Andrei Hagiu1 and Wei Sun2
August 30, 2016
Abstract
We study the eect of market expansion due to exogenous factors on a duopoly's equilibrium prices and pro ts. The total eect on each rm's pro ts 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 pro ts 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 classi cation: 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 MA 02142, [email protected] 2 Harvard Business School, Boston, MA 02163, [email protected]
1 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 2005. Similarly, the cloud infrastructure and computing services oered by Amazon, Google, IBM and Microsoft 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 signi cantly expanded to also include large corporations. What is the eect on rm pro ts of an exogenous market expansion in the absence of new entry? Whereas market expansion always increases monopoly pro ts, the eect on oligopoly pro ts can go either way. At a high level, market expansion has both a direct eect and a competitive eect on each rm's pro t. 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 pro tswe derive precise conditions under which this occurs. Although 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 (in nitesimal) market segment. In this set-up, whether market expansion increases or decreases equilibrium prices and pro ts depends only on the 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. An important and novel result of our analysis is that the sign of the total eect on pro ts can be independent of the
2 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 pro ts. In fact, if the sign of the pro t change is negative, then increasing the new market segment's size exacerbates the negative impact on pro ts. 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 su ciently 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 pro ts. 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 pro ts. 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 brie y 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 2 Related literature
The idea that seemingly bene cial exogenous shocks (i.e. those that would unambiguously raise pro ts for a monopoly rm) may decrease the equilibrium pro ts of oligopoly rms even in the absence of entry dates back to at least Bulow et al. (1985). Speci cally, Bulow et al. (1985) showed that an exogenous cost decrease in one market for a rm that bene ts from economies of scale across two markets can hurt that rm's overall pro tability if it induces a su ciently 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 pro ts 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 pro ts. 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 Bulow 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, Bulow et al. (1985) informally suggest that for the cost decrease to lower total pro ts, the (positive) direct eect has to be small relative to the strategic eect.3 By contrast, we completely characterize the necessary and su cient conditions for market expansion to lower pro ts and show that in our framework, the sign of the total eect of market expansion on pro ts 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. Also 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 pro t
3 Speci cally, sales in the market where the focal rm experiences the cost decrease must be su ciently small relative to sales in the oligopolistic market.
4 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 Bertrand 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 A and B. 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
A M A and B M B , Q (pA, pB,M) = 0 q (pA, pB, m) dm Q (pB, pA,M) = 0 q (pB, pA, m) dm ´ ´
i where M ∈ [0, 1] denotes the number of active market segments, and q (pi, p−i, m) is positive, continuously dierentiable in (pi, p−i), decreasing in pi and increasing in p−i ( rms are competitors)
i for i ∈ {A, B}. This implies that Q (pi, p−i,M) is also positive, continuously dierentiable in
(pi, p−i), decreasing in pi and increasing in p−i for i ∈ {A, B}. Furthermore, we assume the right
i derivative of Q (pi, p−i,M) with respect to M is well-de ned for every M ∈ [0, 1], so that
∂ Qi (p , p ,M) + i −i = qi (p , p ,M) ∂M i −i
5 i In particular, we allow q (pi, 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
i Q (pi, p−i,M) in M is relevant. Note also that we have implicitly assumed m is uniformly dis- tributed over [0, 1], which is without loss of generality: any density function can be simply folded
A B into the underlying demands q (pA, pB, m) and q (pB, pA, m). Intuitively, the market consists of a continuum of segments m ∈ [0, 1]; demands in each segment
A B m are q (pA, pB, m) and q (pB, pA, m). Throughout the paper, we are interested in the eect of increasing the number M of active market segments on equilibrium rm pro ts. 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 A we show that this assumption involves no loss of substance. The pro ts of the two rms are
A B πA = pAQ (pA, pB,M) and πB = pBQ (pB, pA,M) .
The Nash equilibrium with M active market segments is then de ned by
∗ ∗ and ∗ ∗ pA(M) = argmax πA(pA, pB,M) pB(M) = argmax πA(pA, pB,M). pA pB
Throughout the paper, we use * only for equilibrium prices and pro ts. We omit * for quantities: quantities are always evaluated at the prices that appear in the relevant expressions. To ensure well-de ned maximization problems and stability of the solutions to the simultaneous pricing game, we assume that the following conditions hold at all equilibrium values:
2 2 ∂ πA and ∂ πB 2 < 0 2 < 0 ∂pA ∂pB
2 2 2 2 ∂ πA ∂ πB ∂ πA ∂ πB (3.1) ∆ ≡ 2 2 − > 0. ∂pA ∂pB ∂pA∂pB ∂pA∂pB
6 3.1 Decomposing the eect of market expansion
Using the Envelope Theorem for i ∈ {A, B}, we can write the eect of an exogenous market expansion (increase in M) on equilibrium rm pro ts as
d π∗ ∂ π ∂ π d p∗ i = i + i −i . (3.2) d M ∂ M ∂ p−i d M |{z} | {z } direct eect competitive eect As the market expands, the customer base expandsat the old equilibrium prices, both rms can simply sell more and earn more pro t. 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
∂ πi i for . ∂ M = piq i ∈ {A, B} The second term in equation 3.2 is the competitive eect of market expansion. Since ∂ πi = ∂ p−i i ∗ ∂ Q d p−i pi > 0, the sign of the competitive eect is equal to the sign of , i.e. the eect of market ∂ p−i d M 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 pro t. 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 2 ∂ πB > 0) or upward if they are strategic substitutes (i.e. ∂ πB < 0). In the latter case, it is ∂pA∂pB ∂pA∂pB 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 pi , we start with the rst-order conditions ∂ πA = 0 and ∂ πB = 0 (which must d M ∂ pA ∂ pB
7 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
∂2π d p∗ ∂2π d p∗ ∂2π A A + A B + A = 0 ∂pA d M ∂pA∂pB d M ∂pA∂M ∂2π d p∗ ∂2π d p∗ ∂2π B B + B A + B = 0 (3.3) ∂pB d M ∂pA∂pB d M ∂pB∂M
Solving these two equations for d pA and d pB leads to d M d M
∗ 2 2 2 2 d pA 1 ∂ πA ∂ πB ∂ πB ∂ πA = − 2 d M ∆ ∂pA∂pB ∂pB∂M ∂pB ∂pA∂M d p∗ 1 ∂2π ∂2π ∂2π ∂2π B B A A B (3.4) = − 2 , d M ∆ ∂pA∂pB ∂pA∂M ∂pA ∂pB∂M
2 where ∆ > 0 is the determinant de ned in equation 3.1. The term ∂ πi in the expression of ∂pi∂M d p∗ 2 2 i corresponds to the non-strategic price eect described above, whereas the term ∂ πi ∂ π−i d M ∂pi∂p−i ∂p−i∂M corresponds to the strategic price eect.
