A Derivation Of Sraffa's First Equations

1.0 Introduction

Piero Sraffa wrote down his 'first equations' in 1927, for an economy without a surplus. D3/12/5 starts with these equations for an economy with three produced commodities. I always thought that they did not make dimensional sense, but Garegnani (2005) argues otherwise. This post details Garegnani's argument, albeit with my own notation.

There are arguments about how and why Sraffa started on his research project I do not address here. The question is how did he relate what he was doing at this early date to Marx. In addition to Garegnani, DeVivo, Gehrke, Gilibert, Kurz, and Salvadori are worth reading here.

2.0 Givens

I assume an economy in a self-replacing state in which n + 1 commodities are produced.

• c_0,0 is the input of the first commodity used in producing the output of the first industry.
• (c_{., 0})^T = [c_1,0, c_2,0, ..., c_n,0] are the inputs of the remaining n commodities used in producing the output of first industry.
• c₀ = [c_0,1, c_0,2, ..., c_0,n] are the inputs of the first commodity used in producing the output of the remaining industries
• The element c_i,j, i, j = 1, 2, ..., n, of the matrix C is the input of the ith commodity used in producing the output of the jth industry.
• q₀ = is the quantity produced of the first commodity.
• (q)^T = [q₁, q₂, ..., c_n] are the outputs of the remaining n commodities used in producing the output of first industry.

All quantities are given in physical units. I abstract from fixed capital; all inputs are used up in the production of the outputs. Table 1 presents these parameters for the first example in the first chapter in Sraffa 1960.

Table 1: The Example from Sraffa (1960), Chapter 1

Input	Industry
	Iron	Wheat
Iron	c0, 0 = 8 tons iron	c0 = [12 tons iron]
Wheat	c., 0 = [120 quarters wheat]	C = [280 quarters wheat]
Output	q0 = 20 tons iron	q = [400 quarters wheat]

The following must hold for economy to be in a self-replacing state:

q_i = c_i,0 + c_i,1 + ... + c_i,n, i = 0, 2, ..., n

All quantities are non-negative. The economy must hang together in some sense. In Sraffa's terminology, all commodities are basic.

3.0 Coefficients of Production

I like to think of the coefficients scaled for unit output in each industry. Accordingly, define:

a_{0, 0} = c_{0, 0}/q₀

(a_{., 0})_i = (c_{., 0})_i/q_j, i = 1, 2, ..., n

(a₀)_j = (c₀)_j/q_j, j = 1, 2, ..., n

(A)_i,j = (C)_i,j/q_j, i, j = 1, 2, ..., n

4.0 All Quantities Measured in Unit Outputs of the First Industry

The given inputs can be thought of as produced in the previous year. The amount of, say, iron directly used as input in producing other commodities is (a₀ q). Table 2 indicates how much iron is needed as input in all previous years.

Table 2: Iron Inputs for Other Commodities

Year	Iron
0	a0 q
1	a0 A q
2	a0 (A)2q
...	...
n	a0 (A)nq
...	...

Even though my notation picks out the first commodity, there is nothing special about it. Suppose some commodity is selected. Let v₀ be the quantity of this commodity needed directly and indirectly to produce a unit of the first commodity. Let v be the quantities of this commodity needed directly and indirectly to produce each of the remaining commodities. v₀ and v must satisfy the following system of n + 1 linear equations:

v₀ a_{0, 0} + v a_{., 0} = v₀

v₀ a₀ + v A = v

For a non-trivial solution to exist, the determinant of the matrix in Table 3 must be zero, which it is in the case pf the Sraffa example.

Table 3: A Matrix

1 - a0, 0 = (3/5) tons	-a0 = [(-3/100) tons]
-a., 0 = [-6 quarters]	I - A = [(3/10) quarters]

I set v₀ to unity. The amount of this commodity used directly and indirectly in the production of all other commodities is easily found:

v = a₀(I - A)^-1

5.0 Rescaling the Givens

I then rescale the givens.

b_{0, 0} = v₀ c_{0, 0}

b_{i, 0} = v_i (c_{., 0})_i, i = 1, 2, ..., n

b_{0, j} = v₀ (c₀)_j, j = 1, 2, ..., n

b_{i, j} = v_i c_{i, j}, i, j = 1, 2, ..., n

s₀ = v₀ q₀

s_i = v_i q_i, i = 1, 2, ..., n

Table 4 presents Sraffa's example with these calculations. Here, a unit of wheat is 10 quarters. That is, one ton iron is used directly and indirectly in producing 10 quarters of wheat.

Table 4: Sraffa's Example Again

Input	Industry
	Iron	Wheat
Iron	b0, 0 = 8 tons iron	b0 = [12 tons iron]
Wheat	b., 0 = [12 tons wheat]	B = [28 tons wheat]
Output	s0 = 20 tons iron	s = [40 tons wheat]

I then have Sraffa's 'first equations':

b_{0, j} + b_{1, j} + ... + b_{n, j} = s_j, j = 0, 1, ..., n

For the economy to be in a self-replacing state, the following must hold:

b_{i, 0} + b_{i, 1} + ... + b_{i, n} = s_i, i = 0, 1, ..., n

Even though I am adding together, say, quantities of iron and wheat, the dimensions are consistent.

6.0 A Re-interpretation

Suppose the first produced commodity is labor, not iron. c_{0, 0} becomes the amount of labor performed in households (outside the market) to reproduce the labor force. c_{., 0} is the commodity basket paid out in wages when the workers obtain all of the surplus product. a₀ are the direct labor coefficients for each industry, and A is the Leontief input-output matrix. v is the vector of labor valus (also known as employment multipliers). Under the assumptions, prices of production are identical to labor values.

This model is descriptive. The givens do not show how required inputs might decrease with innovation or the formal and real subsumption of labor.

References

• Garegnani, Pierangelo (2005) On a turning point in Sraffa's theoretic and interpretative position in the late 1920s. European Journal of the History of Economic Thouht 12 (3): 453-492.
• Gehrke, Christian, Heinz D. Kurz, and Neri Salvadori (2019) On the 'origins' of Sraffa's production equations: A reply to de Vivo. Review of Ploitical Economy 31 (1): 100-114.