Using entropy statistics we analyse scaling patterns in terms of changes in the ratios among product characteristics of 143 designs in civil aircraft. Two allegedly dominant designs, the piston propeller DC3 and the turbofan Boeing 707, are shown to have triggered a scaling trajectory at the level of the respective firms. Along these trajectories different variables have been scaled at different moments in time: this points to the versatility of a dominant design which allows a firm to react to a variety of user needs. Scaling at the level of the industry took off only after subsequently reengineered models were introduced, like the piston propeller Douglas DC4 and the turbofan Boeing 767. The two scaling trajectories in civil aircraft corresponding to the piston propeller and the turbofan paradigm can be compared with a single, less pronounced scaling trajectory in helicopter technology for which we have data during the period . Management and policy implications can be specified in terms of the phases of codification at the firm and the industry level. • The authors are grateful to Paolo Saviotti for the collaboration and his comments on earlier versions of this paper. The research reported here is partially based on data gathered in a project funded by the ESRC (Saviotti and Bowman 1984; Saviotti and Trickett 1993) . The first author acknowledges financial support from the European Commission TMR-grant ERB4001GT961736; the second author is partially funded by TSER project PL97-1296. •• Corresponding author Introduction Utterback and Abernathy (1975) have proposed the concept of a product life-cycle to describe technological evolution at the level of an industry. At the start of a product life-cycle, a variety of product designs is being developed. The competition between designs is eventually resolved into a dominant design. Hereafter, innovation concentrates on process innovation and incremental product innovation given the dominant design. Nelson and Winter (1977) and Dosi (1982) proposed to describe a series of incremental innovations within a stable design framework as a natural trajectory or technological trajectory, respectively. Along a trajectory, development is guided and constrained by a set of heuristics which make up a technological paradigm. The trajectory concept can be appreciated as the dynamic analogue of the concept of a dominant design. Winter (1977, 1982) and Sahal (1981 Sahal ( , 1985 stressed that trajectories do not only concern periods during which the basic technological principles remain unchanged, but also a stage of incremental scaling of designs. A prime example of a series of scaled models in civil aircraft has been the piston propeller DC-trajectory. The scaling of the engine power, wing span, and fuselage length have led to improvements in speed by a factor of two, and in maximum take-off weight and range by a factor of five from the introduction of the DC3 in 1936 to that of the DC7 in 1956 (Miller and Sawers 1968; Jane's 1978) . The main heuristic of many technological paradigms can be represented in terms of the scaling of designs by means of step-by-step improvements. However, specific design principles function only within a limited range of function levels, outside of which a structural redesign becomes necessary. As Sahal (1985: 62) formulated it: "(t)he point of departure of the theory advanced here is the well-known observation that change in size of an object beyond a certain point requires changes in its form and structure as well." Thus, technological development within a paradigm must come to an end at some 2 point in time. This expectation justifies the idea that technological paradigms exhibit life-cycles. For example, in the case of piston propeller aircraft technology, the life-cycle came to an end when it was understood that the functioning of propellers would decrease rapidly as cruising speed approached the speed of sound. Engineers envisaged that further scaling would become increasingly more difficult to realise due to non-linear rises in vibration and heat generation. Constant (1980) introduced in this context the concept of a presumptive anomaly. In the case of piston propeller technology the presumptive anomaly was circumvented by developing turbopropellers, rocket, jet, and turbofan engines as alternative technologies. The turbofan engine technology based on jet propulsion would eventually provide the basic technology for a new technological paradigm. Similarly, the quest for down-scaling computers led to a series of technological paradigms (vacuum tubes, transistors, integrated circuits). A new paradigm has to compete during its initial phase with the momentum invested in the previous technology. A period of experimentation and recombination can be expected during which established firms may fall behind and new firms enter the market (Anderson and Tushman 1990) . At this point, various firms follow their own trajectories which compete for dominance at the industry level. This competition process is resolved into a new dominant design which lays down a new set of heuristics. However, market segmentation may cause a bifurcation of trajectories as different design trajectories meet segregated market demands (Teubal 1979; Foray and Grübler 1990; Frenken et al. 1999) . In the case of civil aircraft, turbofan