Strategic Buying to Prevent Seller Exit C. Robert Clark and Mattias K. Polborny June 4, 2009
Abstract We consider a dynamic oligopoly model in which a seller may drop out of the market when demand for its product is insu¢ cient in the …rst period. Buyers su¤er some disutility if a seller exits the market and so their …rst period purchase decision not only depends on current period preferences and prices, but also on the potential e¤ect that their behavior has on the probability of seller survival. Speci…cally, some buyers may choose to purchase from the seller with the lower survival probability even though they like the other seller’s product better, a behavior that we call “strategic buying”. We analyze how the incidence of strategic buying depends on parameters and also the implications of the strategic buying motive for sellers’…rst period pricing decisions.
HEC
Montréal,
3000 Côte-Sainte-Catherine,
Montréal,
QC, CANADA
H3T
2A7,
and CIRANO;
[email protected] y University of Illinois at Urbana-Champaign, 483 Wohlers Hall, 1206 S. Sixth Street Champaign, IL, 61820;
[email protected]
1
1
Introduction
The United Kingdom’s new Defence Industrial Strategy calls on the Ministry of Defence to consider the e¤ect of its sourcing decisions on the competitiveness of the markets for certain types of military equipment. In a speech to the Royal United Services Institute in September 2005, the Minister for Defence Procurement, Lord Drayson, characterized “Appropriate Sovereignty” as one of the organizing concepts of this industrial strategy.1 He stressed that “Appropriate Sovereignty” does not mean “having a completely national or onshore industrial base in all areas” or “security of supply in the traditional sense”; rather it implies the need, in a limited number of areas, for the UK to act in such a way as “to maintain realistic global competition” so that it is “not dependent on an overseas monopoly.” We call such a behavior strategic buying: If a buyer can potentially in‡uence whether a seller stays in a market, it may purchase from this seller even if it would otherwise prefer the product of one of its competitors who is less likely to exit the market. Of course, the UK is not the only nation buying from these suppliers of military equipment and so whether or not Britain’s Ministry of Defence’s purchase decision will in fact in‡uence supplier survival depends on the behavior of the defence departments of other nations. These defence departments have similar interests in maintaining competitive markets since they are also better o¤ if there is competition among several suppliers of some types of military equipment than if one seller has a monopoly. Given that the interaction is between multiple buyers and multiple sellers, it is necessary to analyze this situation using a game-theoretic framework. The goal of this paper is to characterize buyer and seller behavior in this type of environment. More precisely, we study a dynamic oligopoly model in which a seller may drop out of the market when demand for its product is insu¢ cient. Customers su¤er should exit occur, so their purchase decisions are in‡uenced not only by prices and current preferences, but also by a desire to prevent seller exit. As a consequence, some buyers may choose to buy strategically from a seller with a lower survival probability in the hope of increasing it even though they like another seller’s product better. In practice, there are several di¤erent reasons why buyers might want to prevent seller exit. Perhaps the most compelling is the desire to maintain competition, as illustrated by the military procurement example above. Another example of strategic buying to maintain competition comes from the market for prisons services. Again, there is a small number of large private providers of prison services; and government agencies from a number of nations must periodically decide from which of these providers to source. Evidence of strategic buying comes again from Britain where Her Majesty’s Prison Service admits to sourcing at least one prison from a more expensive provider in order to assure competitive bidding in the future (O¢ ce of Fair Trading, 2004). More generally, 1
See http://www.mod.uk/DefenceInternet/AboutDefence/People/
Speeches/MinDP/RoyalUnitedServicesInstituterusi12September2005.htm.
2
this type of behavior could occur in any situation where there are multiple government agencies from di¤erent jurisdictions procuring goods or services from a small set of suppliers. The above examples provide evidence of strategic buying in order to maintain competition in the procurement of goods or services by the public sector, but the incentive for strategic buying should also exist in the private sector. Consider, for instance, the market for regional jets.2 Following the exit of Fairchild Dornier in April of 2002, just two players were left in this market: Canada’s Bombardier and Brazil’s Embraer. Regional jets are quite expensive, and the number of planes sold is relatively small (for instance, as of June 2007 there were just 62 CRJ900 aircraft, Bombardier’s 86-seat version in airline service throughout the world, with 36 further …rm orders). Several airline companies are su¢ ciently large relative to the market that their decisions can have a substantial impact on the probability of each manufacturer’s survival. Given this, and the fact that airline companies buying regional jets are much better o¤ if competition between Bombardier and Embraer is maintained than if one of them had a monopoly, airlines have an incentive to engage in strategic buying. In contrast to the examples above, airline o¢ cials do not explicitly admit to strategic buying, but this is not surprising, because doing so could give their rivals a strategic advantage.3 On a smaller scale, similar concerns may also lead to strategic buying in small towns with two (or few) doctors, grocery shops, or gas stations, for example. In each case, buyers must consider the potential e¤ect of their purchase decision on the future market structure. Buyers may also wish to prevent seller exit in order maintain variety if they are unsure about their future preferences. We set up a two-period model in which competing sellers o¤er di¤erentiated varieties of a product. We assume that, in the second period, a seller will remain active if and only if a su¢ cient number of customers bought its product in the …rst period. The two varieties are di¤erentiated in two dimensions. First, each variety has a quality component that all buyers appreciate in the same way. Second, each buyer also has an idiosyncratic preference for one of the goods. This horizontal component is di¤erent for each buyer (but drawn from the same distribution) and is also the buyers’private information. Consider the market for a particular type of military product such as tanks. The quality component could, for instance, be speed since presumably all buyers prefer faster tanks. In contrast, buyers may have idiosyncratic preferences over the type of terrain on which they would like their tanks to perform better. For instance some buyers may prefer tanks that perform better in desert-like conditions, while others may prefer tanks that perform better in rocky terrain. We assume that there are only …nitely many buyers, and so the realized distribution of buyers’horizontal components remains unknown. This generates some social risk as to whether sellers survive. Buyers each purchase from only one seller. When making their selection, they are aware of 2 3
Regional jets are airplanes that seat between 35 and 100 passengers and have a medium range. In the public sector there is often pressure on o¢ cials to justify their actions to the public if they are observed
to buy from a more expensive supplier.
3
their “buying power”, i.e., that they may be pivotal for the survival of a seller. We examine how buyers decide which product to purchase and how the incidence of strategic buying depends on parameters, in particular, on the utility di¤erence in the second period from having two rather than one seller in the market, on the number of buyers in the market, and on the number of buyers that is necessary for a seller to survive. We also analyze how non-myopic sellers set prices in the …rst period in the presence of strategic buyers. Our main results are as follows. If one seller o¤ers a more attractive product than the other seller for the average expected customer (for example, the stronger seller’s product is better or its price is lower than the at-risk seller’s product), then equilibrium buying behavior is such that some buyer types purchase the at-risk seller’s product strategically, even though they prefer the stronger seller’s product in terms of present period utility. However, strategic buying never goes so far as to reverse which seller is more at risk. We also show that, as long as we consider the price as exogenous, strategic buying is underprovided in equilibrium, as it is e¤ectively a public good for buyers. Interestingly, though, when considering the e¤ect of strategic buying on sellers’…rst period pricing behavior, it is possible that buyers would be better o¤ if they could commit ex-ante not to engage in strategic buying. After characterizing buyer behavior taking seller prices as given in Section 3, we proceed to examine the …rst-period pricing behavior of sellers in Section 4. The analysis of optimal …rst period pricing is complicated by the fact that sellers also get a di¤erent payo¤ depending on whether they are a monopolist or one of two duopolists in the second period. This gives rise to a predatory pricing e¤ect: the greater is the monopoly payo¤ relative to the duopoly payo¤, the lower are …rst period equilibrium prices, as both sellers try to price their competitor out of the market. In contrast, the larger is the customers’incentive for strategic buying, the higher are sellers’…rst-period prices, so that sellers e¤ectively take advantage of strategic buying. In certain situations price changes may have an additional e¤ect. The ultimate reason why sellers “need” a certain number of customers in our model is a …nancial constraint that requires that the seller reach a certain revenue level R in the …rst period. For exogenous prices, every level of the revenue constraint corresponds to a particular required number of customers. However, when sellers change prices, the required number of customers changes at certain critical boundaries, where R=p is an integer. Lower prices generate increased concern on the part of buyers as to seller survival since more customers are required. Therefore buyers engage in more strategic buying in order to aid a seller whose price has decreased. This increase in strategic buying can make a price decrease to just below a critical level worthwhile for a seller. Our paper is related to the growing literature on buyer power which examines situations in which buyers are su¢ ciently large that they can in‡uence market outcomes. The focus of this literature has mostly been on how buyer power arises and on the way in which buyers obtain concessions from sellers (see for instance Stigler (1964), von Ungern-Sternberg (1996), Dobson and Waterson (1997), 4
Katz (1987), Inderst and Wey (2007a), and Chen (2007)). More recently there has been some work looking at the long-term consequences of buyer power on the behavior of sellers. In particular, a set of papers has examined the impact of buyer power on the sellers’ incentives to invest and to innovate (Chen (2006), Battigalli, Fumagalli, and Polo (2007), Inderst and Sha¤er (2007), Inderst and Wey (2003, 2007a, 2007b) and Montez (2008)). More speci…cally we are related to a small number of papers that examine buyer and seller behavior in dynamic situations in which buyers seek to a¤ect the structure of the market in future periods and sellers’ optimal current prices are in‡uenced by this behavior. The paper closest to our’s is by Biglaiser and Vettas (2007). In their setup, two sellers of a homogeneous good have …xed two-period capacities. A seller who receives too many customer orders in the …rst period has no remaining capacity in the second period, which makes the rival seller e¤ectively a monopoly in the second period. In equilibrium, multiple buyers randomize over which producer to buy from in order to avoid coordination on a single seller. While this setup is in some sense the opposite of ours (because a seller exits the market if it is too successful in the …rst period), some results are similar to ours. In particular, strategic buying softens price competition, so that buyers would be better o¤ if there were less strategic buying. In Lewis and Yildirim (2002) a single buyer decides in each period from which of two competing sellers to buy. There is learning-by-doing so that a supplier’s costs are decreasing in the number of periods in which it was chosen by the buyer. Furthermore, the buyer has incomplete information about the sellers’costs and wants to minimize the lifetime cost of buying. The buyer faces a tradeo¤ between helping sellers move down their learning curves and maintaining competition. When learning economies are large, the buyer starts by initially alternating between suppliers in order to enhance competition, but ultimately ends up contracting with only one seller who gets further and further ahead of its rival in terms of experience. While the initial alternating behavior resembles strategic buying in our model, there is no risk in Lewis and Yildirim (2002) that the weaker …rm actually exits. Furthermore, since there is only a single buyer in their model, they cannot examine the extent to which strategic buying is a public good. That strategic buying is a public good is an important and novel feature of our paper, and so we are related, at least peripherally to the general literature on the private provision of public goods (see, for example, Bergstrom et al. (1986)). In our model, if both sellers survive, all buyers bene…t and so strategic buying that increases the probability of seller survival, is essentially a public good and as a result there is a free-rider problem.4 In contrast, in Lewis and Yildirim, there is just a single buyer. In Biglaiser and Vettas, there are multiple buyers, but they are assumed to be symmetric and the products o¤ered are assumed to be homogeneous. These features mean that 4
However, strategic buying is only guaranteed to be underprovided from an ex-post perspective; taking into account
the e¤ect of strategic buying on pricing, it is possible that all buyers would be better o¤ if they could commit not to buy strategically.
