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- Publisher
- de Gruyter
- Copyright
- © 2021 Bojan Ristić et al., published by Sciendo
- ISSN
- 2233-1999
- eISSN
- 2233-1999
- DOI
- 10.2478/jeb-2021-0007
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- See Article on Publisher Site
Abstract
This paper aims to identify the possible implications of quantity competition in markets with differentiated products on entry deterrence. If capacity commitments characterise this industry, quantities can be expected as the choice variable of rational players, even in the presence of product differentiation. Different equilibria of a static game occur depending on the degree of asymmetry of players, incumbent and entrant, which will crucially affect the shape of their best response functions. Asymmetry can stem from players’ advantage in demand and costs, their different objective functions, or the first-mover advantage. We will analyse entry where incumbent maximises the weighted average of profit and revenue while entrant is maximising profit. The reduction of asymmetry may intensify competition in the industry and, consequently, reduce entry bar- riers. Our findings provide an insight that could be used for practical recommendations for conducting com- petition policy and other sector-specific regulations, where the introduction and higher intensity of competi- tion are desirable. Keywords: differentiated oligopoly, capacities, quantity competition, asymmetry of players, incumbent, entrant JEL Classification: D43, L13, L21 Bojan Ristić, PhD Assistant Professor Faculty of Economics, University of Belgrade E-mail: bojan.ristic@ekof.bg.ac.rs 1. INTRODUCTION ORCID: 0000-0002-9883-8914 Entry into an industry is an essential aspect of in- dustrial organisation. Empirical facts suggest that Dejan Trifunović, PhD entry is a common phenomenon in many industries. Full Professor Typically, in markets with established incumbents, Faculty of Economics, University of Belgrade entry is on a small-scale basis, and the survival rate of E-mail: dejan.trifunovic@ekof.bg.ac.rs entrants is low. Thus, it is important to understand eco- ORCID: 0000-0003-2125-8142 nomic incentives for entry and how incumbents can use their position to block entry or force the entrant to Tomislav Herceg, PhD (corresponding author) leave the market. Assistant Professor When faced with the potential entrant, the incum- Faculty of Economics and Business, bent should decide whether to block or allow the en- University of Zagreb try. This decision for the incumbent is easy when it can E-mail: therceg@efzg.hr block entry by producing monopolistic output. More Address: Trg J. F. Kennedyja 6, 10000 Zagreb often, the incumbent should produce a larger quantity ORCID: 0000-0001-8869-6775 than the monopolist, and it needs excess capacities to Copyright © 2021 by the School of Economics and Business Sarajevo 84 CAPACITY COMPETITION IN DIFFERENTIATED OLIGOPOLIES: ENTRY DETERRENCE WITH ALTERNATIVE OBJECTIVE FUNCTIONS achieve this objective. Of course, capacity serves the Moreover, this strategy may also reflect limit-pricing purpose only if it cannot be costlessly liquidated, mak- aimed at discouraging entry in the market since the ing a credible commitment visible to the entrant in monopolist who maximises this alternative objective the sense of Dixit (1982). Hence, the capacity expan- function has a larger capacity and lower equilibrium sion before the potential entrance is a signal of entry price than the profit maximising monopolist. deterrence. The other possible incumbent’s strategy The empirical analysis of predatory capacity expan- is presented in the model of Milgrom and Roberts sion was limited because there was only a potential (1982), where entrant does not know the incumbent’s threat of entry in many industries. However, in the casi- marginal cost and knows only the probability distribu- no industry, the entry plans are apparent because the tion for this random variable, while entrant’s cost is entrant has to make contracts with vendors and sup- common knowledge. If the incumbent has low cost, it pliers. Casinos compete in the size of the floor space, will send a signal in the form of a low price to the en- and vendors in the second stage compete in supply- trant that its cost is low, preventing entry in this way. ing casino machines and other products. Investments Also, there are other possible ways to prevent entry, in the casino floor space are irreversible, and the ca- such as sleeping patents, when incumbent patents pacity expansion represents a commitment to deter the technology that it does not intend to use to pre- entry. This type of competition is similar to Kreps and vent an entrant from its usage. Scheinkman (1983) two-stage competition that leads The capacity expansion as a tool for blocking en- to one stage Cournot (1838) outcome. Cookson (2018) try is relevant in industries with large fixed capacities finds that incumbent casinos increase the floor space – like railways, airlines or the casino industry. In the by 4 to 7% after the entry announcement. The higher airline industry, potential entry is more often deterred demand did not trigger this capacity expansion since with limit-pricing in Milgrom and Roberts (1982) spirit this would imply that the entry would be more likely than with capacity expansion. Sweeting, Roberts and to succeed upon incumbent’s increase of capacities. Gedge (2020) study the behaviour of incumbent air- In contrast, Cookson (2018) finds that the likeli- lines in the US and find that when Southwest airlines hood of successful entry was reduced after the ca- (potential entrant) operates at airports that are on a pacity expansion, which implies that the incumbent certain route but do not operate on the route itself, intended to deter entry. The second argument that incumbents see this as the potential threat of entry capacity expansion was strategic can be inferred from and react by price reductions. According to Goolsbee the behaviour of casinos that were further from the and Syverson (2008), these price cuts were up to 20%, entrant’s planned location, and these casinos did not and Morrison (2001) determined that these price cuts increase capacities like the nearby incumbent casinos. lowered traveller’s expenditures on air tickets by 3.3 The third fact is that