Competition, Strategy, and Price Dynamics

RAM C. RAO and FRANK M. BASS*

The authors explore how competition affects dynamic pricing of new products.Dynamics of diffusion, saturation, and cost reduction due to exfjerience are oilconsidered, with emphasis on the lost. The competition omong firms is modeled osa dynamic Nash equilibrium in which each firm chooses its optimal dynamic strategy,correctly anticipating its rivals' strategies. The dynamics of price and morket shareare characterized for an n-firm oligopoly. Empirical examination of price pathsacross eight products in the semiconductor components industry shows them to be

consistent with analytical results.

Competition, Strategy, and Price Dynamics: ATheoretical and Empirical Investigation

Dynamic pricing models, which explicitly consider thefuture consequences of current actions, have ignored theeffects of competitive actions. Though competition isclearly a relevant consideration for managerial deci-sions, incorporating competition into dynamic pricingmodels involves costs due to the complexity of the anal-ysis and the quantity of data needed to parameterize andtest the models. However, market share strategies basedon experience curve effects provide a strong rationale forincorporating comj>etition into dynamic pricing models.

When experience curve effects are present for a prod-uct category, a frequent strategic prescription is to achievea cost advantage by gaining market share, accumulatingexperience, and thus reducing relative costs (Abell andHammond 1979, p. 116). What will happen if all (or asubset of) firms in an industry pursue such aggressivestrategies? This critical strategic question can only beaddressed by explicitly incorporating competition into adynamic pricing model.

"Ram C. Rao is Associate Professor and Frank M. Bass is the Eu-gene McDermott Professor of Management, The University of Texasul Dallas.

The authors thank Randall G, Chapman of the University of Al-berta, Edmonton, Abel P. Jeuland. B. Peter Pashigian. Arnold Zellnerand. in particular. John P. Gould of the Graduate School of Business,University of Chicago, for many valuable discussions during the earlysiages of this research. The article has benefited greatly from the sug-gestions of three anonymous JMR reviewers and from the encourage-meni and insightful comments of Barton A. Weitz, Special Issue Ed-itor, The usual disclaimer applies.

The pricing model presented here incorporates bothdynamic and competitive effects. The dynamic consid-erations pertain primarily to cost reduction through ex-perience curve effects; however, the demand dynamicsof diffusion and eventual saturation also are examined.Competition is modeled by postulating that firms behavenoncooperatively. maximizing their own profits, and eachfirm correctly anticipates its rivals* strategies and the ef-fects of those strategies on the firm's profits.' By mod-eling competition in this manner, we can consider firmsthat have different costs, demand structures, discountfactors, access to market information, and planning ho-rizons (maximization of current profits or the discountedvalue of future profits). Within this rich framework, weexamine the impact of two asymmetries between firms:costs and length of planning horizon.

The article is divided into three sections. In the firstsection, the model is developed and, through mathe-matical analysis, both price and market share dynamicsin an n-firm oligopoly are characterized. Numerical re-sults then are used to obtain potentially useful manage-

'By this modeling approach, a noncooperative Nash equilibrium so-lution can be developed. This solution is often evoked to model theoutcome of multiagent strategic interactions (Prescott and Townsend1979). A Nash equilibrium is defined as follows. Suppose there aren firms and each must choose a strategy s,, / = 1, 2 n. from afeasible set A,. Let V, denote the profit to /, Then s*f.A,, i = 1 ,2 .. . . . fi, is a Nash equilibrium if for each firm i

^ V,{sf s?' s*)

for all s.tA,.

Journal of Marketing ResearchVol. XXII (August 1985), 283-96

284 JOURNAL OF MARKETING RESEARCH. AUGUST 1985

rial insights. In particular, the effect of competition onthe desirability of nonmyoplc strategies and the value ofan initial cost advantage are elucidated. Finally, an em-pirical analysis of semiconductor prices is used to testthe model's predictions. The results suggest that pricepath variations across products are related to the growthrate of demand and price elasticity. These empiricalfindings are in contrast to the Boston Consulting Group(1972) contention that the differences in prices over timeare due to leaming rates and industry shake-out.

A MODEL OF DYNAMIC PRICING IN ANOLIGOPOLY

Leaming on the cost side implies that firms realizelower product costs with increases in cumulative pro-duction. An early study substantiating this effect (Al-chian 1963) in aiiframe production remains a classic. Onthe demand side, learning can be thought of as the im-itation effect alluded to in work on diffusion of inno-vations (Bass 1969; Rogers 1962). This imitation effectsuggests that the rate of adoption of a new product bynonadopters increases as the number of adopters in-creases. In addition to the two leaming dynamics, Bass(1969) suggests that saturation is important in the dy-namics of demand for durables. The general model ofdynamic pricing described in this section incorporates allthree elements; however, the results in the next two sec-tions represent only the effect of cost-side leaming.

In the approach taken in this article, firms are assumedto interact strategically. Each firm is conscious of theeffect of its actions on the profits of competing firms andon its own profits. Quantity, rather than price, is as-sumed to be the firms' decision variable. For many in-dustries this is reasonable, though for some (e.g., pack-aged goods) this modeling approach would beinappropriate. Given that firms decide on quantities, theprice of each firm's product depends on its and its com-petitors' choice of quantities. Here the prices of all fums'products are assumed to be the same, that is, the prod-ucts are not greatly differentiated. For many industrialproducts and some consumer products this kind of modelis very realistic.

Model Formulation

Let Xi, and q,^ denote the cumulative and current outputof firm I = 1, 2, . . . , rt at time t and c,, the marginalcost of production (i.e,, unit variable cost). Cost-sideleaming is modeled by postulating that

( I )

where;

(2) C = dC/dx,, < 0, C" = -, > 0.

Equation 1 states that firm f s unit variable cost of pro-duction depends on its cumulative volume. The first in-equality in equation 2 states that cost decreases with cu-mulative volume X,,. The second inequality states that therate of cost decline is slower as Xi, becomes larger. Math-

ematically, C is a decreasing, convex function.The presence of diffusion and saturation effects im-

plies that the current quantity sold depends on both cur-rent price and cumulative quantity sold. For the undif-ferentiated market studied here, it is more useful to thinkof the inverse demand function, that is. the dependenceof current price p, on current output and cumulative out-put of all firms. The inverse demand function can bewritten as

(3) p, = giX,,Q,)

where X, is industry cumulative sales and Q, current in-dustry sales at time /. Thus, X, = %Xi, and Q, = îqu-To understand better the effect of X on p, it is useful toconsider two polar cases, one in which there is only dif-fusion and the other in which there is only saturation. Inthe presence of diffusion, as X grows, more adopters areattracted to the market. This increased demand meansthat the partial effect of X on p is positive. In contrast,with market saturation, the potential adopter populationshrinks, resulting in a negative partial effect of X on p.Mathematically,

dp/dX > 0 for all X if only diffusion exists (i.e.. no sat-uration) and dp/dX < 0 for all X if only saturation exists,(i.e., no diffusion).