5
no free-rider problem arises and buyers do not consider buying something other than their most preferred alternative. In a related paper, Romano (1991) examines a situation in which consumers may engage in excessive consumption in order to keep a …rm in the market when otherwise it might shut down. Each consumer has an incentive to free-ride on the excessive consumption of others, but there exist Nash consumption equilibria in which some consumers buy strategically (more than they otherwise would). However, Romano looks only at a monopoly supplier and so cannot consider the e¤ect of strategic buying on the price competition of sellers.
2
Model
Two sellers, labeled i = 0; 1, each produce a single variety of a di¤erentiated product at zero marginal cost. N buyers each have a unit demand for the good in each of two periods.5 Before the start of the second period, sellers decide whether to remain in the market or exit. Each seller bases this decision on its …rst period revenue, and stays in the market if and only if its revenue is greater than or equal to R.6 This is a reduced form model meant to capture some capital market frictions. For example, sellers could be …nancially constrained and need revenues to pay back loans and …nance necessary investments. If a …rm has revenue less than R, it goes bankrupt. Creditors may lack the ability to operate the …rm and may therefore prefer to liquidate its assets. Buyers make purchase decisions to maximize their expected lifetime utility, keeping in mind that …rst period purchase decisions may a¤ect seller survival and hence buyers’ second period utility (we describe this relation in more detail in Section 3). We denote by
b
and
s
the buyer and
seller discount factors respectively. We assume that each variety has a quality component, vi , and all buyers appreciate this quality in the same way. Second, each buyer also has an idiosyncratic preference for one of the goods, which we call the horizontal component. The horizontal component is di¤erent for each buyer and is also the buyer’s private information. A buyer who draws horizontal component x and purchases the product of seller 0 at price p0 receives a net utility in this period of v0
p0 5
x. Similarly, if this buyer purchases from seller 1 at price p1 , it receives v1
p1
(1
x).
An alternative formulation is to have buyers purchase multiple units and allow them to split their purchases
amongst di¤erent sellers. A brief analysis of this type of formulation is provided in the working paper version of this paper, and is available from the authors upon request. 6 More generally, we might consider that, from the buyers’perspective, the probability of survival of a particular seller is an increasing function of the seller’s revenue. Strategic buying incentives are likely to be important in this more general setup, too, as long as a buyer’s expected bene…t is a nonlinear function of each seller’s revenue (if it were a linear function of revenue, then the situation is observationally equivalent to buyers having a higher willingness to pay, and in this case there is no strategic interaction between di¤erent buyers). Alternatively, one could assume that a seller’s exit decision depends on the relative pro…ts of the two suppliers, since this signals which is more likely to be at a competitive disadvantage moving forward.
6
We assume that vi > 1 so that all buyers have a positive willingness to pay for each of the products. Furthermore, each buyer’s horizontal component is drawn from a uniform distribution on [0,1]. Since there are only …nitely many buyers, the realized distribution of buyers’horizontal components remains unknown and whether a particular seller survives is, in general, random. When we analyze pricing in Section 4, we assume that sellers set their …rst period prices to maximize total (discounted) pro…ts.
Period 1
Period 2 i. Sellers exit if pi ni < R
i. Sellers set prices ii. Buyers choose from whom to buy
!
ii. Remaining sellers set prices iii. Buyers choose from whom to buy
Figure 1: Timeline
3
Buyer Behavior
3.1
Equilibrium
In this section, we take the sellers’prices as given and examine buyer behavior. We are interested in equilibrium behavior in the subgame in which prices are …xed and buyers play against one another. Note that, for a given price, revenue depends simply on the number of units sold, and so the “critical revenue level” R corresponds to a critical number of customers required. We call this associated critical number of customers n and, in this section, we analyze purchasing decisions, taking n as given.7 We restrict attention to the case where n
N=2, which means that both sellers
can survive (otherwise, strategic buying with the objective to ensure second period competition is not meaningful). Note that since at least one …rm will survive under this assumption, it also implies that a buyer can never be pivotal for the survival of both seller 0 and seller 1. In the second period there are no strategic considerations and so, if both sellers are still in the market, buyers simply purchase whichever product yields higher net utility. A buyer with horizontal component x will purchase from seller 0 if and only if v0
p0
x
v1
p1
(1
x)
0 7
For the relative pro…ts case mentioned in Footnote 5, a seller (say seller 0) exits if sR0 < R1 , where Ri is the
realised revenue of seller i, and s is the critical share for survival (so if s = 2, seller 0 exits if its pro…ts are less than half of seller 1’s). We can rewrite this to …nd a condition on the critical number of customers required. Since R0 = p0 n0 (where n0 is the number of seller 0’s customers and R1 = p1 n1 , we have that seller 0 exits if sp0 n0 < p1 n1 . Using N = n0 + n1 , we can rewrite this as sp0 n0 < p1 (N rest of the analysis proceeds in the same way.
7
n0 ), which we can solve for n0 = p1 N=(sp0 + p1 ). The
and from seller 1 if and only if v1
p1
(1
x)
v0
p0
x
0: If only one seller is still in the market, all buyers purchase from this seller, as long as this gives them a non-negative net consumer surplus. Clearly, the equilibrium price in the second period will be higher and net consumer surplus will be lower if the market is a monopoly than if both sellers survive. In principle, we could solve the second period subgames explicitly and use the di¤erence in the buyers’ payo¤s in the monopoly and duopoly subgames (as functions of the second period parameters) in their …rst period objective functions. We do this in Section 4, but here, in order to keep notation shorter, we just denote this di¤erence by
. This parameter may therefore be interpreted as the di¤erence between the
buyers’second period continuation values in the one seller and two sellers cases:
=
b (SD
SM )
(where SD is the second period expected buyer surplus when there are two sellers on the market, and SM is the expected buyer surplus when there is just one seller on the market). More generally, captures any increase in buyers’utility generated by the survival of a greater number of sellers into the second period. In order to focus on the strategic e¤ect in which we are interested –strategic buying to prevent seller exit–, and for tractability, we assume that
is the same for all buyers. One interpretation
of this assumption is that each buyer receives a new independent draw of x in the second period. Second period pricing decisions of both sellers (if they are active in period 2) then depend only on the second period values of vi , but not on the realized …rst period demand, and so all buyers have ex-ante the same expected consumer surplus. Allowing for correlation between today’s and tomorrow’s values of the horizontal component x would generate di¤erent values of
for di¤erent
buyers, and would complicate the analysis considerably. The reason is that sellers would be able to infer information on buyers’second period types from …rst period sales. Knowing this, buyers have an incentive to in‡uence the sellers’information by strategically altering their …rst period purchase decision. This is an interesting, but complicated, problem, and we therefore choose to switch this e¤ect o¤ in order to focus on strategic buying to prevent seller exit. We now determine the expected demand for each variety in a symmetric equilibrium (i.e., where all buyers play the same strategy). To do so, we calculate the expected payo¤ of a buyer with horizontal component x given that all other consumers follow an x ~–rule, i.e., they buy from seller 0 if their horizontal component x
x ~, and from seller 1 otherwise. We then look for a cuto¤
equilibrium in which all individuals use the same cuto¤ x ~ (which is a function of (v0 ; v1 ; p0 ; p1 )) to guide their purchase decision.8 8
It is natural to consider symmetric equilibria in this context, but there also exist asymmetric equilibria of the
form: buyers with idiosyncratic valuations in some range buy strategically from seller 0 and buyers with idiosyncratic
8
The bene…t for a type x buyer from seller 0’s product, given a price of p0 , is v0
p0
x+
N n
1 n x ~ 1
1
(1
x ~)N
n
:
(1)
Strategic buying incentives enter through the last term as follows: with probability x ~)N
n,
exactly n
1 of the other N
N 1 n 1
x ~n
1 (1
1 consumers buy from seller 0, so that the buyer we consider
is in fact pivotal for the survival of seller 0 and receives the extra utility
. Similarly, the bene…t
of buying seller 1’s product at a price of p1 is v1 N 1 n 1
where
x ~)n
(1
p1
1x ~N n
(1
x) +
N n
1 (1 1
x ~)n
1 N n
x ~
;
(2)
is the probability of being pivotal for the survival of seller 1.
