Prices Of Production And A Wheat Theory Of Value

1.0 Introduction
In this post, I describe a theory of prices that is an alternative to the neoclassical supply-and-demand theory of prices as scarcity indices. In this exposition, I consider the simple case in which the modeled economy does not produce a surplus. In this simple case, prices of production are in the ratios of labor values and of commodity values, for any specified commodity. This post illustrates this claim too.

I do not draw the conclusion from the equivalence illustrated here that labor values have no priority over commodity values. After all, the formal model illustrated here does not include a principal agent problem arising with labor.

2.0 Technology and Advanced Wages
Consider a simple economy in which only three commodities are produced, namely, wheat, iron, and pigs. Each commodity is produced by a specified process requiring (possibly zero) inputs of labor, wheat, iron, and pigs. Suppose these processes are observed to produce the quantities of outputs shown in Table 1 from the inputs shown there. In other words, these processes are observed to operate at the scale shown. No assumption about returns to scale is made here. In particular, it is not necessary for Constant Returns to Scale to prevail.

TABLE 1: Technique in Use

INPUTS	Wheat Industry	Iron Industry	Pig Industry
Labor	1 Person-Year	2 Person-Years	3 Person-Years
Wheat	230 Quarters	70 Quarters	90 Quarters
Iron	12 Tons	6 Tons	3 Tons
Pigs	12 Pigs		12 Pigs
OUTPUTS	450 Quarters	21 Tons	60 Pigs

Suppose wages are advanced at the start of the production period, and that these advanced wages consist of 10 quarters wheat and 6 pigs per person-year. Then one could specify the inputs to the production processes as consisting exclusively of wheat, iron, and pigs, with no labor input (as shown in Table 2). This is now the example from paragraph 2 of Sraffa's Production of Commodities by Means of Commodities. Notice that the outputs can just replace the inputs, including the advanced wages, with no commodity surplus being left over. Capitalists do not make an accounting profit in this economy.

TABLE 2: Production of Commodities by Means of Commodities

INPUTS	Wheat Industry	Iron Industry	Pig Industry
Wheat	240 Quarters	90 Quarters	120 Quarters
Iron	12 Tons	6 Tons	3 Tons
Pigs	18 Pigs	12 Pigs	30 Pigs
OUTPUTS	450 Quarters	21 Tons	60 Pigs

3.0 Prices of Production
With the social division of labor in this economy, firms in each industry at the end of the production period have an inventory of a single commodity. To continue production, they must trade some of that commodity for the other commodities they need as inputs. Prices of (re)production are time-invariant prices that allow these trades to occur and the economy to be smoothly reproduced through the actions of the agents in the economy. For this simple example, prices of production must satisfy three equations:

240 p_w + 12 p_i + 18 p_p = 450 p_w

90 p_w + 6 p_i + 12 p_p = 21 p_i

120 p_w + 3 p_i + 30 p_p = 60 p_p

where:

• p_w is the price of a quarter of wheat,
• p_i is the price of a ton of iron, and
• p_p is the price of a pig.

These equations are linearly dependent. Any multiple of a solution set of prices is also a solution. I arbitrarily pick a quarter of wheat as the numeraire. The solution set of prices is then $1 per quarter wheat, $10 per ton iron, and $5 per pig.

4.0 Labor Values
One can work out a consistent accounting in which the amount of labor time embodied in each commodity is measured. Labor values are found as the solution to the following system of three linear inhomogeneous equations in three unknowns:

1 + 230 v_w + 12 v_i + 12 v_p = 450 v_w

2 + 70 v_w + 6 v_i = 21 v_i

3 + 90 v_w + 3 v_i + 12 v_p = 60 v_p

where:

• v_w is the person-years labor embodied in a quarter of wheat,
• v_i is the person-years labor embodied in a ton of iron, and
• v_p is the person-years labor embodied in a pig.

This system of equations has a unique solution. The labor values for the commodities are 1/40 person-years per quarter wheat, 1/4 person-years per ton iron, and 1/8 person-years per pig.

5.0 Wheat Values
One can also work out a consistent accounting system in which the amount of wheat embodied in each commodity is measured. Wheat values for iron and pigs are found as the solution to the following system of two linear inhomogeneous equations in two unknowns:

90 + 6 w_i + 12 w_p = 21 w_i

120 + 3 w_i + 30 w_p = 60 w_p

where:

• w_i is the quarters wheat embodied in a ton of iron and
• w_p is the quarters wheat embodied in a pig.

The wheat values of commodities are w_i = 10 quarters per ton and w_p = 5 quarters per pig.

Calculating iron values for wheat and pigs and calculating pig values for wheat and iron are left as an exercise to the reader.

6.0 Contrast and Comparison
For any set of values (prices of production, labor values, or commodity values), one can find quantities of each commodities that are valued as equal for that set. Table 3 illustrates by showing the values of specified quantities of each commodity. In this example, the following equation holds:

10 quarters wheat = 1 ton iron = 2 pigs,

whether commodities are valued in terms of dollars, embodied labor, or any commodity value (such as wheat). The equivalence of all these values is a special case. This equivalence works out from considering a pure circulating capital case in which an economic surplus not paid out in wages is not produced.

TABLE 3: Value of Specified Quantities of Commodities

Quantities	Value in
	Prices of Production	Person-Years of Embodied Labor	Quarters of Embodied Wheat
10 Quarters Wheat	$10	1/4 Person-Years	10 Quarters
1 Ton Iron	$10	1/4 Person-Years	10 Quarters
2 Pigs	$10	1/4 Person-Years	10 Quarters