Adam Smith, David Ricardo, And The Labor Theory Of Value

1.0 Introduction

I resolutely am not commenting on unhappy current events.

Smith and Ricardo thought a (simple) LTV was not applicable to capitalism. Prices do not tend to or orbit around labor values. At least that is their claim.

Ricardo had more to say about the LTV.

This argument is not new. Smith confined the LTV to a supposed "early and rude state of society which precedes both the accumulation of stock and the appropriation on land" (WoN, book 1, chapter 6; see also book 1, chapter 8). Ricardo thought this was sloppy reasoning. The LTV does not become nonapplicable merely because of the accumulation of capital and of the division of society into capitalists and workers (Principles, 3rd edition, chapter 1, section III).

2.0 Technology

A simple model of circulating capital can be used to make Ricardo's point. Let a₀ be a row vector of direct labor coefficients. Let A be the Leontief input-output matrix. An element of a₀ and the corresponding column of A specify the labor time and the capital goods needed to operate a process to produce one unit of the output of that industry. The technology satisfies the following common assumptions:

• Some labor is needed to operate every process in each industry.
• Constant returns to scale prevail.
• Each commodity enters, directly or indirectly, into the production of every commodity. Iron, for example enters indirectly into the production of automobiles if iron is needed to produce steel and steel is needed to produce cars.
• The technology is productive. For some level of operation of the processes for each industry, some commodities are left over after reproducing the capital goods used in producing them.

Now for the unusual special case. Let λ be the largest eigenvalue of the Leontief matrix. This eigenvalue is also known as the Perron-Frobenius root of the Leontief matrix. Assume that the vector of direct labor coefficients is a corresponding left-hand eigenvector:

a₀ A = λ a₀ (Display 1)

3.0 Labor Values

Let v be the row vector of labor values. By definition, labor values satisfy the system of equations in Display 2:

v A + a₀ = v (Disp. 2)

The total labor to produce a commodity is the sum of the labor values of the capital goods used in that industry and the direct labor coefficient.

Under the special case assumption, labor values are a multiple of direct labor coefficients:

v = (1/(1 - λ)) a₀ (Disp. 3)

One can check this solution by merely plugging it into the solution in Display 2:

(1/(1 - λ)) a₀ A + a₀ = (λ/(1 - λ)) a₀ + a₀ = (1/(1 - λ)) a₀ (Disp. 4)

Since the above is a one-line proof, I thought I would include it. Labor values are also an eigenvector of the Leontief matrix.

4.0 Prices

Under the usual assumptions, the row vector p of prices satisfies the system of equations in Display 5:

p A (1 + r) + w a₀ = p (Disp. 5)

The scalar w is the wage, and r is the rate of profits.

Let R be the maximum rate of profits, obtained when the wage is zero, and the workers live on air. For the special case, the solution to the price system is quite simple:

R = (1/λ) - 1 = (1 - λ)/λ (Disp. 6)

r = R (1 - w) (Disp. 7)

p = v (Disp. 8)

One can check this solution by plugging it into the system of equations in Display 5.

So Ricardo was correct. The LTV could apply to capitalism under a special case. The failure of the LTV to apply in general is because those special case conditions cannot be expected to arise.

5.0 Conclusion

Ricardo had a point. As any ent would tell you, Smith was too hasty. The simple conflict between the wage and the rate of profits in Display 7 applies more generally than the above special case. The LTV, even when it is not valid, points to theories of the returns to capital.

I suppose reconstructing arguments between Ricardo and Smith with modern economics may conflict with how some historians do history. Even saying that Smith was analyzing capitalism could be considered an anachronism.

Roger Garrison On The Inadequacy Of Hayekian Triangles

Hayek introduced his triangles in his lectures for his book Prices and Production. The first edition was in 1931 and the second in 1935. He attempted a more general treatment of capital theory in his 1941 book, The Pure Theory of Capital. I want to claim that Hayek knew that he could not draw his triangles under these more general assumptions. And that he knew that capital theory needed more development than he was able to give.