The equilibrium cuto¤, x ~; is found by equating (1) and (2), so that v0 If v0
p0
(v1
p0 = v1
p1 ) + 1
2~ x+
N n
1 1
x ~n
1
(1
x ~)N
n
(1
x ~)n
1 N n
x ~
= 0:
(3)
p1 so that both products are equally attractive for the average buyer, then x ~ = 1=2
is the unique solution of (3). To see this, note that the term in square brackets in (3) can be written x ~n
1 (1
x ~)N
n
1
x ~ 1 x ~
N 2n+1
, which is positive for x < 1=2, negative for x > 1=2 and zero
for x = 1=2. Intuitively, if both products are ex-ante equally attractive, and all other buyers behave according to an x ~ = 1=2 rule, then both sellers have the same elimination risk, and thus the e¤ects that might push a buyer into strategic buying from either seller just cancel out. If the two products di¤er in how attractive they are for the average customer, then, in this case, x ~ 6= 1=2. In what follows we suppose (without loss of generality) that v1
p1 > v0
p0 and so, in this case, x ~ < 1=2.
Substituting x ~ = 1=2 in (3) yields a negative expression, so that the zero must be to the left of 1=2. We next note that, if buyers are myopic and do not care about the second period (equivalently, = 0), then the cuto¤ is given by xm =
v0
If buyers do care about the second period (
p0
v1 + p1 + 1 : 2
> 0), then, since x ~ < 1=2, the term in square brackets
in (3) is positive and so the zero of (3) is further to the right than when consumers do not care about the second period. This implies the following proposition. Proposition 1. If the two products di¤ er in how attractive they are for the average customer, then buyers will engage in strategic buying in equilibrium: They are more likely to buy from the at-risk seller than if they were myopic. That is, the equilibrium cuto¤ , x ~, lies between xm and 1=2. valuations in some other range buy strategically from seller 1.
9
In equilibrium, x ~ exceeds xm , so that there are some customer types who would prefer product 1 if they considered only their …rst period utility, but choose to buy product 0. These strategic buyer types are located between xm and x ~ and are willing to sustain a loss of present period utility of up to u ~ = 2(~ x
xm ) to decrease the probability of seller 0’s exit. Intuitively, seller 0 with the (in
expectation) less attractive product, is more likely to be eliminated than seller 1. Strategic buying counteracts the superiority of seller 1’s product, because it is more likely for a buyer to be pivotal for the survival of the “bad” seller 0 than for the survival of the “good” seller 1. However, the fact that x ~ < 1=2 shows that the strategic buying e¤ect does not overcompensate the advantage of seller 1. If, in fact, seller 1’s expected demand were smaller than seller 0’s, then each customer’s strategic buying would have to be in favor of seller 1; hence, this situation cannot be an equilibrium. In principle, there can exist multiple values of x ~ that satisfy (3). The reason is that the derivative of the term in square brackets with respect to x may take any sign, so that the overall derivative may be positive at some values of x. We focus on stable equilibria for which the derivative of (3) is negative at x ~.9 At least one stable equilibrium is guaranteed to exist, because the left hand side of (3) is continuous, positive at x = xm , and negative at x = 1=2. The comparative static e¤ects on stable equilibria can easily be obtained by analyzing how the parameter change a¤ects (3). For example, if a parameter change increases the left-hand side of (3) at x ~, then the corresponding new stable equilibrium must be at a higher value of x. 3.1.1
Discussion
The …rst thing to note is that the …xed prices assumed in this section could arise in a full model with endogenous prices as described below in Section 4. In the Appendix we construct an example in which the sellers are asymmetric in the sense that v1 > v0 , and, in the full equilibrium, prices are such that v1
p1 > v0
p0 and strategic buying occurs.
Second, it is important to clarify that when we characterize buyers as being "strategic" what we mean is "non-atomistic." Buyers believe correctly that they are su¢ ciently big that their actions can in‡uence market outcomes. Therefore unlike small atomistic buyers who would always buy their most preferred option, large buyers may intentionally purchase something that is not optimal for them in the short run, but could have long term bene…ts. This is why measuring strategic buying involves comparing myopia with forward sight. Third, it should be noted that the choice to exit has not been exogenously imposed. The only 9
If the derivative of (3) is positive at some value x0 , then x0 is an equilibrium in the sense that it is optimal for
a player to use this cuto¤ if all other players also use it. However, if the other players use a cuto¤ rule with x0 + " (where " is small, but positive), then an individual located at x + " strictly prefers to buy from seller 0. Hence, any reasonable adjustment process would lead to a further increase of the cuto¤, away from x0 ; in this sense, x0 is an unstable equilibrium.
10
thing that is imposed on one of the sellers in this part of the analysis is an initial weakness (lower quality) relative to the rival seller. This generates an exogenously imposed higher risk of exit, but the outcome is a function of the purchasing behavior of buyers which depends on their randomly assigned locations and their strategic buying behavior if they are not myopic. Finally, it should also be pointed out that although our focus is on whether …rms earn su¢ cient revenue to remain in the market, our model actually captures the features of a setup in which R can be interpreted more generally as the minimum revenue necessary for a seller to engage in some type of behavior that buyers may value in the second period. Besides staying in the market, another example is investment in value-creating innovations. Such innovations should increase the willingness to pay for the second period products and so
can be interpreted as the increase in
utility from the innovation. In this case buyers may engage in strategic buying in order to guarantee that innovation occurs. The crucial features of our setup are that (i) all consumers get the same ;and that (ii)
3.2
does not depend on the identity of the surviving …rm.
Comparative statics
We now analyze the e¤ects of changing various model parameters (speci…cally, the buyers’survival payo¤
, the total number of buyers, N; and the threshold n) on the extent of strategic buying,
i.e., the probability that a buyer engages in strategic buying. Our results are summarized in the following proposition. Proposition 2. Suppose that sellers’prices are given and that v0 comparative static results obtain:
p0 6= v1
p1 : Then the following
1. The extent of strategic buying is strictly increasing in the individual bene…t of having more sellers on the market in the second period,
.
2. The extent of strategic buying in equilibrium is less than the extent that would maximize the buyers’ aggregate utility. 3. A su¢ cient condition for an increase in the number of customers, N , to increase strategic buying is that x ~N < n
1, i.e. the expected number of weak seller customers is less than the
number required to guarantee survival, minus one. 4. A su¢ cient condition for an increase in the required number of customers, n, to decrease the extent of strategic buying is that x ~N < n, i.e. the expected number of weak seller customers is less than the number required to guarantee survival. In what follows we provide intuition for these results (some details are relegated to the Appendix). Without loss of generality, we continue to assume that v0 11
p0 < v1
p1 . The case where
v0
p0 > v1
v0
p0 = v1
p1 would obviously be completely symmetric to the one we analyze below, and if p1 , then there is no strategic buying in any case (~ x = 1=2).
It is important to note that the results of Proposition 2 have not been con…rmed with endogenous prices and that, as we show further on in the paper, prices are a¤ected by the magnitudes on which we conduct the comparative statics. 3.2.1
Value to buyers of seller survival
In a slight abuse of notation, we write x ~( ) to denote the equilibrium cuto¤ as a function of Consider what happens when
increases, say from
1
to
2
>
1.
Because the term in square
brackets in (3) is positive for x < 1=2, the left hand side of (3) becomes positive for evaluated at the old cuto¤ x ~(
1 ).
.
2
when
Since the function given by the left hand side of (3) is decreasing
in x, the new zero must be to the right of x ~(
1 ),
so we have x ~(
2)
> x ~(
1 ).
