Expanded reproduction and the Kalecki principle

In this post, following Trigg’s book on reproduction schemas, I will explore how the work of Michał Kalecki contributes to answering the question where the demand for the surplus part of total output comes from. One can explore in more detail how surplus value is allocated by its use: some of it is used for expanding the capital stock ($dC_i$), some for expanding the workforce $dV_i$) and some for capitalist consumption ($u_i$), the subscript $i$ referring to departments. Also, Department 2 can be disaggregated into two departments, $D_2$ producing consumption goods ($CoG$) for capitalists and $D_3$ producing $CoG$ for workers. Using the same table as in the previous post, with $D_2$ split into two and surplus value allocated by use, we have the table:

Departments	Constant capital $C_i$	Variable capital $V_i$	Surplus value $S_i$	~	~	Total output $W_i$
			$u_i$	$dC_i$	$dV_i$
$D_1$	4000	1000	500	400	100	6000
$D_2$	550	275	220	36 $\frac{2}{3}$	18 $\frac{1}{3}$	1100
$D_3$	950	475	380	63 $\frac{1}{3}$	31 $\frac{2}{3}$	1900
$\sum$	5500	1750	1100	500	150	9000

Table 1: Three-sector (expanded) reproduction schema

The $C_i$ column is the amount of constant capital used up in the given period of production, in this case 5500. The columns that represent surplus value/product are different from the columns $C_i$ and $V_i$, because the former are the final outputs from this period of production, and they are not used as inputs in this period. Since there is accumulation (the capital stock is incremented), the output of $D_1$ ($W_1=6000$) is more than the total input of MoP, $C_1=5500$. If however we add to $C_i$ the additional units of constant capital ($dC_i$), and do the same with $V_i$ and $dV_i$, creating ‘ex post’ variables $C_i$* and $V_i$*, then the row sums and columns match:

Departments	$C_i$*	$V_i$*	Capitalist consumption $u_i$	Total output $W_i$
$D_1$	4400	1100	500	6000
$D_2$	586 $\frac{2}{3}$	293 $\frac{1}{3}$	220	1100
$D_3$	1013 $\frac{1}{3}$	506 $\frac{2}{3}$	380	1900
$\sum$	6000	1900	1100	9000

Table 2: ‘ex post’ three-sector reproduction schema (quantities at the end of period)

The columns $C_i$ and $V_i$ I would interpret as the inputs of MoP and labour (workforce/wages) in the next period, respectively, whereas $u_i$ is the part of the profit received in sector $i$ that is spent on consumption by owners, instead of accumulation. So $C_i$* and $V_i$ are mixed variables in the sense that some of it is the input in the current period and some the increment added in the current, but used in the next period. We have the equalities of outputs by sector to outputs grouped by use:
$W_1 = C_1$* $+ C_2$* $+ C_3$* $= C_1 + dC_1 + C_2 + dC_2 + C_3 + dC_3$
$W_2 = u_1 + u_2 + u_3$
$W_3 = V_1$* $+ V_2$* $+ V_3$* $= V_1 + dV_1 + V_2 + dV_2 + V_3 + dV_3$ $\tag{1}\label{outputs_use}$

Expressed in words:

• $W_1$, total output of MoP (means of production) = MoP needed for replacement and accumulation ($C_i + dC_i$)
• $W_2$, total output of CoG for capitalists = profit extracted by capitalists, spent on personal consumption.
• $W_3$, total output of CoG for workers (replacement and new workforce)’ = total wage bill (of workers in new period, including newly hired workers)

We also have the sectoral equalities:
$W_1 = (C_1 + dC_1) + (V_1 + dV_1) + u_1 \\ W_2 = (C_2 + dC_2) + (V_2 + dV_2) + u_2 \\ W_3 = (C_3 + dC_3) + (V_3 + dV_3) + u_3 \tag{2}\label{sectoral_outputs}$

There is one more rearrangement: defining gross profits $P_i$* as total output minus ex post variable capital, ie. $P_i$* $= W_i - V_i$*. At this point it’s not yet clear how to interpret gross profits meaningfully. Ex post variable capital will be spent in the next period of production from gross revenue, but this is also true for $C_i$*. At any rate, if we reorder terms this way, adding $C_i$* to $u_i$, we get the table with gross profits:

Departments	$V_i$*	$P_i$*	Total output $W_i$
$D_1$	1100	4900	6000
$D_2$	293 $\frac{1}{3}$	806 $\frac{2}{3}$	1100
$D_3$	506 $\frac{2}{3}$	1393 $\frac{1}{3}$	1900
$\sum$	1900	7100	9000

Table 3: Kalecki’s interpretation of the three-sector schema (ex post variable capital and gross profits)

This is the same as grouping the terms in the output as:
$W_i = (V_i + dV_i) + (C_i + dC_i + u_i)$

An equality becomes visible from this table, namely that the surplus of CoG that $D_3$ produces (surplus in the sense that it is in excess of the wage bill of its own workers) is equal to the wage bill of the other two sectors:
$P_3 = V_1$* $+ V_2$* $\tag{3}\label{wage_goods_gross_profit}$

If we then add the gross profits of the other two sectors, $P_1$* and $P_2$*, we get:
$P_1 + P_2 + P_3 = V_1$* $+ P_1 + V_2$* $+ P_2$
$P = W_1 + W_2$
$\tag{4}\label{total_gross_profit_outputs_equal}$

This means that total gross profit in the economy is equal to the (gross) output of the sector producing capital goods ($D_1$) and capitalist CoG ($D_2$). Since gross profit is defined as all spending on capital goods (replacement + incremental) and capitalist consumption, and these are supplied by sectors 1 and 2.

