Technology For A Model Of Extensive And Intensive Rent

Table 3: Model Parameters and Variables

Parameter or Variable	Description
n	The number of produced commodities, with n > 1.
k	The number of types of land, with k > 1.
m	The number of processes available for producing corn on land, with m > k.
a0	A row vector of labor coefficients containing n + m - 1 elements. The element a0, j; j = 1, 2, ..., n + m - 1; is the person-years of labor needed to run the th process at a unit level.
A	A n x (n + m - 1) matrix. The element ai,j; i = 1, 2, ..., n; j = 1, 2, ..., n + m - 1; is the quantity of the ith commodity produced by the jth process when run at a unit level.
B	A n x (n + m - 1) sparse matrix. The element bi,j; i = 1, 2, ..., n; j = 1, 2, ..., n + m - 1; is the quantity of the ith produced commodity needed to run the jth process at a unit level.
C	A k x (n + m - 1) sparse matrix. The element ci,j; i = 1, 2, ..., k; j = 1, 2, ..., n + m - 1; is the acres of the ith type of land needed to run the jth process at a unit level.
t	A column vector, with k elements, of the available lands. The element ti; i = 1, 2, ..., k; is the number of acres of the ith type of land available.
d	A column vector, with n elements, specifying the requirements for use. The element di; i = 1, 2, ..., n; is the required net output of the th produced commodity.
q	A column vector, with n + m - 1 elements, of the level of operation of each process.
p	A row vector, with n elements, of prices.
ρ	A row vector, with k elements, of rents per acre.
r	The rate of profits. If given, the wage is to be found by solving the model.
w	The wage. If given, the rate of profits is to be found by solving the model.

This post continues the exposition of this model. Sraffa posed the problem of demonstrating some claims:

"the order of fertility ... is not defined independently of the rents; that order, as well as the magnitude of the rents themselves, may vary with the variation of r and w." (Sraffa 1960)

I have yet to consider the possibility of more than one agricultural product; I don't know that I solved the last problems in the Kurz and Salvadori (1995) chapter on rent. But I am starting on the first sentence in the following:

"89. More complex cases can generally be reduced to combinations of the two [extensive and intensive rent] that have been considered. The main type of complication arises from the multiplicity of agricultural products." (Sraffa, 1960)

Technology, endowments of land t, requirements for use d, and either the wage or the rate of profits are given parameters for this model. The level of operation q of each process, prices p, rents ρ, and the remaining distributive variable are found as the solution to the model. Each process takes a year to complete, totally uses up its inputs of produced commodities, and exhibits constant returns to scale (CRS).

Technology is characterized by the arrays a₀, A, B, and C. For this to be a model of extensive and intensive rent, these arrays must have more structure than specified in Table 3. Assume all labor coefficients a₀ and available endowments of types of land t are positive. Each industrial process produces exactly one commodity, and all agricultural processes produce the same commodity, called 'corn;. That is, the first (n - 1) rows and (n - 1) columns of the output matrix B comprise an identity matrix. The last elements of those rows are zero. The first (n - 1) elements of the last row, that is, the nth row, are zero and the remaining elements are unity. The first (n - 1) columns of the matrix C are zeros. Each of the last m columns has exactly one strictly positive entry; all other elements are zero. Each row of C has at least one positive element.

Let a⁽ⁱ⁾₀; i = 1, 2, ..., m; be a row vector of labor coefficients with n elements. The first n - 1 coefficients are the first n - 1 coefficients in a₀. The last coefficient is a_{0, n + i - 1}. Let A⁽ⁱ⁾ be the square input matrix with the corresponding columns from A, and let B⁽ⁱ⁾ be the square output matrix with the corresponding columns from B. Notice that B is the n x n identity matrix. In other words, a vector of labor coefficients a⁽ⁱ⁾₀ and a Leontief matrix A⁽ⁱ⁾ correspond to each process for producing corn.

Corn is assumed to be a basic commodity for each Leontief matrix A⁽ⁱ⁾. Suppose by an appropriate ordering of commodities and industries, the Leontief matrix can be decomposed into the form where the first n - l rows and columns is the submatrix F⁽ⁱ⁾_{(n - l) x (n - l)}. The submatrix G⁽ⁱ⁾_{l x (n - l)} is the last l rows for the first n - l colomns. The zero matrix 0⁽ⁱ⁾_{(n - l) x l} is the last l columns for the first n - l rows. The submatrix Â⁽ⁱ⁾_{l x l} is the last l rows and l columns. The subscripts indicate the size of the submatrices. The matrix Â⁽ⁱ⁾_{l x l} is assumed not to be further decomposable, and every column of G⁽ⁱ⁾_{l x (n - l)} is assumed to contain a non-zero element. Then the last l commodities are basic commodities. This ordering of commodities allows corn to be the last commodity in the order. A basic commodity enters either directly or indirectly into the production of all commodities.

The Leontief matrix Â⁽ⁱ⁾ is not decomposable, all elements are non-negative, and at least some elements are strictly positive. Then, by a theorem of Perron and Frobenius, the largest eigenvalue in modulus is real and non-negative and a corresponding eigenvector has all positive elements. This largest eigenvalue, λ_PF(Â⁽ⁱ⁾), is the Perron-Frobenius root of the Leontief matrix. Assume that the maximum eigenvalue for the submatrix of basic commodities exceeds the maximum eigenvalue for the submatrix for non-basic commodities, so to speak:

λ_PF(F⁽ⁱ⁾) < λ_PF(Â⁽ⁱ⁾)

Then the Perron-Frobenius root of the submatrix of basic commodities is the Perron-Frobenius root of the Leontief matrix A⁽ⁱ⁾. Furthermore, assume that this Perron-Frobenius root does not exceed unity:

0 < λ_PF(Â⁽ⁱ⁾) < 1

These assumptions guarantee that a positive level of operation of the industries producing basic commodities exists such that a surplus product remains after replacing the used-up means of production. A Leontief matrix with these properties is a Sraffa matrix (Kurz and Salvadori, 1995).

The maximum rate of profits is related to the Perron-Frobenius root of the Leontief matrix:

R_i = [1 - λ_PF(Â⁽ⁱ⁾)]/λ_PF(Â⁽ⁱ⁾) = [1/λ_PF(Â⁽ⁱ⁾)] - 1; i = 1, 2, ..., m

The assumption that a surplus product can be produced with each system including a corn-producing producing process implies that the maximum rate of profits is positive. Lands that can never enter into a system of production with a surplus product are not considered in this model.

This model of rent remains open. Net output and wages or the rate of profits are determined outside the model. Following Pasinetti (2007), one can say these variables are determined at the institutional level, not at the structural level characterized by this model. Neither the labor force nor the existing endowments of capital goods are constraints in this model. Employment and the amount of produced means of production are found by solving the model. Whether full employment is achieved or capacity is fully used is a realization problem, to be analyzed after solving the model.