Overlapping Ownership, Endogenous Quality, and Welfare

This paper investigates how overlapping ownership a¤ects quality levels, consumer surplus, rms prots and welfare when the industry is a vertically di¤erentiated duopoly and quality choice is endogenous. This issue is particularly relevant since recent empirical evidence suggests that overlapping ownership constitutes an important feature of a multitude of vertically di¤erentiated industries. We show that overlapping ownership while detrimental for welfare, may increase or decrease the quality gap, consumer surplus and rmsprots. In particular, when the overlapping ownership structure is such that the high quality rm places a positive weight on the low quality rms prots, the incentives of the high quality rm to compete aggressively reduce. This may increase the equilibrium quality of the low quality rm, which in turn may lead to higher consumer surplus, despite higher prices. JEL Classication: L13; L41. Keywords: Overlapping Ownership, Vertical Di¤erentiation. Duarte Brito (dmb@fct.unl.pt) gratefully acknowledges nancial support from Fundação para a Ciência e a Tecnologia (UID/ECO/04007/2019). Ricardo Ribeiro (rribeiro@porto.ucp.pt) gratefully acknowledges nancial support from Fundação para a Ciência e a Tecnologia (UID/GES/00731/2019). Helder Vasconcelos (hvasconcelos@fep.up.pt) gratefully acknowledges nancial support from Fundação para a Ciência e a Tecnologia (UID/ECO/04105/2019). All remaining errors are of course our own.

We contribute to this strand of the literature by studying the e¤ects of overlapping ownership on the quality choices, consumer surplus, pro…ts and welfare of a vertically di¤erentiated duopoly. This issue is particularly relevant since recent empirical evidence suggests that overlapping ownership constitutes an important feature of a multitude of vertically di¤erentiated industries. See, for example, Schmalz (2018), Newham, Seldeslachts and Banal-Estanol (2018) and Backus, Conlon and Sinkinson (2019) for evidence on the airline, banking, supermarket and pharmaceutical industries. 2;3 We show that when the overlapping ownership structure is such that the high quality …rm places a positive weight on the low quality …rm's pro…ts, the incentives of the high quality …rm to compete aggressively reduce. This may increase the equilibrium quality of the low quality …rm, which in turn may lead to higher consumer surplus, despite higher prices.
The model, the equilibria and the conclusions are presented in sections 2, 3 and 4, respectively. 1 Brito, Ribeiro and Vasconcelos (2019a) show that overlapping ownership can induce product prices and output levels that are even higher and lower, respectively, than those in a monopoly.
2 For a characterization of the importance of overlapping ownership in those industries please see Tables 2, 3 and 4 in Schmalz (2018), Table 1 in Newham, Seldeslachts and Banal-Estanol (2018) and Figure 12 in Backus, Conlon and Sinkinson (2019). 3 We can identify features of an airline such as bag handling, gate location, connecting layover times, ‡ight schedules, in- ‡ight services, legroom, seat characteristics, and ‡ight frequency with the quality of an airline (Barbot, 2004;Brueckner and Flores-Fillol, 2019). We can identify the probability of failure of a bank with the quality of the bank (Vives, 2016). We can identify features of a supermarket such as product assortment, store location, product availability, car parking space, and opening hours with the quality of a supermarket (Aslan, 2019). We can identify the brand name of a pharmaceutical product (even though generics are legislated to be therapeutically identical to branded products) with the perceived quality of the product (Cabrales, 2003).
All proofs are presented in the online mathematical appendix.

Theoretical Model
We follow Wauthy (1996)'s approach and notation. Two duopolists, …rm 1 and …rm 2, sell products of di¤erent quality to a continuum of consumers that have di¤erent valuations for quality. We assume that each consumer is identi…ed by a parameter that characterizes the utility when purchasing from …rm i = L; H, as follows: u i = s i p i , where s i and p i denote the quality and price of …rm i.
is uniformly distributed over the support [ ; + ], and + = is assumed to be su¢ ciently large so that the market is not covered in equilibrium. We focus on the non-trivial case in which s H > s L , with s H and s L denoting the quality level of the high (H) and low (L) quality …rm, respectively.
The utility of not purchasing any product (outside option i = 0) is normalized to zero: u 0 = 0. 4 We assume that quality is costless and can take values in interval [0; s + ] in the lines of Choi and Shin (1992) and Wauthy (1996). This simpli…es the analysis considerably. Assuming …xed or variable costs of quality as in Motta (1993) makes the model intractable and it is no longer possible to solve explicitly for the equilibrium quality levels. This constitutes a very interesting potential area for future research.
We also assume that, due to overlapping ownership, …rm i's objective function places a weight w i < 1 on …rm j's pro…t (with the weight on own pro…t normalized to 1). These assumptions imply that the objective function of …rm i = L; H is b i = i + w i j = p i D i + w i p j D j , where i and D i denote the pro…t and demand of …rm i. 5