Finally, the demand function is assumed to be down-ward-sloping, that is, the partial effect of Q onp is neg-ative, so that

(4) Bp/dQ < 0.

Each firm / then is faced with the following optimi-zation problem: choose </„, t € (O.oo) such that it maxi-mizes the discounted value of profits. Let r be the rateof continuous discounting in the industry. Firm I's prob-lem can be stated mathematically as

(5)

maximize V, = I exp(-rr)( p, -

(6)

(7)

= q,,

given.

i = 1,2 n.

(• = I , 2 n,

i = 1, 2 n.

where p, is given by equation 3, c , by equation I, andXjQ is an initial condition. The integral in equation 5 rep-resents the sum of discounted net profits. The differen-tial equation 6 states that the cumulative output growsat the rate of current output.

The maximizing problem in equation 5 is substantiallyidentical to that considered by Etolan and Jeuland (1981).Clarke. Marako, and Heineke (1982), and Kalish (1983).In all these cases there is only one firm, that is. the modelis for a monopoly, and the decision variable is price, notquantity. For a monopolist, the solution is invariant tothe choice of decision variable so that equation 5 is anequivalent representation of the models considered by

COMPETITION, STRATEGY, AND PRICE DYNAMICS 285

these autht)rs. All three articles consider various com-binations of cost functions and demand functions and ob-tain a qualitative characterization ofthe optima! dynamicprice strategy, suggesting the conditions for optimal pricepaths to exhibit increasing or decreasing segments. Incontrast. Robinson and Lakhani (1975) focused on quan-tifying the outcomes. On the basis of a numerical ex-ample, they found that if diffusion and cost learning arepresent, dynamic strategies are substantially differentfrom, and greatly more profitable than, myopic strate-gies. Bass and Bultez (1983). however, found that if dif-fusion is a function of time ratber than cumulative vol-ume, a monopolist is unlikely to gain much throughdynamic strategies, at least for the numerical examplesthey considered.

Recently efforts have been made to build competitioninto the model. Spence (1981) considered a problemsimilar to the one addressed here but his interest is intbe implications for industry structure, performance, andwelfare aspects. Thompson and Teng (1980) consideredthe market share type of models for which they com-puted optimal strategies for duopolies and triopoiies. Morerecently, Eliashberg and Jeuland (1983) modeled com-petition when diffusion and product differentiation arepresent, but learning and discounting are absent. Clarkeand Dolan (1984) studied intuitive price strategies in aduopoly through simulation.

Results

To solve the problem in equations 5 through 7, firmI must know the choices of all its competitors. One wayto model such situations is the Nash noncooperative so-lution. Let fl,, denote the vector of quantities chosen byall firms except f. Then a Nash solution implies that firm( chooses its path q^ so that

(8) 1= 1,2 n.

where q'^ is the optimal strategy for firm /. The abovecondition states that given all firms choose their optimalstrategies, qf^ solves the problem in equations 5 through7 for firm /.

The solution to the model involves finding q^ for allfirms simultaneously. The mathematical developments,based on optimal control theory, are given in AppendixA. Here it is enough to note that a necessary conditionsatisfied by q* leads to the following equation for firm/ (equation A12, Appendix A).

(9) p,( 1 + (m,,/Ti,)) = r c, exp ( - r (T - t))dT

- t))dr

where TI, ^ i(iQ,/(ip,)/{Qjp,) is the elasticity of de-mand, px, = dp,/BX,, and m,, is firm /'s share of currentoutput Q,.

Equation 9 relates market share of firm ( at time t to

(1) price p,, (2) elasticity of demand, (3) future costs offirm (• under the first integral of the right side, and (4)effect of industry sales X on price and firm /'s output qîn the future under the second integral of the right side.The first useful outcome of the analysis of competitionis that the n equations 9 can be manipulated to under-stand how market shares behave over time. Previousstudies have focused on price dynamics. Here, interestcenters on price dynamics as well as the ordering of firmsby market share over time. In the following analysis theoptimal output paths ^,,, ( = 1, 2, . . . , n, and thereforep,. are assumed to be differentiable.^ Market share dy-namics are examined for three polar cases, (I) only de-mand saturation, (2) only demand diffusion, and (3) onlycost learning. First, some additional notation is needed.Denote

k = r. — r.K, Cj, tj,

where / andy are two representative firms.

Case I. dCi,/dXi, = 0. px, < 0 (i.e.. Only DemandSaturation)

The specific model of saturation considered is one inwhich industry demand Q is given by

(10) Q = (Mp'' - X)

where a > 0 is a diffusion parameter, M the market po-tential when price equals 1. and p. an elasticity. It iscustomary to assume, for a monopoly, . < - ] . Here,for an n-firm oligopoly, the assumption is p, < - (1 /n ) .Because \x. is negative, the market potential increases asprice decreases. The following result then obtains.

Theorem IFor the demand function of equation 10:

a. Price monotonically declines over time.b. For any two firms ( and j , the lower cost firm has a

higher market share everywhere.c. The difference in sales of / and j is maximized as

/ —> 00.

Proof . ,See Appendix B.

To understand the meaning of the result, let firm /have a lower cost (i.e.. k < 0). Then firm /'s share ofoutput is greater t han / s at all times, so that the orderingof firms by market share is preserved throughout. Thesize difference, as measured by sales levels, will begreatest at the end of the horizon.

The monotonic price decline constitutes skimming, re-sulting from an attempt to dynamically price discrimi-nate. It is sensible given that there are no diffusion orcost learning effects which might encourage penetration.The divergence of sales toward the end can be inter-preted as the high-cost firm trying to saturate the market

assumption witl hold if H,. given all other firms' outputs, isconcave in ^,.

286 JOURNAL OF MARKETING RESEARCH, AUGUST 1985

faster than its low-cost rival, at least toward the end.Such an outcome may prevail at all times but it cannotbe expected with certainty on the basis of the analyticdevelopments.

Case 2. dcjjdxi, = 0, px, > 0 (i.e.. Only DemandDiffusion)

The specific model of diffusion considered is the sep-arable model given by

(11) Q =

with the assumption that/ '(X) > 0 and/"(X) < 0. Thismodel has been studied, for example, by Kalish (1983).The first derivative of/being positive indicates that cur-rent demand increases as cumulative demand grows. Thesecond derivative being negative indicates that the rateof increase itself slows as X becomes large.

