Market Expansion in Duopoly

Transcription

1 Market Expansion in Duopoly ndrei Hagiu 1 and Wei Sun 2 ugust 30, 2016 bstract We study the eect of market expansion due to exogenous factors on a duopoly's equilibrium prices and prots. The total eect on each rm's prots consists of a direct eect, proportional to the demand derived from the new market segment, and a competitive eect, proportional to the change in the rival rm's equilibrium price. We show that the sign of the total eect on prots depends only on rst-order and second-order demand elasticities, and on the ratio between how important the new market segment is relative to the existing segments for the two rms. In particular, this sign can be independent of the magnitude of the (positive) direct eect. The total eect on each rm's price consists of a non-strategic eect (the adjustment induced by the demand from the new market segment) and a strategic eect (the response to the price adjustment by the rival rm). Due to the strategic eect, market expansion to a more price elastic segment for both rms can lead to a higher equilibrium price for one rm. We also derive implications for empirical estimations of the eects of market expansion. JEL classication: L13, M21 Keywords: market expansion, duopoly. We thank Julian Wright and seminar participants at the National University of Singapore for their helpful comments. 1 MIT Sloan School Of Management, Cambridge M 02142, ahagiu@mit.edu 2 Harvard usiness School, oston, M 02163, wsun@hbs.edu 1

2 1 Introduction Oligopolies can see their relevant markets expand unexpectedly due to exogenous factors, such as decreases in the price (or increases in the quality) of an underlying platform or a complementary technology, the appearance of new information or cultural trends that change customer perceptions, etc. For example, as videogames gradually entered the cultural mainstream, the addressable market for makers of home videogame consoles (Microsoft, Nintendo, Sega, Sony) expanded from teenage boys in the late 1980s, to older males throughout the 1990s, and nally to just about everyone else after the release of the Nintendo Wii in Similarly, the cloud infrastructure and computing services oered by mazon, Google, IM anicrosoft initially only appealed to start-ups and small companies. Due to the emergence of virtualization and other complementary technologies that helped alleviate security and reliability concerns for cloud services, the relevant market has signicantly expanded to also include large corporations. What is the eect on rm prots of an exogenous market expansion in the absence of new entry? Whereas market expansion always increases monopoly prots, the eect on oligopoly prots can go either way. t a high level, market expansion has both a direct eect and a competitive eect on each rm's prot. The direct eect is always positive (more customers to sell to), while the competitive eect is negative (respectively, positive) if the competitor's new equilibrium price is lower (respectively, higher). If the competitive eect is negative and outweighs the direct eect, market expansion decreases equilibrium protswe derive precise conditions under which this occurs. lthough we focus on a competitive duopoly, our formal analysis and insights also apply when the two rms are complementors (e.g. Microsoft and Intel in the market for personal computers). In our model, there is a (continuous) set of existing market segments and we conceptualize market expansion as the addition of a new (innitesimal) market segment. In this set-up, whether market expansion increases or decreases equilibrium prices and prots depends only on the rstorder and second-order demand elasticities, and on the ratio between how important the new market segment is relative to the existing segments for the two rms. n important and novel result of our analysis is that the sign of the total eect on prots can be independent of the 2

3 magnitude of the direct eect, which is proportional to the size of the new market segment and is always positive. For instance, for two symmetric rms, the size of the new market segment has no bearing on whether market expansion decreases or increases equilibrium prots. In fact, if the sign of the prot change is negative, then increasing the new market segment's size exacerbates the negative impact on prots. Market expansion can also have a counter-intuitive total eect on equilibrium prices. On the one hand, if the new market segment is more price elastic for the two rms, then, assuming no price discrimination, each rm is inclined to decrease its price for the entire marketthis is the non-strategic eect. However, there is also a strategic eect: each rm's price decrease aects the competitor's demand elasticity, prompting the competitor to further adjust its pricedownward if the two prices are strategic complements, and upward if they are strategic substitutes. Thus, if the prices exhibit suciently strong strategic substitutability, the strategic eect may overturn the non-strategic eect, leading to a counter-intuitive result for one of the two rms, i.e. market expansion to a more price elastic segment leads to a price increase. Our analysis has important implications for empirical studies. In particular, both rst-order and second-order price elasticities must be accurately estimated for the existing market in order to correctly predict the sign of the total eects of market expansion on prices and prots. Furthermore, we show that dierent sources of consumer heterogeneity aect the outcome dierently. Thus, empirical models that accomodate only one type of consumer heterogeneity can yield qualitatively wrong preditions regarding the eect of market expansion. The next section discusses the related literature. Section 3 sets up our model and derives the expressions for the total eects of market expansion on equilibrium prices and prots. To build intuition for interpreting these expressions, Section 4 analyzes and interprets a simpler version of our model, by shutting down the strategic eect. Section 5 returns to the full equilibrium, interprets the expressions obtained in Section 3, and derives additional results. Section 6 briey shows that our analysis also applies to complementors. Section 7 illustrates our main results with simple examples based on linear demand. Section 8 discusses the implications of our theoretical analysis for empirical estimations. Finally, Section 9 concludes. 3