Game, Timing and Equilibrium
Consumers and …rms play the following game. At the beginning, nature draws the valuations of each consumer for quality. Next, …rms address a two-stage decision problem. In the …rst (second) stage, each …rm chooses the quality (price) of its product. Finally, each consumer selects the option (i = H; L; 0) that provides the highest utility. We focus on the sub-game perfect Nash equilibrium (SPNE) of the game and begin by addressing the consumers decision problem. 4 If, alternatively, the market was fully covered, the outcome would be maximum di¤erentiation, regardless of the ownership structure and prices would increase in the presence of overlapping ownership. 5 We are agnostic about whether overlapping ownership is induced by common-ownership, cross-ownership or both and about which particular type of weight is used. See Brito et al. (2018) for a review of the implications of each type of ownership on the objective function of …rms. See Backus, Conlon and Sinkinson (2019) and Brito et al. (2019b) for a discussion of di¤erent alternative weights.

Consumers Decision Problem
It is straightforward to show that consumers with (i) HL = p H p L s H s L will purchase the high quality product; (ii) p L s L = L0 < HL = p H p L s H s L will purchase the low quality product; and (iii) < L0 = p L s L will choose not to purchase. This yields the following demand functions: (1)

Firms Decision Problem
The SPNE of the game involving the …rms' decision problem is obtained by backward induction.
In the second stage, the two …rms (simultaneously) set the prices that maximize their objective function given the quality levels. Lemma 1 presents the corresponding equilibrium.
Lemma 1 The equilibrium prices, as a function of the quality levels, are: Given the quality levels s H and s L , both equilibrium prices increase in w L and w H . The fact that each …rm places a positive weight on the rival's pro…t makes them price less aggressively. The price di¤erence p H p L increases with w H and decreases with w L because own equilibrium price is more a¤ected by an increase in the weight given to the rival than the rival's price.
Having addressed the equilibrium in the pricing stage, we now address the quality stage. The two …rms (simultaneously) set quality levels anticipating the price equilibrium above. Lemma 2 presents the corresponding equilibrium. 6 Lemma 2 In equilibrium, …rms set the following quality levels: Corollary 1 For any (w L ; w H ) 2 (0; 1) 2 , s H is invariant to w H and w L , while s L is increasing in w H and decreasing in w L . 6 The condition for an uncovered market is In order to discuss the implications of Corollary 1 on consumer surplus, …rms'pro…ts and welfare, we begin (for tractability) by analyzing some particular cases before addressing the general case.

Benchmark Case
In the absence of overlapping ownership, Lemmas 1 and 2 imply that: This yields, as in Choi and Shin (1992) and Wauthy (1996), that the lower quality …rm chooses quality and price levels which are 4=7 and 2=7, respectively, of those of the higher quality …rm. As a consequence, HL = 5 12 + and L0 = 1 8 + , which yields that D H = 7 + 12( + ) and D L = 7 + 24( + ) . In turn, consumer surplus is CS = 7s + +2 24( + ) and, since costs are zero, …rms' pro…ts are H = In this case, the overlapping ownership structure is such that only the high quality …rm places a positive weight on the low quality …rm's pro…ts. Lemmas 1 and 2 imply that: This yields, as established by Corollary 1, that the high quality …rm chooses the same equilibrium quality (and price) as in the benchmark case while the low quality …rm has an incentive to increase its equilibrium quality (and price) and narrow the quality gap. The reason being that the high quality …rm will now price less aggressively (given it internalizes the externality imposed on the rival), which makes the demand of the low quality …rm more sensitive to its quality level. 7 As a consequence, HL increases while L0 remains unchanged: HL = 5 w H 12 4w H + and L0 = 1 8 + , which 7 In the benchmark case, the choice of quality level by the low quality …rm is the result of two countervailing e¤ects. The …rst (direct) e¤ect induces the …rm to increase its quality level since higher quality increases demand. The second (strategic e¤ect) induces the …rm to decrease its quality level since higher quality leads the rival (the high quality …rm) to lower its price, which in turn decreases demand for the …rm. In case 1, overlapping ownership increases the …rst e¤ect (since it makes the demand of the low quality …rm more sensitive to its quality level, as the high quality …rm will price less aggressively) and may increase or decrease the second e¤ect (since it can make the price of the high quality …rm more or less sensitive to the quality level of the low quality …rm, depending on the quality levels). The impact on the …rst e¤ect dominates and so the quality level of the low quality …rm, increases.

