3.2 Numerical Methods for Integrations
3.2.2 Temporal Integration
Goodwin’s 1967 classic, ‘A growth cycle’. So the role for Classical/Marxist centres of gravitation13in theory may be adjusted to the conjecture that actual observations may be regarded as near enough to those associated with short-period macroeconomic rest states14 to allow econometric methods to be used to fit them in, say, time series analysis. Here I leave it for the Andrew Harveys of this world to take over.15
APPENDIX 1 THE MAIN ISSUES AND RESULTS OF
accept that in a competitive situation there is a tendency to equality of rates of profit in all activities so that we have to explain the origin and size of the overall, economy-wide r;thenwe need to know before the analysis starts what we mean by a quantity of capital in order that it may be a determi-nant ofr(an exogenous, given variable), and one of the reasons why rmay be high or low relative to wis that we have a ‘little’ or a ‘lot’ of capital. If it is not possible to find such a unit (the debates showed that outside one commodity models, it is not), it is not possible to say rtakes the value it does partly because we have so much ‘capital’ and because ‘its’ marginal product has a particular value.
This aspect of the debate was related to a methodological critique asso-ciated with the distinction between differences and changes. The results of the debate were mostly drawn from comparisons of long-period positions, which reflect differences in initial conditions. It is argued that they can tell us nothing about processes – changes – in particular, the processes of accu-mulation. Joan Robinson (1979 [1974]) was to characterise this critique as
‘history versus equilibrium’. Its implications are reflected in the discussion in the text about the short period and the long period.
The reaction to the criticism of the aggregate production function and the meaning of ‘capital’ in its construction was to try to avoid the use of
‘capital’ and ‘its’ marginal product and make the social rate of return – Irving Fisher’s central concept – a key concept. It provides on the produc-tivity side of the story what the rate of time preference does on the psy-chological side; see Solow (1963).
Parallel with this development was Paul Samuelson’s attempt (1962) to rationalise Solow’s use of J.B. Clark–Frank Ramsey–J.R. Hicks models in growth theory and econometric work; see Solow (1956; 1957). Samuelson attempted to show that the rigorously derived results of the simple model were robust, and that they illuminated the behaviour of more complex het-erogeneous capital models. Lying behind all this was the conceptual under-standing that ‘capital’ and rare related in such a way that the demand curve for ‘capital’ is well-behaved, that is, downward sloping. This result as well as other neoclassical parables derived from the simple model – the negative associations between rand the capital–output ratio and sustainable levels of consumption per head – together with the marginal productivity theory of distribution itself were refuted by the capital-reversing and reswitching results, as Samuelson (1966b) handsomely acknowledged. Capital-revers-ing (the Ruth Cohen curiosum) is that a lessproductive,less capital-inten-sive technique may be associated with a lowervalue of r. The reswitching result is that the same technique, having been the most profitable one for a particular range of values of r and w, could also be most profitable at another range of values of r and w, even though other techniques were
profitable at values in between. Both refute the agreeable (neoclassical) intuition of the results of the simple models and, Pasinetti (1969; 1970) argued, of Solow’s Fisherian approach (Solow, 1963) as well. Solow (1970) did not agree.
APPENDIX 2
The essential methodology of ACMS is as follows: consider the production function
(1a) which, because of constant returns to scale, may be written as
(1b)
that is, qf(k) (2)
Now
(3) and, assuming perfect competition and static expectations,
(4) Equation (4) has an inverse function that relates kto wand, because q f(k), it also allows qto relate to w, say
(5) ACMS turn this procedure around and suppose that the formof the rela-tionship between productivity and the wage rate is known. Let it be
(6) qg*(w)
qg*(w) wf(k)f(k)k Q
Lf(k)f(k)k Q
Kf(k)
where q QL, k KL Q
Lf
KL, 1Q F(K, L)
that is, expression (6) is the general form of the regression equation in the text above. Then, with their assumptions,
(7) which is a differential equation for f(k)with a solution
(8) whereAis a constant of integration. Note that equation (8) is constrained to make f(k)0 and f(k)0.