3.2 Elasticities
We will characterize the eect of market expansion on equilibrium prices and pro ts in terms of elasticities. To do so, we need to introduce the following notation (for conciseness, we omit the arguments (pA, pB,M)):
8 ∂ log(QA) average ( rst-order) self elasticity ε¯A = − positive A ∂ log(pA) ∂ log(qA) marginal ( rst-order) self elasticity εA = − > 0 positive A ∂ log(pA) ∂ log(QA) positive for competitors; ( rst-order) cross elasticity εA = > 0 B ∂ log(pB ) negative for complementors εA ∂ log(εA) second-order self elasticity ε A = A positive A ∂ log(pA) A positive for strategic substitutes; εA ∂ log(ε ) second-order cross elasticity ε A = − A B ∂ log(pB ) negative otherwise Tab. 1: First-order and Second-order Elasticities
Thus, A and B are the rst-order own-price demand elasticities for rms A and B in the ε¯A ε¯B existing market, whereas A and B are the own-price elasticities faced by A and B in the new εA εB market segment. The rst-order own-price elasticities determine the pro t-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, A's second-order self elasticity
εA A quanti es A's ability to move its own rst-order condition by changing its price. The larger εA this elasticity is, the less price adjustment is necessary for A 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 func- tions and by the following lemma (all proofs are in the appendix).
Lemma 1. For any , at the price equilibrium ∗ ∗ we have M (pA (M) , pB (M))
2 A 2 B ∂ π Q εA ∂ π Q εB A A and B B (3.5) 2 = − ∗ εA 2 = − ∗ εB ∂pA pA ∂pB pB 2 A 2 B ∂ π Q ε¯A ∂ π Q εB A A and B B (3.6) = ∗ εB = ∗ εA ∂pA∂pB pB ∂pA∂pB pA ∂2π ∂2π A = qA(1 − εA) and B = qB(1 − εB). (3.7) ∂pA∂M A ∂pB∂M B
The expressions in 3.5 imply that for second-order conditions to hold at the pro t-maximizing
9 prices, the second-order self elasticities must be positive, i.e.4
εA εB A and B εA > 0 εB > 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 equi- librium determine whether the prices of the two goods are strategic substitutes or strategic com- plements for each of the two rms. Speci cally, if increasing A's price makes B's demand more elastic, then B decreases its equilibrium price, so B views the price of A as a strategic substitute to its own price. By contrast, the signs of the rst-order elasticities A and B determine whether the εB εA two rms' products are simple substitutes or complements: here, the two rms are competitors, so A and B , whereas in section 6 we analyze the case in which they are complementors, εB > 0 εA > 0 so A and B . εB < 0 εA < 0 Finally, to interpret equation 3.7 in lemma 1, recall that at the price equilibrium A and ε¯A = 1 B . Thus, when the market expands to a more elastic segment for rm A A A , the ε¯B = 1 εA > 1 = εA average demand elasticity faced by A increases, making it more pro table for rm A 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:
εA εB εB εA A B B A (3.8) ∆ > 0 ⇐⇒ εA εB > εA εB . 4 Our assumptions of zero marginal costs and well-de ned pro t maximization imply that the demand functions εA εB we consider cannot have constant elasticity, which in turn implies A B . In Appendix A, we show how εA εB 6= 0 our model can be modi ed to accomodate positive marginal costs: in that case, demand functions with constant elasticity are possible. 5 To understand this condition, note that at the pro t-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. By continuity, this implies the slope of the own-price elasticity around the equilibrium price must be positive.
10 3.3 Deriving the total eects of market expansion
The total eects of market expansion on equilibrium prices and pro ts can be written as functions of the rst-order and second-order elasticities. To do so, we replace the various terms that determine dp∗ dp∗ A and B in equation 3.4 with their expressions as functions of elasticities from Lemma 1, then dM dM use equation 3.2 (the full derivations are in Appendix C).
Proposition 1. The eect of market expansion on equilibrium prices is
−i i ε −i εi ∗ q −i i q i −i dp i ε−i (1 − εi) + −i ε−i 1 − ε−i i = p∗ Q Q (3.9) i i ε−i ε−i i dM εi −i −i εi εi ε−i − εi ε−i and the eect of market expansion on equilibrium pro ts is
−i i −i εi ε ∗ Q q i −i −i i d π i −i εi 1 − ε−i + εi (1 − εi) i = p∗qi 1 + εi q Q , (3.10) i −i i ε−i ε−i i d M εi −i −i εi εi ε−i − εi ε−i where i ∈ {A, B} and −i ≡ {A, B}\{i}.
