
Figure 1: A Pattern Diagram with Joint Production 
1.0 Introduction
This post completes an example. I analyzed bits of this example
here and
here.
This post may make no sense if you have not read a long series of previous posts
or, maybe, the papers highlighted
here
and here.
I am interested in how and if my approach to analyzing and visualizing variations in the choice
of technique with technical progress extends to joint production. The example suggests fake switch points
do not pose an insurmountable obstacle for such an extension.
2.0 Technology
I repeat the specification of technology.
I postulate an economy in which two commodities, corn and linen, can be produced from inputs of corn, linen, and labor. Managers
of firms know of three processes (Tables 1 and 2) to produce corn and linen. Each process produces net outputs of corn and
linen as a joint product. Inputs and outputs are specified in physical units (say, bushels and square meters) per unit
level of operation of the given process. Inputs are acquired at the start of the year, and outputs are available
for sale at the end of the year.
Table 1: Inputs for The Technology
Input  Process 
(a)  (b)  (c) 
Labor  e^{σ0,1(1  t)}  e^{σ0,2(1  t)}  e^{σ0,3(1  t)} 
Corn  20  20  30 
Linen  20  20  30 
Table 2: Outputs for The Technology
Output  Process 
(a)  (b)  (c) 
Corn  21  23  36 
Linen  27  25  34 
I assume that requirements for use are such that two processes must be operated to satisfy those
requirements. I need to investigate the implications of this assumption further. Apparently, for this example,
it implies that the economy is not on a golden rule steady state growth path, with the rate of profits equal
to the rate of growth. Anyway, with this assumption, three techniques  Alpha, Beta, and Gamma  can be operated.
Table 3 specifies which processes are operated for each technique.
Table 3: Techniques
Techniques  Processes 
Alpha  a, b 
Beta  a, c 
Gamma  b, c 
The technology, as I have defined it, is parameterized. I consider the following specification for
the rate of decrease in labor coefficients.
σ_{0,1} = 2
σ_{0,2} = σ_{0,3} = 5/2
Bidard & Klimovsky's example arises when t is unity.
3.0 Prices and the Choice of Technique
A system of two price equations arises, for each technique. I assume the labor coefficient is treated as a constant
over the period of production  say, a year. With linen as numeraire, these equations for the Alpha technique are:
(20 p_{1} + 20)(1 + r) + [e^{σ0,1(1  t)}] w = 21 p_{1} + 27
(20 p_{1} + 20)(1 + r) + [e^{σ0,2(1  t)}] w = 23 p_{1} + 25
One can these equations for two variables in terms of, say, the rate of profits. For each technique, its wage curve
shows the wage as a function of the rate of profits. One cannot generally base the choice of technique, under joint
production, on figuring out which technique contributes to the outer frontier at a given rate of profits.
Instead, one can calculate profits and losses, with the given rate of profits and a technique's price system
for the processes not in the technique. This exercise only makes sense when the rate of profits, the wage,
and prices are nonnegative for the starting technique.
The technique is costminimizing only if no extra profits can be made with
processes outside the technique.
I deliberately frame this as a combinatorial argument.
Bidard likes what he calls a market algorithm, where, when one identifies a process earning extra profits,
one introduces the process into the technique. In the case of joint production, it is not clear
which process should be dropped. Furthermore, examples exist in which a costminimizing technique
exists but cannot be reached from certain starting points with the market algorithm.
4.0 Patterns
I have constructed the figure at the top of the post to illustrate how the choice of technique
varies with technical progress in this example. The dashed lines highlight features of the
example that do not bear on the choice of technique. The light vertical solid lines
divide time into numbered regions. Table 3 lists the costminimizing techniques, in
order of an increasing rate of profits in each region.
Table 3: Regions
Regions  Techniques 
1  Gamma, No Production, Alpha 
2  Gamma, No Production, Alpha 
3  Gamma, Alpha & Gamma, Alpha 
4  Alpha & Gamma, Alpha 
5  Beta, Alpha & Gamma, Alpha 
I could say a lot more about the example. I will note that in region 1, the wage increases
with the rate of profits, for the Alpha technique, in the interval for the rate of profits where both
wages and the price of corn are positive. In region 2, the wage decreases with the rate of profits, for
the Alpha technique. The division between regions 2 and 3 is associated with that interval for the
rate of profits for Alpha transitioning to have a nonempty intersection with the similar
interval for the Gamma technique.
for
5.0 Conclusion
This post has illustrated that one type of my types of pattern diagrams can apply to joint production. This type
illustrates how the relationship between the choice of technique and distribution varies with
technical progress. It can be constructed even in cases, such as joint production, where the choice of
technique cannot necessarily be based on wagerate of profits curves and their outer frontier.
If fake switch points are not shown, this type of pattern diagram does not depend on the
specification of the numeraire. If the ordinate in Figure 1 were the wage, instead of the
rate of profits, it would be upside down, in some sense. A different numeraire would rescale the
wage. When corn is numeraire, only one fake switch point exists. It, too, would be a horizontal
line segment. But fake switch points are fake precisely because they do not impact the choice
of technique. They can be left off the diagram.
The example also illustrates new types of patterns for dividing adjacent regions. Under joint
production, a technique can be associated with nonnegative prices and a wage for an
interval of the rate of profits that does not include a rate of profits of zero. Both the Alpha
and the Beta technique exhibit this possibility in the example. And we can divide regions
based on when the range of rate of profits in which such a technique becomes costminimizing
comes to include zero
or begins to interact with the range in which another technique is costminimizing
This example also illustrates that the costminimizing technique may not be unique in a
range of rates of profits. I think this nonuniqueness is qualitatively different than
how nonuniqueness can arise in models with only circulating capital. In circulating
capital models, nonuniqueness is associated with two techniques having identical
wage curves. Not so here.
I do not intend to write this example up any more extensively. I have no socalled
paradoxical behavior here, such as reswitching, reverse capitaldeepening, or the
reverse substitution of labor. I may go on to explore where techniques are described
by rectangular matrices, with more produced commodities than processes, and there is
a dependence on the requirements for use.
References
 Bidard, Christian and Edith Klimovsky (2004). Switches and fake switches in methods of production. Cambridge Journal of Economics. 28 (1): 8997.