Finding Rent By Solving A Linear Complementary Problem (LCP)

1.0 Introduction

This post is revisiting this example of extensive and intensive rent. I find writing up examples of rent difficult because they quickly get too complicated. This time, I'm thinking of confining myself to a small range of the rate of profits, where the order of fertility is opposite the order of rentability. This post is the first exposition I know of that explicitly sets out a LCP for rent. Maybe Bidard has something in French. (I should stick to GIFs, not JPEGs as below.)

2.0 Specification of Parameters as Matrices and Vectors

The data for this problem in the analysis of the choice of technique consist of technology, the endowments of lands of three types, and requirements for use. This data is specified by a number of matrices and vectors.

The technology consists of five production processes. Each process takes a year to complete and completely uses its inputs of produced commodities, iron and corn. Corn is an agricultural commodity, grown on a specified type of land. Each type of land comes out of a production process as good as it was when it went into it. Each process exhibits constant returns to scale (CRS), up to the limit imposed by the endowments of land.

Labor inputs are specified by the row vector, a₀, the vector of labor coefficients:

Each labor coefficient is the physical input of labor, measured in, say, person-years, required to operate the corresponding process at unit level. The inputs of iron and corn required to operate each process at unit level are specified by the input matrix A:

Each column of the input matrix is the physical inputs of iron and corn for a process. The columns of the output matrix, B, are the physical outputs of iron and corn for each process, when operated at a unit level:

The matrix of land coefficients, C, is the last matrix for specifying the technology.

Each column of the land coefficients specifies the services of the types of land, in acres, required to operate a process at unit level. Iron is produced without land services, while corn can only be produced on a type of land. The existence of two processes for producing corn introduces the possibility of intensive rent in a model that would otherwise be one of extensive rent.

Another parameter is the available endowments of land, expressed in the column vector t:

Endowments of each type of land are measured in acres.

Requirements for use, d is the final parameter specified in this section:

Requirements for use is also known as final demand. The first element of the above vector is the tons iron in final demand. The second element is the bushels corn in final demand. I am considering the case when both elements are numerically equal. To calculate the order of fertility, I need to consider the choice of technique as final output expands.

3.0 Conditions on Matrices

The above specifications of vectors and matrices is a numerical example of a more general model. Clearly, there can be any number of industrial commodities, types of land, and processes available on each type of land. Here I specify a set of sufficient conditions that the above example satisfies.

Direct labor is needed as an input to operate each process. All of the elements of the vector of labor coefficients are positive. Commodity inputs are assumed to be needed for each process. Thus, each column of A contains some non-zero entries. By assumption, no pure joint production, such as the production of mutton and wool, is possible. Each column of B contains exactly one positive entry. All the rest are zero. Some process produces each (non-land) commodity. Therefore, each row of B contains at least one non-zero entry.

The input and output matrices have a specific structure. Suppose the produced commodities consist of n - 1 industrial commodities and one agricultural commodity, called corn. More specifically, the output matrix has the following structure:

The subscripts represent the size of each submatrix. The upper left submatrix is the identity matrix. The upper right is a matrix of all zeros. The lower left is a row vector of zeros. And the lower right submatrix is a unit row vector. The first n - 1 processes produce the industrial commodities. The remaining processes produce corn.

The input matrix for land is assumed to have a certain structure too. Land is not needed as a direct input to produce the industrial commodities. The elements of the first n - 1 columns of C are all zero. Each process for producing corn requires an input of the services of one type of land. That is, each of the last m - n + 1columns of C contain exactly one non-zero element. Each type of land is used in at least one process for producing corn. Each row of C contains at least one non-zero element.

Each technique is associated with a solving subsystem (Quadrio Curzio & Pellizzari 2010), as defined by an n-element row vector of labor coefficients and a nxn square matrix of input coefficients:

A solving subsystem resembles the vector of direct labor coefficients and the input-output matrix for a model with circulating capital alone. The first (n ― 1) labor coefficients and columns in the solving subsystems are from the industrial processes specified by the technology. The last labor coefficient and last column are from a corn-producing proces. So far, I have only considered solving subsystems for extensive rent.