Before providing the full interpretation of these expressions in Section 5, it is useful to discuss a simpli ed scenario, which is signi cantly 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 learning6it is an intermediate step on the adjustment path
6 This is a special case of the ctitious play introduced by Brown (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 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:
myopic ∗ pi (M, dM) = argmax πA(pi, p−i (M) ,M + dM) pi
Here, the system of equations 3.3 on page 8 still applies, except that we substitute zero for
2 2 ∂ πA and ∂ πB , i.e. each rm ignores the other rm's price change. With a small abuse of ∂pA∂pB ∂pA∂pB dierentiation notation, we have
2 myopic myopic ∗ ∂ πi d pi pi (M, dM) − pi (M) ∂pi∂M ≡ lim = − 2 . dM→0 ∂ πi d M dM 2 ∂pi
We obtain the following corollary to Proposition 1 (the expressions below are derived by setting
εi the cross-price elasticities of the own-price elasticities i in equations 3.9 to zero). ε−i Corollary 1. If rms behave myopically, the eect of market expansion on the price for rm i ∈ {A, B} is dpmyopic qi 1 − εi i ∗ i (4.1) = pi i i dM Q εi εi and the eect of market expansion on pro ts for rm i ∈ {A, B} is
dπmyopic q−iQi 1 − ε−i i ∗ i i −i (4.2) = pi q 1 + −i i ε−i −i d M Q q ε−i ε−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 A (i.e. A A ), so rm A's overall demand elasticity increases. The εA(M) > ε¯A(M) = 1 magnitude of this increase is proportional to the size of the excess elasticity, A , weighted by εA − 1 the importance of the new market segment, qA . Faced with higher overall demand elasticity, rm QA
12 Fig. 4.1: Price Eect of Myopic Play Following Entry into New Market Segment
A 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 pro t-maximization with 0 marginal costs) is
εA inversely proportional to the second-order self elasticity A . The logic of this chain of eects is εA summarized in Figure 4.1. Consequently, the sign of the myopic price eect (given by 4.1) is solely determined by the
εi own-price elasticity of demand coming from the new market segment. Indeed, since i is positive, εi dpmyopic the sign of i is equal to the sign of i for rm . Thus, in the absence of the dM 1 − εi i ∈ {A, B} 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 pro t-maximizing prices). Let us now turn to the eect of market expansion on pro ts, given by (4.2), which we rewrite here from rm B's perspective :
dπmyopic QB qA 1 − εA B = p∗ qB 1 + εB A . (4.3) B A B A εA d M q Q A direct| {z eect} εA | competitive{z eect } direct eect
Under myopic play, rm B's pro t change depends only on the rival rm A's demand elasticity in the new market (i.e. A), but not on B's own elasticity in the new market (i.e. B). Intuitively, εA εB in the absence of the strategic eect, only A's demand elasticity in the new market matters for A's price change and therefore for the competitive eect on rm B's pro ts. Thus, the addition of a more price-elastic segment drags the two rms into a prisoner's dilemma.
The competitive eect on B's pro ts is also proportional to B , the elasticity of rm B's εA > 0
13 pro ts with respect to rm A's price, and to qA QB , which we will henceforth refer to as the bias of QA qB market expansion. When both rms' total demands increase in the same proportion ( qA qB ), QA = QB we say that market expansion is neutral. When rm A's demand increases by a higher percentage than B's ( qA qB ), we say that the market expansion is biased in favor of rm A. Thus, a larger QA > QB elasticity of B's pro ts with respect to A's price or a larger bias in favor of rm A amplify rm A's incentive to modify its price and therefore the competitive eect of market expansion on B's pro ts (which is negative if and only if A ). εA > 1 An important implication is that, having xed the bias of market expansion qA QB , the sign QA qB of the total eect of market expansion on rm pro ts dπB is independent of the magnitude of d M the direct eect ∗ B. This is most clearly illustrated when the two rms' demand functions are pBq almost symmetric.
De nition 1. The two rms' demand functions are almost symmetric if there exists a constant k, such that for any (p1, p2,M),
A B Q (pA = p1, pB = p2,M) = k Q (pB = p1, pA = p2,M).
Almost symmetry means the two rms face isomorphic demands. This implies the following:
For any , there exists a symmetric price equilibrium, i.e. one in which ∗ ∗ . • M pA (M) = pB (M)
• At 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 QA d QB . d M = k d M
• The rst-order and second-order elasticities for the two rms are identical, i.e. for any triplet (p , p , m), 1 2 A B εA (pA = p1, pB = p2, m) = εB(pB = p1, pA = p2, m) B A εA (pA = p1, pB = p2, m) = εB(pB = p1, pA = p2, m)
εA εB ε A (p = p , p = p , m) = ε B (p = p , p = p , m) A A 1 B 2 B B 1 A 2 A B εA εB εB (pA = p1, pB = p2, m) = εA (pB = p1, pA = p2, m).
14 Market expansion is always neutral, that is qA QB . • QA qB = 1
Corollary 2. If the two rms behave myopically and their demands are almost symmetric, then the eect of market expansion on pro ts is
∗ −i dπi ∗ i i 1 − ε−i = pi q 1 + ε−i −i . d M ε−i ε−i
Proof. The rst statement follows directly from 4.3 by setting qA = kqB and QA = kQB.
