1.0 IntroductionI think the analysis of the choice of technique in a steady state is a settled question. (The meaning of Sraffa's equations in
wider contexts can be debated.) One strength of the analysis of the choice of technique is the existence of several methods of analysis, all reaching the same conclusion. If one wanted to overthrow this analysis, one would need to show that one is not attacking just one such method, but all of them  or at least as many as possible. This post illustrates this strength of the analysis by presenting three such methods.
2.0 Example TechnologyI need an example technology (Table 1) to use in stepping through different methods for analyzing the choice of technique. Each process requires the inputs shown to be purchased at the start of the production period (a year) for each unit of output produced and available at the end of the year. Two processes are known for producing steel, and two other processes are likewise known for producing corn. The coefficients are fairly arbitrary. In this example, to produce any net output in a steady state, all commodities  that is, both steel and corn  must be produced.
Table 1: ConstantReturnstoScale (CRS) Production ProcessesInputs  Industry Sector 
Steel Industry  Corn Industry 
First SteelProducing Process  Second SteelProducing Process  First CornProducing Process  Second CornProducing Process 
Labor (PersonYrs)  3220/3321  13930/63099  3115/3321  1 
Steel (Tons)  0  0  1/2  9/20 
Corn (Bushels)  1/18  2752/7011  0  0 
Output  1 Ton  1 Ton  1 Bushel  1 Bushel 
The analysis of the choice of technique calculates which production process would be adopted for each combination of prices and interest rates. For this technology to be compatible with a steady state, at least one process for producing steel and one process for producing corn must be adopted. A "technique" consists of one process from each of the industries in this example. Table 2 defines the four techniques, each named with a greek letter. (I think this convention of using greek letters in this context may have been introduced by Joan Robinson.)
Table 2: Techniques and Production Processes
Technique  SteelProducing Process  CornProducing Process 
Alpha  First  First 
Beta  First  Second 
Gamma  Second  First 
Delta  Second  Second 
A technique, in this case, is expressed by a 2element row vector of direct labor coefficients and a square Leontief InputOutput matrix. For example, the labor coefficients,
a_{0}^{α}, for the first technique are:
a_{0}^{α} = [(3220/3321) (3115/3321)]
The Leontief InputOutput matrix,
A^{α}, for the first technique can be expressed as two columns
a_{.1}^{α} and
a_{.2}^{α}:
A^{α} = [ a_{.1}^{α} a_{.2}^{α}]
The first labor coefficient and the first column in the Leontief InputOutput matrix come from the specified production process from the steel industry for that technique:
a_{.1}^{α} = a_{.1}^{β} = [0, (1/18)]^{T}
The second labor coefficient and the second column in the Leontief InputOutput matrix come from the specified production process from the corn industry for that technique:
a_{.2}^{α} = a_{.2}^{γ} = [(1/2), 0]^{T}
I leave to the reader how to completely specify
a_{0}^{β},
A^{β},
a_{0}^{γ},
A^{γ},
a_{0}^{δ}, and
A^{δ}.
3.0 Direct MethodHeinz D. Kurz and Neri Salvadori refer to this method for analyzing the choice of technique I describe here as the "Direct Method". Before proceeding, I need to introduce some notation. Let
p be a twoelement row vector of prices:
p = [p_{1}, p_{2}]
where
p_{1} is the price of a ton steel and
p_{2} is price of a bushel corn. Let
w be the wage, assumed to be paid at the end of the year for each personyear of labor expended during the year. Let
r be the rate of interest, also called the rate of profits.
I need to introduce a column vector to represent the numeraire. Let
e_{2} be the second column of the 2x2 identity matrix:
e_{2} = [0, 1]^{T}
The assumption that
e_{2} is the numeraire implies the following equation:
p e_{2} = 1
This specification of the numeraite implies that,
p_{2}, the price of a bushel corn is unity.
The problem is to find a pair (
p,
w), given the interest rate
r, such that
 No process can be operated with costs less than revenues.
 For any process that is operated, the costs do not exceed the revenues.
The first condition implies the following four inequalities must hold:
p a_{.1}^{α}(1 + r) + a_{01}^{α} w ≥ p_{1}
p a_{.1}^{γ}(1 + r) + a_{01}^{γ} w ≥ p_{1}
p a_{.2}^{α}(1 + r) + a_{02}^{α} w ≥ p_{2}
p a_{.2}^{β}(1 + r) + a_{02}^{β} w ≥ p_{2}
The conjunction of the requirement that steel be produced with the second condition implies that one of the first two inequalities must be met with a strict equality. The analogous requirement for corn production implies that at least one of the last two inequalities must be met with equality.
These specifications are easily graphed (Figure 1). Given the interest rate, the first two inequalities yield upwardsloping lines in the the figure. The last two inequalities yield the downwardsloping lines. The first condition implies the solution must lie on or above all of the lines in the figure. The second condition implies that the solution must lie on
 At least one of the upwardsloping lines
 At least one of the downwardsloping lines.
The only point that satisfies these conditions is graphed. It lies on the upwardsloping line corresponding to the first steelproducing process and the downwardsloping line corresponding to the second cornproducing process. Thus, this analysis shows that the beta technique is costminimizing at an interest rate of 100%. The solution wage and price of steel can be read off the figure.

