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A note on the diffusion of innovation
TECHNOLOGICAL
FORECASTING
AND
SOCIAL
CHANGE
44, 283-290
(1993)
A Note on the Diffusion of Innovation LUIGI FILIPPINI
ABSTRACT
The paper deals with the process of substitution between technologies in a framework of increasing returns to scale. The approach stresses the interaction between capacity expansion and market demand as explanations of the diffusion of technologies into their niches. The demand and supply sides of the diffusion process are therefore brought together to determine simultaneously patterns of output and prices. It analyzes the dynamics of the substitution path, where a logistic diffusion process for the new technology is assumed, and determines the substitution curve between the old technology and the new one.
Introduction In a series of contributions, J. S. Metcalfe considers the diffusion of innovation. It is concerned with “the process by which new technological forms are integrated into the economy to impose changes upon its structure . . . (and) with how the economic significance of a new technology changes over time” [l, pp. 560-5611. The latter case is closely related to the analysis of the process by which new technologies partially or completely substitute for existing technologies. Therefore it is one dimension of the problem of the transition between different economic equilibria. The approach developed stresses the interactions between capacity expansion and market demand as explanations of the diffusion of technologies into their niches. The demand and supply sides of the diffusion process are therefore brought together to determine simultaneously patterns of output and prices. The assumptions made are a competitive market and a technology with decreasing returns to scale. This note provides an extension to consider increasing returns to scale, and it is shown that most of the results are still valid in the new framework. So the assumptions made in the seminal contributions are restricted and not necessary. The paper is divided into three parts. The first gives the reader an introduction to the problem, namely the process of substitution between two technologies; the second deals with the dynamics of the substitution path, where a logistic diffusion process for the new technology is assumed; the last determines the substitution curve between the old technology and the new one. The Equilibrium Niche J. S. Metcalfe has analyzed the diffusion mechanism for a single innovation in many contributions [l-5]. His central idea is “that the development of the new technology acts LUIGI FILIPPINI is a professor of economics at the Universitl Cattolica in Milan, Italy. His main research field is the theory of production and its applications. Address reprint request to Professore Luigi Filippini, Universitl Cattolica, Largo Gemelli 1, 20123 Milano, Italy. 0
1993 Elsevier Science Publishing
Co., Inc.
OLl40-1625/93/$6.00
284
L. FILIPPINI
as an impulse to growth and structural change and that this impulse can usefully be analysed via a distinction between the long-period niche for the new technology and the process of adjustment into that niche” [3, p. 401. In a partial equilibrium framework he considers the interactions between the growth of demand and the growth of productive potential for a nondurable material assuming that the learning process generates a sigmoid growth curve of demand and the accumulation of capacity to produce the new material depends on the rate of profit. He also presents a theory of substitution between the old and the new technologies, each of which produces a nondurable material employed as an input in subsequent stages of the production process, along the lines of the just-mentioned diffusion mechanism. The model is so general that it could be applied to many different technologies: new materials, new energy sources, new uses for material. The approach followed is based upon separation between the conditions determining the degree of substitution in the long run and the ones determining the pattern of adjustment toward this long-run position. In particular, the long-run position is defined independently of the adjustment path.’ The transition to the long-run position is determined by the interaction between the dynamics of market growth (and decline) and the dynamics of capacity expansion (and contraction) appropriate to the technologies in question. The process of substitution is between two technologies that supply the same productive service to users. As more materials appear, more niches also appear, and a few materials disappear completely. The outcome depends on the demand curve for the productive service, the supply curves, and the qualitative superiority of the new technology. It is assumed that a units (a < 1) of the new material are equivalent from the user’s viewpoint to one unit of the old material so that the two materials are perfect substitutes. In a market closed to foreign trade in the two materials, the new material is introduced at a given point in time when the old material has settled into its long-run niche and supplies the entire market for the productive service. The substitution of the new for the old is analyzed in two stages. In the first stage, the long-run niche for the old material is redefined and a new set of niches for the two technologies is created. In the second stage, the new material might eliminate or share the market for the productive service with the old material. A new long-run position is created which may be quite different in terms of output and price from the previous equilibrium. The two materials also have different technologies of production and correspondingly different equilibrium supply curves. In this paper decreasing returns to scale are assumed for the production of the old material and increasing returns for the new material.2 Increasing returns to scale are of particular theoretical interest because of their preventing entry of new firms. An important instance of that may be the need to acquire a patent in order to start the productive process. Therefore, we assume a single firm is in the market and prices are formed as a cost-plus pricing model.3 The conditions of long-run demand for the productive service are static and given as follows:
’ For the appropriateness for the analysis of the process of competition and technological change, see Metcalfe [l, p. 5801. 2 Decreasing returns to scale imply that the unit cost of the old material increases with quantity produced, whereas increasing returns imply that the unit cost of the new material decreases with the quantity. 3Firms set their prices by adding a certain percentage markup to their average total costs of production.
