Use values and exchange values in Marx’s extended reproduction schemes

Abstract

Marx-Engels’ numerical illustrations of the extended reproduction suggest that a two-sector economy reaches a balanced growth path, from the second period onwards. We explain this surprising result and show that for technical reasons, disproportions between sectors can prevent the system from reproducing itself. But, in Marx’s reproduction schemes, such a crisis is not only due to purely technical factors and one must wonder what role is played by the relative price in the reproduction of the system. The answer is given by comparing two models having a similar structure but quite different rules for the determination of the relative price. In Marx’s model, the price is given by the labour values and thus, it is exogenously fixed. We contrast Marx’s analysis with an endogenous price model in which the price depends on the conditions of the accumulation of capital. The Appendices point out the complete accordance of Engels’ corrections with Marx’s model and Marx’s unfruitful quest for a balanced growth path as a tool for the analysis of crises.

Keywords:

• Marx
• reproduction
• two-sector model
• crises

JEL Classification:

• O41
• B14
• B51

Acknowledgements

We wish to express our gratitude to the anonymous referees for their valuable comments and suggestions which enable us to improve our article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Let us give two significant examples. According to Rosa Luxemburg ([1913] 2003, 335) Marx’s result is a consequence of his assumption of a constant technique: the scheme “assumes that the composition of capital is the same in every year, […i.e., it…] is not affected by accumulation. This procedure would be quite permissible in itself in order to simplify the analysis, but when we come to examine the concrete conditions […] for reproduction, then at least we must take into account […] changes in technique, which are bound up with the process of capital accumulation.” After a detailed analysis of a Marx’s numerical example, Michio Morishima (1973, 120) concludes that “any state of unbalanced growth will disappear in Marx’s economy in a single year.” That is why he modifies some assumptions of Marx’s model. In part 1 and in Appendices 1 and 3, we will show that Marx’s results depend on his numerical examples and are not the logical consequence of his model.

2 Through Mori’s article, we have become aware of the existence of a six-sector reproduction scheme in Marx’s manuscripts. Its interest does not lessen the importance of studying the two-sector scheme.

3 At the beginning of the twentieth century, Marx’s model was used by the participants in the Marxist debates on the crisis of capitalism (namely, R. Luxemburg, O. Bauer, M. Tugan-Baranovsky, K. Kautsky, and R. Hilferding). Later on, the two-sector model is adopted by M. Kalecki, O. Lange, J. M. Keynes, J. Robinson, and by J. Meade and J.R. Hicks in their formalizations of the General Theory. At the beginning of the 1960s, the first systematic analyses of the properties of two-sector models are published. Shinkai’s model (1960), inspired by J. Robinson, opens this line of research. Immediately after, Uzawa (1961) publishes “a neoclassical version of Shinkai’s model” (40, footnote 2). This article and the following (1963) are at the origin of many contributions to the neoclassical theory of the two-sector growth during the 1960s. With regard to Marx’s model, the studies of Morishima (1973) and Nikaido (1983 and 1985) are especially important.

4 In Marx’s terms, “All the various branches of production pertaining to each of these two departments form one single great branch of production, that of the means of production in the one case and that of articles of consumption in the other” (Marx [1885] 1956, 242). There is an “aggregate capitalist” (244) in each sector. The economy behaves as if it were made up of two commodities and two representative capitalists.

5 If $D\hspace{0.17em}>\hspace{0.17em}0,\hspace{0.17em}\hspace{0.17em}{k}_1\hspace{0.17em}>\hspace{0.17em}{k}_2:\hspace{0.17em}$$G_1\hspace{0.17em}>\hspace{0.17em}0\Rightarrow q\hspace{0.17em}>\hspace{0.17em}(1-c)k_2$ and $G_2\hspace{0.17em}>\hspace{0.17em}0\Rightarrow q\hspace{0.17em}<\hspace{0.17em}(1-c)k_1$. Conversely, if $D\hspace{0.17em}<\hspace{0.17em}0,\hspace{0.17em}\hspace{0.17em}{k}_1\hspace{0.17em}<\hspace{0.17em}{k}_2$: $G_1\hspace{0.17em}>\hspace{0.17em}0\Rightarrow q\hspace{0.17em}<\hspace{0.17em}(1-c)k_2$ and $G_2\hspace{0.17em}>\hspace{0.17em}0\Rightarrow q\hspace{0.17em}>\hspace{0.17em}(1-c)k_1.$

