Oligopoly pricing: the role of ﬁrm size and number

This paper examines a homogeneous-good Bertrand-Edgeworth oligopoly model to explore the role of (cid:133)rm size and number in pricing. We consider the price impact of merger, breakup, investment, divestment, entry, and exit. A merger leads to higher prices only when it increases the size of the largest seller and industry capacity is neither too big nor too small post-merger. Similarly, breaking-up a (cid:133)rm only leads to lower prices when it concerns the biggest producer and aggregate capacity is within an intermediate range. Investment and entry (weakly) reduce prices, whereas divestment and exit yield (weakly) higher prices. Taken together, these (cid:133)ndings suggest that size matters more than number in the determination of oligopoly prices.


Introduction
There are many reasons for why market competition may not result in a competitive outcome, i.e., a situation where …rms price at marginal cost.One of these reasons was provided by the Irish scholar Francis Y. Edgeworth (1925), namely the presence of capacity constraints. 1 Indeed, when demand is su¢ciently large in comparison to production capacity, …rms may well be capable of selling all of their supplies at supracompetitive prices; that is, at prices above marginal cost.
For a given level of demand, however, there are at least two distinct factors that a¤ect available capacity, the number of …rms and their size.For instance, capacity constraints tighten both when an incumbent divests part of its business and when it redeploys its assets to leave the industry all together.
In case of exit, there is a reduction in the number of competitors, whereas this need not be true in case of divestment.Though one would expect both events to induce an upward pressure on prices, after all there is less production capacity available, the precise impact on prices is a priori unclear.In particular, one may wonder whether …rm size and number have a comparable e¤ect on oligopoly pricing and, if not, what drives the di¤erence.
The purpose of this paper is to shed light on the role of …rm size and number in capacity-constrained oligopoly pricing.Toward that end, we analyze an n-…rm homogeneous-good Bertrand-Edgeworth oligopoly. 2As is well-known, this model captures three di¤erent types of market situation, each of which can be characterized by the ratio of industry capacity to market demand.
When production capacity is 'small' relative to market demand, there is a symmetric pure-strategy Nash equilibrium which equates aggregate capacity and demand.When capacity is 'large' relative to market demand, there also is a pure-strategy solution, which has …rms pricing at marginal cost.In what follows, we refer to the …rst as the monopolistic zone and the second as the competitive zone.Finally, if industry capacity is 'intermediate', then there is no pure-strategy equilibrium.As pointed out by Edgeworth (1925), prices are indeterminate in this case and oscillate inde…nitely between some lower and upper bound.We refer to this as the Edgeworth zone.
To examine the role of …rm size and number in pricing in a capacity-constrained oligopoly model, we consider the impact of changes in (the allocation of) production capacity.Speci…cally, we explore the price e¤ects of six di¤erent capacity-altering events: merger, breakup, investment, divestment, entry, and exit.In case of merger or breakup, there is a change in the number of sellers, but aggregate capacity is not a¤ected.With investment and divestment, the number of …rms remains constant, but capacity changes.Finally, with entry and exit, both capacity and the number of …rms change.
These events can have two distinct types of e¤ect, both of which may a¤ect pricing.On the one hand, they can have an impact on the span of each of the three zones.This implies that the events can cause the industry to switch zones.For instance, when capacities are large and the market is in the competitive zone, divestment or exit might reduce production capacity to such an extent that the industry moves into the Edgeworth zone.On the other hand, given that the industry is in the Edgeworth zone, the events may also a¤ect the degree of price dispersion.We refer to this as the price ‡uctuation range, i.e., the range of prices that are set with positive probability.
We …nd that a merger either has no e¤ect on price or elevates price.There are two necessary conditions for prices to rise as a result of merger: (i) the merger increases the size of the largest seller (including the possibility of a merger between smaller producers that becomes the new leading …rm post-merger), and (ii) the industry is in the Edgeworth zone or the merger brings the industry into the Edgeworth zone.Mergers that have no e¤ect on the market position or size of the biggest industry player do not a¤ect price.Moreover, if the industry is in the monopolistic or the competitive zone, then a merger only a¤ects prices when it causes the industry to move into the Edgeworth zone.As with mergers, breaking-up a …rm in smaller, separate entities has an impact on price only when it concerns the biggest seller.Moreover, it is only e¤ective, in the sense of leading to lower prices, when the industry is in the Edgeworth zone.In all other cases, splitting-up suppliers in smaller, independent units does not a¤ect price.
Investment and entry increase production capacity and overall have a comparable price reducing e¤ect.If the industry is in the monopolistic zone, then both these events may bring the industry into the Edgeworth zone and, ultimately, into the competitive zone.Roughly speaking, the opposite occurs with divestment and exit, both of which reduce available production capacity.Starting in the competitive zone, these events may lead the industry into the Edgeworth zone and, eventually, into the monopolistic zone.The fact that entry and investment as well as divestment and exit work in the same direction, indicates that production capacity is a more important price determinant than the number of competitors in these types of industry.
As to price dispersion, we …nd that a merger (break-up) may yield a wider (narrower) price ‡uctuation range, but only when it a¤ects the size of the industry leader.Investment by the largest industry player also leads to more price dispersion.Taken together, our …ndings thus suggest that size matters more than number in the determination of oligopoly prices.
The next section introduces the model and provides a detailed speci…cation of the price ‡uctuation range.Section 3 explores the price e¤ects of merger, breakup, investment, divestment, entry, and exit.Section 4 o¤ers a brief discussion of our …ndings and Section 5 concludes.All proofs are relegated to the Appendix.

