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Product introduction decisions in a duopoly
European Journal of Operational Research 152 (2004) 745–757 www.elsevier.com/locate/dsw
Production, Manufacturing and Logistics
Product introduction decisions in a duopoly Gilvan C. Souza
*
The Robert H. Smith School of Business, University of Maryland, College Park, MD 20742-1815, USA Received 26 March 2002; accepted 16 September 2002
Abstract In this paper we model a dynamic environment with two firms that fight for share of industry sales and profit in a market with constant size. They capture share by repeatedly introducing new products. Price changes from period to period reflecting each firmÕs learning. We formulate the problem as a repeated game, where each playerÕs decision at each period is to introduce a new product. We find that, in general, each firmÕs frequency of product introductions (clockspeed) in equilibrium decreases as its fixed product introduction cost increases. Further, we find that a higher rate of manufacturing learning results in a higher clockspeed and in a higher market share, and also results in significantly worse profits for the competitor. The results underscore the importance of manufacturing expertise in a firmÕs ability to introduce new products. Ó 2002 Elsevier B.V. All rights reserved. Keywords: Strategic planning; Research and development; Game theory
1. Introduction In this paper we use a stylized model to address a dynamic competitive environment with two firms that fight for share of industry sales and profit in a market with constant size. They capture share by repeatedly introducing new products. Price changes from period to period reflecting each firmÕs learning. We use dynamic games to study this scenario. Abernathy and Utterback (1978) propose a framework that describes the dynamics of innovation in an industry. As an industry starts, the technology is still evolving, and several firms enter and exit the industry with different design architectures. Once a dominant design emerges, the basis of competition shifts to primarily minor product innovations due to choices in process design, and only a few firms survive. Our research primarily addresses the industry after the emergence of a dominant product design, or the specific (mature) phase in Abernathy and UtterbackÕs framework. Thus, our stylized model assumes a duopoly, a
*
Tel.: +301-405-0628; fax: +301-405-8655. E-mail address: [email protected] (G.C. Souza).
0377-2217/$ - see front matter Ó 2002 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(02)00709-9
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mature market of constant size, and that consumers are indifferent to two equally priced products introduced at the same time. We find that, in general, each firmÕs frequency of product introductions (clockspeed) in equilibrium decreases as its fixed product introduction cost increases, however, there are non-intuitive results. For example, there are instances where a higher fixed product introduction cost for both firms result in an equilibrium where both firms are better off. Further, we find that a higher rate of manufacturing learning results in a higher clockspeed and in a higher market share, and also results in significantly worse profits for the competitor. The results underscore the importance of manufacturing expertise in a firmÕs ability to introduce new products. From a theoretical perspective, this is one of the first papers to formally address strategic product introduction, that is, when and how frequently should a firm introduce new products. There exists an emerging literature on this subject, but it is mainly based on case studies (Fine, 1998), or is empirical (Mendelson and Pillai, 1999). Although there exist papers that use differential games to address capacity and pricing decisions (Gaimon, 1989), we are not aware of published papers that use dynamic games to model competition in product introduction. This paper is organized as follows. In Section 2, we review the related literature. In Section 3, we introduce the mathematical model. In Section 4, we show numerical results for the case of identical competitors; non-identical competitors are analyzed in Section 5. We conclude in Section 6.
2. Literature review Machine replacement models (MRMs) analyze the decision of when to replace an aging machine with a new one based on replacement and maintenance costs, technological forecast, and the state of machine deterioration. The literature in MRM is extensive; for a review see Nair (1995). Our model is similar to MRM in that the introduction of a new product in our model is based on the ages of the firmÕs and the competitorÕs products whereas the replacement decision in MRM is based on machine deterioration. In MRM models, however, there is no competition. A related stream of literature approaches the problem of adopting a new technology from either a decision-theoretic perspective (Balcer and Lippman, 1984; McCardle, 1985; Kornish, 1999), or from a game-theoretic perspective (Reinganum, 1981; Mamer and McCardle, 1987). The former is similar to MRM, except that there is uncertainty about the revenues provided by the new technology. Regarding the latter, Reinganum (1981) proves the existence of a pair of symmetric Nash equilibria on the timing of technology adoption in a duopoly: one firm adopts early and the other firm adopts later––it is never a Nash equilibrium for both firms to adopt the technology at the same time. In our dynamic game, we find that, in general, firms prefer not to introduce new products simultaneously. Regarding the timing of product introduction, a stream of literature analyzes the time-to-market and performance trade-off for a single product development project (Cohen et al., 1996; Bayus, 1997; Bayus et al., 1997). It is not always in a firmÕs best interest to speed up a product development project, because the monetary benefits of being first to market may be outweighed by higher development costs. Another stream of literature analyzes the best time to introduce a product line extension (Wilson and Norton, 1989; Moorthy and Png, 1992). A state-of-the-art survey of decisions in product development projects, including timing, is provided in Krishnan and Ulrich (2001). Our model does not deal with tactical considerations at a single project level. Instead, we assume a fixed time for product development, and focus on strategic product introduction––when and how often to introduce products. Fine (1998) describes how different industries have different clockspeed––the frequency of new product introductions in an industry. He suggests that a firmÕs clockspeed should match its industryÕs, a claim that has some empirical support (Mendelson and Pillai, 1999). Eisenhardt and Brown (1998) emphasize a more
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proactive approach to strategic product introduction, and suggest that the best strategy is time pacing–– introducing a new product at regular time intervals, and not when opportunities arise (event pacing). Carrillo (2001) uses a decision-theoretic model based on diffusion curves to show that high product introduction costs may provide an incentive for a firm to have a lower clockspeed than its industryÕs, a result that is in line with ours. Two closely related papers to this are Morgan et al. (2001) and Souza et al. (2002). Both papers dynamically model a firmÕs decision to introduce new products, however, their models are from a decisiontheoretic perspective: firm A optimizes its product introduction decisions based on an estimate of firm BÕs behavior. Souza et al. (2002) assume random demand, include the simultaneous optimization of production and inventory, and focus on an infinite horizon. Morgan et al. (2001) assume deterministic demand, include the possibility of crashing product development projects (that is, lead time for a project is also a decision variable), and focus on a finite horizon. As in this research, both pieces conclude that a firmÕs clockspeed is highly dependent on its product introduction cost and market characteristics, however, firm AÕs optimal solution may not constitute an equilibrium in a game-theoretic sense. Hence, their approach is better applied to a situation as described in Eisenhardt and Brown (1998), in which firm BÕs product introduction strategy is independent of AÕs. We assume in this paper that both firms are rational, and we seek the market equilibrium, although we do not include production and inventory decisions for our model.
3. Model Consider a competitive stationary market––the market size l is the same each period––comprised of two firms, A and B. This market resembles, for example, demand for large consumer appliances, such as water heaters or refrigerators. The planning horizon is infinite and divided into time periods of equal length. Firms A and B are rational competitors, and compete by regularly introducing new products to grab market share through a mechanism described in Section 3.1. Although we consider the market size fixed, this model can also be applied to a situation where demand is stochastic, but can always be met, for example, in a make-to-order environment, or when there is full backlogging at no penalty. We think of a period here as, say, six months, and this assumption is reasonable for such a long period. Furthermore, the game is intractable if inventory is considered (Souza, 2000). We use superscripts A and B to identify the two firms.
3.1. The market share model Product price and performance are the major factors that influence consumersÕ purchase decisions (Krishnan et al., 1999), and prices typically decay with age (Krishnan et al., 1999). Because we are focusing on product introductions, and to keep the model tractable, we do not treat price as a decision variable, and consider that it decays with the age of a product as follows. Variable cost decreases with age according to a learning curve, and a firmÕs markup is non-increasing with age (that is, firms are able to offer a premium on newer products). For tractability, we assume learning with age (time) as opposed to learning with volume. We note that both types of learning are observed in practice (Cogan and Burgelman, 1989). We assume that a productÕs age is a proxy for its performance. We use a market share attraction model h d (Karnani, 1985), and set the attraction of a product as K ¼ ðpriceÞ ðageÞ , where h > 0 and d > 0 are industry-specific parameters that define the marketÕs elasticity to price and technology, respectively. Denote AÕs (BÕs) product age by iðjÞ. ProductsÕ ages are integers bounded by 1 and n. A product of age 1 is a new product, and a product of age n is the oldest possible product. Firm AÕs (BÕs) variable cost and A B B A B percent cost markup are denoted by vA i and si 1 (vj and sj 1), where si , sj > 1. Firm AÕs variable cost
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G.C. Souza / European Journal of Operational Research 152 (2004) 745–757 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0
1
2
3
4
5
6
7
8
9
10
A
B
*
AB
*
B
A
B
*
AB
*
δ=1
δ=2
Fig. 1. Firm AÕs market share over time when A(B) introduces a new product every 3(2) periods. A
A ln / = ln 2 at age i reflects a learning curve vA , where 0 < /A 6 1 (Yelle, 1979). Firm AÕs product price at i ¼ v1 i A A age i is thus si vi , which is non-increasing in i; similarly for firm B. Consequently, AÕs product attraction is B h A ln /A = ln 2 h d KA ¼ ðsA Þ i ; BÕs product attraction is KB ¼ ðsBj vB1 jln / = ln 2 Þ jd . Firm AÕs market share is i v1 i
cA ij ¼
KA K þ KB A
and firm BÕs market share is 1 cA ij . The notation explicitly indicates that market share in our model is only a function of the ages. Example 1 illustrates the model. Example 1. Suppose firm A(B) introduces a new product every three (two) periods, with /A ¼ /B ¼ 1 (no B learning), h ¼ 1, and sA j ¼ sj for all j. At time t ¼ 0, firms A and B have products of ages 1 and 2, respectively. Fig. 1 plots firm AÕs market share as a function of time for d ¼ 1 and d ¼ 2. We also indicate in the horizontal axis when a new product introduction by firm A, firm B, or both simultaneously, occurs. The curve for d ¼ 2 has a higher amplitude than the curve for d ¼ 1, because the market is more elastic to the difference in product ages when d ¼ 2. At t ¼ 3, a simultaneous product introduction by firms A and B occur, and the market is evenly split. The market is also evenly split at t ¼ 4, when both firms have products of age two. At t ¼ 5, however, firm AÕs market share drops considerably, because firm B has a new product in the market and firm A has an older product of age three. We also note that the productsÕ ages at t ¼ 0 influence each firmÕs market share only until a simultaneous product introduction by A and B occur, and that occurs at most at a t ¼ 6, the smallest common multiple of 2 and 3. A simultaneous product introduction by A and B is a regeneration point for the market. Thus, the share patterns in Fig. 1 repeat every six periods. If firms A and B use randomized product introduction strategies, then fixed cycles may not occur.