We next show that ifthere is this link from the productivity–wage rate relationship to the production function, the regression coefficient,b, is not only the elasticity of productivity with respect of the wage rate, (dq/dw) (w/q), but also the elasticity of substitution of capital for labour,. meas-ures the responsiveness of the capital–labour ratio to changes in the ratios of the marginal products of capital and labour, and, therefore,with perfect competition and static expectations, to changes in relative factor prices.
With constant returns to scale,may be defined as follows:
(9) We already have expressions for : see expression (3) above. may be shown to be: (1/L)(kf(k)), and we know that QLf(k).16
Substituting these expressions in expression (9) gives
(10) Now
(4) so that
(11) From qf(k), we obtain
dq f(k)dk
dwf(k)dkf(k)dkkf(k)dk kf(k)dk
w f(k)f(k)k f(k) (f(k)f(k)k)
kf(k)f(k) Q(2QKL)
QKandQL (QK) (QL)
Q(2QKL) qf(k; A) qg*(qf(k)k)
and, thus
(12) Substituting expression (12) in expression (11), we obtain
and so
(13) Therefore
(14) But (dq/dw) (w/q) is, by definition,b, so that bis an estimate of.
NOTES
1. Write (1)
where 1,Qoutput (income),Llabour and Kcapital.
Thus (2)
and (3)
But
so that (4)
and the share of capital in output (income) is
(5) (6) As is the ratio of the marginal to the average product of capital, it is the elasticity of output with respect to capital. Similarly, it may be shown that wla.
2. Write
(1) Where A(t) is a shift factor reflecting the pull of all forces of technical change.
Differentiating (1) with respect to time, we obtain
Q A (t) f ( K, L )
wk But r Q K, so that
wkrK
Q Q K·K Q
Q K
Q K QKL1K1
Q
K L1K1
QL1K QLK
dq dw
wq f(k) (f(k)f(k)k) kf(k)f(k) dq
dw
f(k) kf(k) dw kf(k) dq
f(k)
dk 1
f(k)dq
(2) where dots over variables indicate derivatives with respect to time.
Dividing by Q, we get
(3) Now
and
so that if factors are paid their marginal products
which is capital’s share in product.
So we may write (3) as
which, ifwkwl1, becomes
(4) or
(5) With Cobb–Douglas, we may write (5) as
(6) Growth in output per head equals the rate of growth of the shift factor (‘technical progress’) plus the rate of growth of capital per man (‘deepening’) with the latter weighted by capital’s share:
(7) and to estimate awe only have to obtain statistics on q*,wkand k*; these either exist or may be constructed.
3. See Appendix 1 for a brief account of the debates and the main results.
4. The ability to play God in the computer age that the analysis in this article allowed me to do led me to succumb to temptation again, this time in the company of two Adelaide colleagues, the late Peter Praetz and Al Watson. As a Trinity we made up a number of worlds of our own in which we knew the true values of key parameters. We then gener-ated, by ‘Monte Carlo’ experiments, observations from these worlds and tested whether the econometric procedures used by our friends, the late Fred Gruen and Alan Powell, to estimate supply responses in Australian agriculture in fact gave unbiased and accurate estimates of the values of the parameters they were interested in (their values/sizes had important implications for policy). Alas, according to us, they did not, though another
‘mate’, Ray Byron, argued that our procedures were as shonky as those of Powell and Gruen’s but for different reasons! It was all good clean fun, of course – see Gruen et al.
1967a, 1967b; Powell and Gruen 1966a, 1966b, 1967, 1968, 1970; Watson et al., 1970a, 1970b; Byron 1970a, 1970b.
aq*wk k*
q*a k*
q*awk k*, Q
QL LA.
(t)
A(t)wk
KKLLQ QA.
