Figure 1: A Bifurcation Diagram |

This post presents another example of bifurcation analysis applied to structural economic dynamics with a choice of technique. This example illustrates:

- Two reswitching examples appear and disappear without a reswitching bifurcation ever occurring, at least on the wage frontier.
- Two bifurcations over the wage axis arise. At the time each bifurcation of this type occurs, another switch point for the same techniques exhibits a real Wicksell effect of zero. Thus, for each, a switch point transitions from being a "normal" switch point to a "perverse" one exhibiting capital-reversing.
- Each of the four types of bifurcations of co-dimension one that I have identified have no preferred temporal order. For example, a bifurcation over the wage axis can add a switch point to the wage frontier. And another such bifurcation can remove a switch point, as time advances.
- The maximum rate of profits approaches an asymptote from below as time increase without bound.

Table 1 specifies the technology for this example, in terms of two parameters, σ and φ. Managers of firms know of one process for producing iron and of three processes for producing corn. Each process is defined in terms of coefficients of production, which specify the quantities of labor, iron, and corn needed to produce a unit output for that process. All processes exhibit constant returns to scale; require a year to complete; and totally use up, in producing their output, the capital goods required as input. I consider the special case in which the rate of decrease of the coefficients of production in the Beta corn-producing process, σ, is 5 percent, and the rate of decrease of coefficients in the Gamma corn-producing process, φ, is 10 percent.

Input | IronIndustry | Corn Industry | ||

Alpha | Beta | Gamma | ||

Labor | 1 | 0.89965 | 0.71733 e^{-σ t} | 1.28237 e^{-φ t} |

Iron | 0.45 | 0.025 | 0.00176 e^{-σ t} | 0.03375 e^{-φ t} |

Corn | 2 | 0.1 | 0.53858 e^{-σ t} | 0.13499 e^{-φ t} |

Three techniques are available for producing a net output of, say, corn, while reproducing the capital goods used as input. The Alpha process consists of the iron-producing process and the corn-producing process labeled Alpha. And so on for the Beta and Gamma techniques.

The choice of technique is analyzed in the usual way. I assume that labor is advanced, and wages are paid out of the surplus product at the end of the year. Corn is taken as numeraire. A wage curve can be drawn for each technique, given the coefficients of production prevailing at a given moment in time. Figure 1 illustrates a case of the recurrence of techniques in the example. The cost-minimizing technique is found by constructing the outer frontier of the wage curves. In Figure 2, the cost-minimizing techniques are Beta, Alpha, Gamma, and Beta, in that order. The switch point at approximately 57 percent exhibits capital-reversing. Around the switch point, a higher wage is associated with the adoption of a more labor-intensive technique. If prices of production prevail, firms will find it cost-minimizing to hire more workers at a higher wage, given net output.

Figure 2: Wage Curves in Region 4 |

Figure 3 illustrates the analysis of the choice of technique for all time. Switch points along the frontier and the maximum rate of profits are plotted versus time. Figure 1, at the top of this post, is a blowup of Figure 3 from time zero to a time of five years. These pictures show which technique is cost-minimizing at each rate of profits, at each moment in time. Bifurcations are also shown. Table 2 lists the cost-minimizing techniques in each region between the bifurcations.

Figure 3: An Extended Bifurcation Diagram |

Region | Cost-MinimizingTechniques | Notes |

1 | Alpha | One technique cost-minimizing. |

2 | Alpha, Beta | "Normal" switch point. |

3 | Beta, Alpha, Beta | Reswitching. Switch pt. at highest r is "perverse". |

4 | Beta, Alpha, Gamma, Beta | Recurrence of techniques. Switch pt. at highest r is "perverse". |

5 | Beta, Gamma, Beta | Reswitching. Switch pt. at highest r is "perverse". |

6 | Gamma, Beta | "Normal" switch point. |

7 | Gamma | One technique cost-minimizing. Maximum r approaches anasymptote. |

I suppose I can extend this example to partition the complete parameter space, as in this example, with an updated write-up here. That analysis will demonstrate, by example, that this sort of bifurcation analysis applies to cases in which multiple commodities are basic in multiple techniques. It is not confined to the special case of the Samuelson-Garegnani model. I am also thinking that I could perform a bifurcation analysis where parameters that vary include the ratio of the rates of profits in various industries, as in these examples of a model of oligopoly. Maybe such an analysis will yield an empirically relevant tale of the evolution of economic duality (also known as segmented markets).

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