DYNAMIC LIMIT A REFORMULATION
PRICING: ~
KENNE~ L. JUDD"~ BRgCE C. PETERSEN ~
Abstract This paper offers a new formulation of the well known dynamic limit pricing problem developed by Darius Gasklns. Criticisms of Gaskins' model center around the lack of a game t~eoretic formulation and the ad hoc fringe expansion equation. In this paper, the expansion equation is based on the importance of internal finance. In the differential game, the dominant firm controls price, thereby determining the available internal finance, and the maximum r a t e of growth, of the fringe. While the results of this study differ from those of Gaskins in a number of ways, dynamic limit pricing is found to be a feasible strategy. I.
INTRODUCTION
In this paper we examine the optimal pricing strategy of a dominant firm or a group of joint profit-maximizing oligopolists facing expansion by a competitive fringe.(1) This problem was first examined by Gaskins (197!) who labeled the pricing strategy of the dominant firm "dynamic limit pricing". We believe a new formulation is in order because of two developments: -I- Gaskins' model has received widespread application at the theoretical, empirical, and policy levels and -2- the strategic assumptions underlying his model have come under telling criticism in'recent years. We believe that our formulation handles the basic criticisms of Gaskins' approach yet continues to yield a rich set of predictions about dominant firm pricing strategy. Gaskins' model has become widely known and used by both economists and non-economists for further theoretical modeling,(2) empirical research in industrial organization,(3) and policy analysis.(4) Some reasons for this wide range of application cad be found in Scherer's (1980, pp. 236-243) excellent description of the model and its predictions. Scherer notes that the model "is compelling not only because it yields rich predictions, but also because these predictions appear to be consistent with a good deal of what we know about American industrial history." (p. 239) The criticisms of Gasklns' model center around the ad hoc nature of the fringe expansion equation and the game theoretic foundation. In particular, Gaskins' fringe expansion equation is not based on any maximization behavior on the part of the fringe. The lac~ of a game theoretic foundation is a problem common to almost all of the limit pricing models in the literature. It has been pointed out by J. Friedman (1979) and Milgrom and Roberts (1982) that under complete information, if established firms' pre-entry actions do not influence post-entry costs or demand, these actions cannot deter entry. The
capital investment decision is one example of s pre-entry action which can affect post-entry conditions.(5) We are aware of no previous explanations, however, for how price could deter either entry or fringe expansion under complete information. Our dynamic limit pricin~ formulation is based on the importance of internal finance (retained earnings) to fringe firms. In this respect our model is related to Spence (1979), in which internal finance plays the crucial role of the constraint on the expansion of later entrants into a new market. Spence, however, chose to examine capacity, not price, as the control variable of the first entrant. We set up the dynamic limit pricing problem as a deterministic, non-cooperative, differential game between the dominant firm and the competitive fringe. The dominant firm controls price while the fringe firms choose their retention ratio. Fringe firms retain all of their income for investment as long as it is in their long-run interest to do so. The connection between current price and expansion is then obvious - current price determines fringe earnings ~ i c h in turn determines the maximum possible rate of expansion of their capital stock. Today's pricing decision then does affect the future circumstances that dominant firms face. It is interesting to note that there are documented examples of dominant firms setting low prices to reduce the current earnings and internal finance of fringe firms, thereby slowing their rate of expansion.(6) The next section of the paper is a brief review and a critique of Gaskins' formulation. In section three we examine the key role of retained earnings as the source of finance for fringe expansion; we then derive the expansion equation in section four. In section five we solve our formulation of the dynamic limit pricing problem and compare the results with Gaskins'. If. GASKINS' NODEL Dynamic limit pricing differs from static limlt pricing in that it allows more general strategies on the part of dominant firms. Firms following a static limit pricing strategy either charge the short-run proflt-maximizing price and allow their market shares to decline, or they set price at the limit price and preclude all entry. Gaskins argues that there is no justification for this dichotomy| rather maximization of the present value of future profits entails a balancing between current profits and future market share. In Gaskins' formulation the optimal pricing strategy maximizes:
V = ~ 0
(p(t)-cd)q(p(t),t)e-rtdt
