level equal to the original external quality, then each customer in market A will place the
7
same value on firm 1’s network service after the quality change as it placed on firm 2’s network service before the quality change, i.e.,
(
*
*
)
(
*
*
)
*
*
pθ1,1A,1 ⋅ θ 1, 2 − θ 1,1 + pθ1,1A, 2 ⋅ θ 2, 2 − θ 1, 2 = p 2, A − p 1, A . Lemma 1 holds because quality and network size determine the value customers receive. The value that each customer would receive firm 1’s network service is determined by the size of firm 1’s network and firm 1’s internal quality, and the size of firm 2’s network and firm 1’s external quality. Likewise, the value that each customer would receive from firm 2’s network service is determined the size of firm 1’s network and firm 2’s external quality, and the size of firm 2’s network and firm 2’s internal quality. If firm 1 adopts an internal quality that is equal to its external quality and an external quality that is equal to firm 2’s internal quality, then each customer values the firms’ services equally. Lemma 2. In the duopoly situation, the price elasticity of demand for the firm with the higher internal quality is more inelastic than for the firm with the lower internal quality. That is to say, if θ i ,i > θ
j, j
, then ε i , A < ε j , A , where firm’s i and j are rivals in market A, ε i, A is the
price elasticity of demand for firm i, ε j , A is the price elasticity of demand for firm j, and price elasticities are expressed as absolute values. Costs for production are separable from costs for quality. Firm i incurs fixed costs K i,m ≥ 0 in market m and a constant marginal cost ci,m > 0 of production. For simplicity, there are no
( )
economies of scope across markets. G i θ i represents firm i’s cost function for quality.1 In the case of physical networks, firm i establishes costly physical connections between customers and
1
For simplicity, I assume that quality costs are independent of the number of customers and the number of network interconnections. Jamison (2001) examines quality choices in a situation where the number of network interconnection affects the cost of quality.
8
networks and chooses service features that customers can use. In the case of software, there may be no extra costs for a technical feature, such as exporting pure text files, to be available for interfacing with extra software packages. To accommodate this range of possibilities, I assume that the cost of interconnection is weakly increasing in quality and is quasiconvex, i.e., Gθi i , j ≥ 0 and Gθi i , j ,θ i , j ≥ 0 for i, j = 1, 2. Each firm takes its rivals' quantity choices as given when it chooses its own quantity levels. Firm 1’s profit maximization problem can be written as:
π1 = max 1 1 q ,θ
∑ ((p B
1, m
m= A
)
)
− c1, m ⋅ q 1,m − K 1,m − G 1
[ ]
θ 1,i ∈ 0,θ
subject to
for i = 1, 2
Q m ≥ q 1,m ≥ 0
(1)
for m = A, B.
Firm 2’s profit maximization problem can be written as:
(
)
max π 2 = p 2, A − c 2, A ⋅ q 2, A − K 2, A − G 2
q 2 , A ,θ 2
subject to
[ ]
θ 2,i ∈ 0,θ QA ≥q
2, A
for i = 1, 2
(2)
≥ 0.
When firm 1 chooses its profit maximizing outputs, its output choice for market m ≠ mˆ reflects the marginal extra-market revenue, p 1q,1m,ˆm ⋅ q 1,mˆ , which is strictly positive and represents the portion of network externalities between markets m and mˆ that firm 1 internalizes when it operates in both markets. Firm 2, which operates in only one market, does not internalize network externalities between markets.
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2.2 Welfare The surplus a customer receives from purchasing depends on the innate value the customer places on the network service, on the internal and external quality choices of the customer's network supplier, and the total number of customers who purchase the network services. In each market, a customer only purchases if he values the service at least as much as the marginal customer does. Integrating over all customers who purchase and summing over all firms and all markets gives the net consumer surplus:
V
net
≡
i ,m 2 q
∑∑ ∫ (p (qˆ , θ )− p (q, θ ))dqˆ B
i ,m
m = A i =1
i
i ,m
i
i,m
−T i ,
0
where qˆ is the vector of all qˆ i ,m , and weighted social welfare: 2
(
)
Z ≡ α ⋅ V net + (1 − α ) ⋅ ∑ π i + T i , i =1
where α = [0, 1] is the weight given to net consumer surplus and T i is a transfer payment from consumers to firm i. If a social planner chooses q and θ to maximize weighted social welfare subject to a non-negative profit constraint for firms, she would: (i) equate the sum of the marginal consumer surplus and the positive network externality to the marginal production cost; and (ii) equate the marginal consumer surplus from quality and the marginal cost of quality.