4 2 Related literature The idea that seemingly benecial exogenous shocks (i.e. those that would unambiguously raise prots for a monopoly rm) may decrease the equilibrium prots of oligopoly rms even in the absence of entry dates back to at least ulow et al. (1985). Specically, ulow et al. (1985) showed that an exogenous cost decrease in one market for a rm that benets from economies of scale across two markets can hurt that rm's overall protability if it induces a suciently aggressive competitive response by its rival in the second market. This counter-intuitive phenomenon can only occur if the strategic eecti.e. the change in prots due to the induced change in the choice of strategic variable (price or quantity) by the competitor in the second marketis negative and outweighs the positive direct eect of the cost decrease in the rst market. The sign of the strategic eect is determined by whether the two rms' products are strategic complements or substitutes. In contrast, a key contribution of our analysis is to show that the strategic eect is not essential for obtaining the counter-intuitive result that market expansion can decrease equilibrium prots. The reason is that the exogenous shock in our framework (market expansion) aects both rms directly. This implies that both rms experience a direct eect and a competitive eect, the latter of which can be further decomposed in a strategic eect and a non-strategic eect. The non-strategic eect can deliver the counter-intuitive result on its own. Meanwhile, in ulow et al. (1985) the exogenous shock is unilateral, i.e. only one rm experiences a direct eect. This implies that the competitive eect is reduced to the strategic eect, which then becomes crucial to the result. Furthermore, ulow et al. (1985) informally suggest that for the cost decrease to lower total prots, the (positive) direct eect has to be small relative to the strategic eect. 3 y contrast, we completely characterize the necessary and sucient conditions for market expansion to lower prots and show that in our framework, the sign of the total eect of market expansion on prots can be independent of the magnitude of the (positive) direct eect. In particular, it is possible that increasing the magnitude of the direct eect exacerbates the negative total eect. lso related is Seade (1985), who shows that in an oligopolistic setting with no entry, an increase in marginal cost common to all competitors (e.g. tax increase) can sometimes lead to higher prot 3 Specically, sales in the market where the focal rm experiences the cost decrease must be suciently small relative to sales in the oligopolistic market. 4

5 for all competitors. While in this set-up all rms face a direct eect, a non-strategic competitive eect and a strategic competitive eect, the mechanism with cost shocks is quite dierent from the one with market expansion that we study. Furthermore, Seade (1985) relies on a Cournot quantity competition model, whereas we use a general ertrand competition model, which yields more tractable conditions and is easier to interpret. Finally, at a high level, our paper is also reminiscent of an earlier literature, which points out that increased competition in an oligopolistic setting (i.e. a higher number of competitors), can result in higher equilibrium pricessee, for example, Salop (1979), Satterthwaite (1979), Rosenthal (1980). We obtain a somewhat similar result, namely that market expansion to a more elastic segment can result in a price increase for one of the competitors, but the underlying mechanisms are very dierent. 3 Set up and derivation of total eects Consider a duopoly with rms and. Throughout sections 3, 4 and 5, the two rms are assumed to be competitors. In section 6, we assume the two rms are complementors (and each of them is a monopolist in its market)the analysis is almost identical, with the exception of changes in the sign of certain eects. Demands for the two rival rms are Q (p, p, M) = M 0 q (p, p, m) dm and Q (p, p, M) = M 0 q (p, p, m) dm, where M [0, 1] denotes the number of active market segments, and q i (p i, p i, m) is positive, continuously dierentiable in (p i, p i ), decreasing in p i and increasing in p i (rms are competitors) for i {, }. This implies that Q i (p i, p i, M) is also positive, continuously dierentiable in (p i, p i ), decreasing in p i and increasing in p i for i {, }. Furthermore, we assume the right derivative of Q i (p i, p i, M) with respect to M is well-dened for every M [0, 1], so that + Q i (p i, p i, M) M = q i (p i, p i, M) 5

6 In particular, we allow q i (p i, p i, M) to be discontinuous in M. Given that we are interested in market expansion throughout the paper, i.e. an increase in M, only the right derivative of Q i (p i, p i, M) in M is relevant. Note also that we have implicitly assumed m is uniformly distributed over [0, 1], which is without loss of generality: any density function can be simply folded into the underlying demands q (p, p, m) and q (p, p, m). Intuitively, the market consists of a continuum of segments m [0, 1]; demands in each segment m are q (p, p, m) and q (p, p, m). Throughout the paper, we are interested in the eect of increasing the number M of active market segments on equilibrium rm prots. This increase can result from an unexpected decrease in the price of an underlying technology or of a complementary product, or from the availability of new information that positively changes customer perceptions. We assume that the rms cannot price discriminate across the dierent market segments. For simplicity, we also assume both rms have zero marginal costs: in appendix we show that this assumption involves no loss of substance. The prots of the two rms are π = p Q (p, p, M) and π = p Q (p, p, M). The Nash equilibrium with M active market segments is then dened by p (M) = argmax π (p, p, M) and p (M) = argmax π (p, p, M). p p Throughout the paper, we use * only for equilibrium prices and prots. We omit * for quantities: quantities are always evaluated at the prices that appear in the relevant expressions. To ensure well-dened maximization problems and stability of the solutions to the simultaneous pricing game, we assume that the following conditions hold at all equilibrium values: 2 π p 2 < 0 and 2 π p 2 < 0 2 π p 2 2 π p 2 2 π 2 π > 0. (3.1) p p p p 6

7 3.1 Decomposing the eect of market expansion Using the Envelope Theorem for i {, }, we can write the eect of an exogenous market expansion (increase in M) on equilibrium rm prots as d π i = π i + }{{} M direct eect π i d p i p i }{{} competitive eect. (3.2) s the market expands, the customer base expandsat the old equilibrium prices, both rms can simply sell more and earn more prot. This is the direct eect of market expansion and corresponds to the rst term in equation 3.2. It is always positive and can be re-written as π i M = p iq i for i {, }. p i Q i p i The second term in equation 3.2 is the competitive eect of market expansion. Since π i p i = > 0, the sign of the competitive eect is equal to the sign of d p i, i.e. the eect of market expansion on the competitor's price. When the market expands to a demand segment with a dierent price elasticity from the average elasticity of the existing segments, rms adjust their prices accordingly. For instance, if the new market segment is more price elastic, each rm has an incentive to accommodate the new customers with lower prices, which has a negative eect on the other rm's prot. This is the non-strategic eect of market expansion on equilibrium prices. However, there is also a strategic eect: each rm anticipates the other's price decrease, therefore further adjusts its own pricedownward if the two prices are strategic complements (i.e. 2 π p p > 0) or upward if they are strategic substitutes (i.e. 2 π p p < 0). In the latter case, it is possible that the strategic eect overwhelms the non-strategic eect for one of the two rms, so that rm ends up with a higher equilibrium price as a result of market expansion to a more elastic segment (Corollary 3 below shows that this cannot hold for both rms). For convenience, gure 3.1 summarizes the taxonomy of the various eects of market expansion. To determine d p i, we start with the rst-order conditions π p = 0 and π p = 0 (which must 7

