The Structure of Classical Theory

Abstract

Classical economics is a term invented by Marx to characterise all economists beginning with William Petty (1623–1687) in England and Pierre Le Pesant de Boisguilbert (1646–1714) in France, and ending with Ricardo in England and Simonde de Sismondi (1773–1842) in France. According to Marx, the focus of classical economists was the determination of the surplus (value), defined as the difference between the value of total output produced and the value of (labour and non-labour) inputs used in production. The evaluation of inputs and outputs is in terms of prices, determined mainly by labour times. Furthermore, classical economists share the view that labour employed in production (in general) is responsible for the creation of surplus. Hence, some caution should be applied because Marx excluded, from his characterisation of classical economics, major economists such as Thomas Malthus (1706–1834) and John Stuart Mill (1806–1873), who not only questioned the validity of the labour theory of (exchange) value but also were eager to replace it with the ephemeral forces of supply and demand. Marx has also used the term ‘vulgar economists’ to refer to those whose analysis was based on the surface phenomena of supply and demand.

Appendices

Appendix A

A1. The Input–Output Analysis

The input–output analysis is a technique, which is used in the study of relations between industries. The major characteristic of input–output tables is that the total input of each industry must be equal to its total output. Quesnay’s Tableau Economique and also Marx’s schemes of reproduction form the basis of modern input–output tables.

The columns of an input–output table describe the inputs of each industry to itself and to the others. The column sum of an input–output table gives the total cost of production. Costs include the rewards of primary inputs also; that is, wages, profits, depreciation, taxes, etc., in short, the value added. The rows of an input–output table refer to the sales of an industry to itself and to the other industries (see also Chap. 1). A portion of the total output produced is absorbed by the final demand; that is, consumption, investment, government expenditures, and exports.¹²

Table A1 Input–outputs

The classical assumption of a given technology implies that if for some reason the output of industry j increases, x _j , it follows that the inputs x _ij of the industry j must increase proportionally. The technological coefficients are determined by the following relationship a _ij =x _ij /x _j , where a _ij s are the technical coefficient or input–output coefficient. Once computed, input–output coefficients are treated as constant (fixed). Moreover, the column sum of technological coefficients for a viable economy must less than one, which is equivalent to saying that the value of output produced must not exceed the value of inputs that were used in its production.

The input–output coefficients can be converted from a descriptive invention to a useful analytical tool with the aid of linear algebra. In the interest of brevity we restrict the analysis to two industries. Thus we have:

$$ {{x_1} = {a_{11}}{x_1} + {a_{12}}{x_2} + {y_1}} $$

$$ {{x_2} = {a_{21}}{x_1} + {a_{22}}{x_2} + {y_2}} $$

where y ₁ and y ₂ are the final demands (consumption, investment, (net) exports, and government expenditures) of industries 1 and 2, respectively. Hence, we have a system of equations which can be written in terms of matrices as follows:

$$ {\left[ {\begin{array}{*{20}{c}} {{x_1}} \hfill \\ {{x_2}} \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}{c}} {{a_{11}}} \hfill & {{a_{12}}} \hfill \\ {{a_{21}}} \hfill & {{a_{22}}} \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}{c}} {{x_1}} \hfill \\ {{x_2}} \hfill \\ \end{array} } \right] + \left[ {\begin{array}{*{20}{c}} {{y_1}} \hfill \\ {{y_2}} \hfill \\ \end{array} } \right]} $$

and in compact form x=Ax+y which solves for the output vector

$$ x = {({\rm I}--{\rm A})^{--1}}y $$

the matrix (Ι–Α)^–1 is known as the Leontief inverse whose columns show the input requirements, both direct, and indirect, on all other producers, generated by one unit of output.

