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
To study the evolution of the economy on a wave train by means of computer simulations, we now have to specify the numerical parameter values. One set of
parameters is given by familiar macroeconomic magnitudes: the capital-output ratio b, the rate of depreciation d and the propensity s to save out of profits. On the basis
of Simon 1990, they are fixed as follows,
b = 1.70 d =
0.0461 d =
0.2670 19
again, see the appendix for further details. The other set of parameters governs technological diffusion and innovation: the annual rate of technical progress l, the
innovation time T the interval between two innovations, the length of the set-up phase T
u
, the autonomous rate of change k at which the capacity share of the BPT increases over this period, and the responsiveness r of investors to differential profits.
Here we put
13
13
Actually, l was derived from a discrete-time rate of growth l 0 =3. That is, l was set such that
explT = q
i
q
i − 1
= 1 + l
0 T, which yields lln 1+l0TT.
l = 0.029
T = 2.00 T
u
= 0.10
k = 0.02
r = 1.00
20 The innovation parameters T, T
u
, and k can be encapsulated in a single parameter n defined as
n = k T
u
T 21
The motivation for this definition and its relationship to Iwai’s 1984 stochastic innovation hypothesis is discussed in Franke 1998a. The remarkable point is that
essentially the same wave train distribution function f· is brought about by different combinations of T, T
u
and k that give rise to identical values of n in Eq. 21
14
. Finally, we take the deflated real wage rate v as the exogenous variable for
income distribution and set v =
0.307 22
Recall v = wT2q
N
from Eq. 18, with N = N
o
= N
T2
. Eqs. 19 and 20 and Eq. 22 may serve as a base scenario. In particular, the special choice of v = 0.307
results in the following values of the wage share, the average profit rate, and the capital growth rate,
6 T = 70.02,
r¯T = 11.22, g¯t = sr¯t = 3.00
23 which prevail on the wave train in the middle between two innovations, at time
t = T2. Eq. 23 shows that the model can be calibrated such as to be roughly
compatible with empirical values of the macroeconomic key variables here involved. The remainder of the paper is concerned with comparative dynamnics. Accord-
ingly, some of the exogenous parameters are varied and the characteristics of the corresponding equilibrium growth paths are studied. In this respect, note first that
variations in the exogenous rate of technical progress l have trivial effects on income distribution, since by construction l determines directly the growth rate of
real wages. In the present setting it is therefore meaningless to compare two economies with different values of l.
A most important case, however, is a change in the responsiveness r of investors to differential profits. Although r certainly has its institutional foundations, it is
also of a psychological nature. Given the economy’s institutions, r may be regarded as a psychological coefficient. Now, it is a centrepiece of the classical theory that
the long-run equilibrium position is independent of such a reaction intensity. In this theory, the only significance of r would be in the disequilibrium adjustment process
toward the steady state. For example, a mathematical analysis of a gravitation process might reveal that this parameter must not be too large if stability is to be
ensured. By contrast, in the present Schumpeterian framework it turns out that the
14
For example, an increase in T together with the corresponding reduction of k or T
u
means a smaller number of ‘active’ techniques i on the wave train. However, the graph connecting the points ln z
i
, f
ln z
i
remains essentially the same, and so will remain the average rates of profit and growth, etc. That the graphs do not exactly coincide is due to the discrete state space of the technological coefficients.
responsiveness r already has a bearing on the long-run equilibrium itself. For example, maintain the other parameter values in 19, 20, 22 and increase the
responsiveness to, say, r = 1.20. Income distribution and growth on the new wave train are then given by
6 T = 66.31,
r¯T = 13.40, g¯t = 3.58
t = T2 24
Comparing these values with Eq. 23, it is seen that a higher responsiveness of investors, or intensified competition, improves the position of profit earners and
also leads to faster growth if workers receive the same real wages. The qualitative changes in the wage share 6 and the average profit rate r¯ are
easily explained. A higher responsiveness r means that the more efficient techniques expand more rapidly than before. Average labour productivity q¯ is therefore higher,
which is tantamount to a lower average cost gap z¯ = a¯a
N
= 1q¯a
N
cf. Eq. 8 and Eq. 14. Recalling Eq. 16 and Eq. 18 and using wa¯ = wq
N
a¯a
N
= vz¯, it remains to note that the wage share is related to the average cost gap by
6 = v
z¯. 25
In the presence of unchanged deflated real wages we therefore have that faster diffusion, which reduces the average cost gap z¯, yields a lower wage share and a
higher average rate of profit. Again we emphasize that this result and the underly- ing mechanism have no counterpart in the classical theory of distribution and
growth.
In the further discussion of the long-run equilibria it will be useful to refer to the original Iwai 1984 model. Iwai characterizes the techniques i by their unit costs i
and postulates that the motions of the capacity shares are directly governed by these unit costs:
k :
i
= − g ln c
i
− ln c
i
k
i
26 where g is a constant parameter that measures the general speed of technological
diffusion Iwai, 1984, p. 328. In Franke 1998a the intuition is confirmed that a higher value of g indeed reduces the cost gap analogously defined on a wave train
of this economy
15
. This result is central to the following discussion of the compar- ative dynamics.