8 Fig. 3.1: Decomposition of the Total Eect of Market Expansion hold for all M) and take the total derivatives with respect to M, obtaining 2 π d p p + 2 π d p p + 2 π p p d p + 2 π p p d p + 2 π p M = 0 2 π p M = 0 (3.3) Solving these two equations for d p and d p leads to d p = 1 ( 2 π p p d p = 1 ( 2 π p p 2 π p M 2 π p 2 2 π p M 2 π p 2 ) 2 π p M 2 π p M ), (3.4) where > 0 is the determinant dened in equation 3.1. The term 2 π i p i M in the expression of d p i corresponds to the non-strategic price eect described above, whereas the term 2 π i p i p i corresponds to the strategic price eect. 2 π i p i M 3.2 Elasticities We will characterize the eect of market expansion on equilibrium prices and prots in terms of elasticities. To do so, we need to introduce the following notation (for conciseness, we omit the arguments (p, p, M)): 8

9 average (rst-order) self elasticity ε = log(q ) log(p ) marginal (rst-order) self elasticity ε = log(q ) > 0 log(p ) (rst-order) cross elasticity ε = log(q ) log(p ) > 0 second-order self elasticity second-order cross elasticity ε ε = log(ε ) log(p ) ε ε = log(ε ) log(p ) positive positive positive for competitors; negative for complementors positive positive for strategic substitutes; negative otherwise Tab. 1: First-order and Second-order Elasticities Thus, ε and ε are the rst-order own-price demand elasticities for rms and in the existing market, whereas ε and ε are the own-price elasticities faced by and in the new market segment. The rst-order own-price elasticities determine the prot-maximizing prices: with zero marginal cost, the rst-order condition in price is equivalent to setting the own-price elasticity equal to 1. Meanwhile, second-order elasticities (last two rows in Table 1 ) determine how much the rst-order conditions change with price. For example, 's second-order self elasticity ( ) ε ε quanties 's ability to move its own rst-order condition by changing its price. The larger this elasticity is, the less price adjustment is necessary for to restore its rst-order condition in response to an exogenous shock (e.g. market expansion). The signs of the elasticities are implied by the assumptions made earlier on the demand functions and by the following lemma (all proofs are in the appendix). Lemma 1. For any M, at the price equilibrium (p (M), p (M)) we have 2 π p 2 2 π = Q ε ε p and 2 π p 2 2 π p p = Q p ε ε and = Q ε ε p (3.5) p p = Q p ε ε (3.6) 2 π p M = q (1 ε ) and 2 π p M = q (1 ε ). (3.7) The expressions in 3.5 imply that for second-order conditions to hold at the prot-maximizing 9

10 prices, the second-order self elasticities must be positive, i.e. 4 ε ε > 0 and ε ε > 0. This is equivalent to Marshall's Second Law of Demand. 5 Expression 3.6 implies that the signs of the second-order cross elasticities evaluated at equilibrium determine whether the prices of the two goods are strategic substitutes or strategic complements for each of the two rms. Specically, if increasing 's price makes 's demand more elastic, then decreases its equilibrium price, so views the price of as a strategic substitute to its own price. y contrast, the signs of the rst-order elasticities ε and ε determine whether the two rms' products are simple substitutes or complements: here, the two rms are competitors, so ε > 0 and ε > 0, whereas in section 6 we analyze the case in which they are complementors, so ε < 0 and ε < 0. Finally, to interpret equation 3.7 in lemma 1, recall that at the price equilibrium ε = ) 1 and ε (ε = 1. Thus, when the market expands to a more elastic segment for rm > 1 = ε, the average demand elasticity faced by increases, making it more protable for rm to decrease pricethis captures the non-strategic eect of market expansion on price. We can use Lemma 1 to write the stability condition 3.1 as a function of elasticities: > 0 ε ε ε ε > ε ε ε ε. (3.8) 4 Our assumptions of zero marginal costs and well-dened prot maximization imply that the demand functions we consider cannot have constant elasticity, which in turn implies ε ε ε ε 0. In ppendix, we show how our model can be modied to accomodate positive marginal costs: in that case, demand functions with constant elasticity are possible. 5 To understand this condition, note that at the prot-maximizing price, each rm's own price elasticity is equal to 1 (due to the rst-order condition). From here, price optimality implies that if a rm increases its price by 1 percent, then its demand must decrease by more than 1 percent, so that the eective price elasticity associated with this price increase must be greater than 1. Similarly, the eective price elasticity associated with a small price decrease must be smaller than 1. y continuity, this implies the slope of the own-price elasticity around the equilibrium price must be positive. 10