Theorem 2For the demand function of equation 11:

a. Price monotonically increases over time.b. For any two firms i and j , the lower cost firm has a

higher market share everywhere.c. The market share difference is minimized as r —> » .

ProofSee Appendix B.

This result is the counterpart of theorem 1 when px,has the opposite sign. It is worth mentioning that theproperties of price paths are preserved when the monop-oly model of Kalish (1983) is extended to the undiffer-entiated oligopoly. Once again, the result agrees withintuition that if diffusion (i.e., word-of-mouth effects) ispresent some penetration pricing will take place. Be-cause market shares exhibit at least one converging pat-tern toward the end, we see that the low-cost firm ac-celerates diffusion more than its high-cost rival over thisconvergent interval. Once again, such behavior may oc-cur everywhere though it may depend on particular sit-uations.

Case 3. Px, = 0, dcn/dx,, < 0 (i.e.. Only CostLearning)

The demand function is

(12) Q = hit)p\

where h(t) is an arbitrary function of time. Costs areassumed to decline with experience in accordance withequations 1 and 2.

Theorem 3For the demand function of equation 12:

a. Price declines monotonically over time.b. For any two firms ( and j , if c,, > c,, and m,, > m ,,

firm i's market share relative toy's is increasing at /.c. Market share order reversals are always followed by

a period in which a high-cost firm's output is largerthan its rival's.

d. Market shares converge as r —» °o.ProofSee Appendix B.

The monotonicity of price decline confirms the valid-ity of the monopoly result for the oligopoly studied here.This price decline is qualitatively similar to that in thecase of only saturation in theorem 1. However, decliningprices here do not constitute skimming, but rather reflectcost decline. In fact, the optimal price is lower than thatwarranted by short-term (i.e., myopic) cost conditions.In other words, some penetration is taking place. Resultsb and c indicate that market share reversals are accom-panied by a high-cost firm pursuing a more aggressivepenetration strategy than its low-cost rival. From resultd, the long-run convergence of market shares suggeststhat, overall, high-cost firms tend to pursue more ag-gressive penetration strategies. The market share dynam-ics are harder to quantify beyond the convergence as t-^ =c. One interesting point to note is that, along an equi-librium path, a high-cost firm may find it optimal to pro-duce more than a low-cost rival so as to reduce futurecosts. Whenever such an outcome occurs, the market shareof the high-cost firm must increase.

Models of competition are sometimes useful to delin-eate situations under which interesting dynamics occur.As shown, market share order is preserved under de-mand diffusion and saturation models considered here.Under cost learning, however, market share reversalscannot be ruled out. The dynamics in the neighborhoodof such reversal are easily characterized. It is known thatthere are situations in which small firms overtake largerrivals. The model developed here does not preclude suchan outcome. In a different context of firms competingfor new geographic markets (rather then temporally), Raoand Rutenberg (1979) have shown that market share re-versals are likely if the low-share firm has a high costrelative to its rival. Thus, the condition for overtakingseems to be somewhat more general.

NUMERICAL EXPERIMENTS IN A DUOPOLY

The numerical experiments described in this sectionexplore two strategic issues related to cost learning, "Howdo competitive considerations affect the relative advan-tage of being nonmyopic?" and "What is the benefit ofhaving a cost advantage over a rival before the latterenters?" The analytical complexity makes numerical ex-periments a useful way to answer such questions.

Numerical solutions to a duopoly are obtained underdifferent assumptions about the firms' behavior and dif-ferent values for key parameters. Three kinds of behav-ior by firms are examined, (I) both firms are non-myopic—the base case, (2) both firms are myopic, and(3) one firm is nonmyopic (firm I in the analysis to fol-low) and the other (firm 2) myopic. In all cases, eachfirm is assumed to know its rival's behavior pattern, thatis, it is not surprised by rival's actions.

The firms were assumed to face the inverse demandfunction - ,.

(13)


where s, is the market size at time t. Firm j ' s marginalcost in period /, c,,. was modeled as

(14) , = 5 + 30x2,

Figure 1COMPARISON OF MYOPIC AND NONMYOPICSTRATEGIES FOR DIFFERENT LEARNING RATES

and

Cost decline due to learning is captured by the learningrate / where / = 2" . For example, if / = .8, the costaffected by learning is reduced to 80% of its originalvalue when cumulative output doubles. 5, is an initialcondition. If S, = I, firm ("s initial cost is 35; 5, - 0.5means that firm I's initial cost is 20. A firm with a lowerS has an initial cost advantage.

Firms are assumed to maximize their discounted prof-its over 20 periods, with the discount factor p set at 0.95.Firm f s problem is to choose q^, qa, . •., qm so as tomaximize V, given by

-

PROFITS OF /NON-MYOPIC F IRM/

WHEN RIVAL 1IS MYOPIC /

/

/

• — —

_BOTH FIRMS MYOPIC

/ ' BOTH FIRMS NON-MYOPIC ~

\pROFITS OF MYOPIC FIRM

\

1

N RIVAL IS NON-MYOPIC

\ |-»-ENTRY UNPROFITABLE FOR MYOPIC FIRM\l I 1 1

(15) i= 1.2.

In the preceding problem both firms ans nonmyopic. Thisis the base case. As in the last section, the solution soughtis a dynamic Nash equilibrium. If both firms are myopic,firm / maximizes current profits Y\\ given by

(16) i= 1,2,

once again the solution being a Nash equilibrium, but astatic one. For the case of firm 1 nonmyopic and firm 2myopic, firm I chooses ^H' <?i2> • • •> 9i2o so as to max-imize V, given (y i, ^22- •••^ Ino- Firm 2, in contrast,chooses its output qi, to maximize ITj given ^,,, again ina Nash fashion.

Three parameters were varied for the numerical ex-amples—the learning rate /. the elasticity T), and thegrowth index .f2o/- i- As / becomes smaller, learning ismore rapid. As TI becomes more and more negative, de-mand is more elastic. Finally, if SIQ/S^ is large, initialdemand is small in comparison with final demand. In allcases s, is assumed to grow linearly from î to 520. Thus,when .SJQ/SX is large, there is significant growth, and ul-timate demand is reached slowly over 20 periods. If.however. SIQ = 5 | , at one extreme, all growth ceasesafter period 1. that is, growth rate is rapid. In the ter-minology used in this article, high growth is the sameas slow growth rate and vice versa. If all else is heldconstant and only y\ is changed, for a given price, de-mand will be greater for smaller f\. For valid compari-sons, the market size was adjusted suitably as •t\ was var-ied. Similarly, the size was adjusted when SIQ/S^ wasvaried. The base case had / = 0.8, y\ = -2, and S2o/s]= 1. Finally, Si and 52 both were set equal to 0.95 sothat each firm's initial cost was 33.5.

