## Tuesday, March 26, 2024

### Perverse Switch Point For Austrian Economics

 Figure 1: The Wage-Rate of Profits Frontier

This post continues a series of posts demonstrating that the change in the economic life of a machine at a switch point is independent of the change of the capital intensity of the technique at a switch point. I here fill in the lower left in a a two-by-two table.

The wage curves above are for the an example with the same structure as in the previous post in this series.This is a 'one-good' model. The manager of firms know three processes to produce a widget, also known as a machine. In the first process, labor and a new widget are used to produce new widgets and a one-year old widget. In the second process, labor and a one-year old widget are used to produce new widgets and a two-year old widget. In the last process, labor and a two-year old widget are used to produce new widgets. The output coefficients, b1,2 and b1,3, specify how many new widgets are produced by the second and third widgets. New widgets are the only commodities that can be consumed. The numeraire is a new widget.

The choice of technique arises because managers of firms can choose different economic lifetimes for the machine. Free disposal is assumed.

One can create a system of equations for the quantity flows for each technique. The net output is one new widget. The solution shows at what level each process is operated. It also shows how many widgets of each age are advanced as capital goods and how much labor is employed, per net output.

One can also create a system of equations for prices of production for each technique. Wages are paid out of the surplus, and the same rate of profits is obtained in operating each process. Figure 1 shows the wage curves for the example, for the specified parameters, b1,2 and b1,3. The cost-minimizing technique is on the outer frontier.

Around the single switch point, the value of capital per worker is higher at a lower rate of profits. This is a negative real Wicksell effect, a formalization of the notion of capital deepening. A greater output per worker is associated with a lower rate of profits. So, as far as real Wicksell effects go and unlike the previous one, this example conforms to obsolete marginalist teaching.

But consider the economic life of a machine. Around the switch point, a lower rate of profits is associated with a shorter economic life of a machine. And operating the machine for a shorter time results in a greater net output per worker. A choice of a shorter life of a machine can be associated with either more or less output per worker.

The machine operates at non-constant efficiency in the example. Whether the efficiency is increasing or decreasing over various years cannot be defined, in general, in models with multiple inputs for the machine. Furthermore, the change in the properties of the analysis of the choice of technique do not need to arise from perturbing parameters for processes in which the machine is operated. These changes can arise from perturbing other processes in multicommodity models.

 Figure 2: A Parameter Space for the Example

Perturbing the parameters b1,2 and b1,3 cannot create an example with a switch point falling into the upper right of my two-by-two table (Figure 2). Thus, I need to consider either a model with a different structure or perturbing additional parameters in ths one-good model.