3.2. The game Define ði; jÞ to be a state of the game. In each period t at state ði; jÞ, firm k decides whether to introduce a new product (pijk ¼ 1) or not (pijk ¼ 0) at a fixed cost of K k . If a product introduction decision is made the new product is available in period t þ 1. Denote by pk the matrix of pijk . The one-period profit by player A at state ði; jÞ is
G.C. Souza / European Journal of Operational Research 152 (2004) 745–757 A A A A rA ðði; jÞ; pijA Þ ¼ cA ij lvi ðsi 1Þ K pij ;
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ð1Þ
similarly for player B. Players maximize total discounted profit in an infinite horizon. Due to the infinitehorizon nature of the game, we assume that K k remains constant. In practice, however, many firms are able to reduce product introduction costs by transferring knowledge between successive product development projects (Eisenhardt and Brown, 1998). Our model does not incorporate this effect, however, one can assess the effect of a reduction in K k in the outcome of the game by solving two separate games for two values of K k . A comparison of the two resulting equilibria provides some insights into the effect of reducing product development costs, although we recognize that this procedure would not fully capture such effects. This dynamic game can be viewed as a stochastic game (see Fudenberg and Tirole, 1991, and Thuijsman, 1992, for a definition of stochastic games), where the history is summarized in a state. Payoffs in the current period and the probability distribution of the state for next period depend on the current state and playersÕ actions in the current period. A strategy is a plan of action––the matrix pk . We allow randomized (behavioral) actions in each state, that is, 0 6 pijk 6 1. Without this possibility, there is no guarantee that the game has an equilibrium (Fudenberg and Tirole, 1991; Thuijsman, 1992). We can interpret 0 < pijk < 1 as the long-run frequency that firm k introduces a product in state ði; jÞ. Denote fi;jk ðpA ; pB Þ player kÕs expected discounted profit in an infinite horizon, for a discount factor a 2 ½0; 1Þ (we omit a from the notation for simplicity), if the initial state of the game is ði; jÞ and ðpA ; pB Þ is the strategy profile. Then, a strategy profile ðpA ; pB Þ is an a-discounted equilibrium of the stochastic game if and only if, for all states ði; jÞ: fi;jA ðpA ; pB Þ P fi;jA ðpA ; pB Þ;
for all pA
fi;jB ðpA ; pB Þ P fi;jB ðpA ; pB Þ;
B
and
ð2Þ
for all p :
We illustrate the concept of equilibrium with Example 2. Example 2. Consider d ¼ h ¼ 1, prices and variable costs constant with age, and K A ¼ K B ¼ 30. The equilibrium is characterized by a probability of product introduction by each competitor at each state, and these are shown in the table in Fig. 2. These equilibrium probabilities generate a Markov chain in the state space. At state ð1; 1Þ, players A and B do not introduce a product. The system moves to state ð2; 2Þ with probability one. At state ð2; 2Þ both players introduce a product with probability 0.147. Thus, with probability 0:02 ¼ 0:1472 , the system moves back to state ð1; 1Þ, with probability 0:125 ¼ 0:147ð1 0:147Þ the system moves to state ð1; 3Þ, with probability 0.125 the system moves to state ð3; 1Þ, and with proba2 bility ð1 0:147Þ the system moves to state ð3; 3Þ. At either state ð1; 3Þ, ð3; 1Þ, or ð3; 3Þ both competitors
1,1 1
2,2 0.125
1,3
1
1
0.02
0.73
1
3,3
0.125
3,1
(only recurrent states are shown)
(i,j)
pij (A)
pij (B)
(1,1)
0.000
0.000
(1,2)
0.000
0.781
(1,3)
1.000
1.000
(1,4)
1.000
1.000
(2,1)
0.781
0.000
(2,2)
0.147
0.147
(2,3)
1.000
1.000
…
…
…
(4,4)
1.000
1.000
Fig. 2. Illustration of equilibrium for Example 2.
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introduce a product with probability one, the system returns to state ð1; 1Þ, and the game continues in this fashion for an infinite horizon. All other states are transient; if the system starts at a transient state it moves to a recurrent state with probability one. The main theoretical result for our game is given in Theorem 1. Theorem 1. For the stochastic game under consideration, there exists a stationary a-discounted equilibrium. Further, the equilibrium can be computed by solving the following non-linear program (NLP) for its global minimum of zero: ( k ) k k n X n X X fi;j rk ðði; jÞ; pijk Þ a½pijA pijB f1;1 þ pijA ð1 pijB Þf1;jþ1 ; min ; ð3Þ k k þ ð1 pijA Þð1 pijB Þfiþ1;jþ1 þð1 pijA ÞpijB fiþ1;1 k2fA;Bg i¼1 j¼1 s:t:
A A fi;jA P rA ðði; jÞ; 1Þ þ aðpijB f1;1 þ ð1 pijB Þf1;jþ1 Þ;
fi;jA fi;jB fi;jB
A
A aðpijB fiþ1;1
B
B aðpijA f1;1
B
B aðpijA f1;jþ1
P r ðði; jÞ; 0Þ þ P r ðði; jÞ; 1Þ þ P r ðði; jÞ; 0Þ þ
þ ð1
þ ð1
B pijA Þfiþ1;1 Þ;
þ ð1
i ¼ 1; . . . ; n; j ¼ 1; . . . ; n;
A pijB Þfiþ1;jþ1 Þ;
ð4Þ
i ¼ 1; . . . ; n 1; j ¼ 1; . . . ; n;
ð5Þ
i ¼ 1; . . . ; n; j ¼ 1; . . . ; n;
B pijA Þfiþ1;jþ1 Þ;
ð6Þ
i ¼ 1; . . . ; n; j ¼ 1; . . . ; n 1;
ð7Þ
0 6 pijA
6 1;
i ¼ 1; . . . ; n 1; j ¼ 1; . . . ; n;
ð8Þ
0 6 pijB
6 1;
i ¼ 1; . . . ; n; j ¼ 1; . . . ; n 1;
ð9Þ
A pnj
¼ 1;
8j;
ð10Þ
pinB fi;jk
¼ 1;
8i;
ð11Þ
unrestricted;
8i; 8j; k 2 fA; Bg:
Proof. See Thuijsman (1992), or Breton (1991) (recall that the action and state space are finite).
ð12Þ
Eq. (4) indicates that the total expected discounted profit fi;jA using the equilibrium strategy must be greater than the total expected discounted profit if player A chooses the pure action pijA ¼ 1 in state ði; jÞ, for all i and j; analogously in (6) for player B. A similar interpretation holds for (5) and (7) for pure action pijk ¼ 0. The complexity of (3)–(12) makes the use of analytical methods to derive a solution very difficult, if not impossible, for a meaningful number of states. Thus, we opt for a comprehensive numerical analysis. We note that numerically finding equilibria for stochastic games of a high state space size is very challenging, as reported in Breton (1991) and Souza (2000). For example, an iterated best-response play algorithm may result in cycling, and is thus not suitable (Souza, 2000). Using MatlabÕs non-linear solver FMINCON (MATLAB, 1996), we were successful in finding equilibria for n ¼ 6 by solving the resulting NLP (102 variables, 192 constraints) for its global minimum of zero, despite difficulties––sometimes, the optimizer converges to a local minimum, necessitating perturbation techniques; the procedure (similar to Breton, 1991) is as follows: An initial guess is given for ðpA ; pB Þ. With this initial guess, fi;jk , i; j; k can be computed as follows. The strategy profile ðpA ; pB Þ generates a Markov chain on the state space, as illustrated in Example 2. Denote the chainÕs transition probability matrix by P. Also, denote the vectors of fi;jk and rk ðði; jÞ; pijk Þ by f k and rk , respectively. Then 1
f k ¼ ðI aPÞ rk :
ð13Þ
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With the initial guess ðpA ; pB Þ and ðf A ; f B Þ, we solve for a global optimum of the NLP, with an objective function value of zero. If the NLP solver converges to a local optimum, we re-start from another guess, or perturb the existing solution, until we find the global optimum. In the next section, we perform a numerical study for the case where the competitors are identical. Then, in Section 5, we consider asymmetric competitors.