11 3.3 Deriving the total eects of market expansion The total eects of market expansion on equilibrium prices and prots can be written as functions of the rst-order and second-order elasticities. To do so, we replace the various terms that determine dp dm and dp dm in equation 3.4 with their expressions as functions of elasticities from Lemma 1, then use equation 3.2 (the full derivations are in ppendix C). Proposition 1. The eect of market expansion on equilibrium prices is dp i dm = p i q i ε ε i i Q i i (1 ε i i) + q i ε εi i i εε i i ε εi Q i i i ε ε i i i ε εi i i ( ) 1 ε i i (3.9) and the eect of market expansion on equilibrium prots is d πi = p i q i 1 + ε i i Q i q i q i ε εi Q i i i ( 1 ε i i ε εi i i εε i i ) + ε ε i ε ε i i i i i (1 ε i i) ε εi i i, (3.10) where i {, } and i {, } \ {i}. efore providing the full interpretation of these expressions in Section 5, it is useful to discuss a simplied scenario, which is signicantly easier to interpret. This also allows us to build intuition that will help in explaining the results for the full equilibrium. 4 Myopic play Myopic play assumes away the strategic eect: in response to an exogenous market expansion (increase in M), each rm is assumed to adjust its price taking the other rm's existing price as given. In other words, rms ignore each other's pricing response to market expansion, so this is not a Nash equilibrium. Nevertheless, there are two reasons for studying this scenario. First, myopic play is used in game theory to model learning 6 it is an intermediate step on the adjustment path 6 This is a special case of the ctitious play introduced by rown (1951) to model learning in games. One interpretation of ctitious play is a belief-based learning rule: in each period, players assume opponents play stationary mixed strategies, form beliefs about opponents' strategies, and behave rationally with respect to these beliefs. Myopic play corresponds to the special case in which every player forms beliefs solely based on the opponents' play in the previous period. 11

12 from the pre-expansion Nash equilibrium to the post-expansion Nash equilibrium. Second, this scenario allows us to shut down the strategic component of the competitive eect and focus on the non-strategic component. Using it as a benchmark, we analyze how the strategic eect interacts with the direct eect of market expansion in Section 5. Formally, the rms respond to a slight market expansion dm by setting prices as follows: p myopic i (M, dm) = argmax π (p i, p i (M), M + dm) p i Here, the system of equations 3.3 on page 8 still applies, except that we substitute zero for 2 π p p and 2 π p p dierentiation notation, we have, i.e. each rm ignores the other rm's price change. With a small abuse of d p myopic i lim dm 0 p myopic i (M, dm) p i (M) dm = 2 π i p i M 2 π i p 2 i. We obtain the following corollary to Proposition 1 (the expressions below are derived by setting the cross-price elasticities of the own-price elasticities ε εi i i in equations 3.9 to zero). Corollary 1. If rms behave myopically, the eect of market expansion on the price for rm i {, } is dp myopic i dm = p i q i Q i 1 ε i i ε εi i i (4.1) and the eect of market expansion on prots for rm i {, } is dπ myopic i = p i q i 1 + q i Q i Q i q i εi i 1 ε i i (4.2) ε ε i i i Let us rst interpret the price eect in (4.1): how does market expansion aect rms' prices under myopic play? Suppose the new market segment is more price elastic than the existing market for rm (i.e. ε (M) > ε (M) = 1), so rm 's overall demand elasticity increases. The magnitude of this increase is proportional to the size of the excess elasticity, ( ε 1), weighted by the importance of the new market segment, q Q. Faced with higher overall demand elasticity, rm 12

13 Fig. 4.1: Price Eect of Myopic Play Following Entry into New Market Segment nds it optimal to decrease its price. The magnitude of the price decrease necessary to bring its own-price elasticity back to unity (required by prot-maximization with 0 marginal costs) is inversely proportional to the second-order self elasticity ε ε. The logic of this chain of eects is summarized in Figure 4.1. Consequently, the sign of the myopic price eect (given by 4.1) is solely determined by the own-price elasticity of demand coming from the new market segment. Indeed, since ε εi i i the sign of dpmyopic i dm is positive, is equal to the sign of 1 ε i i for rm i {, }. Thus, in the absence of the strategic eect, market expansion causes prices to drop if and only if the marginal segment is more price elastic than the average infra-marginal segments (the average elasticity is equal to 1 at the prot-maximizing prices). Let us now turn to the eect of market expansion on prots, given by (4.2), which we rewrite here from rm 's perspective : dπ myopic = p q }{{} direct eect 1 + ε Q q 1 ε q Q. (4.3) ε ε }{{ } competitive eect direct eect Under myopic play, rm 's prot change depends only on the rival rm 's demand elasticity in the new market (i.e. ε ), but not on 's own elasticity in the new market (i.e. ε ). Intuitively, in the absence of the strategic eect, only 's demand elasticity in the new market matters for 's price change and therefore for the competitive eect on rm 's prots. Thus, the addition of a more price-elastic segment drags the two rms into a prisoner's dilemma. The competitive eect on 's prots is also proportional to ε > 0, the elasticity of rm 's 13

14 prots with respect to rm 's price, and to q Q Q q, which we will henceforth refer to as the bias of market expansion. When both rms' total demands increase in the same proportion ( q Q = q Q ), we say that market expansion is neutral. When rm 's demand increases by a higher percentage than 's ( q Q > q Q ), we say that the market expansion is biased in favor of rm. Thus, a larger elasticity of 's prots with respect to 's price or a larger bias in favor of rm amplify rm 's incentive to modify its price and therefore the competitive eect of market expansion on 's prots (which is negative if and only if ε > 1). n important implication is that, having xed the bias of market expansion q Q Q q, the sign of the total eect of market expansion on rm prots dπ is independent of the magnitude of the direct eect p q. This is most clearly illustrated when the two rms' demand functions are almost symmetric. Denition 1. The two rms' demand functions are almost symmetric if there exists a constant k, such that for any (p 1, p 2, M), Q (p = p 1, p = p 2, M) = k Q (p = p 1, p = p 2, M). lmost symmetry means the two rms face isomorphic demands. This implies the following: For any M, there exists a symmetric price equilibrium, i.e. one in which p (M) = p (M). t the symmetric price equilibrium, the ratio between the respective changes in the two rms' total demands due to market expansion is the same as the ratio between the respective demands, i.e. d Q = k d Q. The rst-order and second-order elasticities for the two rms are identical, i.e. for any triplet (p 1, p 2, m), ε (p = p 1, p = p 2, m) = ε (p = p 1, p = p 2, m) ε (p = p 1, p = p 2, m) = ε (p = p 1, p = p 2, m) ε ε (p = p 1, p = p 2, m) = ε ε (p = p 1, p = p 2, m) ε ε (p = p 1, p = p 2, m) = ε ε (p = p 1, p = p 2, m). 14