3. Pricing Below Marginal Cost
In this section I derive conditions under which a profit-maximizing firm operating in multiple markets in a network industry would price below marginal cost in one of its markets. I first develop conditions that would lead the firm to price below marginal cost and identify sufficient conditions for pricing below marginal cost to be predatory. I then examine when
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below-cost pricing would cause a rival to choose to not operate in this market and conditions under which the multimarket firm would prefer that the rival choose to operate in the market. The effect that output in one market has on demand in another market is central to this analysis. Definition 1 provides a useful term for quantifying this effect. Definition 1. The externality elasticity of demand for firm j’s output in market mˆ from firm i’s output in market m is the percentage change in customers’ willingness to pay for firm j’s output in mˆ because of a one percent increase in firm i’s output in m, where i, j = 1, 2 and m, mˆ = A, B. That is to say,
η i ,m , j , mˆ ≡
∂p j ,mˆ q i , m , ⋅ ∂q i ,m p j , mˆ
where η i , m , j ,mˆ is the externality elasticity of demand.
η i , m , j ,mˆ is strictly positive when m ≠ mˆ because an increase in output in market m increases the value that customers place on the network service in market mˆ . When m = mˆ and i = j, η i , m , j ,mˆ is simply the inverse of the firm’s own price elasticity of demand, expressed as a negative number. Assumption 1 ensures an internal solution in market m. Assumption 1: The greatest output that the customers in market m are willing and able to purchase at any price is greater than the product of: (1) the ratio of (i) the product of firm 1’s externality elasticity of demand from market m to market mˆ and its revenue in market mˆ to (ii) firm 1’s market share in market m; and (2) the inverse of the sum of each firm’s ratio of (i) its price in market m minus its market-m price elasticity of demand times its price-cost margin to (ii) its market m price elasticity of demand. That is to say,
11 *
Q
m
>
*
η 1,m ,1,mˆ ⋅ p 1,mˆ ⋅ q1,mˆ s 1,m
*
*
(
2 p i ,m * − ε i , m * ⋅ p i ,m * − c i ,m ⋅∑ * i =1 ε i ,m
)
−1
.
(3)
If there is only one firm in market m, then (3) simplifies to: *
Q
m
>
*
*
ε 1,m ⋅ η 1,m ,1,mˆ ⋅ p 1,mˆ ⋅ q 1,mˆ *
*
(
*
p 1,m − ε 1, m ⋅ p 1, m − c1, m
*
).
Now consider firm 1’s optimal prices in market A. I set aside for the moment whether firm 2 would exit market A if firm 1’s price is below marginal cost. Proposition 1: If Assumption 1 applies to market A, firm 1’s profit-maximizing price in market A will be below its marginal cost if the product of firm 1’s price elasticity of demand for market A and its externality elasticity of demand from its output in market A to market B is greater than the ratio of its market A revenue to its market B revenue, i.e., *
*
1, B *
1, B *
p 1, A ⋅ q 1, A p
⋅q
< η 1, A,1, B ⋅ ε 1, A .