THE DIFFUSION
x0
=
285
OF INNOVATION
c, a constants
c - up,
(1)
where x0 is the quantity of the old material which supplies users with a corresponding unit of productive service and p. is the price of the old material. On the supply side the unit cost of production, G, inclusive of a competitive rate of return on the capital employed, increases with the output of the old material as h,, h,, constants
c, = h, + h,,x, and in equilibrium
(2)
it must be that
On combining eqs. (l-2’), the long-run clearing price, pt, are determined: c - ah,
x,* = 1 + ah,1
and
ho +
niche for the old process, x0*, and the market
cho,
‘O* = 1 + ahol
The introduction of the new material redefines the equilibrium as the two materials are competitors. To determine the new long-run position, let the new material be produced by the following unit cost function subject to increasing returns to scale [6, pp, 18-20, 68-691: c, = h, + h,,/xn and in equilibrium
h,, h,] constants
it must be that
P” = c, For the reason discussed above, in equilibrium, new materials must satisfy the following condition: P,/P~
(4)
(4’) the relative price of the old and the
= a
From eqs. (2), (4), and (5), it is possible to derive the supply curve of the old material and the new one at each price. This curve is discontinuous and may be represented by Figure 1. Three regions are considered: in the first region, the supply is given by the old material; in the second, there is a competition between the old material and the new one; in the third region, the new material prevails. The discontinuity depends on the fact that at -x = h, - ah, 2ab the two materials have the same price in terms of the productivity factor. If the demand schedule is in the first region, only the old material is used; in the second region, the new technology becomes profitable and an incentive to switch to it is built; finally, in the third region, the new material is consolidated. Therefore the coexistence of the technologies depends on the parameters of the demand curve, of the supply curves, and of a. In particular, suppose that pn = ho/a and solve eq. (4). For x, greater than (or equal to) xd(hJ, where&is the demand function, the new technology substitutes the old one if
hnl > c(h,/a - h,) - ah,(h,/a - h,)
286
_----YL L. FILIPPINI
PO
I
I
I
I I
j
1;II
I
h0
hi=
III
Fig. 1. The total
supply curve.
It is interesting to point out two limits of the possible outcomes of the substitution, that is, when the old material cannot survive in the market 07, < h,/a) and a market for the new material cannot be established (h, >~$/a). In fact, in the latter case fixed costs are very large. For the purpose of this paper, the most interesting case is the third region where the new technology displaces the old. The second part, in fact, discusses the third region. For that purpose it is necessary to derive the demand curve for the new material. The demand curve for the new material, D,,(p), is derived by subtracting, at each price, the supply of the old material from the corresponding market demand as, in equilibrium, the total supply of the old and the new materials is equal to the demand for productive service: D,Cp,) = a(c + ho/h,,) - a2(a + l/h&,
(6)
On combining eq. (4’) with eq. (6), the equilibrium material are obtained, satisfying eq. (5): x2, - a[(c + h,/h,,)
- ah,@ + l/h,,)]x,
niche and the price for the new
+ a2hn,(a + l/hoi) = 0
(7)
and a2(a + l/h&Z
- a[c + h,/h,,
+ ah,@ + l/&)lp,
+ hml + ah,(c + h,/h,r)
= 0 (8)
The meaningful cases are summarized as follows: output, x,, (i = 1, 2), and correspondingly for price, < p$/a. For the two positive roots pni, it will suffice values of p. satisfying the equilibrium eq. (5). Generally the long-run demand for the material technologies in the proportion
there exist two positive roots for pni, under the assumption that h, to point out that there exist two might be shared between the two
THE
DIFFUSION
OF INNOVATION
fi = xn,/(ax, + X”i)
287
(9)
where f, is defined as the long-run degree of substitution, that is, the market share for the new material, and it can therefore assume two limiting values 0 and 1. The Transition to the Equilibrium Niche Few innovations are adopted simultaneously. Basically there are two reasons. Firms may expect an increase in demand and may be reluctant to pay the adoption cost before the market has grown sufficiently (product innovation). Firms may expect a decrease in the adoption cost as well as in the uncertainty attached to the decrease of production costs induced by new technology. To determine the path of substitution for the new material, both the dynamics of expansion for the new material and of contraction for the old material are considered. Both effects cannot take place instantaneously. All assumptions still remain on the form of the curves describing the rate of increase of market demand and in the supply of the new material. In particular, users have to acquire information and learn the attributes of the new material before they purchase it. Nonadopters learn by observing the experience of existing adopters. An adequate approach to the demand-generating process based on learning is the logistic curve.4 Thus the rate of increase for the market demand is given by y
= f3x*(t)[Lupn) - Xn(Ol
(10)
where a is a constant adoption coefficient reflecting the information-transmitting mechanism. The mechanism specified in eq. (10) is incomplete in that the equilibrium level of demand will depend on the relative price of the new material and this is likely to change during the diffusion process. Particular patterns of price variations correspond to different logistic curves. To take account of this possibility, an explanation of price changes is required, and this leads to the other second element of the adjustment process, the growth of productive capacity. The supply of material by the new process is limited by the productive capacity, whose rate of increase depends upon the profitability of producing the material. The assumptions made on the supply are as follows: l l
Increasing returns to scale prevail in the production Investment in new capacity occurs as long as the rate rate of return. The flow of funds to finance investment tions by profits generated internally and by external
of the new material. of profit exceeds the normal is generated in fixed proporfunds.