When the two rms are almost symmetric, they assign the new market segment the same im- portance, so the sign of the total eect on rm pro ts is solely determined by elasticities, not by market sizes. In particular, the magnitude of the direct eect on rm 's pro ts, ∗ i, has no i pi q impact on the sign of the total eect of market expansion on rm i's pro ts. 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 su ciently 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 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 A:
d p∗ εA εB εA A ε A ε B qA 1 − εA ε A qB 1 − εB d M = A B A + B B (5.1) ∗ εA εB εB εA A εA εA B εB pA A B B A Q A A Q B εA εB − εA εB εA εA εB | iteration{z factor } | myopic{z eect } | strategic{z eect }
The myopic eect was isolated in the previous section (cf. expression 4.1) and it captures rm A's direct reaction to market expansion, not taking into account rm B's reaction. The strategic price eect of market expansion was entirely shut down in the previous section: given that rm B also reacts directly to market expansion, rm A further adjusts its own price in response to the anticipated change in rm B's price. The sign of the myopic eect is solely determined by the excess price elasticity of demand for rm A coming from the new market segment, whereas the magnitude is proportional to the relative importance of the new market segment for rm A (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 B ( B) and by whether the prices of A and B 1 − εB εA are strategic complements or substitutes from rm A's perspective ( A ). The magnitude of the εB strategic eect is proportional to the relative importance of the new market segment for rm B. To understand the iteration factor in 5.1, note that the new price equilibrium after market expansion can be seen as resulting from in nite iteration of myopic play in the following way. In the rst iteration, each rm only responds to market expansion and therefore the rms increase
A 1−εA B 1−εB their prices by 1 A q A and 1 B q B (these are the price changes under d pA = p A εA d M d pB = p B εB d M Q A Q B εA εB myopic play from 4.1). In the second iteration, each rm observes that its rival's price change has
εA d p1 εB d p1 increased its overall demand elasticity by d ε¯A = −ε A B ε¯A and d ε¯B = −ε B A ε¯B . In response, A B pB A B A pA B the rms further adjust their prices by
A B A B B εA B A A εB d ε¯A/ε¯A q 1 − ε ε d ε¯B/ε¯B q 1 − ε ε 2 A B B and 2 B A A d pA = − A pA = pA B B A d M d pB = − B pB = pB A A B d M εA Q εB εA εB Q εA εB εA εB εA εB εA εB
16 Fig. 5.1: Price adjustment iterations following expansion to new market segment
εA in order to bring their respective overall demand elasticities back to 1. Indeed, A implies εB > 0 the prices of A and B are strategic complements for A, so an increase in pB leads A to also εB εA A 1−εA ε B ε A increase its price. This in turn leads to further adjustments of 3 q A A B and d pA = pA A εA εA εB d M Q A A B εA εA εB εB εA εB εB εA B 1−εB ε B ε A B 1−εB ε B ε B ε A 3 q B A B in the third iteration and 4 q B A A B and d pB = pB B εB εA εB d M d pA = pA B εB εA εA εB d M Q B A B Q B A A B εB εA εB εB εA εA εB εA εB εA εB εA A 1−εA ε A ε B ε A ε B ε A 4 q A B A B in the fourth iteration. And so on. Since A B due to d pB = pB A εA εB εA εB d M εA εB < 1 Q A B A B A B εA εB εA εB εA εB condition 3.8, the amplitude of this sequence of iterations converges to 0 and the total resulting price change for rm A is easily veri ed to be equal to the expression in 5.1.7 And similarly for rm B. Figure 5.1 summarizes the eect of market expansion on prices through the iterations described above. An interesting consequence of the strategic price eect is that, even when the new market segment is more price elastic for both rms (i.e. i for ), one of the new equilibrium εi > 1 i ∈ {A, B} prices might be higher. Indeed, suppose the prices of A and B are strategic substitutes from the
εA perspective of A's pro ts, i.e. A . This means that rm B's propensity to decrease its price εB < 0 7 Speci cally, the sum of the odd-numbered (respectively, even-numbered) iterations converges to the myopic eect (respectively, strategic eect) multiplied by the iteration factor.
17 due to market expansion leads rm A to increase its price through the strategic eect. If this strategic eect is su ciently strong, then it may overwhelm the non-strategic (myopic) eect and A'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 B and A ) then at least one of the two prices must decrease as a result of εB > 1 εA > 1 d p∗ d p∗ market expansion, i.e. A or B . d M < 0 d M < 0
5.2 Eect on equilibrium pro ts
Let us now turn to the discussion of the total eect of market expansion on rm pro ts, given by expression 3.10, which we reproduce here for rm A:
A B εA εB ∗ Q q A B B A d π A B ε 1 − εB + ε 1 − εA A ∗ A q Q A A (5.2) = pAqA 1 + εB A B B A d M εA εB εB εA εA εB − εA εB
Compared to the case of myopic play (expression 4.2), the main dierence is that now, the sign of the total eect on A's pro ts also depends on the demand elasticity of the new market segment for rm A, i.e. A. By contrast, in the case of myopic play, only the rival's new demand elasticity, εA
B, mattered for the sign of d πA . Here, due to the strategic eect, how much rm B changes its εB d M price also depends on how much rm A changes its price, which in turn rests on the elasticity of the new market for rm A. An interesting consequence is that, even if rm A enters a more elastic new market ( A ), this does not necessarily mean that the competitive eect will work εA > 1 against it (recall the competitive eect is the sum of the myopic eect and of the strategic eect).
