w ; t=g
w
wt, where g
w
= l 11
Given an initial number of techniques N
o
, a frequency distribution of their capacity shares k
1
, …, k
N
o
at t = 0 with k
N
o
= 0 and also an initial real wage rate
w0, the evolution of these and the newly arriving techniques is now fully described.
3. Wave trains
Eqs. 1 – 11 specify an evolutionary process that captures the basic Schumpete- rian hypotheses on technological diffusion and innovation. While diffusion consti-
tutes an equilibrating force that tends to steer the economy’s state of technology toward an equilibrium in which all firms use the most efficient production method,
the function of innovation lies precisely in upsetting this equilibrating tendency. The dynamic interaction between the continuous and equilibrating forces of diffu-
sion, and the discontinuous and disequilibrating forces of innovation, may never- theless exhibit a certain regularity. In our deterministic setting for the technological
innovations, the forces of diffusion and innovation balance if, over regular intervals of time, they reproduce exactly the same spectrum of coexisting techniques, only at
a higher scale. This means that the role of technique i at time t is taken over by technique i + 1 at time t + T. Formally. the capacity shares constitute an equi-
librium trajectory. denoted by {k
i
t}, if over time and across techniques they are related by
k
i + 1
t + T = k
i
t, i IN, t ] 0
12 This time path of capacity shares corresponds to what is known in the literature
on partial differential equations with their continuous state space as a travelling wave, or a wa6e train. In the present framework with the discrete unit costs a
i
of successive techniques, a solution of system 1 – 11 is a wave train if there exists a
real function f· such that f
ln a
i
+ g
w
t = k
i
t, i IN, t ] 0
13 cf. Henkin and Polterovich, 1991, p. 556. In fact, compatibility of Eq. 12 and
Eq. 13 follows from the definition of g
w
in Eq. 11 and reformulation of 2 as ln a
i + 1
= − l T + ln a
i
= − g
w
T + ln a
i
, and the relationship k
i + 1
t + T = f[ln a
i + 1
+ g
w
t + T] = f[ − g
w
T + ln a
i
+ g
w
t + g
w
T] = f[ln a
i
+ g
w
t] = k
i
t. By introducing the concept of cost gaps, the frequency distributions of techniques
can be studied independently of time progressing. The cost gap of technique i is given by its unit cost in relation to the unit cost of the best-practice technique N.
Denoting this gap by z
N − i
, we have z
N − i
= a
i
a
N
14
Clearly, the ratios a
i
a
N
remain unaltered if the same number is added to i and N, while the cost gap of a given technique i increases as new techniques are
introduced. The frequency distribution of these cost gaps is directly represented by the wave train function f·. To see this suppose a given cost gap z = z
N − i
is brought about by technique i at a time T years after the introduction of the most
recent technique N = N
t
. Accordingly, let 0 5 t B T, N
t
= N
o
+ m for a suitable
integer number m, and t = mT + t. Then it is easily verified that with respect to some constant number C,
f [ln z
N − i
+ g
w
t + C] = k
i
t, where N = N
t
15 see the appendix. Eq. 15 together with Eq. 12 tells us that it does not matter
whether the cost gap z is brought about by technique i at time t or by technique i + 1 at time t
+
T: the capacity share will be the same
8
. Hence, the distribution of cost gaps remains invariant over time. More precisely, the distribution of the
variable ln z is slightly shifted between two innovations as the term g
wt
increases from 0 to g
wt
, but identical values are assumed if we look at the distribution every T years.
Fig. 1 presents a typical example of the equilibrium evolution of an economy the underlying numerical details are given in the next section. The diagram clearly
shows how the frequency distribution of the techniques’ capacity shares moves regularly over time toward the techniques with lower unit labour requirements
ULR. Correspondingly, as indicated by the bold line on the surface running parallel to the time axis, it is seen how the capacity share of a given technique
Fig. 1. Frequency distributions of techniques on a wave train. Note: Bars indicate frequencies in of techniques with high unit labour requirements ULR at time t = 20.
8
Formally, with N + 1 = N
t
+ 1 = N
t + T
and z = z
N − i
= z
N + 1 − i + 1
, the relationship reads f[ln z + g
w
t + C] = f[ln z
N + 1 − i + 1
+ g
w
t + C] = f[In z
N − i
+ g
w
t + C] = k
i
t = k
i + 1
t + T.
steadily declines in the later stages of its life-cycle. Its rise in the early stages, when it was still among the most efficient techniques, is hidden by the crest of the
distribution surface. The wave train function f· can be read off along the ULR-axis at, for example, t = 20.
The unit costs of the active techniques at that date range from a high of a
l2
= 52.7
to a low of a
12 + 27
= 12.2 see the bold line at t = 20. That is, the new techniques
are more than four times as productive as the old vintages. Though the only purpose of Fig. 1 is to illustrate the properties of a highly stylized economic model,
these differences in efficiencies might not be too extreme. Thus, to emphasize the economic significance of his general approach, Iwai, 1984, p. 324 gives a represen-
tative example of the frequency distribution of an industry in the USA the metal stamp industry where the ratio of payroll to value added covers a range from 0.15
to 0.85
9
. Applying the concept of a wave train to system 1 – 11 raises three questions:
Does a wave train {k
i
t} exist? Is it uniquely determined? Is it globally stable? The latter means that starting from an arbitrary initial frequency distribution {k
i
0}, the resulting solution {k
i
t} of 1 – 11 converges toward the wave train capacity shares {k
i
t} as t gets large. These problems have been subjected to a rigorous mathematical analysis by
Henkin and Polterovich 1991 in another, similarly elementary Schumpeterian model. Under a reasonable set of conditions. the authors prove existence and
uniqueness of a wave train as well as its global stability. Unfortunately, the quite elaborated analysis cannot be readily carried over since the present model differs
in a number of specification details. However, existence and the powerful global stability result were fully confirmed in all our numerical simulations. As we have
found no reason to harbour any doubts, we do not pursue the mathematical issue any further
10
. The property of global stability establishes the economic significance of the
long-run equilibrium notion of a wave train. Actually, the global stability was also exploited to compute the wave trains in the numerical investigations below.