Effect of Learning Rate Variation

Figure 1 depicts the profits to each firm as learningrate is varied from 0.95 to 0.5 under different behaviorpatterns. Profits are shown as an index with 100 refer-ring to the case in which both firms are nonmyopic. Ascan be seen, except when / = 0.95, profits are higherwhen both firms are myopic than when both are non-myopic. Also, the difference in profits for the two strat-egies is maximized for a learning rate of 0.85. and it isabout 30%. To see the real merit of nonmyopic strategiesfrom a managerial viewpoint, compare the profits of thetwo firms when one is myopic and the other is not. If /= 0.8. the myopic firm is precluded from entry, that is,has zero profits. Only when learning is very slow at 0.95is the myopic firm not penalized.

Eêct of Growth Rate Variation

Tbe growth index was varied from 1 to 5, and theprofit index for different strategies is shown in Figure 2.With / = 0.8, regardless of the growth index, a myopicfirm will never enter. Again, firms have higher profitswhen both are myopic than when both are nonmyopic.

Effect of Price Elasticity

Figure 3 shows the profit index as the price elasticityis varied. In highly elastic markets, when T] = —2, entryby the myopic firm is excluded. If demand is not veryelastic, the choice of strategy does not appear to be veryimportant. This finding is understandable because in thiscase the benefits from learning are outweighed by thelow prices resulting from increased output. Interestingly,

288

350

Figure 2

COMPARISON OF MYOPIC AND NONMYOPIC

STRATEGIES FOR DIFFERENT GROWTH RATES

PROFITS OF NON-MYOPIC FIRM WHEN RIVAL IS MYOPIC

150

e 100

so

BOTH FIRMS MYOPIC

BOTH FIRMS NON-MYOPIC

•ENTRY UNPROFITABLE FOR MYOPIC FIRM

PROFITS OF MYOPIC FIRM WHEN RIVAL IS NON-MYOPICI 1 I I 1 1

2.5 3 4 5

myopic and nonmyopic strategies yield almost equalprofits in highly elastic markets.

It is important to understand why under competitionnonmyopic strategies are preferable. Clearly, they areinferior to the situation in which all firms are myopic.The reason is that if firms are all nonmyopic, competi-tion is increased; competition is for this period's market,next period's market, and so on. However, myopic strat-egies are useful only if the rival is also myopic. Oth-erwise, they are unprofitable and will prevent entry. Kal-ish (1983), discussing a monopoly, suggests that eventhough myopic and nonmyopic strategies may not differmuch in terms of profits, the latter are preferable, es-pecially under diffusion effects. Here, we see that, evenin the absence of diffusion, nonmyopic strategies arepreferable in terms of competition. Though nonmyopicstrategies by all constitute heightened rivalry, each firmindividually is better off being nonmyopic. Reducing ri-valry through myopic strategies requires cooperation byfirms. Firms may try to find strategic ways of achievingthis cooperation without collusion. One way might be tomake experience-based lower costs worth less, For ex-ample, if experience is difficult to transfer from oneproduct to another, the threat of a new product, or model,might force firms to reconsider investing heavily in ac-cumulating experience, encouraging them to be moremyopic. Such issues are possibilities for future research.

Effect of Initial Cost Advantage

The final set of experiments centered on the initialconditions. Both firms were assumed to be nonmyopic.Firm 2's initial condition was set at ^2 = 0.95, and Siwas varied from 0.95 to 0.5. Other parameters corre-

JOURNAL OF MARKETING RESEARCH, AUGUST 1985

sponded to the base case. Figure 4 depicts profits to eachfirm as Si is varied. As can be seen, getting an initialadvantage can be profitable. Though firm 2"s profits de-crease almost linearly as firm i's advantage increases,firm I's profits increase in a convex fashion.

Finally, numerical work such as reported here can beused in specific situations to quantify the effects of tim-ing of entry, as well as strategic advantages of travelingdown the learning curve before rival entry. It should helpin answering "what if" questions that often are of inter-est to management.^

EMPIRICAL APPLICATION

In this section we examine price dynamics in the semi-conductor components industry for eight products. Thefour subsections contain hypotheses, model specifica-tion, description of data and demand function estimates,and analysis and discussion of the price dynamics. Thescope of this work must be viewed in the context of thelimited attempts at empirical work in this area, Bass (1980)modeled both demand and price as functions of time withfirms pricing myopically. This approach enabled him toparcel out the time-dependent diffusion dynamics fromthe effect of declining prices on demand shifts. Anotherrecent article (Baden Fuller 1983) addresses technicalproblems of estimating learning parameters on the basis

'A FORTRAN computer program to solve the duopoly problem forseveral demand and cost functions has been developed by the authorsand Is available from the first author.

Figure 3

COMPARISON OF MYOPIC AND NONMYOPIC

STRATEGIES FOR DIFFERENT ELASTICITIES

350

300

200

- 150

50

PROFITS OF 'NON-MYOPIC FIRM/

WHEN RIVAL .IS MYOPIC /

BOTH FIRMS MYOPIC

BOTH FIRMS NON-MYOPIC

PROFITS OF MYOPIC FIRMWHEN RIVAL IS NON-MYOP<C

ENTRY UNPROFITABLEFOfi MYOPIC FIRM

1.25 1.31 1.67 2.00 2 50 3 33


Figure 4EFFECT OF INITIAL COST ADVANTAGES ON PROFITS

{S, = 0.95)

2000 -

1800

1600

</*

m

- 1400o

0.

1200

1000

8001.0 0.9 0.8 0.7 0.6 0.5

of only price data assuming price parallels cost. Day andMontgomery (1983) also elaborate on issues that shouldbe examined through empirical work.

Hypotheses

For the application here, the sole endogenous dynamicelement is cost learning. The products are nondurablesand therefore saturation is not modeled. Demand growthis assumed to be a function of only time (i.e., the de-mand equation is of the equation 12 type). Summingequation 9 across the n firms in this case leads to**

(17) p, - + l))r j i^c] exp(-r(T - t))d-r.