εB Speci cally, if the prices of A and B are strategic substitutes from B's perspective ( B ), then εA < 0 the more elastic the new market for A, the stronger the favorable strategic eect: A's price-cutting to accommodate the new market segment makes B's demand less elastic on average, prompting B 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 Intuitively, looking at the initial decomposition of the total eect on rm pro ts 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 Bulow et al. (1985). However, this need not be the case in our model. Expression 5.2 shows that if the ratio QA qB is held constant, then the qA QB sign of the total eect d πA is independent of the magnitude of the direct eect . This is most d M pAqA 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 dp∗ qi 1 − εi i = p∗ i (5.3) i i i ε−i dM Q εi −i εi − εi and the total eect on pro ts is
d π∗ 1 − ε−i i = p∗qi 1 + εi −i , (5.4) i −i ε−i i d M −i εi ε−i − ε−i
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 −i, just like under myopic 1 − ε−i ε−i play (recall 4.1). The only dierence is the term −i in the denominator of 5.3, which captures εi the strategic eect of market expansion on equilibrium prices. Thus, if the prices of A and B are
εA εB strategic substitutes (i.e. if A B ), then the strategic eect dampens the price change εB = εA < 0 and therefore the competitive part of the total eect on pro ts. More importantly, almost symmetry implies that the sign of the total eect on pro ts is entirely determined by elasticities and independent of the magnitude of the direct eect. In particular, if the new market segment is su ciently elastic (i.e. if A B exceeds 1 by a su ciently large εA = εB 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 pro ts. To understand
19 this, suppose qA becomes larger, i.e. the new market segment is scaled up independently of the existing market and its elasticities. By almost symmetry, the new market segment also increases in relative importance for rm B, so qA and qB increase in parallel by the same amount. Thus, QA QB the larger qA that increases the positive direct eect on A's pro ts is also associated with a larger qB , which makes B's price reduction more aggressive (we continue to assume A B ), which QB εA = εB > 1 in turn hurts A's pro ts. Thus, if the elasticity of the new market segment is su ciently large relative to 1, then the increase in qA actually exacerbates the negative eect on A's pro ts.
5.3 Eect on equilibrium pro t ratio
Finally, when the two rms' pro ts 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 pro ts, i.e. A d π πB . To do so, we use ∗ ∗ i and dM πi = pi Q
∗ ∗ ∗ d πi ∂ πi ∂ πi d p−i ∗ i ∗ i d p−i = + = pi q + pi Q−i d M ∂ M ∂ p−i d M d M to obtain
∗ πA ∗ ∗ ∗ dπA dπB A B A ∗ B ∗ d π∗ B πB dM dM q q QB dpB QA dpA ∗ = ∗ − ∗ = A − B + A − B dM πA πA πB Q Q Q dM Q dM | direct{z eect } | competitive{z eect }
π∗ d A π∗ Thus, the sign of B is determined by two terms. The rst term corresponds to the direct dM eect of relative market size gains on the pro t split. If the relative market size gain for rm A due to market expansion ( qA ) is larger than the relative market size gain for rm B, then the direct QA 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 A to raise its price and rm B to drop its price, then the competitive eect on the ratio between A's pro ts and B's pro ts is negative (and vice versa when
20 rm A drops its price and rm B raises its price). If market expansion leads both rms to raise their prices, then the competitive eect on πA is positive whenever the percentage market size (or πB pro t) increase for rm A due to rm B raising its price is larger than the percentage market size (or pro t) increase for rm B due to rm A raising its price.
We can then plug in the expressions of dpA and dpB from 3.9 to obtain the following corollary. dM dM
Corollary 5. The eect of market expansion on the revenue split is
∗ B A A A B B πA q B A εA B εA q A B εB A εB ∗ A B d π∗ π q q QB 1 − εB εBεA − εAεB − QA 1 − εA εAεB − εBεA B B = − + . ∗ A B εA εB εB εA dM πA Q Q A B B A εA εB − εA εB
Let us focus on the competitive eect. For simplicity, assume that the new market segment is less elastic for rm B ( B ) and of unit elasticity for rm A. This assumption allows us to 1 − εB > 0 B εA εA focus only on the rst part of the competitive eect: q B A A B A . If there was QB 1 − εB εBεA − εAεB no strategic eect, then A would unambiguously gain (in relative pro t terms) from B raising its
B εA price: q B A A . Things are more complicated when the strategic eect is taken into QB 1 − εB εBεA > 0 εA account . If B raises its price and A views the two prices as strategic complements (i.e. A ), εB > 0 then A wants to increase its price, which in turn bene ts B. This bene t is larger when B is more substitutable for A (i.e. B is larger). Thus, for B to bene t proportionately more than ε¯A εA A εA A does ε A εA εA B , the strategic complementarity A must be strong enough relative to B /εA > B/εA εB εA A's second-order self elasticity A or consumers must be more willing to substitute from A to B εA than from B to A.
6 Complementors
Suppose now that instead of being competitors, rms A and B are complementors, i.e. are selling complementary rather than substitute products. Formally, the only modi cation is that
i i q (pi, p−i, m) is now assumed to be decreasing in p−i , which implies that Q (pi, p−i, m) is also decreasing in for , so that we now have A and B . p−i i ∈ {A, B} εB < 0 εA < 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 pro ts can still
21 be decomposed in the same way:
d π∗ ∂ π ∂ π d p∗ i = i + i −i . (6.1) d M ∂ M ∂ p−i d M |{z} | {z } direct eect complementary eect
i ∂ πi ∂ Q The only dierence is that, since = pi < 0, the sign of the complementary eect on ∂ p−i ∂ p−i d p∗ rm 's pro ts is now the opposite of the sign of −i . In other words, whenever market expansion i d M leads one rm to increase its price, the complementary eect on the other rm's pro ts is now negative. On the other hand, the signs of the two components of the complementary eectnon- strategic (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 brie y illustrate some of our main results.
7.1 Symmetric competitors
Consider the following symmetric and linear demands
1 − α (m) p + β (m)(p − p ) if m ∈ [0,M ] i i −i −i 0 q (pi, p−i, m) = if γ0 (1 − α0pi + β0 (p−i − p−i)) m ∈ [M0, 1]
for i ∈ {A, B}. All parameters (α (m) , β (m)) for m ∈ [0,M0] and (α0, β0, γ0) are positive.
Note that β (m) > 0 measures the intensity of competition (substitutability) between the two
rms within market segment m. The interval [0,M0] represents the existing market segments, whereas γ0 measures the size of the new segments. We are interested in the eect on equilibrium pro ts of adding a segment from[M0, 1]. To do so, we will derive the symmetric equilibrium pro ts
∗ π (M) for M ∈ [M0, 1], take the derivative in M and evaluate it at M = M0.
