Sequential entry, industry structure and welfare

Received July 1986, final version received April 19 In this paper we reexamine the relationship between potential corn ustry structure and welfare in a market subject to a sequential entry threat whe rms can make quantity commitments and have acess to .r. constant returns technology. It is show that, provided entry is not blockaded, the incumbent(s, will either prevent entry or allow all the potential emrants in. Furthermore, actual entry may decrease but social weifare never goes down with more entrants. Nevertheless, with a finite number of potential entrants, policies which lower the cost of entry need not be beneficial in weli’are terms.

.

nt

ctio

ecent research has suggested the possibility of a counterintuitive relationship between potential competition and welfar;:. Samuelson (1984) has argued that increasing the number of potential entrants in a market may cause social welfare to decline since this increase reduces the potential entrant’s incentive to enter. Dixit and Shapiro (1984) found some evidence, examining entry dynamics with mixed strategies, that expected profits of an incumbent firm tend to rise with the number of potential entrants. Bernheim (1984) argues in a sequential enry model that some traditional policies to foster competition may have perverse effects. Stiglitz (1981) argued with an example that potential competition may lower welfare. The issue is: will a larger pool of potential entrants reduce the level of welfare in a market? The answer to this qclestion is not only of academic interest since in many countries the public authority has measures in place to fo ging the pool of potential entrants into an industry. Thes e the dissemination of information aboqrt new sectors in particular the most techn opportunities, couragemen series of incentive schemes to facilitate pot atutes, an anon nk Richard Gilbert, Carmen Conference for y acknowled ennsylvania, is gra

142921/88/$3.50 $J 1988, Elsevier Science Publisher3 G.V. (North-

at t esear

X. Woes, Sequential entry, industry structure a

1672

respective research into market opportunities or like encouraging firms opt flexi& technologies which can be readily adopted to the pro tion of new products. t in this paper a very simple mode market with three main features: t is, there is sequential sol of entrants is ordered in a sequent entry into the industry,’ s which determine the (2) firms set quantities and commit to output ccording to the demand market supply; a market clearing price obta schedule and, s to scale technology and (3) all firms have access to the same constant ret have to pay entry cost if they decide to produce a positive amc,unt. firms are quicker than others to respond to a profitable Earlier entrants make a quantity commitment and anticipate the behavior of the remaining potential entrants. We could think of an agricultural market where different producers plant the seeds at different times, maybe because of the diverse weather conditions to which they are subject. A farmer knows the amount planted by previous farmers and tries to forecast lanted by those who follow. Once the output is obtained the ing it to an auctio eer (maybe a government agency) which arket according to the demand schedule. That is, firms choose capacities of production first and then a competitive, market clearing, stage fohIows.”

ypically, prospect.

some

s and compete in supply functions [see Grossman ( 1981) and Singh an there are a few potential if there are many, the ere is a critical number of

the established fir

ts may mean less actual

of e irit of Prescott and Visscher alensee (1978) and Ju

Other models which use the

arginal costs are constant. It entrants, n. In fact they are increasing in n till entry is prevented. At this point market outcome. With more actual entry t the decrease in producer surplus (profits) indu The analysis in the paper can be exte

cost, the number of incumbents and the num entry and entry preventing equilibria coexist.

of potential entrants, both evertheless the comparative

entrants are similar to the case of a single incumbent. The assumption that firms are able to commit to quantities facilitates greatly the analysis and allows us to obtain sharp results.4 odels with sequential choice tend to be analytically hard and welfare results are not easy to obtain. The quantity commitment model can be seen as a reference point which we can handle or even, in some circumstances as the work of Kreps and Scheinkman (1983) shows, as a reduced form of a more complex multistage game where firms choose capacities of production and prices. Section 2 presents the basic model and results. Section 3 deals wit case of multi p?e inc*umbents. Several extensions are considered in section 4 and concluding remarks follow.

Consider a market for a homogenous product with linear demand p = a -X, where p is the price, X total output and a is a positive constant. Our incumbent firm has a cost function C,(x) =S+ vx, where S is a sunk cost and v the constant marginal cost. There is a set of A? potential entrants Iv= {1,. . . , otherwise. F(F>O) is the entry cost. and consider prices net of marginal cost.