4. Numerical results: Identical competitors 4.1. Experimental design In our numerical study, we use n ¼ 6, since for n ¼ 7 we had significant practical difficulties in finding equilibria: long solution times (90 min or more on a Sun Ultra station), and, in most cases, the optimizer converges to a local minimum. Thus, the maximum age for a product is six periods; if we think of a period as six months, then a product is in the market for at most three years. We have evidence, however, that our insights are robust to the choice of n; we comment further on that in Section 4.2. For convenience, consider that firm kÕs percent cost markup ski decreases with age according to a learning curve with rate xk : k ski ¼ sk1 iln x = ln 2 . B A B We use the following parameter values: l ¼ 3, vA 1 ¼ v1 ¼ 20, s1 ¼ s1 ¼ 3, h ¼ 1, and a full factorial experimental design for the other parameters (A and B identical), shown in Table 1. We justify these choices as follows. The parameters l, sk1 and vk1 can be chosen arbitrarily, however, the value of K should be consistent with the choice: if K is too high relative to l, sk1 and vk1 , we have empirically observed that the resulting equilibrium is for both firms to introduce a product every n periods (maximum); if K is too low the resulting equilibrium is for both firms to introduce a product every period. Thus, for the choices of l, sk1 and vk1 above, we consider four different values of K, as shown in Table 1, which capture the intermediary equilibria between introduce a product every period and every n periods. We fix h ¼ 1, and consider three different values of d; these three combinations are chosen to simulate different markets: d ¼ 0:5 represents a price-elastic market, d ¼ 1:0 represents a market where price and technology have the same elasticity, and d ¼ 2:0 represents a technology-elastic market. The two different values of /k in Table 1 capture common learning rates found in practical applications––0:75 6 /k 6 1 (Yelle, 1979). We consider two values of xk : if xk ¼ 1:0, the new product does not command a price premium (that is, ski remains constant with age), and if xk ¼ 0:8, then ski decreases with age. Regarding the discount factor a, other analysis not reported here indicates that equilibrium solutions are relatively insensitive to a; we thus use a ¼ 0:9. A desired property of a game is that it contains a unique equilibrium. We have empirically verified that our game, however, may present multiple equilibria. This (perhaps disappointing) result helps understand why there exist so many different competitive strategies for rational firms. Unfortunately, the theory provides little or no guidance on identifying the number of equilibria. We focus on focal equilibria (Schelling, 1960): pure strategy equilibria, and, in the case of identical firms, symmetric equilibria. Solving Table 1 Experimental design for game with identical competitors Parameter
Values
xA ¼ xB ¼ x d /A ¼ /B ¼ / KA ¼ KB ¼ K
0.8, 1.0 0.5, 1.0, 2.0 0.8, 1.0 10, 30, 50, 70
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the NLP (3) through (12), however, may result in asymmetric equilibria. We search for equilibria where pijA ¼ pjiB . 4.2. Results Table 2 presents summarized results for the experimental design from Table 1. Summarized results include, for each firm, average profit per period and expected time between product introductions (ETPI). These measures can be computed through standard Markov chain techniques (for details, see Souza, 2000). ETPI is a measure of the firmÕs clockspeed: a higher ETPI means a lower clockspeed. We also report ETPI for the market (or marketÕs clockspeed), which informs whether the competitors introduce products simultaneously. For example, for fx; d; /; Kg ¼ f1; 0:5; 0:8; 70g, each firmÕs ETPI is 5.0 (Table 2), whereas the marketÕs ETPI is 2.5. This means that firms A and B never introduce products simultaneously. Detailed equilibria results, which include the equilibrium profile ðpA ; pB Þ, are available upon request. We have also performed tests to assess the stability of the equilibria obtained. Consider in Table 2 equilibria in which A(B) introduce a products if i P 4 (j P 4). These equilibria were obtained using n ¼ 6 in Table 2 Summarized results for game with identical competitors: symmetric equilibria d
/
0.5
1.0
2.0
a b
K
x ¼ 1:0
x ¼ 0:8
Average profit
ETPI (firm)
ETPI (market)
Average profit
ETPI (firm)
ETPI (market)
1.0
10 30 50 70
–a 52.4 51.5 –a
2.00 3.94 5.89 6.00
–a 1.97 3.02 –a
50.0 36.6 28.9 –a
1.00 2.00 3.91 4.00
1.00 1.00 1.96 –a
0.8
10 30 50 70
50.0 40.2 35.5 –a
1.00 4.00 5.00 6.00
1.00 2.00 2.50 –a
50.0 31.8 21.1 15.2b
1.00 2.00 4.00 5.10
1.00 2.00 4.00 3.41
1.0
10 30 50 70
50.0 48.5 46.4 42.5
1.00 2.60 3.67 4.00
1.00 2.31 1.84 4.00
50.0 36.0 28.9 24.3b
1.00 2.00 2.37 3.07
1.00 2.00 1.31 1.56
0.8
10 30 50 70
50.0 39.0 35.6 31.3
1.00 2.00 3.33 4.03
1.00 2.00 1.68 2.01
50.0 31.7 22.9 18.1
1.00 1.56 2.83 4.70
1.00 1.33 1.41 2.35
1.0
10 30 50 70
50.0 30.0 35.0 29.3
1.00 1.00 2.00 2.28
1.00 1.00 1.00 1.49
50.0 30.0 26.0 22.1b
1.00 1.00 2.00 2.42
1.00 1.00 2.00 1.45
0.8
10 30 50 70
50.0 30.0 29.0 24.4
1.00 1.00 2.00 2.34
1.00 1.00 2.00 1.44
50.0 30.0 22.4 17.6b
1.00 1.00 2.00 2.00
1.00 1.00 1.82 1.00
Equilibrium results in a multi-chain Markov chain; the stationary probability distribution depends on initial state. e-Equilibrium, where e 6 0:7.
G.C. Souza / European Journal of Operational Research 152 (2004) 745–757
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(b) Profit
(a) ETPI 50.0
4.00
45.0 3.00 40.0 35.0
2.00 1.00
d=1
30.0
d=1
d=2
25.0
d=2
0.00
20.0 10
30
K
50
70
10
30
K
50
70
Fig. 3. Effect of K on ETPI and profit for x ¼ 1:0 and / ¼ 1:0.
the NLP (3)–(12). If the equilibria are stable, we would expect to find the exact same equilibria by solving the NLP with n ¼ 4, and that is precisely what we have observed. For example, consider fx; d; /; Kg ¼ f1; 2; 1; 50g, in which the equilibrium is for both firms to introduce a product every two periods (Table 2). This result can be obtained using either n ¼ 4 or n ¼ 6 in the NLP (3)–(12). Hence, the results suggest that the choice of n is not so restrictive. In general, the results of Table 2 are in line with intuition. They also conform to results obtained if this problem is viewed from a decision-theoretical perspective (Souza, 2000), when one firm optimizes its actions based on an estimate of the other firmÕs behavior. That is, as K increases, for a fixed x; d, and /, so does ETPI for both firms, as well as for the market, in equilibrium. For example, consider x ¼ 1, d ¼ 1 and / ¼ 1:0 in Table 2: each firmÕs ETPI is 1.00, 2.60, 3.67, and 4.00, for K ¼ 10, 30, 50 and 70, respectively. As new product introduction becomes costlier, each firm has a unilateral incentive to increase its ETPI, and the equilibrium reflects this incentive. These results support the resource-based-view of the firm (Collis and Montgomery, 1995), in that a firmÕs strategy should be consistent with its capabilities. It is also consistent with Schmidt and Porteus (2000), who suggest that a firmÕs cost leadership (product introduction cost here) is necessary to sustain technological innovation (clockspeed). As d increases, for a fixed K, x, and /, ETPI in equilibrium decreases. For example, consider x ¼ 1, / ¼ 1, and K ¼ 30: ETPI for each firm is 3.94, 2.60, and 1.00 for d ¼ 0:5, 1.0, and 2.0, respectively. Firms competing in technology-elastic markets have individual incentives to decrease their ETPI––introducing new products more often––and again, the equilibrium reflects this incentive. One result from Table 2, however, is paradoxical. Consider the equilibria at fx; d; /; Kg ¼ f1; 2; 1; 30g and f1; 2; 1; 50g, with average profits of 30.00 and 35.00, respectively. For easier visualization, we plot them in Fig. 3(b). Here, a higher K implies higher profits in equilibrium, so both firms would rather have a higher fixed product introduction cost than a lower one! This paradox can be reasoned by verifying that ETPI at K ¼ 30 is the same as at K ¼ 10, for d ¼ 2 (Fig. 3(a)). We would expect ETPI to decrease monotonically with K, but in this case it remains constant as K increases from 10 to 30. So, the equilibrium at K ¼ 30 is inefficient. The effect of the learning curve on the ETPI is unclear in a symmetric game. As the learning curve becomes steeper (/ decreases), ETPI may increase or decrease, depending on the values of K and d. For a game with different competitors, however, as we will see in Section 5, learning rates play a crucial role.
5. Numerical results: Different competitors We now consider a situation where the competitors are different. Without loss of generality, consider A to be the most efficient competitor in terms of product introduction costs, that is, K A < K B . We use the
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same parameter values as before, including the experimental setup shown in Table 1, except that now we fix two scenarios. In the first scenario, there is no learning by either firm, that is, /k ¼ xk ¼ 1 for k 2 fA; Bg. In the second scenario, firm A is also the most efficient competitor in terms of manufacturing, that is, /A ¼ xA ¼ 0:8, and /B ¼ xB ¼ 1. Table 3 presents summarized results (detailed results are available upon request). A very interesting (and desirable) feature of the results in Table 3 is that all but four equilibria are
Table 3 Summarized results for game with different competitors d
A
A
x ¼ / ¼ 1:0 xB ¼ /B ¼ 1:0
xA ¼ /A ¼ 0:8 xB ¼ /B ¼ 1:0
a
0.5
KA
KB
Average profit
Average market share
ETPI
A
B
A
B
A
B
ETPI (Market)
10
30
61.6
45.9
0.55
0.45
2.00
4.00
2.00
10 10 30 30 50
50 70 50 70 70
65.8 65.9 56.9 57.1 –a
40.9 37.4 47.3 42.9 –a
0.59 0.59 0.54 0.54 –a
0.41 0.41 0.46 0.46 –a
2.00 2.00 4.00 3.60 6.00
6.00 6.00 6.00 6.00 6.00
1.50 2.00 2.40 2.57 –a
1.0
10 10 10 30 30 50
30 50 70 50 70 70
60.0 71.5 71.3 52.0 57.6 48.6
35.0 26.0 23.8 36.3 31.1 41.4
0.58 0.68 0.66 0.56 0.60 0.51
0.42 0.32 0.34 0.44 0.40 0.49
1.00 1.00 1.22 2.00 2.10 4.00
2.00 4.00 4.21 3.00 4.11 4.00
1.00 1.00 1.22 1.50 1.60 2.00
2.0
10 10 10 30 30 50
30 50 70 50 70 70
50.0 68.0 84.2 48.0 48.6 66.4
30.0 17.0 8.3 17.0 23.1 25.2
0.50 0.65 0.79 0.65 0.54 0.69
0.50 0.35 0.21 0.35 0.46 0.31
1.00 1.00 1.00 1.00 1.86 3.00
1.00 2.00 4.00 2.00 2.17 6.00
1.00 1.00 1.00 1.00 1.86 2.00
0.5
10
30
61.6
40.9
0.60
0.40
1.00
4.00
1.00
10 10 30 30 50
50 70 50 70 70
65.8 65.8 46.2 46.3 35.1
35.9 32.6 35.0 31.6 30.7
0.63 0.63 0.64 0.64 0.65
0.37 0.37 0.36 0.36 0.35
1.00 1.00 1.50 1.50 2.40
6.00 6.00 5.96 6.00 6.00
1.00 1.00 1.20 1.20 1.71
1.0
10 10 10 30 30 50
30 50 70 50 70 70
60.0 71.5 75.2 48.1 55.3 43.2
35.0 26.0 20.8 29.3 22.2 23.1
0.58 0.68 0.71 0.65 0.72 0.71
0.42 0.32 0.29 0.35 0.28 0.29
1.00 1.00 1.00 2.00 1.50 2.00
2.00 4.00 5.00 4.00 6.00 6.00
1.00 1.00 1.00 1.33 1.20 1.50
2.0
10 10 10 30 30 50
30 50 70 50 70 70
50.0 68.0 84.2 48.0 64.2 45.0
30.0 17.0 8.3 17.0 8.3 18.5
0.50 0.65 0.79 0.65 0.79 0.70
0.50 0.35 0.21 0.35 0.21 0.30
1.00 1.00 1.00 1.00 1.00 2.00
1.00 2.00 4.00 2.00 4.00 4.00
1.00 1.00 1.00 1.00 1.00 1.33
Equilibrium results in a multi-chain Markov chain; the stationary probability distribution depends on initial state.