where V is the present value of the dominant firms' profit stream, p(t) is product price, c d equals average total cost of production (assumed to be constant over time), q(p(t),t) is the dominant firms' output, and r is the dominant firms' discount rate. Gasklns assumes that the level of dominant firms' current sales can be decomposed into additive unlvariate functions of price and
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time, such thnt:
<2>
q(p(t), t) = f(p(t))e Yt - x(t)
where f(p(t)) is the market demand curve, Y is the market growth rate, and x(t) is the output of thc competitive fringe which is assumed to be fixed at any po~nt in t~me. Th~ net effect of fringe expansion, ~, is to shift the dominant firm's residua] demand curve laterally. Gaskins argues that if fringe firms view current product price as a PrOxY for future price then expansion will be a monotonically nondecreasing function of current price. He then assumes that expansion is a linear function of current price, given by: x(t) = koeYt(p(t)-~)
x(O) - Xo, P~Cd
<3>
where ~ is the limit price, k0 is the response coefficient at time 0 (k>O), and x 0 is the initial output of the competitive fringe. Gaskins also assumes that the response coefficient k(t) = k 0 eu t is a growing exponential function of time. He argues that increasing disposable income should cause a proportional increase in the quantity of resources available to the fringe for investment in any particular market. Equations , <2>, and <3> allow the optimal pricing strategy of dominant firms to be solved analytically using the mathematic s of optimal control. The objective is to choose p(t) to maximize subject to <2> and <3>, where x(t) is the state variable.(7) The necessary conditions(8) for an optimal p(t) can be used to obtain a system of differential equations describing the time path of prices and fringe market shares. If w(t)=x(t)e "Yt is the normalized size of the fringe, the resulting system of equations is: w(t) = ko(P(t)-~)
~(t) -
- yw(t),
w(O)= Xo,
kO(P-=d) - r(f(p)-w(t)+f'(p)(p(t)-cd)) -2f'(p)
- f''(p)(p(t)-c
d)
<4>
+ yw(t) 9
<5>
Equations <4> and <5> define two possible optimal price trajectories, depending on the initial size of the fringe, its cost disadvantage vis-a-vis the dominant firm, and other factors. If the dominant firm is initially large, it will price initially above the steady-state level and lower it gradually over time, thereby causing Judd and P e t e r s e n
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the fringe to gain market share until the steady-state is reached. This is the strategy which is consistent with a number of corporate histories described by Scherer (1980). If the dominant firm is initially small, it initially sets price below the steady-state level and raises it gradually over time, thereby causing the fringe to lose market share until the steady-state is reached. For further details , we refer the reader to the original paper. The weak point i n Gaskins' formulation is that fringe firms (the entrants) are not treated as rational, maximizing economic agents. As Milgrom and Roberts (1982, p. 444) point out, this is co~mon to most of the existing limit pricing literature. In addition, a number of issues can be raised about the exact specification of the fringe expansion equation, ~(t) -koeVt(p(t)-p). One issue is the response coefficient, kO . A priori nothing is known about this parameter which is unfortunate since x(t), ~(t), and the steady-state values of market share and price critically depend on its magnitude.(9) A second issue is the justification for the response coefficient growing at an exponential rate u . Gaskins' justification, that increasing disposable income should cause a proportional increase in resources available to fringe firms in all industries, seems tenuous in an economy where new industries are emerging and competing for resources, some have matured, and others are declining. Another issue is why fringe expansion does not depend on the present size of the fringe, as well as price. One would expect that the larger the fringe, the greater ~, other things equal. Finally, is there any justification for a positive, much less a linear, relationship between fringe expansion and price? We return to these issues in section IV. Ill. INTFRNAL VERSUS EXTEP~NAL FINANCE Similar to Spence (1979) t h e availability of internal finance is the constraint on the rate of expansion of the fringe in our formulation of dynamic limit pricing. Internal finance, particularly for small firms, has been the dominant source of finance historically(lO) as well as during the post World War II era. We review below first the explanations for the dominance of retentions and then some relevant statistics pertaining to the different sources of corporate finance. Corporations may finance expansion with internal finance or with debt and new shares issues, sources of external finance. In terms of comparative dollar values, retained earnings are largest, debt next, and new share issues are quite unimportant. There are a number of explanations for this pattern of finance, including corporate income taxation, flotation costs, costs of financial distress, agency costs, and limited capital markets. These explanations are briefly discussed b e l o w . One important reason for the small amount of new share issues on the part of unregulated firms is the design of the corporate tax system. The United States and a number of other countries employ what is known as a "classical" tax system. Among the provisions of this system is that capital gains are taxed at the personal level at a favorable rate compared to dividend and interest income. A number Judd and Petersen