incumbents did not increase the billion USD in 1998. Sweeting, Roberts and Gedge capacity after entrants underwent construction works, (2020) determine in their model that incumbent air- which means that incumbents did not use larger ca- lines should use limit pricing rather than a capacity pacities to accommodate entry but to prevent it. expansion to deter entry. This result is also confirmed In this paper, we will use the approach when the by Masson and Shaanan (1982), who find that in 37 entry is blocked with excess capacities. The setup of industries, incumbents use limit pricing more often our model is similar to Dixit (1979, 1980). We assume than capacity expansion. However, empirical evidence that there is one incumbent with a potential monop- of Snider (2009) and Williams (2012) suggests that in- olistic position and one entrant. The firms sell differ - cumbent airlines use capacity expansion to deter po- entiated products and are faced with linear demand tential small-scale entry and limit pricing to deter po- functions. The incumbent has some initial level of ca- tential large-scale entry. pacities, and if it produces above this level, it should The capacity expansion as a means for entry deter- build additional capacities, and its marginal cost rence was identified by Crozet and Chassagne (2013) jumps to a higher level. in French high-speed rail. Namely, the incumbent The contribution of our paper is to analyse entry SNCF expanded the capacity by launching a high- when incumbent maximises the weighted average speed train operator, Ouigo, that prevents potential of profit and revenue. The evidence that justifies us- entry into the low-cost service segment. Cherbonnier ing this objective function can be found in practice et al. (2017) assume that the monopolistic rail opera- since managers’ compensations depend not only on tor maximises the weighted average of its profit and the company’s profit but also on the sales volume and consumer’s surplus due to the exogenous regula- consequently market share (measured by revenue). tory constraints. This objective function explains the There are several possibilities for how a firm’s size is price distortion that exists in the monopoly market. measured: quantity produced, total revenue or market South East European Journal of Economics and Business, Volume 16 (1) 2021 85 CAPACITY COMPETITION IN DIFFERENTIATED OLIGOPOLIES: ENTRY DETERRENCE WITH ALTERNATIVE OBJECTIVE FUNCTIONS share (see Cornes and Itaya 2016). We have chosen The considerable improvement of the entry deter- total revenue as the measure of firm size since it is a rence model with capacities as a means for preventing standard measure of the company size and company entry is provided by Dixit (1979), where fixed costs of classification (micro, small, medium, large) used in entry represent a barrier to entry. Figure 1 illustrates the European Union, CEFTA and most other countries. this model. On the other hand, the entrant only maximises profit Figure 1. Capacity competition and entry deterrence with since its entry decision depends only on the sign of its fixed costs profit (positive or negative). If an entrant enters the market, the post-entry sub- game can result either in Cournot or Stackelberg equi- 2 librium. Since incumbent maximises the weighted av- erage of profit and revenue in a Cournot equilibrium, it produces more than in equilibrium where both firms maximise profit, and entrant produces less. However, the total quantity produced is larger, and the average price for consumers is lower. In Stackelberg equilibri- um, the incumbent produces a larger quantity than in the pure profit maximising equilibrium, while entrant produces less. As in the case of the Cournot mecha- nism, the total quantity produced is larger while con- sumers pay a lower average price. We then consider the first stage of the game when M B Z Q 1 1 1 1 incumbent can block or allow the entry. Compared to the standard situation of profit maximising firms, Source: Dixit, 1979 it is more likely that the incumbent can block entry by producing the quantity that maximises its alter- native objective function, making entry less likely. The line M Q is the reaction function of the in- 1 2 Moreover, when entry occurs, the entrant chooses a cumbent, and M Q is the reaction function of the en- 2 1 smaller capacity level than in a pure profit maximising trant. If the incumbent is the monopolist, it produces equilibrium. at point M . At Q , the entrant has zero production, 1 1 The rest of the paper is organised as follows. In the and its profit is zero. The Cournot-Nash equilibrium is second part, we provide the literature review about obtained at point N, while Stackelberg equilibrium is capacity expansion as a means of blocking entry. In obtained at point S, where the incumbent is the lead- the third part, we derive the equilibrium of the post- er, and the entrant is the follower. entry subgame in the case of the alternative objective If the entrant has fixed costs of entry, its profit be - function of the incumbent, and in the fourth part, we comes zero to the left of Q , for example, at point B, analyse the incumbent’s decision whether to block where the incumbent produces B . The exact posi- or to allow entry. In the last section, we conclude the tion of B depends on the level of fixed costs. When discussion. these costs are higher, this point moves to the origin. Entrant’s reaction function contains two segments, M B and the segment that coincides with the horizon- tal axis, B Q . If entrant’s fixed costs are so high that 2. LITERATURE REVIEW 1 1 the point B is to the left of M , there is no possibility 1 1 Entry deterrence with excessive capacities was for entry, and the incumbent will produce the monop- analysed by Spence (1977) with the idea that invest- oly quantity. ments in irreversible capacities represent a credible The point Z in Figure 1 is the quantity produced threat. In this model, the incumbent produces the by the incumbent at the point where its iso-profit line monopoly output but invests in the large capacity to trough point S meets the horizontal axis. If entrant’s produce the competitive output. If entrant believes fixed costs are small, such that B is located to the in that threat, he will not enter the market. According right of Z , the incumbent will obtain a