(4)
Furthermore, if firm 1’s price in market A is below marginal cost and the inequality in (4) is reversed, then firm 1’s prices could be considered predatory because firm 1’s profit maximizing price would be above marginal cost. The left-hand side of (4) represents the relative value of the markets to firm 1. Expressing the right hand side of (4) as − η
1, A,1, B
η 1, A,1, A
shows that it is a ratio of the relative
effects of a change in firm 1’s output in market A on prices in markets B and A. If (4) and Assumption 1 apply, firm 1 is willing to price below marginal cost in market A because the below-cost price stimulates sales in market A, which stimulates demand in market B. This higher demand increases 1’s profits in market B by an amount that is greater in absolute value than firm
12
1’s greater losses in market A. Consumers are better off with this pricing arrangement than with higher prices in market A because the below-cost pricing increases output in all markets. The following corollaries extend Proposition 1. Corollary 1A: If Assumption 1 holds for market A, then a higher externality elasticity of demand from market A to market B results in a lower price in market A. Furthermore, higher revenue in market B results in a lower price in market A. From Corollary 1A, prices below marginal cost occur more when customers place higher value on network externalities than when they place lower value on these externalities and when revenue from the other market is greater. Both higher value for network externalities and higher revenue from market B increase the firm 1’s profits from choosing higher output in market A to stimulate demand in market B. Corollary 1B: If Assumption 1 holds for market A, then a more price-elastic demand in market A results in a lower price in market A. Prices below marginal cost occur more when demand is more price elastic. This is because higher output decreases revenue less in this situation. This decreases the cost (in terms of lost profits in market A) of stimulating demand in market B. If Assumption 1 does not hold for market A, such that the inequality in (3) is reversed, the firms would choose to serve the entire market demand in A. The profit maximizing price for firm 1 in market A is below its marginal cost if and only if (4) holds and the customer that places the lowest value on the network service has a willingness to pay for firm 1’s service that is below marginal cost. If firm 1 could increase its price in market A without decreasing the output that customers are willing to purchase, then its price could be considered predatory because firm 1’s profit-maximizing price is higher than the prevailing price. It does not matter whether
13
Assumption 1 holds for market B. Even if firm 1 serves the entire market demand in B, such that pricing below marginal cost in market A does not increase sales in market B, higher output in market A still increases firm 1’s profits in market B because it increases market-B customers’ willingness to pay. Two elements of the analysis so far raise issues regarding rivals. Proposition 2 below addresses the first issue, namely, whether prices below marginal cost in market A would cause a rival to exit the market. Proposition 3 addresses the second issue, namely, whether a firm would prefer that a rival operate in market A to provide greater market A output to stimulate demand in market B. First consider firm 2’s preference for serving market A. Firm 2 may choose to not serve market A because, as a single-market firm, it does not benefit when higher output in market A stimulates demand in market B. To receive non-negative profits in market A, firm 2’s price/cost margin must be at least as great as the sum of firm 2’s fixed cost and quality cost, divided by output. Proposition 2 explains that firm 2 must have lower marginal operating costs than firm 1, that the marginal customer in market A must place a higher value on firm 2’s network service than it does firm 1’s network service, or both for firm 2 to choose to operate in market A. Before presenting Proposition 2, I consider how quality choices affect the value the marginal customer places on firm 2’s output relative to firm 1’s output in market A. Consider the relationship between a firm’s preferred internal quality and external quality. Jamison (2001) demonstrates that when firms i and j increase the external quality between their networks, firm i increases its output if Assumption 2 holds. Firm i decreases its output if the inequality in (5) is reversed.
14
Assumption 2: An increase (respectively, decrease) in external quality increases (respectively, decreases) the firm's marginal revenue by an amount that is greater in absolute value than the decrease (respectively, increase) in the firm's marginal extra-market revenue. That is to say, *
*
*
*
*
pθj ,jm,i + p qj ,jm, m ,θ j ,i q j ,m > p qj ,jm, mˆ ,θ j ,i q j ,mˆ .
