This post continues the treatment of this model.
A number of conditions must be satisfied by quantity flows and prices to allow the economy to undergo smooth reproduction. The net output of each produced commodity meets the requirements for use:
(B - A) q = d
The amount of each quality of land that is farmed cannot exceed the available quantity:
C q ≤ t
A vector is greater than or equal to another if each element of the vector is greater or equal to the corresponding element of the other vector. The level of operation of each process is non-negative:
qi, i = 1, 2, ..., n + m - 1
The costs of each process cannot fall below the revenues for that process:
p A (1 + r) + ρ C + w a0 ≥ p B
In other words, no process returns supernormal profits. The required net output is the numeraire:
p d = 1
The price of each produced commodity is non-negative:
pj ≥ 0; j = 1, 2, ..., n
The rent on each type of land is nonnegative:
ρj ≥ 0; j = 1, 2, ..., k
The specification of prices of production includes a duality condition, the rule of non-operated processes. The following display shows the application of this rule to this model:
[p B - p A (1 + r) - ρ C - w a0] q = 0
According to the fourth display above, the quantity in parenthesis above is non-positive. Thus, if costs exceed revenues in a process, that process is not operated.
Consider a non-trivial solution to the price and quantity systems, with a given rate of profits, that also satisfies the rule of non-operated processes. The first three equations specify the quantity system. The next four equations specify the price system. Such a solution consists of a cost-minimizing technique, the wage, and prices of production. The prices of production include the rents on each type of land. Kurz and Salvadori (1995) prove an existence theorem for a more general model of rent that encompasses this model.
The next step in my development of this model would be to find some interesting numerical examples. I do not expect to be fast about doing this.
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