lower profit to Tirole (1988), the lower the depreciation rate of by preventing entry (incumbent would be on a high- the capital and the more specific it is, the more cred- er iso-profit line). Hence, if B > Z , entry is allowed, 1 1 ible is the commitment to deter entry with capacity and incumbent chooses point S as the Stackelberg expansion. leader. As we have explained before, when B < M , 1 1 86 South East European Journal of Economics and Business, Volume 16 (1) 2021 CAPACITY COMPETITION IN DIFFERENTIATED OLIGOPOLIES: ENTRY DETERRENCE WITH ALTERNATIVE OBJECTIVE FUNCTIONS Figure 2. Capacity competition and entry deterrence the incumbent can prevent entry by producing the monopoly quantity. If M < B < Z , the incumbent can 1 1 1 move from the iso-profit line through S to the lower iso-profit line with larger profit by moving from Z to B . In fact, the incumbent can prevent entry by pro- ducing slightly above B , which is the limit output that prevents entry. 2 Concerning the comparative statics in Dixit’s (1979) model, any change of parameters that increases B (for example, smaller entrant’s fixed costs) makes entry more likely. Any change of parameters that T increases M and Z makes entry less likely. One pa- 1 1 rameter in comparative statics is the absolute advan- tage of a firm: the difference between the intercept of the inverse demand function and marginal cost. The increase of the incumbent’s absolute advantage T M V M Q 1 1 1 1 1 increases M while B is unaffected, which makes en- 1 1 try less likely. The second parameter in the compara- Source: Dixit, 1980 tive statics analysis is the degree of product differen- tiation; when it is optimal for the incumbent to allow entry, its profit increases when products are more dif- level of the incumbent’s capacity is k ≤ T . In that 1 1 ferentiated. When it is optimal for the incumbent to case, the equilibrium in the post-entry subgame is at block entry by producing the limit output, its profit T. If the initial level of incumbent’s capacity is k ≥ T , 1 1 increases when products become less differentiated. the equilibrium in the post-entry subgame is at V. If The last result is intuitive; it is easier to prevent entry the initial level of incumbent’s capacity belongs to the if the entrant has a similar product. Finally, Dixit (1979) interval T ≤ k ≤ V , the equilibrium will be located on 1 1 1 demonstrates that entry is less likely when the incum- the TV segment of the entrant’s reaction function. The bent has excess capacities. incumbent will fully employ the capacity such that Further discussion and some additional insights x =k , while entrant will choose its production level as 1 1 about entry were discussed by Dixit (1980). The in- the Stackelberg follower for the given level of produc- cumbent has installed the capacity of k , where r is tion of the leader. 1 1 the cost of capacity, c is the marginal cost of produc- The entrant’s profit decreases when its produc - ing one unit of output, and F represent other fixed tion decreases. In Figure 2, the movement from T to costs. If the incumbent produces up to the capacity, V implies the reduction of entrant’s profit. If entrant’s the total cost of the incumbent is: profit is negative at T, then entry is not possible, and the incumbent will simply choose the monopoly pro- C c x rk F . (1) duction level at M . If entrant’s profit is positive at V, 1 1 1 11 1 the entry cannot be prevented. The incumbent will If the incumbent produces above the capacity, it choose the production level where his iso-profit line C () c rx F 1 1 1 1 1 C c x rk F 1 1 1 11 1 needs additional capacity, and its total cost is: is tangent to the entrant’s reaction function on the TV segment. If the tangency occurs to the right of V, the p qq , 1 11 2 C () c rx F (2) incumbent will be at point V (the corner solution). If 1 1 1 1 1 p qq . 2 22 1 entrant’s profit is positive at T and negative at V, there p The en qq trant has no installed capacit , y, and its cost is some point B (like in Figure 1) on the line segment 1 11 2 qk C rk c q F for . 1 11 1 1 1 11 is C = (c + r ) x + F (Notice the similarity with the TV where entrant’s profit is 0. At this point, the level 2 2 2 2 2 p qq . 2 22 1 previous expression of firm 1 costs). Hence, the in- of the incumbent’s capacity is B , and if it sets the ca- C () r cq F qk for . cumben 1 1 t’s mar 11 ginal c 1 ost jumps disc 11 ontinuously at the pacity above this level, it can prevent entry. If B < M , 1 1 qk C rk c q F for . 1 11 1 1 1 11 capacity level k , and it has two reaction functions, as the incumbent can prevent entry by choosing the mo- C () r cq F for q 0 . shown in Figure 2. nopoly level of output. If B > M , the incumbent faces 2 2 22 2 2 1 1 C () r cq F qk for . 1 1 11 1 11 The reaction function M Q is relevant if the in- the trade-off. If its profit is higher when it blocks entry, 1 2 cumbent has to increase the capacity above k , and it will choose the capacity level slightly above B . If its ( q q ) q () r c q F . 1 1 2 2 2 12 2 2 2 2 C () r cq F for q 0 . ’ ’ 2 2 22 2 2 the reaction function M Q is relevant when the in- profit is higher when it allows entry, the incumbent 1 2 cumbent uses the capacity up to the level k . The en- will choose a point where its iso-profit line is tangent ( q q ) q ( rk c q F ), 1 11 12 1 1 1 1 1 1 ( q q ) q () r c q F . tran 2 t’s reac 2 tion func 2 12 tion is 2 M Q 2 . Suppose the initial 2 2 on the entrant’s reaction function to the left of B . 2 1 1 ( q q ) q () r cq F . 11 12 1 1 1 1 1 ( q q ) q ( rk c q F ), 11 12 1 1 1 1 1 1 (1) () qq q ( r k c q F ) () qq q , South E 1 ast European Journal of E 11 conomics and B 2 1 usiness 1 1 , Volume 16 (1) 2021 1 1 1 11 2 1 87 ( q q ) q () r cq F . 11 12 1 1 1 1 1 (1) () qq q (( r c ) q F ) () qq q . 1 11 2 1 1 1 1 1 11 2 1 (1) () qq q ( r k c q F ) () qq q , 1 11 2 1 1 1 1 1 1 11 2 1 c Rq () (1) () qq q; q (( r c ) q F ) () qq q . 12 1 11 2 2 1 1 1 1 1 11 2 1 cr 1 11 Rq () q . c 12 2 Rq () q ; 12 2 cr 1 11 cr Rq () 2 22 q . 12 2 R () qq 21 1 cr 2 22 2 ( c ) ( c r ) 2( c r ) ( c ) ** 11 2 2 2 2 2 2 1 1 R () qq qq 21 1 ;. 44 2 ( c ) ( c r ) 2( c r ) ( c ) ** 11 2 2 2 2 2 2 1 1 2 ( cr ) ( c r ) 2( c r ) ( c r ) qq ** ;. 1 11 2 2 2 2 2 2 1 1 1 qq 22 ;. 44 22 44 2 ( cr ) ( c r ) 2( c r ) ( c r ) ** 1 11 2 2 2 2 2 2 1 1 1 2( cr ) ( c r ) qq l ;. 