X. Vives, Sequential entry, industry structure and welfare

1674

the incumbent firm makes its productio decision. At stage k, ial entrant chooses whether or not to enter and uced by as fixed the outputs ere are still -k pQtentia1 nous a firm is only interested in t ategy for a potential entrant is thus a n level to any possible cumulated output of earlier firms in the sequence. ones of the potential iven the outputs o he incumbent, x0 script means that the be total output. entrants, xi (j E N), let output of firm i is not included in the sum. Ignoring the sunk cost S of the incumbent, profits of the incumbent are given by ~t&~, X +-,)= (a -X)x,, and refits of potential entrant j by Zj(Xj,X-j)=(a-X)XjF, jE N. Given tuple of strategies of the potential entrants (qj)jeN let Q&Z) be the total output of the last k firms w en they follow the strategies firms produce Z. & ) is defined recursively l

QrW=qn( k

&Z)+&+1-&Z),

k=2 ,..., n.

Nash equilibria, where empty or example, potential entrants cannot promise to choices after they succeed in ally, a subgame perfect Nash e is a nonnegative number, x ,-j(qj(Z)+Z))=max,,,+

njCY9z+Qn-jfY+Z))9

.HN.

ant’s strategy qj( 0) to yield a t of already established firms ture entrants. Condition (i) ofits taking into account the

iscussion of subgame

This is not a general result but a consequence of linear de genous product and constant marginal costs. Lemma 1. If there is no entry cost the Cournot best response functio max (0, a-Z/2}, is the equm’libriumstrategy for any potential entrant. Proof: When F =0 no entry prevention is possible and I-(- ) is the optimal ow if firms j+t,...,n response function of firm n, the last potential entrant. use r( ) and earlier firms produce 2, total output of the last n-j firms is (for a>Z) l

Qn- j(z)=( 1 - l/2”-“)(a - Z). Revenue of firm j with output j when earlier firms produce Z is (a -(Z i-y + Q.E.D. Qn- j( Z + y))y. This expression is maximized at r(Z). Wh:n F >O the entry preventing output is the solution to (a-(Y+r(Y)))r(Y)=F

in [0, a] and equals potential entrant are and only if it can equilibrium strategies

max (0, a -2@}. That is, if Z > Y then profits of the nonpositive. Assuming that a potential entrant enters if make positive profits, Lemma 2 c t of t otential entrants.

aract erizes

Lemma 2. Let qj( . ), j E then for any j qj( ) is a sele

rium strategies of the potenti

l

,(Z) = r(Z) j(Z)=P(Z) = -

if Z < Y and zero otherwise. ifZ,_jzZzO

X. Viva, Sequential entry, industry structure and weljbre

1676

l--A&},

e Sest

k=O,

l,...,

n, with

Ak=

res onse corres ondences ulj( jE N) d only if, given that total 0

us y is best for firm j when firms j+l,...,n satisfying the sequential rationality requirement. i, is not a ce of firm j, best response c 0-j (fig. t he characteristics of the best response corres

explained (a formal proof of Lemma 2 is provided in the ap fact is that firm j will not allow a downstream firm io enter the industry and prevent entry. Firm j makes more profits preventing entry itself since ginal costs are constant and total output will be at least Y acyv~y. refore if fir j allows entry it must be the case that everyone else downstream is going to allow entry too. The last potential entrant py _*ces according to the Cournot reaction function r( 9) if it makes positive profits and otherwise it stays out. Consider potential entrant j, j y7. If Z is smaller but larger tha Z,_j, where Z,_j is the critical Z for which firm j rent between allowi g and preventing entry of the rest of the potenants, entry is prevented. or z’s smaller or equal to Z,- j entry

indifferent

between allowing

and prove

Proposition I. ssume that a potential ent make positive profits7 then (x0,4) is an selection of Yj forall j and