6 If $D\hspace{0.17em}>\hspace{0.17em}0,\hspace{0.17em}\hspace{0.17em}{G}_2\hspace{0.17em}>\hspace{0.17em}0\Rightarrow q\hspace{0.17em}<\hspace{0.17em}{k}_1$ but $G_1\hspace{0.17em}>\hspace{0.17em}0\cancel{\Rightarrow }q\hspace{0.17em}>\hspace{0.17em}{k}_2$. Conversely, if $D\hspace{0.17em}<\hspace{0.17em}0,\hspace{0.17em}\hspace{0.17em}{G}_1\hspace{0.17em}>\hspace{0.17em}0\Rightarrow q\hspace{0.17em}<\hspace{0.17em}{k}_2$ but $G_2\hspace{0.17em}>\hspace{0.17em}0\cancel{\Rightarrow }q\hspace{0.17em}>\hspace{0.17em}{k}_1$

7 The demonstration would be the same if, like Marx, we had taken the values without making a distinction between prices and physical quantities of goods.

8 The instability can only be avoided if the initial proportion is von Neumann’s. In such a case, both accumulation rates are equal and the economy is in equilibrium from the start (see Bidard and Klimovsky 2006, chap. 10).

9 Since $G_1\hspace{0.17em}<\hspace{0.17em}1/a_{11}$, we have $a_{22}-D{G}_1\hspace{0.17em}>\hspace{0.17em}0$ as it can be verified by replacing D by its expression $a_{11}{a}_{22}-a_{21}{a}_{12}.$ Note that if $q\le q_{\max }(G_1)$, condition (3) is satisfied: if $D\hspace{0.17em}<\hspace{0.17em}0$, we have $q_{\max }(G_1)\hspace{0.17em}<\hspace{0.17em}{k}_2$ and if $D\hspace{0.17em}>\hspace{0.17em}0$, we have $q_{\max }(G_1)\hspace{0.17em}<\hspace{0.17em}{k}_1.$

10 D is eliminated in the second equality since ${\mu }^{-1}$ is an eigenvalue of A. Therefore, by definition, it verifies the characteristic equation ${\mu }^{-2}-(a_{11}\hspace{0.17em}+\hspace{0.17em}{a}_{22}){\mu }^{-1}\hspace{0.17em}+\hspace{0.17em}D=0$.

11 Such difficulties are related to the information held by agents, to the nature of their expectations, etc.

12 The indetermination can be eliminated by adding the equality $s_1=s_2$ which, in this model, is compatible with the asymmetrical relationship between sectors. Furthermore, it has the advantage of entailing a balanced growth path with a uniform rate of profit (see subsection 3.1.2).

13 See, for instance, Morishima’s critique of Marx’s scheme (1973, 114 and 122). As we have already seen, in Morishima’s formalization, the profits are not necessarily invested in the sector in which they are generated.

14 Notice that a change in $s_1$ has no effect on the price ceiling: according to Equation (16)(15) $$\begin{array}{c}0\le g_1\le \min (r_1,r_2)\\ 0\le s_1\le \min \left(1,\frac{r_2}{r_1}\right)\end{array}$$(15) , the product $q_{\max }[1-a_{11}(1\hspace{0.17em}+\hspace{0.17em}{s}_1{r}_1)]$ is equal to $a_{21}(1\hspace{0.17em}+\hspace{0.17em}{r}_2),$ which is a constant since it only depends on the fixed price. A change in $s_1$ is compensated by a change in the same direction of $q_{\max }$.

15 A calculation shows that the difference $q_{\max }(G_1)-q_{\max }(s_1,p)$ has the same sign, positive or negative, as $R_1-G_1$.