4 Plotting usage of a new technology against time yields a sigmoid curve. The logistic curve is just one of the whole class of curves that may be labeled sigmoid. See, for example, Stoneman [7] and for a critical approach Davies [8]. It is, moreover, shown that the S-shaped diffusion curve can be predicted when profitability of innovations and strategic interaction across firms are subject to some incomplete information [9]. Firms have the option to conduct experiments and adopt or reject innovation or postpone it. The decision to conduct additional experiments depends on whether the accumulated experimental results have increased the estimate of profitability above a cutoff level that is chosen so as to achieve a balance between the risk of an incorrect decision and the cost of continued experimentation. The function identifying the cumulative proportions of firms adopting the innovation as in the sketched model nearly matched the logistic curve.
L. FILIPPINI
288
.
The rate of profit is defined by the ratio of profits to capital invested, that is, the difference between the price of the new material and its long-run unit costs of production over the capital:output ratio
It follows that the rate of increase in the supply of the new material y
=
X+@.(t) - h”
is given by5
- h,,/x,(t)]
(11)
where k is the supply diffusion coefficient, that is, the inverse of the amount of profit required to expand capacity by one unit, given that capital funds to finance investment come from internal and external sources. For arbitrary pnr eqs. (10) and (11) generate different time paths for demand and capacity. In a closed economy, market expansion and capacity growth cannot grow inconsistently. To maintain equality between them at all points in time, the price of the material must adjust over time and a situation of equilibrium expansion path is achieved. It follows that
dxn(t) = X”(W/A)[l dt
- @/B)xr@)] - C/A
where A = 1 + a2(3k(a + l/h,,) B = aD[c + h,/h,r - ah& + l/h,,)] > 0 C = a*ph,,(a + l/h,,) Equation (12) is a Riccati equation, and the general solution is attained by integration of a linear differential equation of second order or by some equivalent algorithm [lo, Ch. 31. But if any particular solution is known, then it is possible to obtain the complete solution by means of quadrature. A particular integral of eq. (12) is a solution of eq. (7),6 then its general solution is
xn(t) = xnl + [l/@/p
- 2x”,) + Kexp(2pxJA
- B/A)t]-’
(13)
where K depends on the level of output of the new material at its innovation date. In this specification the equilibrium path of expansion is also a logistic curve. Of the two solutions of eq. (7), only the greater is globally stable. So there exists a unique and globally stable long-run niche x,,. The long-run niche is, moreover, independent of both coefficients b and k. 5 In a formal way the assumptions on the supply curve may be represented as follows: The new material is produced under increasing returns to scale as in eq. (4). l The growth rate of capacity, g(t), is proportional to the rate of profit, r(t): l
(1.a)
g(r) = x(1 + MO
where rt is the fraction of the internally generated profits and p is the ratio of external to internal funds invested. l The rate of profits is then defined by r(r) = [p.(r) - h. - h,,/w,(r)l/v
(1.b)
where Y is the capitaLoutput ratio for the new material. Combining eqs. (1.a) and (1.b) we have the rate of increase in the supply of the new material. 6 The following proposition is in fact proved 111, p. 641: A constant coefficient Riccati equation dy/dx + au’ + by + c = 0 has a solution of the form y = m, m a constant, bm + c = 0.