15 Market expansion is always neutral, that is q Q Q q = 1. Corollary 2. If the two rms behave myopically and their demands are almost symmetric, then the eect of market expansion on prots is dπi = p i q i 1 + ε i 1 ε i i i. ε ε i i i Proof. The rst statement follows directly from 4.3 by setting q = kq and Q = kq. When the two rms are almost symmetric, they assign the new market segment the same importance, so the sign of the total eect on rm prots is solely determined by elasticities, not by market sizes. In particular, the magnitude of the direct eect on rm i's prots, p i q i, has no impact on the sign of the total eect of market expansion on rm i's prots. Consequently, if the sign of the total eect is negative (because the new market segment is very price elastic and the elasticity of substitution between the two rms' products is suciently large), then an increase in the magnitude of the direct eect (which is always positive) makes the total eect even more negative! This somewhat surprising feature is not an artifact of the myopic play assumed in this sectionit also holds in the proper Nash equilibrium determined by Proposition 1, as we discuss in the next section. 5 Full equilibrium We now return to the Nash equilibrium and interpret the expressions obtained in Proposition 1. 15

16 5.1 Eect on equilibrium prices Consider rst the eect of market expansion on equilibrium prices from expression 3.9, which we reproduce here for rm : d p p ( ) ε ε = ε ε q 1 ε ε ε ε ε ε ε ε ε }{{ Q ε ε }}{{ } iteration factor myopic eect + εε ε ε q ( ) 1 ε Q ε ε }{{ } strategic eect (5.1) The myopic eect was isolated in the previous section (cf. expression 4.1) and it captures rm 's direct reaction to market expansion, not taking into account rm 's reaction. The strategic price eect of market expansion was entirely shut down in the previous section: given that rm also reacts directly to market expansion, rm further adjusts its own price in response to the anticipated change in rm 's price. The sign of the myopic eect is solely determined by the excess price elasticity of demand for rm coming from the new market segment, whereas the magnitude is proportional to the relative importance of the new market segment for rm (just like in the previous section). In contrast, the sign of the strategic price eect is determined by the excess elasticity of demand for the rival rm (1 ε ) and by whether the prices of and are strategic complements or substitutes from rm 's perspective (ε ε ). The magnitude of the strategic eect is proportional to the relative importance of the new market segment for rm. To understand the iteration factor in 5.1, note that the new price equilibrium after market expansion can be seen as resulting from innite iteration of myopic play in the following way. In the rst iteration, each rm only responds to market expansion and therefore the rms increase their prices by d p 1 q 1 ε = p Q ε ε and d p 1 q 1 ε = p Q ε ε (these are the price changes under myopic play from 4.1). In the second iteration, each rm observes that its rival's price change has increased its overall demand elasticity by d ε = εε d p 1 p ε and d ε = εε the rms further adjust their prices by d p 1 p ε. In response, d p 2 = d ε / ε p = p q 1 ε ε ε Q ε ε ε ε ε ε and d p 2 = d ε / ε ε ε p = p q Q 1 ε ε ε ε ε ε ε 16

17 Fig. 5.1: Price adjustment iterations following expansion to new market segment in order to bring their respective overall demand elasticities back to 1. Indeed, ε ε > 0 implies the prices of and are strategic complements for, so an increase in p leads to also ( ) increase its price. This in turn leads to further adjustments of d p 3 = p q 1 ε Q ε ε ε ε and ε ε ε ε ε ε ( ) ( d p 3 = p q Q 1 ε ε ε d p 4 = p q Q 1 ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ( ε ε ε ε ε ε ε ε in the third iteration and d p 4 = p q 1 ε Q ε ε ) in the fourth iteration. nd so on. Since ε ε ε ε ε ε ε ε ε ε ε ε εε ε ε ε ε ε ε ) and < 1 due to condition 3.8, the amplitude of this sequence of iterations converges to 0 and the total resulting price change for rm is easily veried to be equal to the expression in nd similarly for rm. Figure 5.1 summarizes the eect of market expansion on prices through the iterations described above. n interesting consequence of the strategic price eect is that, even when the new market segment is more price elastic for both rms (i.e. ε i i > 1 for i {, }), one of the new equilibrium prices might be higher. Indeed, suppose the prices of and are strategic substitutes from the perspective of 's prots, i.e. ε ε < 0. This means that rm 's propensity to decrease its price 7 Specically, the sum of the odd-numbered (respectively, even-numbered) iterations converges to the myopic eect (respectively, strategic eect) multiplied by the iteration factor. 17