(5)
Jamison (2001) also demonstrates that firm i would prefer a higher external quality with firm j than internal quality if the effect of an increase (respectively, decrease) in firm i's external quality on firm i's revenue is greater than the effect of an increase (respectively, decrease) in i's internal quality. For example, firm 1 would prefer a higher external quality than internal quality when firm 2’s higher output in market A stimulates demand in market B more than firm 1’s higher market-A output would, and this higher market-B demand makes up for firm 1’s lower profits in market A. This might hold if firm 1 had a small market share in market A and low internal quality relative to its external quality. Furthermore, firm 2 would prefer a higher external quality than internal quality when firm 1’s higher output in market B stimulates demand in market A and the higher market-A demand more than makes up for firm 2’s loss of market share in market A. Now consider the relationship between firm i’s internal quality and firm j’s preferred external quality. Assumption 3: The effect of an increase (respectively, decrease) in firm i's external quality on firm i's revenue is greater than the effect of an increase (respectively, decrease) in j's internal quality on firm j’s revenue. That is to say,
(
*
∂ p i ,m ⋅ q i , m ∑ ∂θ i, j m = A, B
*
)>
(
*
*
)
∂ p j ,m ⋅ q j ,m , ∑ ∂θ j , j m = A, B
15
for i ≠ j. Lemma 3: If Assumption 3 holds, firm i‘s preferred external quality is higher than firm j’s preferred internal quality. Sufficient conditions for Assumption 3 include: (i) firm j's output is greater than firm i's output and (5) is an equality for firm j; and (ii) where i = 1 and j = 2, firm 2's output is at least as great as firm 1's output and firm 2's higher output increases firm 1's revenue in market B more than it decreases firm 1's revenue in market A. Proposition 2: If (4) and Assumption 1 hold for market A, then firm 2 chooses to operate in market A if the higher (or lower) value that the marginal customer places in firm 2’s output relative to firm 1’s output, plus the difference between firm 2’s and firm 1’s marginal operating costs, is at least as great as the sum of firm 2’s average fixed costs, firm 2’s quality cost divided by output, and the absolute value of firm 1’s per-unit loss in market A, i.e., *
∆p + ∆c ≥ p 1, A − c 1, A +
K 2, A + G 2 q 2, A
*
*
,
(6)
where ∆p ≡ p 2, A − p 1, A (recall Lemma 1) is the difference between the marginal customer’s willingness to pay for firm 2’s service and firm 1’s service and ∆c ≡ c1, A − c 2, A . The right hand side of (6) is strictly positive, so the left-hand side must also be strictly positive for firm 2 to choose to operate in market A. Corollaries 2A and 2B explain the relationship between firm 2’s output and ∆p and ∆c. Corollary 2A: If ∆p > 0, then firm 2’s output in market A is greater than firm 1’s combined output in both markets, i.e., if ∆p > 0, then q2,A * > q1,A * + q1,B *.
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Corollary 2A holds because a firm chooses higher quality when the number of customers that its customers can communicate with at that quality is higher. From Lemma 1, ∆p is strictly positive only if firm 2’s internal quality is higher than firm 1’s internal quality, which means that firm 2’s output must be higher than firm 1’s output. Corollary 2B: If (6) holds, then firm 2’s marginal production costs are lower than firm 1’s marginal production costs. Corollary 2B holds because firm 2 must have lower marginal production costs than firm 1 for firm 2 to have higher output than firm 1, which is necessary for firm 2 to have a higher price than firm 1 in market A. As a result, if it is optimal for firm 1 to price below its marginal production costs in market A, then potential rivals in market A will not enter the market unless they have lower marginal production costs. Rearranging terms in (6) shows that the difference between the successful rival’s marginal production costs and firm 1’s marginal production costs must be at least as great as the sum of the absolute value of firm 1’s price/cost margin and the rival’s average fixed and quality costs, less ∆p. Proposition 3: If Assumption 1 holds for market A, then firm 1 would prefer that firm 2 provide strictly positive output in market A, as opposed to exiting market A, if the absolute value of the ratio of (i) the externality elasticity of demand from firm 2’s output to firm 1’s market A output to (ii) the externality elasticity of demand from firm 2’s output to firm 1’s market B output, is less than the ratio of firm 1’s revenue in markets B and A, i.e., *
*
η 2, A,1, A p 1, B ⋅ q 1, B . < * * η 2, A,1, B p 1, A ⋅ q 1, A
(7)
17
Proposition 3 holds because strictly positive output by firm 2 in market A stimulates more profits for firm 1 in market B than it decreases firm 1’s profits in market A. In a sense, firm 2 absorbs some of the negative effects of lower prices in market A, which benefits firm 1. The lefthand side of (7) is not equal to 1 only if the marginal customer in market A values communication with customers in market A differently than does the marginal customer in market B. This could hold if customers in the two markets are innately different in how they value the network service or if the two markets are identical in terms of customer preferences, but the output is greater in one market than in the other. If the marginal customers are identical in the two markets, so that the left hand side of (7) is equal to 1, then Proposition 2 holds as long as firm 1’s revenue is higher in market B than in market A, which would seem likely because firm 1’s market share would be lower in market A. Although not the same, (7) is consistent with (6). Consider the situation where the markets have identical demands. If (6) holds, then firm 1’s revenue in market A would be small compared to its revenue in market B, which increases the right hand side of (7). Furthermore, firm 2’s marginal operating costs are lower than firm 1’s (recall Corollary 2B), which means that output in market A could be greater than the output in market B. (Jamison, 2001; Bergstrom and Varian, 1985) As a result, the marginal customer in market A places lower value on the network service than the marginal customer in market B, which means that the left-hand side of (7) is less than 1. Therefore, firm 1’s profits may be higher if it has a large rival in market A than if it is a monopolist in market A. Firm 1 may be interested in accommodating the entry of such a rival and, if Assumption 1 and (4) hold, may also be willing to enter market A to compete with a large rival even if firm 1 receives negative profits in the market.