1 11 2 2 2 q ; 44 2 42 4( c r ) 2 ( cr ) ( c r ) f 2 2 2 11 1 2 2 2 2( cr ) ( c r ) q 1 11 2 2 2 . q 2 ; 2 84 42 4( c r ) 2 ( cr ) ( c r ) f 2 2 2 11 1 2 2 2 q . 84 Dixit (1980) considers three extensions of the pre- decides to enter the market (and how much to pro- vious entry model. In the first extension, the entrant duce) or stay out. The entrant has the fixed cost of becomes the leader, while in the second, there is a entry, and all firms have the exact marginal costs. A discrete number of choices for the incumbent’s initial certain level of output blocks entry and incumbents capacity k . In the last extension, the post-entry sub- should decide whether to produce this output or to game includes Bertrand competition with differenti- allow entry. With several incumbents, if one of them ated products. The graphical analysis is similar, but the produces the entry deterring quantity, others might reaction functions have a positive slope. free ride on its decision, and entry deterrence be- Dixit’s (1980) model is extended by Schmalensee comes a public good. However, since marginal costs (1981) who assumes that instead of fixed entry cost, are constant and the profit of each incumbent in- there is some minimum efficient scale of capacity, k , creases with the production up to the level of output below which firms cannot profitably operate. Since that blocks entry, all incumbents want to produce the empirical evidence suggests that the efficient this output. The comparative statics result shows that scale is typically lower than 10% of the aggregate de- when the number of incumbents increases, entry de- mand in many industries, the efficient scale is not an terrence becomes more profitable for each incumbent effective barrier to entry. compared to entry allowance. The impact of uncertainty on the entry deterrence with capacities was studied by Maskin (1999), who in- troduced the mean preserving increase in the spread 3. THE MODEL for the intercept of the inverse demand function. When demand is high, measured by the intercept of In our model, we will assume that firm 1 is the in- C c x rk F the inverse demand function, the incumbent produc- cumbent, and firm 2 is the en . trant, and products are 1 1 1 11 1 es at the level of capacity but could produce more if differentiated as in Dixit (1979) with the following sys- it had installed larger capacity. When demand is low, tem of inverse demand functions, where p denotes C () c rx F . 1 1 1 1 1 the incumbent produces at less than full capacity. Due price and q quantity: to the capacity constraint, the increase in price for p qq , 1 11 2 the higher realisation of demand intercept is larger (3) p qq . 2 22 1 in absolute value than the fall in price due to the low demand. Consequently, if entrant’s profit is zero with The parameter γ measures the degree of differenti- C rk c q F for qk . 1 11 1 1 1 11 no uncertainty, its expected profit is positive under ation. When its value increases, products become less uncertainty. To effectively deter entry, the incumbent differentiated (more homogenous), and when its val- C () r cq F for qk . should install a larger capacity under uncertainty than ue 1 decr 1 eases 11 , produc 1 ts 11 are more differentiated. This under certainty to equalise positive and negative parameter measures cross-price effects, i.e. how the C () r cq F for q 0 . price fluctuations for different realisations of demand demand for one product changes when the price of 2 2 22 2 2 and reduce entrant’s profit to 0. Besides, Maskin (1999) its substitute changes. For the sake of simplification, concludes that with uncertainty compared to com- it is assumed that the direct price effects are equal ( q q ) q () r c q F . 2 2 2 12 2 2 2 2 plete certainty, the incumbent is more likely to switch to 1 and that they are higher than the cross ones, so from the strategy of entry deterrence to the strategy 0 < γ < 1. Concerning the intercept of the inverse de- ( q q ) q ( rk c q F ), 11 12 1 1 1 1 1 1 of entry accommodation. mand functions, α , there are two possibilities. The first ( q q ) q () r cq F . 11 12 1 1 1 1 1 Huisman and Kort (2015) consider a similar setup one is that both firms are symmetric, and the second, with a linear demand function exposed to random that there is a more significant willingness to pay for (1) () qq q ( r k c q F ) () qq q , 1 11 2 1 1 1 1 1 1 11 2 1 shocks. When the uncertainty increases, the entrant incumbent’s products (α > α ) due to its established 1 2 waits for more to see the resolution of uncertainty, brand name. The symmetry in demand conditions will C ( c1x ) rk () F qq . q (( r c ) q F ) () qq q . 1 1 1 11 1 1 11 2 1 1 1 1 1 11 2 1 which leaves more time to the incumbent to benefit be assumed in this model. from a monopoly position. This increases the incen- The incumbent has the installed capacity of k and C () c rx c F . 1 1 1 1 1 tive for entry deterrence. Hence, the increase of uncer- other fix Rq () ed costs of F q . I ;f it produces up to the installed 12 2 tainty increases the likelihood of entry deterrence in capacity, it has the cost of each unit of capital equal to p qq , cr 1 11 2 1 11 this dynamic model of entry, in contrast to the static r , and the marginal cost for each produced unit of c . Rq 1 () q . 1 12 2 p qq 22 . 2 22 1 model of Maskin (1999). Formally, The valuable extension of the entry deterrence cr C rk c 2 q 22 F for qk . model refers to the case with several incumbents con- (4) 1 11 1 1 1 11 R () qq 21 1 sidered by Gilbert and Vives (1986). In the first stage of the game, the incumbents decide on their capac- C If the incumben () r cq F t produces above the installed ca- for qk . 1 1 11 1 11 2 ( c ) ( c r ) 2( c r ) ( c ) ** 11 2 2 2 2 2 2 1 1 ity and production. In the second stage, the entrant pacity, its cost function becomes: qq ;. 44 C () r cq F for q 0 . 2 2 22 2 2 2 ( cr ) ( c r ) 2( c r ) ( c r ) 88 ** South E 1 ast E11 uropean Journal of E 2 conomics and B 2 2 usiness, Volume 16 (1) 2021 2 2 2 1 1 1 qq ;. ( q q ) q () r c q F . 12 2 2 2 12 2 2 2 2 44 ( q q ) q ( rk c q F ), 11 12 1 1 1 1 1 1 2( cr ) ( c r ) 1 11 2 2 2 q ; ( q q ) q () r cq F . 11 12 42 1 1 1 1 1 4( c r ) 2 ( cr ) ( c r ) f 2 2 2 11 1 2 2 2 q . 2 (1) () qq q ( r k c q F ) () qq q , 1 11 2 1 1 1 1 1 1 11 2 1 84 (1) () qq q (( r c ) q F ) () qq q . 1 11 2 1 1 1 1 1 11 2 1 c Rq () q ; 12 2 cr 1 11 Rq () q . 