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of recent studies(ll) h~ve e x a m i n e d t h e c o s t of e q u i t y finance ( r e t e n t i o n s an~ new s h a r e i s s u e s ) u n d e r t h e c l a s s i c a l t a x s y s t e m . I n each study, retained earnings i s shown t o d o m i n a t e new s h a r e i s s u e s as a source of finance. The b a s i c ~ n t u i t i o n is that no t a x s a v i n g s o c c u r f r o m t h e i s s u e o f new s h a r e s , w h i l e t a x s a v i n g s do o c c u r when e a r n i n i s a r e r e t a i n e d b e c a u s e a d~v~dend t a x is avoided for a lower t a x on c a p i t a l g a i n s . Given the t y p i c a l s h a r e h o l d e r s ' margina] tax rate, the t a x a d v a n t a g e o f r e t e n t i o n s o v e r new s h a r e i s s u e s appears
to be quite large. Flotation costs are a second reason why internal finance is a lower cost source than external finance. This is particularly true for small issues of debt or new shares - and therefore especially relevant to fringe firms - because the transaction costs tend to be largely fixed costs. ]~e usual explanations given for the low debt-equity ratios employed by unregulated firms in the United States are the costs of financial distress and agency costs. Financial distress refers to the set of problems that arise whenever a firm has difficulties in meeting its principal and interest obligations, with bankruptcy being the most extreme form of financial distress.(12) Agency costs arise from the efforts of creditors of the firm to ensure that the firm honors its contractual obligations. Narginal agency costs tend to rise with the debt-equity ratio because stockholder and bondholder interests' become more diverse the greater the degree of financial leverage. Table I reports the average retention ratios for corporations broken down by asset categories for the last decade.(13) It is apparent that the percentage of income retained by small firms is very high - firms under I0 million dollars in assets retained on average approximately 80% of their income. We emphasize that these are average numbers. It is certainly true that many small firms in declining markets or in industries with limited investment opportunities retain little or no income. A sizable percentage of small firms, then, must be retaining virtually 100% of their income. TABLE I RATES OF RETENTION BY ASSET CLASSES, 1970-1979 (assets of classes in millions)
Under 1 82.6%
1-5 83,5%
5-10 78.9%
10-25 73.0%
25-50 67.6%
50-100 62.0%
100-250 50.4%
over 250 31.9%
Table I does not preclude the possibility that external finance is a large portion of total finance for some size categories. It turns out that this is not the case. New share issues accounted for only 8% of all new equity finance for nonfinancial corporations (a large part originating from public utilities) over the period 1970-79.(14) Furthermore the debt/capital ratio in manufacturing in 1981 was under 20% and varied very little across firm size.(15)
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IV.
THE EXPANSION EQUATION
The importance of i n t e r n a l f i n a n c e i s c l e a r l y a reason why c u r r e n t expansion of the f r i n g e (and f u t u r e o u t p u t ) i s a f u n c t i o n of current price. The g r e a t e r the p ( t ) e s t a b l i s h e d by the dominant f i r m , the g r e a t e r the a v a i l a b l e c u r r e n t i n t e r n a l f i n a n c e for the purchase o f c a p i t a l and the expansion of o u t p u t . The income of the f r i n g e a v a i l a b l e f o r expansion i s ( p ( t ) - c f ) x ( t ) , where c f i s the n o n - c a p i t a l c o s t s of p r o d u c t i o n up to the c a p a c i t y c o n s t r a i n t x ( t ) . ( 1 6 ) (We assume, as does Gaskins, c o n s t a n t r e t u r n s to s c a l e and a c a p a c i t y c o n s t r a i n t . ) The f r i n g e o b v i o u s l y w i l l not r e t a i n IOOZ of i t s income i n a l l time p e r i o d s e v e n t u a l l y i t w i l l c o l l e c t i v e l y want t o pay some d i v i d e n d s . Let u ( t ) be t h e f r a c t i o n of e a r n i n g s r e t a i n e d by t h e f r i n g e . (As we w i l l see i n t h e next s e c t i o n , u w i l l be the c o n t r o l v a r i a b l e of t h e f r i n g e . ) Then t h e expansion e q u a t i o n o f the f r i n g e can be w r i t t e n a s :
x(t) = [(p(t)-cf)x(t)].J.u(t)
<6>