18
4. Lerner Index In this section I examine how network externalities between markets affect a well-known measure of the intensity of competition in a market, the Lerner index. The Lerner index for firm i = 1, 2, for market m is defined by the usual formula: i ,m
L
p i ,m − c i ,m . ≡ p i ,m
As is well known, the Lerner index for a firm at market equilibrium can be calculated as the inverse of the elasticity of demand for the firm’s output. That is to say, L2, A =
1
ε
2, A *
.
(8)
(8) does not apply to firm 1 because firm 1’s Lerner index should reflect the network externalities that exist between markets and that firm 1 internalizes. This leads to Lemma 4. Lemma 4. If a firm serves more than one market for a network service and there are network externalities between the markets, the firm’s Lerner index for market m is equal to sum of (i) the ratio of the firm’s market share to the market price elasticity of demand in market m, and (ii) for each other market in which the firm operates, the externality elasticity of demand from the firm’s output in market m to this other market, times the ratio of the firm’s revenue in this other market to the firm’s revenue in market m. That is to say, 1, m
L
=
1
ε 1,m
*
−η
1, m ,1, mˆ *
⋅
*
*
*
*
p 1,mˆ ⋅ q 1,mˆ p 1,m ⋅ q 1,m
.
(9)
In contrast with a non-network firm or a single-market network firm, a multi-market network firm’s Lerner index for market m reflects more than just the market structure in m. The index is decreasing in the value its customers in market mˆ place on the network externalities
19
from its customers in market m and is decreasing in the firm’s revenues from other markets. These relationships hold because increases in value for network externalities and increases in demand in other markets make it profitable for the firm to stimulate demand in its other markets by lowering its price in market m. Furthermore, the term η
1, m ,1, mˆ *
⋅
*
*
*
*
p1,mˆ ⋅ q 1,mˆ p1,m ⋅ q 1,m
in (9) is repeated
for each additional market in which firm 1 operates. As a result, an increase in the number of other markets lowers firm 1’s Lerner index for market m by increasing the opportunities for the firm to profit by lowering its price in market m. In this situation, or when Proposition 1 applies, firm 1’s Lerner Index for market m would be negative even if firm 1 is a monopoly.
4. Conclusion
In this paper, I analyze a firm’s incentive to price below marginal cost in market A to stimulate demand in market B in a network industry. I find that a firm will have below-cost prices in market A more when demand is price elastic than when it is price inelastic, and when revenue in market A is low relative to revenue in market B. I also find conditions under which the firm will prefer that a rival also operate in market A and conditions under which the rival would choose to operate in market A even though the firm has prices below marginal cost. These two sets of conditions are compatible. Lastly, I describe how a well-known gauge of market competition, the Lerner Index, is affected by multimarket firms in a network industry. This research calls into question some concerns of predatory pricing in network industries, but further research is needed. I have not incorporated the dynamic effects analyzed by Fudenberg and Tirole (2000) and Farrell and Katz (2001). I have also not considered the
20
situation where the two markets are actually two customer types in the same market and the firm must develop optional tariffs to entice customers to reveal their types. Lastly, while I show that the Lerner Index is affected by this analysis and that it can give erroneous results, I do not develop an alternative index.