Figure 1: Direct Method Illustrated At r = 100% 
The direct method is easily generalized to any finite number of techniques. Each additional production process results in an additional line in the figure, upwardsloping for the steel industry and downwardsloping for the corn industry. The method also generalizes for any finite number of commodities. Each additional commodity results in the introduction of another dimension to the figure. Although the figure quickly becomes unvisualizable, the mathematics generalizes.
4.0 Indirect MethodThe indirect method generalizes to cases in which an uncountably infinite number of techniques are available. It is based on constructing the wagerate of profits frontier as the outer envelope of the wagerate of profits curve for each technique (Figure 2). I illustrate how to construct the wagerate of profits curve for the Alpha technique.

Figure 2: Indirect Method Illustrated 
The condition that the same rate of profits be earned for each process comprising a technique yields a system of two equations:
p A^{α}(1 + r) + a_{0}^{α} w = p
One also has the equation setting the price of the numeraire to unity:
p e_{2} = 1
For a given interest rate
r, the above is a linear system of three equations for three variables (
p_{1},
p_{2}, and
w). One can solve the system to express each of these three variables as a function of the interest rate. The wage, for example, can be found as:
w = 1/(a_{0}^{α} [ I  (1 + r)A^{α}]^{1}e_{2})
One knows, from a theorem due to Perron and Frobenius, that the inverse exists between an interest rate of zero and some maximum interest rate.
Figure 2 shows the wagerate of profits curves for each of the four techniques. The costminimizing technique at each rate of interest is the one with the highest wage. Points at which the raterate of profits curves for two or more techniques interesect on the outer frontier are known as switch points. The two switch points are shown in the example. The Gamma technique is costminimizing for a very low interest rate. For a somewhat larger interest rate, the Delta technique is costminimizing. Finally, the Beta technique is cost minimizing for larger interest rates. My exposition illustrates that the direct and indirect methods give the same conclusion. For example, the wagerate of profits frontier shows that the Beta technique is costminimizing for an interest rate of 100%.
5.0 Cost Minimization AlgorithmThis method I take from J. E. Woods. He provides an algorithm for finding the costminimizing technique(s), given the interest rate.
 Pick an initial technique. (For illustration, I start with the Beta technique in the example.)
 Solve the equations specifying the wagerate of profits curve for the selected technique. So you now have a price vector p and the wage w.
 Using p and w, calculate the cost of producing a ton steel with each of the known production processes (Figure 3).
 If the steelproducing process in the selected technique is cheapest, go to Step 6. Otherwise go to Step 5. (In the example, one would go to Step 5 for low interest rates and to Step 5 for higher interest rates.
 Replace the steelproducing process in the technique analysed in Step 2 with the cheapest steelproducing process identified in Step 4. Solve the equations specifying the wagerate of profits curve for the newly selected technique. Use the resulting p and w in Step 6. (In the example, one would calculate the wagerate of profits curve for the Delta technique for a sufficiently low interest rate.)
 Using the specified p and w, calculate the cost of producing a bushel corn with each of the known production processes (Figure 4).
 If cost of producing corn could be found in Step 6 for the technique selected in Step 2 and the cornproducing process in that technique is cheapest, then stop. You have identified the costminimizing technique. Otherwise, replace the cornproducing process in the technique analyzed in Step 6 with the cheapest cornproducing process identified in Step 6. (In the example, the algorithm would terminate in onepass for a sufficiently high interest rate, with the Beta technique identified as the costminimizing technique.)
 Go to Step 2.

Figure 3: Costs of Producing Steel with Prices for the Beta Technique 

Figure 4: Costs of Producing Corn with Prices for the Beta Technique 
Figures 3 and 4 suggest that for a sufficiently low interest rate, the technique consisting of the second steelproducing process and the first cornproducing process, that is, the Gamma technique, is costminimizing. For a somewhat higher interest rate, the technique consisting of the second steelproducing process and the second cornproducing process, that is, the Delta technique, seems to be costminimizing. And, as noted above, the algorithm terminates with the Beta technique identified as the costminimizing technique at an even higher interest rate. In other words, the graphs suggest that the above algorithm converges to the same solution as the indirect method.
6.0 ConclusionsI have not exhausted the methods available for analyzing the choice of technique. For example, I have not formulated any Linear Programs above. Nor have I presented the diagram in my 2005
Manchester School paper. Furthermore, I have glossed over many interesting mathematical questions, such as proving the existence of solutions and proving that all methods give the same result. But this post is already too long.
References Heinz D. Kurz and Neri Salvadori (1995) Theory of Production: A LongPeriod Analysis. Cambridge University Press.
 J. E. Woods (1990) The Production of Commodities: An Introduction to Sraffa, Humanities Press.