On endogenous growth and increasing returns: modeling learning-by-doing and the division of labor

Journal of Economic Behavior & Organization Vol. 46 (2001) 39–55

On endogenous growth and increasing returns: modeling learning-by-doing and the division of labor Nicola De Liso a,∗ , Giovanni Filatrella b,1 , Nick Weaver c,2 b

a University of Lecce, Idse-Cnr Milan, and ISUFI-Lecce, Italy INFM Unit of Salerno and Faculty of Science, University of Sannio, Benevento, Italy c School of Economic Studies, University of Manchester, Manchester, UK

Received 10 August 1999; received in revised form 10 November 2000; accepted 18 November 2000

Abstract This paper discusses those sources of endogenous growth arising from labor as labor. It uses a production function which models the returns to scale as a function of the division of labor and learning. Smithian analysis of the labor process constitutes the basis upon which we build our own approach. © 2001 Elsevier Science B.V. All rights reserved. JEL classification: D24; J24; O41 Keywords: Endogenous growth; Returns to scale; Division of labor; Learning-by-doing

1. Introduction Endogenous growth has become a widely used term, but different authors have used it to mean different things. Sometimes, it has been used synonymously with increasing returns to scale, at other times it has been used to describe endogenously created technological change, which in turn leads to increasing returns. The processes involve various explanations of the roles of human capital and R&D. All of them basically concentrate on a factor that can be accumulated; that is capital in a broad sense. As Mankiw has put it: [C]apital is a much broader concept than is suggested by the national income accounts. In the national income accounts, capital income includes only the return to physical ∗

Corresponding author. Present address: Facolt`a di Giurisprudenza, University of Lecce, via per Monteroni, 73100 Lecce, Italy. Fax: +39-0832-321283. E-mail address: [email protected] (N. De Liso). 1 Permanent address: Facolt` a di Scienze, Universit`a del Sannio, Via Port’arsa 11, 82100 Benevento, Italy. 2 Permanent address: School of Economic Studies, University of Manchester, M13 9PL Manchester, UK. 0167-2681/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 6 8 1 ( 0 1 ) 0 0 1 8 6 - X

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capital. . . More generally, however, we accumulate capital whenever we forgo consumption today in order to produce more income tomorrow. In this sense, one of the most important forms of capital accumulation is the acquisition of skills. Such human capital includes both schooling and on-the-job-training (Mankiw, 1995, p. 293). The role of capital, broadly defined, has been carefully examined in the context of endogenous growth. Indeed, it explains a good deal of endogenous growth. In the present work, however, we look at the sources of endogenous growth in labor rather than capital. That is, we will highlight the role of labor in a narrow sense by deliberately avoiding issues related to human capital. In this context, we can identify two sources of increasing returns to scale: the division of labor and learning-by-doing. Both these phenomena have been recognized as important since Adam Smith (1776), and have been repeatedly referred to by many authors, from Young to Arrow and others. Our perspective is microeconomic and we will refer to those forms of division of labor and learning-by-doing which can be developed within a given firm or organization. It is worth remembering that there may be a trade-off between the exploitation of the available knowledge and the need for expanding and modifying that knowledge (Marengo, 1993), and that dynamic capabilities are necessary to cope with the shifting character of the markets (Teece and Pisano, 1994). The mechanisms which will be referred to later in this work belong in the ones which contribute to extract as much as possible from a given body of knowledge 3 , and this helps explain why, in such a context, neither the division of labor nor learning-by-doing can go on forever. The paper is organized as follows: Section 2 considers Arrow’s analysis of learning-by-doing and points out some difficulties which arise from it; Section 3 contains the analysis and the model that we propose in order to deal with increasing returns to scale — we use a production function in which labor is the only variable; Section 4 illustrates the analytical properties of the mapping obtained in the previous section; Section 5 contains a maximization exercise characterized by our production function and a context in which wages are constant, while price decreases; finally, Section 6 contains the conclusions. 2. Arrow’s analysis In 1962, Arrow published his famous article on ‘The economic implications of learningby-doing’. His explicitly declared aim was to suggest an “endogenous theory of the changes in knowledge which underlie intertemporal and international shifts in production functions” (Arrow, 1962, p. 155, emphasis added). He made use of an aggregate production function in which both capital and labor were used, taking as a starting point examples found in both economic and technical literature. The examples he uses, in which learning has occurred, and on which firms’ managers can rely, include the production of particular types of airframe and the so-called “Horndal effect” observed by Lundberg in the Horndal iron works in Sweden. In the former case Arrow reports T.P. Wright’s study, whose main result was that “the number of labor-hours expended in the production of an airframe. . . is a decreasing function of the total number of 3