where J i s the p h y s i c a l o u t p u t - d o l l a r v a l u e o f c a p i t a l r a t i o . ( 1 7 ) k u s e f u l way t o t h i n k about t h e expansion e q u a t i o n i s t h a t i f K(t) i s the d o l l a r v a l u e of the c a p i t a l s t o c k o f t h e . f r i n g e a t time t , then x ( t ) m K ( t ) J , and thus ~ ( t ) m K ( t ) J . K ( t ) i s j u s t t h e term i n b r a c k e t s i n e q u a t i o n <6>when u - 1. The above expansion e q u a t i o n c o m p l e t e l y excludes sources of external finance. The assumption of no e x t e r n a l f i n a n c e i s s t r o n g e r than n e c e s s a r y . Rather, what i s needed i s t h a t e x t e r n a l f i n a n c e t i n p a r t i c u l a r new share i s s u e s , be s u f f i c i e n t l y more c o s t l y than internal finance. The absence of d e b t f i n a n c e i n the expansion e q u a t i o n i s n o t a s u b s t a n t i v e l i m i t a t i o n . I f debt can be i n c r e a s e d by some f i x e d f i n i t e amount f o r e v e r y a d d i t i o n a l d o l l a r of new i n t e r n a l f i n a n c e , then a m u l t i p l i e r e q u a l t o the r a t i o o f (debt e q u i t y ) / e q u i t y could be i n c l u d e d w i t h o u t any change i n the r e s u l t s . We do n o t i n c l u d e debt f i n a n c e i n the expansion e q u a t i o n , but n o t e , h a t i t could be e a s i l y i n c o r p o r a t e d i n J . Before proceeding t o t h e s o l u t i o n , i t i s a p p r o p r i a t e t o compare our f r i n g e expansion e q u a t i o n with t h a t of Gaskins ! , x ( t ) m kneYt (p(t) -p--). Gaskins assumed a linear relationship between ~ and ~. Interestingly enough, our expansion e q u a t i o n has a linear r e l a t i o n s h i p between t h e s e v a r i a b l e s - q u i t e simply, f r i n g e income v a r i e s p r o p o r t i o n a t e l y with p f o r any x ~ O. I t i s also true, of c o u r s e , t h a t f r i n g e income v a r i e s p r o p o r t i o n a t e l y w i t h x f o r any p. Contrary t o Gssk/ns w f o r m u l a t i o n , t h e n , ~ i s a f u n c t i o n of x, as one might e x p e c t . Indeed, we can r e w r i t e e q u a t i o n <6> as a r a t e o f expansion: xCt)IxCt) (p(t) cf)Ju. We noted that Gasklns' response coefficient, k(t) - k n eTt , is completely unspecified. Something analogous to GaskinsV'k can be found in our formulation. :The partial derivative of equation <6> with respect to p(t), ~x(t)/~ p(t) - k(t) m x(t)Ju, represents the locally linear relationship between expansion and price, just as k did in m
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Gaskins' formulation. Our "response coefficient" depends on the current size of the fringe, the fringe retention rate, and the physical output-capital ratio. What is especially important is that the parameter J is knowable a priori - that is, for individual industries one could determine what the response coefficient is at any moment in time. It is apparent that our response coefficient will increase over time as long as ~(t) > 0. Our for'lulation does not, however, provide any economic justification for Gaskins' assumption that k(t) grows exponentially over time in every industry at some common rate u V.
SOLUTION
We shall determine the nature of the equilibrium in our dynamic game between the dominant firm and the competitive fringe. W e take an open-loop approach to this game. (See Intrilligator (1971) for a forma] discussion of the various equilibrium concepts.) This is a dynamic generalization of the usual price leadership model. In this game the dominant firm chooses a price path, p(t), and the fringe firms choose their reinvestment rate, u(t). Since all fringe firms have access to the same constant returns to scale technology, we can assume without loss of generality that each fringe firm chooses the same u(t). Equilibrium is any p(t) and u(t) such that each is a best reply to the other. That is, in equilibrium, the u(t) chosen by the fringe is optimal for them given the p(t) chosen by tile dominant firm and the p(t) chosen by the dominant firm is Optimal given the fringe's u(t),(18) Both players make their choices in order to maximize discounted profits, with the dominant firm taking into account its impact on fringe capacity. Following Gaskins' notation, we let f(p) be demand at t = 0 and x0 the fringe capacity at t = O. We assume that the before-tax interest rate is r > 0 and that market demand grows at the rate of ~ ) 0 . Recall that cf is the variable marginal cost for the fringe up to the capacity constraint x, where the absolute capacity constraint x can be increased by J units per dollar of profits, c d is the marginal and average cost of production for the dominant firm.(19) We assume that the fringe and dominant firms' costs are not too dissimilar. In particular cf + rJ" is assumed less than the dominant firm's monopoly price. This avoids the trivial case where the fringe firms are pushed out of the market even if the dominant firm acts like a monopolist. Recall that w is the fringe capacity expressed as a proportion of market size, that is, w ( t ) = x(t)e -?t. w is the state variable of interest to both players, the dominant firm wanting to keep it l o w and the fringe possibly wanting to increase it. The evolution of w is given by .
(p -
cf)~uJ
-
"fw
<7>
which is derived from the fringe expansion equation, <6>. The dominant firm's problem is Judd and Petersen
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Max f~
e~t(f(p)
- w)(p - c d ) e ' r t d t
p(t) s.t.
~, = (p - c f )
~,uJ - "yw
where F ~ m is the time at which the ga~e ends. Let ~ be the dominant firm's shadow price for w. By the Pontrya~in tlaximun Principle = rn + p - c d - ~u(p - c f ) J and p ( t ) i s c h o s e n t o maximize t h e c u r r e n t - v a l u e - T as the discount rate) H(w,p,~) = (p - Cd)(f(p)
<8> Hamiltonian (using r
- w) + n ( ( p , c f ) w u J - y w ) ,
<9>
implying that
0 = ( P C d) f' + f(p) - v + , u w .
Each fringe firm will maximize the present value of its net cash flow, taking prices as given. Since each firm is a price-taker, the fringe acts in the aggregate as a profit-maximlzing price-taker. Therefore, the competitive fringe collectively solves the problem
Max
f ~ eYt(p - c f ) w(1 - u ) e "rtdt
uCt) s [ o , 1 ] S.to
w = (P - c f ) w u J - y w
That is. the fringe sets its reinvestment ratio u such that the present-value of its profit stream is maximized. If l is the shadow .price for w from the point of view of a fringe firm. then its evolution is described by = r~ - ( p -
cf)
(1 - u ) - X ( p -
cf)uJ
<11>
and the decision rule for s fringe firm is I
~ >j-I <12>
.
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The fringe's decision rule is bang-ban!~ since both t h e payoff and equation of motion are linear in the control, u. While a complete analysis of equilibrium is beyond the scope of this paper (see. 3udd and Petersen (1984) for the complete analysis), the crucial features are immediately aPl~arent. First, price must drop to the f r i n ~ fJrm~' foals-run average cost, cf § r J-l, at some f~nite ti:~lu. If th~s did not occur, then t h e return on Investment would always exceed the epportunJty cost of investment for the fringe firms, implYinz that they would continually fully reinvest their profits, (p.- cf)w. Therefore, w = (p - cf)wJ - yw > (r - y )w since p > cf + rJ-land w would grow without bound s~nce r > Y , which would be absurd s~nce unbounded srowth in w would ultimately cause fringe supply to exceed total possible demand. Second, in the long-run steady-state, the fringe is only reinvesting enough to maintain market share implying u < 1 in the long-run. Third, at the first time that price equals fringe firm average cost, the fringe firm Capacity is less than the long-run fringe size. Otherwise, the fringe would drop discontinuously from full reinvestment to the partial rate of reinvestment observed in the steady-state, causing the dominant firm to discontinuously increase the price according to its decision rule, (I0). Such an increase however would be inconsistent with attaining the long-run price at that time. Therefore, the long-run price is attained at some finite time and the fringe continues to grow thereafter. Since p = cf-l- rJ -l after some finite time, we next examine the _ approach to this point. Before that time, price exceeds c + urJ"l and u = I since average future price exceeds average cost. ~ e n = i, the present-value Hamiltonian for our~formulation is: H(w,p,n) " (p(t) - cd)(f(P) - w(t)) e-rt + n(t)e-rt[(p(t) For purposes of comparison, the Gaskins' formulation is given by:
HO(w,p,z)
- (p(t) - Cd)(f(p)
- cf)w(t)a
- yw(t)]
present-value
Hamiltonian
- w(t))e -rt + z(t)[ko(P(t)
for
- P) -.Tv(t)]
The first term in either Hamiltonian is the present value accruing from current sales while the second term reflects the effect of current entry on future profits. Differences between the two Hamiltonians occur only in the second terms and arise because of the different expansion equations. Note in particular that the value of the second, dynamic, term is proportional to the value of w(t) in our Hamiltonian but not in Gaskins' because our rate of expansion, w(t), is proportional to w(t). This implies that our game-theoretic equil~brium analysis yields higher initial prices when fringe shares are small. This is expected since maximizing the Hamiltonian with respect to price involves a balancing of the first and second terms, and the value of our second term becomes small as fringe output becomes small. Since our terminal price is lower, and price attains this lower value at some finite time, the average rate of decline in Judd and Petersen
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price whr the, initial w ~s small must be greater in our game analysis th:m in Ca:d.i,~s' model. This is intuitive since the reduced long-run effectiv~nos.~: of limit pricing In t h e game analysis encoura:.'es the dordnant fir,:D to be more a[Igressive in acquiring profits throu~,5 h~;h prices in the initla] states when it has a ~,reater market share. The necessary condition,s for a maximm~ value for tJ,e dominant fir~.'s problem for e i t h ( , r formulation] c~nJ br written as a system of differential eq,,at~ons ~n p(t) and w(t). The systt,:, of equations for Gas]r formulation was -,iver~ in Sectior, ]I, equations <4> and <5>. The ~ and ~ equations for our mode] are determined by differentiatin~ the price equations with u = 1. and are given by(20)
w(t) - (p)t) - cf)w(t)J - Yw(t)
(cf - Cd)W(t)J - (r - Y)[f(p) - v(t) + f (p)(p(t) - cd] + Yw(t)
kt)
-2f'(p) - f " C p ) C p ( t )
A comparison of
- c d)
Gaskins w ~(t) equation with
only the numerators differ. The d i f f e r e n c e s response c o e f f i c i e n t i s endogenous and depends Therefore Gaskins' analysis is very similar to However, as a l r e a d y n o t e d u cannot a l w a y s be one irrational for the fringe firms. The c r u c i a l
ours indicates t h a t arise because o u r linearly on w ( t ) . o u r w = 1 phase. s i n c e t h i s would be difference is that