Accordingly, we initialized the dynamic process 1 – 11 with an arbitrary distribu- tion of capacity shares {k
i
0} at t = 0, and let it run for so long until Eq. 12 was approximately satisfied. Then, setting the wage rate back to w = w0, the process
was rerun with the capacity shares just found as the initial frequency distribution. Against the background of the classical theory of income distribution, another
feature of interest are the wages and profits on a wave train. Since there is now no longer a uniform rate of profit, we refer to the average profit rate in order to
characterize the position of profit receivers by a single number. Denoting by 6 the share of total wages in gross national income 6 = wLY, we have
9
Of course, these differences cannot be directly compared to that of Fig. 1 since we have no information about the probably varying capital-output ratios in Iwai’s example.
10
Strictly speaking, our results could be compatible with the existence of another, but unstable, wave train, although such as a phenomenon would have no practical relevance even if it existed.
y t = wta¯t
16 cf. the remark on Eq. 7. Using Eq. 6, the average rate of profit can be written
as
11
r¯t = 1 − 6t
b − d =
1 − wta¯t b
− d .
17 Since by hypothesis real wages wt grow at the same rate at which the unit
labour requirements of the BPT decline, wta¯t does not systematically rise or fall in the long-run. That is, 6t + T = wt + Ta¯t + T = wta¯t = 6t for all t,
while between two innovations 6t may fluctuate on a small scale these intermedi- ate fluctuations are neglected in the following discussion. By the same token, it is
ensured that the average rate of profit remains constant in the course of the process. Hence, if an initial real wage rate w0 is exogenously given, it has associated with
it a wave train solution {k
i
t} of the evolution of capacity shares, and these in turn determine the average rate of profit r¯ as well as the overall growth rate of the
economy, g¯ = sr¯. In this sense, the degree of freedom from the classical theory of income distribution is maintained.
On the other hand, also the average rate of profit r¯ may be chosen as the exogenous variable. Observe that, by virtue of Eq. 17, this is tantamount to fixing
the wage share 6. In the computation of the corresponding wave train, we can then replace Eq. 11 with Eq. 16 and set directly wt = 6a¯t.
Knowledge of the real wage rate w0 on a wave train is not yet sufficient to assess the position of workers on this long-run equilibrium growth path; we also
have to know the general level of technology at that time, t = 0. To discuss variations of income distribution, it is therefore appropriate to deflate the real wage
by an index of labour productivity. However, the weights constituting such an index should be independent of the frequency distribution of techniques, which will
generally change from one wave train to another. This requirement rules out average productivity q¯ = 1a¯ as a deflator. Instead, it appears natural to employ the
labour productivity of the BPT for this purpose. Looking at wages in the middle between two innovations, define
v = wT2 q
–
N
= wT2a
N,
where N = N
o
= N
T2
18 It will be clear by now that on a wave train we have wt
m
q –
Nm
= v for all
m IN, where t
m
mT + T2, Nm N
mT + T2
. This means that if we study changes in income distribution, we exogenously fix the deflated real wage v and set
the initial real wage w0 such that, on the equilibrium path, wT2 = vq
N
comes about N = N
o
. The wave train distribution function f· is subsequently used to compute the corresponding average rate of profit
12
.
11
In all what follows it will be understood that averages like r¯, a¯ and z¯ the average cost gap are based on the capacity shares k
i
of a wave train.
12
Conversely, we may fix r¯ or 6 = 1 – br¯ + d, respectively, obtain the associated wave train including a¯t, and then from wt = 6a¯t compute the deflated real wage as v = wt
m
q
Nm
for some integer number m.
Before we illustrate the construction of a wage-profit frontier in this way, we should stress an important conceptual difference from income distribution in the production
price systems of the classical theory. In this type of analysis, the study of income distribution, given the ‘book of blueprints’ of techniques, requires the computation
of relative prices, while at least in single production systems quantities do not interfere. In the present context of a one-good economy, relative prices are no problem
they would reappear if we tried to set up a two-sectoral version of the model. However, given the path of technical progress, the determination of the distributional
variables v and r¯ requires the computation of the equilibrium frequencies of the techniques’ capacity shares. As already indicated in the introductory section, the
classical dichotomy between prices and quantities therefore no longer obtains. Despite the highly simplified assumptions on technology, a study of income distribution in
an evolutionary economy has to integrate the quantity side right from the outset.
It has already been noted above that if, instead of the deflated real wage v, the wage share 6 is taken as the exogenous variable, the average rate of profit can be
directly determined by Eq. 17 without further knowledge of the capacity shares. However, quantities will then interfere as soon as the assumption of a one-good
economy is dropped. More importantly, the interdependence between the price and the quantity side under a given wage share occurs, almost trivially, already at a much
more elementary level, in basic multisectoral production price systems with a uniform rate of profit and best-practice techniques only cf. Franke, 1998b, 1999.
This inconvenience is certainly one of the reasons why classical analysis normally chooses real wages, rather than the wage share, to characterize the position of
workers. As we wish to discuss the classical theory within its own preferred setting, we here follow this tradition.
4. Determinants of income distribution