Oligopoly Model
We examine a Bertrand-Edgeworth oligopoly comprising a given set of pro…t-maximizing …rms: N = f1; 2; : : : ; ng.Each …rm i 2 N has a production capacity k i > 0 and produces a homogeneous good on demand at a common marginal cost c 0. Without loss of generality, it is assumed that k 1 k 2 : : : k n .Total industry capacity is denoted by K = P n i=1 k i and K i is the combined production capacity of all …rms other than i.
The market demand function is D : R + 7 !R + .It is assumed that D (c) > 0 and that D (p) is twice continuously di¤erentiable, with D 0 ( ) < 0 and D 00 ( ) 0 at any market price p 0. Sellers select prices simultaneously from [0; 1) so that [0; 1) n is the set of all possible price pro…les.As products are perfect substitutes, consumers prefer to purchase from a lowest-priced seller.Since these suppliers may be capacity-constrained, however, it is possible that only part of them is served in which case higher-priced sellers might still face demand.
To specify individual (residual) demand, let (p i ) = fj 2 N jp j = p i g and (p i ) = fj 2 N jp j < p i g denote the set of …rms that price at and below p i , respectively. 3Furthermore, let p i = (p 1 ; :; p i 1 ; p i+1 ; :; p n ) indicate the prices of all …rms other than i.Demand for …rm i's product is then given by D i (p i ; p i ) = D (p i ) when all its competitors charge a strictly higher price.If there is at least one other seller setting the same price, then its demand is: : Finally, if …rm i sets the strictly highest price in the industry, then its demand is: Thus, (i) customers …rst visit the cheapest seller(s) at the set prices, (ii) at equal prices, demand is allocated in proportion to production capacity, and (iii) rationing is e¢cient. 4…rm's pro…t is then given by: Hence, if …rm i sets the strictly highest price in the industry, then its pro…ts are given by i (p i ; p i ) = (p i c) max fD(p i ; p i ) K i ; 0g.To facilitate our analysis, we write: to indicate the corresponding residual pro…t-maximizing price.Note that, by concavity of the residual pro…t-function, there is a unique p i provided that there is a p i for which D(p i ; p i ) > K i .
The next two results establish some properties that will prove useful in the ensuing analysis.The …rst states that the residual pro…t-maximizing price is increasing with …rm size.This is so because a …rm's residual demand, ceteris paribus, depends positively on its production capacity.The second sheds light on the relation between a …rm's production capacity and market demand at the residual pro…t-maximizing price.
Lemma 1 Fix K and suppose that D (c) > K n .For all i; j 2 N and i 6 = j, if k i > k j , then p i > p j .
Lemma 2 For all i; j 2 N and i 6 = j, D (p i ) does not depend on k i .Moreover, Finally, let p be the price for which total industry capacity equals market demand, i.e., K = D p .We refer to p as the market-clearing price.
As is well-known, the existence of a pure-strategy Nash equilibrium in the above game critically depends on the relation between demand and available production capacity. 5 Speci…cally, there is a pure-strategy solution when total industry capacity is su¢ciently small (i.e., K D (p 1 )) and when it is su¢ciently large (i.e., K D (c) + k 1 ).In the following, we refer to the …rst case as the monopolistic zone and to the second as the competitive zone.In the monopolistic zone, there is a symmetric purestrategy equilibrium in which all …rms set the market-clearing price p > 0. In the competitive zone, there too is a symmetric pure-strategy solution which has all …rms price at marginal cost. 6 joint industry capacity is intermediate (i.e., D (p 1 ) < K < D (c) + k 1 ), then there is no purestrategy equilibrium.As envisaged by Edgeworth (1925), price stability is unlikely in this case and prices are expected to oscillate inde…nitely between some lower and upper bound.We refer to this as the Edgeworth zone.