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in pure strategies, that is, for each experimental cell, pijk 2 f0; 1g, for all i, j, and k. Because of this property, we consider the results in Table 3 to be focal equilibria. First, as d increases, for a fixed K A and K B , ETPI (for both firms, and, consequently, the market) decreases. In equilibrium, each firmÕs product introduction strategy ‘‘fits’’ the market it is competing at––a firmÕs frequency of product introductions is higher in a technology elastic market than in a price elastic market. For example, consider xA ¼ /A ¼ 1:0, K A ¼ 30, and K B ¼ 50. The ETPI for (firm A, firm B, market) for d ¼ 0:5, 1.0 and 2.0 is (4.00, 6.00, 2.40), (2.00, 3.00, 1.50), and (1.00, 2.00, 1.00), respectively. For d ¼ 0:5, detailed results (not reported here) show that the game cycles as follows: (1,4)–(2,5)–(3,6)– (4,1)–(1,2)–(2,3)–(3,4)–(4,5)–(1,6)–(2,1)–(3,2)–(4,3)–(1,4); thus there is a new product in the market every 2.4 periods ðð3 þ 1 þ 4 þ 1 þ 3Þ=5Þ and the two firms never introduce products simultaneously. For d ¼ 1:0, detailed results show that the game cycles as follows: (1,1)–(2,2)–(1,3)–(2,1)–(1,2)–(2,3)–(1,1); thus there is a new product in the market every 1.5 periods ðð2 þ 1 þ 1 þ 2Þ=4Þ and the two firms introduce products simultaneously every six periods. Finally, for d ¼ 2:0, the game cycles as follows: (1,1)–(1,2)–(1,1); thus there is a new product in the market every period, and the two firms introduce products simultaneously every two periods. Firm B (highest K) always has a lower market share and lower profits than AÕs. This can be seen directly in Table 3, by comparing the respective columns for A and B. Also, firm B has ETPI no lower than AÕs, for each game. This result goes against published recommendations that a firmÕs clockspeed should match its industry (Fine, 1998). Here, a less efficient firm (firm B) competes at a lower clockspeed than its more efficient rival, firm A. Firm A introduces products no less often when there is (manufacturing) learning, and consequently, attains a higher (or equal) market share. This can be easily seen by comparing, for a given, K A and K B , AÕs ETPI and market share when xA ¼ /A ¼ 0:8 (learning by A, bottom of Table 3) versus xA ¼ /A ¼ 1:0 (no learning by A, top of Table 3). For example, for d ¼ 1, K A ¼ 30 and K B ¼ 70, firm AÕs ETPI with learning is 1.50 versus 2.10 with no learning; firm AÕs market share with learning is 0.72 versus 0.60 with no learning. Firm AÕs average profit is slightly lower with learning at 55.3 versus 57.6 without learning, but this is because some of the gains are passed to consumers––firm AÕs price and margin are lower with learning when it has a product at age 2 for example. Firm BÕs average profit (ETPI), however, is substantially lower (higher) at 22.2 (6.00) when there is learning by A versus 31.1 (4.11) when there is no learning by A. Thus, when firm A experiences manufacturing learning, it has a much stronger competitive position in the market: a higher market share because of newer and cheaper products on average, and a weakened competitor, with much lower average profits and an older and more expensive product. Firm A gains additional market share because of its cheaper product, a result of learning, and a higher market share results in higher revenues, which can be used to introduce products more often. The opposite happens with firm B. With a lower market share, firm B has lower revenues, and, consequently, cannot introduce products as often because of the fixed costs of product introduction. It is clear from these results how manufacturing learning is important to a firmÕs competitiveness. First, manufacturing learning allows a firm to introduce products more often. This result brings a theoretical justification for IntelÕs amazing rate of product introduction, which has a significant contribution from learning in manufacturing (Fine, 2000). Second, learning not only affects a firmÕs own behavior, but it also affects the competitorÕs behavior: learning by firm A causes firm B to decrease its rate of product introduction, attaining significantly lower profits.
6. Conclusion In this paper, we have studied the game-theoretical solution of a stylized model where two firms compete over time by introducing new products at a fixed cost, and the market demand is constant (or stochastic and
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stationary, if we assume that it can always be met). The market is split according to a market-share attraction model. In this context, we define a state as an age profile ði; jÞ, corresponding to the ages of AÕs and BÕs products, respectively. At each state, each firm makes a decision regarding the introduction of a new product. These two decisions move the system to another state, where similar decisions are made, and so forth. This can be modeled as a non-zero sum stochastic game. When players maximize total discounted profit over an infinite horizon, there exists at least one equilibrium. Equilibria can be found by solving a non-linear program for its global optimum. Through numerical analysis of the underlying game, we reached the following conclusions. First, in general, the equilibria reflect the most important results obtained from a decision-theoretic perspective by Souza (2000). That is, a firmÕs pace of new product introduction increases with increasing market sensitivity to technology and decreasing fixed product introduction costs. Second, the firm with the highest fixed product introduction cost has a lower market share, lower profits, and competes at a lower clockspeed than the more efficient firm. This result goes against published recommendations that a firmÕs clockspeed should match its industry (Fine, 1998), however, it is line with the resource-based-view of the firm (Collis and Montgomery, 1995), in that a firmÕs strategy should be consistent with its capabilities. It is also consistent with Schmidt and Porteus (2000), who suggest that a firmÕs cost leadership (product introduction cost here) is necessary to sustain technological innovation (clockspeed). Finally, we find that a higher rate of manufacturing learning for a firm results in a higher clockspeed and in a higher market share, and also results in significantly worse profits for the competitor. The results underscore the importance of manufacturing expertise in a firmÕs ability to introduce new products. Our paper is one of the first to formally address strategic product introduction, that is, when and how frequently should a firm introduce new products. This research is not without its limitations. First, this is clearly a stylized model, which assumes constant demand, a duopoly, constant product introduction costs, and constant prices. Second, we use numerical analysis to derive our insights, due to the intractability of the model. Third, we consider that demand can always be met. Relaxing our assumptions are natural directions for future research.
Acknowledgements I am grateful to two anonymous referees for their helpful comments on the first version of this paper.
References Abernathy, W., Utterback, J., 1978. Patterns of industrial innovation. Technology Review 80 (7), 40–47. Balcer, Y., Lippman, S., 1984. Technological expectations and adoption of improved technology. Journal of Economic Theory 34, 292–318. Bayus, B., 1997. Speed-to-market and new product development trade-offs. Journal of Product Innovation Management 14, 485–497. Bayus, B., Jain, S., Rao, A., 1997. Too little, too early: Introduction timing and new product performance in the personal digital assistant industry. Journal of Marketing Research XXXIV, 50–63. Breton, M., 1991. Algorithms for stochastic games. In: Raghavan, T.E.S. (Ed.), Stochastic Games and Related Topics. Kluwer Academic Publishers, The Netherlands. Carrillo, J., 2001. Industry clockspeed and the pace of new product development, Working Paper, University of Florida. Cogan, G., Burgelman, R., 1989. Intel Corporation: Strategy for the 1990s, Graduate School of Business Case, Stanford University. Cohen, M., Eliashberg, J., Ho, T., 1996. New product development: The performance and time-to-market tradeoff. Management Science 42 (2), 173–186. Collis, D., Montgomery, C., 1995. Competing on resources: Strategy in the 1990s. Harvard Business Review July–August, 118–128. Eisenhardt, K., Brown, S., 1998. Time pacing: Competing in markets that wonÕt stand still. Harvard Business Review March–April, 59–69.