16 By eliminating p between Equations (v) and (vi)(11) $$\begin{array}{l}q=a_{11}q(1+g_1)+a_{21}(1+g_2)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(i\right)\\ 1=a_{12}q(1+g_1)+a_{22}(1+g_2)+c_1+c_2\hspace{1em}\left(ii\right)\\ pq=(a_{11}p+a_{12})q(1+g_1)+c_1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(iii\right)\\ 1=(a_{21}p+a_{22})(1+g_2)+c_2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(iv\right)\\ (1+r_1)(a_{11}p+a_{12})=p\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(v\right)\\ (1+r_2)(a_{21}p+a_{22})=1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(vi\right)\\ s_1{r}_1=g_1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(vii\right)\\ s_2{r}_2=g_2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(\mathit{\text{viii}}\right)\end{array}$$(11) of system (11), one obtains a decreasing relation between the factors of profit: R₂ = (1 − a₁₁R₁)/(a₂₂ − DR₁). Therefore, R₂ can be eliminated in the expression of q_max(1,p) in Table 1.

17 Note that the numerator is positive, since $a_{11}-D=a_{11}(1-a_{22})\hspace{0.17em}+\hspace{0.17em}{a}_{12}{a}_{21}\hspace{0.17em}>\hspace{0.17em}0$.

18 Note that if $s_1\hspace{0.17em}<\hspace{0.17em}1$, the price of production is not always the price allowing the largest absorption of the means of production. If $D\hspace{0.17em}>\hspace{0.17em}0$ and $s_1$ is large enough, the differential increase $a_{11}q{s}_{1}d{r}_1$ in the amount of the means of production accumulated in sector I as a result from a price increase can exceed the differential decrease $a_{21}d{r}_2$ of the means of production accumulated in sector II.

19 We are very grateful to Michael Gaul for pointing out to us Marx’s erroneous calculation.

20 Obviously, no balanced growth path is found in Marx’s manuscript. Nevertheless, Gehrke is not entitled to deduce that “a steady state cannot be inferred from Marx’s original discussion” (2017, 16). As we saw the balanced path logically belongs to Marx’s model.

21 These critical proportions are the same if the price is endogenous. As we have seen, in Marx’s model with exogenous price, the maximum value of $G_1$ is not $\mu$ but $\min (R_1,R_2).$

22 We do not examine the major influence of Quesnay’s Tableau économique on Marx’s reflections on reproduction. Marx was literally fascinated by Quesnay’s “zig-zag” (see Marx [1861–1863] 1977, 656; [1862–1864] 2013, 398–404).

23 By eliminating p between Equations (v)(11) $$\begin{array}{l}q=a_{11}q(1+g_1)+a_{21}(1+g_2)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(i\right)\\ 1=a_{12}q(1+g_1)+a_{22}(1+g_2)+c_1+c_2\hspace{1em}\left(ii\right)\\ pq=(a_{11}p+a_{12})q(1+g_1)+c_1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(iii\right)\\ 1=(a_{21}p+a_{22})(1+g_2)+c_2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(iv\right)\\ (1+r_1)(a_{11}p+a_{12})=p\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(v\right)\\ (1+r_2)(a_{21}p+a_{22})=1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(vi\right)\\ s_1{r}_1=g_1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(vii\right)\\ s_2{r}_2=g_2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(\mathit{\text{viii}}\right)\end{array}$$(11) and (vi)(11) $$\begin{array}{l}q=a_{11}q(1+g_1)+a_{21}(1+g_2)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(i\right)\\ 1=a_{12}q(1+g_1)+a_{22}(1+g_2)+c_1+c_2\hspace{1em}\left(ii\right)\\ pq=(a_{11}p+a_{12})q(1+g_1)+c_1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(iii\right)\\ 1=(a_{21}p+a_{22})(1+g_2)+c_2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(iv\right)\\ (1+r_1)(a_{11}p+a_{12})=p\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(v\right)\\ (1+r_2)(a_{21}p+a_{22})=1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(vi\right)\\ s_1{r}_1=g_1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(vii\right)\\ s_2{r}_2=g_2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(\mathit{\text{viii}}\right)\end{array}$$(11) of system (11), one obtains a decreasing relation between the factors of profit: $R_2=\frac{1-a_{11}{R}_1}{a_{22}-D{R}_1}$. Therefore, $R_2$ can be eliminated in the expression of $q_{\max }\left(1,p\right)$.