Figure 1
Demand Impacts in Case 1 yields that some demand will now be diverted from the high quality …rm to the low quality …rm, as depicted in Figure 1. Proposition 1 discusses the impact on consumer surplus, …rms pro…ts and welfare.
Proposition 1 If w L = 0: higher price, but are more than compensated by the higher quality, while the latter now purchase a lower quality, but are more than compensated by the lower price.
We now address the impact on …rms'pro…ts. The pro…t of the high quality …rm decreases with w H because some demand is diverted from the high quality …rm to the low quality …rm while p H does not change. The high quality …rm accepts this loss in pro…t since it now places a positive weight on the pro…t of the low quality …rm, which increases with w H because some demand is diverted from the high quality …rm to the low quality …rm while p L increases.
Finally, we address the impact on welfare. Despite the positive impact on consumers, welfare decreases with w H as the result of the following trade-o¤ (since price is irrelevant in terms of welfare). On the one hand, some consumers continue to buy the low quality product, whose quality increases with w H . On the other hand, some consumers switch from the high quality product to the low quality one. The second e¤ect, which is felt by consumers with a higher valuation for quality, dominates.
3.2.3 Case 2: w H = 0 and w L > 0 In this case, the overlapping ownership structure is such that only the low quality …rm places a positive weight on the high quality …rm's pro…ts. Lemmas 1 and 2 imply that: This yields, as established by Corollary 1, that the high quality …rm chooses the same equilibrium quality (and a higher price) while the low quality …rm has an incentive to decrease its equilibrium quality (and increase price for w L < 0:25605/decrease price for w L > 0:25605), widening the quality gap. The reason being that such lower quality level bene…ts the high quality …rm, which is now internalized by the low quality …rm. 8 As a consequence, HL and L0 increase: HL = (c) Welfare decreases with w L .
We begin by addressing the impact on consumer surplus. Now, when comparing the equilibrium decisions when w L = 0 with those when w L > 0 one can divide consumers into …ve distinct groups, 8 In case 2, overlapping ownership may increase or decrease the two e¤ects discussed in footnote 7 (since it can make the demand of the low quality …rm and the price of the high quality …rm more or less sensitive to the quality level of the low quality …rm, depending on the quality levels). Moreover, it introduces a third (direct) e¤ect, whichdominates and -induces the low quality …rm to decrease its quality level, since a lower quality increases the demand for the high quality …rm, now internalized by the low quality …rm. The high quality …rm responds to this increased demand by increasing its price. We now address the impact on …rms' pro…ts. The pro…t of the high quality …rm increases with w L since the decrease in s L allows the …rm to increase p H , which more than compensates the resulting decrease in demand. The pro…t of the low quality …rm may increase or decrease with w L because demand decreases (the diversion to the outside option more than compensates the diversion from the high quality …rm) while p L may increase or decrease.
Finally, we address the impact on welfare. Welfare decreases with w L because (since price is irrelevant in terms of welfare) some consumers switch from the low quality product to the outside option, some consumers continue to buy the low quality product, whose quality decreases, and some consumers switch from the high quality product to the low quality one.

Case 3: w H
In this case, both …rms place a positive and symmetric weight on the rival pro…t, which combines the two previous cases. Lemmas 1 and 2 imply that: This yields that the high quality …rm chooses the same equilibrium quality (and a higher price) while the low quality …rm has an incentive to decrease its equilibrium quality (and increase price for w < 0:569 86/decrease price for w > 0:569 86). This, in turn, suggests that the dominating e¤ect when both …rms places a positive and symmetric weight on the rival pro…t is the one resulting from w L , which is, in fact, established in Proposition 3.
Proposition 3 If w H = w L = w: (a) Consumer surplus decreases with w; (b) The high (low) quality …rm's pro…t increases (can either increase or decrease) with w; (c) Welfare decreases with w:

General Case
The cases above illustrate that overlapping ownership (a) may increase or decrease consumer surplus (in particular, it decreases consumer surplus if w L is signi…cantly di¤erent from zero); 9 (b) may increase or decrease …rms'pro…ts; and (c) decreases welfare.

Conclusions
In this paper, we have analyzed the implications of overlapping ownership in a standard vertical di¤erentiation duopoly model. We have shown that overlapping ownership while detrimental for welfare, may increase or decrease the quality gap, consumer surplus and …rms'pro…ts. In particular, when overlapping ownership leads the manager of the high quality …rm to place some weight on the low quality …rm's pro…ts, the low quality level increases and consumers will bene…t from this. The reason being that when the rival prices less aggressively, quality di¤erentiation is not as relevant and the low quality …rm narrows the quality gap.

Online Mathematical Appendix
In this mathematical appendix, we present the proofs of the results presented in the main text.

Proof of Lemma 1
The …rst-order conditions for maximization of each …rm's objective function are: from where the result follows directly.

Proof of Corollary 1
The proof follows directly from: The denominator is always positive and the second term in the numerator is an inverted parabola with a minimum when wL = 0 or wL = 1. In the former case it is equal to 1 and in the latter case it is equal to 8 (1 wH ). Therefore, @s L @w H > 0.
As for @s L @w L we have that: The denominator is always positive and the numerator is an inverted parabola with an unconstrained maximum at wH = 25w L 9w 2 L +3w 3 L +13 4(w L +1)(4w L w 2 L +1) > 1. Therefore, the numerator is maximized at wH = 1 and takes value 6 (1 wL) 3 < 0, meaning that @s L @w L < 0.

Proof of Proposition 2
Given the equilibrium price and quality expressions one can easily compute consumer surplus, …rms'pro…ts and welfare, which are presented below divided by s + +2 : ; with derivatives: < 0: