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Dynamic limit pricing: Optimal pricing under threat of entry
JOURNAL
OF ECONOMIC
THEORY
Dynamic
3, 306-322 (1971)
Limit Pricing: Optimal under Threat of Entry DARIUS W.
Received
JR.
GASKINS,
July
31,
Pricing
1970
INTRODUCTION
This paper attempts to determine the optimal pricing strategy for a dominant firm or a group of joint profit maximizing oligopolists faced by potential entry into the product market. The models explored here have broader applications, but we limit our discussion to a dominant firm able to deduce a residual demand schedule through knowledge of the output of a well-behaved competitive fringe. Contemporary writings on this subject for the most part contend that the dominant firm will maximize its present value by either charging the short-run profit maximizing price and allowing its market share to decline or by setting price at the limit price and precluding all entry.l A firm practicing short-run profit maximization would have to ignore continually the reality of induced entry. Conversely, a firm charging the limit price has to be convinced that its prevailing market share is optimal. There has been no analytic justification for this strategic dichotomy, and intuition suggests that the optimal strategy would entail a balancing between current profits and future market share. The basic premise of this study is that the rate of entry of rival producers into a particular market is a function of current product price. Mansfield has shown that the variation in rate of firms entering or exiting an industry is positively correlated with the level of industry profits.2 It follows that a dominant firm with high current price and profit levels is sacrificing some future profits through erosion of its market share. The dependence of future market share on the current price level means that the dominant firm’s pricing strategy can only be determined in a dynamic framework.
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PRICING
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THE BASIC MODEL
In general terms, the optimal pricing strategy will maximize the present value of firm profits, as given by Eq. (1) below:
where V = present value of firm’s profit stream, p(t) = product price, c = average total cost of production (assumed to be constant over time), @‘p(8), 1) = dominant firm’s output, Y = dominant firm’s discount rate. The specific functional dependence of saies on time is determined by the nature of the entry phenomenon. We assume that the level of the dominant firm’s current sales can be decomposed into additive univariate functions of price and time, respectively,
wheref(p) = initial demand curve and x(t) = the level of rival sales. The residual demand curve q(p(t), t) at any specific instant is found by subtracting the output of the competitive fringe from the total market demand. Equation (2) indicates that the net effect of rival entry into the product market is to shift the dominant firm’s residual demand curve laterally. This would be the case if rival producers’ short-run supply Curve is completely inelastic. 3 The rate of entry of rival producers [9(i)] is surely determined by their expected rate of return. If potential entrants view current product pride as a proxy ‘for future price, the rate of entry will be a monotonically nondecreasing function of current pribe. We assume that this relationship between rate of entry and current price is linear, as shown by Eq. (3)
&) = k[p(t) - PI,
$9 = x0 2
B 3 c,
C%
p = limit price (a constant),5 k = response coefficient > 0, x0 = initial output of the competitive fringe. The limit price p is defined in this model as that price level at which net entry is equal to zero. The model implies therefore that pricing below p will cause negative entry or exit from the product market by rival producers. The difference between the limit price and the dominant firm’s average total cost is a measure of the cost advantage enjoyed by the dominant firm. It seems reasonable to restrict the limit price to be greater than or equal to the dominant firm’s average total cost. While there may be examples of dominant firms with cost disadvantages, their fate is obvious with or without optimal pricing.
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To expedite the analysis further, we assume that the dominant firm’s initial demand schedule f(p(t)) is d ownward sloping and twice differentiable with respect to output. With these simplifying assumptions, it is possible to determine the optimal pricing strategy analytically using the mathematics of optimal control. In the language of modern control theory, we wish to maximize the functional V = Jrn [p(t) - c][f(p)
- x(t)] e-rt dt
(4)
x(0) = x0 .
(3)
0
subject to w
= &J(t)
- PI,
In the control theory framework, x(t) is the state variable and p(t) is the control variable. We may derive necessary conditions for the optimal path by using Pontryagin’s maximum principle. The Hamiltonian for this problem is given by H = (p(t) - 4CfW
- x(t)) et
+ ~0) JWQ
- PI.
(5)
The adjoint variable z(t) appearing in the Hamiltonian is equal to a V/&X (t) and can be interpreted as the shadow price of an additional unit of rival entry at any point in time.‘j The first term of Eq. (5) (the integrand of Eq. (4)) is the change in present value accruing from current sales. The second term which is the product of &(t) and z(t) reflects the effect of current entry on future profits. Intuitively, maximizing the Hamiltonian with respect to p(t) involves balancing the flows of present value from present and future sales. The maximum principle states that for a maximum V to exist, it is necessary that there exists a z(t) such that: (i)
2*(t) = k@*(t) - p),
x*(o) = x0 ;
(ii)
2*(t) = - +$ (x*(t), z*(t),p*(t),
t);
= (p*(t) - c) e-Tt, ‘,i% z*(t) = 0; + (iii)
H(x*W,
z*(t>,~*tt),
t> = ,yy ~ . . $(-x*(Q,
z*(t), p(t), 1).
The superscript * denotes variables along the optimal easily shown that condition (iii) implies
trajectory.
It is
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PRICING
as long as where the prime denotes a/Q.
(P - C)“?(P) < 0,
2f’CPlf
This basic inequality will always hold if profit is a smooth concave function of price along the initial demand curve. We assume that such is the case. This assumption which assures the concavity of H(x(c), z(t),p(t), t) with respect to p and x Is sufficient to guarantee the existence of an optimal path.’ The necessary conditions produce the two simultaneous ordinary differential equations: i*(t)
= k(p"(t)
- li>,
x*(o)
i*(t)
= (p*(t) - c) e-Tt,
z*(t)
= (x*(t)
lim
*&KC
= x0 ) z*(f)
(51 0,
=
63
where -f(P)
-
(P*O)
-
c)f’(~*N
e-r*
(71
k
We can eliminate the adjoint variable z(t) from these equations and write the necessary condition as the simultaneous differential equations in p(t) and x(t): k(t) = k(p(f) - a>,
40) = x0 ,
9(t) = W - 4 + dx -f(p) - (P -V’(P) - (P - 4fYP)
(8)
cIf’(~)l .
(93
These two equations generate a family of trajectories in the x - p plane. Unfortunately the terminal condition lim z*(t) = 0, as t + co, cannot translated into an explicit condition of x(t) or p(t) and therefore it is not immediately obvious which of these trajectories satisfies (satisfy) the necessary conditions. We argue from a phase-plane portrait of Eqs. (8) and (9) that there is a unique trajectory meeting all the necessary conditions. Figure 1 presents p(t)
FIG.
1.
Phase plane
for the basic model.
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the locii of 9 = 5 = 0 in the x - p plane. These two locii divide the x - p plane into four distinct regions. It is apparent that the intersection of these locii denoted (a,$) is a saddlepoint. Any trajectory entering or originating in regions II or IV will remain in that region. Further, x andp will increase without bound along any trajectory in region II and decrease without bound along a path in region IV. Inspection of the dominant firms objective function V(p(t), x(t), t) indicates that no trajectory in region II or IV could be an optimal path. In region II ever increasing x(t) and p(t) would generate an increasing stream of losses after some point in time. Similarly, in region IV a steadily decreasing p(t) and x(t) will eventually generate an ever increasing stream of negative profits (i.e., when p(t) < c). We now verify that any path reaching the equilibrium point (a,$) will satisfy the terminal boundary condition lim z*(t) = 0, as t + co. At (g, $) both 3 and $ are zero. Equation (7) indicates that at this point z*(t) = (a -f(J) -f’($)($ - c) e-Tt)/k is an exponential function of time that declines to zero, therefore a trajectory reaching this equilibrium point will trivially satisfy the terminal boundary condition. Since the functions c?(x,p) and &,p) are both continuous and continuously differentiably in the x - p plane (2f’(p) + f”(p)(p - c) < 0), there will be only two trajectories (labeled (1) and (2) in the figure) which terminate at (2, $). All other trajectories originating in region I or III will eventually enter either region II or IV. The optimal pricing strategy will therefore be to move along either trajectory (1) or (2) depending on the original size of the competitive fringes output x0 . For x,, < 9 we conclude that the dominant firm will maximize its present value by gradually lowering product price towards the limit price p. Such a pricing policy will induce rival entry and continually reduce the dominant firm’s market share. Conversely, when x0 is greater than 9 the optimal strategy is to price below the limit price, continuously driving out rivals. We are able to prove that the optimal price level will always be below the short run profit maximizing price at every point along the optimal path. If the dominant firm has been moving along the optimal path, instantaneous profit is given by the function T(P) = (p(t) - W(P)
The myopic profit maximizing p
712
= f’!pm)(pm(t)
- X*(o).
(10)
price pm(t) is the solution to the equation -
c> + f(Pm)
- x*0>
= 0.
DYNAMIC
LIMIT PRICING
We have previously seen that maximization point along the optimal path implies that -z(t)
= (f’(p*)(p*(t)
31n
of the Hamiltonian
- c) +f(p*3
at every (12)
- x*(t)> +lk
where z(t) the shadow price of additional rival entry is necessarily negative. Ry prior assumption a2r(p)/a2pnz < 0 and therefore, Eqs. (11) and (12) are both satisfied only if pm(t) > p*(t).
COMPARATIVE
STATICS
AND
DYNAMICS
OF THE OPTEMAL
TRAJECTOKY
We now attempt to establish the effects of variation in the model parameters, k, c, 3, r and x0 on the optimal path. The equilibrium level of rival output 2 is seen to be an important characteristic of the optimal ricing strategy. Setting Eqs. (8) and (9) equal to zero and solving simultaneously for 2 we find that
The market share of the dominant firm at any point in time is s(t) = (f(p) - x(t))lf(p). It is clear that as p(t) and x(t) approach $ and $ respectively, s(t) will approach an equilibrium t. The optimal strategy in fact may be viewed as pricing to achieve a long-run optimal market share equal to:
Differentiating
Eqs. (13) and (14) we establish the following results:
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The first two of these conditions jointly indicate that the dominant firm will price to allow less entry as its cost advantage (p - c) increases. We argue that p will vary directly with the average level of rival costs. A very large ( p - c) will justify pricing to rationalize the relevant industry by driving out “inefficient” producers (2 < x0). An important practical case is the dominant firm which enjoys no cost advantage over rival producers. We model such a case by setting p = c. In this case Eq. (13) indicates that in response to optimal pricing by the dominant firm the output of the competitive fringe will asymptotically approachf(c), the total industry output. The dominant firm in this case prices itself out of the market in the long-run. While it is acknowledged that our model will lose its validity at some point as the dominant firm’s market share declines the conclusion remains that dominant firms with little or no cost advantage decline if they strive to maximize their present value. Condition (c) indicates that as the dominant firm’s discount rate increases it will sacrifice a portion of its long-run market share. A higher discount rate indicates that future profits become relatively less important. Condition (d) indicates that the more rapidly rivals respond to price signals the larger will be the dominant firms long-run market share. This result is at first glance counter-intuitive and would seem to question the efficacy of any public policy designed to increase the responsiveness of potential entrants. We demonstrate below that a concomitant result is that an increase in k will necessarily lower the optimal price trajectory at least in the short run. Because r, k, and c don’t affect p the change in the dominant firms equilibrium market share in response to variation in these parameters will be of the opposite sign of the change in 9. A change in p however will affect f( p) as well as 2 and therefore complicate the determination of the sgn df/dj. Condition (e) indicates that the sign of df/d@ depends on the sign of an expression which contains only one term which can be negative, i.e., -f”( p)( p - c). W e conclude that if either the curvature of the demand curve near p is relatively small or that the dominant firm has a small cost advantage the dominant firms long run market share will increase as p increases. Condition (f) merely restates the obvious point that the long-run optimal price level and market share are independent of the initial output of the competitive fringe. We now attempt to establish the effect of variation of the model parameters on the optimal trajectory away from the equilibrium point. Because our model still includes the general functionf(p) it is not possible to find the explicit form of the whole optimal path. The phase-plane can be
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used however to obtain certain comparative dynamic results without further model specification. Figure 2 indicates the effect of a positive increase in the response coefficient k on the phase portrait. We see that the new equilibrium
FIG. 2. Comparative dynamic effect of a change in the response coefficient.
point (9, , p) has been moved to the left in accordance with condition (d) above. It is easily shown that the slope of trajectories in region I will be increased at every point by an increase in k. The slope of any trajectory in the x - p plane designated G(x, p) is equal to
GA!&=“= By differentiation
52
k(p - 4 + r(x -f(p) - ~‘(P>(P- 4) k(p - PX-V’(P) - ~“(P)(P - 4) = 05’
we find that
dG -a?=
-4x - f(P) (P - P)(-2f’(P>
- f’(P)(P -f”(P)(P
- 4) - 4) .
Recalling from Eq. (7) that the shadow price of entry z*(t) is proportional to the quantity (X -f(p) --f’(p)@ - c)) CZ-?~,we argue that the rmmerator of Eq. (16) is positive since an additional unit of rival entry at any time will surely lower the present value of the dominant firm. previous assumption (-2f’(p) --f”(p)(p - c)) > 0, and we conclude that sgn dGjdk = sgn(p - p). If we examine the situation in region I of the phase plane it is apparent that the new optimal path (2) must lie below the original optimal trajectory (1). At points along the old path the slope has been increased by the increase in k and therefore any trajectory passing through a point on (1) must move above (1) as x increases.No such path could possibly reach the new equilibrium point (gz , p). In a similar fashion, we can
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GASKINS
demonstrate that if the original optimal path was in region III (i.e., x0 > 2,) the optimal trajectory will move downward as k increases. We have shown that the optimal trajectory in the x - p plane will always be lowered by an increase in the responsiveness of rival producers to price signals. While it follows that the initial portion of the dominant firm’s optimal price trajectory will be lowered by increasing k we are not assured that the whole time path p*(t) has been lowered.g By similar arguments we are able to establish the following comparative dynamic results:
(h) g(t) (i)
$$
> 0.
if fn(p)+
0:
(t) < 0, 0
Each of these conditions indicates the short term effect of variation of a particular model parameter. In addition, since variation in x0 the initial output of the competitive fringe only affects the starting point of the optimal path, the last condition holds for the whole time path of p*(t). Unfortunately, it is not possible to assess the impact of small changes in @ on the initial portion of the optimal path in this general model. We have demonstrated elsewhere that for a linear demand curve the response of the optimal price at any point in time is such that: (j)
--dP* (t) < 0 @
(see Ref. 10).
Conditions (h) and (j) jointly indicate that the dominant firm will lower its price in response to an increase in its relative cost advantage. We have the somewhat ironic result that a policy which lowers existing barriers to entry (e.g., mandatory licensing of a vital patent) will raise prices in the short run if dominant firms price according to this model. The economic justification for this seemingly perverse result is that the optimal long-run market share of the dominant firm is increased as its cost advantage increases (we previously saw that di/d( p - c) < 0). As the dominant firm becomes increasingly efficient it will strive to drive out more of the inefficient producers and it does this by lowering the product price in the short run. We observe however that if the change in the cost advantage is due to an increase in p, the long run price will be increased.
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The most sanguine conclusion from this model remains that dominant firms with little or no market power ultimately decline. We fmd much to our chagrin however that growth of the product market destroys this consoling result.
MARKET
GROWTH
MODEL
The American economy with its growing population and increasing per capita income is not consistent with the static product market assumed by the basic model. More realistically dominant firms are faced by markets which are growing as the economy expands. We now explore a model in which the output of the dominant firm is assumed to be of the form
where y = the market growth rate. It is postulated that product demand is increasing exponentially. This particular growth model which has the property that the price elasticity at a given price remains constant over time seems to be consistent with steady growth of disposable income rather than the secular growth of any single product market.ll Consonant with growing disposable income, we also assume that the entry response coefficient ii is now the growing exponential function of time k(t) = koeYt.
WJ
An increase of disposable income should c+se a proportional increase in the quantity of resources available to potential entrants for investment in any product market. Under this growth model, the dominant firm attempts to maximize the present value of its stream of profits
subject to 2(t) = k,,eyt(p(t) - j?), The necessary conditions differential equationP3:
for a maximum
k*(t) = k,eYQ*(t) - j?), s*(t) = (p”(t) - c) et, 6421313-7
x(0) = x, . V generate the simultaneous
x”(0) = xg ) p+z z”(t) = 0,
421) (22)
316
GASKINS
where z*(t) = (x*(t) e-yt - f(p*)
- (p*(t) - c)f’(p*))
ecrt/k.14
This system of differential equations can be converted into an autonomous system by making the substitution w(t) = x(t) e-Yt and eliminating z(t). The resulting equations are
w(O) = x0 ,
GO>= ko(dt) - i-4 - w(t),
- ~0) + I’ l%t>= ko(P - 4 - WP) (--2f’ho - f”mP(t)
- cl>+ w(t) . - 4)
(23)
(24)
The phase plane portrait of these equations is shown in Fig. 3. It is clear from the figure and our previous arguments about the phase-plane portrait of the basic model that there is a unique optimal path which will take one of four possible shapes determined by the values of & and w(0) = x0 . P(t)
G
FIG.
\
I 0
3. Phase plane for the growth model.
The trajectory labeled (1) in the figure will be the optimal path when x0 < z.5< 0. Such a case is of no interest for our particular problem since the initial rival output x0 is surely nonnegative. Trajectory (2), which occurs when x0 > ZZ< 0, is similarly not a valid result under this model since it implies that x(t) = w(t) ert will eventually become negative along the optimal path. A negative output by rival producers is not a feasible solution.lfi We conclude that the trajectories (3) and (4) are the only optimal paths of interest and that the equilibrium optimal price $ is necessarily greater than p. We see that just as under the basic model the optimal pricing strategy results in a constant long-run market share for the
DYNAMIC
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dominant firm. For this model the dominant at any time is given by
317
firms market
share s(c)
It is clear that as p(t) and w(t) approach their equilibrium levels $ and zir the dominant firm’s market share approaches the constant s”= 1 - [ti/f( $)I. To find the equilibrium values 6 and $ we set Eqs. (23) and (24) equal to zero which yields the simultaneous equations
YG= Mb - PI, (Y+ r>25= Q-(b)+f’@>(b- 4) - MF - cl.
P-33
(27)
Equation (27) clearly indicates that for y > 0, ~6is strictly less thanS($>. In contrast to the basic model we now find that a dominant firm with no cost advantage (i.e., p = c) will not price itself out of the market asymptotically. It can be shown that if the curvature of the demand curve at the equilibrium price is not too great (f”( 8) 5 0) an increase in the growth rate y will always increase E.16 This is a disturbing result because growth of the product market not only raises the long run price level ($ > jj) but it also allows dominant firms with insignificant cost advantages to maintain a constant market share over the long haul. It is of some interest therefore to establish t quantitative effects of growth on a specific model. Table I below presents the long-run optimal price and market share of a dominant firm selling in a growing product market. The model analyzed here postulates that the residual demand curve is linear specifically: q(t) = (100 - p(l)) eYt - x(t), and that c = 10, Y = 0.1, k, = 1, x0 = 0.17 The table presents the equilibrium values of s^and J!J for a dominant firm with a substantial cost advantage ( p = 15) and for a firm with no cost advantage ( j = 10 = c)~ TABLE
I
Long Run Optimal Price and Market Share with Market Growth (Linear Demand Curve) y=o
y = 0.02
y = 0.04
y = 0.08
4 = 15
$ = 15 9 = 0.647
j = 15.48 f = 0.716
$ = 15.76 1 = 0.778
5 = 16.22 1 = 0.817
fi = 10 (no cost advantage)
6 = 10 f=O
$ = 11.45 f = 0.181
6 = 12.43 9 = 0.306
$ = 13.67 i = 0.468
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318
The first column represents the case of a static product market. The quantitative difference between the results shown in that column and the other columns are solely attributable to growth of the product market. The striking conclusion from this table is a moderate rate of growth (y = 0.04) enables a dominant firm with no cost advantage to set the long run price 24 % above the limit price and still maintain 30 % of the total market. Further, we see that increasing the rate of growth increases the equilibrium price level and raises the long-run market share of the dominant firm.
COMPARATIVE STATICS AND DYNAMICS OF THE GROWTH MODEL
Differentiation of Eqs. (25), (26), and (27) with respect to various structural parameters of the growth model yields the following results: (k)
$ > 0,
(1) g < 0, (m) Cd
2 < 0, df Sgn dko ==
w
[WB)
+f”($)($ - C)+ $ (-f+)
m> k, ___ $ - c + ___ $- P f(d) ( r Y )I (0)
g
> 0,
(P) dk, 4 -=C0, (4) f
>o,
(r)
4 > 0,
(s)
1 > g > 0.
The first three of these conditions are identical to the results under a static product market. An increase in the dominant firm’s cost advantage
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or a decrease in the discount rate will both increase the long-run optimal market share. Condition (n) indicates that the sign of &/dk, depends upon a host of model parameters. For the specific model considered in Table 1 above it can be shown that d$/dk,, will be positive if the dominant firm enjoys a relatively larger cost advantage ($ > c) and becomes negative as p approaches c. It is also clear that the term k,/r(( p - c>/($ - p)) dominates the sign of d?/dk, for small values of y because ($ - C) approaches zero and ($ - p)/y = G/k remains bounded as y approaches zero. This result is consistent with the comparative static result of the basic model which indicated that d$/dk was proportional to (p - c). We conclude that if the dominant firm enjoys a large cost advantage that an increase in the response of rivals to price signals will likely increase its long-run market share. Under the growth model we find that the equilibrium price $ varies with each of the model parameters. We are disturbed to observe that an increase in the rate of market growth will always increase $. Conversely, the existence of any positive growth rate is sufficient to prevent an increase in @ from being fully reflected in the equilibrium price level. Further, under this growth model an increase in the responsiveness of rival producers to price signals (k,) will result in a lower long-run product price. The sign of d$/dc is consistent with the result under a naive static profit maximizing model. Pt is possible to demonstrate by phase plane analysis of the optimal paths the following general comparative dynamic results: (t)
g
(t) < 0, 0
(u)
q
(4 g
(t)
>
J
(t) > 0, iff”(p)
5 0,
(w) $g (t) < 0. Just as under the basic model it is not possible to determine the sign of dp*/dj (t) for a general demand curve. It is true, however, that for a linear demand curve an increase in the limit price at any point in time will result in a lower optimal price in the short run.17 We conclude that the comparative dynamics for the growth model are qualitatively identical to the results found under the basic model.
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The major substantive changes in the optimal pricing strategy resulting from growth of the product market are that the equilibrium price is raised above the limit price, and that dominant firms with no cost advantage no longer price themselves out of the market in the long-run.
MODEL
IMPLICATIONS
These models have mixed implications for economic policy. Initially we were reassured to find that optimizing dominant firms will ultimately decline if they have no substantial long-run cost advantage. Steady growth of the product market unfortunately mitigates the decline of dominant firms and causes the long-run product price to be above the average cost of production. By-products of faster economic growth are increased concentration levels and higher prices in dominated industries. The comparative statics and dynamics of these models have conflicting implications for patent policy. If, for example, the effectiveness of existing patents was weakened by mandatory licensing the immediate effect would be to lower p the limit price in dominated industries. This would decrease the long-run market shares and product prices but probably would raise prices in the short-run. A policy to increase the responsiveness of potential entrants to price signals by improving the information flow or eliminating capital market imperfections would lower prices both in the short-run and long-run. But such a policy would ultimately lead to larger market shares for dominant firms, if the dominant firms enjoyed substantial long-run cost advantages. ACKNOWLEDGMENT I am especialling indebted to Frederick M. Scherer for seminal discussions of this topic. In addition, I would like to thank Steven Goldman, Harl Ryder, Lester Taylor, and Sidney Winter for useful comments and suggestions on previous drafts.
REFERENCES 1. See, for example, DALE OSBORNE, The role of entry in oligopoly theory, J. Political Econ. LXXII (1964), 396. 2. E. MANSFIELD, Entry, Gibrat’s Law, innovation and the growth of firms, Amer. Econ. Rev. LII(1962), 1026 3. The residual demand curve will also shift laterally if a more general rival supply curve shifts laterally as rival producers enter or exit.
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4. This can be viewed as a first-order approximation of a more complicated functional relationship between 3t:and (p(t) - 3). It can be shown that if rival producers, attempting to maximize the present value of their profit stream, expect the product price to remain constant and are faced by an increasing marginal cost of expansion, then the rivals will expand at a rate proportional to (p - 3)~ (where y > 0). A quadratic entry model which allows asymmetric entry and exit and holds that the rate of entry increases more than proportionally as price increases has been explored in my dissertation “Optimal Pricing by Dominant Firms” unpublished Ph.D. dissertation, University of Michigan, 1970. This more realistic entry model did not aiter qualitatively any of the following results, 5. Cases involving continuous variation and discrete jumps in 6 have been considered in my dissertation, Ref. 4, Chap. 3. 6. Strictly speaking this interpretation requires finding a twice differentiable solution to the Hamilton-Jacobi partial differential equation for this problem. This has been done for a linear version of this problem and may be found in Ref. 4, Chap. 4. 7. Formally, there are problems involved in optimizing a functional when it is an indefinite integral. The major difficulty for our purposes is that most existence theorems have been proved only for optimization over finite time intervals. This diiIiculty is avoided in this particular case by making the substitutions; (1) 7 = 1 - e&t, (2) WI = (La -3)
8. 9. IO. 11. 12. I3. 14.
edt> d is an arbitrary positive constant less than K
The fust substitution maps the time interval (0, a) into the 7 interva1 (0, I) and therefore transforms the original indefinite integral into a definite integral with respect to 7. The second substitution is required to maintain a well-behaved differential constraint over the interval (0 < T < 1). We may now use the existence theorem due to LAMBERTO CESARI in Existence theorems for optimal solutions in Pontryagin and Lagrange problems, SIAM J. Control 3 (1966) 478-79. This existence theorem also requires that the state variable x(r) and control variable S(t) of the transformed problem are contained in compact sets. We can satisfy this by imposing constraints on x(t) and S(t), respectively. As long as trajectories satisfying necessary conditions for optimality do not exceed these constraints we may apply this existence theorem. We note that S(t) is bounded only if lim t+oO(p*(t) - 5) edt is finite, and that p*(f) must approach 3 at least as fast as a declining exponential. The strict inequalities will hold only when the dominant firm has no cost advantage, i.e., 3 = c. The velocity j*(t) along the new path has also changed and it cannot be shown that it will always be less than the velocity along the original optima1 path This result is demonstrated in Ref. 4, p. 13. It also may be shown that this result holds for a demand curve of constant unitary elasticity. No counterexamples have been found. A growth model under which price elasticity steadily declines is explored in Ref. 4, Chap. 4. The assumption that the growth rate of disposable income, y is less than the discount rate I is required to guarantee convergence of the present value integral. We are unable to prove existence of an optimal path in this case. The method of transformation used to prove existence for the basic model is inappropriate for this model because the new control variable S(t) is unbounded since limp*(t) # 3. We again assume that f”(p)(p(t) - c) + 27(p) < 0, which guarantees an interior solution.
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15. This result which is also possible under the basic model leads us to consider a model under which the state variable x(t) is constrained to be positive. Such a model is analyzed in Ref. 4, Chap. 5. 16. Differentiating B with respect to y we find that
Since zi, < f(&, the numerator of the second term is strictly less than f($)Cf’($) + fu($)($ - c) - k,/r). This expression will certainly be negative if f”(j) 2 0. The denominator of dtjdy is negative by prior assumption and we conclude that f”fi
,< 0 a di/dy
> 0.
17. The optimal price trajectory for this model is found in Ref. 4, Chap. 4.
OF ECONOMIC
THEORY
Dynamic
3, 306-322 (1971)
Limit Pricing: Optimal under Threat of Entry DARIUS W.
Received
JR.
GASKINS,
July
31,
Pricing
1970
INTRODUCTION
This paper attempts to determine the optimal pricing strategy for a dominant firm or a group of joint profit maximizing oligopolists faced by potential entry into the product market. The models explored here have broader applications, but we limit our discussion to a dominant firm able to deduce a residual demand schedule through knowledge of the output of a well-behaved competitive fringe. Contemporary writings on this subject for the most part contend that the dominant firm will maximize its present value by either charging the short-run profit maximizing price and allowing its market share to decline or by setting price at the limit price and precluding all entry.l A firm practicing short-run profit maximization would have to ignore continually the reality of induced entry. Conversely, a firm charging the limit price has to be convinced that its prevailing market share is optimal. There has been no analytic justification for this strategic dichotomy, and intuition suggests that the optimal strategy would entail a balancing between current profits and future market share. The basic premise of this study is that the rate of entry of rival producers into a particular market is a function of current product price. Mansfield has shown that the variation in rate of firms entering or exiting an industry is positively correlated with the level of industry profits.2 It follows that a dominant firm with high current price and profit levels is sacrificing some future profits through erosion of its market share. The dependence of future market share on the current price level means that the dominant firm’s pricing strategy can only be determined in a dynamic framework.
306
DYNAMIC
LIMIT
PRICING
307
THE BASIC MODEL
In general terms, the optimal pricing strategy will maximize the present value of firm profits, as given by Eq. (1) below:
where V = present value of firm’s profit stream, p(t) = product price, c = average total cost of production (assumed to be constant over time), @‘p(8), 1) = dominant firm’s output, Y = dominant firm’s discount rate. The specific functional dependence of saies on time is determined by the nature of the entry phenomenon. We assume that the level of the dominant firm’s current sales can be decomposed into additive univariate functions of price and time, respectively,
wheref(p) = initial demand curve and x(t) = the level of rival sales. The residual demand curve q(p(t), t) at any specific instant is found by subtracting the output of the competitive fringe from the total market demand. Equation (2) indicates that the net effect of rival entry into the product market is to shift the dominant firm’s residual demand curve laterally. This would be the case if rival producers’ short-run supply Curve is completely inelastic. 3 The rate of entry of rival producers [9(i)] is surely determined by their expected rate of return. If potential entrants view current product pride as a proxy ‘for future price, the rate of entry will be a monotonically nondecreasing function of current pribe. We assume that this relationship between rate of entry and current price is linear, as shown by Eq. (3)
&) = k[p(t) - PI,
$9 = x0 2
B 3 c,
C%
p = limit price (a constant),5 k = response coefficient > 0, x0 = initial output of the competitive fringe. The limit price p is defined in this model as that price level at which net entry is equal to zero. The model implies therefore that pricing below p will cause negative entry or exit from the product market by rival producers. The difference between the limit price and the dominant firm’s average total cost is a measure of the cost advantage enjoyed by the dominant firm. It seems reasonable to restrict the limit price to be greater than or equal to the dominant firm’s average total cost. While there may be examples of dominant firms with cost disadvantages, their fate is obvious with or without optimal pricing.
308
GASKINS
To expedite the analysis further, we assume that the dominant firm’s initial demand schedule f(p(t)) is d ownward sloping and twice differentiable with respect to output. With these simplifying assumptions, it is possible to determine the optimal pricing strategy analytically using the mathematics of optimal control. In the language of modern control theory, we wish to maximize the functional V = Jrn [p(t) - c][f(p)
- x(t)] e-rt dt
(4)
x(0) = x0 .
(3)
0
subject to w
= &J(t)
- PI,
In the control theory framework, x(t) is the state variable and p(t) is the control variable. We may derive necessary conditions for the optimal path by using Pontryagin’s maximum principle. The Hamiltonian for this problem is given by H = (p(t) - 4CfW
- x(t)) et
+ ~0) JWQ
- PI.
(5)
The adjoint variable z(t) appearing in the Hamiltonian is equal to a V/&X (t) and can be interpreted as the shadow price of an additional unit of rival entry at any point in time.‘j The first term of Eq. (5) (the integrand of Eq. (4)) is the change in present value accruing from current sales. The second term which is the product of &(t) and z(t) reflects the effect of current entry on future profits. Intuitively, maximizing the Hamiltonian with respect to p(t) involves balancing the flows of present value from present and future sales. The maximum principle states that for a maximum V to exist, it is necessary that there exists a z(t) such that: (i)
2*(t) = k@*(t) - p),
x*(o) = x0 ;
(ii)
2*(t) = - +$ (x*(t), z*(t),p*(t),
t);
= (p*(t) - c) e-Tt, ‘,i% z*(t) = 0; + (iii)
H(x*W,
z*(t>,~*tt),
t> = ,yy ~ . . $(-x*(Q,
z*(t), p(t), 1).
The superscript * denotes variables along the optimal easily shown that condition (iii) implies
trajectory.
It is
DYNAMIC
LIMIT
309
PRICING
as long as where the prime denotes a/Q.
(P - C)“?(P) < 0,
2f’CPlf
This basic inequality will always hold if profit is a smooth concave function of price along the initial demand curve. We assume that such is the case. This assumption which assures the concavity of H(x(c), z(t),p(t), t) with respect to p and x Is sufficient to guarantee the existence of an optimal path.’ The necessary conditions produce the two simultaneous ordinary differential equations: i*(t)
= k(p"(t)
- li>,
x*(o)
i*(t)
= (p*(t) - c) e-Tt,
z*(t)
= (x*(t)
lim
*&KC
= x0 ) z*(f)
(51 0,
=
63
where -f(P)
-
(P*O)
-
c)f’(~*N
e-r*
(71
k
We can eliminate the adjoint variable z(t) from these equations and write the necessary condition as the simultaneous differential equations in p(t) and x(t): k(t) = k(p(f) - a>,
40) = x0 ,
9(t) = W - 4 + dx -f(p) - (P -V’(P) - (P - 4fYP)
(8)
cIf’(~)l .
(93
These two equations generate a family of trajectories in the x - p plane. Unfortunately the terminal condition lim z*(t) = 0, as t + co, cannot translated into an explicit condition of x(t) or p(t) and therefore it is not immediately obvious which of these trajectories satisfies (satisfy) the necessary conditions. We argue from a phase-plane portrait of Eqs. (8) and (9) that there is a unique trajectory meeting all the necessary conditions. Figure 1 presents p(t)
FIG.
1.
Phase plane
for the basic model.
310
GASKINS
the locii of 9 = 5 = 0 in the x - p plane. These two locii divide the x - p plane into four distinct regions. It is apparent that the intersection of these locii denoted (a,$) is a saddlepoint. Any trajectory entering or originating in regions II or IV will remain in that region. Further, x andp will increase without bound along any trajectory in region II and decrease without bound along a path in region IV. Inspection of the dominant firms objective function V(p(t), x(t), t) indicates that no trajectory in region II or IV could be an optimal path. In region II ever increasing x(t) and p(t) would generate an increasing stream of losses after some point in time. Similarly, in region IV a steadily decreasing p(t) and x(t) will eventually generate an ever increasing stream of negative profits (i.e., when p(t) < c). We now verify that any path reaching the equilibrium point (a,$) will satisfy the terminal boundary condition lim z*(t) = 0, as t + co. At (g, $) both 3 and $ are zero. Equation (7) indicates that at this point z*(t) = (a -f(J) -f’($)($ - c) e-Tt)/k is an exponential function of time that declines to zero, therefore a trajectory reaching this equilibrium point will trivially satisfy the terminal boundary condition. Since the functions c?(x,p) and &,p) are both continuous and continuously differentiably in the x - p plane (2f’(p) + f”(p)(p - c) < 0), there will be only two trajectories (labeled (1) and (2) in the figure) which terminate at (2, $). All other trajectories originating in region I or III will eventually enter either region II or IV. The optimal pricing strategy will therefore be to move along either trajectory (1) or (2) depending on the original size of the competitive fringes output x0 . For x,, < 9 we conclude that the dominant firm will maximize its present value by gradually lowering product price towards the limit price p. Such a pricing policy will induce rival entry and continually reduce the dominant firm’s market share. Conversely, when x0 is greater than 9 the optimal strategy is to price below the limit price, continuously driving out rivals. We are able to prove that the optimal price level will always be below the short run profit maximizing price at every point along the optimal path. If the dominant firm has been moving along the optimal path, instantaneous profit is given by the function T(P) = (p(t) - W(P)
The myopic profit maximizing p
712
= f’!pm)(pm(t)
- X*(o).
(10)
price pm(t) is the solution to the equation -
c> + f(Pm)
- x*0>
= 0.
DYNAMIC
LIMIT PRICING
We have previously seen that maximization point along the optimal path implies that -z(t)
= (f’(p*)(p*(t)
31n
of the Hamiltonian
- c) +f(p*3
at every (12)
- x*(t)> +lk
where z(t) the shadow price of additional rival entry is necessarily negative. Ry prior assumption a2r(p)/a2pnz < 0 and therefore, Eqs. (11) and (12) are both satisfied only if pm(t) > p*(t).
COMPARATIVE
STATICS
AND
DYNAMICS
OF THE OPTEMAL
TRAJECTOKY
We now attempt to establish the effects of variation in the model parameters, k, c, 3, r and x0 on the optimal path. The equilibrium level of rival output 2 is seen to be an important characteristic of the optimal ricing strategy. Setting Eqs. (8) and (9) equal to zero and solving simultaneously for 2 we find that
The market share of the dominant firm at any point in time is s(t) = (f(p) - x(t))lf(p). It is clear that as p(t) and x(t) approach $ and $ respectively, s(t) will approach an equilibrium t. The optimal strategy in fact may be viewed as pricing to achieve a long-run optimal market share equal to:
Differentiating
Eqs. (13) and (14) we establish the following results:
312
GASKINS
The first two of these conditions jointly indicate that the dominant firm will price to allow less entry as its cost advantage (p - c) increases. We argue that p will vary directly with the average level of rival costs. A very large ( p - c) will justify pricing to rationalize the relevant industry by driving out “inefficient” producers (2 < x0). An important practical case is the dominant firm which enjoys no cost advantage over rival producers. We model such a case by setting p = c. In this case Eq. (13) indicates that in response to optimal pricing by the dominant firm the output of the competitive fringe will asymptotically approachf(c), the total industry output. The dominant firm in this case prices itself out of the market in the long-run. While it is acknowledged that our model will lose its validity at some point as the dominant firm’s market share declines the conclusion remains that dominant firms with little or no cost advantage decline if they strive to maximize their present value. Condition (c) indicates that as the dominant firm’s discount rate increases it will sacrifice a portion of its long-run market share. A higher discount rate indicates that future profits become relatively less important. Condition (d) indicates that the more rapidly rivals respond to price signals the larger will be the dominant firms long-run market share. This result is at first glance counter-intuitive and would seem to question the efficacy of any public policy designed to increase the responsiveness of potential entrants. We demonstrate below that a concomitant result is that an increase in k will necessarily lower the optimal price trajectory at least in the short run. Because r, k, and c don’t affect p the change in the dominant firms equilibrium market share in response to variation in these parameters will be of the opposite sign of the change in 9. A change in p however will affect f( p) as well as 2 and therefore complicate the determination of the sgn df/dj. Condition (e) indicates that the sign of df/d@ depends on the sign of an expression which contains only one term which can be negative, i.e., -f”( p)( p - c). W e conclude that if either the curvature of the demand curve near p is relatively small or that the dominant firm has a small cost advantage the dominant firms long run market share will increase as p increases. Condition (f) merely restates the obvious point that the long-run optimal price level and market share are independent of the initial output of the competitive fringe. We now attempt to establish the effect of variation of the model parameters on the optimal trajectory away from the equilibrium point. Because our model still includes the general functionf(p) it is not possible to find the explicit form of the whole optimal path. The phase-plane can be
DYNAMIC
LIMIT
PRICING
313
used however to obtain certain comparative dynamic results without further model specification. Figure 2 indicates the effect of a positive increase in the response coefficient k on the phase portrait. We see that the new equilibrium
FIG. 2. Comparative dynamic effect of a change in the response coefficient.
point (9, , p) has been moved to the left in accordance with condition (d) above. It is easily shown that the slope of trajectories in region I will be increased at every point by an increase in k. The slope of any trajectory in the x - p plane designated G(x, p) is equal to
GA!&=“= By differentiation
52
k(p - 4 + r(x -f(p) - ~‘(P>(P- 4) k(p - PX-V’(P) - ~“(P)(P - 4) = 05’
we find that
dG -a?=
-4x - f(P) (P - P)(-2f’(P>
- f’(P)(P -f”(P)(P
- 4) - 4) .
Recalling from Eq. (7) that the shadow price of entry z*(t) is proportional to the quantity (X -f(p) --f’(p)@ - c)) CZ-?~,we argue that the rmmerator of Eq. (16) is positive since an additional unit of rival entry at any time will surely lower the present value of the dominant firm. previous assumption (-2f’(p) --f”(p)(p - c)) > 0, and we conclude that sgn dGjdk = sgn(p - p). If we examine the situation in region I of the phase plane it is apparent that the new optimal path (2) must lie below the original optimal trajectory (1). At points along the old path the slope has been increased by the increase in k and therefore any trajectory passing through a point on (1) must move above (1) as x increases.No such path could possibly reach the new equilibrium point (gz , p). In a similar fashion, we can
314
GASKINS
demonstrate that if the original optimal path was in region III (i.e., x0 > 2,) the optimal trajectory will move downward as k increases. We have shown that the optimal trajectory in the x - p plane will always be lowered by an increase in the responsiveness of rival producers to price signals. While it follows that the initial portion of the dominant firm’s optimal price trajectory will be lowered by increasing k we are not assured that the whole time path p*(t) has been lowered.g By similar arguments we are able to establish the following comparative dynamic results:
(h) g(t) (i)
$$
> 0.
if fn(p)+
0:
(t) < 0, 0
Each of these conditions indicates the short term effect of variation of a particular model parameter. In addition, since variation in x0 the initial output of the competitive fringe only affects the starting point of the optimal path, the last condition holds for the whole time path of p*(t). Unfortunately, it is not possible to assess the impact of small changes in @ on the initial portion of the optimal path in this general model. We have demonstrated elsewhere that for a linear demand curve the response of the optimal price at any point in time is such that: (j)
--dP* (t) < 0 @
(see Ref. 10).
Conditions (h) and (j) jointly indicate that the dominant firm will lower its price in response to an increase in its relative cost advantage. We have the somewhat ironic result that a policy which lowers existing barriers to entry (e.g., mandatory licensing of a vital patent) will raise prices in the short run if dominant firms price according to this model. The economic justification for this seemingly perverse result is that the optimal long-run market share of the dominant firm is increased as its cost advantage increases (we previously saw that di/d( p - c) < 0). As the dominant firm becomes increasingly efficient it will strive to drive out more of the inefficient producers and it does this by lowering the product price in the short run. We observe however that if the change in the cost advantage is due to an increase in p, the long run price will be increased.
DYNAMIC LIMIT PRICIN6;
315
The most sanguine conclusion from this model remains that dominant firms with little or no market power ultimately decline. We fmd much to our chagrin however that growth of the product market destroys this consoling result.
MARKET
GROWTH
MODEL
The American economy with its growing population and increasing per capita income is not consistent with the static product market assumed by the basic model. More realistically dominant firms are faced by markets which are growing as the economy expands. We now explore a model in which the output of the dominant firm is assumed to be of the form
where y = the market growth rate. It is postulated that product demand is increasing exponentially. This particular growth model which has the property that the price elasticity at a given price remains constant over time seems to be consistent with steady growth of disposable income rather than the secular growth of any single product market.ll Consonant with growing disposable income, we also assume that the entry response coefficient ii is now the growing exponential function of time k(t) = koeYt.
WJ
An increase of disposable income should c+se a proportional increase in the quantity of resources available to potential entrants for investment in any product market. Under this growth model, the dominant firm attempts to maximize the present value of its stream of profits
subject to 2(t) = k,,eyt(p(t) - j?), The necessary conditions differential equationP3:
for a maximum
k*(t) = k,eYQ*(t) - j?), s*(t) = (p”(t) - c) et, 6421313-7
x(0) = x, . V generate the simultaneous
x”(0) = xg ) p+z z”(t) = 0,
421) (22)
316
GASKINS
where z*(t) = (x*(t) e-yt - f(p*)
- (p*(t) - c)f’(p*))
ecrt/k.14
This system of differential equations can be converted into an autonomous system by making the substitution w(t) = x(t) e-Yt and eliminating z(t). The resulting equations are
w(O) = x0 ,
GO>= ko(dt) - i-4 - w(t),
- ~0) + I’ l%t>= ko(P - 4 - WP) (--2f’ho - f”mP(t)
- cl>+ w(t) . - 4)
(23)
(24)
The phase plane portrait of these equations is shown in Fig. 3. It is clear from the figure and our previous arguments about the phase-plane portrait of the basic model that there is a unique optimal path which will take one of four possible shapes determined by the values of & and w(0) = x0 . P(t)
G
FIG.
\
I 0
3. Phase plane for the growth model.
The trajectory labeled (1) in the figure will be the optimal path when x0 < z.5< 0. Such a case is of no interest for our particular problem since the initial rival output x0 is surely nonnegative. Trajectory (2), which occurs when x0 > ZZ< 0, is similarly not a valid result under this model since it implies that x(t) = w(t) ert will eventually become negative along the optimal path. A negative output by rival producers is not a feasible solution.lfi We conclude that the trajectories (3) and (4) are the only optimal paths of interest and that the equilibrium optimal price $ is necessarily greater than p. We see that just as under the basic model the optimal pricing strategy results in a constant long-run market share for the
DYNAMIC
LIMIT PRICING
dominant firm. For this model the dominant at any time is given by
317
firms market
share s(c)
It is clear that as p(t) and w(t) approach their equilibrium levels $ and zir the dominant firm’s market share approaches the constant s”= 1 - [ti/f( $)I. To find the equilibrium values 6 and $ we set Eqs. (23) and (24) equal to zero which yields the simultaneous equations
YG= Mb - PI, (Y+ r>25= Q-(b)+f’@>(b- 4) - MF - cl.
P-33
(27)
Equation (27) clearly indicates that for y > 0, ~6is strictly less thanS($>. In contrast to the basic model we now find that a dominant firm with no cost advantage (i.e., p = c) will not price itself out of the market asymptotically. It can be shown that if the curvature of the demand curve at the equilibrium price is not too great (f”( 8) 5 0) an increase in the growth rate y will always increase E.16 This is a disturbing result because growth of the product market not only raises the long run price level ($ > jj) but it also allows dominant firms with insignificant cost advantages to maintain a constant market share over the long haul. It is of some interest therefore to establish t quantitative effects of growth on a specific model. Table I below presents the long-run optimal price and market share of a dominant firm selling in a growing product market. The model analyzed here postulates that the residual demand curve is linear specifically: q(t) = (100 - p(l)) eYt - x(t), and that c = 10, Y = 0.1, k, = 1, x0 = 0.17 The table presents the equilibrium values of s^and J!J for a dominant firm with a substantial cost advantage ( p = 15) and for a firm with no cost advantage ( j = 10 = c)~ TABLE
I
Long Run Optimal Price and Market Share with Market Growth (Linear Demand Curve) y=o
y = 0.02
y = 0.04
y = 0.08
4 = 15
$ = 15 9 = 0.647
j = 15.48 f = 0.716
$ = 15.76 1 = 0.778
5 = 16.22 1 = 0.817
fi = 10 (no cost advantage)
6 = 10 f=O
$ = 11.45 f = 0.181
6 = 12.43 9 = 0.306
$ = 13.67 i = 0.468
GASKINS
318
The first column represents the case of a static product market. The quantitative difference between the results shown in that column and the other columns are solely attributable to growth of the product market. The striking conclusion from this table is a moderate rate of growth (y = 0.04) enables a dominant firm with no cost advantage to set the long run price 24 % above the limit price and still maintain 30 % of the total market. Further, we see that increasing the rate of growth increases the equilibrium price level and raises the long-run market share of the dominant firm.
COMPARATIVE STATICS AND DYNAMICS OF THE GROWTH MODEL
Differentiation of Eqs. (25), (26), and (27) with respect to various structural parameters of the growth model yields the following results: (k)
$ > 0,
(1) g < 0, (m) Cd
2 < 0, df Sgn dko ==
w
[WB)
+f”($)($ - C)+ $ (-f+)
m> k, ___ $ - c + ___ $- P f(d) ( r Y )I (0)
g
> 0,
(P) dk, 4 -=C0, (4) f
>o,
(r)
4 > 0,
(s)
1 > g > 0.
The first three of these conditions are identical to the results under a static product market. An increase in the dominant firm’s cost advantage
DYNAMIC
LIMIT
PRICING
319
or a decrease in the discount rate will both increase the long-run optimal market share. Condition (n) indicates that the sign of &/dk, depends upon a host of model parameters. For the specific model considered in Table 1 above it can be shown that d$/dk,, will be positive if the dominant firm enjoys a relatively larger cost advantage ($ > c) and becomes negative as p approaches c. It is also clear that the term k,/r(( p - c>/($ - p)) dominates the sign of d?/dk, for small values of y because ($ - C) approaches zero and ($ - p)/y = G/k remains bounded as y approaches zero. This result is consistent with the comparative static result of the basic model which indicated that d$/dk was proportional to (p - c). We conclude that if the dominant firm enjoys a large cost advantage that an increase in the response of rivals to price signals will likely increase its long-run market share. Under the growth model we find that the equilibrium price $ varies with each of the model parameters. We are disturbed to observe that an increase in the rate of market growth will always increase $. Conversely, the existence of any positive growth rate is sufficient to prevent an increase in @ from being fully reflected in the equilibrium price level. Further, under this growth model an increase in the responsiveness of rival producers to price signals (k,) will result in a lower long-run product price. The sign of d$/dc is consistent with the result under a naive static profit maximizing model. Pt is possible to demonstrate by phase plane analysis of the optimal paths the following general comparative dynamic results: (t)
g
(t) < 0, 0
(u)
q
(4 g
(t)
>
J
(t) > 0, iff”(p)
5 0,
(w) $g (t) < 0. Just as under the basic model it is not possible to determine the sign of dp*/dj (t) for a general demand curve. It is true, however, that for a linear demand curve an increase in the limit price at any point in time will result in a lower optimal price in the short run.17 We conclude that the comparative dynamics for the growth model are qualitatively identical to the results found under the basic model.
320
GASKINS
The major substantive changes in the optimal pricing strategy resulting from growth of the product market are that the equilibrium price is raised above the limit price, and that dominant firms with no cost advantage no longer price themselves out of the market in the long-run.
MODEL
IMPLICATIONS
These models have mixed implications for economic policy. Initially we were reassured to find that optimizing dominant firms will ultimately decline if they have no substantial long-run cost advantage. Steady growth of the product market unfortunately mitigates the decline of dominant firms and causes the long-run product price to be above the average cost of production. By-products of faster economic growth are increased concentration levels and higher prices in dominated industries. The comparative statics and dynamics of these models have conflicting implications for patent policy. If, for example, the effectiveness of existing patents was weakened by mandatory licensing the immediate effect would be to lower p the limit price in dominated industries. This would decrease the long-run market shares and product prices but probably would raise prices in the short-run. A policy to increase the responsiveness of potential entrants to price signals by improving the information flow or eliminating capital market imperfections would lower prices both in the short-run and long-run. But such a policy would ultimately lead to larger market shares for dominant firms, if the dominant firms enjoyed substantial long-run cost advantages. ACKNOWLEDGMENT I am especialling indebted to Frederick M. Scherer for seminal discussions of this topic. In addition, I would like to thank Steven Goldman, Harl Ryder, Lester Taylor, and Sidney Winter for useful comments and suggestions on previous drafts.
REFERENCES 1. See, for example, DALE OSBORNE, The role of entry in oligopoly theory, J. Political Econ. LXXII (1964), 396. 2. E. MANSFIELD, Entry, Gibrat’s Law, innovation and the growth of firms, Amer. Econ. Rev. LII(1962), 1026 3. The residual demand curve will also shift laterally if a more general rival supply curve shifts laterally as rival producers enter or exit.
DYNAMIC
LIMIT
PRICING
321
4. This can be viewed as a first-order approximation of a more complicated functional relationship between 3t:and (p(t) - 3). It can be shown that if rival producers, attempting to maximize the present value of their profit stream, expect the product price to remain constant and are faced by an increasing marginal cost of expansion, then the rivals will expand at a rate proportional to (p - 3)~ (where y > 0). A quadratic entry model which allows asymmetric entry and exit and holds that the rate of entry increases more than proportionally as price increases has been explored in my dissertation “Optimal Pricing by Dominant Firms” unpublished Ph.D. dissertation, University of Michigan, 1970. This more realistic entry model did not aiter qualitatively any of the following results, 5. Cases involving continuous variation and discrete jumps in 6 have been considered in my dissertation, Ref. 4, Chap. 3. 6. Strictly speaking this interpretation requires finding a twice differentiable solution to the Hamilton-Jacobi partial differential equation for this problem. This has been done for a linear version of this problem and may be found in Ref. 4, Chap. 4. 7. Formally, there are problems involved in optimizing a functional when it is an indefinite integral. The major difficulty for our purposes is that most existence theorems have been proved only for optimization over finite time intervals. This diiIiculty is avoided in this particular case by making the substitutions; (1) 7 = 1 - e&t, (2) WI = (La -3)
8. 9. IO. 11. 12. I3. 14.
edt> d is an arbitrary positive constant less than K
The fust substitution maps the time interval (0, a) into the 7 interva1 (0, I) and therefore transforms the original indefinite integral into a definite integral with respect to 7. The second substitution is required to maintain a well-behaved differential constraint over the interval (0 < T < 1). We may now use the existence theorem due to LAMBERTO CESARI in Existence theorems for optimal solutions in Pontryagin and Lagrange problems, SIAM J. Control 3 (1966) 478-79. This existence theorem also requires that the state variable x(r) and control variable S(t) of the transformed problem are contained in compact sets. We can satisfy this by imposing constraints on x(t) and S(t), respectively. As long as trajectories satisfying necessary conditions for optimality do not exceed these constraints we may apply this existence theorem. We note that S(t) is bounded only if lim t+oO(p*(t) - 5) edt is finite, and that p*(f) must approach 3 at least as fast as a declining exponential. The strict inequalities will hold only when the dominant firm has no cost advantage, i.e., 3 = c. The velocity j*(t) along the new path has also changed and it cannot be shown that it will always be less than the velocity along the original optima1 path This result is demonstrated in Ref. 4, p. 13. It also may be shown that this result holds for a demand curve of constant unitary elasticity. No counterexamples have been found. A growth model under which price elasticity steadily declines is explored in Ref. 4, Chap. 4. The assumption that the growth rate of disposable income, y is less than the discount rate I is required to guarantee convergence of the present value integral. We are unable to prove existence of an optimal path in this case. The method of transformation used to prove existence for the basic model is inappropriate for this model because the new control variable S(t) is unbounded since limp*(t) # 3. We again assume that f”(p)(p(t) - c) + 27(p) < 0, which guarantees an interior solution.
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15. This result which is also possible under the basic model leads us to consider a model under which the state variable x(t) is constrained to be positive. Such a model is analyzed in Ref. 4, Chap. 5. 16. Differentiating B with respect to y we find that
Since zi, < f(&, the numerator of the second term is strictly less than f($)Cf’($) + fu($)($ - c) - k,/r). This expression will certainly be negative if f”(j) 2 0. The denominator of dtjdy is negative by prior assumption and we conclude that f”fi
,< 0 a di/dy
> 0.
17. The optimal price trajectory for this model is found in Ref. 4, Chap. 4.