aircraft with swept wings has become the dominant technology, but turbopropeller aircraft with straight wings is still used in short-range airline operations. While single products can be expected to exhibit a life-cycle at the level of the firm, families of products are expected to exhibit a business cycle at the level of the industry (Foray and Garrouste 1991). The frequencies of these cycles are of different orders of magnitude (Simon 1969) . The methodological problem is how to account for developments on both the micro-level and the macro-level. Below, we 3 develop a scaling measure based on information theory which will enable us to analyse the data at the level of individual firms, at the industry level, and in relation to each other. Methods Statistical studies on patterns in technological change during the product life-cycle are based on a variety of data sources. Jane's (1978 Jane's ( , 1989 Jane's ( , 1998 , Chante (1990), and Green and Swanborough (1982) . A description of the characteristics is provided in Table 1 . 1 These data sources include prices, sales, entry-exit and patent data (Gort and Klepper 1982) , innovation counts, sales, entry-exit and productivity data (Klepper and Simons 1997), product characteristics, sales and entry-exit data (Tushman and Anderson 1986; Anderson and Tushman 1990) , patent data (Malerba and Orsenigo 1996) , and product characteristics (Sahal 1981 (Sahal , 1985 Saviotti 1996; Frenken et al. 1999 ). 4 The process of product design contains an ongoing search between users and producers for an optimal 2 The characteristic approach to technological innovation can be considered an adaptation of Lancaster's (1966) demand approach. While Lancaster needed only a set of service characteristics, a requires a 5 6 match between technical and service specifications. In the case of product innovation, uncertainty prevails on the side of both the producer and the user (Clark 1985; Andersen 1991) . Users are expected to select particular designs on the basis of functions, but new user wants can be envisaged when new combinations of technical characteristics become possible. Therefore, the assumption of a stable market environment selecting upon a variation of techniques solely on the basis of prices, can no longer be taken for granted. For example, the engine power needed to fly an aircraft of a certain size is bounded by technical principles, but its fuel consumption bears on the operating costs taken into account by users. The relative technical and commercial importance of engine power is thus at variance over time. The distinction between technical and service characteristics is not always easy to apply. The scaling measure as developed below takes into account all relations between characteristics, whether defined as a technical or a service characteristic, thus solving the definition problem without losing its conceptual meaning. Scaling trajectories can be expected when the definitions of and the relationships between various characteristics of a product become stabilised ("closure"). Innovations aiming at scaling may be motivated by signals internal to the technology; the scaling in some parts of a technology can generate imbalances in other parts which in turn call for adjustments (Rosenberg 1969) . During scaling processes, however, the overall design architecture and the set of functions of a technology remain largely invariant over prolonged periods of time. Rather, technical and service characteristics can then be expected to co-evolve (Windrum and Birchenhall 1998) . A quantitative empirical analysis should enable us to indicate these periods of relative stability. representation of the supply side as a set of technical characteristics. See also, Saviotti (1996) . Designs make up the interface between supply and demand as it is expressed in the various trade-offs among characteristics. Scaling then, is indicated when these trade-offs remain stable over time. We will model the trade-offs by using the ratios among all six characteristics, thus taking into account both the relations among and between technical and service characteristics. Scaling may affect all these ratios. If none of the trade-offs is changed, the scaling is not innovative. For example, for the product characteristics of the Douglas DC3 we have: engine power = 1636 kwatt, The set of ratios can be considered as a probability distribution (p 1 ,...,p 30 ) by dividing each ratio by the sum of the ratios. In this manner, we obtain a probabilistic representation for each aircraft. In order to analyse the scaling development in subsequent product designs, one is in need of a distance measure between the representations in terms of probability distributions. Using information theory, one is able to calculate on the basis of a distribution the expected information content contained in the message that the distribution has changed as a next design was introduced on the market using the following formula (Theil 1967 (Theil , 1972 : The expected information content of the a posteriori distribution (q 1 ,...,q 30 ) given the a priori distribution (p 1 ,...,p 30 ) can also be considered as an information-theoretical distance between product designs in terms of scaling. If none of the trade-offs was changed, the probability distribution has remained the same. Compared with the