Price Fluctuation Range
Before examining the price e¤ect of changes in …rm size and number, it is useful to distinguish two types of impact.First, a change in capacity allocation may a¤ect the size of the Edgeworth zone and therefore the size of the monopolistic and competitive zones).Second, given that joint capacity K is such that the industry is in the Edgeworth zone, it may impact what we refer to as the price ‡uctuation range, i.e., the di¤erence between the maximum and minimum prices that are set with positive probability.
In order to characterize the price ‡uctuation range, de…ne: as …rm i's iso-pro…t price.That is, b p i is the price below p i at which …rm i sells at capacity and obtains the same pro…t as when it is the single highest-priced …rm pricing at p i .Figure 1 graphically illustrates the concept for the industry leader (i.e., …rm 1) with marginal cost normalized to zero.
Firm 1's residual pro…t maximizing and iso-pro…t price for c = 0.
The next result relates the iso-pro…t price to production capacity.It shows that the iso-pro…t price positively depends on …rm size when industry capacity is given.Roughly speaking, larger …rms have a higher residual pro…t per unit of capacity.Since unit costs are the same, this implies that a …rm's iso-pro…t price is increasing in its capacity, all else unchanged.
Proposition 1 Suppose that D (p 1 ) < K < D (c) + k 1 and …x K.For all i; j 2 N and i 6 This …nding implicitly de…nes the lower bound of the price ‡uctuation range to be given by b p 1 .To see this, …rst note that none of the …rms has an incentive to put mass on prices in excess of their residual pro…t-maximizing price.Moreover, one can easily construct cases where charging the residual pro…t-maximizing price is a best-response.Taken together, and following Lemma 1 above, this implies that p 1 constitutes the upper bound of the price ‡uctuation range.
Suppose then that …rm 1 sets p 1 .Since this is optimal only when …rm 1 has residual demand at that price, it holds that D (p 1 ; p 1 ) > K 1 .That is, all its rivals set a lower price and are capacityconstrained.Yet, in that case, each of its rivals can raise their price arbitrarily close to p 1 and still sell at capacity.If that happens, then it is a best-response of the largest seller to undercut the price of its competitors by the smallest possible amount.This holds true, since it goes from producing below capacity to producing at capacity while selling at (approximately) the same price.In turn, this induces its rivals to cut their prices slightly.This hypothetical downward spiral stops when a …rm prefers hiking its price to cutting it further.By Proposition 1, this is the largest seller.
Yet, if the largest seller does not put mass on prices below b p 1 , then neither will any of its rivals.
Because …rm 1 has no incentive to cut price further, there is nothing to gain by pricing below b p 1 .In fact, following the above logic, they have an incentive to follow the price hike by …rm 1, which then induces a new Edgeworth price cycle.Combining the above then leads to the following conclusion.

Results
We now proceed by studying three scenarios that may alter the size distribution of …rms: (i) Merger and Breakup, (ii) Investment and Divestment, and (iii) Entry and Exit.For all these cases, we examine the impact on the Edgeworth zone and price ‡uctuation range, respectively.

Merger and Breakup
The …rst possibility that we consider is a change in the size distribution of …rms that results from merger or breakup.In the …rst case, the number of …rms is reduced, whereas it increases in the second case.
Importantly, in both cases it is assumed that total industry capacity K remains constant.