22 Aggregate demands for M ∈ [M0, 1] are then
i Q (pi, p−i,M) = (1 + γ0 (M − M0)) − (α + γ0α0 (M − M0)) pi + β + γ0β0 (M − M0) (p−i − pi) , where we have denoted
M0 and M0 α ≡ 0 α (m) dm β ≡ 0 β (m) dm. ´ ´
The symmetric equilibrium is then easily calculated to be
1 + γ (M − M ) p∗ (M) = 0 0 2α + β + γ0 (2α0 + β0)(M − M0)
(1 + γ (M − M ))2 α + β + γ (α + β )(M − M ) π∗ (M) = 0 0 0 0 0 0 2 2α + β + γ0 (2α0 + β0)(M − M0) From here, it is straightforward to obtain
∗ dπ α + β 2α + β − 2α + 3β (α0 − α) − β β0 − β |M=M0 = γ0 3 . dM 2α + β
First, note that the sign of the pro t change does not depend on γ0, the size of the new market segment, consistent with Corollary 4.8 Second, the sign of the pro t change depends on the excess sensitivities of the new market segment in price relative to the average sensitivities in the existing market, i.e. α0 − α and β0 − β. The larger either of these relative sensitivities, the more likely the pro t change to be negative. If the new market segment has the same elasticities as the average existing market segments, i.e. if α0 = α and β0 = β, then the pro t 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 for all M), but pro ts increase because demand is now larger. 2α+β
8 It is worth emphasizing that the derivative must be evaluated at M = M0. If we were to evaluate it at M > M0, then a change in γ0 would also change the size of the existing market (not just of the marginal segment), so unsurprisingly, the sign of the pro t change would also depend on γ0.
23 7.2 Asymmetric 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 As a result, we focus on the case in which the two rms are complementors and the linear demands are given by 1 − p − p ; if m ∈ [0,M ] i i −i 0 q (pi, p−i, m) = if 1 − βipi − p−i m ∈ [M0, 1]
for i ∈ {A, B}, where βi > 1.
The price equilibrium for M ∈ [M0, 1] is easily calculated to be
2 ∗ 2M (M0 + β−i (M − M0)) − M pi (M) = 2 . 4 (M0 + βi (M − M0)) (M0 + β−i (M − M0)) − M
The change in price due to the addition of a small segment from [M0, 1] is then
∗ dpi 2 |M=M0 = (1 − 2βi + β−i). dM 9M0
dp∗ dp∗ For example, if and , then A B . Thus, even though βA = 2 βB = 5 dM |M=M0 > 0 > dM |M=M0 both rms enter a more elastic market, their prices move in dierent directionsone rm actually
2 increases its price. However, note that for all (βA, βB) ∈ (1, +∞) , 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).
A 9 dQ pA Recall that A's rst order self-elasticity is − / A . With linear demand, the numerator is a constant. Thus, dpA Q when B increases its price, QA increases (because the two rms are competitors), and therefore A's rst order self-elasticity decreases. This incentivizes A to increase its price as well.
24 8 Implications for empirical estimations
As 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 pro ts right. Some commonly used empirical models for estimating demand, while convenient, are not su ciently 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 (Ackerberg et al., 2007). This is why the BLP framework (Berry 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 Appendix E. Consider two rms A and B facing symmetric demands. Following Lancaster (1971), we adopt the discrete choice framework: each consumer buys from A or B (not both), or does not buy anything at all. Consumer 's net utility from buying rm 's product is k k, i k ui(k) = θi − αipk + i where and the A's and B's are independently and identically distributed, following a k = A, B i i 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 A
over B, i.e. A H L B, whereas type II consumers have an intrinsic preference θI = θ > θ = θI for B over A, i.e. B H L A . Price sensitivities are equal, i.e. . θII = θ > θ = θII αI = αII = 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 ∗ ∗ , and iii) each rm's conditional market share among the type of consumers pA = pB = 1 by whom it is preferred varies from 40% to 80%.
25 2. Heterogeneity in willingness-to-pay: type I consumers value A and B more highly than type
II consumers, i.e. A B H L A B . Price sensitivities remain equal, i.e. θI = θI = θ > θ = θII = θII
αI = αII = 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 ∗ ∗ , and iii) the pA = pB = 1 conditional market share of each rm among type I consumers varies from 40% (this occurs
when θH = θL) to 50%.
3. Heterogeneity in price sensitivity: type I consumers are more price-sensitive than type II
consumers, i.e. H L . In this case, we assume A B A B . αI = α > α = αII θI = θI = θII = θII = 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 ∗ ∗ , and iii) the pA = pB = 1
ratio of αH /αL varies from 1 to 4.
We assume the addressable market for rms A and B expands through the addition of a new market segment, in which the rst-order self elasticities at the prevailing prices ( ∗ ∗ ) are pA = pB = 1 A B . We are interested in determining the eect of this market expansion for each case. εA = εB = 2.5 We have imposed the constraint that each rm's aggregate market share remains xed at 40% and the prevailing price equilibrium is always ∗ ∗ in order to keep the three models pA = pB = 1 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. k and for all and in our θi = θ αi = α i k ε¯B ε¯A symmetric model, so the elasticities A, B and A are always estimated to have the same value, ε¯B εB εB regardless of the underlying demand structure. As a result, in all three cases, the Logit model predicts that a 1% market expansion results in a 0.29% decrease in pro t. 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% pro t decrease to a 1% pro t increase, as shown in the rst panel of Figure 8.1. As consumer preferences are more dierent (in a horizontal
26 sense), the competitive eect of market expansion decreases, eventually leading to a positive total eect on pro ts. At the extreme, preferences are so dierent that consumers no longer view A and B as substitutes and each of them only considers either A or B. 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 pro ts.
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% (θH = θL) to 50% , the real eect of a 1% market expansion varies from a 0.29% pro t decrease to a 0.38% pro t decrease (see the second panel in Figure 8.1).