Denote exp(-r) by p, Then equation 17 can be writtenas

(18) p.=

Denote (iri/(mn + 1)) 2, c , by Z,. For sufficiently smalltime intervals, equation 18 can be approximated by

(19)

The interpretation of Z, is that it is the myopically op-timal price for the oligopoly at time /. So, the dynami-cally optimal price p, is a weighted combination of themyopically optimal price and the price next period.Equation 19 is a representation of price dynamics underthe assumption that firms are making optimal decisions.An observed price path will refiect myopic behavior if0 = 0 where ( is an estimated weight in an empiricalversion of equation 19. Such an outcome is possible fortwo reasons, (1) firms actually behave myopically (i.e.,nonoptimally) or (2) the optimal sequence of prices p^pi*. . . is close to (i.e.. largely mimics) the sequence of priceswhich, in fact, would result from myopic behavior. Inboth cases, an observer will be unable to impute non-myopic behavior given the observations on prices. Sta-tistically, this means that an observer could not rejectthe null hypothesis that $ = 0, Here, firms are assumedto behave optimally. However, we show that dependingon the values of key product-market parameters, dynam-ically optimal prices will mimic myopic prices more orless. We argue that cross-sectionally (i.e., across prod-ucts), 0 will systematically differ, with lower values cor-responding to situations where a priori, on the basis ofthe key parameters, optimal prices can be expected tobe close to myopic prices.^

The question now is, "Is it possible to say what factorslead to myopic and optimal price paths being 'close'?"To answer, consider the cost learning rate. Dynamic op-timality requires that in order to benefit from the learningcurve, current output should be larger (i.e.. current pricelower) than warranted by myopic considerations. Be-cause the cost function C is convex, a given level ofoutput produces more learning benefits when cumulativeoutput is lower (i.e., cost higher). In other words, ob-taining lower costs becomes increasingly difficult as cu-mulative output increases and costs go down. Once costshave been reduced, optimal strategies will be more likemyopic strategies. Consequently, if learning is rapid,fewer periods will yield optimal prices different frommyopic prices in comparison with a slow learning situ-ation. This outcome indicates something about the du-ration over which optimal and myopic prices differ. Asecond aspect is the magnitude of the difference. Iflearning is extremely low. or absent in the extreme, de-partures from myopic strategies will be muted. In fact,if learning is very slow or very rapid, observed priceswill appear to be consistent with myopic behavior (i.e.,p = 0). Thus, intermediate learning rates would yieldoptimal strategies that are very different from myopicstrategies. This effect has also been observed by Spence(1981).

^Because diffusion and saturation are absent, teims containindrop out.

'This would be valid only for products in a given industry so thatthere are no systematic differences in the discount rates across prod-ucts. Across industries, p will also depend on the risk characteristicsof the industry. For the application here, all products are in a singleindustry.


Next, from equation 19 we see that the weight p isassociated with next period price. Equation 19 can bewritten recursively as

(20) p,= Z, + i\

-H p^(Z, . : - Z , ) + . . . 1 .

Because of convexity of the cost function the differences(ZT - Z,-i), T = r + l , r + 2, . . . converge to zero, andalso future cost reductions will be smaller than before.Market growth, however, can compensate for this slow-ing of cost decline. In growing markets, current priceswill reflect events farther into the future, and so 0 willbe different from zero.

Finally, from equation 19 it is obvious that p, is smallerthan Z, because /7,4i < P,- Current profits are sacrificedfor future benefit resulting from increased experience. Inhighly elastic markets, a small sacrifice in price will re-sult in large increases in quantity sold and greater cu-mulative experience. This implies that Z, - p, will besmall in highly elastic markets, or 0 will be close to zeroreflecting the small difference in myopic and optimalprices.

Putting together the preceding arguments, we can statethe hypotheses to be tested: (1) extremes of slow andrapid learning, (2) absence of market growth, and (3)high elasticity all point to the likelihood of observing |close to zero. Intermediate learning rates, market growth,and low elasticity all would correspond to high ^.

These hypotheses can be illustrated also through nu-merical experiments in a duopoly by varying learningrates, growth rates, and elasticities. Two mieasures of thedifference between myopic and nonmyopic prices werecomputed. Define

(21) = 1,2,

where T is the length of the planning horizon for thefirms. The quantity Nr is a normalized measure of thedifference between optimal and myopic prices. Becausep, < Z,, Nj < 0. For a myopic strategy p, = Z, every-where, so Nj- = 0. Thus, as |A 7-| gets bigger, higher val-ues of 1 are likely. A second measure of the extent ofnonmyopicity is also useful. Empirical work consists ofsampling prices over some horizon. Even though opti-mal prices may be very different from myopic prices, itis necessary to know how many observations in the sam-ple reflect this difference. A useful measure is

(22) D,= t<T.

Now D,e (0,1). Suppose T = 20, and it is found that Djis 0.85. This means that 85% of the difference betweenmyopic and nonmyopic prices is to be found in 15% ofthe observations. The remaining 85% of the observationscontain prices that on average are less than 1% awayfrom myopic prices. In other words, high values of D^would suggest 1 closer to zero relative to lower values.

The values of A2o and D3 were computed for a duopoly

identical to the one in the last section. Results are re-ported in Table I for various learning rates, growth rates,and elasticities. From Table 1, looking at learning rateswe see that {Njol is maximized for the intermediate rateof 85%, consistent with the previous conjecture. Sec-ond, Dj is minimized for learning of 90%. For growth,Di is minimized and A2o 's maximized when there is sig-nificant market growth. Finally, there is a weak rela-tionship of less elastic markets having lower values ofDi than highly elastic markets. The numerical experi-ments thus generally confirm the hypotheses. Intuitively,^ is a measure of how close optimal and myopic pricesare. The estimate ^ is then used, in conjunction withestimates of relevant model parameters, to examine thehypotheses.

Model Specification

The basic model of price dynamics is equation 19. Itcan be rewritten as

(23) p, - p/7r+i = in-r\/(,nri + 1))(1 - P)C, + ii,+ i

where:

(24) C, -•

Table 1COMPARISON OF MYOPIC AND OPTIMAL PRICE PATHS

FOR DIFFERENT PARAMETER VALUES

Fixedparameters

ElasticityT) = -2.0

Growth indexSx/s, - 1.0

-,

Elasticityf\ = -2.0

Learning rate/ = 80%

Growth indexS-K/S, = 1.0

Learning rate/ = 80%

Parametervaried

Learning

slow

n iid

Growth

none

significant

Elasticitylow

•

high

Learningrale

95'908S807570656050

Growthindex

1.0001.1251.6672.5005.000

-Vl.n1.25!.431.672.002.503.33

-0.455-1.275--1.648-1.600-1.397-1.092-0.855-0.622-0.475

-1.600-1.714-1,932-2.052-2.693

---

-

.456

.495

.582

.612

.536

.652,391

-0.263-0,603-0,844-0,971-0,988-0.912-0.798-0.662-0.475

-0,971-1.002-1.055-1.129-1,272

-0,845-0,869-0,905-0,937-0.926-1,028-1,036

Dj(%)

57,847.351,260.770,783,593.3

100.0100.0

60,758.554.655,047.2

58.058.157.258.160.362.274.5


Tbe error term Um is addressed subsequently. The quan-tity C, is the average of the marginal costs of the firmsin the industry. The c,, are assumed to satisfy equation2. Clearly, the model of competition suggests that bothn. the number of firms, and the actual costs cv of thedifferent firms influence the path of prices. Unfortu-nately, neither of these figures is available for tbe ap-plication at hand. As is often the case, cost data are con-fidential. However, if n is constant over the period ofobservations, it is possible to show, after some algebra,that C, will be a decreasing, convex function given theassumptions on c^,. A parsimonious representation of C,using the only observable X, is^

(25) 7<0,

where y is the industry learning parameter and X, =2U'i Qk- The specification preserves tbe convexity of C,.Using equation 25, the model of price dynamics be-comes

(26) p, - ^p,î = 6n + 8|X?' -t- u,_,.

