Saturday, July 06, 2013

Sraffa Prices As A Linear System

Figure 1: Two Equivalent Block Diagrams for a Linear System

1.0 Introduction

I have previously gone on about complex, out-of-equilibrium phenomena arising in certain non-linear models for economics. This post provides a contrast, by defining linear. Sraffa's system of equations for prices of production, from a certain perspective1, is an example of a linear system.

I regard the mathematical manipulations expressed in this post as fairly trivial. Nevertheless, it will not surprise me if some find it difficult to read. I do not think any such reading difficulties result solely from defects in my expository powers. Rather, I am trying to echo the sort of abstract reasoning typical of advanced mathematics courses taught at many universities. I think I only gesture here at the advantages provided by such abstractions.

2.0 Definition of "Linear"

Functions can be characterized as linear or non-linear. A function, f(), maps elements in some set to elements in another, possibly different, set. The set of possible arguments2 for a function is known as the domain of the function. The set that elements of the domain are mapped into is known as the range of the function. One assumes that elements of the domain can be added together, in some sense, to obtain another element of the same set. Furthermore, each element of the domain can be multiplied by a scalar3. Last, one makes the same assumptions about the elements of the range.

The function f is linear if the following two conditions are met:

f(x1 + x2) = f(x1) + f(x2)
f(a x) = a f(x)

These equations are illustrated, respectively, by Figure 1 above and Figure 2 below. The first condition states that when a linear function is applied to the sum of two elements, the summation can equally well be calculated after applying the function to the elements being summed. The second condition states that the order of scalar multiplication and the application of the function can likewise be interchanged, with no effect on the output.

Figure 2: Two More Equivalent Block Diagrams for a Linear System

Maybe the simplest example of a linear system is the equation of a straight line going through the origin:

y = f(x) = m x,

where x and y are real numbers4.

3.0 Sraffa's Price Equations

The above definition would not be worth much if the only example of a linear function was a straight line through the origin in a two-dimensional Cartesian space. Accordingly, I will describe an example for a function whose argument is a vector.

Suppose an economy is observed at a point in time. And, in this economy, at the observed scale, firms have adopted n processes to produce n commodities. The j-th process is characterized by its inputs and outputs. Its inputs consist of a0,j person-years of labor, a1,j units of the first commodity, a2,j units of the second commodity, and so on. Its outputs consist of b1,j units of the first commodity, b2,j units of the second commodity, and so on5. A common rate of profits, r, is also among the givens in this model. These givens allow one to set up the following system of equations for the wage, w, and prices of production6, p1, p2, ..., pn:

(p1 a1,1 + ... + pn an,1)(1 + r) + a0,1 w = p1 b1,1 + ... + pn bn,1
(p1 a1,2 + ... + pn an,2)(1 + r) + a0,2 w = p1 b1,2 + ... + pn bn,2
(p1 a1,n + ... + pn an,n)(1 + r) + a0,n w = p1 b1,n + ... + pn bn,n

The above system of n equations in n + 1 unknowns can be conveniently expressed in matrix form:

p A(1 + r) + a0w = p B,

where a0 and p are row vectors and A and B are square vectors. Some manipulations yields the following matrix equation:

p [B - A(1 + r)] - a0w = 0

These manipulations suggest the definition of a linear function.

3.1 A Linear Function

Accordingly, consider the following function:

f(p, w) = p [B - A(1 + r)] - a0w

This function maps a vector space with the dimension n + 1 to an n-dimensional vector space. Figure 3 illustrates for the case where n is two. The components of the vector calculated by this function are the extra profits earned in each process in use. Two, almost one-line, proofs demonstrate the linearity of this function.

Figure 3: A Linear Function for a Two-Commodity Economy

3.1.1. Proof of the First Condition

By definition, the value of the function for the sum of two elements of its domain is:

f(p + q, u + v) = (p + q) [B - A(1 + r)] - a0(u + v)


f(p + q, u + v) = p [B - A(1 + r)] - a0u + q [B - A(1 + r)] - a0v

Or, by the definition of the function:

f(p + q, u + v) = f(p, u) + f(q, v),

which was to be shown.