12 2 cr 2 22 R () qq 21 1 2 ( c ) ( c r ) 2( c r ) ( c ) ** 11 2 2 2 2 2 2 1 1 qq ;. 44 2 ( cr ) ( c r ) 2( c r ) ( c r ) ** 1 11 2 2 2 2 2 2 1 1 1 qq ;. 44 2( cr ) ( c r ) 1 11 2 2 2 q ; 42 4( c r ) 2 ( cr ) ( c r ) f 2 2 2 11 1 2 2 2 q . 84 1 C c x rk F . 1 1 1 11 1 C () c rx F . 1 1 1 1 1 p qq , 1 11 2 p qq . 2 22 1 C c x rk F 1 1 1 11 1 C rk c q F for qk . 1 11 1 1 1 1 C c x rk F . C () c rx F 1 1 1 11 1 . 1 1 1 1 1 C () r cq F for qk . 1 1 11 1 11 C () c rx F . C c x rk F . p qq , 1 1 1 1 1 1 11 2 1 1 1 11 1 C c x rk F . 1 1 1 11 1 p qq . 2 22 1 C () r cq F for q 0 . 2 2 22 2 2 p qq , C () c rx F . 1 11 2 1 1 1 1 1 C () c rx F . C c x rk F . 1 1 1 11 1 1 1 1 1 1 p qq . qk C rk c q F for . 2 22 1 1 11 1 1 1 11 C c x rk F . ( q q ) q () r c q F . 1 1 1 11 1 2 2 2 12 2 2 2 2 p qq , 1 11 2 p qq , C () c rx F 1 11 2 1 1 1 1 1 C rk c q F for qk . C () r cq F qk p qq . 1 11 1 1 1 11 for . 1 1 11 1 11 2 22 1 C () c rx F . ( q q ) q ( rk c q F ), 1 1 1 1 1 p qq . 11 12 1 1 1 1 1 1 2 22 1 C c x rk F 1 1 1 11 1 p qq , ( q q ) q () r cq F . 1 11 2 11 12 1 1 1 1 1 C () r cq F for qk . C rk c q F for qk . C () r cq F for q 0 . 1 1 11 1 11 1 11 1 1 1 11 2 2 22 2 2 p qq , C rk c q F for qk . 1 11 2 p qq . 1 11 1 1 1 11 2 22 1 C () c rx F . 1 1 1 1 1 1 p qq . 2 22 1 (1) () qq q ( r k c q F ) () qq q , 1 11 2 1 1 1 1 1 1 11 2 1 C () r cq F for q 0 . ( q q ) q () r c q F . C () r cq F for qk . (5) The first reaction function is relevant when q < k , 2 2 22 2 2 2 2 2 12 2 2 2 2 1 1 1 1 11 1 11 C () r cq F for qk . C rk c q F for qk . 1 11 1 1 1 11 1 1 11 1 11 and the second one when q ≥ k . It is evident that for (1) () qq q 1 (( r 1 c ) q F ) () qq q . p qq , 1 11 2 1 1 1 1 1 11 2 1 C1 c11 x rk 2 F . C rk c q F for qk . 1 1 1 11 1 C c x rk F . 1 11 1 1 1 11 1 1 1 11 1 ( q q ) q () r c q F . The entrant has no other options but to build the r> 0, the incumben ( q q ) q t’ s r (eac rk tion func c q F tion when it pr ), o- C () r cq F for q 2 0 . 2 2 12 2 2 2 1 2 11 12 1 1 1 1 1 1 2 2 22 2 2 p qq . 2 22 1 C () r cq F q 0 C () r cq F qk for . for . 2 2 22 2 2 1 1 11 1 11 capacity for the intended scale of production, and its duces less than k is above its reaction function when ( q q ) q () r cq F . c 11 12 1 1 1 1 1 C C () () r c cq rx F F . qk 11 for . C () c rx F . 1 1 1 1 1 1 1 11 1 11 Rq 1 ()1 1 1 1 q ; cost function is: it produces more than k . Therefore, the incumbent’s ( q q ) q ( rk c q 12 F ), 2 ( q q ) q () r 11 c q 12 F . 1 1 1 1 1 1 2 2 2 12 2 2 2 2 C rk c q F qk 22 for . 1 11 1 1 1 11 ( q q ) q () r c q F . C () r cq F for q 0 . reac 2 tion func 2 2 tion is disc 12 on 2tinuous a 2 2 t k 2 , while entrant 2 2 22 2 2 1 ( q q ) q () r cq F . 11 12 1 1 1 11 (1) () cr qq q ( r k c q F ) () qq q , p qq , 1 1 11 11 2 1 1 1 1 1 1 11 2 1 C () r cq F for q 0 . p qq , 1 11 2 (6) has the unique reaction function obtained by maxim- 2 2 22 2 2 Rq 1 () 11 2 q . 12 2 ( q q ) q ( rk c q F ), 11 12 1 1 1 1 1 1 C () r cq F qk for . ( q q ) q ( rk c q F ), p1 1 qq 11 . 1 11 ( q q ) q () r c q ising its pr F p . qq ofit giv.en by (7): 2 22 1 11 (1) 12 () qq 1 1 q1 (( 1r 1 c )1q F ) () qq q . 2 22 1 2 2 2 12 2 2 2 2 1 11 2 1 1 1 1 1 11 2 1 ( q q ) q () r cq (1)F () . qq q ( r k c q F ) () qq q , 1 11 2 1 1 1 1 1 1 11 2 1 11 12 1 1 1 1 1 ( q q ) q () r c q F . We will assume that the en trant always maximises 2 2 2 12 2 2 2 2 ( q q ) q () r cq F . 11 12 1 1 1 1 1 cr 2 22 C () r cq F for q 0 . profit, and its objective function is given by: (11) C rk c q F for qk . C R () qq rk c q F for qk . 2 2 22 2 2( q q ) q ( rk c q F ), . 1 11 1 1 1 11 (1) () qq q (( r c ) q F ) () qq q . 121 11 1 1 1 11 1 11 1 12 11 1 1 1 2 1 1 1 1 1 1 1 1 c 11 2 1 11 Rq () q ; (1) () qq q ( r k c q F ) () qq q , ( q q ) q ( rk c q F ), 12 2 1 11 2 1 1 1 1 1 1 11 2 1 11 12 1 1 1 1 1 1 ( q q ) q () r cq F . (1) () qq q ( r k c q F ) () qq q , 22 11 12 1 1 1 1 1 1 11 2 1 1 1 1 1 1 11 2 1 (7) We have two Cournot equilibr ia, depending on the ( q q ) q () r c q F . C () r cq F qk C () r cq F qk 2 2 2 12 for 2 . 2 2 2 for . ( q q ) q () r cq F . 1 1 11 1 11 c 1 1 11 1 11 11 (1) 12 () qq 1 q 1 1(( r1 c 1) q F ) () qq q . cr 2 ( c ) ( c r ) 2( c r ) ( c ) 11 1 1 11 2 1 1 1 1 1 11 ** 2 1 11 1 11 2 2 2 2 2 2 1 1 Rq () q ; incumben qq Rq () (1) t’s pr () oduc qq tion r q ela q . tiv (e t (;. r o its installed capaci c ) q F ) () - qq q . 12 2 12 12 1 11 2 2 1 1 1 1 1 11 2 1 22 22 (1) () qq q ( r k c q F ) 44 () qq q , 1 1 11 2 1 1 1 1 1 1 11 2 1 If the incumbent has the same objec tive function, ties. In the case of the incumben t’s production, which ( q q ) q ( rk c q F ), C 11 () r cq 12 F 1 for q 1 1 0 . 1 1 1 C () r cq F for q 0 . cr 2 2 22 2 2 2 (12) 22 () qq 2 2q ( r k c q F ) () qq q , c 1 11 2 1 1 1 1 1 11 1 1 11 2 1 11 we have the following two profit functions for the in- leave excess capacities, at the intersection of func- C c x rk F . Rq () q . Rq 1 () 1 1 11 1 q ; (1) () qq q (( r c ) q F )c () qq q . 12 2 ( q q ) q () r cq F . 12 2 1 11 2 1 1 1 1 1 11 11 2 1 cr 11 12 1 1 1 1 1 2 ( cr ) ( c r ) 2( c r ) ( c r ) Rq ** () 2 22 q ; 22 1 11 2 2 2 2 2 2 1 1 1 cumbent depending on whether it produces below or tions 12 R (q ) and R (q 2 ), the equilibrium corresponds to R () qq (1) () qq q (( r c ) q F ) () qq qq q 1 . 2 2 1 . ;. C c x rk F . C c x rk F . 12 21 1 1 11 2 1 1 1 1 1 11 2 1 22 1 1 1 11 1 ( q q ) q22 () r c q F . 1 1 1 11 1 ( q q ) q () r c q F . 2 2 2 12 2 2 2 2 2 2 2 12 2 2 2 2 22 44 cr above the capacity level, respectively: point V in Figure 2 that can be defined as: 1 11 C () c rx F Rq 1 () 1 1 1 1 q . cr C c x rk F c . 12 2 cr 1 11 (1) () qq q ( r k c q F ) () qq q , 1 1 1 11 1 11 2 22 1 11 2 1 1 1 1 1 1 11 Rq () 2 1 q . 22 Rq () q ; R () qq 12 2 12 2 . 21 1 C () c rx c F . C () c rx F . 2 ( c ) ( c r ) 2( c r ) ( c ) ( q 22 q ) q ( rk c q F ), 1 1 1 1 1 1 ( 1 q 11 1 1 q ) 1q ( rk c q22 F ), ** 11 2( 11 12 cr )1 2 ( c 1 21 r 21 1 ) 1 2 2 2 1 1 11 12 1 1 1 1 1 1 l 1 11 2 2 2 Rq () q ; (8) qq ;. 12 2 q ; (1) () qq q (( r c ) q F ) () qq 12 q . p qq , 1 1 11 2 1 1 1 1 1 11 2 1 2 1 11 2 cr 44 C () c rx F . ( q q42 ) q () r cq F . 1 11 1 1 1 1 1 ( q cr q ) q () r cq F . 11 12 1 1 1 1 1 (12) 11 12 1 1 1 1 1 2 22 Rq () q . 12 2 R () qq cr p qq cr , . p qq , p qq . 2 ( c ) ( c r ) 2( 2 c 22 r ) ( c ) 21 1 1 2 11 221 2 11 1 ** 11 22 2 2 2 2 2 2 1 1 1 11 2 R () qq . Rq () 22 q qq . ;. 