
Figure 1: A Partition of PriceWage Space for a TwoCommodity Reswitching Example 
1.0 Introduction
This post generalizes my approach in Vienneau (2005). In that article, I present a Linear Programming (LP)
problem for the firm. In the case of an economy that produces two commodities, one can present a graphical
display that clarifies how Sraffa's equations arise. The dual LP is important in this development. Here,
I show how that approach can work for a case in which rates of profits systematically vary among industries.
I was pleased that this approach works out for markup pricing. In a sense, this post derives
both a direct and an indirect approach for analyzing the choice of technique, in the context of
a model of markup pricing.
2.0 The Model
To begin with, consider a model of the production of N commodities from labor and these
commodities. This is a model with circulating capital and no joint production. Assume that managers
of firms know of U_{j} processes for producing the output of that industry.
Each process is defined by:
 a_{0, j}(u), u = 1, 2, ..., U_{j},
the personyears of labor needed to produce one unit of the jth commodity.
 a_{., j}(u) = [a_{1, j}(u) ..., a_{N, j}(u)]^{T}, the inputs of each commodity needed to produce one unit of the jth commodity.
Each process exhibits constant returns to scale (CRS), requires a year to complete, and use up all their inputs.
I also take a set of weights for industries, 1/s_{1}, ..., 1/s_{N}, as givens.
Let prices be p = [p_{1}, ..., p_{N}].
Also, let e = [e_{1}, ..., e_{N}]^{T} be the numeraire, so that:
p e = 1
I should have some assumptions on coefficients ensuring that the economy can be productive by a suitable
choice of technique.
I introduce some variables as abbreviations:
k_{j}(u) = p a_{., j}(u)
c_{j}(u) = p a_{., j}(u) + w a_{0, j}(u)
π_{j}(u) = p_{j}  c_{j}(u)
r_{j}(u) = π_{j}(u)/k_{j}(u)
2.1 The Firm's LP
The managers of a firm take the wage, w and prices p
as given. Let ω = [ω_{1}, ..., ω_{N}]^{T} be the firm's
inventory of each commodity at the start of the year. Let q_{j}(u) be
the quantity of the jth commodity that the firm produces with the uth process known for producing
that commodity. Let q_{N + 1} be the value of inventory not used for purchasing inputs into production.
Each year the managers of the firm choose how much to produce of each commodity and with which process so
as to maximize the weighted increment of value:
(1/s_{1})[π_{1}(1) q_{1}(1) + π_{1}(2) q_{1}(2) + ... + π_{1}(U_{1}) q_{1}(U_{1})]
+ (1/s_{2})[π_{2}(1) q_{2}(1) + ... + π_{2}(U_{2}) q_{2}(U_{2})]
...
+ (1/s_{N})[π_{N}(1) q_{N}(1) + ... + π_{N}(U_{N}) q_{N}(U_{N})]
Such that the firm can purchase all of the inputs into production needed at the beginning of the year:
k_{1}(1) q_{1}(1) + k_{1}(2) q_{1}(2) + ... + k_{1}(U_{1}) q_{1}(U_{1})
+ k_{2}(1) q_{2}(1) + k_{2}(2) q_{2}(2) + ... + k_{2}(U_{2}) q_{2}(U_{2})
...
+ k_{N}(1) q_{N}(1) + k_{N}(2) q_{N}(2) + ... + k_{N}(U_{N}) q_{N}(U_{N}) ≤ p ω
For all j, u:
q_{j}(u) ≥ 0
The weights formalize the concept that managers find some industries more desirable or easier to invest in than others.
It works out that an industry that managers are less willing to contest or expand production in has a larger
rate of profits, in the system of prices of production.
2.2 The Dual LP
The above LP has a dual problem. It is to choose r to minimize:
p ω r
Such that for all j, u:
p a_{., j}(u) (1 + rs_{j}) + w a_{0, j}(u) ≥ p_{j}
r ≥ 0
When a decision variable is positive in a solution to the primal LP, the corresponding constraint is met with equality in the dual LP.
Suppose the solution of the primal LP leads to each commodity being produced by a specific process in each industry.
The price system defined by the technique composed of those process will be satisfied. The economy will be on the wage
curve for that technique.
3.0 Solution of the Primal LP
The solution to the primal LP is illustrated by Table 1. In a solution, only basis variables are positive
The table specifies the value of each basis variable, when only it is positive in the solution,
and conditions that must hold for it to be in the basis.
The decision variable q_{N + 1} is a slack variable, introduced
to convert the inequality constraint in the primal LP into an equality.
It represents the value of inventory carried over, without supporting production.
The conditions for when a decision variable is in the basis are intuitive. Consider the first row.
A given commodity is produced with a given process only if the rate of profits made in other
processes producing that commodity do not exceed the rate of profits made in the given process. Furthermore,
the markedup rate of profits in producing other commodities must not exceed the markedup rate of profits
in the given process.
Finally, the (undiscounted) cost of producing a the given commodity must not exceed the revenue made from selling iron.
(I am aware that there is some redundancy in how I have stated conditions in the table.)
Table 2: Solution of Primal LP
Variable in Basis  Value  When Optimal 
q_{J}(V)  p ω/k_{J}(V)  For u = 1, 2, ...,U_{J} [p_{J}  w a_{0, J}(V)]/k_{J}(V) ≥ [p_{J}  w a_{0, J}(u)]/k_{J}(u) 
For all j, u (1/s_{J})r_{J}(V) ≥ (1/s_{j})r_{j}(u) 
c_{J}(V) ≤ p 
q_{N + 1}  p ω  For all j, u c_{j}(u) ≥ p 
The solution to the primal LP, in a twocommodity example, is easily visualized. The second commodity is the
numeraire, and the price of the first commodity is graphed on the ordinate.
Figure 1 partitions the space formed from the price of iron and the wage.
A single decision variable enters the basis inside each region in Figure 1.
Each region is labeled by that decision variable, in an obvious notation.
On the boundaries, a solution to the LP can be formed from a linear combination of decision variables.
In the example, both commodities must be produced for the economy to be selfsustaining.
Firms are willing to produce both only if prices lie along the heavy locus.
The figure shows that this is a reswitching example. One technique is adopted at low and high wages,
while the other technique is used at intermediate wages.
The figure also illustrates that the wage cannot exceed a maximum.
4.0 Conclusion
I have thought about how this LP approach generalizes. In a general joint production framework, it is not
immediately obviously how to assign processes to industries. So I do not see how to define the weights. I
suppose one could have a weight for each process, instead of for each industry.
Land presents another difficulty. One would like to impose additional constraints in the primal LP
to specify that overall production cannot require that more than a given quantity of some inputs
cannot be used in production. Then multiple processes would be used, in a model of extensive rent,
in certain industries. But should not such constraints be imposed above the level of the firm?
That is, if a firm's production meets the constraints, they might still be violated in the
economy as a whole.
But, I suppose, this LP approach applies to cases of fixed capital, where joint production is such
that firms in an industry can choose to operate multiple processes, each jointly with a machine
of a specific age.
Reference