Edgeworth Zone
As pointed out above, the Edgeworth zone is characterized by the following condition: D (p 1 ) < K < D (c) + k 1 .This condition captures the range of industry capacity for which prices may expected to be unstable.Notice that both the lower bound D (p 1 ) and the upper bound D (c) + k 1 critically depend on the size of the largest seller.In fact, these boundaries are a¤ected only when there is a change in the production capacity of the industry leader.This implies that the Edgeworth zone remains unaltered as long as k 1 is largest and una¤ected by merger or breakup.
Suppose then that there is a merger that becomes the new industry leader in terms of production capacity.Since k 1 has increased post-merger, the upper bound D (c) + k 1 increases.Moreover, since K is constant, the (new) largest …rm faces more residual demand.Following Lemma 1, this implies that p 1 increases, so that the lower bound D (p 1 ) decreases.Taken together, this implies an expansion of the Edgeworth zone when a merger leads to an increase of the largest supplier's size.It is worth highlighting that a merger will never induce a shift from the Edgeworth zone to the monopolistic or competitive zone, whereas the opposite can occur. 7That is, if the industry is in the monopolistic or competitive zone, then a merger may bring the industry into the Edgeworth zone, thereby leading to more price dispersion.
This conclusion is reversed in case of a breakup of …rms in smaller, separate, entities.As with merger, both the lower bound D (p 1 ) and the upper bound D (c) + k 1 remain una¤ected when the split-up does not involve the largest seller.If the breakup concerns the biggest industry player, however, then the upper bound decreases.The decline in capacity moreover means that the residual pro…t-maximizing price of the biggest …rm decreases, which implies an increase of the lower bound D (p 1 ).Therefore, the Edgeworth zone contracts if, and only if, the leading …rm is split-up in separate units.Notice that, in that case, there may be a shift from the Edgeworth zone into the monopolistic or competitive zones.
In other words, if the industry is in the Edgeworth zone, then breakup of the biggest industry player might lead to lower prices and potentially to more price stability.

Price Fluctuation Range
Let us now examine the e¤ect of a merger or breakup on the price ‡uctuation range.Starting with a merger, the next result shows that when the merger is the industry leader in terms of production capacity, the price ‡uctuation range shifts upward.In stating this result, let the merger be indicated with the subscript 's'.
Proposition 2 Suppose that D (p 1 ) < K < D (c) + k 1 and that there is a subset S N of …rms that merge.If this merger is the biggest …rm post-merger, then p s > p 1 and b p s > b p 1 .That is, the price ‡uctuation range shifts upward.
The intuition behind this result is as follows.Since industry capacity remains unaltered, the (new) largest …rm faces more residual demand, which means it has a higher residual pro…t-maximizing price (Lemma 1).Since the industry leader's residual pro…ts are higher post-merger, this implies an increase of the highest iso-pro…t price (Proposition 1).Such a merger, therefore, leads to an upward shift of the price ‡uctuation range.
Since both the upper bound and the lower bound increase when the merger is the biggest seller, it is a priori unclear whether the ‡uctuation range expands or contracts.Assuming linear demand, the next example shows that it may expand: Example 1 Assume that market demand is given by D (p) = a bp and that a bc > 0. Assume further that D (p 1 ) < K < D (c) + k 1 () a bp 1 < K < a bc + k 1 .Now suppose that …rm i is the highest priced seller facing residual demand: with a > K i and b > 0. Its residual pro…t-maximizing price p i is then given by: with corresponding residual demand and pro…ts: Firm i's iso-pro…t price is given by: Now consider the ‡uctuation range: Taking the derivative with respect to k 1 gives: We conclude that the ‡uctuation range is expanding with …rm 1's capacity.
una¤ected (since K 1 does not change).As …rm 1's residual pro…ts are una¤ected by the merger, the iso-pro…t (b p 1 c) k 1 should also stay the same.Since k 1 does not change in this case, this implies that b p 1 does not change either.Hence, the price ‡uctuation range is not a¤ected by a merger when the merger is not the biggest …rm in the industry.
Finally, as to the impact of a breakup, suppose that …rm 1 is split-up in smaller, separate entities.
This implies that there is a new industry leader that is smaller than before the breakup.That …rm faces less residual demand when setting the highest price so that the upper bound of the price ‡uctuation range decreases.By Proposition 1, the same holds for the iso-pro…t price.Hence, a breakup of the largest seller leads to a downward shift of the price ‡uctuation range.Moreover, the price ‡uctuation range remains una¤ected when the breakup concerns a smaller supplier.