For an analysis of the firm as a knowledge-creating entity see Nonaka et al. (2000).

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airframes of the same type previously produced. Indeed, the relation is remarkably precise; to produce the Nth airframe of a given type. . . the amount of labor required is proportional to N−1/3 ” (Arrow, 1962, p. 156). In the latter case, productivity rose by 2 percent per annum despite the fact that no investment had occurred in the 15 years considered. Incidentally, let us note that also Kaldor had something to say about productivity growth without investment: It may be assumed that some increases in productivity would take place even if capital per man remained constant over time, since there are always innovations — improvements in factory lay-out and organization, for example — which enable production to be increased without additional investment (Kaldor, 1957 [reprint 1980], pp. 265–266). 4 Arrow uses as his index of experience gross cumulative investment. This is problematic. Firstly embodied and disembodied forms of technical change are subsumed under a single heading — embodied technical change implies a change in physical capital while disembodied technical change does not. Secondly, Arrow’s concept of “learning” is a peculiar one since he argues that the choice of cumulative production of capital goods is motivated by the fact that: “each new machine produced and put into use is capable of changing the environment in which production takes place, so that learning is taking place with continually new stimuli [and that. . . at] any moment of new time, the new capital goods incorporate all the knowledge then available, but once built their productive efficiency cannot be altered by subsequent learning” (Arrow, 1962, p. 157). Thus, he is referring to two different forms of technological change, rather than learning per se: the one developed in capital goods industry and the one developed in the firms using new vintages of capital goods; neither form, however, can be properly classified as “learning-by-doing”. Learning has also been tackled by Rosenberg (1982), who distinguishes between learningby-doing, which is a characteristic of human beings, and learning-by-using, which is a form of disembodied technical change, concerning the use of machines produced by others. Learning-by-using involves the same capital goods performing better and better as their properties become more known; typical examples are the better exploitation of machine tools and a reduction in the amount of maintenance required by machinery. Rosenberg’s point confutes Arrow’s assertion that once built new capital goods’ efficiency cannot be altered — this view, by the way, is also contradicted by the Horndal effect. Learning occurs, often unintentionally, not only in the production of physical goods, but also in the production of services. For instance, as early as 1832 Charles Babbage stressed the role of learning-by-doing in the calculation of logarithms from 1 to 200,000. The calculation process was organized by dividing the job into three phases; in the first a handful of pure mathematicians elaborated formulae as general as possible; in the second a few persons having some knowledge of mathematics produced some examples; in the third, persons whose knowledge of mathematics did not go beyond the four basic mathematical 4

Kaldor’s analysis considered explicitly a — macroeconomic — technical progress function, and took into account the existence of increasing returns to scale engendered by both static and dynamic economies of scale; he also explicitly recalled the relationship between increasing returns, technical progress and learning-by-doing.