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Fine, C., 1998. Clockspeed: Winning Industry Control in the Age of Temporary Advantage. Perseus Books, Reading, MA. Fine, C., 2000. Clockspeed-based strategies for supply chain design. Production and Operations Management 9 (3), 213–221. Fudenberg, D., Tirole, J., 1991. Game Theory. MIT Press, Cambridge, MA. Gaimon, C., 1989. Dynamic game results of the acquisition of new technology. Operations Research 37 (3), 410–425. Karnani, A., 1985. Strategic implications of market share attraction models. Management Science 31 (5), 536–547. Kornish, L., 1999. On optimal replacement thresholds with technological expectations. Journal of Economic Theory 89, 261–266. Krishnan, V., Ulrich, K., 2001. Product development decisions: A review of the literature. Management Science 47 (1), 1–21. Krishnan, T., Bass, F.M., Jain, D.C., 1999. Optimal pricing strategy for new products. Management Science 45 (12), 1650–1663. Krishnan, V., Singh, R., Tirupati, D., 1999. A model-based approach for planning and developing a family of technology-based products. Manufacturing and Service Operations Management 1 (2), 132–156. Mamer, J.W., McCardle, K.F., 1987. Uncertainty, competition, and the adoption of new technology. Management Science 33 (2), 161– 177. MATLAB, 1996. Matlab Reference Guide. The MathWorks, Inc., Natick, MA. McCardle, K.F., 1985. Information acquisition and the adoption of new technology. Management Science 31 (11), 1372–1389. Mendelson, H., Pillai, R.R., 1999. Industry clockspeed: Measurement and operational implications. Manufacturing and Service Operations Management 1 (1), 1–20. Moorthy, K.S., Png, P.L., 1992. Market segmentation, cannibalization, and the timing of product introductions. Management Science 38 (3), 345–359. Morgan, L.O., Morgan, R.M., Moore, W.L., 2001. Quality and time-to-market trade-offs when there are multiple product generations. Manufacturing and Service Operations Management 3 (2), 89–104. Nair, S.K., 1995. Modeling strategic investment decisions under sequential technological change. Management Science 41 (2), 282–297. Reinganum, J., 1981. On the diffusion of new technology. Review of Economic Studies XLVIII, 395–405. Schelling, T., 1960. The Strategy of Conflict. Harvard University Press, Cambridge, MA. Schmidt, G., Porteus, E., 2000. Sustaining technology leadership can require both cost competence and innovative competence. Manufacturing and Service Operations Management 2 (1), 1–18. Souza, G.C., 2000. A theory of product introduction under competition. Doctoral Dissertation. The University of North Carolina at Chapel Hill. Souza, G.C., Bayus, B.L., Wagner, H.M., 2002. The optimal pacing of new product introductions. Working Paper, Smith School of Business, University of Maryland. Thuijsman, F., 1992. Optimality and equilibria in stochastic games, The Netherlands, CWI Tract 82. Centrum voor Wiskunde en Informatica. Wilson, L., Norton, J., 1989. Optimal entry timing for a product line extension. Marketing Science 8 (1), 1–17. Yelle, L., 1979. The learning curve: Historical review and comprehensive survey. Decision Sciences 10, 302–328.
Production, Manufacturing and Logistics
Product introduction decisions in a duopoly Gilvan C. Souza
*
The Robert H. Smith School of Business, University of Maryland, College Park, MD 20742-1815, USA Received 26 March 2002; accepted 16 September 2002
Abstract In this paper we model a dynamic environment with two firms that fight for share of industry sales and profit in a market with constant size. They capture share by repeatedly introducing new products. Price changes from period to period reflecting each firmÕs learning. We formulate the problem as a repeated game, where each playerÕs decision at each period is to introduce a new product. We find that, in general, each firmÕs frequency of product introductions (clockspeed) in equilibrium decreases as its fixed product introduction cost increases. Further, we find that a higher rate of manufacturing learning results in a higher clockspeed and in a higher market share, and also results in significantly worse profits for the competitor. The results underscore the importance of manufacturing expertise in a firmÕs ability to introduce new products. Ó 2002 Elsevier B.V. All rights reserved. Keywords: Strategic planning; Research and development; Game theory
1. Introduction In this paper we use a stylized model to address a dynamic competitive environment with two firms that fight for share of industry sales and profit in a market with constant size. They capture share by repeatedly introducing new products. Price changes from period to period reflecting each firmÕs learning. We use dynamic games to study this scenario. Abernathy and Utterback (1978) propose a framework that describes the dynamics of innovation in an industry. As an industry starts, the technology is still evolving, and several firms enter and exit the industry with different design architectures. Once a dominant design emerges, the basis of competition shifts to primarily minor product innovations due to choices in process design, and only a few firms survive. Our research primarily addresses the industry after the emergence of a dominant product design, or the specific (mature) phase in Abernathy and UtterbackÕs framework. Thus, our stylized model assumes a duopoly, a
*
Tel.: +301-405-0628; fax: +301-405-8655. E-mail address: [email protected] (G.C. Souza).
0377-2217/$ - see front matter Ó 2002 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(02)00709-9
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mature market of constant size, and that consumers are indifferent to two equally priced products introduced at the same time. We find that, in general, each firmÕs frequency of product introductions (clockspeed) in equilibrium decreases as its fixed product introduction cost increases, however, there are non-intuitive results. For example, there are instances where a higher fixed product introduction cost for both firms result in an equilibrium where both firms are better off. Further, we find that a higher rate of manufacturing learning results in a higher clockspeed and in a higher market share, and also results in significantly worse profits for the competitor. The results underscore the importance of manufacturing expertise in a firmÕs ability to introduce new products. From a theoretical perspective, this is one of the first papers to formally address strategic product introduction, that is, when and how frequently should a firm introduce new products. There exists an emerging literature on this subject, but it is mainly based on case studies (Fine, 1998), or is empirical (Mendelson and Pillai, 1999). Although there exist papers that use differential games to address capacity and pricing decisions (Gaimon, 1989), we are not aware of published papers that use dynamic games to model competition in product introduction. This paper is organized as follows. In Section 2, we review the related literature. In Section 3, we introduce the mathematical model. In Section 4, we show numerical results for the case of identical competitors; non-identical competitors are analyzed in Section 5. We conclude in Section 6.
2. Literature review Machine replacement models (MRMs) analyze the decision of when to replace an aging machine with a new one based on replacement and maintenance costs, technological forecast, and the state of machine deterioration. The literature in MRM is extensive; for a review see Nair (1995). Our model is similar to MRM in that the introduction of a new product in our model is based on the ages of the firmÕs and the competitorÕs products whereas the replacement decision in MRM is based on machine deterioration. In MRM models, however, there is no competition. A related stream of literature approaches the problem of adopting a new technology from either a decision-theoretic perspective (Balcer and Lippman, 1984; McCardle, 1985; Kornish, 1999), or from a game-theoretic perspective (Reinganum, 1981; Mamer and McCardle, 1987). The former is similar to MRM, except that there is uncertainty about the revenues provided by the new technology. Regarding the latter, Reinganum (1981) proves the existence of a pair of symmetric Nash equilibria on the timing of technology adoption in a duopoly: one firm adopts early and the other firm adopts later––it is never a Nash equilibrium for both firms to adopt the technology at the same time. In our dynamic game, we find that, in general, firms prefer not to introduce new products simultaneously. Regarding the timing of product introduction, a stream of literature analyzes the time-to-market and performance trade-off for a single product development project (Cohen et al., 1996; Bayus, 1997; Bayus et al., 1997). It is not always in a firmÕs best interest to speed up a product development project, because the monetary benefits of being first to market may be outweighed by higher development costs. Another stream of literature analyzes the best time to introduce a product line extension (Wilson and Norton, 1989; Moorthy and Png, 1992). A state-of-the-art survey of decisions in product development projects, including timing, is provided in Krishnan and Ulrich (2001). Our model does not deal with tactical considerations at a single project level. Instead, we assume a fixed time for product development, and focus on strategic product introduction––when and how often to introduce products. Fine (1998) describes how different industries have different clockspeed––the frequency of new product introductions in an industry. He suggests that a firmÕs clockspeed should match its industryÕs, a claim that has some empirical support (Mendelson and Pillai, 1999). Eisenhardt and Brown (1998) emphasize a more
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proactive approach to strategic product introduction, and suggest that the best strategy is time pacing–– introducing a new product at regular time intervals, and not when opportunities arise (event pacing). Carrillo (2001) uses a decision-theoretic model based on diffusion curves to show that high product introduction costs may provide an incentive for a firm to have a lower clockspeed than its industryÕs, a result that is in line with ours. Two closely related papers to this are Morgan et al. (2001) and Souza et al. (2002). Both papers dynamically model a firmÕs decision to introduce new products, however, their models are from a decisiontheoretic perspective: firm A optimizes its product introduction decisions based on an estimate of firm BÕs behavior. Souza et al. (2002) assume random demand, include the simultaneous optimization of production and inventory, and focus on an infinite horizon. Morgan et al. (2001) assume deterministic demand, include the possibility of crashing product development projects (that is, lead time for a project is also a decision variable), and focus on a finite horizon. As in this research, both pieces conclude that a firmÕs clockspeed is highly dependent on its product introduction cost and market characteristics, however, firm AÕs optimal solution may not constitute an equilibrium in a game-theoretic sense. Hence, their approach is better applied to a situation as described in Eisenhardt and Brown (1998), in which firm BÕs product introduction strategy is independent of AÕs. We assume in this paper that both firms are rational, and we seek the market equilibrium, although we do not include production and inventory decisions for our model.
3. Model Consider a competitive stationary market––the market size l is the same each period––comprised of two firms, A and B. This market resembles, for example, demand for large consumer appliances, such as water heaters or refrigerators. The planning horizon is infinite and divided into time periods of equal length. Firms A and B are rational competitors, and compete by regularly introducing new products to grab market share through a mechanism described in Section 3.1. Although we consider the market size fixed, this model can also be applied to a situation where demand is stochastic, but can always be met, for example, in a make-to-order environment, or when there is full backlogging at no penalty. We think of a period here as, say, six months, and this assumption is reasonable for such a long period. Furthermore, the game is intractable if inventory is considered (Souza, 2000). We use superscripts A and B to identify the two firms.