Investment and Divestment
In the preceding subsection, we have studied changes in the size distribution of suppliers under the assumption that aggregate capacity K is …xed.However, the size allocation may also change when there are modi…cations in industry capacity.In this subsection, we consider production capacity investments and divestments by incumbents.That is, we examine the impact of changes in K assuming that the number of producers remains constant.The question of what happens when the number of …rms changes too is taken up in the next subsection.

Edgeworth Zone
Recall that the Edgeworth zone is de…ned by D (p 1 ) < K < D (c) + k 1 .Starting with the upper bound (i.e., D (c) + k 1 ), it is clear that it increases when the biggest industry player invests in additional production capacity and decreases when this …rm divests.The upper bound may also increase when there is a smaller supplier that invests to such a degree that it becomes the new industry leader in terms of capacity.
Since investments and divestments directly a¤ect total industry capacity K, there may be a shift from the Edgeworth zone to the competitive zone.This can occur when a smaller seller invests, for example.The reverse cannot happen, because divestment by the biggest seller (or any other …rm) reduces total industry capacity by the same amount.That is to say, the inequality K < D (c) + k 1 is immune to divestments.
As to the lower bound (i.e., D (p 1 )), it is a¤ected only when the residual pro…t-maximizing price of the largest producer changes.This price is e¤ectively determined by the combined capacity of this …rm's rivals (i.e., K 1 ).Notice that K 1 is una¤ected by …rm 1's investments or divestments as long as it maintains its leading position.The lower bound does change in response to investment or divestment by a smaller supplier, however, since this directly a¤ects K 1 .For instance, divestment by a smaller producer yields a reduction of K 1 .This implies an increase of p 1 and, therefore, a decrease of the lower bound D (p 1 ).Similarly, investment by a smaller supplier leads to an increase of K 1 .This induces a lower p 1 and, consequently, an increase of the lower bound D (p 1 ).
Since total industry capacity is a¤ected, it is then possible that the industry shifts from the monopolistic zone to the Edgeworth zone (e.g., when the leading industry player invests) and vice versa (e.g., when the leading industry player divests).

Price Fluctuation Range
Let us now direct our attention to the potential impact of investment or divestment on the price ‡uctuation range.The following result considers the case where the largest seller invests in extra capacity.
Proposition 3 Suppose that D (p 1 ) < K < D (c) + k 1 .The price ‡uctuation range p 1 b p 1 is increasing in the size of the largest industry player k 1 .
As an increase in k 1 does not a¤ect K 1 , the residual pro…t-maximizing price p 1 remains una¤ected.
Since residual demand also stays the same, residual pro…ts do not change either.By the very de…nition of the iso-pro…t price, this implies that (b p 1 c) k 1 should also stay the same.However, since k 1 increases, this requires a reduction of the iso-pro…t price b p 1 .Taken together, this means that the price ‡uctuation range p 1 b p 1 expands downward when the industry leader invests in additional production capacity, all else equal.
The reverse holds when the largest supplier divests as long as it remains the industry leader.Like with investment, the residual pro…t-maximizing price p 1 does not change.Yet, to keep (b p 1 c) k 1 constant, the iso-pro…t price rises to compensate for the decline in k 1 .In case divestment is severe so that …rm 1 looses its leading position, the new leader (…rm 2) faces more residual demand than …rm 1 when setting the highest price.This induces an increase of the price ‡uctuation range upper bound.
Since …rm 2's capacity remains unaltered, this implies that its iso-pro…t price increases too.Hence, in this case, the ‡uctuation range shifts upward.
Let us now consider the impact of growth by a …rm that is not the industry leader.When a smaller …rm invests without taking over the leading role, it reduces …rm 1's residual demand and residual pro…tmaximizing price p 1 .This implies a reduction of …rm 1's residual pro…ts.Since k 1 does not change, it follows that the iso-pro…t price b p 1 reduces too.Notice that the e¤ect is similar when a smaller …rm invests to such a degree that it becomes the new industry leader.We conclude that the price ‡uctuation range shifts downward whenever a smaller supplier invests.
As to divestment by a smaller producer, this has a direct negative e¤ect on K 1 .Since this increases …rm 1's residual demand, the residual pro…t-maximizing price p 1 rises.The resulting rise in residual pro…ts implies that the iso-pro…t (b p 1 c) k 1 should increase as well.Since k 1 has not changed in this case, the iso-pro…t price b p 1 increases too.Hence, divestment by a smaller supplier leads to an upward shift of the price ‡uctuation range.