=p it qi is a

(i) Y 5 a/2 and x0 = a/2. Entry is blockaded. (ii) a/2 < Y g m and x0 = X Entry is pretiented. (iii, Y 2 Y, and x0 =a/2. Entry is allowed. Remark 1. When entry is allowed all otential entrants enter, producing according to their Cournot best response functions. As we ave noted this is a consequence of the constant retums technology: it always incumbent to be the entry preventer if total output is to -be the This way entry is allowed only when no firm wants to prevent entry. according to Lemma 2 and Z+r(Z) is increasing in Z. 52 Y >u/2. In this case the unique S.P.E. is :or the incumbent to set Y entrant to use 4( *) where q(Z) =r(Z) if Z< Y and 0 otherwise. Therefore the incu entry. However the incumbent producing zero and the entrant us&Z) =a for Z>O is a Nash equilibrium where the i~~urn~~t does the entrant produces the monopoly output. The threat of ~rod~~~g produces a positive output is clearly not credible. ‘To derive the proposition we have assumed that a otential entrant enters if and only if It can make strictly positive profits. We can dispe equilibrium potential entrant j, j 2 2, will j, j 2 2, were to enter making zero p the total output of the previous firms Z in s its output decreases j- 1 profits wo since firm j- 1 must have an opti to enter making zero

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X. Cues, Sequential entry, industry structure and welfare

Remark 2. For limit outputs less than but close to m (the entry preventing output which makes the incumbent indifferent betwee allowing and preventtotal entry) the equilibrium total output (equal to ) is larger t put if entry were allowed, Xg. It is easily checked that Xz=( i/2”+ ‘)a. Xg is less than Ynsince when Y = Y,, profits of the incumbent preventing and allowing entry are equal, (a - Y,)Ym = (a - Xz)a/2, and Yn> a/2. This means that for limit outputs less t an but close to Y”the equilibrium tot (preventing entry) is larger than if entry were allowed. When entry is allowed equilibriu y>x;. 2.2. Comparative statics Comparative statics with respect to the number of potential entrants n follow easily from roposition 1. The critical entry preventing output Y” increases with n, converging to a as n goes to infinity. Therefore the region where entry is allowed shrinks as n grows. When there are a lot of potential entrants the incumbents are better off keeping them out, when there are only a few it is too costly (relatively) to prevent entry and they are allowed in. Suppose that Y > a/2 so that entry is not blockaded. Let ti(Y) solve (treating n as a continuous variable) Y”= I! We find fi= -log( l -(2Y/a - 1)2)/lag 2. or n > I?(Y) entry is prevented. ore potential entrants make the incumbent deter entry but total output is tal output with entry, XE, increases with n, (2) is a change of regime, from allowing to preventing entry, total reases since if entry was allowed at n=n, we know that therefore if entry is prevented at n= n, + 1 es up (from X& to Y), and (3) when entry is prevented total Total surplus TS is also nondecreasing in u. Again provided entry is not

to n’and for n> ti it stays constant at the

X. Vives,Se~~e~Zia~e

cbure and we#iare

Proposition 2. Fix there is a constala larger th is pye~e~t increasing in n and total output and total surplus are nondecreasing in n.

We have thus that more potential emtrants may mean less actual entry in

the industry but never lower welfare. This result contrasts wit entry models where increasing the number o c~trants ma welfare. This comes about because a larger umber of otential entrants reduces entry probabilities when looking at symmetri mixed strategy equilibria. If identical players use the same strategy then firms must randomize the entry decision when there is no room in the market for all t potential entrants. In a model presented by Samuelson expected total surplus is decreasing in n when n is larger than the optimal number [Samuelson (1984)]. Dixit and Shapiro examine entry dynamics with mixed strategies and find in simulations with a linear demand model that the expected profits if incumbents tend to rise with the number of potential entrants.

What would happen to our original model if instead of an incumbent firm we would have m incumbents? In that case we would have a simultaneous move by the m incumbents at stage 0. In Gilbert and Vives (1986) we have analyzed the public good aspect of entry deterrence with m incumbents and one potential entrant in a model with nonlinear demand for an homogenous product. Combining the analysis of section 2 with the one in Gilbert and Vives the subgame perfect equilibria of the game with m incumbents and n potential entrants can be characterized. Proposition 3 gives the result. The Cournot output of an incumbent is a/(m+ 1) and the total Cournot output of the m incumbents is (m/(m+ 1))a. The largest entry preventing output for which to prevent entry is profitable for all incumbents is denoted by Ym,n= The entry preventing output that makes incumbent i indifferent between preventing or allowing entry given that the other incumbents Cournot levels is denoted by -Y, 9,,. After some computations it ca derived that

,n-

- m+

m+l

n4

.