if and only if m is a root of the quadratic
equation
am* +
THE DIFFUSION
OF INNOVATION
289
The behavior of the price equation follows by substitution of eq. (12) into eq. (11). Thus the relation between p.(t) and xn(t) along the balanced growth is as follows: pn(l) = (/&/A)
+ h, - /@x&)/A
- [(K/A)
- h,,]/x,(t)
(14)
From this and from eq. (4), p,,(t) and costs fall as the output of the new material along the balanced diffusion path grows. The sign of the difference between price and costs along the equilibrium diffusion path is equal to zero for the two equilibrium quantities. Passing from the lower quantity to the greater one, the sign of this continuous function in that interval must be positive and is preserved along the path. In fact there is no other zero root and the greater root is globally stable. This result follows from the intermediate value theorem [12, p. 881. The Substitution Curve The substitution curve provides an index of the state of the selection process across competing technologies. And it is the reflection of the process of competition between the old and the new materials as reflected in their relative profitability. To determine the substitution curve for the new technology corresponding to the equilibrium growth paths of output, one must decide how market shares are to be measured. Because the materials are physically substitutes, then the market share for the new material at time f can be computed as f(t) = X”(Mo.x&)
+ xn(t)l
(19
It is useful to point out that the physical market share is equivalent in the long run to the measure of the market shares in terms of their long-run relative prices. On differentiating eq. (15) we have @U)/dt
= f(Nl
- f(01]&(0
(16)
- &(~)I
where g is the growth rate of output of the old material (go,)and the new one (g,J, which vary over the substitution process. Along the diffusion growth path, g,(t) is determined from eq. (12). The solution of eq. (16) follows as f(t) = f’[l
-
-I
R
1 + Xni[ l/(B/(3
- 2Xni) + Kexp(2SxJA
- B/A)t] I
(17)
where R = CI_Y~/(UX~+ x,). R is the ratio of the total outputs in the pre- and the postinnovation long-run positions, that is, a measure of the long-run impact of the new technology upon the market for the productive service. The substitution curve is reduced to a logistic curve, and substitution tends to the long-run level f,. In addition, the rates of diffusion of innovation and of substitution depend on the part in brackets in eq. (13), since it is the dynamics of the new technology that drives the process. To infer the effect of parameter changes upon the substitution process, two examples will suffice. An improved process to produce the new material has the effect of being associated with a higher fi, R, and the rate of diffusion, so that any substitution level is reached sooner. Similar implications follow in a higher value for p that does increase diffusion substitution to take place.
290
L. FILIPPINI
Conclusions Some general policy implications derive from the analysis of diffusion discussed in this paper. The first is related to the distribution between the technological and the performance characteristics associated with the new technology. In the case discussed, the new technology is radical when analyzed in terms of its technical characteristics. During the developing period, the old technology performs in a better way. Therefore the function of supporting the new technology often falls to government. Moreover, the diffusion of the new technology requires an environment where the profit mechanism efficiently operates; in particular, the distribution of the profits is according to the relative performance of the competing technologies. Profits provide the incentives to invest in capacity, in R&D, and in training activities. Government has also a role in supporting research, developing skills, and supplementing funds. I am indebted to E. Agliardi, C. Beretta, A. D’Agata, G. Fiorentini, J. S. Metcalfe, A. Vaglio, and the referees for helpful comments and to M. Cimoli, G. Colangelo, D. Delli Gatti, M. Morroni, and S. Perzolli for valuable conversations on an earlier draft of the paper. I gratefully acknowledge support from C.N.R. and M. U.R.S.T., Rome.
References 1. Metcalfe, J. S., The Diffusion of Innovations: An Interpretative Survey, in Technical Change andEconomic Theory, G. Dosi, C. Freeman, R. Nelson, G. Silverberg, and L. Soete eds., Pinter Publishers, London, 1988, pp. 560-589. 2. Metcalfe, J. S., Impulse and Diffusion in the Study of Technological Change, Futures 13, 347-357 (1981). 3. Metcalfe, J. S., Technological Innovation and the Competitive Process, in Technology, Im~ovotion and Economic Policy, P. Hall ed., Philip Allan, Oxford, 1986, pp. 35-64. 4. Metcalfe, J. S., and Gibbons, M., On the Economics of Structural Change and the Evolution of Technology, in StructuralChange, Economic Interdependence and World Development: Vol. 3: Structural Change and Adjustment in the WorldEconomy, L. L. Pasinetti and P. Lloyd, eds., Macmillan, London, 1987, pp. 91-102. 5. Cameron, H. M., and Metcalfe, J. S., On the Economics of Technological Substitution, Technological Forecasting and Social Change 32, 147-162 (1987). 6. Varian, H. R., Microeconomic Analysis, 2nd ed., Norton, New York, 1984. 7. Stoneman, P., The Economics Analysis of Technological Change, Oxford University Press, Oxford, 1983. 8. Davies, S., The Diffusion of Process Innovations, Cambridge University Press, Cambridge, 1979. 9. Bhattacharya, S., Chatterjee, K., and Samuelson, L., Sequential Research and the Adoption of Innovation, in Strategic Behoviour and Industrial Competition, D. J. Morris, P. J. N. Sinclair, M. D. E. Slater, and J. S. Vickers, eds., Clarendon Press, Oxford, 1986, pp. 219-243. 10. Davis, H. T., Introduction to Nonlinear D!@erentiol and Integral Equations, Dover, New York, 1962. 11. Kreider, D. L., Kuller, R. G., and Ostberg, D. R., Dt@zrential Equations, Addison-Wesley, Reading, MA, 1968. 12. Binmore, K. G., Mathematical Analysis, 2nd ed., Cambridge University Press, Cambridge, 1985. Received 6 May 1992; revised 21 October 1992
FORECASTING
AND
SOCIAL
CHANGE
44, 283-290
(1993)
A Note on the Diffusion of Innovation LUIGI FILIPPINI
ABSTRACT
The paper deals with the process of substitution between technologies in a framework of increasing returns to scale. The approach stresses the interaction between capacity expansion and market demand as explanations of the diffusion of technologies into their niches. The demand and supply sides of the diffusion process are therefore brought together to determine simultaneously patterns of output and prices. It analyzes the dynamics of the substitution path, where a logistic diffusion process for the new technology is assumed, and determines the substitution curve between the old technology and the new one.