4( c r ) 2 ( cr ) ( c r ) 21 1 f 2 2 2 11 1 2 2 2 12 2 12 c q 2 ( cr 22 ) ( c r ) 2( c r ) . ( c r ) 11 22 p qq . 44 ** p qq . 2 1 11 2 2 2 2 2 2 1 1 1 p qq , 2 Rq 2 () The c 22 ontribution of our paper is t 1 q ; o consider how 2 1 22 11 12 qq (1) () qq q ( r k;. c q F ) () qq q , 12 (1) () qq 2 q ( r k c q F ) () qq 1 q , 11 2 184 1 1 1 1 1 11 2 1 1 11 2 1 1 1 1 1 1 11 2 1 22 22 cr C rk c q F for qk . 44 1 2 11 ( 1c 1)1( c 11 r ) 2 22 2( c r ) ( c ) p qq . diff** erent objective functions affect the entry game in On the other hand , at the intersection of functions 11 2 2 2 2 2 2 1 1 R () qq 2 22 1 . qq 21 ;. 1 2 ( c ) ( c r ) 2( c r ) ( c ) 12 cr ** (1) () qq q (( r c ) q F ) () qq q . cr 22 2 ( cr ) ( c r ) 11 2( c 2r ) 2 ( c 2 r ) 2 2 2 1 1 (1)2 () 22 qq q ** (( r c ) q F ) () qq 1 q . 11 2 1 1 1 1 1 11 2 1 C rk c 1 q 11 F for qk . 1 22 11 2 2 2 qq 2 2 2 1 ;. 1 1 C rk c q F for qk . 1 11 2 1 1 1 1 1 11 2 1 this market. We will 44 , therefore, assume that incumben t R (q ) and R (q ), the equilibrium, which corresponds 11 R 1 () qq 11 1 1 1 11 12 1 11 1 1 1 Rq () q qq . . 1 ;. 2 2 1 22 21 112 1 12 2 22 44 2( cr ) ( c r ) 22 22 44 l C () r cq F qk 1 11 2 2 2 for . maximises the linear combination of profit and rev - to point T in Figure 2, can be defined as: 1 1 11 1 11 C rk c q F for qk . q ; 1 11 1 1 1 11 2 ( c ) ( c r ) 2( c r ) ( c ) ** c 42 11 2 2 2 2 2 2 1 1 enue, 2wher (e cr the c w )eigh ( ct for r the rev ) enue 2( is λ. W e c will r ) ( c r 11 ) C () r cq F for qk . C () r cq F for qk . ** 1 11 qq 2 2 2 2 2 ;. 2 1 1 1 1 1 11 11 1 1 1 11 1 11 11 Rq () q ; qq ;. 22 12 2 ( cr ) 2 ( c r ) 2( c r ) ( c r ) Rq () q ; 2 ( c ) ( c r ) 2( c r ) ( c ** ) C12 ** 12 c x rk cr F 2 1 11 2 2 2 2 2 2 1 1 1 11 . 2 22 2 22( 44 cr 2) 2( c 2 r 1 ) 1 1 1 1 2 11 22 1 1 11 2 2 2 22 assume that the level of capacities k is the same as qq 4( c r ) 2 ( cr ;. ) ( c r ) C qq () r cq 44 F for q 0 . ;. 22 12 R () qq q 1 f ; 2 2 2 11 1 2 2 2 12 2 2 22 2 2 . 22 C () r cq F for qk . 22 21 11 1 1 11 1 11 q 44 . 44 42 2 22 cr in the pure profit-maximizing equilibrium. A more in- (13) cr 1 11 C () r cq F q 0 84 C () r cq F for q 0 . for . 1 11 2 2 22 2 2 Rq () q . 2 2 22 2 2 Rq () q . 12 2 2 ( cr ) ( c r ) 2( c r ) ( c r ) C12 () c rx F . ** 2 1 11 2 2 2 2 2 2 1 1 1 volved analy 2( cr sis could include a higher lev ) ( c r 4() c el of capaci r ) 2 ( cr - ) ( c 22 r ) 1 1 1 1 1 l ( q q ) q () r c q F . 1 11 22 qq 2 f 2 2 2 2 2 11 ;. 1 2 2 2 2 2 2 12 12 2 2 2 2 q ; 22 C () r cq F for q 0 . 2( cr ) ( c r ) 2 2 ( ( c cr ))( c ( cr q r) ) 2(2(c cr ) r )( c ( c ) r ) . 2 2 22 2 2 1 l ** ** ’ 2 2 1 11 2 2 2 11 1 11 2 22 2 2 2 44 2 22 22 2 1 1 1 1 1 ties, k > k , when incumbent maximizes the alterna- q ; qq qq 1 1 42 ;. ;. 84 1 12 12 2 ( q q ) q () r c q F . ( q q ) q22 () r c q F . 22 2 2 2 12 2 2 2 2 2 2 2 12 2 2 2 2 42 44 44 cr tiv p e objec qq tiv e func , tion. Thus, we have the following 2 22 cr 1 11 2 ( q q ) q ( rk c q F ), 4( 2 c 22 r ) 2 ( cr ) ( c r ) R () qq If the post-entry subgame r . esults in a Stackelberg 11 12 1 1 1 1 1 1 f 2 2 2 11 1 2 2 2 21 1 ( q q ) q () r c q F . R () qq 2( . cr ) ( c r ) 2 2 2 12 2 2 2 2 q 21 l 1 . 4( c r ) 2 ( cr ) ( c r ) objective functions for the incumben 1 11t depending on 2 2 2 2 22 2 2 11 1 2 2 2 2 f p qq . 2 q ; 2 ( 22 q q 1 ) q ( rk c q F ), equilibr q ium, as in point S in Figure 1, the incumbent . ( q q ) q ( rk c q F ), 1 ( q q ) q () r 84 cq F . 2 11 2( 12 cr ) 1 ( c 1 1r 1 1) 1 2 11 12 1 1 1 1 1 1 11 2 ( 12 cr )1 ( c 1 r 11 ) 1 2( c r ) ( c r ) 2 ** 1 11 2 2 2 42 1 11 2 2 2 2 2 2 1 1 1 how much capacity it uses: 84 q qq ;;. 1 12 as the leader (l ) will maximise its profit by choosing 22 2 ( q q ) q () r cq F . ( ( qq qq) q ) q () (r rk cq c q F F .), 42 2 ( c ) ( c r ) 2( c r ) ( c ) 44 11 12 1 1 1 1 1 1 11 12 1 1 1 1 1 ** 11 12 1 1 1 1 1 11 2 2 2 2 2 2 1 1 4( c r ) 2 ( cr ) ( c r ) 2 ( c ) ( c r ) 2( c r ) ( c ) C** rk 11 c q F for 2 qk f 2 2 . 2 2 2 2 2 11 2 qq 1 1 1 2 2 2 ;. the point where his lowest iso-profit line is tangent to 1 11 1 1 1 11 22 qq ;. q . 12 (1) () qq 2 q ( r k c q F ) () qq q , 22 44 ( q q ) q () r cq F . 1 4( c 11 r ) 2 2 (1cr 1 1 ) 1 1 ( c 1 r 11 )2 2 1 11 12 1 1 1 1 1 f 2 2 2 11 1 2 2 2 44 84 entrant’s reaction function given by (11). Therefore, in q . 2 2( cr ) ( c r ) l 2 (1 1 ) () 11 qq 2 q 2 ( r k 2 c q F ) (9) () qq q , (1) () qq q ( r k c q F ) () qq q , 1 11 2 1 1 1 1 1 1 1 11 11 2 1 2 84 1 1 1 1 1 1 11 2 1 q ; (1) () qq q (( r c ) q F ) () qq q . 1 the subgame perfect equilibrium, we have the follow- C () r cq F for qk . 1 11 2 2 1 1 1 1 1 11 2 1 1 1 11 1 11 2 ( cr ) ( c r ) 2( c r ) ( c r ) 42 ** 1 11 2 2 2 2 2 2 1 1 1 2 ( cr ) ( c r ) 2( c r ) ( c r ) (1) () qq q ( r k c q F ) () qq q , ** qq ;. 1 11 2 2 2 2 2 2 ing quan 1 1 tities f 1 or the leader and the entrant – as the 1 11 2 1 1 1 1 1 1 11 2 1 12 (1) () qq q (( r c ) q F ) () qq q . 22 (1) () qq q (( r c ) q F ) () qq q . qq ;. 1 11 2 1 1 1 1 1 1 11 211 1 2 1 1 1 1 1 11 2 1 12 22 44 4( c r ) 2 ( cr ) ( c r ) f 2 2 44 2 11 1 2 2 2 follower ( f ) in this game, respectively: c C q () r cq F for q 0 . . (1) () qq q (( r c ) q F ) 2 2 () 2 qq 11 22 q . 2 2 1 11 2 1 1 1 1 1 11 2 1 Rq () q ; 12 2 84 22 c c 2( cr ) ( c r ) 11 l (14) 11 1 11 2 2 2 Rq () 2( cr )q ( ; c r ) q ; Rq () q ; l 1 11 2 2 2 12 2 12 2 1 q cr ; ( q q ) q () r c q F . 22 22 1 The post 1 -en 11 try subgame can result in either 42 c 2 2 2 12 2 2 2 2 2 11 Rq () q . 42 12 2 Rq () q ; 12 2 cr Cournot or S cr tackelber g equilibrium, depending on 1 11 4( c r ) 2 ( cr ) ( c r ) 1 11 f 2 2 2 11 1 2 2 2 Rq () q . Rq () q . 4( c r ) 2 ( cr ) ( c r ) 12 2 12 2 q . f 2 2 2 11 1 2 2 2 ( q q ) q ( rk c q F ), 2 the values of the parameters. Recall from Figure 2 that 11 12 22 1 1 1 1 1 1 22 q . cr 2 84 1 11 2 Rq () q . cr 84 12 2 Cournot equilibr 2 22 ium at T or V is the corner solution of ( q q ) q () r cq F . R () qq 22 11 12 1 1. 1 1 1 21 1 cr cr the Stackelber 22 g equilibr ium. We will first determine In the post-entry Cournot equilibrium (the corner 2 22 2 22 R () qq R () qq . 21 1 21 1 the Cournot equilibrium when both firms maximise solution of the Stackelberg equilibrium), where the cr (1) () qq q ( r k c q F ) () qq q , 2 22 1 11 2 1 1 1 1 1 1 11 2 1 R () qq 2 ( c ) ( c r ) 2( c r ) ( c ) pr ** ofits as the 11 benchmar 2 k 2 case. 2From (8), we 2can 2obtain 2 1incumben 1 t maximises the linear combination of profit 21 1 qq ;. 