Entry and Exit
Finally, let us consider the possibility of a simultaneous change in aggregate capacity and the number of …rms.This may happen in case of entry or exit.As before, we …rst discuss the impact on the Edgeworth zone and then examine how the price ‡uctuation range is a¤ected.

Edgeworth Zone
Suppose that industry capacity expands as a result of entry.If the newcomer is the largest …rm in the industry, then the upper bound of the Edgeworth zone (i.e., D (c) + k 1 ) increases.This is so since there is a new 'number one' post-entry, which implies an increase of k 1 .At the same time, however, the lower bound (i.e., D (p 1 )) increases too.Indeed, the new industry leader faces less residual demand than its predecessor so that p 1 is lower and D (p 1 ) is higher post-entry.
If the newcomer is smaller than the largest …rm in the industry, then this has no e¤ect on the upper bound of the Edgeworth zone since k 1 does not change.Yet, also in this case, entry leads to a reduction of residual demand for the biggest …rm and, consequently, to a reduction of p 1 .Hence, entry always leads to an increase of the lower bound of the Edgeworth zone D (p 1 ).
Exit by the industry leader reduces the upper bound D (c)+k 1 .It also implies that the new industry leader (i.e., …rm 2 pre-exit) faces more residual demand when setting its residual pro…t-maximizing price, which therefore increases.As a result, D (p 1 ) decreases so that the Edgeworth zone shifts leftward.That is, the monopolistic zone becomes smaller, whereas the competitive zone expands.If the exiting …rm is not the industry leader, then the upper bound remains unaltered.Yet, the leading …rm faces more residual demand post-exit so that p 1 increases and, therefore, D (p 1 ) decreases.Consequently, the Edgeworth zone expands in this case.

Price Fluctuation Range
Entry and exit also a¤ect the price ‡uctuation range.The next two results show that entry leads to a downward shift of the price ‡uctuation range, whereas exit induces an upward shift.In stating these results, let the prime indicate the situation post-entry.The logic behind these results is as follows.If a newcomer enters the industry that is smaller than the market leader, then this causes a drop in residual demand for the largest …rm.Consequently, the residual pro…t-maximizing price p 1 is lower post-entry.Since the largest seller's production capacity remains unaltered, this implies a drop of the iso-pro…t price b p 1 .The same holds when the entrant is the new industry leader.First, its residual demand is lower than the residual demand of the preceding leader.This has the implication that the upper bound of the price ‡uctuation range (p 1 ) decreases.
Since residual pro…ts are lower and the industry leader's capacity is larger, the corresponding iso-pro…t price (b p 1 ) is lower post-entry.As a result, entry induces a downward shift of the price ‡uctuation range.
Exit mirrors the situation with entry.If a …rm leaves the industry, then there is more residual demand for the (new) industry leader, i.e., p 1 increases.Since residual pro…ts are higher and the largest …rm is weakly smaller post-exit, the iso-pro…t price (b p 1 ) increases too.Consequently, exit creates an upward shift of the price ‡uctuation range.