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operations performed the actual calculations. Babbage (1832) noted how this third group of workers quickly improved the speed and accuracy of their calculations. Nilsson (1995) added to the concept of learning-by-doing the role of workers’ cumulative experience in promoting innovation rather than merely speeding up existing processes. He calls this innovating-by-doing, and gives three examples: standardization of components in the US armories during the early 1800s; the importance of pilots in suggesting procedures to the aircraft industry; the introduction of ‘thin slab technology’ in the steel industry in the late 1980s. Nilsson’s work — we can note — complements recent work on the evolution of capabilities and core-competencies. 5 3. Labor and endogenous growth 3.1. Prolegomena to a model The aim of our analysis is to complement the, by now, traditional concern with explanations of endogenously created increasing returns due to human capital accumulation, or R&D activity, or returns to physical capital 6 , with a discussion of the role of labor as labor. We use a “new” production function, in which labor is the only input. Physical capital is either non-existent or can be subsumed under the constant A. We use this function as purely a micro-theoretical tool. However, we think that this is a useful abstraction because it allows us to examine some of the “origins of endogenous growth” 7 which have not hitherto been formally highlighted. Our starting point is a sort of Cobb–Douglas production function in which only one commodity is produced and labor is the only input: Y = ALZ

(1)

where Y is total production, A is a positive constant, L the labor input and Z represents the returns to scale. Labor is, at the beginning, homogeneous and undifferentiated. If Z is greater, equal to, or smaller than 1 we will have, increasing, constant, and decreasing returns to scale, respectively. We assume that the labor supply is unlimited. Our principal aim is to explain why Z can have different values at different times, and, in particular why it might become greater than 1 over time. The first step is to write a dynamic function: t Yt = ALZ t

(2)

and to recognize that the exponent itself is some function of the labor input, the division of labor and learning-by-doing. To highlight the link between the returns and output, the returns to scale can also be written as the labor/output elasticity: Zt =

Lt ∂Yt Yt ∂Lt

(3)

5 See the works already quoted by Marengo (1993), Teece and Pisano (1994), Nonaka et al. (2000) as well as the work by Prahalad and Hamel (1990). 6 The standard references are Aghion and Howitt (1998) and Barro and Sala-i-Martin (1995). 7 This is the title of Romer’s 1994 article.

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Whenever, we add units of labor to a function such as (2) organizational adjustments, which in this case are synonymous with an increased division of labor, are necessary. We will emphasize the role of a more detailed division of labor and of learning-by-doing. Such a function is dynamic according to two different criteria: because it takes time into account, and because it shows irreversibilities. Once certain phenomena have occurred, they become a permanent feature of the productive system. That is, once we have learnt how to produce more efficiently, we cannot forget it — though this does not mean that output could not be restrained voluntarily. We should be able to observe, at least conceptually, three distinct effects which individually and jointly affect total production and the productivity of the individual worker; that is total production can be superadditive because: (i) of a more detailed division of labor in the work process; (ii) there occurs learning-by-doing at the individual level, so that each component becomes more productive as time goes by; (iii) there occurs learning-by-doing at the level of the firm as a whole; an improvement in collective competence. The three effects can be observed in the dynamics of Zt . It is important to emphasize the fact that we will concentrate on Zt as our ‘index of experience’, and not on cumulative output directly, as more ‘traditional’ works do (cf. Fellner, 1969). The reasons for doing this are threefold: first of all, we are specifically interested in the evolution of Z as time goes by; secondly, as Eq. (3) shows, Zt and Yt are linked, so that there is always a relationship between the two, i.e. Z can capture the ‘cumulative-output effect’, while, thirdly, at the same time, by giving us the level of the returns Zt , takes into account the effect of time on learning. In other words, the level of the returns today depends on their level of the previous period(s), while capturing the contribution of learning-by-doing which grows, up to a certain maximum, as time passes; learning-by-doing affects the returns, depending on how long one has been performing the same task. We do not put any time index on A, as we are not interested in exogenous variations in total output. Of course, this does not mean that all of the factors which can affect A are unimportant; in particular, we can expect a ‘contagion effect’ going from the variable part of the function to the constant term A itself. 8 We want to write explicitly a function for Z in order to take into account both (i) the increased division of labor and (ii) learning. Both phenomena are characterized by certain features and limits that we can briefly describe as follows. 1. With respect to the division of labor, we must note that the work process can be ‘divided’ up to a certain point, i.e. there exists a technologically-determined upper limit to the number of operations into which the process itself can be split. In order to model this division-of-labor effect we make use of a logistic equation which possesses two properties: on the one hand, it allows for a process of smooth division of labor to take place, while, on the other hand, it also contains a ‘damping term’ which keeps the process from going to infinity. To identify the upper limit to the process of division of labor is quite difficult: as F.W. Taylor taught us, the limits to subdivision of labor do not finish at the ratio 1/1, i.e. one person per task, as what we see today as the simplest task might be further split into sub-operations at some future time. 8 Early studies on the (aggregate) production function actually considered the opposite direction through the functional form Y = A(t)f (K, L) (Solow, 1957).