3. Heterogeneity in price sensitivity: as the ratio of price sensitivitiesαH /αL varies from 1 (everyone is equally price sensitive) to 4, the real eect of a 1% market expansion varies from a 0.29% pro t decrease to a 2.07% pro t decrease (see the third panel in Figure 8.1).
Heterogeneous intrinsic preference Heterogeneous willingness to pay Heterogeneous price sensitivity 1.0 −0.5 0.8 −0.30 0.6 −1.0 0.4 −0.34 0.2 % Profit change % Profit change % Profit change −1.5 −0.2 −2.0 −0.38 0.4 0.5 0.6 0.7 0.8 0.40 0.42 0.44 0.46 0.48 0.50 1.0 1.5 2.0 2.5 3.0 3.5 4.0 α α Market share among consumers of favorable type Market share among high−type consumers Ratio of price sensitivities H L
Fig. 8.1: Pro t 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 A and B. However, it yields an excessively optimistic prediction if consumers mainly dier in their willingness to pay for A and B 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 dierences in intrinsic preferences and/or willingness-to-pay, but not dierences in price sensitivity. The prediction of such a model would always be excessively optimistic: if only the rst two types of consumer heterogeneity are included, then the worst prediction for a 1% market expansion is a 0.38% decrease in pro t, whereas the correct model could yield a pro t decrease of more than 2%. Thus, to accurately predict the eect of market expansion, an empirical demand model needs to accomodate the type of consumer heterogeneity that is most relevant to the market studied.
9 Conclusion
Our analysis of the eects of exogenous market expansion on a duopoly's equilibrium prices and pro ts has several clear managerial implications. First and most straightforward, whereas an exogenous expansion of the addressable market is always good news for a monopolist, it can be bad news for rms selling competing products in an oligoplistic marketeven in the absence of new entryif the newly added segment is more price elastic relative to the average existing segment. Similarly, market expansion can be bad news for rms selling complementary products if the newly added segment is less price elastic relative to the average existing segment. Second, whether rms' pro ts increase or decrease as a result of market expansion does not depend on the absolute size of the newly added market segment, but rather on the ratio between the relative sizes of the new market segment to the existing segments for the two rms. Third, even if the new market segment is more (respectively, less) price elastic for both rms, it may be optimal for one rm to increase (respectively, decrease) its price if rms' prices are strategic substitutes. This is true both when rms are competitors and when they are complementors. Fourth, we have shown that adopting su ciently exible empirical frameworks is crucial to reaching accurate forecasts of the eect of market expansion. However, implementing such empirical frameworks can require prohibitively expensive data-science resources. If such resources are not available, then managers should rst determine the main source of dierences among their customers, then perform the analysis with a simple Logit model, and nally adjust the results based on the main source of customer heterogeneity based on our analysis in Section 8. For example, if customers dier mainly in terms of their price sensitivity, then the real eect is likely more negative than the predicted
28 eect. At a more abstract level, we have shown that the mechanism through which market expansion leads to unexpected eects on rm pro ts can be obtained and explained by relying on traditional substitutability/complementarity and own-price elasticities. In particular, while strategic substi- tutability/complementarity does aect the magnitude of the eects on pro ts, it is not the crucial factor determining their sign. There are several promising directions in which our work here can be extended. First and most straightforward, one could apply the same methodology we have employed here to study the impact of exogenous shocks other than market expansion (e.g. uniform demand shifts, new entry) in a similarly general framework. Second, one could introduce network eects in consumer demand and investigate how their magnitude changes the eect of market expansion on rm prices and pro ts.
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A Generalizing to non-zero marginal cost
For rm i ∈ {A, B} with marginal cost ci, we rede ne residual demand as a function of the premium or mark up (p˜i = pi − ci) the rm charges:
˜i i Q (p ˜i, p˜−i,M) = Q (pi =p ˜i + ci, p−i =p ˜−i + c−i,M).
With this transformation, all of our calculations go through, provided demand functions are replaced with residual demand functions, and prices are replaced with markups. Note that all elasticities are now de ned in terms of markups and residual demands as well. Thus, even if the original demand function has constant elasticity, the eective elasticity de ned in terms of markups and residual demands may not be constant.