Since all buyers
whose horizontal component is in (xm ; x ~( )) engage in strategic buying, the probability and extent of strategic buying increase as seller survival becomes more important for consumers. In the limit, as
goes to 1, x ~ approaches 1=2. To see this, recall that x ~ = 1=2 is the only value for which the
term in square brackets in (3) is 0. Intuitively, if the overriding concern of buyers is the survival of both sellers, then they will distribute themselves in a way that makes the survival of both sellers most likely. It is also interesting to compare the equilibrium cuto¤ with the cuto¤ rule that would be jointly optimal for all buyers, given prices. To do this, we could set up a social planner’s problem, but it is easy to see directly that each buyer who buys strategically exerts a positive externality on each other buyer equal to its own strategic buying bene…t, and the jointly optimal cuto¤ rule takes this into account. The socially optimal cuto¤,10 x ~SO , is therefore given by v0
p0
(v1
p1 ) + 1
2~ xSO + N
N n
1 1
x ~nSO1 (1
x ~SO )N
n
Because the e¤ect of multiplying by N is equivalent to an increase in
(1
x ~SO )n
1 N n x ~SO
= 0:
,x ~SO > x ~. As a public good
for all buyers, strategic buying is underprovided in equilibrium. Note, however, that this result may depend on prices being exogenous to the analysis in this section. In Section 4, we endogenize prices and show that it is possible that buyers would be better o¤ if they could commit ex-ante not to engage in strategic buying, since sellers take advantage of strategic buying by raising prices. 3.2.2
Number of buyers N
The Appendix contains the equilibrium cuto¤ condition when there are N + 1 buyers which is then used to calculate the change in the equilibrium cuto¤ from an increase in the total number of buyers from N to N + 1. If N is already very large, a further increase in N reduces the extent 10
“Socially” optimal here refers to the group of all buyers and disregards the sellers’pro…ts.
12
to which an individual buys strategically. Intuitively, in this case the survival of both sellers is already virtually assured and a further increase in N makes it even less likely that one buyer would be pivotal for a seller’s survival, so that buyers increasingly decide based on their present period utility. Furthermore, not only do increases in N , starting from some already high level of N decrease the extent of strategic buying, but as N becomes very large (goes to in…nity), x ~ converges to xm , the equilibrium cuto¤ when buyers are myopic. However, the most interesting result is that in some cases an increase in N can actually increases strategic buying. A su¢ cient condition for this to occur is x ~N < n
1, i.e. the expected number
of customers of seller 0 is lower than the minimum number required for survival, minus one. What happens is that, for …xed x ~, the increase in N makes it more likely for any individual buyer to be pivotal for the survival of seller 0 and, at the same time, it becomes less likely to be pivotal for the survival of seller 1. Hence, individual buyers become more inclined to purchase the product of seller 0 and thus the set of strategic buyer types expands. It is also interesting to contrast the results of this section to those from the costly voting literature (see Ledyard (1984), Palfrey and Rosenthal (1983), Börgers (2004), Krasa and Polborn (2009)). There, an increase in the number of voters always leads to a lower cost threshold above which people cease voting, and the participation rate goes to zero as the number of citizens goes to in…nity. Our model generates the same limit result in the sense that as the number of buyers increases, each buyer’s probability of buying the product that it likes less goes to zero. However, in contrast to the costly voting literature, the relation between the number of players and the incidence of strategic buying is not monotonic in our model: an increase in the number of buyers can make each buyer more likely to engage in strategic buying. The reason for the di¤erence is that the incentive for a voter to participate in an election is the possibility of being pivotal, which happens when there is a tie between the voting supporters of both candidates. The critical number of voters therefore depends on the number of voters who participate, and the event of being pivotal becomes less likely as the number of citizens grows. In contrast, in our model, the incentive for strategic buying is based on reaching a …xed target level n, and this may become more or less likely as N increases. 3.2.3
Customer threshold n
The Appendix contains the equilibrium cuto¤ condition when the number of customers required is n + 1 which is then used to calculate the change in the equilibrium cuto¤ from an increase in the customer threshold from n to n + 1. A su¢ cient condition for x ~ to be decreasing in n is that Nx ~ < n. Hence, if the expected number of seller 0’s customers is lower than the minimum number required for survival, then an increase in n reduces the extent of strategic buying. Intuitively, the increase of n makes it less likely that the critical number of buyers is (almost) reached, and so
13
decreases each individual’s incentives to engage in strategic buying.
4
Pricing in the presence of strategic buyers
How can sellers manipulate their prices in the …rst period to take advantage of the fact that buyers wish to prevent exit? The answer to this question is complicated by the fact that, in addition to the strategic buying on the part of the buyers, there are other in‡uences on sellers’pricing behavior. In particular, sellers derive a greater pro…t when they are the only surviving seller than when their rival also survives. In general, seller 0’s total expected pro…t is 0
= p0 N x ~ + Prob(both sellers survive)D0 +Prob(only seller 0 survives)M0 ;
(4)
where D0 and M0 are the expected second period pro…ts if both sellers or only seller 0 survives, respectively. Seller 1’s pro…t is de…ned analogously. There are two fundamentally di¤erent ways that pricing in‡uences the probabilities of survival in equation (4). First, an increase in price has a marginal e¤ect, shifting x ~ such that the expected number of customers for the seller decreases. Second, starting from a situation in which an integer multiple of the price is equal to the revenue that is required for a seller to stay in the market, R, a small price decrease increases the number of customers required to survive by 1. In general, seller 0 remains in the market if its realized number of customers, n0 , satis…es p0 n 0
R; so that the critical number of customers required for seller 0 survival, is n = d pR0 e (where
dye denotes the smallest integer that is greater than or equal to y). Seller 0 exits if fewer than n
buyers purchase its product. If the equilibrium cuto¤ is x ~, this occurs with probability d pR e 1 0 X
i=0
N i x ~ (1 i
x ~)N i :
Moreover, x ~ is the solution to v0
p0
(v1
p1 ) + 1
2~ x+
lN m 1 R p0
lN m 1 R p1
1
d pR e 1
x ~
1
(1
Unless we are at a critical “boundary”point where d pR0 e =
R p0
0
(1
N d pR e
x ~)
0
d pR e 1 N d pR e
x ~)
1
x ~
or d pR1 e =
1
R p1 ,
(5) = 0:
small price changes do
not change n, the number of customers required for survival. The only e¤ect of pricing on survival probabilities is the marginal e¤ect through x ~. In Section 4.1, we solve for a symmetric non-boundary equilibrium and analyze this marginal e¤ect. We examine how prices change with respect to exogenous changes in Mi , Di and
. We also
derive comparative static results for the primitive parameters vi (the quality component), 14
b
and
(buyer and seller discount factors respectively), and N , since these will simultaneously a¤ect
s
M , D, and
. However, the comparative static results of Proposition 3 are essential in order to
logically distinguish the e¤ect of strategic buying on pricing from the predatory pricing e¤ect that arises because …rms have an incentive to price the other …rm out of the market. Variations in capture how important the strategic buying incentive for consumers is. In contrast, M and D in‡uence only the …rms’predatory pricing incentives. Section 4.1 focuses on the case where, in equilibrium, marginal price changes do not a¤ect n. We consider the case where a price change does a¤ect n in Subsection 4.2.
4.1
Symmetric non-boundary equilibrium
We start by considering the marginal e¤ect of strategic buying on pricing while assuming that the equilibrium is such that R=p is not near or at a boundary (an integer), so that marginal changes of p0 and p1 do not change n0 and n1 . Moreover, we assume that v0 = v1 = v, and look for a symmetric equilibrium in which both sellers charge the same price (so that n0 = n1 = n). Seller 0’s expected total pro…t over both periods is equal to
0
= N p0 x ~ + D0
N Xn i=n
N i x ~ (1 i
x ~)N
i
+ M0
N X
i=N n+1
N i x ~ (1 i
x ~)N i :
(6)
Seller 0 gets duopoly pro…ts if it has at least n customers so that it survives, but no more than N
n customers such that seller 1 has at least n customers and stays on the market. Seller 0 gets
monopoly pro…ts if it has at least N
n + 1 customers, since this implies that seller 1 has less than
the minimum number of customers required for survival, n. We obtain the following comparative static results for symmetric non-boundary equilibria. Proposition 3. Suppose that a symmetric equilibrium exists in which the equilibrium price satis…es d Rp e 6=
R 11 p.
Then the following comparative static results obtain:
1. An increase in
increases the equilibrium price.
2. An increase in Mi decreases the equilibrium price. 3. A change in Di does not a¤ ect the equilibrium price. Proof. See Appendix. 11
Examples of such symmetric equilibria are provided below. Note that the existence of pure strategy equilibria
is guaranteed for su¢ ciently small
, Di , and Mi . Since the objective function is clearly concave when there are no
strategic buying or predatory pricing incentives (so when seller 0’s objective function is just given by N p0 xm (p0 ; p1 )), for small enough
, Di , and Mi the objective function must still be concave.