21
REFERENCES Bergstrom, T., and Varian, H. R. “When are Nash Equilibria Independent of the Distribution of Agents’ Characteristics.” Review of Economic Studies, Vol. 52 (1985), pp. 715-718. Cr-mer J., Rey, P., and Tirole, J. "Connectivity in the Commercial Internet." Journal of Industrial Economics, Vol. 48 (2000), pp. 433-472. Dixit, A. “Comparative Statics for Oligopoly.” International Economic Review, Vol. 27 (1986), pp. 107-122. Evens, D. S., and Schmalensee, R. “A Guide to the Antitrust Economics of Networks.” Antitrust, Vol. 10 (1996), pp. 36-40. Farrell, J., and Katz, M. L. “Innovation, Rent Extraction, and Integration in Systems of Markets,” Journal of Industrial Economics ***need cite*** Farrell, J., and Katz, M. L. “Competition or Predation? Schumpeterian Rivalry in Network Markets.” Mimeo, Competitive Policy Center, University of California, Berkeley, 2001. Fudenberg, D., and Tirole, J. “Pricing a Network Good to Deter Entry.” Journal of Industrial Economics, Vol. 48 (2000), pp. 373-390. Gabel, D. “Competition in a Network Industry: The Telephone Industry, 1894-1910.” Journal of Economic History, Vol. 54 (1994), pp. 543-572. Gilbert, R. J., and Katz, M. L. “An Economist’s Guide to U.S. v Microsoft.” Mimeo, Competitive Policy Center, University of California, Berkeley, 2001. Jamison, M.A. “Network Externalities, Mergers, and Industry Concentration.” Mimeo, Public Utility Research Center, University of Florida, 2001. Katz, M. L., and Shapiro, C. “Network Externalities, Competition, and Compatibility.” American Economic Review, Vol. 75 (1985), pp. 424-440.
22
APPENDIX
Proof of Proposition 1. Up to two firms, 1 and 2, operate in market m and firm 1 also operates in market mˆ . From (1), if Q m binds the firms’ optimal choices of output, then at least one of the firms could increase its profits if Q m were not binding. Therefore, combining the firms’ first order conditions for choosing output in market m gives: *
*
*
*
*
*
*
*
p1, m + p 1q,1m, m ⋅ s 1,m ⋅ Q m − c1,m + p 1q,1m,ˆm ⋅ q 1,mˆ + p 2,m + p q22, m, m ⋅ s 2,m ⋅ Q m − c 2,m > 0 . where si,m represents firm i’s market share in market m. Rearranging terms gives: *
*
*
p 1,m − c 1,m + p 2,m − c 2,m + p q1,1m,ˆm ⋅ q 1, mˆ −p
1, m * q1 , m
⋅s
1, m *
*
−p
2,m * q 2,m
⋅s
1
*
p
*
p 1,m − c1,m + p 2,m − c 2,m + p 1q,1m,ˆm ⋅ q 1,mˆ
*
⋅
q
1, m *
⋅ s 1,m +
ε −c
1, m
Recalling that η i ,m , j , mˆ ≡
p
1, m *
ε
2,m *
⋅
q
2, m *
*
p
p
>Qm, *
1, m *
*
1, m *
p
2,m *
*
⋅ s 2,m
ε
Qm
1
*
>Qm,
2,m *
p 1,m − c 1,m + p 2,m − c 2,m + p q1,1m,ˆm ⋅ q 1, mˆ 1, m *
*
+p
2,m *
1, m *
1, m *
−c
*
+
2,m
p
2,m *
ε 2,m +p
> Q m , and
*
1, mˆ * q1 , m
⋅q
1, mˆ *
>
p 1,m
ε 1,m
*
*
+
p 2, m
ε 2, m
*
*
.
∂p j ,mˆ q i , m and rearranging terms gives: ⋅ ∂q i ,m p j , mˆ
ε 1,m * − 1 1,m 2,m * − 1 2,m η 1, m ,1,mˆ * ⋅ p 1, mˆ * ⋅ q 1,mˆ * − c + p 2,m * ε −c + >0, * ε 1,m * ε 2, m * s 1, m ⋅ Q m
23 *
*
η 1,m ,1,mˆ ⋅ p 1,mˆ ⋅ q1,mˆ *
s 1,m ⋅ Q m *
s 1,m ⋅ Q m *
*
η 1,m ,1,mˆ ⋅ p 1,mˆ ⋅ q1, mˆ
*
*
1, m * 2,m * *1− ε + c1,m + p 2,m * 1 − ε + c 2,m , > p 1, m ε 1,m * ε 2,m *
p 1,m * − ε 1, m * ⋅ p 1,m * + ε 1, m * ⋅ c1,m p 2,m * − ε 2,m * p 2,m * + ε 2,m * c 2,m
q 1,m
*
q 1,mˆ
*
,
*
∂p 1,mˆ ∂q 1,m q 1, m , − 1,m ⋅ 1,m > * ∂q ∂p q 1, mˆ *
*
*
*
*
q 1,m ∂p 1,mˆ q 1,m p 1,mˆ ∂q 1, m p 1, m q 1,m , − 1,m ⋅ ⋅ ⋅ ⋅ ⋅ > * * 1, m * * * ∂q p 1,mˆ q 1,m ∂p q 1,m p 1,m q 1,mˆ
η
1, m ,1, mˆ *
⋅ε
1, m *
⋅
p 1,mˆ
*
p 1,m
*
>
q 1,m
*
q 1,mˆ
*
,
24
η
1, m ,1, mˆ *
*
⋅ εˆ
p 1,m ⋅ q 1, m
εˆ
1, m *
1, m *
>
*
*
*
*
, and
*
*
p 1, m ⋅ q 1,m p 1, mˆ ⋅ q 1,mˆ
* *
< η 1,m ,1,mˆ ⋅ p 1,mˆ ⋅ q 1,mˆ ,
which proves Proposition 1. Proof of Corollary 1A. Consider the first order condition given in (A1). An exogenous increase in η 1, m ,1,mˆ implies a higher p 1q,1m,ˆm . This means that firm 1 would choose a higher output to decrease its marginal revenue in market m. Proof of Lemma 3. From (1) and (2), firm j’s first order conditions for optimally choosing output include: p j ,m + p qj ,jm, m q j , m − c j ,m + p qj ,jm,ˆm q j ,mˆ = 0 , where, in the case of firm 2, pqj ,jm,ˆm q j ,mˆ = 0 . Totally differentiating the first order conditions gives: pθji,,mj + p qj ,jm, m ,θ i , j q j ,m + p qj ,jm,ˆm ,θ i , j q j ,mˆ dq j ,m , and =− 2 p qj ,jm, m + p qj ,jm, m ,q j , m q j ,m + p qj ,jm,ˆm ,q j , m q j ,mˆ dθ i , j pθji,,mˆj + p qj ,jm,ˆmˆ ,θ i , j q j ,mˆ + p qj ,jm, mˆ ,θ i , j q j ,m dq j ,mˆ . =− 2 p qj ,jm,ˆmˆ + p qj ,jm,ˆmˆ ,q j , mˆ q j ,mˆ + p qj ,jm, mˆ ,q j , mˆ q j ,m dθ i , j The sign of firm j's reaction in its output to a change in external quality depends on the sign of the numerator because 2 p qj ,jm, m + p qj ,jm, m ,q j , m q j , m is the slope of firm j's marginal revenue curve with respect to its own output, which is negative by assumption, and p qj ,jm, mˆ , q j , mˆ q j ,m is the slope of its marginal extra-market revenue, which is also negative by assumption.
25
Firm i’s first order conditions for optimally choosing θ i,i and θ i,j include:
∑p
m = A, B
∑
q
i ,m * θ i ,i
*
q i ,m = Gθi i ,i , and
(A2)
j,A j,B i ,m * * ∂q i , m * ∂q pθ i , j + p qi ,mj , A + pq j ,B ∂θ i , j ∂θ i , j
i ,m *
m = A, B
= Gθi i , j .
(A3)
From (A2) and (A3), if firm i‘s preferred external quality is higher than firm j’s preferred
( )
( ), or
*
j, j *
internal quality, then Gθi i , j θ i , j > Gθj j , j θ
∑
q
i ,m *
m = A, B
j,A j,B i ,m * * ∂q i , m * ∂q pθ i , j + p qi ,mj , A + pq j ,B ∂θ i , j ∂θ i , j
* * > ∑ pθj ,jm, j q j ,m . m = A, B
This confirms Lemma 3. Proof of Proposition 2. From (2), firm 2 earns non-negative profits in market A if and only if:
(p
)
2, A *
p
*
*
− c 2, A q 2, A ≥ F 2, A + G 2 , or 2, A *
−c
2, A
F 2, A + G 2
≥
q 2, A
*
*
.
