A Linear Program for Markup Pricing

Figure 1: A Partition of Price-Wage Space for a Two-Commodity 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 person-years 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/s₁, ..., 1/s_N, as givens. Let prices be p = [p₁, ..., p_N]. Also, let e = [e₁, ..., 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 = [ω₁, ..., 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/s₁)[π₁(1) q₁(1) + π₁(2) q₁(2) + ... + π₁(U₁) q₁(U₁)]

+ (1/s₂)[π₂(1) q₂(1) + ... + π₂(U₂) q₂(U₂)]

...

+ (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:

k₁(1) q₁(1) + k₁(2) q₁(2) + ... + k₁(U₁) q₁(U₁)

+ k₂(1) q₂(1) + k₂(2) q₂(2) + ... + k₂(U₂) q₂(U₂)

...

+ 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 marked-up rate of profits in producing other commodities must not exceed the marked-up 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
qJ(V)	p ω/kJ(V)	For u = 1, 2, ...,UJ [pJ - w a0, J(V)]/kJ(V) ≥ [pJ - w a0, J(u)]/kJ(u)
		For all j, u (1/sJ)rJ(V) ≥ (1/sj)rj(u)
		cJ(V) ≤ p
qN + 1	p ω	For all j, u cj(u) ≥ p

The solution to the primal LP, in a two-commodity 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 self-sustaining. 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

• Robert L. Vienneau (2005). On Labour Demand and Equilibria of the Firm. The Manchester School 73 (5): 612-619.