Groucho Marxism

An input–output model is a quantitative economic model that represents the inter-dependencies between different sectors of an economy. The idea dates back to the 18th century French economist François Quesnay, who developed a simple input-output model which he referred to as a ‘tableau économique’. Karl Marx’s economic analysis provided another early example involving a set of tables where the economy consisted of two interlinked departments. The Russian economist Alexander Bogdanov has been credited with originating the modern concept of an input-output model in 1921. However it was another Russian economist, Wassily Leontief, who did most to develop the field of input-output analysis; he was awarded the Nobel Prize in Economics for this work in 1973.

The basic input-output model depicts inter-industry flows within an economy, showing how output from one industrial sector may become an input to another industrial sector. Suppose that we have an economy with m sectors, each of which produces a quantity q_i units of single homogenous commodity. Suppose further that A_ij units from sector i in order to produce 1 unit from sector j. If Q_i is the quantity of goods demanded from sector i then we must have q_i = ∑_j A_ijq_j +Q_i for each i. This can be written in matrix form as q = Aq+Q, which after re-writing becomes (I-A)q = Q, where I is the identity matrix. If the matrix I-A is invertible then we can write q = (I-A)^-1Q, and q can be shown to be positive under a relatively weak assumption on the matrix I-A known as the Hawkins-Simon condition.

In addition to the quantity model just described, there is also a ‘value’ model that examines the relationship between commodity values and inter-industry flows within an economy. In this model, the value of commodity j can be expressed as v_j = ∑_i v_iA_ij +V_j, where V_j is the value added per unit output of sector j. Thus can be written in matrix form as v = vA+V, which after re-writing becomes v(I-A) = V. Similarly to the quantity model described above, if the matrix I-A is invertible then v = V(I-A)^-1, and v can be shown to be positive provided the matrix I-A satisfies the Hawkins-Simon condition. Note that we have vq = vAq+vQ = vAq+Vq, and therefore vQ = Vq. In other words, the total value added is equal to the value of the demand vector Q.

There is also a third ‘price’ model that examines the relationship between commodity prices and inter-industry flows. In this model, the price of commodity j can be expressed as p_j = (1+r)∑_i p_iA_ij +(1+r)wL_j, where r, w, and L_j are the profit rate, wage rate, and labour inputs in sector j, respectively. This can be written in matrix form as p = (1+r)(pA+wL). Rearranging gives p[I-(1+r)A] = (1+r)wL, and similarly to the models described above, if the matrix I-(1+r)A is invertible we can write p = (1+r)wL[I-(1+r)A]^-1, which is positive provided the matrix I-(1+r)A satisfies the Hawkins-Simon condition.  In Marxian economics, wages are usually assumed to be set at subsistence level, so w = pb for some subsistence commodity vector b. We then have p = (1+r)p(A+bL), so p is an eigenvector of A+bL with eigenvalue 1/(1+r).

A well-known result called as the Perron-Frobenius theorem says that under a weak condition on the matrix A+bL, such a p and r exist and are unique. Furthermore, as w = pb, if the subsistence bundle b is uniquely determined, then wages are uniquely determined too. (We can ensure that b is uniquely determined by assuming that if there is more than one subsistence bundle, wages will be determined by the one with the lowest price.) In general there is no simple relationship between these prices and the values v defined above. Moreover, there is no logical reason why the values added in production, V, should be equal to the labour inputs, L; that is, there is no logical reason why the labour theory of value should necessarily hold. This is something that is usually just assumed in Marxian economics.