15
Intuitively, an increase of
makes it possible for sellers to take advantage of strategic buying
by charging higher prices. If seller 0, say, charges a higher price than seller 1, then in expectation fewer people will buy from seller 0 than from seller 1. Consequently, seller 0’s exit probability is larger. This induces strategic buying in favor of seller 0, and hence a price increase does not lead to the same demand reduction as in a standard Hotelling model with myopic buyers. Knowing that buyers will support at-risk sellers therefore allows sellers to increase their price. An increase in the pro…ts to a seller from being the only one on the market in the second period provides it with an incentive to lower its price in order to drive its rival out of the market. Therefore, the higher is Mi , the lower is the equilibrium price. The size of the duopoly pro…t Di plays no role for …rst period pricing in a symmetric equilibrium. To see why, note that the probability that seller 0 is in a duopoly situation in the second period does not change with x ~, in the neighborhood of x ~ = 1=2: an increase of x ~, for example, makes it less likely that seller 0 drops out in the second period, but also (and by the same amount) more likely that seller 1 drops out so that seller 0 becomes the monopolist. The probability of the third event (that seller 0 is one of two duopolists) remains constant.12 We now analyze how Proposition 3 translates into comparative static results for the primitive model parameters v, N ,
b
and
s.
In particular,
is a function of the buyer discount factor
b
and also the quality component v. Speci…cally, as mentioned above, it is the discounted di¤erence between second period expected buyer surplus when there are two sellers on the market, SD , and expected buyer surplus when there is just one seller on the market, SM . That is,
=
b (SD
SM ):
In the Appendix, we show that the …rst order condition of the optimization problem (6) can be written @x ~ @x ~ N 1 + p0 + M0 2 @p0 @p0 n
1 2
1 1
N 1
= 0:
(7)
Furthermore, applying the implicit function theorem to (5) in a symmetric non-boundary equilibrium (i.e., substituting x = 1=2 after di¤erentiation) yields @x ~ = @p0 2+
1 1 N 3 [N 2
N 1 n 1
= +1
2n]
@x ~ From (7) we can solve for p0 by substituting for @p using (8) and for 0
and M0 =
s N (v
(8)
and M0 using
=
b (v
7 4)
1) (see Appendix)
p0 = 1 + Note that
@x ~ : @p1
N n
1 N 1 N 1 2 n 1
1 1
1 2
N 1
2
b
v
7 4
(N + 1
2n)
s N (v
1) :
(9)
is the probability that a buyer is pivotal in a symmetric equilibrium.
Ceteris paribus, the smaller this pivot probability, the closer is the equilibrium price to 1, which is 12
Note that this reasoning suggests that in an asymmetric equilibrium (for example with v0 6= v1 ), Di will in
general in‡uence pricing since the above cancelling argument requires symmetry. See Section 7.1 for an example of an asymmetric non-boundary equilibrium.
16
the equilibrium price in a standard Hotelling model (given that the transportation cost parameter is normalized to 1). Also note that, henceforth we restrict v
2 so that if there is only one seller, it
is optimal for it to price such that all types buy. Under this restriction
and M0 are both positive.
The two terms inside the curly brackets correspond to the strategic buying e¤ect and the predatory pricing e¤ect, respectively. (This is a legitimate interpretation since setting
b
= 0
makes all buyers myopic and hence switches o¤ the strategic buying e¤ect; and setting
s
= 0
switches o¤ the predatory pricing e¤ect.) The predatory pricing e¤ect always decreases equilibrium prices, and, since n
N=2, the strategic buying e¤ect always increases prices in a symmetric non-
boundary equilibrium. Note that (disregarding the e¤ect of the …rst two terms, which are the pivot probability), the strategic buying e¤ect is smallest when N + 1
2n is small, that is, if n is close
to N=2. Remember that the strategic buying incentive in favor of seller 0 is proportional to the di¤ erence between the probability for being pivotal for the survival of seller 0 and the corresponding probability for seller 1. If N + 1
2n is small, customers are concerned about both sellers, and
these two strategic buying e¤ects (one in favor of seller 0, and the other one in favor of seller 1) are of similar magnitude and di¤erent direction.13 We can now investigate the e¤ect of changes in b , s , v, and N on the equilibrium price. First, dp0 dp0 > 0 (since v 2), and < 0. These e¤ects are consistent with the results it is clear that d b d s from Proposition 3 since is increasing in b and Mi is increasing in s . Next, we determine how v in‡uences …rst period prices. Note that, in a standard Hotelling model, the equilibrium price is independent of the customers’ willingness to pay, v, but depends only on the transportation cost (or di¤erentiation) parameter that is normalized to 1 in our model. Hence, any non-zero e¤ect of v in our model is a consequence of the strategic buying and predatory pricing e¤ects. Moreover, the e¤ect of v is not a priori clear, because v increases both linearly, and from Proposition 3 we know that increases in directions. Speci…cally, an increase in
and Mi
and Mi cause p0 to move in di¤erent
causes equilibrium price to increase, while an increase in
Mi causes it to fall as sellers reduce their prices in an e¤ort to drive their competitors out of the market. Di¤erentiating (9) with respect to v yields dp0 = dv
N n
1 1
1 2
N 1
[2 b (N + 1
2n)
N
s] :
(10)
The terms in square brackets correspond to the strategic buying e¤ect and the predatory pricing e¤ect, respectively. Both e¤ects are linear in v, so that an increase in v increases p0 if and only if the strategic buying e¤ect is larger than the predatory pricing e¤ect. As discussed above, the strategic buying e¤ect diminishes when (N + 1 2n) is close to zero. If the discount factor of buyers 13
Compare also equation (8) where (N +1 2n) being small has the same e¤ect as
being small, namely decreasing
the strategic buying e¤ect and bringing the derivative closer to 1=2, which is the derivative in a standard Hotelling model.
17
and sellers is the same, (10) is positive if and only if N > 4n
2, and negative if the inequality
is reversed. Hence, when N is large relative to n, an increase in v leads to an increase in the …rst period price p.14 Next, we analyze the e¤ect of N on equilibrium pricing. Let us denote by Z(N ), the terms in curly brackets in (9), and by Piv(N), the pivot probability. Note that, if Z(N ) > 0, the strategic buying e¤ect outweighs the predatory pricing e¤ect (so that the equilibrium price is larger than 1 in this case) and vice versa. The change in the equilibrium price when N increases is dp = Z dPiv + Piv dZ. One can show that the pivot probability goes down as N increases (so dPiv < 0), but since both Z and dZ can be positive or negative the equilibrium price can increase or decrease as N changes. In the Appendix we characterize the equilibrium condition corresponding to (9) when there are N + 1 buyers and analyse the e¤ect of increasing the number of buyers on p0 . An increase in N reduces the probability that each …rm is just getting enough customers to survive, which in turn reduces their competitor’s predatory pricing incentive, thus increasing the equilibrium price. On the other hand, the e¤ect of an increase in N through the strategic buying channel is ambiguous. For most parameter values this e¤ect is negative. As N grows large, buyers are less likely to purchase strategically in an e¤ort to prevent exit and therefore sellers can no longer price high to take advantage of this behavior. However, when n is close to N=2, buyers are concerned about the survival of both sellers, and so for these parameter values the strategic buying e¤ects in favor of seller 0 and in favor of seller 1 roughly o¤set one another. For …xed n, when N increases the survival of both sellers is no longer threatened and so strategic buying may increase.
4.2
Symmetric boundary equilibrium
We now turn to the case where the revenue constraint is such that small price changes a¤ect the number of customers a seller needs for survival. A price decrease that causes the number of customers required for survival to increase will generate a discrete jump in strategic buying on the part of consumers. More speci…cally, consider a situation where the sellers’ prices are such that each seller just satis…es its revenue constraint, then a deviation to a slightly lower price is always pro…table for a seller since it leads to a discrete increase in demand. Unfortunately, these discontinuities in the pro…t function make a general analysis very di¢ cult. For this reason, we consider a numerical example that illustrates the e¤ect just described and in 14
It is also interesting to examine what happens if the vertical quality of one of the two products marginally
increases. In the standard Hotelling model
@pi @vi
= 13 . In our setup, the only additional e¤ect on price of the increase
in quality is through the predatory pricing e¤ect (as vi increases, monopoly pro…ts increase and therefore so does the incentive to drive the other seller out of the market and so …rst period prices fall). In the neighbourhood of is no strategic buying e¤ect, and so if
s
= 0 such that there is no predatory pricing motive,
18
@pi @vi
= 13 .
1 2
there
which we can solve for a symmetric boundary equilibrium. In order to focus on the strategic buying e¤ect, we set
s
= 0 in our example such that Di = Mi = 0: (This is so that …rms have no predatory
pricing motive. If we allow for positive monopoly pro…ts in the example, the result is unchanged.) Example 1. Suppose that N = 4,
s
= 0,
b
= 1 and v0 = v1 = v = 11=4.