A.1.1 Price Determination

In what follows we introduce prices p ₁ and p ₂ which correspond to the outputs of industries 1 and 2, respectively. The total revenues from the sales of each sector will be:

Revenues of industry 1: p ₁ x ₁

Revenues of industry 2: p ₂ x ₂

Τhe total cost of each industry is:

Cost of industry 1: (p ₁ a ₁₁+p ₂ a ₂₁)x ₁

Cost of industry 2: (p ₁ a ₁₂+p ₂ a ₂₂)x ₂

We further assume a uniform profit rate r and we get: ${p_1} = {p_1}{a_{11}} + {p_2}{a_{21}} + r({p_1}{a_{11}} + {p_2}{a_{21}}) = (1 + r)({p_1}{a_{11}} + {p_2}{a_{21}})$

$$ {p_2} = {p_1}{a_{12}} + {p_2}{a_{22}} + r({p_1}{a_{12}} + {p_2}{a_{22}}) = (1 + r)({p_1}{a_{12}} + {p_2}{a_{22}}) $$

or in matrix form:

$$ {\left[ {\begin{array}{*{20}{c}} {{p_1}} \hfill \\ {{p_2}} \hfill \\ \end{array} } \right] = (1 + r)\left[ {\begin{array}{*{20}{c}} {{a_{11}}} \hfill & {{a_{21}}} \hfill \\ {{a_{12}}} \hfill & {{a_{22}}} \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}{c}} {{p_1}} \hfill \\ {{p_2}} \hfill \\ \end{array} } \right]} $$

The problem now is to find the rate of profit and prices which are consistent with the givens of this economy. Clearly, the profit rate will correspond to the maximum eigen value of our hypothetical economy while equilibrium prices will correspond to the associated eigenvector. If we therefore write the above system in compact form, we will have:

$$ p\prime (1/1 + r) = A\prime p\prime \,or\,p\gamma = p{\rm A}, $$

where the eigenvalue 1/1+r and where the eigenvector p corresponds to the vector of positive relative prices.

A.1.2 A Numerical Example

For the better understanding of the preceding analysis let us take a realistic input–output table in order to present a series of questions that we referred to above. The data are presented below:

Table A.2 Aggregated input–output table

The input–output coefficients of the above input–output table are estimated if we divide each of the inputs of each sector by the total output produced. Thus we have:

$$ A = \left[ {\begin{array}{*{20}{c}} {8969/71189} & {32919/244519} & {489/114875} \\ {8180/71189} & {84296/244519} & {16661/114875} \\ {3034/71189} & {18286/244519} & {9350/114875} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}{c}} {0.125} & {0.134} & {0.004} \\ {0.114} & {0.344} & {0.145} \\ {0.042} & {0.074} & {0.081} \\ \end{array} } \right] $$

We observe that the sum of no column or row of matrix Α exceeds one and as a result the economy produces surplus and thus it is capable of reproduction (for details see Passineti 1977).

A.1.3 The Marxian Theory of Value and Direct Prices

If we symbolise the row vector of values of produced commodities by λ, the row vector of direct labour coefficients by a _ο the indirect labour, that is, the labour contained in the inputs which are used in the current production of commodity j by λ _j a _ij and the value of depreciation, that is the wear and tear of the fixed capital invested in every production period, by λ _j d _ij , then we will have:

$$ {[{\lambda_1},{\lambda_2},...,{\lambda_n}] = [{a_{o1}},{a_{o2}},...,{a_{on}}] + [{\lambda_1},{\lambda_2},...,{\lambda_n}]\left[ {\begin{array}{*{20}{c}} {{a_{11}} + {d_{11}}} \hfill & {{a_{12}} + {d_{12}}} \hfill & {...} \hfill & {{a_{1n}} + {d_{1n}}} \hfill \\ {{a_{21}} + {d_{21}}} \hfill & {{a_{22}} + {d_{22}}} \hfill & {...} \hfill & {{a_{2n}} + {d_{2n}}} \hfill \\ \cdots \hfill & \cdots \hfill & \cdots \hfill & \cdots \hfill \\ {{a_{n1}} + {d_{n1}}} \hfill & {{a_{n2}} + {d_{n2}}} \hfill & \cdots \hfill & {{a_{nn}} + {d_{nn}}} \hfill \\ \end{array} } \right]} $$