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It is worth emphasizing that the process of division of labor is affected by the existing capital which in our production function is not a variable; in particular, we may expect (constant) capital to affect the value of λ, i.e. the asymptotic value that can be reached by the division of labor, of the logistic component referred to in Eq. (4) — see Section 3.2. 9 Furthermore, as Adam Smith pointed out long ago, the division of labor may depend on the extent of the market and this is an economic limit. The relationship between the division-of-labor and the extent-of-the-market deserves some attention. Young (1928) argued it is bi-directional: a more detailed division of labor favoring a more extended market, while a more extended market is likely to lead to a more detailed division of labor. Following this reasoning, we arrive at the conclusion that the division of labor depends on the division of labor, which is something more than a tautology: in fact, we have here an endogenous force capable of generating different forms of structural economic dynamics, characterized by the growth of total output, evolution of labor skills and performance, and decreasing prices. 2. Learning capabilities are not unlimited, particularly when they are applied to a particular work process; it can be useful to think of forms of learning which develop within a specific trajectory. Learning, whatever the length of period during which we have been doing a certain operation, or whatever the cumulative output we have produced, cannot go beyond a certain limit. Also, with respect to the way in which individuals and organizations ‘remember’, we can reasonably expect that only a certain percentage of what has been learnt can be retained, and the more we go forward in time, the more we have forgotten of past production. Such a process can be modeled by means of a ‘map with memory’ — more of this later. A key point in pushing farther the limits of learning is constituted by work organization. Often ‘simple’ reorganization of work has a great influence on the productivity of individuals and whole firms. Whether a firm is organized along Taylorist lines or not makes a difference, and more and more studies have been performed on the way in which firms see themselves and on how they change their internal organization given the existing technology (Clegg et al., 1996). Furthermore, as Kaldor pointed out, reorganization may create fresh opportunity for further reorganization (Kaldor, 1957). 10 However, despite the fact that there exist technological and economic forces which tend to push farther the limits of both the division of labor and learning, limits themselves cannot be pushed farther forever. 3.2. The model We can now write an explicit functional form for Z capable of taking into account increased forms of division of labor as well as learning: 9

However, our reference to Taylor exemplifies what we mean: one of the main reasons of Taylor’s fame consisted of his capability to speed up production processes of given plants. 10 A critical point is emphasised by Weick and Westley (1996, p. 440): “to learn is to disorganize and increase variety. To organize is to forget and reduce variety. In the rush to embrace learning, organizational theorists often overlook this tension. . . ”.