Suppose all produced commodities are basic commodities and that the technique represented by each of the matrices in solving subsystems is productive. A commodity is basic if it enters, directly or indirectly, into the production of all commodities. For a matrix representing a productive technology, there exists some level of operation of these processes such that a surplus product remains after the replacement of the commodity inputs.

What about solving subsystems for intensive rent? Consider two processes that produce corn on the same type of land. These would be the last two processes in the numeric example above. Sraffa (1960) shows how a linear combination of these two processes can eliminate the input of the land services. Let this linear process be the last column in a matrix where the first n - 1 columns for the input matrix are the processes for producing the industrial commodities. The corresponding vector of labor coefficients and this matrix constitute the solving subsystem for a technique that pays intensive rent.

Consider the special case in which all the elements of the solving subsystems for techniques that pay intensive rent are non-negative. Furthermore, as with the solving subsystems for extensive rent, all produced commodities are basic commodities, and all of these matrices are productive. I speculate that under these assumptions, a cost-minimizing technique, possibly combining intensive and extensive rent, exists at a given rate of profits, if a feasible technique exists. And I claim such a solution is unique, except at switch points and for a fluke case when a type of land is just fully farmed. The numeric example demonstrates that this special case is non-empty.

4.0 Block Structured Matrices

Following Erreygers (1995) and Kurz & Salvadori (1995), the choice of technique can be formulated in terms of certain block structured matrices. The new matrix of input coefficients is:

The new matrix of output coefficients is:

The new matrix of final demand is:

The new row vector of prices is:

The vector p is the n-element vector of prices of produced commodities, iron and corn in the example. The vector ρ is the k-element vector of rents for the different types of land.

5.0 The Choice of Technique

The problem of the choice of technique is to find a m-element column vector q of levels of operation of the processes and a (n + k)-element row vector of prices that satisfy a certain set of inequalities and equalities.

The quantity of each commodity must meet or exceed the requirements for use:

The above inequality includes the condition that the use of land services cannot exceed those available from endowments. Each process must be operated at a non-negative level:

The two inequalities complete the quantity system.

Two inequalities also specify the price system. The cost of no process falls below revenues. In other words, no extra profits are possible:

In this formulation, a unit of labor is the numeraire. All prices are non-negative:

Finally, the cost-minimizing technique must satisfy two duality conditions. The law of free goods holds. Any commodity in excess supply, including a type of land, has a price of zero:

The law of non-operated processes holds. Any process in which costs exceed revenues is not operated:

Given the rate of profits, a solution to this combination of equalities and inequalites is a cost-minimizing technique.

6.0 Mapping Block-Structured Matrices to a Linear Complementary Problem

I now consider a mapping of these block-structured matrices and vectors to a LCP. The top of one column vector is the excess supply of commodities and land:

The bottom of this column vector is the excess of costs over revenues for each process:

A vector is defined that contains both levels of operation of each process and the prices of each commodity:

A vector is defined for the right-hand side of an equation in the LCP:

The LCP can be specified in terms of these vectors and matrices.

7.0 The Choice of a Technique as a Linear Complementary Problem

The LCP is to find column vectors x and z such that:

All the elements of x and z are non-negative:

Elements of x are positive only when elements of z are zero and vice-versa:

The solution of the LCP specifies:

• Which processes are operated
• The levels of operation of these processes
• The prices of produced commodities
• The rents of land, including rents of zero for non-scarce land.

A large amount of theory exists for this problem in mathematical programming.

8.0 Conclusion

The above LCP differs from my previous exposition of a LCP for the choice of technique in that:

• It supports a theory of rent and is in terms of block matrices that include land.
• It does not allow for a positive steady-state rate of growth.

I know of solutions for my numeric example. The LCP, as I see it, allows one to find bounds on final demand and the rate of profits for which a solution is optimal. I do not know that I need iterate through an algorithm yet to solve thje LCP.