For these parameters, (25) implies that
= 1, and assuming that n = 1, the …rst order
condition (7) simpli…es to 1 2
2+
p0 3 1 0 2 [4
+1
2]
= 0;
which we can solve for p0 = 1:75. If R is positive, but su¢ ciently small, then the assumption above that n = 1 is satis…ed and there is a symmetric non-boundary equilibrium with p0 = p1 = 1:75. However, consider now what happens if R = 1:505 and prices can only be charged in multiples of 0.01.15 Again, if both sellers charge pi = 1:75, then a single buyer is su¢ cient for each seller to survive. We now argue that decreasing the price to just below R = 1:505 may be attractive for a seller. The reason is that at this point a greater number of customers is required for survival, which implies that a lower price generates increased strategic buying by customers to aid the “high risk” seller with the lower price. This increase in strategic buying can potentially justify the price decrease. In the example, suppose that we …x seller 1’s price to p1 = 1:75 and consider seller 0’s optimal response. We claim that, rather than also charging p0 = 1:75, seller 0 can do better by lowering its price to 1:5, which is the highest price such that one customer is not enough to secure its survival. If p0 = 1:5, then the indi¤erent consumer satis…es the following equation v
p0
x ~+
3 x ~(1 1
x ~)2 = v
The term on the left hand side that multiplies
p1
(1
x ~) +
3 3 x ~ : 0
(11)
is a buyer’s pivot probability for seller 0, since a
buyer is pivotal, if, of the other three customers, one has chosen to buy from seller 0 and two have chosen to buy from seller 1. The term that multiplies
on the right hand side is a buyer’s pivot
probability for seller 1, since a buyer is pivotal for that seller if and only if all three other customers buy from seller 0. If seller 1 charges p1 = 1:75 and seller 0 charges p0 = 1:5, then the cuto¤ is x ~ to an expected pro…t of
3:78. This is bigger than the expected pro…t of
0
0
0:630, leading =
4 2
1:75 = 3:5
that seller 0 would get if it also charged a price of 1:75. To see the increased e¤ect of strategic buying in this example, consider what would happen if seller 0 charged just a little bit more than the revenue constraint, say p0 = 1:51. In this case, demand would be determined by v 15
p0
x ~+
(1
x ~)3 = v
p1
(1
x ~) +
x ~3 ;
(12)
The assumption of a discrete price grid is made to avoid non-existence problems that arise when sellers optimize
on open sets (e.g., try to set “the smallest price that is strictly greater than 1.5”).
19
as both sellers only need one customer to survive. For p1 = 1:75 and p0 = 1:51, this can be solved for x ~ = 0:568. Hence, a further decrease of p0 by only 0:01 leads to a more than 10% jump in demand, from 0:568 to 0:630, as a result of the increase in strategic buying. Of course, our arguments so far only show that (p0 ; p1 ) = (1:75; 1:75) is not an equilibrium if R = 1:505, because seller 0 has a pro…table deviation. One can show that, for these parameters, seller 1’s optimal response to p0 = 1:5 is also p1 = 1:5, so that this pair of strategies constitutes an equilibrium. Hence, even though there is no strategic buying in equilibrium in this example (since both sellers charge the same price), the possibility of strategic buying o¤ the equilibrium path in‡uences equilibrium prices. So both at R close to zero, and at R = 1:505, we have a unique equilibrium. This suggests that if R is increased from low values to 1:505; at some point there is a transition where the type of equilibrium switches. In fact we can show that there is an interval of revenue constraints [R0 ; R1 ] such that for R’s below this interval, there is a unique equilibrium where both sellers set p = 1:75 (price is high so that n = 1, and R is su¢ ciently low that it is not worthwhile to lower price to convince buyers to buy strategically), for R’s above this interval, there is a unique equilibrium where both sellers set R
" (so that n = 2 and buyers must engage in more strategic buying),
and for R’s in the interval there are two equilibria: i) both sellers setting p = 1:75, ii) both sellers setting R
4.3
". In the example R0
0:835 and R1
1:285.
Consumer welfare and seller pro…ts
There are a number of interesting observations related to the welfare and pro…t e¤ects of strategic buying in both the symmetric boundary and non-boundary cases. First, note that in the symmetric boundary example above, both sellers and buyers are better o¤ when R is su¢ ciently low that (p0 ; p1 ) = (1:75; 1:75) is the equilibrium than when R is 1.505 so that (p0 ; p1 ) = (1:5; 1:5). This is obvious for sellers, as the equilibrium price is higher. For buyers, note that the probability of two sellers surviving is for R = 1:505, while this probability is 1 this probability di¤erence of 1=2 with
(1=2)4
4 2
(1=2)4 = 6=16 in the equilibrium
(1=2)4 = 14=16 when R is low. Multiplying
= 1 shows that, when R increases from a low value to
R = 1:505, the utility loss for consumers from the higher probability of exit is greater than the utility gain from lower prices. More generally, what can be said about seller pro…ts? Here we investigate whether sellers are better o¤ when they face myopic consumers. We assume that
s
= 1 so that sellers care equally
about both the …rst and second period. In this case, if consumers are myopic (set
= 0), and
using the same parameter values as in Example 2 (with R su¢ ciently small that n = 1), equation 7 yields …rst period prices of (p0 ; p1 ) = ( 18 ; 18 ) and so each seller earns pro…ts of the other hand
=
4 2
1 8
= 14 . If, on
= 1, then in the …rst period sellers charge higher prices (p0 ; p1 ) = ( 78 ; 78 ) which
20
yields pro…ts of
=
4 2
7 8
=
7 4
for each seller. The probability of exit is the same and so second
period pro…ts are the same whether consumers are myopic or not. Therefore, expected pro…ts are higher when consumers are myopic since this e¤ectively lowers the …rst period price elasticity. We also investigate whether sellers are better o¤ when they (both) face a survival constraint than when they do not. If there is no survival constraint (n = 0), equation 7 yields …rst period prices of (p0 ; p1 ) = (1; 1) and so each seller earns …rst-period pro…ts of
= 2. Second-period pro…ts
are also equal to 2 since both …rms stay in the market. If, on the other hand, n = 1, then in the …rst period sellers charge prices (p0 ; p1 ) = ( 78 ; 78 ) which yields total pro…ts for each seller of 3
=
4 7 4X 4 1 i ( ) (1 + 2 8 2 i 2
1 4 ) 2
i=1
=
63 16
i
4 X 4 1 i +7 ( ) (1 i 2 i=4
1 4 ) 2
i
and so in this case total pro…ts are lower when there is a survival constraint than when there is not. The predatory pricing motive causes sellers to charge a much lower price in the …rst period than they would if there were no constraint, and so even though second period pro…ts are higher (since there is some chance of monopoly pro…ts), overall pro…ts fall.16 It is also interesting to note that sellers might prefer to commit to prices for both periods if they could (assuming they survive into the second period). Seller 0’s expected pro…t over both periods from committing to a price p0 (for both periods) is
0
= N p0 x ~ + N xm p0
N Xn
N i x ~ (1 i
i=n
N i
x ~)
+ N p0
N X
i=N n+1
N i x ~ (1 i
x ~)N i :
(13)
and so the total-pro…t maximizing price (using the parameter values from above) is (p0 ; p1 ) = 112 ( 112 25 ; 25 ) which yields pro…ts of
( 87 ; 78 )
which yields pro…ts of
for a total of
=
161 32
<
448 25 :
=
= 7 4
448 25
for each seller. If they cannot commit, they set (p0 ; p1 ) =
in the …rst period and pro…ts of
=
105 32
in the second period
This is because by committing to keep prices constant, they remove
their incentive to predatory price since being a monopolist is no longer as attractive. Finally, note that from the buyers’ point of view, less strategic buying would be bene…cial in the symmetric boundary example given above. Suppose, for instance, that buyers could ex-ante commit to not buy strategically. Then the model of …rst period behavior would just be a standard Hotelling model with an equilibrium price of 1, and since at this lower price each seller still needs two customers in order to survive, the probability of exit would be exactly the same as in the equilibrium in the example. Hence, in contrast to the case of exogenous prices discussed above in Section 3 where we showed that strategic buying is underprovided, it is actually overprovided once 16
This example assumes symmetry; namely that both …rms face a survival constraint. If only one …rm faces a
survival constraint, the incentive on the part of the seller without a constraint to predatory price should be even greater.
21
we take into account the e¤ect of strategic buying on the sellers’ pricing behavior. This is true more generally so long as R is such that the price cut that accompanies the decrease in strategic buying is not so big as to cause the number of customers necessary for survival to change (in this case it is the e¤ect of strategic buying on consumer welfare is ambiguous since without strategic buying prices fall, but the probability of exit is higher).
