How To Draw Hayekian Triangles In A Model Of The Production Of Commodities

1.0 Introduction

This post revisits an analysis of Hayekian triangles in the context of a circular flow of production. Here I go through the mathematics to show how to construct the "triangle".

2.0 The Technique and Net Output

A technique is specified by a row vector a₀ of direct labor coefficents of a square Leontief input matrix A. Each labor coefficient and corresponding column of the Leontief matrix specify a process to produce one unit of the good produced by that industry. All coefficients are specified in physical units, such as barrel oils per kilowatt. I assume:

• The economy is in a stationary state.
• Constant returns to scale (CRS) prevail.
• All direct labor coefficients are positive.
• Every good enters, either directly or indirectly, into the production of every good.
• The Leontief matrix specifies a productive technique in that a suplus product can be produced.
• Full employment is assumed. More generally, the units in which labor is measured is scaled such that employment is unity.
• Wages are paid at the end of the year, not advanced with the payments for capital goods at the beginning of the year.

These assumptions are stronger than needed.

The proportions of net output are assumed to be specified by a column vector d. This vector is also a numeraire. The level of net output y is specified by the scalar c:

y = c d

This formulation allows for specifying any number of techniques, all with the same numeraire and composition of net output, but at different levels.

3.0 Quantity Flows

The net output vecotr y and the gross output vector q are related as:

y = q - A q = (I - A) q

Total employment is unity:

a₀ q = 1

These equations have a solution. Consumption per worker is:

c = 1/[a₀ (I - A)^-1 d

Gross quantities are:

q = c (I - A)^-1 d = c d + c A d + c A² d + ...

The first term in the infinite expansion on the right-hand side is the net product available at the end of the given year. The second term is the quantities of capital goods being produced in the current year to support the production of the net output in the next year. The third term is the capital goods being produced in the current year to eventually produce the net output two years hence. Note that all of these vectors, of consumption goods, specific capital goods, and so on are heterogeneous.

4.0 Labor and Capital Flows in the Hayekian Triangle

The labor l_i expended in the current year, with previously produced capital goods, to produce the net output, that is, goods of the first order, is defined as:

l₁ = c a₀ d

The labor l_i expended in the current year to produce goods of each of the higher orders is:

l_i = c a₀ A^{i - 1} d, i = 2, 3, ...

The sum of these quantities of labor is unity, that is, the labor force employed in the current year.

The capital goods expended in the current year to produce goods of each order is:

k_i = c Aⁱ d, i = 1, 2, ...

The sum of these quantities of capital goods used in the current year is A q.

5.0 Value Flows in the Hayekian Triangle

Let z_i be the addition in value for each stage in the Hayekian triangle. This is merely the value added by original factors of production, properly time discounted for each stage:

z_i = w(r) l_i (1 + r)^{i - 1}, i = 1, 2, ...

The notation reflects the interdependence of the wage 𝑤(𝑟) and the interest rate 𝑟 in a stationary state.

For goods of first order, the length of this step in the Hayekian triangle is:

z₁ + z₂ + ... = c p(r) d

where p(r) is a row vector of prices. For goods of the second order, the length of the step in the Hayekian triangle is:

z₂ + z₃ + ... = c p(r) d - z₁

For goods of the third order, the length of the step in the Hayekian triangle is:

z₃ + z₄ + ... = c p(r) d - (z₁ + z₂)

These steps can be continued. This completes one derivation of the lengths of the steps in a Hayekian triangle.

The length of the ith step can also be expressed as:

z_i + z_{i + 1} + ... = p(r) k_i (1 + r)ⁱ + w(r) l_i (1 + r)^{i - 1}, i = 1, 2, ...

With a couple of substitutions and factoring, the above becomes:

z_i + z_{i + 1} + ... = c [p(r) A (1 + r) + w(r) a₀] A^{i - 1} d (1 + r)^{i - 1}

Or:

z_i + z_{i + 1} + ... = c p(r) A^{i - 1} d (1 + r)^{i - 1}, i = 1, 2, ...

The Hayekian triangle, with an infinite number of steps, has now been derived, in two ways, from the circulating capital case of a model of the production of commodities by means of commodities.

6.0 Conclusion

The above derivations assume knowledge of the solutions of the price system for the technique. A more complete exposition would present that solution. It would also show that the Hayekian triangle approaches one constructed with a geometric series, as the order of goods increases. The composition of capital goods approaches that of Sraffa's standard system.