Introduction In a series of contributions, J. S. Metcalfe considers the diffusion of innovation. It is concerned with “the process by which new technological forms are integrated into the economy to impose changes upon its structure . . . (and) with how the economic significance of a new technology changes over time” [l, pp. 560-5611. The latter case is closely related to the analysis of the process by which new technologies partially or completely substitute for existing technologies. Therefore it is one dimension of the problem of the transition between different economic equilibria. The approach developed stresses the interactions between capacity expansion and market demand as explanations of the diffusion of technologies into their niches. The demand and supply sides of the diffusion process are therefore brought together to determine simultaneously patterns of output and prices. The assumptions made are a competitive market and a technology with decreasing returns to scale. This note provides an extension to consider increasing returns to scale, and it is shown that most of the results are still valid in the new framework. So the assumptions made in the seminal contributions are restricted and not necessary. The paper is divided into three parts. The first gives the reader an introduction to the problem, namely the process of substitution between two technologies; the second deals with the dynamics of the substitution path, where a logistic diffusion process for the new technology is assumed; the last determines the substitution curve between the old technology and the new one. The Equilibrium Niche J. S. Metcalfe has analyzed the diffusion mechanism for a single innovation in many contributions [l-5]. His central idea is “that the development of the new technology acts LUIGI FILIPPINI is a professor of economics at the Universitl Cattolica in Milan, Italy. His main research field is the theory of production and its applications. Address reprint request to Professore Luigi Filippini, Universitl Cattolica, Largo Gemelli 1, 20123 Milano, Italy. 0
1993 Elsevier Science Publishing
Co., Inc.
OLl40-1625/93/$6.00
284
L. FILIPPINI
as an impulse to growth and structural change and that this impulse can usefully be analysed via a distinction between the long-period niche for the new technology and the process of adjustment into that niche” [3, p. 401. In a partial equilibrium framework he considers the interactions between the growth of demand and the growth of productive potential for a nondurable material assuming that the learning process generates a sigmoid growth curve of demand and the accumulation of capacity to produce the new material depends on the rate of profit. He also presents a theory of substitution between the old and the new technologies, each of which produces a nondurable material employed as an input in subsequent stages of the production process, along the lines of the just-mentioned diffusion mechanism. The model is so general that it could be applied to many different technologies: new materials, new energy sources, new uses for material. The approach followed is based upon separation between the conditions determining the degree of substitution in the long run and the ones determining the pattern of adjustment toward this long-run position. In particular, the long-run position is defined independently of the adjustment path.’ The transition to the long-run position is determined by the interaction between the dynamics of market growth (and decline) and the dynamics of capacity expansion (and contraction) appropriate to the technologies in question. The process of substitution is between two technologies that supply the same productive service to users. As more materials appear, more niches also appear, and a few materials disappear completely. The outcome depends on the demand curve for the productive service, the supply curves, and the qualitative superiority of the new technology. It is assumed that a units (a < 1) of the new material are equivalent from the user’s viewpoint to one unit of the old material so that the two materials are perfect substitutes. In a market closed to foreign trade in the two materials, the new material is introduced at a given point in time when the old material has settled into its long-run niche and supplies the entire market for the productive service. The substitution of the new for the old is analyzed in two stages. In the first stage, the long-run niche for the old material is redefined and a new set of niches for the two technologies is created. In the second stage, the new material might eliminate or share the market for the productive service with the old material. A new long-run position is created which may be quite different in terms of output and price from the previous equilibrium. The two materials also have different technologies of production and correspondingly different equilibrium supply curves. In this paper decreasing returns to scale are assumed for the production of the old material and increasing returns for the new material.2 Increasing returns to scale are of particular theoretical interest because of their preventing entry of new firms. An important instance of that may be the need to acquire a patent in order to start the productive process. Therefore, we assume a single firm is in the market and prices are formed as a cost-plus pricing model.3 The conditions of long-run demand for the productive service are static and given as follows:
’ For the appropriateness for the analysis of the process of competition and technological change, see Metcalfe [l, p. 5801. 2 Decreasing returns to scale imply that the unit cost of the old material increases with quantity produced, whereas increasing returns imply that the unit cost of the new material decreases with the quantity. 3Firms set their prices by adding a certain percentage markup to their average total costs of production.