Substituting gives: p
2, A *
−p
1, A *
−c
2, A
+c
1, A
∆p + ∆c ≥ p
≥c
1, A *
1, A
−c
−p
1, A
+
1, A *
+
F 2, A + G 2 q 2, A
F 2, A + G 2 q 2, A
*
*
*
, and
*
,
which confirms Proposition 2. Proof of Corollaries 2A and 2B. The equality in Lemma 1 can also be expressed as: p 1, A
*
q
1, A
+q
1, B
=q
2 , A*
,q
2, A
=q
1 , A*
+q
1, B *
,θ
1 ,1
=θ
2 , 2*
= p 2, A .
(A4)
26
The left-hand side of (A4) indicates that firm 1’s customers would now access q2,A customers with a quality of θˆ1, 2 instead of q1,A + q1,B customers. This increases (respectively, decreases) value for the customers of firm 1 if q2,A > q1,A + q1,B (respectively, q2,A < q1,A + q1,B). The left-hand side of (A4) also indicates that firm 1’s customers would now access q2,A customers with a quality of θ 2, 2 instead of q1,A + q1,B customers with a quality of θ 1,1 . Because quality and the number of customers accessed are strategic complements, this increases (respectively, decreases) value for the customers of firm 1 if q2,A > q1,A + q1,B (respectively, q2,A < q1,A + q1,B). As a result, p2,A* > p1,A * if and only if q2,A > q1,A + q1,B. This confirms Corollary 2A. Because firm 1 internalizes network externalities that firm 2 does not, firm 2 has greater output than firm 1 in market A only if firm 2 has lower marginal production costs than firm 1 in market A. This confirms Corollary 2B. Proof of Proposition 3. From (1), the effect of an exogenous increase in firm 2’s output has a positive effect on firm 1’s profits if and only if: p 1q,2A, A ⋅ q 1, A − p
1, B q2, A
⋅q
1, B
(
)
dq 1, A 1, A dq 1, B 1, A 1, A 1, A p p c + − + ⋅ p 1,1A, B ⋅ q 1, A + 1, A ⋅ q q dq 2, A dq 2, A q
(
)
dq 1, B dq 1, A − 2, A p 1, B + p 1q,1B, B ⋅ q1, B − c1, B + 2, A ⋅ p 1q,1B, A ⋅ q 1, B > 0 dq dq
.
Substituting from firm 1’s first order conditions for choosing its optimal output gives: p 1q,2A, A ⋅ q 1, A − p
1, B q2, A
⋅q
1, B
dq 1, A 1, B 1, B dq 1, B ⋅ p q1, A ⋅ q + 2, A ⋅ p 1q,1A, B ⋅ q 1, A + 2, A dq dq
dq 1, B dq 1, A 1, B 1, B 1, A 1, A − 2 , A ⋅ p q1 , B ⋅ q + 2 , A ⋅ p q1 , A ⋅ q > 0 dq dq p 1q,2A, A ⋅ q 1, A + p q1,2B, A ⋅ q 1, B > 0 ,
,
27
η
2 , A,1, A
p 1, A ⋅ q 1, A p 1, B ⋅ q 1, B 2 , A,1, B ⋅ +η ⋅ > 0 , and q 2, A q 2, A −
η 2, A,1, A p 1, B ⋅ q 1, B . < η 2, A,1, B p 1, A ⋅ q 1, A
This confirms Proposition 3. Proof of Lemma 4. Dividing by firm 1’s price in market m and rearranging terms in (A1) gives: p 1,m − c 1,m q 1, m q 1,mˆ 1, m 1, mˆ , and p p = − − 1, m ⋅ 1, m ⋅ q q p 1,m p 1,m p 1,m
η 1,m ,1, mˆ p1, mˆ ⋅ q1, mˆ 1 p 1,m − c 1,m = − ⋅ 1, m 1, m . ε 1, m ε 1,mˆ p 1,m p ⋅q This confirms Lemma 4.