18 due to market expansion leads rm to increase its price through the strategic eect. If this strategic eect is suciently strong, then it may overwhelm the non-strategic (myopic) eect and 's new price might end up being higher than before. However, this cannot hold for both rms at the same time due to the stability condition 3.8 on page 10. This result is formalized in the following corollary. Corollary 3. If the new market segment is more price elastic than the existing market for both rms (i.e. if ε > 1 and ε > 1) then at least one of the two prices must decrease as a result of market expansion, i.e. d p < 0 or d p < Eect on equilibrium prots Let us now turn to the discussion of the total eect of market expansion on rm prots, given by expression 3.10, which we reproduce here for rm : d π = p q 1 + ε Q q ( ) q ε ε Q 1 ε ε + ε ε ε ε ε ε ε ε ε ( ) 1 ε (5.2) Compared to the case of myopic play (expression 4.2), the main dierence is that now, the sign of the total eect on 's prots also depends on the demand elasticity of the new market segment for rm, i.e. ε. y contrast, in the case of myopic play, only the rival's new demand elasticity, ε, mattered for the sign of d π. Here, due to the strategic eect, how much rm changes its price also depends on how much rm changes its price, which in turn rests on the elasticity of the new market for rm. n interesting consequence is that, even if rm enters a more elastic new market (ε > 1), this does not necessarily mean that the competitive eect will work against it (recall the competitive eect is the sum of the myopic eect and of the strategic eect). Specically, if the prices of and are strategic substitutes from 's perspective (ε ε < 0), then the more elastic the new market for, the stronger the favorable strategic eect: 's price-cutting to accommodate the new market segment makes 's demand less elastic on average, prompting to increase its price, or at least to reduce the amount by which it cuts its price in direct response to expansion into a more elastic market. 18

19 Intuitively, looking at the initial decomposition of the total eect on rm prots given by expression 3.2 on page 7, one would expect that the direct eect of market expansion should determine both the magnitude and the sign of the total eect, i.e. a larger direct eect should make the total eect more likely to be positive, as in ulow et al. (1985). However, this need not be the case in our model. Expression 5.2 shows that if the ratio Q q sign of the total eect d π q Q is held constant, then the is independent of the magnitude of the direct eect p q. This is most clearly seen when the two rms are almost symmetric. Corollary 4. If the two rms are almost symmetric, then the eect of market expansion on equilibrium prices is and the total eect on prots is dp i dm = p i q i Q i ε εi i i 1 ε i i ε ε i i i (5.3) d πi = p i q i 1 + ε i i 1 ε i i ε ε i i i ε εi i i, (5.4) Proof. These expressions are derived directly from (3.9) and (3.10) by imposing almost symmetry. With almost symmetric rms, the sign of the change in rm i's equilibrium price and therefore of the competitive eect in 5.4 is once again equal to the sign of ( 1 ε i), just like under myopic play (recall 4.1). The only dierence is the term ε ε i i i in the denominator of 5.3, which captures the strategic eect of market expansion on equilibrium prices. Thus, if the prices of and are strategic substitutes (i.e. if ε ε = ε ε and therefore the competitive part of the total eect on prots. < 0 ), then the strategic eect dampens the price change More importantly, almost symmetry implies that the sign of the total eect on prots is entirely determined by elasticities and independent of the magnitude of the direct eect. In particular, if the new market segment is suciently elastic (i.e. if ε = ε exceeds 1 by a suciently large amount), then the total eect is negative and an increase in the magnitude of the (positive) direct eect contributes to exacerbating the negative eect of market expansion on prots. To understand 19

20 this, suppose q becomes larger, i.e. the new market segment is scaled up independently of the existing market and its elasticities. y almost symmetry, the new market segment also increases in relative importance for rm, so q Q and q Q increase in parallel by the same amount. Thus, the larger q that increases the positive direct eect on 's prots is also associated with a larger q, which makes 's price reduction more aggressive (we continue to assume ε Q = ε > 1), which in turn hurts 's prots. Thus, if the elasticity of the new market segment is suciently large relative to 1, then the increase in q actually exacerbates the negative eect on 's prots. 5.3 Eect on equilibrium prot ratio Finally, when the two rms' prots move in the same direction, one may want to know which rm gains or loses more (in relative terms) from market expansion. Thus, we are interested in determining the sign of the eect of market expansion on the ratio between the two prots, i.e. ( ) d π π. To do so, we use dm π i = p i Q i and d πi = π i M + π i d p i p i = p i q i + p i Q i d p i i to obtain ( ) π d π π dm π = dπ dm π dπ dm π ( ) q = Q q Q }{{ } direct eect ( Q + ) dp Q dm Q dp Q dm }{{} competitive eect Thus, the sign of d ( ) π π dm is determined by two terms. The rst term corresponds to the direct eect of relative market size gains on the prot split. If the relative market size gain for rm due to market expansion ( q Q ) is larger than the relative market size gain for rm, then the direct eect is positive. This eect is easily understood. The second term captures the dierence between the respective percentage market size changes due to changes in the rival rm's price (competitive eect). If market expansion leads rm to raise its price and rm to drop its price, then the competitive eect on the ratio between 's prots and 's prots is negative (and vice versa when 20

21 rm drops its price and rm raises its price). If market expansion leads both rms to raise their prices, then the competitive eect on π π is positive whenever the percentage market size (or prot) increase for rm due to rm raising its price is larger than the percentage market size (or prot) increase for rm due to rm raising its price. We can then plug in the expressions of dp dm and dp dm from 3.9 to obtain the following corollary. Corollary 5. The eect of market expansion on the revenue split is ( ) π d π π dm π ( ) q = Q q + Q ( ) ( ) ( ) ( ) q Q 1 ε ε ε ε ε ε ε q Q 1 ε ε ε ε ε ε ε. ε ε ε ε ε ε ε ε Let us focus on the competitive eect. For simplicity, assume that the new market segment is less elastic for rm (1 ε > 0) and of unit elasticity for rm. This assumption allows us to focus only on the rst part of the competitive eect: ( ) ( q Q 1 ε ε ε ε ε ε ε ). If there was no strategic eect, then would unambiguously gain (in relative prot terms) from raising its price: ( ) q Q 1 ε ε ε ε > 0. Things are more complicated when the strategic eect is taken into account. If raises its price and views the two prices as strategic complements (i.e. ε ε > 0), then wants to increase its price, which in turn benets. This benet is larger when is more substitutable for (i.e. ε is larger). Thus, for to benet proportionately more than ( ) ( ) does ε ε /ε ε > ε /ε, the strategic complementarity ε ε must be strong enough relative to ( ) 's second-order self elasticity ε ε or consumers must be more willing to substitute from to than from to. 6 Complementors Suppose now that instead of being competitors, rms and are complementors, i.e. are selling complementary rather than substitute products. Formally, the only modication is that q i (p i, p i, m) is now assumed to be decreasing in p i, which implies that Q i (p i, p i, m) is also decreasing in p i for i {, }, so that we now have ε < 0 and ε < 0. Thus, the entire formal analysis conducted in the previous four sections (including Proposition 1) goes through unchanged. The total eect of market expansion on equilibrium prots can still 21

22 be decomposed in the same way: d π i = π i + }{{} M direct eect The only dierence is that, since π i p i = p i Q i p i π i d p i p i }{{} complementary eect. (6.1) < 0, the sign of the complementary eect on rm i's prots is now the opposite of the sign of d p i. In other words, whenever market expansion leads one rm to increase its price, the complementary eect on the other rm's prots is now negative. On the other hand, the signs of the two components of the complementary eectnonstrategic (myopic) and strategicof market expansion on equilibrium prices are determined in the same way as before, so Corollary 3 holds unchanged. 7 Examples with linear demands In this section, we provide simple examples with linear demands in order to briey illustrate some of our main results. 7.1 Symmetric competitors Consider the following symmetric and linear demands 1 α (m) p q i i + β (m) (p i p i ) if m [0, M 0 ] (p i, p i, m) = γ 0 (1 α 0 p i + β 0 (p i p i )) if m [M 0, 1] for i {, }. ll parameters (α (m), β (m)) for m [0, M 0 ] and (α 0, β 0, γ 0 ) are positive. Note that β (m) > 0 measures the intensity of competition (substitutability) between the two rms within market segment m. The interval [0, M 0 ] represents the existing market segments, whereas γ 0 measures the size of the new segments. We are interested in the eect on equilibrium prots of adding a segment from[m 0, 1]. To do so, we will derive the symmetric equilibrium prots π (M) for M [M 0, 1], take the derivative in M and evaluate it at M = M 0. 22

23 ggregate demands for M [M 0, 1] are then Q i (p i, p i, M) = (1 + γ 0 (M M 0 )) (α + γ 0 α 0 (M M 0 )) p i + ( β + γ 0 β 0 (M M 0 ) ) (p i p i ), where we have denoted α M 0 0 α (m) dm and β M 0 0 β (m) dm. The symmetric equilibrium is then easily calculated to be p (M) = 1 + γ 0 (M M 0 ) 2α + β + γ 0 (2α 0 + β 0 ) (M M 0 ) π (M) = (1 + γ 0 (M M 0 )) 2 ( α + β + γ 0 (α 0 + β 0 ) (M M 0 ) ) ( 2α + β + γ0 (2α 0 + β 0 ) (M M 0 ) ) 2 From here, it is straightforward to obtain ( ) ( ) ( ) dπ α + β 2α + β 2α + 3β dm (α0 α) β ( β 0 β ) M=M 0 = γ 0 ( ) 3. 2α + β First, note that the sign of the prot change does not depend on γ 0, the size of the new market segment, consistent with Corollary 4. 8 Second, the sign of the prot change depends on the excess sensitivities of the new market segment in price relative to the average sensitivities in the existing market, i.e. α 0 α and β 0 β. The larger either of these relative sensitivities, the more likely the prot change to be negative. If the new market segment has the same elasticities as the average existing market segments, i.e. if α 0 = α and β 0 = β, then the prot change is positive as expected. In this case, demands are isomorphic, so adding the additional market segments does not change the equilibrium price (p (M) = 1 2α+β for all M), but prots increase because demand is now larger. 8 It is worth emphasizing that the derivative must be evaluated at M = M 0. If we were to evaluate it at M > M 0, then a change in γ 0 would also change the size of the existing market (not just of the marginal segment), so unsurprisingly, the sign of the prot change would also depend on γ 0. 23

24 7.2 symmetric complementors We now use a linear demand example to illustrate the possibility that market expansion into a more price elastic segment for both rms may result in a higher equilibrium price for one of the two rms. However, the linear demand assumption constrains prices to be strategic complements when the two rms are competitors, so the scenario discussed prior to Corollary 3 cannot occur. 9 s a result, we focus on the case in which the two rms are complementors and the linear demands are given by 1 p q i i p i ; if m [0, M 0 ] (p i, p i, m) = 1 β i p i p i if m [M 0, 1] for i {, }, where β i > 1. The price equilibrium for M [M 0, 1] is easily calculated to be p i (M) = 2M (M 0 + β i (M M 0 )) M 2 4 (M 0 + β i (M M 0 )) (M 0 + β i (M M 0 )) M 2. The change in price due to the addition of a small segment from [M 0, 1] is then dp i dm M=M 0 = 2 (1 2β i + β i ). 9M 0 For example, if β = 2 and β = 5, then dp dm M=M 0 > 0 > dp dm M=M 0. Thus, even though both rms enter a more elastic market, their prices move in dierent directionsone rm actually increases its price. However, note that for all (β, β ) (1, + ) 2, at least one of the two prices must decrease, consistent with Corollary 3 (recall that this Corollary also applies to the case when the two rms are complementors). 9 Recall that 's rst order self-elasticity is dq dp / p Q. With linear demand, the numerator is a constant. Thus, when increases its price, Q increases (because the two rms are competitors), and therefore 's rst order self-elasticity decreases. This incentivizes to increase its price as well. 24

