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.