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t

Zt+1


λ = + γ ωj Zt−j 1 + β e−αLt+1

(4)

j =0

This is our fundamental equation, and we are going to explain what its main characteristics and implications are. The exponent Z is written as the sum of two components: (i) the first which shows the pattern along which labor can be divided, represented by a logistic equation, while (ii) the second component describes the way in which past returns, via past production, affect present returns and thus production. 1. For the first component, the logistic is chosen as it is capable of describing the transition between two asymptotic values, ranging from the most limited to the best possible form of division of labor: formally, this component of the returns, would reduce to λ/(1 + β) when or L → 0 (or L  1/α), and to λ when or L → ∞ (or L  1/α). 2. For the second component, the formalization of learning must take into account that: (a) learning is limited; (b) only a certain percentage of what is learnt can be retained; (c) the more we go back in time, the less the effect of that production is felt today. Put it another way, the cumulative effect is finite. The formalization proposed, with both γ and ω < 1, takes into account these characteristics 11 — but let us elaborate on this. We can write our function in a slightly different way, so that we can make use of the mapping technique. The first step consists of the assumption that the labor force employed in the work process is increased by L0 each period: Zt+1 =

λ 1 + β e−αL0 (t+1)

+γ

t


ωj Zt−j

(5)

j =0

The second step consists in introducing a new variable which we call W, and we will write it as Wt+1 =

t


ωj Zt−j =

j =0

t


ωj Zt−j + Zt

(6)

j =0

which can be rewritten as Wt+1 = ω

t


ωj −1 Zt−j + Zt

and, by setting i = j − 1

(6 )

j =0

Wt+1 = ω

t−1


ωi Zt−i−1 + Zt = ωWt + Zt

(6 )

i=0

Thus, substituting (6 ) into (5) we get  λ Z + γ (Zt + ωWt ) t+1 = −αL 0 (t+1) 1+βe  Wt+1 = ωWt + Zt 11

In order to avoid confusion, we have to point out that ω’s superscript j is a genuine exponent.

(7)

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This is a linear non-autonomous two-dimensional map 12 capable of describing the value of the returns at time t + 1 as a function of t. The returns of each period are weighted by a parameter ω, while γ ‘translates’ the learning effects into production, so that the higher is γ , the more the learning affects Z and the function as a whole. Given the assumption on the way in which production is ‘remembered’ we expect both ω and γ to be smaller than 1. It is convenient to introduce a map because while Eq. (5) retains memory of all the previous steps (namely, the returns to scale at each time step, t), Eq. (7) gives the dynamics in terms of the configuration one step before. The ‘price’ that must be paid is the introduction of a new variable, Wt , and the map becomes two-dimensional — and yet it is much more convenient than a one-dimensional map with memory. We will look at the values of the parameters in the next section. Before we do this, let us point out that we are aware of the fact that the quantitative results we get depend on our assumptions; however, the basic assumptions we make, i.e. that the division of labor and learning within a specific work process cannot go on forever, do seem to respect what we have seen throughout the history of technology — at the end of the next section we will reproduce the Horndal effect. We thus believe that the model we propose is an interesting way to formalize the dynamics of the returns within the limits described earlier in this section.

4. Analytical properties of the map First of all we consider the system’s asymptotic behavior, so that given t sufficiently large one gets:  Z = λ + γ Z + ωγ W (8) W = Z + ωW and after solving:  λ(1 − ω)     Z = 1−ω−γ    W =

(9)

λ 1−ω−γ

We need to check that the stability conditions are fulfilled, that is we have to calculate the eigenvalues of the Jacobian (cf. the Appendix A) and we have to impose the condition that their modulus be smaller than 1. Thus, we obtain ω+γ <1

(10)

We can now run a simulation with the following values: λ = 0.6,

β = 0.8,

α = 0.1,

ω = 0.8,

γ = 0.1,

L0 = 0.1

12 A full analysis of the mapping techniques, including nonlinear cases, can be found in Thompson and Stewart (1986).

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Fig. 1.

Fig. 1 shows how returns change as we add labor units within the work process. The solid line corresponds to the case in which there also occurs learning (γ > 0, i.e. γ = 0.1), while the dashed line shows what happens if learning is zero. It is very interesting to compare the two lines, as with γ > 0 returns, after some time, become increasing, while with γ = 0 they do not. Of course, the higher is γ the greater will be the difference between the two paths. It is important to underline that in either case there exists an asymptotic value for Z, which means that neither the division of labor, nor learning-by-doing can have positive effects for ever, i.e. there exists an upper limit for both of them. In Fig. 2, the relationship between labor input and total output is shown. As one can see the presence of positive learning is quite important.