previous product, such a design would be a perfectly scaled version. In that case, every q i is equal to its corresponding p i , so that I vanishes, since log 2 (1) = 0. It can be shown that otherwise I is positive (Theil 1972, pp. 59f.): the message that change has occurred is expected to contain information or, in other words, a probabilistic entropy is generated. In the following, we shall use I as a measure of the degree of scaling between two product designs: the lower the value of I, the more similar are the ratios between two product designs and the more the latter design can be considered as a scaled version of the former design. Note that our approach differs from that of Sahal's (1981 Sahal's ( , 1985 who used parametric tests on scaling constants over long periods of time. We do not assume that scaling has a unequivocal direction throughout the process of scaling. For example, if we want to compare the distribution of ratio values of the Douglas DC4 with the Douglas DC3, we take the DC3 as the a priori distribution (p 1 ,...,p 30 ), and the DC4 as the a posteriori distribution (q 1 ,...,q 30 ). As done for the DC3 above, one is able to calculate the ratios between the product characteristics of the DC4, and divide these ratios by the sum of ratios to obtain the envisaged representation of the DC4. Using formula (1), we can then calculate the scaling distance between these two product designs, that is, the I (DC4 | DC3). 8 The advantage of the algorithmic approach becomes clear when we compare three instances in a series like the sequence of three products A-B-C. In a geometrical representation of differences between three products designs in an Euclidean space, the distance between A and C will be smaller than the sum of the distances between A and B and between B and C (Theorem of Pythagoras). In contrast, the informationtheoretical distance between two products A and C being I ( C | A ) is not necessarily smaller than the sum of the distance between products A and B being I ( B | A ) and between products B and C being I ( C | B ). Three designs A, B, and C, and their respective distances I are depicted in Figure 1 . Design A precedes B, and B precedes C. Using the dynamic information measure, it is possible that the sum of the intermediate distances between A and B, and B and C, will be smaller than the distance between A and C. In this case: This formulation is equivalent to: If this inequality is confirmed, the transition from design A to design C via design B can be considered a critical transition. The inequality enables us to evaluate the function of the intermediary. In the normal case, one expects C | A) . From the perspective of design C, the "signal" sent by design A was amplified by design B as an auxiliary transmitter to such an extent that design B "boosts" the signal from A (Leydesdorff 1992 (Leydesdorff , 1995 . Thus, the transition is critical for the further path of the development. We use the term "critical transition" rather than "path-dependent transition", because the concept of path-dependency was defined by Arthur (1989)  The probability distribution of ratios provides us with a unified representation of each product design in the dataset. This enables us to measure scaling patterns in a series of product designs. For example, we will analyse below the scaling trajectories in the Douglas and Boeing companies. The trajectory notion, however, also refers to an industry-wide convergence in scaling patterns. In order to measure scaling trajectories at this level of aggregation, one needs to compare each product design with all other designs relevant in the competition. Although each observation maintains a distance from all other observations in the dataset, not all observations are relevant for the comparison. We are interested in the development and competition in certain periods. The population thus has to be determined dynamically. The data show that the major development cycles of individual firms have been the DC3-MD11 series with an average frequency of a new product each 5.6 years, the Boeing 707-777 series with an average of 6.2 years, and the Fokker Fseries with an average of 8.0 years. Since pre-war product development cycles were usually considerably smaller than post-war cycles, we have simplified our computations by using a time window of five years after the year of introduction. Thus, we compare the design of an aircraft introduced in, say, 1936, with all designs introduced during the period [1937][1938][1939][1940][1941]. However, we have tested our results for their robustness by changing the time-frame to ten years (e.g. , 1937-1946). This did not affect our conclusions. On the one hand, a test of scaling at the industry level needs to take into account the rate of diffusion of the ratios between product characteristics. On the other hand, the codification of design principles associated with the emergence of a dominant design also implies a convergence of particular design principles that have been developed in the past (Dosi 1982). Thus, the coming into existence of a scaling trajectory at the industry level is essentially a two-sided phenomenon. It refers both to the diffusion of design principles, and to the convergence of design principles. These phenomena are different: the some aircraft firms may scale a dominant