Tbe error term u,+ i results for two reasons. One is theapproximation in going from equation 18 to 19. A sec-ond reason is that though p, is a weighted combinationof current cost and future price, the latter is stochastic.The observed price will be different from the forecastprice on which current decisions are based.

As will be seen, equation 26 is used to estimate -y andp. The two demand parameters, the growth rate andelasticity, are obtained by estimating a demand function.

The Data and Demand Function Estimation

Tbe empirical work reported here covers a class ofsemiconductor components over part of or the entire pe-riod 1955-1972. Some of the earlier model develop-ments anticipated the particular application. For exam-ple, the model of a market with undifferentiated productswould be acceptable for semiconductor components. Ifnot exactly alike, different firms' components tend to behighly substitutable. We use data comprising time seriesof quantity and price at the level of product class. Tbemissing components of the data are (1) the changes inthe number of firms over time, (2) the changes in in-dividual product designs over time. (3) changes in size/characteristics mix in tbe product class over time, and(4) the changing role of imports. Some of these com-ponents are more important than others. A more detaileddiscussion of the quality of data and potential conse-quences of inadequacies is given by Bass and Rao (1983).

Tbe data used are from a study by Webbink (1977)conducted for the Federal Trade Commission and con-tained in the Staff Report on the Semiconductor Industry.

The data interval is annual as this was the only one re-ported. Three basic pieces of information are provided,,(1) the quantity of devices shipped, including all do-mestic shipments as well as exports, (2) the average valueof shipments as appearing on invoices, a proxy for price,and (3) the value of shipments of all electronic goods,a proxy for business activity in the electronic goods sec-tor of the economy. All the information is in tbe pub-lication of the Electronics Industries Association, Elec-tronic Market Data Book, 1975 edition. Webbink alsoprovides the manufactured goods component of thewholesale price index abstracted from Economic Reportof the President, 1976. Tbe data in the Webbink rept^rtcover 10 components: electron receiving tubes, germa-nium transistors, silicon transistors, germanium diodes,silicon diodes, Zener diodes, chassis mounted rectifiers,linear rectifiers, digital integrated circuits, and linear in-tegrated circuits. Of these, two are excluded from thestudy, electron receiving tubes, which have been aroundsince the 1930s and did not seem appropriate for the study,and germanium transistors, which had a relatively shortlife and were quickly replaced by silicon transistors.

The demand function estimated was straightforward.In addition to price, two explanatory variables were in-cluded, time from introduction, to capture growth, aphenomenon of interest, and the shipment of all elec-tronic goods, to capture the effect of business cycles.The specification for the quantity demanded Q, in yearf is

inQ, = a,/m + , + '(\lnp, + e,

where:

''To the extent that this parsimonious specification fails to capturethe behavior of C,. there is the adverse consequence of specificationbias.

/ is the time in years from date of introduction,G, is tbe value of shipments of all electronic goods in

year t, andp, is the price in year t,

with G, and p, measured in constant (1967) dollars. Thechoice of specification recognizes that the two main in-fluences in the electronic components markets have beendeclining prices and the steady increase in new appli-cations or sales. These two influences are captured by p,and t. Omitted are the prices of substitutes and comple-ments. Sometimes substitutes are identifiable, for ex-ample, silicon transistors may be substitutes for ger-manium transistors. Even here, however, one silicontransistor may compete with another. Webbink's studyreports unsuccessful attempts to capture the effect ofsubstitutes. Tbe demand for complements is probably animportant determinant of demand for individual com-ponents. Tbe identification of particular complementaryrelationships between products is not always clear. How-ever. G, at least partly captures the effect of comple-ments and therefore this omission may not be serious.

The method of estimation used was either ordinary leastsquares (OLS) or a maximum likelihood autoregressiveprocedure (ARl) due to Beach and MacKinnon (1978).

292 JOURNAL OF AAARKETING RESEARCH, AUGUST 1985

Table 2GLOSSARY OF SYMBOLS AND NOTATION

STEGDIESDIEZDECRELie

LREDie

NOBQt

G

PXy1

DWOLSWLSARl

Silicon transistorsGermanium diodesSilicon diodesZener diodesehassis mounted rectifiersLinear integrated circuitsLinear rectifiersDigital integrated circuitsNumber of observationsQuantity shippedTime from date of introductionValue of shipments of all electronic goods output inconstant (1967) dollarsPrice in constant (1967) dollarseumulative outputLearning parameterLearning rate, defined to be 2"'Discount factorDurbin-Watson statisticOrdinary least squaresWeighted least squaresAutoregressive least squares, maximum likelihood.

both procedures being available on TSP.^ Table 2 is aglossary of symbols and Table 3 reports estimation re-sults. Of the eight components, all but one have signif-icant price elasticities with demand being least elastic forsilicon diodes and Zener diodes and most elastic for dig-

ital integrated circuits. The first five products in Table3 are classified as "significant growth" because they have&! significantly greater than zero. The last three are char-acterized by no growth. The usual r-statistics, Durbin-Watson statistic, R', and f-statistic are all reported inTable 3. With the exception of germanium diodes, theregressions are all meaningful.

Analysis of Industry Price Paths

Having estimated the demand functions, we now canexamine the price paths. The analysis has two steps. Thefirst is to estimate the pair (7,^) using the model of equa-tion 26. Then ^ is compared across products in light ofthe estimates a, f| from the demand functions and y fromthe model of equation 26. This comparison is made tosee if the data call for a rejection of the hypotheses.

Equation 26 was estimated by either weighted leastsquares (WLS) or ARl, a procedure alluded to previ-ously. The WLS procedure is OLS with a weightingmechanism to account for heteroscedasticity. The left sideof equation 26 is monotonically declining, approxi-mately as p,. We therefore decided to use a weight thatwould reflect a price decline of 70% with each doublingof X,. The weight used was X"^'' ' ' . The same weightwas applied for all products estimated with WLS. In onecase where serial correlation was present in u the ARlestimation procedure was used.

Estimation consisted of a search for p in the half-openinterval [0,1) and for -y < 0.^ The grid for p was 0.02.