Trigg writes that Kalecki interpreted this ‘ex post identity’ as following: ‘[T]he money expenditures by capitalists upon consumption and investment that generate the resultant volume of profits’, or shortly ‘capitalists earn what they spend’. It is easy to see that ‘capitalists earn what they spend’ is true in the case of simple reproduction where the personal consumption of owners provides the demand for the surplus product, and returns to them as profit, but it less clear for expanded reproduction. In expanded reproduction some of the surplus product takes the form of new constant capital and new workers, so the demand must come from advanced amounts of money by the ‘departments’ (companies) themselves, not capitalists as consumers. To clearly see how this is possible, I need to write explicitly the sequence of payments/sales and production.

For now I’ll go on with the derivation as in the book. The three-sector schema can be represented in an input-output table as below:

	$D_1$ (inputs)	$D_2$ (inputs)	$D_3$ (inputs)	$S_i$	$S_i$	$S_i$	$W_i$
				$dC$ (addit. constant)	$dV$ (addit. variable)	$u$ (capit. consumption)	$W_i$ (total output)
$D_1$ (outputs by use)	4000	550	950	500			6000
$D_2$						1100	1100
$D_3$	1000	275	475		150		1900
$S_i$	1000	275	475				1750
$W_i$	6000	1100	1900				9000

As in the previous post (new/net) investment is classified here as final demand (it is not an input in this period of production), specifically as additional constant ($dC$) and variable ($dV$) capital. Total personal consumption of capitalists is in column $u$, under $S$.

The equality of Equation \eqref{wage_goods_gross_profit} reads in terms of the I/O table as:
$P_3 = W_3-(V_3+dV_3)=(V_1+dV_1)+(V_2 + dV_2)$

The wage terms are $V_i = X_{3i}$, so then the equation is:
$W_3 - (X_{33} + dV_3) = (X_{31} + dV_1) + (X_{32} + dV_2)$
and since none of the output of $D_1$ and $D_2$ can take the form of $dV$, therefore $dV_1 = dV_2 = 0$, then this simplifies to:
$W_3 - (X_{33} + dV_3) = X_{31} + X_{32} \tag{5}\label{use_output_3}$

In other words, all the output of $D_3$ not used by itself for replacement and expansion of its workforce is, logically, used by the other two sectors. Also, it is more clearly visible that
$S = u + dC + dV = u + I \tag{6}\label{surplus_investm_capcons}$
ie. surplus value (total profit) = capitalist consumption + investment (accumulation). This equation is not the same as \eqref{total_gross_profit_outputs_equal}, because there gross profit P* is defined as all spending on capital goods and capitalist consumption, whereas here it is only the part spent on accumulation (new capital goods and labour) and capitalist consumption. This is interpreted by Trigg/Kalecki as ‘capitalists cast money into circulation as aggregate demand on capitalist consumption and investment, which is realized as surplus value’ (p. 35). (There is again a nagging question of causality here left unanswered.)

This can be rendered as a ‘Kalecki multiplier’, by defining capitalist consumption $u$ as made up of a constant part $B_0$ and a part dependent on profits as a fraction $\lambda$ of total profits $P$:
$u = B_0 + \lambda P \tag{7}\label{kalecki_u_lambda}$

Where $P$ is total profits. Since $P = u + I$,
$P = B_0 + \lambda P + I \tag{8}\label{eq8}$
and therefore $P = \frac{B_0 + I}{1-\lambda} \tag{9}\label{eq9}$

Equation \eqref{eq9} represents a ‘multiplier’ relationship between total profits and total ‘exogenous’ expenditure by capitalists ($B_0 + I$: constant capitalist consumption + new investment), with a multiplier effect of $1/(1-\lambda)$.

Since profits consist of capitalist consumption and investment, the components of the final demand $f$ (see previous post, Equations 5 and 13), this can be substituted into the macro-income (‘Keynesian’) multiplier $y = \frac{1}{e}f$, yielding

$$y = \frac{1}{e (1-\lambda)} (B_0 + I) \tag{10}\label{kalecki_multipl}$$

So that the ‘Keynesian’ multiplier now has the terms $e$, the share of surplus value, and the ‘Kalecki multiplier’ $1/(1-\lambda)$, with $\lambda$ being the fraction of profits spent on consumption.

Once more, we have a macro-identity here, but the question of whether the necessary amount of money is there to sell all outputs (and if yes where does it come from) in the case of expanded reproduction still hangs over these identities. This question can be addressed in the framework of the monetary circuit, which I will discuss in my next post.

Reference

Andrew Trigg, Marxian reproduction schema: money and aggregate demand in a capitalist economy. Routledge, 2009

Written on October 28, 2018

[ political-economy  economic-models  classical-political-economy  marxian-economics  kalecki  demand  ]