rational firms will reduce their growth rate when their investment needs become smaller than available revenues. The phase-plan portraits of the ~ and ~ equations for Gasklns' formulation and our formulation appear in Figures I A a n d IB below. In both cases the ~ = 0 and ~ = 0 loci divide the phase-plane into four regions. The steady-state values for p and w would be ~ and if the fringe firms always fully reinvested earnings. A major difference between the two figures is that in Gaskins' formulation, the slope of the ~ = 0 locus is equal to 7 , the market rata of growth, while in our formulation the ~ = 0 locus is a horizontal line. This means that in Gaskins' formulation, the steady-state price which is approached only asymptotically, will always exceed fringe marginal cost unless the market is not growing. Gaskins fends this to be a "disturbing result" (p. 3]7) and provides numerical examples which show rather large deviations of price over marginal cost. We find instead that the steady-state price always equals fringe marginal cost and that it is reached in a finite period of time. This is represented in IB by the trajectory ABC. Initially. our solution has price and w following the flows in Figure IB along A-~. However, when price falls to the steady-state price, paS , the Judd and Petersen
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pric(: dec]ine c(~oses, a~' mov:in~" f r o m B t o ~.
w g r o d u a l l y ~ncren~em, as
indicated
L. w~O &
P
.3
-I
A
w F i g u r e IA
Judd a n d P e t e r s e n
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Gaskins Phase Plane
170
by
F
r
L
~r j ~ - p ~ .|
Cf*YJ
=0
&
=p
A
w
Figure IB
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Dynamic Game Phase Plane
171
CONCLUSION We have reexamined the usual approach to dynamic limit pricing. Whereas Gaskins assumed mechanical expansion behavior by the fringe, we have profit-maximizing fringe decision-malting. Dynamic limit pricing is still valuable to the dominant firm however when we make the assumption that fringe firms rely mostly on retained earnings for growth, an assumption which casual examination of small growing firms seems to validate. In examining the outcome of competition between the dominant price-setting firm and its fringe competitors we find features both similar to and different from the original Gaskins analysis. In particular, while prices decline here also, they decline more rapidly and attain a lower-long run level. Also, we find prices to be higher than in Gaskins' model when the fringe is small. The reason for both differences is that the rationality of the fringe reduces the value of dynamic limit pricing and encourages the dominant firm to price higher when it has a large market share. We believe that this is a first ste~ towards 8 more complete analysis of limit pricing. Further development will hopefully provide guidance as to the welfare implications of dynamic limit pricing and appropriate policy responses.
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Ftr,TNOTF~q This paper has greatly benefitted from many suggestions and comments by blark Sattertbwaite, Ron Braeutigam, Richard Caves, Steven Hatthews, and John Panzar, and also from the research assistance of Scott HcShan. We also gratefully acknowledge the financial support of the National Science Foundation and the J.L. Kellogg Graduate School of Management at Northwestern University (Ken Judd) and the Slonn Foundation. *
9-~ Department of Managerial Economics and Decision Sciences, J.L.Kellogg Graduate School of Management, Northwestern University, 2001 Sheridan Road, Evanston, I11Jnois 60201. ~@ Department Illinois 60201. (I)
(2)
(3)
(4)
(5) (6)
of
Economics,
Northwestern
University,
Evanston,
This problem is of considerable interest because most concentrated industries consist of a large number of fringe firms alongside one or more dominant firms. For some examples of highly concentrated industries with a large number of fringe firms, see Scherer (1980, p. 62). Some examples from Scherer of industries in 1972 with four firm concentration ratios of 90 or greater with a large number of fringe firms include flat glass (11), cereal breakfast foods (34), turbines and turbine generators (59), and electric lamps (103). To cite but a few of the theoretical extensions of Gaskins' model, Brock (1975) includes technological progress, Lee (1975) adds non-price policies and learning by doing, DeBondt (1977) includes scale effects and Encaoua and Jacquemin ~1980) incorporate non-price policies. At the empirical level, Gaskins' model clearly demonstrates the the possibility of a feedback relationship between price and market structure - the choice of a pricing policy affects market share over time, as well as market share determining pricing policy. While the vast majority of industrial organization studies continue to be cross-sectional, a few recent studies are dynamic, and more are likely to follow. Brock (1975), for example, estimates Gsskins' model econometrically for the computer industry; w h i l e ~rtin (1979) includes a concentration equation based on Gaskins' model in a system of simultaneous equations. Martin finds that a dynamic specification of concentration is critical to the specification of the profitability equation. Gaskins' model has seen application at the policy level, including frequent citations in law Journals. It appears that a number of lawyers as well as economists interested in antitrust issues are familiar with the madel, inclnding Dunfee and Stern (1975), Easterbrock (1981), and Kaplow (1982). For an analysis of capital investment as a deterrent to entry, see Dixit (1980). One recent example, dealing with I.B.M. in competition with plug compatible manufactures, is discussed by McAd&~s (1982).
