Reswitching and capital models

Reswitching of techniques was shown to be possible in Leontief’s input–output model with flow and stock matrices. The finding is unearthed and related to the capital debate based on Sraffian

that the neoclassical assumption of a decreasing relationship between factor intensity and factor reward need not hold. The source of the trouble is that capital consists of produced commodities, such as machinery and equipment. There is an aggregation issue, which neoclassical economists assume away. A symptom of the aggregation issue is that as the rate of return on capital increases, a capital-intensive technology may be substituted out-as neoclassical economists would predict since the rate of return is the price of capital for entrepreneurs-but the capital-intensive technology may reemerge at a still greater rate of return. This phenomenon of reswitching of techniques contradicts the construct of a macroeconomic production function and its use in the distribution of income. Loud criticaster and defender of the neoclassical school of thought were Joan Robinson and Bob Solow, respectively; for a review of the capital controversies, see Cohen and Harcourt (2003).
However, the best intellectuals among the neoclassical opponents and their opponents, Paul Samuelson and Luigi Pasinetti, respectively, were basically in agreement. In fact, in an early paper Samuelson himself provided an example of reswitching. Samuelson (1966) did so in the context of an Austrian circulating-capital model, where capital is input that is in the pipeline of production. Labor-intensive and time-intensive techniques coexist, and reswitching may occur.
In the simplest input-output analytic terms the Austrian model of circulating capital can be based on x(t) = Ax(t + 1) + y(t), where t is time, x is the gross output vector, y the net output vector, and A the matrix of input-output coefficients. This equation subsumes a single technique for every product, with a unitary lag of production. To produce output, the inputs must be available one unit ahead of time. Distributive lags in production have been analyzed and estimated by ten Raa (1986), who solves for the path of gross outputs that sustains a path of net outputs, and ten Raa et al. (1989), who estimate different production lags. Alternative techniques can be introduced by working directly with the use and make tables (which underlie the A matrix): Vs = Us + y, where s is the vector of activity levels, U the use table and V the make table. U and V are rectangular, reflecting that there may be more activities than products, a source of substitution. Production time lags may vary as in ten Raa et al. (1989) and this modification would yield a full-blown input-output model of Austrian circulating capital.
Leontief, however, modeled capital not as circulating but as fixed, using a matrix of capital coefficients B. The row vector of product prices fulfills p = pA + rpB + l, normalizing the price of labor at 1, denoting the rate of return on capital by r, and assuming perfect competition (zero profit). Leontief (1986) considered the possibility of reswitching an empirical issue and investigated if the phenomenon occurs in the U.S. economy. In this paper Leontief mentioned that a numerical test example can easily be constructed of a simple, say, three sector input-output system in which the cooking recipes are such that some recipes would drop out if the rate of return goes up from 5% to say 10%, but reappear again when it rises to 15%. He did not spell out an example. We corresponded on this paper. I wrote that reswitching can be shown in an even simpler input-output system, with only two sectors. He wrote the paper was rejected by "our leading economic journal" (the American Economic Review). "It might amuse you to know that the editor rejected it explaining that it would be of no interest to its readers. " In this note I present the example and position Leontief 's model in the capital literature.

The example
Leontief 's input-output system consists of a matrix of input-output coefficients, A, a matrix of capital coefficients, B, and a row vector of labor coefficients, l. Matrices A and B have dimension n × n and row vector l has dimension 1 × n, where n is the number of sectors. The system (A, B, l) constitutes a technology. An alternative technology is indicated by (α, β, λ). My example consists of A = 0, B = 0 0 0 1 , l = (1 1) and α = 0, β = 0 5 0 0 , λ = (1 0). Only sector 2 features technical change. The first economy is decomposed, with sector 1 using labor and sector 2 using labor and home-made capital. The second economy is integrated, with sector 2 using capital produced by sector 1. Solving for the product prices, p = l(I -A -rB) −1 . In the first economy this reduces In the second economy this reduces to p The change of technology (in sector 2) does not impact the price of product 1 (as sector 1 uses no product of sector 2, neither intermediate input nor capital). Therefore, the second technology will be adopted if and only if the second price of product 2 is less than the first price, 5r < (1 -r) −1 . This condition for technical change is fulfilled for rates of return on the interval (0 0.28) and the interval (0.72 1) (rounded). At intermediate rates of return the first technology is adopted. The relationship between the rate of return on capital and the choice of technique is not monotonic in the example. Reswitching is possible. An empirically relevant example of reswitching has been found by Han and Schefold (2006). The choice of technology depends on the product prices. Since the product prices are continuous with respect to the input-output coefficients, the reswitching result holds for positive input-output coefficients. Hence reswitching is not only possible for perhaps rather unusual numerical values.

Alternative models and their connection
The Sraffian literature and the recent input-output literature on reswitching, e.g., Zambelli (2018), are based on a different price equation, namely p = pA + rpA + l. In words, Leontief assumes capital is fixed, while Sraffa c.s. assume capital is circulating. The two points of view are consistent if the flow coefficients (A) are proportional to the stock coefficients (B). Remarkably, this is precisely Bródy's (1970) capital condition. Moreover, ten Raa (1997) has shown that Bródy's capital condition is required for Leontief 's dynamic input-output model to be consistent with a temporal distributed use-make framework.
Anyway, reswitching has been established in Sraffa's model and in Leontief 's model. In the simplest case of an economy with two production sectors this has been