design with respect to maximum take-off weight, others with respect to speed, and still others with respect to range. Hence, to test the dominant design hypothesis, one needs to distinguish between the diffusion of design principles through time and the convergence of design principles that can be observed in retrospect. In our information-theoretical framework, these different dynamics call for a change in the a priori and the a posteriori frames of reference. In the case of diffusion of a design through time, the frame of reference is a particular product design as the a priori expectation of future designs. The diffusion of a product design can then be measured by its distance I following formula (1) The hypothesis that a scaling trajectory is characterised by common heuristics among firms, holds that ratios between product characteristics will tend to converge. The degree of convergence, then, is indicated by the distances between all the products in a population that preceded a product, when taking the latter as the frame of reference with hindsight ( Figure 2b) . Thus, the single design introduced later in time is taken as a yardstick, while the population of products preceding this design during five years is considered the set of events relevant for the evaluation.  FIGURE 2 ABOUT HERE A stylised history of civil aircraft Before turning to the computational results, let us briefly return to the expectation. We have used the civil aircraft industry since the history of its technological developments is one of the best documented. Both historians and economists have analysed these technological development in considerable detail (Miller and Sawers 1968; Constant 1980; Mowery and Rosenberg 1982; Sahal 1985; Gardiner 1986a Gardiner , 1986b Vincenti 1990; Bilstein 1996) . Although these studies differ in perspectives and in the levels of their quantitative and qualitative measurements, there is general consensus on the following findings: 1. The early history of civil aircraft, covering roughly the period between 1920 and 1935, is characterised by a large variety of designs and a small market demand. The main bottleneck concerned the limited range of flight, which rendered the competitiveness of this technology low compared to the services of trains and boats. In the thirties, new product designs in the United States allowed for long-range airline operations at reasonable speed levels. As a result, aircraft out-competed trains in terms of speed, comfort, and price over long distances (e.g., coast-to-coast flights). The Douglas DC3, an all-metal, monocoque piston propeller monoplane with engines placed under the wings, is commonly considered as the dominant design. Because of the rapid increase in market demand for cheap air-traffic, learning curves of the DC were particularly steep. The total production of the DC3 was over 10,000 models, including military production (Jane's 1978). During the fifties, market demand in European countries increased rapidly. Some European producers followed the successful American piston propeller aircraft, while other firms used turbopropeller aircraft (e.g., Vickers, Fokker). Early attempts of DeHavilland and Aerospatiale to commercialise jet engine technology in civil aircraft failed. In the late 1950s, a second revolution in civil aircraft design took place as the Boeing company 13 successfully introduced the Boeing 707, a long-range turbofan aircraft that had been redesigned from this company's bomber line. The introduction of the Boeing 707 was followed by a series of turbofan aircraft labelled the 700-series. In the late sixties, McDonnell Douglas and Lockheed succeeded in introducing long-range turbofan aircraft at competing price levels. Only a few European and Sovjet aircraft designers developed civil turbofans, while the majority focused on short-range turbopropeller planes. Finally, during the seventies and eighties, turbofan technology diffused throughout the market for medium and long-range distance flights. Most recently, within the segment of short-range aircraft which has traditionally been covered by turbo-propeller aircraft, turbofan technology has also penetrated, rendering the diffusion of this new paradigm nearly complete (Jane's 1995). In summary, two major breakthroughs have taken place which revolutionised the civil aircraft industry. The advent of the Douglas DC3 in the mid-thirties, which opened up a mass market and imposed a dominant design in piston propeller aircraft, and the introduction of the Boeing 707 in the late fifties, which radically extended payload, speed and range capabilities using turbofan engine technology and swept wings. The production and operating costs of these designs and their subsequent follow-up models (the DC-series and the 700-series, respectively) decreased rapidly as a result of high sales figures which allowed for steep learning curves. Both the DC3 and the Boeing 707 are said to have reoriented design principles at the industry-wide level. In the remainder of this article we will focus on the questions of whether these two designs have generated critical transitions, and whether their introduction forced the industry to follow a common scaling trajectory. Critical transitions at the firm level 14 Using the inequality given in formula (3), product sequences including the Douglas