# The mathematics of the linear model of production

In the interest of brevity, we hypothesize a fixed capital model with depreciation. In our estimations, we excluded the wages advanced because such capital is relatively very small and usually is obtained by businesses through their available credit lines and the same is true, to a lesser extent, with the materials advanced which require turnover times." So we may write

where upper-case bold-faced letters refer to square matrices, lower-case bold-faced letters refer to vectors of dimensions conformable to pre- or post-multiplication by matrices; finally, scalars are indicated by lower-case letters in italics. The notation in Equation 4.1 is as follows,

*7t =* lx_{w} vector of relative PP defined up to multiplication by a scalar *X =* 1 x« vector of prices of unit values 1 = 1 x/i vector of employment coefficients A = *nXn* matrix of input—output coefficients D = «x« matrix of depreciation coefficients

I = «X/) identity matrix of the same dimensions with the matrix A К = nX/j matrix of fixed capital coefficients

e = lx« vector of ones or market prices. This vector also serves as the summation vector x = i?xl vector of gross output r = the economy-wide rate of profit and *w =* the wage rate

After some manipulation of Equation 4.1, we arrive at the equation of relative PP, which need to be normalized. Thus, we have

or

or

It is important to note that the vector of unit labor values, *X,* is normalized through its multiplication by the monetary expression of labor time (MELT), which in our case is the ratio of the column vector of gross output, x, evaluated at market prices (ex) over the same gross output evaluated in unit labor values *Xx.* Thus, the monetary expression of labor values called “direct prices” (Shaikh 1977) and symbolized by v are defined as follows:

We post-multiply Equation 4.2 by the standard commodity, C, which is derived from the right-hand-side (r.h.s.) eigenvector of the matrix of vertically integrated capital coefficients H (see Shaikh 1998):

where R = 1 / Я,_{11ах} is the maximum rate of profit or the reciprocal of capital- output ratio derived from the maximal eigenvalue, A,_{mx}, of Equation 4.3 and

<7 the column vector of output proportions corresponding to the maximal eigenvalue. The standard proportions or standard commodity must be normalized when multiplied by the ratio of gross output evaluated in market prices (MP) over the DP multiplied by the output proportions such that

and in so doing, we establish the following equality (see Shaikh 1998):

The next step is to fix the relative prices by the normalized standard commodity, **s, **and derive the normalized row vector of PP, **p,**

with the property

So, Equation 4.2 can be rewritten as

We post-multiply 4.4 by the normalized standard commodity, and we get It follows that

We divide through by **vs, **and we end up with
which solves for the linear WRP curve

where *p = r / R,* with 0 < *p <* l.^{3} By taking into account Equation 4.4, we arrive at

We know that the maximum eigenvalue of the matrix HR equals to one and, therefore, the matrix H*Rp* has an eigenvalue less than one, which makes it a convergent matrix. Hence, the matrix H represents the vertically integrated fixed capital coefficients and, if data are available, could include the circulating (besides the fixed) capital advanced, that is, both materials and wages. The lack of data, but also the simplicity of presentation, limits our exposition to matrix of fixed capital coefficients. In the case of the circulating capital model, the matrix H consists of the intermediate inputs per unit of output, **A, **as well as the wage goods input per unit of output, bl, or

We post-multiply Equation 4.6 by the inverse of the diagonal matrix of DP, v, and we get

where p^{-1} denotes the ratio of PP to labor values or DP. Equation 4.7 can be restated in the case of a single industry /, as follows:

where is the *j*-th column of matrix H, pHy / *Vj* is the so-called capita intensity of the vertically integrated industry producing commodity *j,* an< R^{-1} is the capital intensity of the standard system (which is independent о prices and distribution). Finally, the derivative of Equation 4.8 w.r.t. *p* gives

or

where *kj =* (pH */v) ..* We factor out *kj* from the bracketed term, and we get

The change in PP/DP w.r.t. *p* depends on the elasticity of industry’s *j* capital intensity w.r.t. the relative rate of profit, *p* (the first bracketed term of Equation 4.9), which can be positive, negative or even zero; this term might be called “Sraffian or Wicksellian effect”. According to Ricardo and Marx, the sign of change in PP/DP w.r.t. *p* depends mainly on the second bracketed term, which for obvious reasons, we call the “Ricardo-Marx effect”. Hence, relative prices move according to industry’s capital intensity relative to the economy’s average.^{4} Sraffa’s great contribution is the introduction and the accounting of the complex price effects, which may not let the Ricardo-Marx effect to play its dominant role. It is possible, therefore, the Sraffian effect to enhance, lessen or even supersede the Ricardo-Marx effect for particularly high relative rates of profit. These possibilities become extremely important in the capital theory controversies, whose theoretical implications were presented in Chapter 3, while their empirical dimensions are discussed in the next sections.

For the two extreme and hypothetical relative rates of profit, Equation 4.9 becomes

and for p = 1

From Equations 4.10 and 4.11, we may conclude the following:

- 1 If the Ricardo-Marx effect is positive, then the capital intensity of industry
*j*is higher than the standard ratio, while the Sraffian effect can be positive, negative or even zero. If it is positive, it may strengthen the upward direction of the price movement; however, if the elasticity is negative, then the outcome depends on the relative strength of these two effects. - 2 The elasticity of capital intensity w.r.t.
*p*is very close to zero (especially for low values of p) and so the likelihood for the change in the direction of price movement for realistic p depends on how close is the industry’s capital intensity, fe,, to the standard capital intensity, R^{_l}. - 3 The capital intensities of industries in the fixed capital model are usually quite distant from the standard capital intensity; therefore, a counteractive elasticity effect cannot make anything quite different.
- 4 The situation becomes less rigid, when we move to circulating capital models, where the elasticity of capital intensity w.r.t. p remains low, but the differences in capital intensities of industries relative to the standard capital intensity are much smaller than those of the fixed capital model. As a consequence, we cannot rule out the case that a change in p may give rise to an elasticity of capital intensity with a sign opposite to that of the Ricardo-Marx effect; furthermore, the value of this elasticity may exceed that of the Ricardo-Marx effect and thus may even overturn the direction of the price trajectory displaying extremes. Thus, one may not exclude results, which are opposite to those theorized by Ricardo and Marx and for low values of p, a vertically integrated industry may start as capital- (labor-) intensive industry relative to the standard ratio,
*R,*but as p increases, it may be transformed to a labor- (capital-) intensive industry.

From the above discussion, it follows that the claims ofSraffa (1960) and his followers for the development of quite complex price-feedback effects because of income redistribution that change the characterization of an industry is most likely an exaggeration. It goes without saying that, to a great extent, this is an empirical question, which we explore in the sections below.