or in compact form

$$ \lambda = {\alpha_o} + \lambda ({\rm A} + D) $$

where D is the matrix of depreciation coefficients. The above equation solves for:

$$ \lambda = {a_o}{[{\rm I}--({\rm A} + D)]^{ - 1}} $$

Using the above input–output numerical example together with the corresponding matrix of depreciation coefficients and the vector of the labour input coefficients, that is,

$$ D = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 \\ {0.012} & {0.019} & {0.029} \\ {0.006} & {0.004} & {0.002} \\ \end{array} } \right]\,and{ }{a_o} = \left[ {\begin{array}{*{20}{c}} {0.592} & {0.198} & {0.422} \\ \end{array} } \right] $$

and after the appropriate substitutions we get:

$$ \lambda = \left[ {0.784 0.547 0.567} \right] $$

The row vector λ gives the quantity of homogenised labour contained (directly and indirectly) in the output of each sector. The notion of value in Marx is, as we know, monetary. Thus we have to transform the above quantities of direct and indirect labour time to direct prices, which we symbolise by d _j . For the estimation as well as for the comparison of direct prices with the market prices, we consider that the market price of each unit of output of a sector is equal to 1 monetary unit. We stipulate the following normalisation condition dλx=λex and so the vector of labour values is transformed to direct prices as follows:

$$ d = \lambda \frac{{e \cdot x}}{{\lambda \cdot x}} = [0.784{ }0.547{ }0.567] \cdot 1.68 $$

so we get:

$$ d = \left[ {0.846 1.025 1.064} \right] $$

We observe that the vector of direct prices is extremely close to the vector of market prices, which we symbolise by e and which is the row unit vector. This can be identified by using one of the usual measures of deviations¹³:

• The mean absolute deviation is 0.147 or 14.7%.

• The mean absolute weighted by the output deviation is 0.098 or 9.8%.

A.1.4 Prices of Production

In Marxian analysis prices of production are defined as the prices that incorporate the economy’s general profit rate. For their estimation, we need the real wage of the workers, that is, the basket of goods that the workers spend their wage money on, i.e.,

$$ w = pb $$

where w is the wage money and b is the (nx1) vector of real wage goods. In addition, we take into account the following matrices that correspond to the above numerical example:

$$ < t > = \left[ {\begin{array}{*{20}{c}} {0.0019} & 0 & 0 \\ 0 & {0.1291} & 0 \\ 0 & 0 & {0.0771} \\ \end{array} } \right],\,K = \left[ {\begin{array}{*{20}{c}} 0 & 0 & 0 \\ {1.136} & {0.480} & {1.8918} \\ {0.0272} & {0.0410} & {0.08083} \\ \end{array} } \right]\,and\,b = \left[ {\begin{array}{*{20}{c}} {0.0046} \\ {0.0124} \\ {0.0118} \\ \end{array} } \right] $$

where <t> is the diagonal matrix of indirect tax coefficients, K is the matrix of fixed capital stock coefficients and b is the vector of wage goods. Thus prices of production are defined as

$$ {p = pb{a_o} + pA + pD + p < t > + rpK} $$

where ba _o is a new matrix that represents the quantity of commodity i which is required for the consumption of workers in order to produce commodity j. The above relation after some manipulation gives the following eigenequation:

$$ {p(1/r) = pK{{[I - (ba + A + D + < t > )]}^{ - 1}}} $$

According to Perron–Frobenious theorem only the maximal eigenvalue is associated with a unique positive left-hand side eigenvector which gives the vector of relative prices. In terms of our numerical example we get the rate of profit r= 0.136 and the normalised vector of prices of production which will be:

$$ p = \left[ {1.054 0.971 1.027} \right] $$

Once again we invoke the statistics of deviation:

• The mean absolute deviation is 0.036 or 3.6%.

• The mean absolute weighted deviation is 0.032 or 3.2%.