25 8 Implications for empirical estimations s Proposition 1 has shown, correctly estimating the rst-order and second-order price elasticities in the existing market is essential for getting the sign of the total eects on prices and prots right. Some commonly used empirical models for estimating demand, while convenient, are not suciently exible for properly estimating these elasticities, which can lead to incorrect predictions, both quantitatively and qualitatively. In Section 7.2 we have already pointed out that linear demand constrains competitors' prices to be strategic complements. The limitations of the Logit model (cf. Lancaster, 1971, McFadden, 1974, 1980) are well known to empirical researchers (ckerberg et al., 2007). This is why the LP framework (erry et al., 1995, 2004) is often adopted in order to accomodate consumer heterogeneity. In this section, we show that the discrete choice model leads to qualitatively dierent estimations of the eect of market expansion depending on the underlying sources of consumer heterogeneity. We summarize the key building blocks of this example and the results it entails in the remainder of this section, while details are provided in ppendix E. Consider two rms and facing symmetric demands. Following Lancaster (1971), we adopt the discrete choice framework: each consumer buys from or (not both), or does not buy anything at all. Consumer i's net utility from buying rm k's product is u i (k) = θi k α i p k + ɛ k i, where k =, and the ɛ i 's and ɛ i 's are independently and identically distributed, following a Logit distribution. Suppose there are only two types of consumers, each making up 50% of the total population, which is normalized to 1. We focus on three separate cases, each with a dierent source of consumer heterogeneity. 1. Heterogeneity in intrinsic preferences : type I consumers have an intrinsic preference for over, i.e. θi = θ H > θ L = θi, whereas type II consumers have an intrinsic preference for over, i.e. θii = θh > θ L = θii. Price sensitivities are equal, i.e. α I = α II = α. We vary the demand parameters such that i) each rm's market share in the aggregate consumer population is 40% (i.e. 20% of consumers buy nothing), ii) the price equilibrium is p = p = 1, and iii) each rm's conditional market share among the type of consumers by whom it is preferred varies from 40% to 80%. 25

26 2. Heterogeneity in willingness-to-pay : type I consumers value and more highly than type II consumers, i.e. θ I = θ I = θ H > θ L = θ II = θ II. Price sensitivities remain equal, i.e. α I = α II = α. We vary demand parameters such that i) each rm's market share in the aggregate consumer population is 40%, ii) the price equilibrium is p = p = 1, and iii) the conditional market share of each rm among type I consumers varies from 40% (this occurs when θ H = θ L ) to 50%. 3. Heterogeneity in price sensitivity : type I consumers are more price-sensitive than type II consumers, i.e. α I = α H > α L = α II. In this case, we assume θi = θi = θii = θ II = θ. Here, we vary demand parameters such that i) each rm's market share in the aggregate consumer population is 40%, ii) the price equilibrium is always p = p = 1, and iii) the ratio of α H /α L varies from 1 to 4. We assume the addressable market for rms and expands through the addition of a new market segment, in which the rst-order self elasticities at the prevailing prices (p = p = 1) are ε = ε = 2.5. We are interested in determining the eect of this market expansion for each case. We have imposed the constraint that each rm's aggregate market share remains xed at 40% and the prevailing price equilibrium is always p = p = 1 in order to keep the three models comparable. This also implies that the Logit model will always estimate the eect of market expansion to be the same across all three cases. Indeed, the Logit model assumes that all consumers have the same baseline utility and price sensitivity, i.e. θ k i = θ and α i = α for all i and k in our symmetric model, so the elasticities ε, ε ε and ε ε are always estimated to have the same value, regardless of the underlying demand structure. s a result, in all three cases, the Logit model predicts that a 1% market expansion results in a 0.29% decrease in prot. Now compare this prediction with the real eect of market expansion for each of the three sources of consumer heterogeneity: 1. Heterogeneity in intrinsic preferences : as each rm's conditional market share among the type of consumers by whom it is preferred varies from 40% (no dierence in preferences) to 80%, the real eect varies from a 0.29% prot decrease to a 1% prot increase, as shown in the rst panel of Figure 8.1. s consumer preferences are more dierent (in a horizontal 26

27 sense), the competitive eect of market expansion decreases, eventually leading to a positive total eect on prots. t the extreme, preferences are so dierent that consumers no longer view and as substitutes and each of them only considers either or. In this case, there is no competitive eect and the conditional market share is 80% (recall that 20% of consumers buy nothing), so a 1% market expansion leads to a 1% increase in prots. 2. Heterogeneity in willingness-to-pay : as each rm's conditional market among consumers of the high type (i.e. type I) varies from 40% (θ H = θ L ) to 50%, the real eect of a 1% market expansion varies from a 0.29% prot decrease to a 0.38% prot decrease (see the second panel in Figure 8.1). 3. Heterogeneity in price sensitivity : as the ratio of price sensitivitiesα H /α L varies from 1 (everyone is equally price sensitive) to 4, the real eect of a 1% market expansion varies from a 0.29% prot decrease to a 2.07% prot decrease (see the third panel in Figure 8.1). Heterogeneous intrinsic preference Heterogeneous willingness to pay Heterogeneous price sensitivity % Profit change % Profit change % Profit change Market share among consumers of favorable type Market share among high type consumers Ratio of price sensitivities α H α L Fig. 8.1: Prot Change for Dierent Sources of Consumer Heterogeneity Thus, the exact source of consumer heterogeneity matters: the standard Logit model yield an excessively pessimistic prediction if consumers mainly dier in terms of their favorite product among and. However, it yields an excessively optimistic prediction if consumers mainly dier in their willingness to pay for and or in their price sensitivity. Now consider a market where consumers mainly dier in their price sensitivity, but are quite similar in terms of their intrinsic preferences and willingness-to-pay. Suppose we used an empirical model that only accomodates 27