Fig. 2.

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The difference between the solid line (with learning-by-doing) and the dashed line (without learning-by-doing) grows continuously, as it is magnified by the exponent of the production function. Thus, the solid line shows increasing returns to scale, while the dashed line shows decreasing returns. A specific comment must be made about β. In the previous simulations, we have shown the results and we have considered a positive β, which is the parameter affecting the ‘division-of-labor effect’, as it is part of the denominator of the logistic, i.e. the first component of Z. A positive sign means that if we add labor units, there always exists a positive, though decreasing, effect. Were we to make use of a negative β, as we added labor units the effect on the returns would be immediately negative. Such a phenomenon can be partially compensated by learning-by-doing, so that the net effect on the returns could be compensated: this is a good example of a dynamics in which two forces work in opposite directions. We can now point out the links between the parameters and the ‘real’ production system, so that we can observe the time pattern along with the higher level of the returns, due, respectively, to ‘dimension’ (i.e. we observe what happens when we add labor L0 units) and ‘learning’. First of all let us define τ1 =

1 αL0

(11)

which depicts the pattern along which the asymptotic value concerning organization and dimension ZD , λ, λ/(1 + β) is reached (see Fig. 3). Similarly, we can also define τ2 = −

1 log (ω)

(12)

which depicts the transitional time between the two levels of productivity due to learning-bydoing ZL ; τ 2 describes the way in which workers experience a change in productivity,

Fig. 3.

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Fig. 4.

the two levels of which depend on the parameters γ and ω, but also on the value of the initial conditions; the two productivity values thus are λ, λ(1 − ω)/(1 − ω − γ ) (see Fig. 4). To conclude this section, we can apply our function to a simulated — and heavily adapted — Horndal effect, in order to get a certain growth of average productivity, once a certain dimension and a ‘sufficient’ amount of time has passed. 13 The time span we consider in our simulation concerns 220 periods starting from the 80th period, i.e. the ‘sufficient time’ for the plant to adjust in terms of dimension and for learning to take place. In order to make the simulation closer to the original Horndal effect we can transform our 220 periods into the 15 years referred to in the original example, which means that the solar year would be represented by 14.6 ‘months’. Given the set of values indicated in Fig. 5, we get — during the 80-to-300 period — a growth rate of average productivity of 2 percent per annum, as indicated in the original Horndal example. 14 In Fig. 5, the dashed line represents the best linear fit of growth of productivity itself. It is important to underline that productivity grows exponentially only for a limited amount of time, and this is consistent with the fact that both learning and division of labor cannot expand for ever; in early stages productivity does not grow (and can actually decrease because Z is smaller than 1), while at later stages Z stabilizes itself taking its asymptotic value (i.e. it becomes a constant exponent).

13 Of course, this simulation contains a high degree of arbitrariness as it is based ‘only’ on the simulation and not the actual data — yet we believe that we get interesting results. 14 It is important to emphasize that we check what the parameters of the model are when confronted with a ‘real’ situation. Of course, it would be possible to get a certain rate of growth by means of simpler mathematical relationships, which, however, would not explain the complex relationships between production, division of labor and learning.

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Fig. 5.