'All models were estimated by both OLS and ARL Results of theformer are reported in cases where the serial correlation, as measuredby the r-statistic computed by ARl, is not significant.

''An alternative to searching for p is to regress price on lagged priceand X^. This procedure would estimate 1/p. TTiis method was notfound to be effective because it is required that p e [0,1).

Table 3ESTIAÂTES OF DEMAND FUNCTIONS

(Model: ttiQ = ao + a^tnt 4- ajInG + •f\lnP + 0)

Product

STE

ZDE

SDIE

GDIE

Lie

eRE

LRE

Die

'Significant"Significant"Significant

Estifftotionmethod

ARl

OLS

OLS

ARl

OLS

OLS

ARl

OLS

at the .01 level.at the .05 level.at the ,1 level.

NOB

18

14

16

16

9

12

12

9

-2.85(-0.79)-2.98

(-0.75)-3.99

(-0.85)-2.99

(-0.49)-10.66'(-8.90)

4.67'(1.89)

-1.07(-0.23)-4,49

(-1.24)

Estimatesit-statistics)

a,

2.03'(10.79)

0.90"(3,92)0.87'(3.70)0.63'(1.49)1.07'

(16.67)0.03

(0,23)-0.13

(-0.30)-0.84

(-0.94)

0.32(0.85)0.49

(1.16)0.78"

(1,54)0,76

(111)1.28'

(10.21)-0.11

(-0.40)0,49(0.99)1.20"

(3.16)

V

-0.73'(-6.05)-0.43'

(-3.71)-0.44*'

(-2.49)0.24

(0.55)-0.98'(34.3)-1.37'

(-9.88)-1,09"

(-2,23)-2.25'

(-3.65)

DW

1.24

1.83

1.31

1.43

2.98

1.47

1,18

1.71

,.

0.972

0.983

0.982

0.351

0.999

0.987

0.728

0.998

F

176

247

272

3.7

19776

287

811

1432


For 7, the rate / = 2^ was varied in steps of 1% from99% to as low as was needed. The criterion was to min-imize the sum of squared residuals when WLS was usedand to maximize the likelihood function over (p,3.7) whenARl was used, where p is the serial correlation coeffi-cient of a,. Last, in the final estimates reported here 80is set to zero in all cases except one, chassis mountedrectifiers. This procedure was consistent with the obser-vation that lim p, appeared to be zero for the other prod-

ucts. One less parameter was particularly helpful withrelatively few observations.''

For two of the products in the sample, silicon tran-sistors (STE) and Zener diodes (ZDE), it was not pos-sible to locate a maximum for the likelihood functionover 3 in the interval [O.I), that is, $ appeared to belarger than one. For the remaining six products the re-sults are reported in Table 4. In all cases the model fitis good, and S| is significant at the \% level or better.Moreover, the DW statistic is in the acceptable range.Though 8,) was omitted for all products except chassismounted rectifiers, in the final estimated model. So ifincluded is not significantly different from zero for thoseproducts for which it was omitted. The leaming rateranged from a low (slow) of 78% for digital integratedcircuits to a high (rapid) of 38% for chassis mountedrectifiers. All leaming rates are in the range often re-ported for semiconductors except the 38% for chassismounted rectifiers. Appropriately, in that instance ^ =0 suggesting that, with such rapid leaming, observedprices are close to myopic prices. For the rest, exami-

Table 5COMPARISON OF PRICE PATHS ACROSS PRODUCTS

ProductCRELieLREDieSDIEGDIEZDESTE

GrowthRapidSlow"RapidRapidSlowSlowSlowSlow

Elasticity-1.37-0.98-1.09-2.25-0.44

—-0.43-0.73

Learningratef%J38717578706688"88"

Estimateddiscountfactor

<^>

0.000.160.360.380.620.86>I>1

•"A question arises of whether y should be estimated using the dis-count i'actor p employed by the industry or in conjunction with p, theimputed weight. Fortunately. 7 is not sensitive lo the choiee of p. andthe details are given by Bass and Rao (1983).

"Appears to be misclassified.'Corresponds to p = 0.7. 0.8. or 0.9.

nation of $ must be made in conjunction with the de-mand parameters.

Table 5 summarizes the cost and demand parametersacross products along with 0. Products are listed in orderof increasing (i. The hypotheses suggest that little growth,high elasticities, and learning rates at either extreme, lowor high, should all indicate lower ^. For three of theproducts with no growth, 0 is indeed less than 0,4. Inparticular, for chassis mounted rectifiers with an ex-tremely high leaming rate, (5 = 0. Of the three productswith market growth, two have ^ = 0.6. One which isnot price elastic has the higher ^, again consistent withthe theoretical prediction. There appears to be one prod-uct. Lie, which despite growth has a (5 = 0.16. It isprice elastic, however, and the leaming rate is higherthan for two other products with no growth. Overall, thisproduct appears to be misclassified with respect to growth.Finally, for the two products for which ^ > 1, leamingis intermediate at 88%—the slowest rate for any product

Table 4ESTIMATES OF PRICE PATHS

(Model: p, - pp,+ i = 5o + 61X/ 4-

ProductSDIE

GDIE

U C

CRE

LRE

Die

EstimationmethodWLS

WLS

WLS

WLS

ARl"

WLS

NOB

14

14

7

10

10

7

0.62

0.86

0.16

0.00

0.36

0.38

So—

—

—

0.558(28.27)

—

—

Estimates'(t-statistics)

6.894(11.06)

2.317(7.58)10.839

(40.49)43.50

(15.24)2.505

(143.53)7.663

(55.35)

y

-0.515

-0.599

-0.494

-1.396

-0.415

-0.358

Implied1%

70

66

71

38

75

78

DW

2.10

1.71

1.97

2.04

1.91

2.49

0.856

0.721

0.994

0.963

0.999

0.994

"All coefficients (8(, and 8,) significant at the .01 level."Estimate of serial correlation coefficient, p = -0.894 (-7.39).


in the sample—and moreover they are characterized bymarket growth, In summary, at least from rank order of( it appears that the hypotheses developed previouslycannot be rejected. However, because only eight prod-ucts were examined, stronger claims must await moreempirical studies. The variations in price paths acrossproducts depend on the demand parameters—growth andelasticity and the cost parameter learning rate.

Though the empirical findings are consistent with thetheoretical analysis, it is clear that entry and exit mustbe handled explicitly. For example, the current modelsuggests that the number of firms and the distribution ofcosts across firms should enter the analysis. Includingother features in future research should provide sharperinsights into dynamic price behavior.