22 12 44 2 ( c ) ( c r ) 2( c r ) ( c ) 2 ( c ) ( c r ) 2( c the incumben r )( 1 ( c) () t)’s r qq eac tion func q (tion depending on the ( r c ) q F ) () qq and r ev q enue g . iven by (9), the incumbent’s reaction ** ** 11 2 2 2 2 2 2 11 1 1 2 2 2 2 2 2 1 1 1 11 2 1 1 1 1 1 11 2 1 qq ;. qq ;. 12 22 2 level of capacity it uses: functions depending on the level of its production are 44 44 2 ( c ) ( c r ) 2( c r ) ( c ) ** 11 2 2 2 2 2 2 1 1 qq ;. 2 ( cr ) ( c r ) 2( c r ) ( c r ) ** as follows: 22 1 11 2 2 2 2 2 2 1 1 1 c 44 qq ;. Rq () q ; 12 2 ( cr 44 ) 2( c r ) 2( c r ) ( c r ) 2 ( cr ) ( c r ) 2( c r ) ( c r ) ** c (1) ** 1 11 2 2 2 2 2 2 1 1 1 1 11 2 2 2 2 2 2 1 1 1 22 11 qq ;. qq ;. 12 Rq () q ; 22 12 2 (10) 44 44 (15) 2 ( cr ) ( c r ) 2( c r ) ( c r ) cr ** 1 11 2 2 2 2 2 2 1 1 1 1 11 qq ;. Rq () 2( cr )( c q. r ) 12 l 22 12 1 11 2 2 2 2 c (1 ) r (1) q ; 44 1 1 1 R () q q . 42 12 2 2( cr ) ( c r ) 2( cr ) ( c r ) l 1 11 2 2 2 l 1 11 2 2 2 q ; q ; 1 2 4( c 42 r) 2 ( cr ) ( c r ) 42 2( cr ) ( c r ) f 2 2cr 2 11 1 2 2 2 1 11 2 2 2 2 22 q . q ; R 2 () qq 1 . 2 2 21 1 2 c (1 ) ( c ) ( cr ) 4( c r ) 2 (84 cr ) ( c r ) 11 11 2 2 2 4( c r 42 ) 2 ( cr ) ( c r ) * 2 222 2 11 1 2 2 2 f 2 2 2 11 1 2 2 f 2 q ; South East European Journal of Economics and Business, Volume 16 (1) 2021 89 q . q . 1 2 2 4 84 84 4( c r ) 2 ( cr ) ( c r ) f 2 2 2 11 1 2 2 2 q . 2 ( c ) ( c r ) 2( c r ) ( c ) 2 ** 11 2 2 2 2 2 2 1 1 2 ( cr ) c (1 ) * 2 22 1 1 84 qq ;. q . 44 4 2 ( cr ) ( c r ) 2( c r ) ( c r ) ** 1 11 2 2 2 2 2 2 1 1 1 2 c (1 ) r (1 ) ( cr ) qq ;. 11 1 2 2 2 * q ; 44 2 4 2 ( c r ) c (1) r (1) * 2 22 1 1 1 2( cr ) ( c r ) l q . 1 11 2 2 2 q ; 4 42 4( c r ) 2 ( cr ) ( c r ) f 2 2 2 11 1 2 2 2 q . 2 c (1 ) r (1 ) ( cr ) 2 11 1 2 2 2 l 84 q ; 42 4 ( cr ) 2 ( c (1 ) r (1 ) ) ( c r ) 2 22 1 1 1 2 2 2 f q . 84 1/ 2 cr 2F 2 2 2 2 B cr (1 ) (1 ) 11 1 M c (1 ) 11 M The first reaction function is relevant when q < k , Just as in the Cournot equilibrium, a similar 1 1 and the second one when q ≥ k . It is evident that comparative static analysis can be conducted for 1 1 the quantity produced by incumbent increases with Stackelberg equilibrium. It is interesting to observe λ – the relative importance of revenue to profit in its that the incumbent’s quantity produced increases objective function. Hence, when λ increases, both ver- with λ, which means that incumbent who maximises c (1) sions of the incumbent’s reaction functions given by the alternative objective function produces a larger Rq () q ; 12 2 (15) move upward, which is intuitive since it becomes quantity than the incumbent who maximises profit. more critical for the incumbent to maximise revenue On the other hand, the entrant produces a lower c (1 ) r (1) 1 1 1 R () q q . c (1) 12 2 than profit. Just in the original setting with λ = 0, we quantity than in the benchmark model of profit maxi- Rq () 22 q ; 12 2 have two equilibria depending on the incumbent’s misation. The result of these opposing movements c (1) c (1 ) r (1) production lev 11 el. From the intersection of R (q2) and is the increase in the total quantity produced in the 2 1 c (1 1 ) ( 1 c ) ( cr ) Rq () q ; 1 * 11 11 2 2 2 R () q q . 12 2 q 12 2 ; lf R (q ) we have 22 (notice that the reaction function of Stackelberg industry. Formally , 22 ( qq )/ 0 2 1 4 12 c (1 ) r (1) Firm 2 remains unaffected by alterations related to which is always the case for positive values of c and r , 1 1 1 1 1 2 ( cr ) c (1 ) R () q q . c (1) * 2 22 1 1 12 2 11 2 c (1 ) ( c ) ( cr ) firm 1 objective function): and f q or the 0 < γ < 1. Based on the same assumptions . , Rq () 22 q ; * 11 11 2 2 2 12 2 q ; 1 4 the average price of leader’s and follower’s products 4 c (1 ) r (1) 2 1 c (1 1 ) ( 1 c ) ( cr ) decreases with the increase of λ, which has a positive * 11 11 2 2 2 2 ( cr ) c (1 ) R () q q . * 2 22 1 1 q 12 2 ; 2 c (1 ) r (1 ) ( cr ) 2 1 q 11 1 2. 2 2 2 * influenc 2 e on consumer surplus and the competition 4 q ; 4 2 4 (16) conditions in this industry. 2 ( cr ) c (1 ) * 2 22 1 1 2 c (1 ) ( c ) ( cr ) q . * 11 11 2 2 2 2 (cc (1r) ) c (1) r (1) 2 2 * 2 11 22 1 1 1 q ; 2 c (1 ) r (1 ) ( cr ) 1 4 q Rq () q ; . * 11 1 2 2 2 2 12 2 4 q ; 1 22 4 4 Furthermore, from the intersection of R (q2) and c (1 ) r (1) 2 ( cr ) c (1 ) 1 4. INCUMBENT’S DECISION AND λ VALUE 2 22 1 1 1 1 1 * 2 c (1 ) r (1 ) ( cr ) R () q q . q* 11 1 2 . 2 2 2 ( c r ) c (1) r (1) 2 12 2 2 22 1 1 1 R (q ) we have: 2 * q ; 1 2 1 4 q 2 c (1 )22 r (1 ) ( cr ). 11 1 2 2 2 l 2 4 q 4 ; The value of B from Figure 1 is obtained at the in- 1 2 42 2 ( c r ) c (1) r (1) tersection of entrant’s iso-profit line when long-run * 2 22 1 1 1 2 c (1 ) ( c ) ( cr ) 2 c (1 ) r (1 ) ( cr ) 11 11 2 2 2 * q . * 11 1 2 2 2 2 q ; q ; 1 4 ( cr ) 2 ( c (1 ) r (1 ) ) ( c r ) 1 4 profit is pr ecisely zero and its reaction function. The 2 2 2 c ( 22 1 ) r (1 1) ( cr 1 ) 1 2 2 2 f 11 41 2 2 2 l 4 q . q ; (17) following expression shows tha 2 t B is invariant to the 84 42 2 ( cr ) c (1 ) 2 ( c r ) c (1) r (1) * 2 22 1 1 * 2 22 1 1 1 changes of q λ. . q 2 c (1 ) r (1 ) ( cr ) . 2 2 11 1 2 2 2 l 4 ( cr 4 ) 2 ( c (1 ) r (1 ) ) ( c r ) 4 q ; 2 22 1 1 1 2 2 2 1 f 1/ 2 q . cr 2F 42 2 2 2 2 2 B 84 (19) 2 c (1 ) r (1 ) ( cr ) * 11 1 2 2 2 4( cr ) 2 ( c (1 ) r (1 ) ) ( c r ) 2 c (1 ) r (1 ) ( cr ) By check 2 ing the sig 22 n of der 1 ivatives with r 1 espec 1t to 2 2 2 f 11 1 2 2 2 q ; l q . q ; 2 1 4 2 1/ 2 λ or, just, by the simple comparison of (16) and (17) However, when the incumbent produces above ca- 84 cr 2F 42 2 2 2 2 cr (1 ) (1 ) B 11 1 with (12) and (13), respectively, it is evident that the pacities 2 (, the monopoly out c r ) c (1) c ome is obtained fr r (1) om R M 2 22 1 1 1 * 1 1 q . 4 ( cr ) 2 ( c (1 ) r (1 ) ) ( c r ) 2 2 2 22 1 1 1 2 2 2 f 1/ 2 quantity pr oduc c (1 ed b )y the incumben t is always larger for q =0. 4 11 2 q . cr 2F Rq () q ; 2 2 2 2 2 12 2 B 84 than in equilibr 1 ium when it maximises profit, while cr (1 ) (1 ) 11 1 c (1 ) 11 M . M the entrant is producing less. Moreover, the increase (20) 1 1 c (1 ) r (1) 2 c (1 ) r (1 ) ( cr ) 1 1 1 11 2 2 1 2 2 2 l R () q 1/ 2 q . 