THE DIFFUSION
x0
=
285
OF INNOVATION
c, a constants
c - up,
(1)
where x0 is the quantity of the old material which supplies users with a corresponding unit of productive service and p. is the price of the old material. On the supply side the unit cost of production, G, inclusive of a competitive rate of return on the capital employed, increases with the output of the old material as h,, h,, constants
c, = h, + h,,x, and in equilibrium
(2)
it must be that
On combining eqs. (l-2’), the long-run clearing price, pt, are determined: c - ah,
x,* = 1 + ah,1
and
ho +
niche for the old process, x0*, and the market
cho,
‘O* = 1 + ahol
The introduction of the new material redefines the equilibrium as the two materials are competitors. To determine the new long-run position, let the new material be produced by the following unit cost function subject to increasing returns to scale [6, pp, 18-20, 68-691: c, = h, + h,,/xn and in equilibrium
h,, h,] constants
it must be that
P” = c, For the reason discussed above, in equilibrium, new materials must satisfy the following condition: P,/P~
(4)
(4’) the relative price of the old and the
= a
From eqs. (2), (4), and (5), it is possible to derive the supply curve of the old material and the new one at each price. This curve is discontinuous and may be represented by Figure 1. Three regions are considered: in the first region, the supply is given by the old material; in the second, there is a competition between the old material and the new one; in the third region, the new material prevails. The discontinuity depends on the fact that at -x = h, - ah, 2ab the two materials have the same price in terms of the productivity factor. If the demand schedule is in the first region, only the old material is used; in the second region, the new technology becomes profitable and an incentive to switch to it is built; finally, in the third region, the new material is consolidated. Therefore the coexistence of the technologies depends on the parameters of the demand curve, of the supply curves, and of a. In particular, suppose that pn = ho/a and solve eq. (4). For x, greater than (or equal to) xd(hJ, where&is the demand function, the new technology substitutes the old one if
hnl > c(h,/a - h,) - ah,(h,/a - h,)
286
_----YL L. FILIPPINI
PO
I
I
I
I I
j
1;II
I
h0
hi=
III
Fig. 1. The total
supply curve.
It is interesting to point out two limits of the possible outcomes of the substitution, that is, when the old material cannot survive in the market 07, < h,/a) and a market for the new material cannot be established (h, >~$/a). In fact, in the latter case fixed costs are very large. For the purpose of this paper, the most interesting case is the third region where the new technology displaces the old. The second part, in fact, discusses the third region. For that purpose it is necessary to derive the demand curve for the new material. The demand curve for the new material, D,,(p), is derived by subtracting, at each price, the supply of the old material from the corresponding market demand as, in equilibrium, the total supply of the old and the new materials is equal to the demand for productive service: D,Cp,) = a(c + ho/h,,) - a2(a + l/h&,
(6)
On combining eq. (4’) with eq. (6), the equilibrium material are obtained, satisfying eq. (5): x2, - a[(c + h,/h,,)
- ah,@ + l/h,,)]x,
niche and the price for the new
+ a2hn,(a + l/hoi) = 0
(7)
and a2(a + l/h&Z
- a[c + h,/h,,
+ ah,@ + l/&)lp,
+ hml + ah,(c + h,/h,r)
= 0 (8)
The meaningful cases are summarized as follows: output, x,, (i = 1, 2), and correspondingly for price, < p$/a. For the two positive roots pni, it will suffice values of p. satisfying the equilibrium eq. (5). Generally the long-run demand for the material technologies in the proportion
there exist two positive roots for pni, under the assumption that h, to point out that there exist two might be shared between the two
THE
DIFFUSION
OF INNOVATION
fi = xn,/(ax, + X”i)
287
(9)
where f, is defined as the long-run degree of substitution, that is, the market share for the new material, and it can therefore assume two limiting values 0 and 1. The Transition to the Equilibrium Niche Few innovations are adopted simultaneously. Basically there are two reasons. Firms may expect an increase in demand and may be reluctant to pay the adoption cost before the market has grown sufficiently (product innovation). Firms may expect a decrease in the adoption cost as well as in the uncertainty attached to the decrease of production costs induced by new technology. To determine the path of substitution for the new material, both the dynamics of expansion for the new material and of contraction for the old material are considered. Both effects cannot take place instantaneously. All assumptions still remain on the form of the curves describing the rate of increase of market demand and in the supply of the new material. In particular, users have to acquire information and learn the attributes of the new material before they purchase it. Nonadopters learn by observing the experience of existing adopters. An adequate approach to the demand-generating process based on learning is the logistic curve.4 Thus the rate of increase for the market demand is given by y
= f3x*(t)[Lupn) - Xn(Ol
(10)
where a is a constant adoption coefficient reflecting the information-transmitting mechanism. The mechanism specified in eq. (10) is incomplete in that the equilibrium level of demand will depend on the relative price of the new material and this is likely to change during the diffusion process. Particular patterns of price variations correspond to different logistic curves. To take account of this possibility, an explanation of price changes is required, and this leads to the other second element of the adjustment process, the growth of productive capacity. The supply of material by the new process is limited by the productive capacity, whose rate of increase depends upon the profitability of producing the material. The assumptions made on the supply are as follows: l l
Increasing returns to scale prevail in the production Investment in new capacity occurs as long as the rate rate of return. The flow of funds to finance investment tions by profits generated internally and by external
of the new material. of profit exceeds the normal is generated in fixed proporfunds.