3.1.2. Proof of the Second Condition

By definition, the value of the function for an argument consisting of the product of a scalar and an element of the domain of the function is:

f(c p, c w) = (c p) [B - A(1 + r)] - a0(c w)


f(c p, c w) = c { p [B - A(1 + r)] - a0w}

Or, by definition,

f(c p, c w) = c f(p, w)

which, again, was to be shown.

3.2 Observations and Questions

Consider all the elements of the domain of a function that map into the zero element in the range. This subspace of the domain is called the null space7. Figure 4 illustrates a null space for a linear function that, generically, does not arise for the Sraffa model. The three dimensions in the figure represent the domain of the function. For a linear model, the origin is in the null space. In this case, two non-zero independent vectors, represented by the two heavy arrows not along any of the three axes, map to zero. So the plane in which these two vectors lie represents the subset of the domain which maps to zero.

Figure 4: The Subspace of Zeros of a Linear Function

Wages and prices of commodities are positive in an economically meaningful solution to Sraffa's model. Thus, the null space should contain a ray leading from the origin through the first quadrant. Furthermore, if the extension of such a ray is all of the null space, the solution of this model is unique, up to multiplication by a constant. Choosing a numeraire for measuring prices and the wage specifies a location on this ray.

The economic setting of this model suggests conditions8 that might lead to the desired properties of the null space:

  • No coefficients of production are negative, while direct labor inputs are always positive.
  • Every process requires some commodities as inputs, and produces at least some commodities.
  • Every commodity is produced as an output by at least some process.
  • The economy hangs together, in some sense. One cannot find two or more sets of commodities where, for instance, no commodities from the first set enter as inputs into the second set and vice versa.
  • The production processes are all distinct, in some sense. Technically, no production process is a linear combination of the other processes.
  • The economy produces a surplus. The quantities of commodities required as inputs can be replaced out of the outputs, with some commodity output left over.
  • With a notional rescaling of processes, a set of commodities can be found that, in some sense, enter into the production of all commodities and that are being produced at a same rate of surplus production.
  • The rate of profits does not exceed that maximum rate of surplus production.

More is going on here than a counting of equations and variables.

4.0 Conclusion

Sraffa, in his book, does not present his sequence of models in these abstract terms. But many comments and sections, such as the appendix on "beans", demonstrate that he was aware of the mathematical issues arising with his models. One can read Sraffa as having an interest in computability not shown in my exposition.

Finally, this post proves that the use of models in which the solution illustrates the mutual interdependence of a system of equations is simply insufficient to demonstrate that economists think of the economy as a complex, non-linear system.

  1. If one took the wage, instead of the rate of profits, as the independent variable, Sraffa's equations would define a non-linear system. Furthermore, since Sraffa's model is open, it is consistent with non-linearities in economic relationships not in the model, such as provided by Increasing Returns to Scale.
  2. In this section, arbitrary elements of the domain are represented by x, x1, and x2.
  3. Technically, the domain and the range are each examples of a vector space, also known as a linear space. The scalars are from a field. The sets of real and complex numbers are canonical examples of a field.
  4. Although the graph of an affine function, y = m x + b, is a straight line, an affine function is, technically, non-linear when the y-intercept is non-zero.
  5. Since more than one commodity can be produced as output for each process, this is a model of joint production. See Chapter VII, Sections 50-52 of Production of Commodities By Means of Commodities.
  6. Prices of production allow for the outputs to be redistributed among industries such that the economy can continue (re)production undisturbed.
  7. For a linear function, the null space is a linear space.
  8. Such conditions are more obvious for the special case of no single-product industries. I do not fully understand the issues for joint production, especially when the processes in use are chosen from a larger set of possible processes.

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