Recapitulation
The preceding analysis presents many di¤erent situations in which there may or may not be a price e¤ect.In the following, we present three tables (one for each zone) that provide an overview of the impact of the six events on pricing.In these tables, the '"' and '#' mean that prices go up or go down, respectively.The '-' sign means that there is no impact on price and the 'x' indicates that the situation cannot occur.Each of the three tables distinguishes between 'same zone' and 'switch zone'.'Same zone' means that the industry remains in the respective zone after the event, whereas 'switch zone' means that the event causes a shift to an adjacent zone.Lastly, 'regular' means that the industry leader (in terms of production capacity) is not involved, whereas 'leader' means that the largest …rm takes part in the event.Starting in the monopolistic zone (Table 1), a merger has a price e¤ect only when it induces a switch to the Edgeworth zone.In that case, there is a 'jump' from a single market clearing price to a higher range of prices.This requires an increase of the production capacity of the largest player.If the biggest …rm increases in size, then a merger may result in higher prices and create price dispersion.Breakup has no e¤ect and cannot lead to a shift in zones.The same holds for divestment and exit, i.e., these events cannot move the industry from the monopolistic zone to the Edgeworth zone.Yet, since they lead to a reduction of total industry capacity, they positively a¤ect price levels.The opposite holds for investment and entry, which increase industry capacity and lead to a reduction of prices.This holds true independent of whether these events trigger a switch to the Edgeworth zone. 8rge Capacity Same Switch Zone Regular Leader Regular Leader Turning to the competitive zone (Table 2), the e¤ect of a merger is comparable to when the industry is in the monopolistic zone.It a¤ects price only when it results in a switch in zones, which requires an increase in size of the industry leader.When this happens, prices rise.Breakup cannot induce a shift in zones and neither can capacity expanding events like investment or entry.Divestment and exit may bring the industry into the Edgeworth zone and, therefore, induce a price increase.Taken together, there may be a price increasing e¤ect, but only when (1) there is a reduction in the number of …rms or production capacity, and (2) the industry leaves the competitive zone.

Intermediate Capacity Same Zone
Switch Zone Regular Leader Regular Leader Finally, let us summarize the potential price e¤ects when the industry is in the Edgeworth zone.
Starting with the events that a¤ect available industry capacity, divestment and exit always drive prices upward independent of whether there is a switch in zones.The reverse holds for investment and entry.
Breaking-up …rms that are not the industry leader has no e¤ect on price, whereas breaking-up the largest …rm leads to lower prices.A merger does not induce a shift to the competitive or monopolistic zone, but can have a price increasing e¤ect when the largest industry player has grown post-merger.

Discussion
Recent With few exceptions, these and similar studies take the size distribution of …rms as given.Our analysis highlights the price impact of changes in …rm size and number.In particular, we have shown how changes in market structure may a¤ect the degree of price dispersion by making an industry more or less competitive.With an eye on the potential welfare implications, this warrants more (empirical) research on oligopoly pricing under capacity constraints.

Conclusion
We have analyzed an n-…rm homogeneous-good Bertrand-Edgeworth model to examine the impact of …rm size and number on oligopoly pricing.We considered six di¤erent events that have an e¤ect on the number of …rms (merger and breakup), available production capacity (investment and divestment), or both (entry and exit).We showed that a merger leads to higher prices only when it increases the size of the largest seller and industry capacity is neither too big nor too small post-merger.Similarly, breakingup a …rm only leads to lower prices when it concerns the biggest producer and aggregate capacity is within an intermediate range.Investment and entry (weakly) reduce prices, whereas divestment and exit yield (weakly) higher prices.The overall takeaway from our explorations is, therefore, that size matters more than number in the determination of oligopoly prices.
We see several avenues for future research.One is to conduct a similar study assuming that products are less than perfect substitutes.Another is to consider cost (in)e¢ciencies that may result from merger.
This may well create an (additional) downward or upward pressure on prices.As to the latter, we exclusively considered unilateral e¤ects and ignored potential coordinated e¤ects.Clearly, the upward price e¤ect of a merger may well be stronger in case it facilitates a 'mutual meeting of the minds'.Funding: Not applicable.
Data Availability Statement: Not applicable.
6 Appendix: Proofs Proof of Lemma 1.Consider some …rm i 2 N .Its residual pro…t-maximizing price, p i , is the price that solves: Notice that, for a given industry capacity K, the LHS is decreasing in p i and the RHS is decreasing in k i .Hence, 8i; j 2 N and i 6 = j, if k i > k j , then p i > p j .
Proof of Lemma 2. Following the proof of Lemma 1, the residual pro…t-maximizing price of …rm i is implicitly de…ned by: D(p i ) K i + (p i c) @D (p i ; p i ) @p i = 0: Rearranging gives: D(p i ) = K i (p i c) @D (p i ; p i ) @p i : Observe that the RHS does not depend on k i .
Proof of Proposition 1.Consider some …rm i 2 N .Its iso-pro…t price is given by: Notice that, by the de…nition of the residual pro…t-maximizing price p i , it holds that: (D (p i ; p i ) K + k i ) + (p i c) @D (p i ; p i ) @p i = 0: Hence, the derivative This leaves the case where a subset of smaller suppliers merge and becomes the biggest …rm in the industry.In that case, this new industry leader faces more residual demand when setting the highest price so that the upper bound of the ‡uctuation range increases (i.e., p s > p 1 ).Moreover, by Proposition 3, the iso-pro…t price is increasing with capacity when total industry capacity is constant.Since k s > k 1 , this implies that also in this case it holds that b p s > b p 1 post-merger.Now suppose there is an increase in k 1 .To assess the impact on the ‡uctuation range, notice that: In this case, it holds that dp 1 dk 1 = 0 since K 1 remains unaltered with an expansion of k 1 .Hence, the preceding equality simpli…es to: (p 1 c) (D(p 1 ; p 1 ) K 1 ) k ; which is positive.We conclude that the price ‡uctuation range is increasing in the size of the largest industry player.