3.1. The market share model Product price and performance are the major factors that influence consumersÕ purchase decisions (Krishnan et al., 1999), and prices typically decay with age (Krishnan et al., 1999). Because we are focusing on product introductions, and to keep the model tractable, we do not treat price as a decision variable, and consider that it decays with the age of a product as follows. Variable cost decreases with age according to a learning curve, and a firmÕs markup is non-increasing with age (that is, firms are able to offer a premium on newer products). For tractability, we assume learning with age (time) as opposed to learning with volume. We note that both types of learning are observed in practice (Cogan and Burgelman, 1989). We assume that a productÕs age is a proxy for its performance. We use a market share attraction model h d (Karnani, 1985), and set the attraction of a product as K ¼ ðpriceÞ ðageÞ , where h > 0 and d > 0 are industry-specific parameters that define the marketÕs elasticity to price and technology, respectively. Denote AÕs (BÕs) product age by iðjÞ. ProductsÕ ages are integers bounded by 1 and n. A product of age 1 is a new product, and a product of age n is the oldest possible product. Firm AÕs (BÕs) variable cost and A B B A B percent cost markup are denoted by vA i and si 1 (vj and sj 1), where si , sj > 1. Firm AÕs variable cost
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G.C. Souza / European Journal of Operational Research 152 (2004) 745–757 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0
1
2
3
4
5
6
7
8
9
10
A
B
*
AB
*
B
A
B
*
AB
*
δ=1
δ=2
Fig. 1. Firm AÕs market share over time when A(B) introduces a new product every 3(2) periods. A
A ln / = ln 2 at age i reflects a learning curve vA , where 0 < /A 6 1 (Yelle, 1979). Firm AÕs product price at i ¼ v1 i A A age i is thus si vi , which is non-increasing in i; similarly for firm B. Consequently, AÕs product attraction is B h A ln /A = ln 2 h d KA ¼ ðsA Þ i ; BÕs product attraction is KB ¼ ðsBj vB1 jln / = ln 2 Þ jd . Firm AÕs market share is i v1 i
cA ij ¼
KA K þ KB A
and firm BÕs market share is 1 cA ij . The notation explicitly indicates that market share in our model is only a function of the ages. Example 1 illustrates the model. Example 1. Suppose firm A(B) introduces a new product every three (two) periods, with /A ¼ /B ¼ 1 (no B learning), h ¼ 1, and sA j ¼ sj for all j. At time t ¼ 0, firms A and B have products of ages 1 and 2, respectively. Fig. 1 plots firm AÕs market share as a function of time for d ¼ 1 and d ¼ 2. We also indicate in the horizontal axis when a new product introduction by firm A, firm B, or both simultaneously, occurs. The curve for d ¼ 2 has a higher amplitude than the curve for d ¼ 1, because the market is more elastic to the difference in product ages when d ¼ 2. At t ¼ 3, a simultaneous product introduction by firms A and B occur, and the market is evenly split. The market is also evenly split at t ¼ 4, when both firms have products of age two. At t ¼ 5, however, firm AÕs market share drops considerably, because firm B has a new product in the market and firm A has an older product of age three. We also note that the productsÕ ages at t ¼ 0 influence each firmÕs market share only until a simultaneous product introduction by A and B occur, and that occurs at most at a t ¼ 6, the smallest common multiple of 2 and 3. A simultaneous product introduction by A and B is a regeneration point for the market. Thus, the share patterns in Fig. 1 repeat every six periods. If firms A and B use randomized product introduction strategies, then fixed cycles may not occur.
3.2. The game Define ði; jÞ to be a state of the game. In each period t at state ði; jÞ, firm k decides whether to introduce a new product (pijk ¼ 1) or not (pijk ¼ 0) at a fixed cost of K k . If a product introduction decision is made the new product is available in period t þ 1. Denote by pk the matrix of pijk . The one-period profit by player A at state ði; jÞ is
G.C. Souza / European Journal of Operational Research 152 (2004) 745–757 A A A A rA ðði; jÞ; pijA Þ ¼ cA ij lvi ðsi 1Þ K pij ;
749
ð1Þ
similarly for player B. Players maximize total discounted profit in an infinite horizon. Due to the infinitehorizon nature of the game, we assume that K k remains constant. In practice, however, many firms are able to reduce product introduction costs by transferring knowledge between successive product development projects (Eisenhardt and Brown, 1998). Our model does not incorporate this effect, however, one can assess the effect of a reduction in K k in the outcome of the game by solving two separate games for two values of K k . A comparison of the two resulting equilibria provides some insights into the effect of reducing product development costs, although we recognize that this procedure would not fully capture such effects. This dynamic game can be viewed as a stochastic game (see Fudenberg and Tirole, 1991, and Thuijsman, 1992, for a definition of stochastic games), where the history is summarized in a state. Payoffs in the current period and the probability distribution of the state for next period depend on the current state and playersÕ actions in the current period. A strategy is a plan of action––the matrix pk . We allow randomized (behavioral) actions in each state, that is, 0 6 pijk 6 1. Without this possibility, there is no guarantee that the game has an equilibrium (Fudenberg and Tirole, 1991; Thuijsman, 1992). We can interpret 0 < pijk < 1 as the long-run frequency that firm k introduces a product in state ði; jÞ. Denote fi;jk ðpA ; pB Þ player kÕs expected discounted profit in an infinite horizon, for a discount factor a 2 ½0; 1Þ (we omit a from the notation for simplicity), if the initial state of the game is ði; jÞ and ðpA ; pB Þ is the strategy profile. Then, a strategy profile ðpA ; pB Þ is an a-discounted equilibrium of the stochastic game if and only if, for all states ði; jÞ: fi;jA ðpA ; pB Þ P fi;jA ðpA ; pB Þ;
for all pA
fi;jB ðpA ; pB Þ P fi;jB ðpA ; pB Þ;
B
and
ð2Þ
for all p :
We illustrate the concept of equilibrium with Example 2. Example 2. Consider d ¼ h ¼ 1, prices and variable costs constant with age, and K A ¼ K B ¼ 30. The equilibrium is characterized by a probability of product introduction by each competitor at each state, and these are shown in the table in Fig. 2. These equilibrium probabilities generate a Markov chain in the state space. At state ð1; 1Þ, players A and B do not introduce a product. The system moves to state ð2; 2Þ with probability one. At state ð2; 2Þ both players introduce a product with probability 0.147. Thus, with probability 0:02 ¼ 0:1472 , the system moves back to state ð1; 1Þ, with probability 0:125 ¼ 0:147ð1 0:147Þ the system moves to state ð1; 3Þ, with probability 0.125 the system moves to state ð3; 1Þ, and with proba2 bility ð1 0:147Þ the system moves to state ð3; 3Þ. At either state ð1; 3Þ, ð3; 1Þ, or ð3; 3Þ both competitors
1,1 1
2,2 0.125
1,3
1
1
0.02
0.73
1
3,3
0.125
3,1
(only recurrent states are shown)
(i,j)
pij (A)
pij (B)
(1,1)
0.000
0.000
(1,2)
0.000
0.781
(1,3)
1.000
1.000
(1,4)
1.000
1.000
(2,1)
0.781
0.000
(2,2)
0.147
0.147
(2,3)
1.000
1.000
…
…
…
(4,4)
1.000
1.000
Fig. 2. Illustration of equilibrium for Example 2.
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introduce a product with probability one, the system returns to state ð1; 1Þ, and the game continues in this fashion for an infinite horizon. All other states are transient; if the system starts at a transient state it moves to a recurrent state with probability one. The main theoretical result for our game is given in Theorem 1. Theorem 1. For the stochastic game under consideration, there exists a stationary a-discounted equilibrium. Further, the equilibrium can be computed by solving the following non-linear program (NLP) for its global minimum of zero: ( k ) k k n X n X X fi;j rk ðði; jÞ; pijk Þ a½pijA pijB f1;1 þ pijA ð1 pijB Þf1;jþ1 ; min ; ð3Þ k k þ ð1 pijA Þð1 pijB Þfiþ1;jþ1 þð1 pijA ÞpijB fiþ1;1 k2fA;Bg i¼1 j¼1 s:t:
A A fi;jA P rA ðði; jÞ; 1Þ þ aðpijB f1;1 þ ð1 pijB Þf1;jþ1 Þ;
fi;jA fi;jB fi;jB
A
A aðpijB fiþ1;1
B
B aðpijA f1;1
B
B aðpijA f1;jþ1
P r ðði; jÞ; 0Þ þ P r ðði; jÞ; 1Þ þ P r ðði; jÞ; 0Þ þ
þ ð1
þ ð1
B pijA Þfiþ1;1 Þ;
þ ð1
i ¼ 1; . . . ; n; j ¼ 1; . . . ; n;
A pijB Þfiþ1;jþ1 Þ;
ð4Þ
i ¼ 1; . . . ; n 1; j ¼ 1; . . . ; n;
ð5Þ
i ¼ 1; . . . ; n; j ¼ 1; . . . ; n;
B pijA Þfiþ1;jþ1 Þ;
ð6Þ
i ¼ 1; . . . ; n; j ¼ 1; . . . ; n 1;
ð7Þ
0 6 pijA
6 1;
i ¼ 1; . . . ; n 1; j ¼ 1; . . . ; n;
ð8Þ
0 6 pijB
6 1;
i ¼ 1; . . . ; n; j ¼ 1; . . . ; n 1;
ð9Þ
A pnj
¼ 1;
8j;
ð10Þ
pinB fi;jk
¼ 1;
8i;
ð11Þ
unrestricted;
8i; 8j; k 2 fA; Bg:
Proof. See Thuijsman (1992), or Breton (1991) (recall that the action and state space are finite).