DC3 as the intermediate design (B in Figure 1 ) and the Douglas DC4 as the follow-up design (C) were used for testing whether the DC3 marked a dominant design for the Douglas company. As preceding designs (A), we selected thirteen historically important precursors including the Boeing 247D and Fokker's Trimotor, which have been identified as the DC3's major competitors at the time (Miller and Sawers 1968) . Table   2 shows the results of the tests on critical transitions in product sequences that include the DC3 and the follow-up DC4 model, as defined in formula (3) We also tested the chronological DC/MD-series which succeeded the emergence of the DC3 for critical transitions. These results are listed in Table 3 . Except for DC7-DC8-DC9 sequence, no critical transitions have taken place. The DC8 implied a fundamental reorientation of design principles, since from then (1967) onwards, Douglas used turbofan engines instead of piston propeller engines. Obviously, turbofan technology implied a set of new design principles (including swept wings which allowed for a smaller wing span in relation to fuselage length). The ratios between product characteristics hence were organised in a new manner and scaled according to new heuristics. In the history of the Douglas company, we can thus distinguish a piston propeller and a turbofan trajectory. During all other transitions, no reorientation in design principles took place within the series. All models developed before the DC8 concerned piston propeller aircraft, which all built upon the DC3-design, but in different respects. For example, the DC4 was a up-scaled version of the DC3, while the DC5 was a down-scaled version of the DC3. The DC6 was a faster version of the DC4 made possible by its pressurised cabin which allowed for higher altitudes. The DC7 was marked by its long range. The models developed after the DC8 concern turbofan aircraft. Again, scaling pursued different directions from the DC8 onwards: the DC9, MD 80 and MD 85 are all down-scaled aircraft in all product dimensions. The DC10 is also smaller than the DC8 but has a higher engine power and maximum takeoff weight. Finally, the MD11 is the largest aircraft developed by McDonnell Douglas exceeding all previous aircraft in terms of engine power, length, maximum take-off weight and speed.  In 1957, Boeing introduced the 707 as a new civil aircraft using turbofan engine technology. This aircraft has allegedly served as a dominant design for the subsequent 700-series. Using our methodology, we tested product sequences including the Boeing 707 as the dominant design (B in Figure   1 ) and the Boeing 727 as the follow-up design (C). 4 As preceding designs (A), we used the latest 3 Confer Gardiner's (1986b : 143) distinction between robust and lean designs : "a robust design is one that brings together several new divergent lines of development to form a new 'composite' design, which is then internally adjusted to form a new 'consolidated' design, which is then further developed as a variety of 'stretched' design. Lean designs fail at one, two, or more often all three of these stages". Gardiner mentions the DC3 and the Boeing 707 as examples of robust designs in the history of civil aircraft. 4 The Boeing 717 has never been used for civil services. Testing the trajectory development within the Boeing 700-series (Table 5) , we found that only the most recent transition is critical, while all other transitions were found non-critical. The result for the most recent transition suggests that it was not the development of the 707, but the development of the 767 that laid the basis for the 777. Indeed, the 777, the largest two-engine aircraft ever built, elaborates on the two-engine lay-out incorporated in the 757 and 767, as different from the four-engined Boeing 747. As was the case in Douglas DC3, Boeing's dominant 707 design has been elaborated in various directions. The 727 and the 737 were down-scaled models using three and two engines, respectively, instead of the four-engine set-up of the 707. The 747 is a up-scaled wide-body version with four engines, while the 757, 767 and 777 all incorporated two engines.  18 In the previous section, we have analysed developments in aircraft design in terms of product sequences. The results indicate that both the DC3 and the Boeing 707 reoriented design principles when compared with their precursors. The results so far suggest only that the DC3 and the Boeing 707 have functioned as dominant designs at the firm level. These designs, however, are not necessarily the dominant designs that diffused throughout the industry. This question must be analysed in terms of dynamics at the level of the entire population of products. As explained in section 2, one is able to calculate an average value of the expected information content for each product design in relation to relevant successive designs ("diffusion") and in relation to its precursors ("convergence"). As an indicator of the rate of the diffusion of a design, the average I-value
Probabilistic entropy statistics and scaling trajectories in 143 civil aircraft designs have been analysed in terms of changes in the product characteristics REF .
Scaling Trajectories in Civil Aircraft (1913-1997)'
Research Policy 29(3) (2000) 331-348
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