CONCLUSIONS

We have extended the analysis of dynamic pricing ina monopoly to an undifferentiated oligopoly by tacklingthe problem analytically, numerically, and empirically.The main analytical results pertain to market share dy-namics in the three polar cases of only demand diffusion,saturation, or cost learning. In practice, of course, thesefeatures may be present simultaneously. We see that,though market share reversal is not possible when onlydemand diffusion or saturation is present, it cannot beruled out under cost learning. Very possibly, this is oneof the reasons for the emphasis on both market share andexperience by many consulting companies. The analyt-ical developments also provide a theoretical model spec-ification to be used in empirical work.

The numerical experiments, restricted to a duopoly,provide some managerial implications. Previous workhas resulted in an inconclusive debate on the value ofnonmyopic strategies over myopic ones for a monopo-list. We show that once competition is introduced, non-myopic strategies clearly dominate. Another, somewhattentative, finding is that strategic thinking might betterconcentrate on actions before rival entry rather than afterentry.

The empirical analysis confirms that industry price pathsare explained on the basis of not just the cost learningparameter, but also the demand parameters of growthand elasticity. Given the general paucity of empirical workin this area, a useful beginning appears to have beenmade.

Clearly, many issues remain to be resolved. Withinthe present model it may be interesting to characterizemore fully the market share dynamics and reversals.Mathematical developments of sufficient conditions fordynamic equilibrium are reported by Bass and Rao (1983).These, too, might be strengthened. The most importantinnovation would be explicit models of entry and exit.The former have received some attention by Eliashbergand Jeuland (1983). Another useful direction would beto include product differentiation in the present model.Other marketing mix variables such as advertising alsoremain to be integrated. Finally, the all-pervasive factor

of uncertainty must be recognized more explicitly in suchmodels.

APPENDIX ADERIVATION OF EQUATION 9

The problem faced by the representative firm / givenby equations 5 through 7 can be solved by using optimalcontrol theory. To see how this can be done, first formthe current value Hamiltonian //y, given by

(Al) //, -

where X,, and ili, are the adjoint variables associated withthe n differential equations 6. From the Pontryagin Max-imum Principle {see, e.g.. Sethi and Thompson 1981),the necessary conditions for solving /'s problem are givenby equation 6 and

(A2)

(A3)

(A4)

(A5)

dH./dq,, = 0

d\i/dt = -dHjdxu +

) = 0 and lim exp(-r/)»t(,(/) = 0.

Condition A2 corresponds to the requirement that qi,maximize H,. The equality obtains for an interior solu-tion, implicitly assumed here. Conditions A3 and A4 arethe adjoint equations with the usual interpretation that \and IIJ, represent the rate of change of V, with respect toXi, and Xj,, respectively. Conditions A5 are the transver-sality conditions for the infinite horizon discounted prob-lem.

The differential equation A3 has the following solu-tion, which also satisfies A5.

(A6) i = exp(rr) c,,exp(-rf)

Similarly, the solution to A4 is

(A7) ill, = exp(r/) cxpi-rT)(qi,dp/dXj.,)di.

Condition A2 yields, for the problem at hand.

(A8) p, - c,, + K + q^dp/dq^, = 0.

Putting A6 in A8 results in

(A9) p, + qu

-f-exp(-r(T - t))dT

,T)exp(-r(T -

COMPETITION, STRATEGY, AND PRICE DYNAMICS

There are n equations of the type A9 corresponding tothe n tlrms. For the demand function considered here.

dp/dq, = dp/dQ,

i, = dp/dX,

/ = 1, 2 n,

1= 1,2, . . . , n .

and

Denote

(A 10)

(All)

when T), is the elasticity of demand. Then A9 can bewritten as

(A12) p,(l + (m,i,/f\,)) = r i C;, exp(-r(T -

*?,TP.VT exp(-r(T -

where m^ is firm Ts share of current output Q,.

APPENDIX BPROOF OF THEOREM 1

1. Summing the n equations (2-9), for this case, re-sults in

(Bl)

where:

A = n 0. (.• \L < - , and

To establish the monotonicity of/>. first note that Ap> K everywhere, and as / —• =o, p, is declining. Now forp to have any increasing segments, it is required that atp = 0, p > 0. Proof consists of showing that p = 0 =>p > 0. Differentiating B1 with respect to /,

(B2) Ap+Ap =

where:

rAp- rK

A =

or(B3) A = (p/p)l\ - \i.(A ~ n)] - a(A - n).

Putting B3 in B2 gives

(B4) Bp = (Q/iiMp'^-') + rAp - rK + a(A - n)p

where:

B =A+ I - {Q/aMpn > 0 (.• Q < aMp" = X).

Differentiate B4 with respect to / and let p - 0 to get

Bp = -ag/jiA/p^"' - (r + a)a(A - n)p

> 0 (•.• A - n < 0 ) = ^ p > 0

2. Denote v = (m,, - m,,). Let k = c, -from equation 12,

(B5) >

295

c) < 0. Then,

As / -> X, y > 0, because t], k < 0. Differentiate B5with respect to t and let >' = 0. Then

(B6) y = r\i.aMp*''^k/Q<0.

Thus if V, = 0 anywhere, y < 0. T > r. But y^ > 0. So>' > 0 everywhere.

3. The difference in sales of / and 7 is given by Qy.Multiply both sides of B5 by Q. Note that both p andthe terms in the bracket are negative and minimized ast —* ^. So Qv is maximized as / —> (».

Q.E.D.

PROOF OF THEOREM 2

1. The strategy of proof here is identical to that oftheorem 1. and is not repeated. Denote a = (11/(1 +nr\)) > 0. Then note that

(B7) p =

With /* = 0,

+ rp- raK.

B e c a u s e / i > O a s r — > o o , p > 0 e v e r y w h e r e .2 . It can b e s h o w n tha t

111* +L Ji

(B8) y=(\/p)\

and

(B9) y = iy/j\)f{X)p''(a - j]) + (r/p)[aKy - -^k].

Thus y < 0 at y = 0 because TI, it < 0. This implies y> 0 everywhere.

3. In B8 the terms in the brackets on the right sideare minimized as r —» » and p is maximized as f —> =0.So y is minimized as / ^» =».

PROOF OF THEOREM 3

1. For this case

(BIO) p ^ ar I K exp(-r(T - t))d'T.

Because K., is monotonically declining in t,

p, < aK,.

From BIO

(BID p= -arK, + rp<Q,

2 y = {-^/pV C K exp(-r(T - t))dT. DifferentiatingBll with respect to t.

(B12) y=y{r

If fc, > 0 and y > 0, clearly y > 0.


3. Next, if market share ordering changes, y = 0, ^sign (y) = sign (k,) :^ higher cost firm's share increasesrelative to rival's.

4. Finally, as f —»•=», A:, ^ 0 because of convexity ofC.

Q.E.D.

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Documents

Competition, Strategy, and Price Dynamics