Judd and P e t e r s e n
17~
(7)
The lhmd]toninn fur dclslcins' modcl is given by R = (p(t)-cd)(f(p)eTt-x(t))e-rt+z(t)koeYt(p(t)_~)
(8)
where z(t), the costate variable, is the shadow price of an adc]itiona] unit of rival entry at any point in rig, . The first term in the equation is the change in present value accruin~ from current sale.~:. The second term is the product of z(t) and ~(t) which is the effect of current entry on future profits. The necessary conditions in Gashins' formulation are: (1)
x*(t) = koeYt(p*(t).~),
(il)
z*Ct) - - ~H -
(ill)
(9)
(i0) (ii) (12)
(13) (14) (15)
(16) (17)
~H ap(t)
x (O),'Xo;
(x * (t),z * (t),p * (t),t);
(p*(t)-~d)e-rt ,
llm
z (t)=0;
t~ )e_rt§ " ((fCp)e ~ t -x * (t))+(p , (t)-c)f'
t -0.
Gaskins provides a numerical example at the end of his paper for a given demand curve and a given response coc fficient. We recomputed the steady-state values of market sh~.re and price, along with the price trajectories for a range of response coefficients and demand parameters. We find that the results are very sensitive to the selection of the response coefficient and the demand parameters. Plausible results for any given demand curve can be obtained only by experimenting with the selection of k See Butters an~ Lintner (]945) for a review of the historical importance of retentions as a source of finance for expansion. For a review of the studies see Auerbach (1983). There are several possible types of costs arising from financial distress short of bankruptcy including lawyers' and accountants fees, lost sales, higher costs of production, reduced output, foregone or delayed investment, higher financing costs, and general disruption of firms activities. For an in-depth discussion, see Haley and Schall (1979, p. 377). Retention ratios for corporations by asset size appear in the Internal Revenue Service, Statistics of Income, Corporate Income Tax Returns, 1970-1979, Table 5. See King and Fullerton (1984), Table 6.15. After corrections for inflation, King and Fullerton (1984, p. 239) report a debt/capital ratio of .198 for 1981. Furthermore, the debt/capital ratio varies little across firms in different size categories. This type of cost function is commonly used in theoretical work in industrial organization. See for example Spence (1977) and Dixit (1980). It should be noted that I/(Jp), not I/J is the conventional capital-value of output ratio. Since ~(t) is expressed in physical units of output, not in dollar value of output, J must also be expressed in physical units of output per dollar of
Judd a n d P e t e ~ s e n
174
(is)
(]9)
(2o)
capital per period of time. This p r e s e n t s no p r o b l e m f o r applicotions as long as the distinction between I/(Jp) and I/.I i s kept i n mind. h:; a s examl.]C , s u p p o s e t h e a f t e r - t a x income o f the friars is $]5,r and p ffi $lO,(g}O (e.g. output is auLo,~obiles) and ]/Jp = 3 (the average value in the U.S.), then J = I/.r ,and therefore ~ = 500. Tn ex~,~li n i n~; t h i s o p e n - ] oop eqn~ 1 ~ hr~ urn, w<, a r o imld j cJ f l y assu,nin[; that at some initial time the players sir~ultaneously make irreversible decisions concern~n,2 p(t) and u(t). Nhile this assumption is somewhat unrealistic, it is the only known tractab]e approach. c may be i n t e r p r e t e d i n a number o f ways. I f i n v e s t m e n t i s irreversible, but t h e d o m i n a n t f i r m ' s s a l e s a r e alwa.vs g r o w i n g , then the irreversibility is not binding and c d is the long-run marginal cost, i.e. 0 short-run marginal cost plus the opportunity cost of capital. If the dominant firm's sales is declining, then cd is the marginal variable cost, since the capita] costs sr~. stink and unrecoverable. The necessary conditions for a maximum value of the dominant firm's problem generates the simultaneous differential equations:
(i)
v*(t) = (p*(t)-~f) g e ( t ) u * ( t ) J
(li)
~*(t)
=
(pe(t)-cd)e-rt
-~*(t)
+ n*Ct)Cp(t)-cf)Ju(t)
This systen o f differential equations can be converted into autonomous system in the paper by eliminating ~ (t).