30 B Proof of Lemma 1
Applying B B QB and B at equilibrium, we have QB = −εB pB ε¯B = 1
2 B B B B B ∂ p Q ∂ ε Q B ∂ πB B B B B B B ∂εB B B B εB εB B 2 = QB + = QB − = QB − Q − εBQB = −εB Q ∂pB ∂pB ∂pB ∂pB pB B B Q εB = − εB pB
Similarly,
2 B B B B B ∂ πB B ∂ pBQB B ∂ εBQ B B B ∂εB B ∂εB = QA + = QA − = QA 1 − εB − Q = −Q ∂pA∂pB ∂pA ∂pA ∂pA ∂pA
B B Q εB = εA pA
Finally,
2 A A ∂ πA ∂ pAq A A ∂q A A A A A = = q + p A = q − q εA = q (1 − εA) ∂pA∂M ∂pA ∂p
C Proof of Proposition 1
Starting with the expression of dpA from equation 3.4 and replacing the various terms with their dM expressions as functions of elasticities from Lemma 1, we obtain:
A εA B εB A εB B εA A Q A B B Q B A A q B A q A B dp B ε q 1 − εB + B ε q 1 − εA A ε 1 − εA + B ε 1 − εB = p B p B = pA Q B Q B . A B A B B A A B B A dM Q Q εA εB εB εA εA εB εB εA pApB εA εB − εA εB εA εB − εA εB
And similarly for dpB . dM We can then write the total eect on pro ts:
31 B d πB ∂ πB ∂ πB d pA p B B d pA = + = pBqB + A Q εA d M ∂ M ∂ pA d M p d M A εB B εA B q ε B 1 − εA + q ε A 1 − εB p B B A QA B A QB B B = pBqB + A Q εA p A B B A p εA εB εB εA εA εB − εA εB A B εB εA q Q ε B 1 − εA + ε A 1 − εB B qB QA B A B B = pBqB 1 + εA A B B A . εA εB εB εA εA εB − εA εB
D Proof of Corollary 3
εA A Assume A and B throughout. If in addition A , then d p (from expression 5.1 εA > 1 εB > 1 εB > 0 d M < 0 εB B on page 16) and, similarly, if B , then d p . Suppose therefore that both rms view prices εA > 0 d M < 0 εA εB as strategic substitutes, i.e. A and B . Then εB < 0 εA < 0
εB ! d pA qA QB 1 − εA ε B sign = sign 1 + A B , B A B εA d M q Q 1 − εB A εB εA ! d pB qB QA 1 − εB ε A sign = sign 1 + B A . A B A εB d M q Q 1 − εA B εA
However, we have
εB εA εA εB qA QB 1 − εA ε B qB QA 1 − εB ε A ε A ε B A B · B A = A B > 1, B A B εA A B A εB εB εA q Q 1 − εB A q Q 1 − εA B B A εB εA εA εB where the last inequality is implied by the stability condition 3.8 on page 10. Thus, since both
εB εA A B 1−εA ε B B A 1−εB ε A q Q A B and q Q B A are negative, we conclude that at least one of them must be smaller B A B εA A B A εB q Q 1−εB A q Q 1−εA B εB εA than -1, so d pA or d pB . d M < 0 d M < 0
E Details of the numerical example used in section 8
Recall that consumer 's net utility from buying rm 's product is k k, where i k ui(k) = θi − αipk + i and the A's and B's are independently and identically distributed, following a Logit k = A, B i i
32 distribution. There are two types of consumers, each making up 50% of the total population, which is normalized to 1. The conditional market shares of the two rms within the population of type I consumers are
A A exp(θI − αI pA) sI = A B 1 + exp(θI − αI pA) + exp(θI − αI pB) B B exp(θI − αI pB) sI = A B . 1 + exp(θI − αI pA) + exp(θI − αI pB)
And symmetrically for the conditional market shares of the two rms, A and B , within the sII sII population of type II consumers. The aggregate market shares in the total consumer population are then A 1 A A and B 1 B B . s¯ = 2 sI + sII s¯ = 2 sI + sII In this setting, given that equilibrium prices are ∗ ∗ , we can derive the rst and pA = pB = 1 second-order elasticities as functions of the conditional market shares realized in equilibrium:10
E αsB(1 − sB) p 1 B B B B B B (E.1) ε¯B = B = B B αI sI 1 − sI + αII sII 1 − sII E [s ] sI + sII
E αsBsA p ε¯B = A . (E.2) A E [sB]
2 B B B ε¯B E α 1 − 2s s (1 − s ) ε B = 2 − . (E.3) B E [sB]
B A 2 B B A ε¯B −E αs s + E α 1 − 2s s s ε B = . (E.4) A E [sB]
Now consider each of the three dimensions of heterogeneity in turn. In the model with hetero- geneous intrinsic preferences, A B H L B A and . Recalling that θI = θII = θ > θ = θI = θII αI = αII = α in equilibrium the rst-order conditions in price implies A B , we can use the expression ε¯A =ε ¯B = 1 of B from equation (E.1) to determine the value of corresponding to a given set of conditional ε¯B α
ε¯B ε¯B 10 To derive the expressions of B and B , we also use that B at the equilibrium prices, so B B εB εA ε¯B = 1 E αs (1 − s ) = E sB.
33 market shares A B A B A. We then use the value of to determine the other sI = sII , sII = sI = 0.8 − sI α elasticities from (E.2), (E.3) and (E.4). Finally, we obtain the resulting pro t change from expres- sion (5.4). Applying this procedure as A varies from 40% to 80%, we obtain the rst panel in sI Figure 8.1. The methodology for the model with heterogeneity in willingness-to-pay is identical. The only dierence is that now A B H L B A , so A B and A B A. Here, θI = θI = θ > θ = θII = θII sI = sI sII = sII = 0.8 − sI we vary A from 40% to 50%. sI In the model with heterogeneity in price sensitivities, we have A B A B and θI = θI = θII = θII = θ H L . Here, the procedure runs in the following two steps. First, for each αH , αI = α > α = αII k ≡ αL the system of equations
1 kαLsB 1 − sB + αLsB 1 − sB = 1 0.8 I I II II L exp(θ−kα ) B 1+2 exp(θ−kαL) = sI
exp(θ−αL) B L = sII 1+2 exp(θ−α ) B B sI + sII = 0.8 admits a unique solution in L B B . Second, for each , we use the L B B deter- α , θ, sI , sII k α , θ, sI , sII mined in the frst step to calculate the remaining elasticities according to expressions (E.2), (E.3) and (E.4). Then use expression (5.4) to derive the resulting pro t change. Finally, consider the Logit model, which ignores the heterogeneity of consumer types and therefore implicitly assumes that the conditional market shares are always equal to the aggregate market share (recall the latter is kept equal to 40%). The elasticities estimated by the Logit model at equilibrium prices are therefore given by
B B ε¯B = α 1 − s¯
B A εA = αs¯
B εB 2 B B εB = 2 − α 1 − 2¯s 1 − s¯
B ε¯B A B εA = αs¯ −1 + α 1 − 2¯s .
34 From B , the Logit model estimates 1 , which can then be used to determine B, ε¯B = 1 α = 1−s¯B εA εB ε¯B B and B . Using expression (5.4), this leads to an estimated pro t decrease of 0.29% for a 1% εB εA market expansion. Note that this estimate is independent of the real conditional market shares.
35