5. A maximization exercise Let us now propose a maximization exercise, the starting point being the usual objective function, i.e. max Π = pY − wL

(13)

where Π is profit, p price, Y the level of production, w wage and L the labor input. Various cases can be considered, depending on the assumptions about the values of the returns to scale, price and wage rate. Should p and w be constant and the returns increasing, the more we produce the higher the profits. The latter statement can be easily demonstrated by referring to the second derivative of the profit function which, if positive, implies that the value of the function is increasing at an increasing rate. In fact given that Y = ALz , we can write 15 L = A−1/z Y 1/z so that the profit function becomes w Π = pY − 1/z Y 1/z (14) A and from Eq. (14) we can calculate d2 Π wY(1−2z)/z = (z − 1) dY 2 A1/z z2

(15)

which is always positive as long as z > 1 — and thus the function does not have a maximum, although it can show a minimum, as will be shown later in this section. In the following, however, we consider the ‘classical’ case in which returns are increasing, the wage rate is constant over time, there is not labor shortage, and demand matches supply 15

Here, we consider the case in which Z has reached its asymptotic value (cf. Eq. (9)), so that Z = z.

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as long as the price decreases as production grows. The latter assumption can be summarized as follows: p˜ D = f (p) = Y ⇒ p = (1−ν) , where 0 < ν < 1 (16) Y Given the latter set of assumptions, the key dynamics thus concern the rate at which returns increase and price decreases. If the price decreases monotonically with production we can rewrite our objective function as max Π = [p(Y )]Y − wL and, we can observe what happens with the above indicated value for p:   p˜ w w Π= Y − 1/z Y 1/z = pY ˜ ν − 1/z Y 1/z A A Y (1−ν) Now we can calculate dΠ w ν−1/z =0⇒Y = 1/z dYY =Y A zpν ˜

(17)

(18)

(19)

In order to check that there exists a maximum, we can calculate the second derivative and see when it is negative:   d2 Π 1 w ν−2 1/z−2 = pν(ν ˜ − 1)Y − Y <0 (20) − 1 dY 2 A1/z z z As ν < 1, and by using our Eq. (19) we get Y

ν−1/z

>

1/z − 1 1 w(1/z − 1) ⇒1> ⇒ν< ν−1 z − 1)

A1/z zpν(ν ˜

(21)

We thus reduce the maximization problem to the evaluation of two cases, that is 1 z 1 ν> z

ν<

or

νz < 1

(22)

or

νz > 1

(23)

In case Eq. (22) holds, we have one ‘simple’ maximum point. On the other hand, if Eq. (23) holds, things are more complex. Let us briefly examine both cases. 1. If Eq. (22) holds profits unsurprisingly increase with p˜ and decrease with w; in fact, by denoting with Π maximum profits which occur when Y = Y , we get ν/(1/z−ν) (1/z)/(1/z−ν)   w w w Π (p, ˜ w) = p˜ − (24) A1/z zpν ˜ A1/z A1/z zpν ˜ In order to show how price and wage affect the profit function we can simplify Eq. (24) as follows:  1/z ν/(1/z−ν) ˜ A zpν Π (p, ˜ w) = p˜ [1 − νz] (25) w

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Fig. 6.

Thus, when νz < 1 the term in the square brackets will be positive as will be the exponent ν/(1/z − ν), so that the higher the price, the higher the profits, while the higher the wage the lower the profits themselves according, respectively, to the following rules:  νz/(1−νz) 1 Π ∝ p˜ 1/(1−νz) , Π∝ (26) w 2. If Eq. (23) holds we can actually have minimum — i.e. negative — profits in a first stage, until production expands beyond a certain level. Thus, a large value of Z does not necessarily imply higher profits from the very beginning of production. Both situations are depicted in Fig. 6 in which the solid line represents the first case, i.e. νz < 1, while the dashed line shows the second case, i.e. νz > 1. As one can see, it is only after a production level of 150 that profits become zero and positive afterwards. In this second case, after the indicated level of production, the more we produce the higher the profits, which means that the rate at which price decreases is more than compensated by the rate of growth of productivity, i.e. we are in situation similar to the one indicated at the very beginning of this section. 16 6. Conclusion Since, the late 1980s endogenous growth has become central to economic analysis, and it has been widely explained by means of the role played by physical and human capital and/or R&D activities. Economists, however, have been conscious for a long time of the existence of two more forces which contribute to growth, i.e. the division of labor and learning-by-doing. 16

The values of the parameters were chosen in order to show the two profit functions in one picture on the same scale. They are of course arbitrary, but since the qualitative results hold this should overcome the usual criticism of ad hocness.