4 Plotting usage of a new technology against time yields a sigmoid curve. The logistic curve is just one of the whole class of curves that may be labeled sigmoid. See, for example, Stoneman [7] and for a critical approach Davies [8]. It is, moreover, shown that the S-shaped diffusion curve can be predicted when profitability of innovations and strategic interaction across firms are subject to some incomplete information [9]. Firms have the option to conduct experiments and adopt or reject innovation or postpone it. The decision to conduct additional experiments depends on whether the accumulated experimental results have increased the estimate of profitability above a cutoff level that is chosen so as to achieve a balance between the risk of an incorrect decision and the cost of continued experimentation. The function identifying the cumulative proportions of firms adopting the innovation as in the sketched model nearly matched the logistic curve.
L. FILIPPINI
288
.
The rate of profit is defined by the ratio of profits to capital invested, that is, the difference between the price of the new material and its long-run unit costs of production over the capital:output ratio
It follows that the rate of increase in the supply of the new material y
=
X+@.(t) - h”
is given by5
- h,,/x,(t)]
(11)
where k is the supply diffusion coefficient, that is, the inverse of the amount of profit required to expand capacity by one unit, given that capital funds to finance investment come from internal and external sources. For arbitrary pnr eqs. (10) and (11) generate different time paths for demand and capacity. In a closed economy, market expansion and capacity growth cannot grow inconsistently. To maintain equality between them at all points in time, the price of the material must adjust over time and a situation of equilibrium expansion path is achieved. It follows that
dxn(t) = X”(W/A)[l dt
- @/B)xr@)] - C/A
where A = 1 + a2(3k(a + l/h,,) B = aD[c + h,/h,r - ah& + l/h,,)] > 0 C = a*ph,,(a + l/h,,) Equation (12) is a Riccati equation, and the general solution is attained by integration of a linear differential equation of second order or by some equivalent algorithm [lo, Ch. 31. But if any particular solution is known, then it is possible to obtain the complete solution by means of quadrature. A particular integral of eq. (12) is a solution of eq. (7),6 then its general solution is
xn(t) = xnl + [l/@/p
- 2x”,) + Kexp(2pxJA
- B/A)t]-’
(13)
where K depends on the level of output of the new material at its innovation date. In this specification the equilibrium path of expansion is also a logistic curve. Of the two solutions of eq. (7), only the greater is globally stable. So there exists a unique and globally stable long-run niche x,,. The long-run niche is, moreover, independent of both coefficients b and k. 5 In a formal way the assumptions on the supply curve may be represented as follows: The new material is produced under increasing returns to scale as in eq. (4). l The growth rate of capacity, g(t), is proportional to the rate of profit, r(t): l
(1.a)
g(r) = x(1 + MO
where rt is the fraction of the internally generated profits and p is the ratio of external to internal funds invested. l The rate of profits is then defined by r(r) = [p.(r) - h. - h,,/w,(r)l/v
(1.b)
where Y is the capitaLoutput ratio for the new material. Combining eqs. (1.a) and (1.b) we have the rate of increase in the supply of the new material. 6 The following proposition is in fact proved 111, p. 641: A constant coefficient Riccati equation dy/dx + au’ + by + c = 0 has a solution of the form y = m, m a constant, bm + c = 0.