Jack Birner knows this, although I do not know a reference off-hand. I want to say that Roger Garrison knows this too, that he presents his diagrams of trianlges in his 2001 book, Time and Money: The Macroeconomics of Capital Structure, mainly for their heuristic and pedagogical value. He says something like this in the following:

"...The widely recognized but rarely understood Hayekian triangle, introduced in his 1931 lectures at the London Schoold of Economics, were subsequently published (in 1931 with a second edition in 1935) as Prices and Production. The triangle, described in the second lecture ... is a heuristic device that gives analytical legs to a theory of business cycles first offered by Ludwig von Mises... Triangles of different shapes provide a convenient but highly stylized way of describing changes in the intertemporal pattern of the economy's capital structure.

In retrospect, we see that the timing of Hayek's invitation to lecture at the London School of Economics takes on a special significance. We learn from the preface of the subsequent book that had the invitation come earlier, he couldn't have delivered those lectures; had it come later, he probably wouldn't have delivered them.

(The invitation) came at a time when I had arrived at a clear view of the outlines of a theory of industrial fluctuations but before I had elaborated it in full detail or even realized all the difficulties which such an elaboration presented. (Hayek, [1935] 1967: vii)

Hayek mentions plans for a more complete exposition and indicates that his capital theory would have to more developed in much greater detail and adapted to the complexities of the real world before it could serve as a satisfactory basis for theorizing about cyclical fluctuations.

A decade after the London lectures, the more complete exposition took form as The Pure Theory of Capital (1941). In this book Hayek fleshed out the earlier formulations and emphasized the centrality of the 'capital problem' in questions about the market's ability to coordinate economic activities over time. The 'pure' in the title meant 'preliminary to the introduction of monetary considerations.' Through some 450 pages in length, the book only achieved only the first half of the original objective. The final sixty pages of the book did contain a 'condensed and sketchy' (p. viii) treatment of the rate of interest in a monetary economy, but the task of retelling the story in Prices and Production in the context of the Pure Theory was put off and ultimately abandoned. The onset of the war was the proximate reason for cutting the project short; Hayek's exhaustion and waning interest in business-cycle issues - and his heightened interest in the broader issues of political philosophy - account for his never returning to the task. In later years, he acknowledged that Austrian capital theory effectively ended with his 1941 book and lamented that no one else has taken up the task that he originally set for himself (Hayek, 1994: 96).

More fully developing the Austrian theory of the business cycle came to be synonymous with writing the follow-on volume to Hayek's Pure Theory. Many a graduate student has imagined himself undertaking this very project, only to abandon the idea even before the enormity of the task was fully comprehended. Thus, while the comparatively simple relationships of capital-free Keynesian theory captured the attention of the economics profession, the inherently complex relationships of Austrian theory languished.

Time and Money is not the sequel to Hayek's Pure Theory. Rather, the ideas and graphical construction in the present volume take the original Hayekian triangle of Prices and Production to be the more appropriate point of departure for creating a capital-based macroeoconomics. The trade-off between simplicity and realism is struck in favour of simplicity. Hayek's triangles allow us to make a graphical statement that there is a capital structure and that its intertemporal profile can change. This statement enables the Austrian theory to make a quantum leap beyond the competing theories that ignore capital altogether or that treat capital as a one-dimensional magnitude." -- Roger Garrison (2001: 10-11).

I think J. R. Hicks' Value and Capital is an attempt to formalize some ideas from Hayek's capital theory. Garrison does not mentioned the possibility that maybe the formalization of a general theory along Austrian lines is impossible.

Anyways, here is an attempt to fairly shorten the quotation:

Hayek ... indicates that his capital theory would have to more developed in much greater detail ... before it could serve as a satisfactory basis for theorizing about cyclical fluctuations... The final sixty pages of [Hayek 1941] did contain a 'condensed and sketchy' (p. viii) treatment of the rate of interest in a monetary economy, but the task of retelling the story in Prices and Production in the context of the Pure Theory was put off and ultimately abandoned... More fully developing the Austrian theory of the business cycle came to be synonymous with writing the follow-on volume to Hayek's Pure Theory... Time and Money is not the sequel to Hayek's Pure Theory. Rather, the ideas and graphical construction in the present volume take the original Hayekian triangle of Prices and Production to be the more appropriate point of departure... -- Roger Garrison (2001: 10-11).