1680

X. Vives, Sequential entry, industry

structure

and

welfare

and rnz 2, the second is strict if n 2 1. [See Vives (1983) for a derivation of these expressions and a proof of Proposition 3. Let M be the set of incumbents, x = xi (i E M), q = qj (jE N) and assume as before that a potentia entrant enters if and only if it can make positive profits. Proposition 3. (x, q) is an equilibrium if and all j 1 Iand

only ifqj

is a selection of ‘yi for

Ysma/(m+l) and xi= aJ(m + 1), i E M. Entry is blockaded. 0i +l)=I,, t n and Xi= a/(m + l), i E M. Entry is allowed. All potential enter producing to their Cournot best response functions, Y(O).

ma 2: when t e total output of all is efine as in incumbents except firm i equals Z, then firm i is illdifferent between allo&ng and preyenting entry. ark.

n

When entry is prevented there is typically a continuum of entry preventing equilibria with incumbents producing on the hyperplane CipMxi= Y Since I$ “> Y, ,, with an established oligopoly we have a region of entry preventing outputsl [Ym,,, -Y,,J, where both to prevent entry and to allow entry are ossible equilibrium outcomes. As in Gilbert and Vives the entry preventing uilibria are Pareto dominated in terms of profits by the entry equilibrium. _Y,,. and Ym,& are both increasing in n and they converge to a as n goes to parative statics with respect to n are similar than before except re are two constants 6( Y m) and IZ(Y m), vZ>n ~0, such that if n +c6 to ahow entry is an equilibrium and if n> n to prevent entry is an equilibrium provided entry is not blockaded. fi solves in nrm9. = Y and n solves Ym, n= Y ti=n when m=l. (Fig. 2 shows t?(Y,0) and II(Y l) for given Y) g=-

(1 -((m+

l)(Yfa)-m)2)

log 2 a

ential entry,

in

0

n Fig. 2.

and total surplus with respect to the number of potential entrants. When entry is prevented there is typically a continuum of entry preventing equilibria but we know that for r@z,fi] the entry equilibrium Pareto dominates in terms of profits the entry preventing equilibria. Therefore if for purposes of comparison we take the symmetric entry preventing equilibria profits of incumbent i are nonincreasing in n. Proposition 4 summarizes the comparative statics with respect to n. Proposition 4. Fix Y and m and suppose that entry is not blockaded (Y > ma/(m + I), then there are constants ii and II, ii> 5 > @ such that jar n 5 ii entry is allowed and for n&z entry is prevented. Total output and total surplus are nondecreasing in n and restricting attention to symmetric entry preventing equilibria the profits of incumbent i are nonincreasing in n.

alfrey, analyzing a model of spatial equilibrium in olitical co suggested the possibility of a model with mo incumbent and alfrey (1984, p. 155)]. At least in the whether we consider one inc

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X. Vhes, Sequential entry, industry structure and welfare

4.1. Quadratic payoffs

The formal analysis can be extended easily to cover cases where the best response function of any firm (when there is no entry cost) is linear. to be the case we need payoffs to be quadratic. We could have linear for a homogenous product, as before, and quadratic costs. Nevertheless with increasing marginal costs and for large n the incumbent may let other firms enter and prevent further entry if to produce the limit output is too costly. Another possible specification would be a linear demand system for symmetrically differentiated products of the type pi = a - flxi - Tcj+ ixj, where #I> r>O and x>O,i=Q,..., n. In these cases it is no longer true that the monopoly output is equal to the Stackelberg output of the incumbent as in s is sb?J?x= gto\skd section 2, where both VB’w-2equal to a/2, hi 2X marginal costs are constant. 4.2. Nonlinear payoffs Nonlinear demand may cause problems even with a homogenous good. Supnose that the product is homogenous and inverse demand is given by p= k(X). P is very nice: smooth downward sloping, concave and cutting the axes. Under these conditions (and constant margianl cost, say) we know that e Cournot best response function, r,, is continuously differentiable with - 1c I$CO. Ignore the entry cost for the moment and consider prices net of marginal cost. Firm n is going to use r,. Firm n - 1 will net, by producing y, z,,_ ,(y, z) = P(Z + y + r,(Z + y))y if earlier firms in the sequence produce 2. A sufficient condition for ‘it,_1(. ,Z) to be strictly concave in y is that r, be convex. Let P,__1 be the best response function of firm n - 1. If we go back another stage to x,,_~ we find that a sufficient condition for ~t,,_~ to be strictly concave in the output of firm n-2 is that r, and r,,_ 1 be convex. To have strict concavity of the profit function of firm j we need to assure that the best response function of the remaining n-j firms are convex. The n-j+ 3th derivative of or example, a sufficient condition for r, to be convex edness) of profit functions ably more complicated in results follow as before, ts are constant if entry is if entry is to be allowed all