5
Managerial implications
In the Introduction we described a number of situations in which incentives for strategic buying exist, such as government procurement of military equipment and prisons, and airline purchases of regional jets. Our model provides several insights for managers of both buyers and sellers in these and other institutions. We start with the implications for sellers. Strategic buying is most likely to lead to a signi…cant change in customer behavior if the probability that a customer is pivotal is relatively large. Thus, the e¤ects we describe apply mostly to markets in which a few buyers purchase big-ticket items (often investment goods, as in the examples described above). In contrast, strategic buying is unlikely to be very signi…cant in industries where each buyer makes up only a negligible share of the market. If a seller is at risk of exit, buyers may shift their current purchases towards the at-risk …rm in order to avoid facing a monopolistic market in the future. Managers of …rms selling in such situations should be aware of this forward-looking behavior on the part of consumers and can take advantage of it. Fundamentally, strategic buying lowers the price elasticity of demand and gives sellers the opportunity to charge higher prices. Note that a change in the elasticity of demand a¤ects the optimal price for all …rms, not just the one that faces a risk of exit, but also its competitors who may themselves be relatively secure in their continued existence. The fact that customers may engage in strategic buying is valuable for a seller as it softens price competition and, ceteris paribus, increases the quantity of demand that an at-risk …rm faces. Thus, it is important for sellers not to reveal their intention to stay in the market, as this removes the buyers’incentive to engage in strategic buying from that …rm. For example, it may be worthwhile for sellers to delay investment or to avoid undertaking any other activity that would reveal a commitment to remain in the market in the future. In the standard Hotelling model, the equilibrium price depends on the di¤erentiation parameter (the “transportation cost”that is normalized to 1 in our model), but not on customers’total valuation of the product (i.e., v). Thus, when …rms can in‡uence customer perceptions by advertising in such a setup, their only objective is to increase the perceived di¤erentiation between their own variey and that of their competitor. In contrast, with strategic buying, markups depend positively on buyer valuation for the product, so that sellers also have an incentive to increase this valuation. 22
For example, while advertising that only increases customers willingness to pay for the product class (as opposed to the seller’s particular variety) has no purpose in the standard Hotelling setup, it would have some value in a strategic buying setup. We now turn to the managerial implications for buyers. First, the act of strategic buying involves a trade-o¤ between the choice that would be optimal for the current period (either in terms of price, or in terms of usefulness for the buyer) on the one hand, and an investment in the future supply market structure on the other hand. Managers’willingness to forego current period bene…ts from buying the cheaper and/or more appropriate product by engaging in strategic buying should depend positively on the probability that their purchase decision has a positive in‡uence on the future market structure, and on the size of the bene…t of having more sellers to choose from in the future. In many cases, buyers and sellers may have an ongoing relationship over many years. When considering the future value of keeping a particular seller alive, buyers will often also focus on the costs and bene…ts of having a stock of that particular variety going into the future. This is an incentive for strategic buying in addition to the one that we model formally. For example, having purchased a regional jet from a particular seller in the past, an airline might be very interested in its ability to have the jet serviced should there be any mechanical trouble. The exit of the seller that provided the jet would make this very di¢ cult. This concern generates both incentives for and against buying from at-risk sellers. On the one hand, having purchased a large number of items from a producer in the past increases the amount that is at stake if the seller exits the market, making the buyer more willing to “invest”in strategic buying so as to avoid this outcome. On the other hand, the worry that any warranty obligation on the products that would be bought today might not be met should generate an incentive to shy away from at-risk sellers that con‡icts with the strategic buying motive. Finally, sellers must consider the pricing response to strategic buying. Although strategic buying is underprovided for …xed prices in the sense that buyers would like more strategic buying to take place, we have shown that the full equilibrium predicts that buyers may actually be harmed by the incentive to engage in strategic buying, since it allows sellers to increase their prices. Thus, if it were possible for buyers to commit against strategic buying, this might be bene…cial for them in our framework. Practically, however, such a bene…cial commitment may not be easy to achieve, as any long-term commitment is likely not just to a¤ect the option of “strategic buying”, but in general, the ability to switch suppliers in the future, which may open up a buyer to exploitation by the locked-in supplier. For this reason, a commitment to not buy strategically may well be a cure that is worse than the disease.
23
6
Conclusions
In this paper, we build a dynamic oligopoly model to examine situations in which buyers care about whether a seller stays in the market in the next period. Buyers have two main reasons for caring about future market structure: if a seller exits, the market price may be higher and/or less variety may be available. In our model, a seller exits if it has less than n customers. Our analysis predicts that in such situations, buyers sometimes engage in strategic buying: they purchase from a seller that …ts them worse today in the hope that their purchase decision will make a di¤erence as to whether this seller survives. We also show that non-myopic sellers take advantage of strategic buying to charge higher prices. Our focus has been on oligopoly situations, but many of the same incentives for buyers and sellers exist in the context of an at-risk monopolist. Buyers may choose to purchase from the seller rather than not at all in order to keep it alive,17 or (as in Romano’s 1991 study of the monopoly situation) buy more from it than they otherwise would. Although in many ways the monopoly situation is much more tractable, the oligopoly setup is, in most cases, more realistic and yields richer results. In particular, our setup allows us to study in detail the pricing game played by competing sellers in the face of strategic buyers. Clearly, monopolists have no predatory pricing motive. Relative to Romano (1991) our analysis yields a number of di¤erent results. For instance, the seller pricing strategies we uncover are such that buyers would like to commit to not buy strategically, while in his setup buyers are never worse o¤ from their excessive buying. One important issue left for future research is to develop empirical tests of strategic buying. The most easily testable implication of our model (relative to the standard Hotelling model) comes from the comparative static pricing results in Section 4. As mentioned above, in the standard Hotelling model, the markup of price over cost is independent of the consumers’net valuation parameter v, while (in most circumstances) in our model, shocks to v a¤ect equilibrium markups. Our model also predicts that the price elasticity of demand is smaller, the higher is the pivot probability (see equation (8)). As the pivot probability is not directly observable, one would have to estimate it simultaneously from the data.18 This may be possible by looking at a cross section 17
Consider a similar setup to our’s in which only seller 0 is in the market and where buyers can purchase from it
or not at all (in which case they get utility of 0), buyers may engage in strategic buying: buy from seller 0 when they prefer not to buy. The indi¤erent agent in this case is given buy the x ~ that solves ! N 1 n 1 v p0 x ~+ x ~ (1 x ~)N n = 0: n 1 It is clear from this equation that x ~
xm = v
(14)
p0 since the last term on the LHS is greater than or equal to zero.
In other words, strategic buying may occur. 18 For tests of the costly voting model of participation in elections a somewhat related problem exists (see, for example, Shachar and Nalebu¤ (1999)). The theory predicts that more individuals should vote when the expected election margin is close. Therefore, many studies have examined the correlation between voter turnout and (ex post
24
of markets within the same industry. Another potential testing opportunity is that some markets may consist of some agents who expect that they will also buy in the next period while others know that they are one-time buyers (
b
= 0 in our terminology). If an observable characteristic allows to assign individuals to one of
these groups, then a direct comparison of di¤erent individuals’behavior would be possible.
References [1] Battigalli,P., C. Fumagalli, and M. Polo, 2007 “Buyer Power and Quality Improvements”, Research in Economics 61: 45-61 [2] Bergstrom, T., L. Blume and H. Varian, 1986, “On the Private Provision of Public Goods”, Journal of Public Economics, 29: 25–49. [3] Biglaiser, G. and N. Vettas, 2007, “Dynamic Price Competition with Capacity Constraints and Strategic Buying”, mimeo. [4] Börgers, T., 2004, “Costly Voting”, American Economic Review, 94: 57-66. [5] Chen, Z., 2007, “Buyer Power: Economic Theory and Antitrust Policy”, Research in Law and Economics, 22: 17-40. [6] ________2006, “Monopoly and Product Diversity: The Role of Retailer Countervailing Power”, mimeo, Carleton University. [7] Dobson, P., and M. Waterson, 1997, “Countervailing Power and Consumer Prices”, Economic Journal, 107: 418-30. [8] Inderst, R. and G. Sha¤er, 2007, “Retail Mergers, Buyer Power, and Product Variety”, The Economic Journal, 117: 45-67. [9] Inderst, R. and C. Wey, 2003, “Market Structure, Bargaining, and Technology Choice in Bilaterally Oligopolistic Industries”, RAND Journal of Economics, 34, 1-19. [10] __________________2007a, “Buyer Power and Supplier Incentives”, European Economic Review, 51: 647-667. [11] ___________________2007b, “Countervailing Power and Dynamic E¢ ciency”, mimeo. or –better– ex ante) election closeness. A di¤erence is that it is easier to determine whether an election is “close” than when a …rm is “close to exiting”.
25
[12] Katz, M., 1987, “The Welfare E¤ects of Third Degree Price Discrimination in Intermediate Goods Markets”, American Economic Review, 77: 154-167. [13] Krasa, S., and M. Polborn, 2009, “Is mandatory voting better than voluntary voting?”, Games and Economic Behavior, 66: 275-291. [14] Ledyard, J., 1984, “The Pure Theory of Large Two-Candidate Elections”, Public Choice, 44: 7–41. [15] Lewis, T., and H. Yildirim, 2002, “Managing Dynamic Competition”, American Economic Review, 92: 779-797. [16] Montez, J., 2008, “Downstream Mergers and Producer’s Capacity Choice: Why Bake a Larger Pie When Getting a Smaller Slice”, forthcoming RAND Journal of Economics. [17] Palfrey, T. and H. Rosenthal, 1983, “A Strategic Calculus of Voting”Public Choice, 1983, 41: 7–53. [18] Romano, R., 1991, “When Excessive Consumption is Rational”, American Economic Review, 81: 553-564. [19] Stigler, G., 1964, “A Theory of Oligopoly”, Journal of Political Economy, 62: 44-66. [20] von Ungern-Sternberg, T., 1996, “Countervailing Power Revisited”, International Journal of Industrial Organization, 14: 507-19.