(t) A(t)wkK
KwlL L Q
K K Qwk A(t) f
L Q L A(t) f
K Q K Q
QA. (t)
A(t)A(t) f K
K QA(t) f
L L Q QA(t)f(K, L)A(t) f
KKA(t) f LL
5. Nevertheless, as Thomas Michl (2002, 53, 29/5/02, letter to G.C. Harcourt) reminded me (in this case, a euphemism for bringing it to my attention for the first time), ‘Empirical research guided by the neoclassical growth model has consistently found that the appar-ent elasticity of output with respect to capital exceeds its predicted value, typically taken to be the share of profit in national income’.
6. Samuelson (1966a, 444–5) wrote: ‘Until the laws of thermodynamics are repealed, I shall continue . . . to believe in production functions . . . a many-sectored neoclassical model with heterogeneous capital goods and somewhat limited factor substitutions can fail to have some of the simple properties of the idealised J.B. Clark neoclassical models.
Recognising these complications does not justify nihilism’.
7. I think it is at this point that Franklin Fisher and I may still part company, see his comment on this in Fisher (2005).
8. Thomas Michl (29/5/02; Letter to G.C. Harcourt) thought this a useful way forward and suggested that engineers could usefully be asked too. His own conjecture ‘is we could find the best-practice firms are already working with the most automated technique’.
9. See, for example, Granger (1993). The link between Granger’s article and what is in the text may be more clear to me than to my readers. I heard Granger give the paper at the 1992 RES conference and we discussed the theory underlying it afterwards.
10. Dennis Robertson (1956, 16, emphasis in original) mentions the two concepts of the long period Marshall had in mind: ‘one in which it stands realistically for any period in which there is time for substantialalterations to be made to the size of the plant, and one in which it stands conceptually for the Never-never land of unrealised tendency’.
11. It is significant that Anwar Shaikh (2005) makes use of Goodwin’s classic 1967 prey-predator Marxian model in his analysis.
12. The following statement by Joan Robinson (1962; 1965, 100) is a theoretical counterpart of Kalecki’s insight: ‘The short period is here and now, with concrete stocks of means of production in existence. Incompatibilities in the situation . . . will determine what happens next. Long-period equilibrium is not at some date in the future; it is an imaginary state of affairs in which there are no incompatibilities in the existing situation, here and now’.
13. At least four different versions of the concept of centres of gravitation (c.g.) may be identified. Three are analogies drawn from physics, the fourth, from meteorology. The first relates to a frictionless pendulum always swinging, but always passing the same minimum point on its path to and fro. The second is a pendulum, the motion of which eventually ends because of friction. It settles at the minimum point of its path, which is also the c.g. of the first version. The third (due to the late David Champernowne) con-cerns a dog running towards its master, who is riding a bike. The bike is the c.g., always moving. The dog’s direction of movement at any point of time may be predicted by the knowledge of where the bike (master) is at that point of time. The fourth is akin to average temperatures (of seasons and places), which are predicted by their relationships to the average values of relevant factors. The averages are good predictors over time but not day by day; see Harcourt (1982, 205–21).
14. Nor is the existence of short-period rest states inconsistent with the view of the world associated with Allyn Young (1928), Kaldor and Myrdal of virtuous and vile cumula-tive causation processes, as opposed to the stable long-period equilibrium positions of, say, Chicago mainstreamers.
15. Foley and Michl (1999) have introduced the concept of the fossil production function.
It allows the econometric application of the Cobb–Douglas function to be salvaged by giving a different meaning to the economic processes producing the data. Taking their lead from Ricardo, Joan Robinson and Kaldor, they suggest that capital-using technical progress occurs over time, with only one best-practice technique being available at any moment of time. Empirical observations, either cross-section or time series, reflect current and past vintages, with average productivity rising over time. Michl (2002) shows that this procedure helps to make sense of evidence taken from OECD countries, and of the findings of discrepancies between the values of capital–output elasticities and profit shares.
16. This can be shown as follows:
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