ð12Þ
Eq. (4) indicates that the total expected discounted profit fi;jA using the equilibrium strategy must be greater than the total expected discounted profit if player A chooses the pure action pijA ¼ 1 in state ði; jÞ, for all i and j; analogously in (6) for player B. A similar interpretation holds for (5) and (7) for pure action pijk ¼ 0. The complexity of (3)–(12) makes the use of analytical methods to derive a solution very difficult, if not impossible, for a meaningful number of states. Thus, we opt for a comprehensive numerical analysis. We note that numerically finding equilibria for stochastic games of a high state space size is very challenging, as reported in Breton (1991) and Souza (2000). For example, an iterated best-response play algorithm may result in cycling, and is thus not suitable (Souza, 2000). Using MatlabÕs non-linear solver FMINCON (MATLAB, 1996), we were successful in finding equilibria for n ¼ 6 by solving the resulting NLP (102 variables, 192 constraints) for its global minimum of zero, despite difficulties––sometimes, the optimizer converges to a local minimum, necessitating perturbation techniques; the procedure (similar to Breton, 1991) is as follows: An initial guess is given for ðpA ; pB Þ. With this initial guess, fi;jk , i; j; k can be computed as follows. The strategy profile ðpA ; pB Þ generates a Markov chain on the state space, as illustrated in Example 2. Denote the chainÕs transition probability matrix by P. Also, denote the vectors of fi;jk and rk ðði; jÞ; pijk Þ by f k and rk , respectively. Then 1
f k ¼ ðI aPÞ rk :
ð13Þ
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With the initial guess ðpA ; pB Þ and ðf A ; f B Þ, we solve for a global optimum of the NLP, with an objective function value of zero. If the NLP solver converges to a local optimum, we re-start from another guess, or perturb the existing solution, until we find the global optimum. In the next section, we perform a numerical study for the case where the competitors are identical. Then, in Section 5, we consider asymmetric competitors.
4. Numerical results: Identical competitors 4.1. Experimental design In our numerical study, we use n ¼ 6, since for n ¼ 7 we had significant practical difficulties in finding equilibria: long solution times (90 min or more on a Sun Ultra station), and, in most cases, the optimizer converges to a local minimum. Thus, the maximum age for a product is six periods; if we think of a period as six months, then a product is in the market for at most three years. We have evidence, however, that our insights are robust to the choice of n; we comment further on that in Section 4.2. For convenience, consider that firm kÕs percent cost markup ski decreases with age according to a learning curve with rate xk : k ski ¼ sk1 iln x = ln 2 . B A B We use the following parameter values: l ¼ 3, vA 1 ¼ v1 ¼ 20, s1 ¼ s1 ¼ 3, h ¼ 1, and a full factorial experimental design for the other parameters (A and B identical), shown in Table 1. We justify these choices as follows. The parameters l, sk1 and vk1 can be chosen arbitrarily, however, the value of K should be consistent with the choice: if K is too high relative to l, sk1 and vk1 , we have empirically observed that the resulting equilibrium is for both firms to introduce a product every n periods (maximum); if K is too low the resulting equilibrium is for both firms to introduce a product every period. Thus, for the choices of l, sk1 and vk1 above, we consider four different values of K, as shown in Table 1, which capture the intermediary equilibria between introduce a product every period and every n periods. We fix h ¼ 1, and consider three different values of d; these three combinations are chosen to simulate different markets: d ¼ 0:5 represents a price-elastic market, d ¼ 1:0 represents a market where price and technology have the same elasticity, and d ¼ 2:0 represents a technology-elastic market. The two different values of /k in Table 1 capture common learning rates found in practical applications––0:75 6 /k 6 1 (Yelle, 1979). We consider two values of xk : if xk ¼ 1:0, the new product does not command a price premium (that is, ski remains constant with age), and if xk ¼ 0:8, then ski decreases with age. Regarding the discount factor a, other analysis not reported here indicates that equilibrium solutions are relatively insensitive to a; we thus use a ¼ 0:9. A desired property of a game is that it contains a unique equilibrium. We have empirically verified that our game, however, may present multiple equilibria. This (perhaps disappointing) result helps understand why there exist so many different competitive strategies for rational firms. Unfortunately, the theory provides little or no guidance on identifying the number of equilibria. We focus on focal equilibria (Schelling, 1960): pure strategy equilibria, and, in the case of identical firms, symmetric equilibria. Solving Table 1 Experimental design for game with identical competitors Parameter
Values
xA ¼ xB ¼ x d /A ¼ /B ¼ / KA ¼ KB ¼ K
0.8, 1.0 0.5, 1.0, 2.0 0.8, 1.0 10, 30, 50, 70
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the NLP (3) through (12), however, may result in asymmetric equilibria. We search for equilibria where pijA ¼ pjiB . 4.2. Results Table 2 presents summarized results for the experimental design from Table 1. Summarized results include, for each firm, average profit per period and expected time between product introductions (ETPI). These measures can be computed through standard Markov chain techniques (for details, see Souza, 2000). ETPI is a measure of the firmÕs clockspeed: a higher ETPI means a lower clockspeed. We also report ETPI for the market (or marketÕs clockspeed), which informs whether the competitors introduce products simultaneously. For example, for fx; d; /; Kg ¼ f1; 0:5; 0:8; 70g, each firmÕs ETPI is 5.0 (Table 2), whereas the marketÕs ETPI is 2.5. This means that firms A and B never introduce products simultaneously. Detailed equilibria results, which include the equilibrium profile ðpA ; pB Þ, are available upon request. We have also performed tests to assess the stability of the equilibria obtained. Consider in Table 2 equilibria in which A(B) introduce a products if i P 4 (j P 4). These equilibria were obtained using n ¼ 6 in Table 2 Summarized results for game with identical competitors: symmetric equilibria d
/
0.5
1.0
2.0
a b
K
x ¼ 1:0
x ¼ 0:8
Average profit
ETPI (firm)
ETPI (market)
Average profit
ETPI (firm)
ETPI (market)
1.0
10 30 50 70
–a 52.4 51.5 –a
2.00 3.94 5.89 6.00
–a 1.97 3.02 –a
50.0 36.6 28.9 –a
1.00 2.00 3.91 4.00
1.00 1.00 1.96 –a
0.8
10 30 50 70
50.0 40.2 35.5 –a
1.00 4.00 5.00 6.00
1.00 2.00 2.50 –a
50.0 31.8 21.1 15.2b
1.00 2.00 4.00 5.10
1.00 2.00 4.00 3.41
1.0
10 30 50 70
50.0 48.5 46.4 42.5
1.00 2.60 3.67 4.00
1.00 2.31 1.84 4.00
50.0 36.0 28.9 24.3b
1.00 2.00 2.37 3.07
1.00 2.00 1.31 1.56
0.8
10 30 50 70
50.0 39.0 35.6 31.3
1.00 2.00 3.33 4.03
1.00 2.00 1.68 2.01
50.0 31.7 22.9 18.1
1.00 1.56 2.83 4.70
1.00 1.33 1.41 2.35
1.0
10 30 50 70
50.0 30.0 35.0 29.3
1.00 1.00 2.00 2.28
1.00 1.00 1.00 1.49
50.0 30.0 26.0 22.1b
1.00 1.00 2.00 2.42
1.00 1.00 2.00 1.45
0.8
10 30 50 70
50.0 30.0 29.0 24.4
1.00 1.00 2.00 2.34
1.00 1.00 2.00 1.44
50.0 30.0 22.4 17.6b
1.00 1.00 2.00 2.00
1.00 1.00 1.82 1.00
Equilibrium results in a multi-chain Markov chain; the stationary probability distribution depends on initial state. e-Equilibrium, where e 6 0:7.
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753
(b) Profit
(a) ETPI 50.0
4.00
45.0 3.00 40.0 35.0
2.00 1.00
d=1
30.0
d=1
d=2
25.0
d=2
0.00
20.0 10
30
K
50
70
10
30
K
50
70
Fig. 3. Effect of K on ETPI and profit for x ¼ 1:0 and / ¼ 1:0.
the NLP (3)–(12). If the equilibria are stable, we would expect to find the exact same equilibria by solving the NLP with n ¼ 4, and that is precisely what we have observed. For example, consider fx; d; /; Kg ¼ f1; 2; 1; 50g, in which the equilibrium is for both firms to introduce a product every two periods (Table 2). This result can be obtained using either n ¼ 4 or n ¼ 6 in the NLP (3)–(12). Hence, the results suggest that the choice of n is not so restrictive. In general, the results of Table 2 are in line with intuition. They also conform to results obtained if this problem is viewed from a decision-theoretical perspective (Souza, 2000), when one firm optimizes its actions based on an estimate of the other firmÕs behavior. That is, as K increases, for a fixed x; d, and /, so does ETPI for both firms, as well as for the market, in equilibrium. For example, consider x ¼ 1, d ¼ 1 and / ¼ 1:0 in Table 2: each firmÕs ETPI is 1.00, 2.60, 3.67, and 4.00, for K ¼ 10, 30, 50 and 70, respectively. As new product introduction becomes costlier, each firm has a unilateral incentive to increase its ETPI, and the equilibrium reflects this incentive. These results support the resource-based-view of the firm (Collis and Montgomery, 1995), in that a firmÕs strategy should be consistent with its capabilities. It is also consistent with Schmidt and Porteus (2000), who suggest that a firmÕs cost leadership (product introduction cost here) is necessary to sustain technological innovation (clockspeed). As d increases, for a fixed K, x, and /, ETPI in equilibrium decreases. For example, consider x ¼ 1, / ¼ 1, and K ¼ 30: ETPI for each firm is 3.94, 2.60, and 1.00 for d ¼ 0:5, 1.0, and 2.0, respectively. Firms competing in technology-elastic markets have individual incentives to decrease their ETPI––introducing new products more often––and again, the equilibrium reflects this incentive. One result from Table 2, however, is paradoxical. Consider the equilibria at fx; d; /; Kg ¼ f1; 2; 1; 30g and f1; 2; 1; 50g, with average profits of 30.00 and 35.00, respectively. For easier visualization, we plot them in Fig. 3(b). Here, a higher K implies higher profits in equilibrium, so both firms would rather have a higher fixed product introduction cost than a lower one! This paradox can be reasoned by verifying that ETPI at K ¼ 30 is the same as at K ¼ 10, for d ¼ 2 (Fig. 3(a)). We would expect ETPI to decrease monotonically with K, but in this case it remains constant as K increases from 10 to 30. So, the equilibrium at K ¼ 30 is inefficient. The effect of the learning curve on the ETPI is unclear in a symmetric game. As the learning curve becomes steeper (/ decreases), ETPI may increase or decrease, depending on the values of K and d. For a game with different competitors, however, as we will see in Section 5, learning rates play a crucial role.