26
7
Appendix
7.1
Example with vertical quality di¤erences
Example 2. Suppose that N = 3,
s
= 1,
b
= 1, v0 = 2, v1 = 11=5, and that R su¢ ciently small
such that n = 1. We are interested in the full non-boundary equilibrium. Seller 0’s expected total pro…t over both periods is equal to
0
= 3p0 x ~ + D0
2 X 3 i x ~ (1 i
x ~)3
i
+ M0 x ~3 ;
x ~)i x ~3
i
+ M1 (1
(15)
i=1
while seller 1’s expected pro…t is
1
= 3p1 (1
x ~) + D1
2 X 3 (1 i
x ~)3 :
(16)
i=1
We need to solve the second period problem in order to get values for Di , Mi , and
. If both sellers
are still present in the market, the equilibrium cuto¤ is given by x ~=
v0
p0
(v1 2
p1 )
2 5
=
p0
p1 2
Sellers choose prices to maximize pro…ts
D1 = 3p1 (1
p0
2 5
D0 = 3p0 x ~ = 3p0
p1 2
x ~) = 3p1
3 p0 p 1 + 5 2
:
Solving the corresponding …rst order conditions yields equilibrium prices of p0 = sales of x ~=
7 15 ,
14 15
and p1 =
16 15 ,
and pro…ts of D0 =
98 128 and D1 = : 225 225
If only seller 0 is still on the market, it earns monopoly pro…ts of M0 = 3(2 seller 1 is still on the market, it earns monopoly pro…ts of M1 = 3(11=5
1) = 3; while if only
1) = 18=5: To calculate
, we must determine consumer surplus in the cases of monopoly and duopoly respectively. If there is only one seller on the market (either seller 0 or 1), consumer surplus is Sm = 12 : If both sellers are present, consumer surplus is given by Z x~ Z 1 SD = (v0 x p0 )dx + (v1 =
Z
0
(1
x)
p1 )dx
x ~
0
7 15
(2
x
14 )dx + 15
Z
1
( 7 15
27
11 5
(1
x)
16 383 )dx = : 15 450
(17)
Thus,
=
383 450
1 2
=
79 225 .
Using (5) to solve for x ~=
259
225(p0 608
p1 )
and substituting this and the values for Mi , Di , and
;
calculated above into (15) and (16) yields
…rst-period equilibrium prices of p0 = 0:38 and p1 = 0:42; equilibrium cuto¤ of x ~ = 0:441; and expected pro…ts of 0
= 1:54 and
1
= 2:40:
Note that we have v1 11 5
p1 > v0
p0
0:42 > 2
0:38
and buyers engage in strategic buying since if they were myopic (
= 0), we would have
xm = 0:412 < x ~:
7.2
E¤ect of an increase in the number of buyers on strategic buying
An increase of the total number of buyers from N to N + 1 changes the equilibrium condition corresponding to (3) to v0 Using
p0
(v1
N n 1
=
N n
1 1
p1 ) + 1
2~ x+
N 1 N n 1 N n+1 ,
1
N
N n
1
x ~n
1
(1
x ~)N
n+1
(1
x ~)n
1 N n+1
x ~
= 0:
(18)
we can subtract (3) from (18) to get
Nx ~ (1 n+1
x ~)n
1 N n
x ~
1
N (1 x ~) x ~n N n+1
1
(1
x ~)N
n
:
(19)
If the term in curly brackets is positive so that the left hand side of (18) exceeds the left hand side of (3), then the new x ~(N + 1) lies to the right of x ~(N ), so that strategic buying increases with N . If, instead, the term in curly brackets in (19) is negative, then an increase in N shifts x ~ to the left so that there is less strategic buying. The expression in (19) can in general be positive or negative, depending on parameters. First, if N grows large for …xed n, then the term in curly brackets goes to (1 28
x ~)n x ~N
n
x ~n (1
x ~)N
n,
which is negative in equilibrium since x ~ < 1=2. Consequently, if N is already very large, a further increase in N reduces the extent to which an individual buys strategically. As N goes to in…nity, inspection of (3) shows that, for each interior value of x ~, the factor that multiplies
goes to zero.
Hence, not only do increases in N , starting from some already high level of N decrease the extent of strategic buying, but as N becomes very large, x ~ converges to xm , the equilibrium cuto¤ when buyers are myopic. Next note that (19) may actually be positive for some parameters, so that an increase in N increases strategic buying. A su¢ cient condition for this is x ~N < n brackets can be written
N x n+1 N n+1
1. The second term in square
and therefore is negative under the condition above. The …rst
term in square brackets is positive, since it can be written as the …rst inequality uses x ~N < n
N (1 x ~) (n 1) N n+1
1 and the second one uses n
>
N 2(n 1) N n+1
> 0, where
N=2. Hence, in summary, if
x ~N
n
1, then an increase in N leads to more strategic buying.
7.3
E¤ect of an increase in the number of customers necessary for survival on strategic buying
When the number of customers required for seller survival increases from n to n+1, the equilibrium equation corresponding to (3) becomes v0
p0
(v1
p1 ) + 1
2~ x+
Subtracting (3) from (20) and using N n
1 1
(N
n) n
x ~ (1
x ~)
N
1 n
N 1 n
1 x ~n
1
= (1
x ~n (1
x ~)N
N 1 N n n n 1
x ~)N
n
1 n
(1
x ~)n x ~N
1 n
= 0:
(20)
yields (N
n) (1 n
x ~)
1 (1
x ~
x ~)n
1 N n
x ~
:
When this expression is negative, the function given by the left hand side of (20) lies below the function given by (3), and consequently the zero of (20) is at a lower value of x. The second term in square brackets is positive, since N n > n and 1 x ~>x ~. Therefore, a su¢ cient condition for h i ~ < n. x ~ to be decreasing in n is that N n n (1 x~ x~) 1 < 0, which is equivalent to N x
7.4
Proof of Proposition 3
Note …rst that
@ @x
PN
n N i=n i
x ~i (1 x ~)N
i
= 0, since the term in brackets is a symmetric function
around x = 1=2 and therefore has a local extremum at x = 1=2. Consequently, its derivative there must be zero. Di¤erentiating (6) with respect to p0 , taking into account the above result, and evaluating at x ~ = 1=2, we get d 0 @x ~ @x ~ = N=2 + N p0 + M0 dp0 @p0 @p0
N X
i=N n+1
29
N i
1 2
N 1
[2i
N ]:
(21)
Dividing (21) by N and simplifying the term that multiplies M0 yields equation (7) in the text. The third claim of the proposition follows immediately from the fact that (7) is independent of D0 . @x ~ @p0 ,
To determine
we apply the implicit function theorem to (5) to get @x ~ = @p0 2+
where
= (N
1) x ~n
(n
n) x ~n
2 (1
x ~)N
x ~)N
1 (1 n
+ (1
n 1
x ~)n
x ~)n
+ (1
2x ~N n
1 N 1 n 1
;
(22)
1x ~N n 1
. In a symmetric equilibrium such that x ~ = 1=2, this
simpli…es to equation (8) in the text. Since n
N=2, the term in square brackets in the denominator
of (8) is positive, which implies that the denominator is larger than 2 and that an increase in decreases the absolute value of @ x ~=@p0 . Suppose that the equilibrium price for (
0 ; M 0 ; D0 ) i i
is p0 . Consider an increase of
decreases the absolute value of @ x ~=@p0 ; hence, the left hand side of (7), evaluated at
to p0
00 .
This
is positive,
implying that the new equilibrium price must be higher than p0 . Now consider an increase in Mi to Mi00 . Given that @ x ~=@p0 < 0, this means that the left hand side of (7), evaluated at p0 , is now negative, implying that the new equilibrium price must be lower than p0 .
7.5
Expressions for
and Mi
In the case of two sellers in the second period, the model is a standard Hotelling model with a price of 1 (given that the “transportation cost parameter” is normalized to 1). Therefore, Z 1=2 Z 1 5 : SD = (v x 1)dx + (v (1 x) 1)dx = v 4 0 1=2 If there is only one seller and v buy, hence at pM = v
(23)
2, then it is optimal for the seller to price such that all types
1. Expected consumer surplus is therefore Z 1 (v x (v 1))dx = 1=2:
(24)
0
And so, 7 ): (25) 4 A monopolist sells to N customers at a price of v 1, hence the discounted monopoly pro…t is =
Mi =
b (v
s N (v
1):
(26)
If both sellers are active in the second period, they each charge a price of 1 and sell to half of the market, therefore Di = 30
s
N : 2
(27)
7.6
E¤ect of an increase in the number of buyers on equilibrium pricing
The equilibrium condition corresponding to (9) when there are N + 1 buyers is p0 (N + 1) =
N n
1 2
1
N
2 b (v
Subtracting (9) from (28), using p0 (N + 1) b
v
N n 1
=
7 )((N + 1) 4 N 1 N n 1 N n+1 ,
2n + 1)
s (N
+ 1)(v
1) + 1:
(28)
and simplifying we get
N (N + 1 2n) N 1 1 N 1 1) s (v 2 2(N n + 1) n 1 N (N + 2 4n) + (4n 2)(n 1) (N n + 1)
p0 (N ) = 7 4
(29)
The e¤ect of a change in N on prices depends on the two terms in curly brackets in (29). The …rst one of these corresponds to the predatory pricing e¤ect, while the second one corresponds to the strategic buying e¤ect.
31