Now, the profit rate differentials in the present model can be expressed as r
i
− r¯ = − 6ba
i
− a¯a¯. With Eq. 25, the upper part of the diffusion Eq. 10
can thus be rewritten as k
:
i
= − rv
z¯ b
a
i
− a¯
a¯ k
i
27 As the percentage deviations of unit labour requirements a
i
in 27 correspond to Iwai’s log differences in unit costs c
i
in 26, we have the following approximate
15
By using an approximation procedure, Iwai derives an explicit mathematical formula for his stochastic economy that produces the same result. In other respects, however, his approxiamation
procedure is problematic; see Franke 1998a.
relationship between the concept of the technological speed of diffusion and some central parameters in our model,
g : g rv
z¯ b
= r6
b 28
Let us also call g the speed of technological diffusion. Before going on, it is worth mentioning that, had the investment function 9
been specified as g
i
= s[r¯ + rr
i
− r¯]
9a Eq. 28 would become g : srvz¯b. In this case, income distribution on the wave
trains would also be dependent on the propensity to save out of profits, another result which is alien to the production price systems of the classical theory.
Evidently, an increase in s has then the same effect as an increase in r. Hence, under Eq. 9a a higher savings propensity would lead to a lower wage share and
a higher average rate of profit. It is also worth mentioning that the growth rate g¯ of the economy would rise more than proportionately.
While in Iwai’s model the speed of diffusion is an exogenous parameter, Eq. 28 reveals that it is endogenous in our setting if the real wage rate v is chosen as the
exogenous income distribution variable
16
. Nevertheless, despite the interference of the average cost gap z¯ a ceteris paribus increase in the responsiveness r to profit
rate differentials will also cause the general speed of technological diffusion g to rise. Although, as indicated above, the average cost gap z¯ decreases, the
direct influence of r remains dominant, so that the product rz¯ in 28 increases. In fact, if rz¯ were to decrease, the lower value of g, according to the remark
on Eq. 26, would be associated with a higher value of z¯, which would be a contradiction.
This line of reasoning can be readily carried over to ceteris paribus changes in the deflated real wage rate v. Just like r, a rise in v speeds up the diffusion of new
techniques and thus lowers the average cost gap z¯, but again not so much as to diminish g in Eq. 28. Since by 25 the higher product vz¯ means a higher share
of wages 6, we can conclude that an increase in real wages v induces a fall of the average rate of profit r¯ in the new long-run equilibrium. In this sense, the classical
inverse relationship between real wages and profits is re-established.
The income distribution effects just discussed are summarized in Fig. 2, which shows the response surface of the average rate of profit r¯ on the respective wave
trains under variations of the deflated real wage rate v and the responsiveness coefficient r. Over the range of parameters considered, the relationship between r¯
and v is nearly linear. On the other hand, r¯ is a convex function of r, i.e. the increments in the profit rate induced by a further rise in r become smaller with
higher values of r.
16
g will again be exogenous if instead the average profit rate, or equivalently the wage share 6, is
chosen as the exogenous distributional variable.
Fig. 2. Average rate profit r¯ on a wave train under variations of the deflated real wage rate v and the responsiveness of investors r. Note: the reference stick indicates the base scenario.
Having discussed the effects of v and r, the parameter n of Eq. 21 above, which characterizes the introduction of new techniques, should not be neglected. Since a
rise in n corresponds to a faster diffusion of the best-practice techniques in their set-up phase, the effect will be expected to be similar to an increase in r. This is
certainly true as regards the average profit rate or the wage share. To give an example, raising n in the base scenario from n = 0.0010 to n = 0.0030 increases the
average profit rate by about 4, while reducing n to n = 0.0003 lowers r¯ by almost 4 vis-a-vis r¯ = 11.22 and n = 70.02 in the base scenario, the precise figures are
r¯ = 15.23 and 6 = 63.21 in the first case, and r¯ = 7.32, 6 = 76.66 in the second case.
However, it should also be noted that a change in n affects the general speed of technological diffusion g as it was specified in Eq. 28. The decrease of the cost
gap z¯ that is associated with a rise of n means. perhaps some what surprisingly. that a faster introduction of new techniques reduces the overall speed of diffusion g.
While g = 0.412 in the base scenario, we obtain, g = 0.372 for n = 0.0030 and g =
0.451 for n = 0.003
17
.
17
The possible effects from the set-up phase of innovations tends to be underestimated in the literature. For example, Silverberg and Lehnert 1993 and Silverberg and Verspagen 1997 in their
stochastic and otherwise more detailed models do not even make n or a similar parameter explicit. They just mention that the initial capacity share of new techniques is ‘small’ and do not discuss any variations
in this respect.
5. A note on the law of the falling rate of profit