if and only if m is a root of the quadratic
equation
am* +
THE DIFFUSION
OF INNOVATION
289
The behavior of the price equation follows by substitution of eq. (12) into eq. (11). Thus the relation between p.(t) and xn(t) along the balanced growth is as follows: pn(l) = (/&/A)
+ h, - /@x&)/A
- [(K/A)
- h,,]/x,(t)
(14)
From this and from eq. (4), p,,(t) and costs fall as the output of the new material along the balanced diffusion path grows. The sign of the difference between price and costs along the equilibrium diffusion path is equal to zero for the two equilibrium quantities. Passing from the lower quantity to the greater one, the sign of this continuous function in that interval must be positive and is preserved along the path. In fact there is no other zero root and the greater root is globally stable. This result follows from the intermediate value theorem [12, p. 881. The Substitution Curve The substitution curve provides an index of the state of the selection process across competing technologies. And it is the reflection of the process of competition between the old and the new materials as reflected in their relative profitability. To determine the substitution curve for the new technology corresponding to the equilibrium growth paths of output, one must decide how market shares are to be measured. Because the materials are physically substitutes, then the market share for the new material at time f can be computed as f(t) = X”(Mo.x&)
+ xn(t)l
(19
It is useful to point out that the physical market share is equivalent in the long run to the measure of the market shares in terms of their long-run relative prices. On differentiating eq. (15) we have @U)/dt
= f(Nl
- f(01]&(0
(16)
- &(~)I
where g is the growth rate of output of the old material (go,)and the new one (g,J, which vary over the substitution process. Along the diffusion growth path, g,(t) is determined from eq. (12). The solution of eq. (16) follows as f(t) = f’[l
-
-I
R
1 + Xni[ l/(B/(3
- 2Xni) + Kexp(2SxJA
- B/A)t] I
(17)
where R = CI_Y~/(UX~+ x,). R is the ratio of the total outputs in the pre- and the postinnovation long-run positions, that is, a measure of the long-run impact of the new technology upon the market for the productive service. The substitution curve is reduced to a logistic curve, and substitution tends to the long-run level f,. In addition, the rates of diffusion of innovation and of substitution depend on the part in brackets in eq. (13), since it is the dynamics of the new technology that drives the process. To infer the effect of parameter changes upon the substitution process, two examples will suffice. An improved process to produce the new material has the effect of being associated with a higher fi, R, and the rate of diffusion, so that any substitution level is reached sooner. Similar implications follow in a higher value for p that does increase diffusion substitution to take place.
290
L. FILIPPINI
Conclusions Some general policy implications derive from the analysis of diffusion discussed in this paper. The first is related to the distribution between the technological and the performance characteristics associated with the new technology. In the case discussed, the new technology is radical when analyzed in terms of its technical characteristics. During the developing period, the old technology performs in a better way. Therefore the function of supporting the new technology often falls to government. Moreover, the diffusion of the new technology requires an environment where the profit mechanism efficiently operates; in particular, the distribution of the profits is according to the relative performance of the competing technologies. Profits provide the incentives to invest in capacity, in R&D, and in training activities. Government has also a role in supporting research, developing skills, and supplementing funds. I am indebted to E. Agliardi, C. Beretta, A. D’Agata, G. Fiorentini, J. S. Metcalfe, A. Vaglio, and the referees for helpful comments and to M. Cimoli, G. Colangelo, D. Delli Gatti, M. Morroni, and S. Perzolli for valuable conversations on an earlier draft of the paper. I gratefully acknowledge support from C.N.R. and M. U.R.S.T., Rome.
References 1. Metcalfe, J. S., The Diffusion of Innovations: An Interpretative Survey, in Technical Change andEconomic Theory, G. Dosi, C. Freeman, R. Nelson, G. Silverberg, and L. Soete eds., Pinter Publishers, London, 1988, pp. 560-589. 2. Metcalfe, J. S., Impulse and Diffusion in the Study of Technological Change, Futures 13, 347-357 (1981). 3. Metcalfe, J. S., Technological Innovation and the Competitive Process, in Technology, Im~ovotion and Economic Policy, P. Hall ed., Philip Allan, Oxford, 1986, pp. 35-64. 4. Metcalfe, J. S., and Gibbons, M., On the Economics of Structural Change and the Evolution of Technology, in StructuralChange, Economic Interdependence and World Development: Vol. 3: Structural Change and Adjustment in the WorldEconomy, L. L. Pasinetti and P. Lloyd, eds., Macmillan, London, 1987, pp. 91-102. 5. Cameron, H. M., and Metcalfe, J. S., On the Economics of Technological Substitution, Technological Forecasting and Social Change 32, 147-162 (1987). 6. Varian, H. R., Microeconomic Analysis, 2nd ed., Norton, New York, 1984. 7. Stoneman, P., The Economics Analysis of Technological Change, Oxford University Press, Oxford, 1983. 8. Davies, S., The Diffusion of Process Innovations, Cambridge University Press, Cambridge, 1979. 9. Bhattacharya, S., Chatterjee, K., and Samuelson, L., Sequential Research and the Adoption of Innovation, in Strategic Behoviour and Industrial Competition, D. J. Morris, P. J. N. Sinclair, M. D. E. Slater, and J. S. Vickers, eds., Clarendon Press, Oxford, 1986, pp. 219-243. 10. Davis, H. T., Introduction to Nonlinear D!@erentiol and Integral Equations, Dover, New York, 1962. 11. Kreider, D. L., Kuller, R. G., and Ostberg, D. R., Dt@zrential Equations, Addison-Wesley, Reading, MA, 1968. 12. Binmore, K. G., Mathematical Analysis, 2nd ed., Cambridge University Press, Cambridge, 1985. Received 6 May 1992; revised 21 October 1992