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What we have attempted here is the formalization of a production process characterized by endogenous growth generated by the two forces just mentioned. Thus, rather than focusing, as is conventional, on the roles of both physical and human capital, we have focused our attention on ‘simple’ labor. Taking as a starting point a production function with labor as the only variable input, we have pointed out three main reasons that can lead to endogenously created increasing returns, namely a more detailed division of labor, learning-by-doing concerning the individual and learning-by-doing concerning the organization as a whole. We have then divided the exponent of the production function, Z, into two varying entities, the first being related to the process of division of labor, the second being related to learning. Neither the division of labor nor learning, within a given plant or work process, can grow forever. Mathematically, the first component can be approximated by a logistic equation, while the second, which subsumes both the individual and the organization as a whole, is approximated by a difference equation relating the stock of knowledge, via the exponent Z, to cumulative output. Now, despite the fact that we had to make some assumptions on the functional forms and we made use of specific values for the parameters, we have to point out that the qualitative results do not depend on such choices; furthermore, the formalization proposed, explains why Z can have different values at different times. As we have tried to clarify, there exist technological and economic reasons which justify the fact that the returns experience a change from decreasing to increasing, but the same forces set an upper limit to the growth of returns. The division of labor cannot be pushed beyond a certain point, while the speed that can be reached by each individual is limited by their physical and mental abilities. The organization as a whole, will be thus influenced by each of the members and, if rigidly set, the whole process will evolve according to the slowest member. Each operation, in fact, can lend itself to a different degree of learning not only because of human limitations but also because of the intrinsic difficulties. Economically, we have recalled that the division of labor — and, we may add, learning — is limited by the extent of the market. Conversely, the extent of the market is limited by the degree of division of labor and the learning that has taken place. This leads us to a virtuous circle leading to specialization and interdependence. A comment is needed to underline the changes that might have occurred in the labor force. If at the beginning of the production process we have undifferentiated labor, after a few periods as a result of specialization we will have different kinds of labor involved in different tasks. If one of the members of the work process has to be changed, the organization will loose their experience acquired through the simple process of doing.

Acknowledgements The authors wish to thank Bernhard Böhm, Riccardo Leoncini, Stan Metcalfe, Bernard Walters and the anonymous referees for their comments and suggestions. The present work was partly developed within the Progetto di Interesse Nazionale “Infrastrutture, competitività e livelli di governo” co-ordinated by Gilberto Antonelli, University of Bologna, Department of Economics. Support from the CNR Progetto Speciale on “Technological systems, research evaluation and innovation policies” is also acknowledged.

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Appendix A As we have seen in Section 4 the stability of our map requires that ω + γ < 1. In order to reach this result we need first of all to calculate the eigenvalues of the Jacobian. To do this we compute the eigenvalues µ by setting to zero the characteristic equation ∂Wt+1 ∂Wt+1 ∂Wt − µ ∂Zt det =0 ∂Zt+1 ∂Zt+1 − µ ∂W ∂Zt t (ω − µ)(γ − µ) − ωγ = 0 µ(µ − ω − γ ) = 0 Solving the equation we get for the two eigenvalues the simple expressions µ1 = 0 µ2 = ω + γ The condition |µ1 | < 1 is always true, while for the second eigenvalue, for a positive γ and ω, it reads simply ω + γ < 1. In Fig. 7, we show a sketch of the behavior of Z as a function of γ + ω. The branch in the positive plane is always stable, while the negative branch is always unstable. This is not a problem as a negative exponent Z is economically meaningless.

Fig. 7. A sketch of the behavior of Z as a function of γ + ω.

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