Proposition 4 < p 1 and b p 0 1 < b p 1 .Proposition 5
Suppose that D (p 1 ) < K < D (c) + k 1 .If a new …rm enters the industry, then p 0 1 That is, the price ‡uctuation range shifts downward.Suppose that D (p 1 ) < K < D (c) + k 1 .If an incumbent exits the industry, then p 0 1 > p 1 and b p 0 1 > b p 1 .That is, the price ‡uctuation range shifts upward.
Author Contributions: Validation, I.B. and I.B. and M.M.; Formal analysis, I.B. and I.B. and M.M.; Investigation, I.B. and I.B. and M.M.; Writing-original draft, I.B. and M.M..All authors have read and agreed to the published version of the manuscript.

( 1 ) 2 i;
reduces to:db p i dk i = (p i c) (D (p i ; p i ) K) k which is positive since K = D p and p i > p.We conclude that if k i > k j , then b p i > b p j ,for all i; j 2 N and i 6 = j.nProof of Proposition 2. Consider a merger S N that involves …rm 1.The claims that p s > p 1 and b p s > b p 1 follow directly from Lemma 1 and Proposition 3.

Table 1 -
Price E¤ects in the Monopolistic Zone.

Table 2 -
Price E¤ects in the Competitive Zone.

Table 3 -
Price E¤ects in the Edgeworth Zone.
years have seen a renowned interest in oligopoly pricing under capacity constraints (see, e.g., Bos and Vermeulen (2021), Montez and Schutz (2021) and Edwards and Routledge (2022)) and, in particular, in providing rationales for Edgeworth price cycles in markets with perfect and less than perfect substitutes (see, e.g., Myatt and Ronayne (2019) and Gabszewicz, Marini Zanaj (2022)).Such a revamped interest comes from several sources, perhaps the most important one being the existing empirical evidence.Price dispersion is shown to be persistent in both physical and online shops (see, e.g., Aas, Wulfsberg and Moen (2018) and Gorodnichenko, Sheremirov and Talavera (2018)).Edgeworth price cycles are also well documented.For example,Eckert (2003)andNoel (2007a, 2007b)provide evidence of sawtooth pricing paths in Canadian retail gasoline market.Wang (2008)explores empirically the existence of collusive price cycles in the Australian retail gasoline market.Similarly, Foros and Steen (2013) and Linder (2018) describe Edgeworth-like price cycles in European retail gasoline markets.Furthermore, such pricing dynamics has also been documented for pharmaceutical markets and search-engine advertising (see, e.g., Hauschultz and Munk-Nielsen (2020) and Zhang and Feng (2011)).Finally, it is noteworthy that similar pricing patterns have been observed in controlled laboratory experiments (see, e.g., Kruse, Rassenti, Reynolds, and Smith (1994) and Fonseca and Normann (2013)).
) (D (p i ; p i ) K i ) k i + c:Holding total industry capacity K …xed, we now show that b p i increases with k i .Taking the …rstderivative with respect to k i :dk i (D (p i ; p i ) K + k i ) + (p i c) dp i dk i k i (D (p i ; p i ) K + k i ) + (p i c) @D(p i ;p i) @p i (p i c) (D (p i ; p i ) K) k 2 i :