12 2 q ; of the incumbent’s quantity is more significant than cr 2F 2 2 2 22 2 cr (1 ) (1 ) B 11 1 42 the fall of entrant’s quantity, which means that the For λ=0 expression (20) reduces to M = (α – c – r )/2, M 1 c (1 ) 1 1 1 1 11 2 . M 1 4 ( cr ) 2 ( c (1 ) r (1 ) ) ( c r ) total quantity produced increases. As a result, the which is identical to M from Figure 2. For all other val- 2 c (1 ) ( c ) ( cr ) 2 22 1 1 1 2 2 2 1 f 2 * 11 11 2 2 2 q λ . q ; 1 2 average price of incumbent and entrant’s products ues of λ in the range 0 < λ < 1, we have M > M . Similar cr (1 ) (1 ) 1 1 84 11 1 4 c (1 ) 11 M M 1 is declining with the increase of λ. All the mentioned conclusions could be reached for the monopoly out- 2 ( cr ) c (1 ) * 2 22 1 1 comparative statics results indicate a more competi- come when the incumbent’s production is below in- q . 1/ 2 2 cr 2F 2 2 2 2 4 λ B tive industry than in a situation where both players stalled capacities, which is obtained from R for q =0 c (1 ) 1 1 2 11 M . maximise profits. as: 2 c (1 ) r (1 ) ( cr ) * 11 1 2 2 2 When we have the Stackelberg equilibrium in the q ; 2 cr (1 ) (1 ) 11 1 4 post-entry sub-game where incumbent maximises (21) M the alt 2er ( nativ ce objec r )tiv e func c (1) tion, the quan r (1) tities pro- * 2 22 1 1 1 q . duced by the two firms are: 4 c (1 ) 11 M . 2 c (1 ) r (1 ) ( cr ) 11 1 2 2 2 l q ; 42 (18) 4 ( cr ) 2 ( c (1 ) r (1 ) ) ( c r ) 2 22 1 1 1 2 2 2 f q . 84 1/ 2 cr 2F 2 2 2 2 B 90 South East European Journal of Economics and Business, Volume 16 (1) 2021 cr (1 ) (1 ) 11 1 M c (1 ) 11 M . Evidently, with the introduction of λ into the mod- and his survival upon entry is a major priority for the el, we have positive shifts of both monopoly out- authority responsible for competition issues on high- comes based on the incumbent’s alternative reaction speed rails. Yardstick competition models like this can functions given by (15). monitor and eventually prevent incumbent’s preda- We have shown that the point M is to the right of tory behaviour in such industries as a precondition for M , while the position of B is unaffected, which im- imposing sustainable competition. The Infrastructure 1 1 plies that it is more likely that entry could be blocked manager should use auction-based allocation of train λ λ with M , when M >B . This makes entry more diffi- paths to discourage incumbent’s entry-deterring ca- 1 1 1 cult since the incumbent can block entry by merely pacity expansion. Also, the government should orga- producing the quantity M that maximises its alter- nise auctions for public service obligation contracts native objective function. This result is intuitive since instead of direct negotiations with the incumbent, incumbent cares not only about profit but also about which provides the additional possibility for prevent- the firm’s size and is more willing to block entry than ing incumbent’s capacity expansion. to allow it. Besides, the introduction of λ > 0 also af- Finally, our model can be validated empirically. fects the price drop in the post-entry subgame for However, it is challenging to identify entry intentions both Cournot and Stackelberg equilibrium, thus re- in some industries and to distinguish capacity expan- ducing the profit attractiveness of entry for the new sion due to the larger demand, from capacity expan- competitor. sion aimed at entry deterrence. All these possibilities open a variety of possibilities for further research. 5. CONCLUSION REFERENCES We have reconsidered Dixit (1979, 1980) model of preventing entry with capacity expansion. The con- Cherbonnier, F., Ivaldi, M., Muller-Vibes, C., and Van Der tribution of our approach is to assume different ob - Straeten, K. 2017. Competition for versus in the mar- jective function for the incumbent, consistent with ket of long-distance passenger rail services. Review of contemporary principal-agent relationships, where Network Economics 16 (2): 203-238. managers pursue the objectives of profit maximisa- Cookson, J. A. 2018. Anticipated entry and entry deter- tion and the increase of a firm’s size. Entry in an in- rence: Evidence from the American casino industry. dustry under such conditions is more difficult since it Management Science 64 (5): 2325-2344. is more likely that the incumbent can block entry by Cornes, R., Itaya, J. 2016. Alternative objectives in an oligop- producing the quantity that maximises its alternative oly model: An aggregative game approach. CESifo work- objective function. Our model might better explain ing paper no. 6191. recent empirical facts that entry is typically on a small- Cournot, A. 1838. Researches into the Mathematical scale basis which coincides with our result that en- Principles of the Theory of Wealth. New York: The trant’s quantity falls when incumbent pursues the al- Macmillan Company. ternative objective function. Our model also explains Crozet. Y., and Chassagne, F. 2013. Rail access charges in the empirical fact that the survival rate of entrants is France: Beyond the opposition between competition low. In an environment with massive incumbents that and financing. Research in Transportation Economics 39 care about their profits and size, small entrants have a (1): 247-254. diminished likelihood of survival. Dixit, A. 1979. 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Spence, M. 1977: Entry, capacity, investment and oligopolis- Economic Theory 14 (2): 429-437. tic pricing. The Bell Journal of Economics 8 (2): 534-544. Masson, R. T., Shaanan, J. 1982. Stochastic-dynamic limiting Sweeting, A., Roberts, J. W., Gedge, C. 2020. A model of dy- pricing: An empirical test. The Review of Economics and namic limit pricing with an application to the airline in- Statistics 64 (3): 413-422. dustry. Journal of Political Economy 128 (3): 1148-1193. Milgrom, P., Roberts, J. 1982. Limit pricing and entry un- Tirole, J. 1988. The Theory of Industrial Organization. der incomplete information: An equilibrium analysis. Cambridge, MA: MIT Press. Econometrica 50 (2): 443-459. Williams, J. W. 2008. Capacity investments, exclusionary be- Morrison, S. A. 2001. Actual, adjacent, and potential com- havior, and welfare: A dynamic model of competition in petition estimating the full effect of Southwest Airlines. the airline industry. Empirical Studies of Firms & Markets. 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Journal
South East European Journal of Economics and Business – de Gruyter
Published: Jun 1, 2021
Keywords: differentiated oligopoly; capacities; quantity competition; asymmetry of players; incumbent; entrant; D43; L13; L21