5. Numerical results: Different competitors We now consider a situation where the competitors are different. Without loss of generality, consider A to be the most efficient competitor in terms of product introduction costs, that is, K A < K B . We use the
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same parameter values as before, including the experimental setup shown in Table 1, except that now we fix two scenarios. In the first scenario, there is no learning by either firm, that is, /k ¼ xk ¼ 1 for k 2 fA; Bg. In the second scenario, firm A is also the most efficient competitor in terms of manufacturing, that is, /A ¼ xA ¼ 0:8, and /B ¼ xB ¼ 1. Table 3 presents summarized results (detailed results are available upon request). A very interesting (and desirable) feature of the results in Table 3 is that all but four equilibria are
Table 3 Summarized results for game with different competitors d
A
A
x ¼ / ¼ 1:0 xB ¼ /B ¼ 1:0
xA ¼ /A ¼ 0:8 xB ¼ /B ¼ 1:0
a
0.5
KA
KB
Average profit
Average market share
ETPI
A
B
A
B
A
B
ETPI (Market)
10
30
61.6
45.9
0.55
0.45
2.00
4.00
2.00
10 10 30 30 50
50 70 50 70 70
65.8 65.9 56.9 57.1 –a
40.9 37.4 47.3 42.9 –a
0.59 0.59 0.54 0.54 –a
0.41 0.41 0.46 0.46 –a
2.00 2.00 4.00 3.60 6.00
6.00 6.00 6.00 6.00 6.00
1.50 2.00 2.40 2.57 –a
1.0
10 10 10 30 30 50
30 50 70 50 70 70
60.0 71.5 71.3 52.0 57.6 48.6
35.0 26.0 23.8 36.3 31.1 41.4
0.58 0.68 0.66 0.56 0.60 0.51
0.42 0.32 0.34 0.44 0.40 0.49
1.00 1.00 1.22 2.00 2.10 4.00
2.00 4.00 4.21 3.00 4.11 4.00
1.00 1.00 1.22 1.50 1.60 2.00
2.0
10 10 10 30 30 50
30 50 70 50 70 70
50.0 68.0 84.2 48.0 48.6 66.4
30.0 17.0 8.3 17.0 23.1 25.2
0.50 0.65 0.79 0.65 0.54 0.69
0.50 0.35 0.21 0.35 0.46 0.31
1.00 1.00 1.00 1.00 1.86 3.00
1.00 2.00 4.00 2.00 2.17 6.00
1.00 1.00 1.00 1.00 1.86 2.00
0.5
10
30
61.6
40.9
0.60
0.40
1.00
4.00
1.00
10 10 30 30 50
50 70 50 70 70
65.8 65.8 46.2 46.3 35.1
35.9 32.6 35.0 31.6 30.7
0.63 0.63 0.64 0.64 0.65
0.37 0.37 0.36 0.36 0.35
1.00 1.00 1.50 1.50 2.40
6.00 6.00 5.96 6.00 6.00
1.00 1.00 1.20 1.20 1.71
1.0
10 10 10 30 30 50
30 50 70 50 70 70
60.0 71.5 75.2 48.1 55.3 43.2
35.0 26.0 20.8 29.3 22.2 23.1
0.58 0.68 0.71 0.65 0.72 0.71
0.42 0.32 0.29 0.35 0.28 0.29
1.00 1.00 1.00 2.00 1.50 2.00
2.00 4.00 5.00 4.00 6.00 6.00
1.00 1.00 1.00 1.33 1.20 1.50
2.0
10 10 10 30 30 50
30 50 70 50 70 70
50.0 68.0 84.2 48.0 64.2 45.0
30.0 17.0 8.3 17.0 8.3 18.5
0.50 0.65 0.79 0.65 0.79 0.70
0.50 0.35 0.21 0.35 0.21 0.30
1.00 1.00 1.00 1.00 1.00 2.00
1.00 2.00 4.00 2.00 4.00 4.00
1.00 1.00 1.00 1.00 1.00 1.33
Equilibrium results in a multi-chain Markov chain; the stationary probability distribution depends on initial state.
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in pure strategies, that is, for each experimental cell, pijk 2 f0; 1g, for all i, j, and k. Because of this property, we consider the results in Table 3 to be focal equilibria. First, as d increases, for a fixed K A and K B , ETPI (for both firms, and, consequently, the market) decreases. In equilibrium, each firmÕs product introduction strategy ‘‘fits’’ the market it is competing at––a firmÕs frequency of product introductions is higher in a technology elastic market than in a price elastic market. For example, consider xA ¼ /A ¼ 1:0, K A ¼ 30, and K B ¼ 50. The ETPI for (firm A, firm B, market) for d ¼ 0:5, 1.0 and 2.0 is (4.00, 6.00, 2.40), (2.00, 3.00, 1.50), and (1.00, 2.00, 1.00), respectively. For d ¼ 0:5, detailed results (not reported here) show that the game cycles as follows: (1,4)–(2,5)–(3,6)– (4,1)–(1,2)–(2,3)–(3,4)–(4,5)–(1,6)–(2,1)–(3,2)–(4,3)–(1,4); thus there is a new product in the market every 2.4 periods ðð3 þ 1 þ 4 þ 1 þ 3Þ=5Þ and the two firms never introduce products simultaneously. For d ¼ 1:0, detailed results show that the game cycles as follows: (1,1)–(2,2)–(1,3)–(2,1)–(1,2)–(2,3)–(1,1); thus there is a new product in the market every 1.5 periods ðð2 þ 1 þ 1 þ 2Þ=4Þ and the two firms introduce products simultaneously every six periods. Finally, for d ¼ 2:0, the game cycles as follows: (1,1)–(1,2)–(1,1); thus there is a new product in the market every period, and the two firms introduce products simultaneously every two periods. Firm B (highest K) always has a lower market share and lower profits than AÕs. This can be seen directly in Table 3, by comparing the respective columns for A and B. Also, firm B has ETPI no lower than AÕs, for each game. This result goes against published recommendations that a firmÕs clockspeed should match its industry (Fine, 1998). Here, a less efficient firm (firm B) competes at a lower clockspeed than its more efficient rival, firm A. Firm A introduces products no less often when there is (manufacturing) learning, and consequently, attains a higher (or equal) market share. This can be easily seen by comparing, for a given, K A and K B , AÕs ETPI and market share when xA ¼ /A ¼ 0:8 (learning by A, bottom of Table 3) versus xA ¼ /A ¼ 1:0 (no learning by A, top of Table 3). For example, for d ¼ 1, K A ¼ 30 and K B ¼ 70, firm AÕs ETPI with learning is 1.50 versus 2.10 with no learning; firm AÕs market share with learning is 0.72 versus 0.60 with no learning. Firm AÕs average profit is slightly lower with learning at 55.3 versus 57.6 without learning, but this is because some of the gains are passed to consumers––firm AÕs price and margin are lower with learning when it has a product at age 2 for example. Firm BÕs average profit (ETPI), however, is substantially lower (higher) at 22.2 (6.00) when there is learning by A versus 31.1 (4.11) when there is no learning by A. Thus, when firm A experiences manufacturing learning, it has a much stronger competitive position in the market: a higher market share because of newer and cheaper products on average, and a weakened competitor, with much lower average profits and an older and more expensive product. Firm A gains additional market share because of its cheaper product, a result of learning, and a higher market share results in higher revenues, which can be used to introduce products more often. The opposite happens with firm B. With a lower market share, firm B has lower revenues, and, consequently, cannot introduce products as often because of the fixed costs of product introduction. It is clear from these results how manufacturing learning is important to a firmÕs competitiveness. First, manufacturing learning allows a firm to introduce products more often. This result brings a theoretical justification for IntelÕs amazing rate of product introduction, which has a significant contribution from learning in manufacturing (Fine, 2000). Second, learning not only affects a firmÕs own behavior, but it also affects the competitorÕs behavior: learning by firm A causes firm B to decrease its rate of product introduction, attaining significantly lower profits.
6. Conclusion In this paper, we have studied the game-theoretical solution of a stylized model where two firms compete over time by introducing new products at a fixed cost, and the market demand is constant (or stochastic and
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stationary, if we assume that it can always be met). The market is split according to a market-share attraction model. In this context, we define a state as an age profile ði; jÞ, corresponding to the ages of AÕs and BÕs products, respectively. At each state, each firm makes a decision regarding the introduction of a new product. These two decisions move the system to another state, where similar decisions are made, and so forth. This can be modeled as a non-zero sum stochastic game. When players maximize total discounted profit over an infinite horizon, there exists at least one equilibrium. Equilibria can be found by solving a non-linear program for its global optimum. Through numerical analysis of the underlying game, we reached the following conclusions. First, in general, the equilibria reflect the most important results obtained from a decision-theoretic perspective by Souza (2000). That is, a firmÕs pace of new product introduction increases with increasing market sensitivity to technology and decreasing fixed product introduction costs. Second, the firm with the highest fixed product introduction cost has a lower market share, lower profits, and competes at a lower clockspeed than the more efficient firm. This result goes against published recommendations that a firmÕs clockspeed should match its industry (Fine, 1998), however, it is line with the resource-based-view of the firm (Collis and Montgomery, 1995), in that a firmÕs strategy should be consistent with its capabilities. It is also consistent with Schmidt and Porteus (2000), who suggest that a firmÕs cost leadership (product introduction cost here) is necessary to sustain technological innovation (clockspeed). Finally, we find that a higher rate of manufacturing learning for a firm results in a higher clockspeed and in a higher market share, and also results in significantly worse profits for the competitor. The results underscore the importance of manufacturing expertise in a firmÕs ability to introduce new products. Our paper is one of the first to formally address strategic product introduction, that is, when and how frequently should a firm introduce new products. This research is not without its limitations. First, this is clearly a stylized model, which assumes constant demand, a duopoly, constant product introduction costs, and constant prices. Second, we use numerical analysis to derive our insights, due to the intractability of the model. Third, we consider that demand can always be met. Relaxing our assumptions are natural directions for future research.
Acknowledgements I am grateful to two anonymous referees for their helpful comments on the first version of this paper.
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