X. Vives, ~e~~e~tiQ~entry, in stry structure and we/fare

1683

4.3. Relaxing quantity commitme Id happen if fir

not commit to output levels but only to capacity levels and afterwards compete in quantities (con nt on the chosen capacity levels)? This would b a model in the spirit Suppose that firm i once has installed capacity kj has a cost function of the capacity limit type, Ci(Xi, ki) = WXi+ rki + F

if

Xi5 ki

(where w and Yare positive) and that costs are infinite otherwise. If potential entrant i now decides to enter it sinks capacity ki knowing the capacities y sunk by previous entrants and anticipating the decisions of the firms

in the sequence. Once all capacities are installed a Cournot game is played. This model has been analyzed by Gilbert (1986), Ware (1984) and Eaton and Ware (1985) (E-W for short).

The first thing to note is that given that demand is linear and therefore marginal revenue for a firm is decreasing in the output of the other firms [like in Dixit (1980)] we should expect no excess capacity in equilibrium (subgame perfect equilibrium). Capacity is costly and potential entrants will not be fooled by capacities which will not be used in case entry occurs and in case of no entry an incumbent will produce up to the capacity limit due to the marginal revenue assumption. In fact this is proved in E-W (Proposition 3). Furthermore these authors consider the free entry equilibria of the model described above (that is equilibria where the number of potential entrants is always larger than the number of entrants in equilibrium) and conclude that the aggregate output in equilibrium is no smaller than the Bain limit output (E-W, Proposition 5) and that ‘when the number of firms is “large”, for all practical purposes the output is the limit output. Thus, in a sense, we are resurrecting the limit output model’ (E-W, p. 15). This means that for n ‘large’, say n&i in Proposition 2, the simple quantity commitment model is a very good approximation of the more sophisticated capacity investment model in terms of predicting the market price. An obvious diflerence is t in the quantity commitment model entry prevention is Carrie incumbent only, while in the capacity investment model feasible in the sense that the required ca n this case entry prevention has t in the industry. n the new co number of potential e

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X. Vives, Sequential mtry, industry structure and

welfare

only one potential entrant. In this case total output never goes down when oing from one to two firms8 other possibility is to consider a cost functio endent of the capital investment (1980)]. In t ase excess capacity will play a established firms will have an incentive to expand capacity to gain more market share if entry is to occur although an incumbent may also try to ‘delegate’ the burden of entry deterrence to a second entrant if it finds the ean and task too castly. First steps to analyze this problem are taken in iordan (1985) and Schwartz and Baumann (E986) although a complete welfare analysis is still lacking. Still another variation would be to consider the capacity limit type cost function but with price competition at the market stage. In general, then, firms would use mixed strategies in prices and the analysis would become complex. reps and Scheinkman (1983) have considered this situation with a duonolv equilibm _ facing no threat of entry and concluded that the Cournot rium is the unique equilibrium outcome (with a specific rationing rule for unsatisfied demand). This gives support to the view of the quantity setting model as a reduced form of a more complicated multistage game Obviously this is not to say that our quantity commitment model is the reduced form of the capacity-price model when capacity choices are sequential. Relaxing quantity commitments enriches the model but the analysis becomes very complex and welfare results are hard to obtain. The results in our paper are an indication nevertheless that there is a reasonably large class of models where welfare is monotonic with respect to the number of otential entrants.