Judd and Petersen
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the
|~ibliogra phy Auerbach, A.J 9 " T-aX ation, Corporate Financial Policy and the Cost of Capital," Journa3 of Economic Literature ~ 21 (Sept. 1983): 9f)5-940. Brock, G., The U.S. Co~Tuter Industry, Cambridge, M~ss.:Ballinger, 1975. Butters, J.K., and J. Lintner, Effects of Federa] Taxes on Grow~nq Enterprises, Harvard University, 1945. De Bondt, R. "On the Effects of Retarded Entry," European Economic Review 8 (August 1c~77): 361-71. Dixit, A. "The Role of Investment in Entry-Deterrence," Economic Journal, 90 (March 19~0): 95-105. Dunfee, T.~. and L.W. SLern, "Potential Competition Theory as an Anti-Herger Tool under Section 7 of the Clayton Act: A Decision Model," Northwestern University Law Review, 69,(1975): 821-871. Easterbrook, F.ll., "Haximum Price Fixing," The University of Chica~o Law Review, 48 (19S1): 386-910. Encaoua, D. and A.P. Jaequemin, "Degree of Monopoly, Indices of Concentration and Threat of Entry," International Economic Review, 31 (1980) 87-105. Friedman, J., "On Entry Preyenting Behavior and Limit Price Models of Entry," in Applied Games Theory, eds. S.J. grams, Warzburg, Vienna: Springer-Verlag: 236-53. Gaskins, D.V:. "Dynamic Limit Pricing: Optimal Pricing under Threat of Entry," Journal of Economic Theory 3 (September 1971): 306-22~ Haley, C.W., and L.D. Schall, The Theory of Financial Decisionst 2nd ed. New York: McGraw-Hill Book Company, 1979. Ireland, N.J., "Concentration and the Growth of Market Demand: A Comment on Gaskins' Limit Pricing t~odel," Journal of Economic Theory 5 (1972) 303-305. Internal Revenue Service, Statistics of Income, Corporation Income Tax Re%u~ns, U.S. Government Printing Office, Washington, D.C., 1970-1979. Intrilligator, M.D., Mathematical Optimization and Economic Theoryt Prentice-Hall, Englewood Cliffs, N. J., 1971. Kamien, H.I., and N.L. Schwartz, "Limit Pricing and Uncertain Entry." ~conpmetrica 39 (~'~y 1971): 441-455, Judd, K.L. and B.C. Petersen, "Dynamic Limit Pricing and Internal Finance," Discussion Paper No. 603S, ~ e Center for Mathematical Studies in Economics and Management Science, Northwestern University. Kaplow, L. "The Accuracy of Traditional Market Power Analysis and a Direct Adjustment Alternative," Harvard Law Reviewp 95 (June 1982): 1817-1848. King, M. and D. Fullerton, The Taxation of Income from Capital. The University of ChiCago Press, 1984. Lee, W., "Oligopoly and Entry," Journal of Economic Theory, 13 (August 1975): 35-54. I,~rtin, S., "Advertising, Concentration, and Profitability: The Simultaneity Problem," Bell Journal of Economics r I0 (Autumn 1979):639-47. McAdams, A. "The Computer Industry" in W. Adams, The Structure of Judd and P e t e r s e n
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Amcric:in ]ndustr~,, sJ xtlx edition, .~Ic,~iJ llnn Pub]ishlng C~)., 1982. M~Igro:.' F. and J. Roberts. "LJrait Pricin8 and Entry Under Incomplete Information: An Eqt.ilibrium Ann]ys]s," Economot.r$ca, 50 (Plarch, ]9,~32): 44"~-459. Scherer, F.~l., Industrial H:irl:et Structure and Econo:,sic Performance, P,nnd ,~Ic~,'u]]y Pub]ishin~ Co., ]98'). Sp~nce, ~!. "Entry, ~nvestmc.nt and O ] ~ o p o ] J s t i c P r i c i n g , " Bell J o u r n a l of Economics, ~ (Autumn 1977): 5 ~ - 4 4 . Spence. A.~I., "Investr~ent Strategy and Growth in a New ~nrket," I~ell Journal of Econor~ics. I0 (Sprin[; 1979): 1-10.
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