e have argued in a very simple model of sequential entry into an stry with one or more established firms that actual entry may decrease but social welfare never goes down otential entrants. hen there are many potential them out t in order to do so they olicies that lower the r our mar et with a single y a public authority

uctionw

is smal

nough with respect to the cost of tential entrant are ere ti=a--(w+r).

competitive outco is not to put a zero entr

imperfection in the market (n= 00) then it cost but this is not the case if, for whatevcz entrants is limited. The bottom line of our discursion is clear: the public authority has to be careful when desig ote competition when there is an incentive for firms ic entry dete activitir >. Our model may give some guidance towards policies to enlarge the potential entrants (n). For example, it is clear that it may pay at expand n only up to a point fi in our model. Potential entrants in e. ess of fi have no effect in the market outcome. Nevertheless, it is worth notic g that in the paper we have taken as given the entry cost F and the number of potential entrants n when, in some circumstances, policies to lower the entry cost will have an effect in the pool of potential entrants. The analysis of this phenomenon and some of the extensions mentioned in section 4 will have to await further research.

Proof of Lemma 2. Potential entrant j, given Z, the cumulative output of all firms up to j- 1, and anticipating the responses of subsequent entrants will respond in the following manner. If Z is larger than or equal to the limit output I: firm j will stay out. Let Z. solve Z +r(Z) = Y if Y >a/2 and be zero otherwise. Z0 =max{0,2Y-a}. Note that if Y>Z>Z,, then Z+r(Z)> Y and firm j will blockade entry by producing r(Z), as no subsequent firm would want to enter the market. In deciding whether to allow or prevent entry firm j has to compare the profits derived from allowing entry of all the remaining firms [using r(a)] with the profits derived from preventing entry. Allowing entry of all the remaining n-j firms [according to r( . )I firmj gets a maximum revenu.e of

eve

try it

gets

Z) be the unique y lar

y (if ot

X. Vioes, Sequential entry, industry structure and w

1686

et

n-j

SOlVe f (’ -j)(Z)

+ 2 =

Y if the solution is

sitive and be zero

otherwise. 2Y -(l + A,_Ja l-A,_,

eventing entry. For _j firm j is indifferent between allowing a will allow entry _j firm j will prevent entry and for 2 2, for k=O, l,...,n e r(Z). It is easily checked that Zk,. 1 + ces the remaining so that by producing r(Z) when 2 <2,-j firm j i ding to its Cournot firms to enter. Notice again that firm j will produce a best response function r( 9) whenever entry is not p pute Y,, first and We complete the proof by backwards inducti Of lyj+~;..*, ‘Y, erive Yj assumi:ig that firms j+ 1,. . . , n use any se (j=l,... , n - 1). Since Y is the output for which it Z)sO if ZZY and =r(Z) if ZcY and firm Menters only if it can make positive profits, 1) and let firms equals zero otherwise. Consider potential entra j+ l,..., n use any selection of vSj+l,. . . , V,, where the correspondences are given as in the statement of the lemma, then ds the best reply of firm j. Q.E.D. a 3. The interval [@Km), ii( Km)] contains at and m such that Y > ma/(m+ 1).

e show that Ym,n c _Ym,n + 1 for al $‘,Y, ,J, where ,J then Y# [_Ym get that ‘(let A,

ost one integer f.~~ral/ Y

This means that if

llows since A,< A,+l a

ernheim, 1984, Strategic deterrence of sequential entry 1 0 an industry, The Rand Journal of Economics 15, l-l 1. ty of natural monopoly: ee entry and the sustai and J.A. &heir&man. 1983 rand revisited by Cournot, in: Evans, ed., ell: Essays on industrial organization and regulation (Northlland, Amster ixit, A., J980, The role of invest nt in entry deterrence, Economic Journal, arch. ixit, A. and C. Shapiro, 1984, ixed strategies, in: LG. Thomas, ed., Strategic planning (Lexington 63-79. egic investment in an industry with a co etitive product market, Journal onomics, 115-l 3 1. are, 1985, A theory of market structure wit sequential entry, imeo. industrial organization

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R. and X.

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