Thursday, August 31, 2017

A Fluke Switch Point With A Real Wicksell Effect Of Zero

Figure 1: A Fluke Switch Point
1.0 Introduction

A switch point in which the wage curves for two techniques are tangent to one another at the switch point is a fluke. Likewise, a switch point that occurs at a rate of profits of zero is a fluke. This post presents a two-commodity example with a choice between two techniques, in which the single switch point is simultaneously both types of flukes. The wage curves are tangent at the switch point, and the switch point occurs at a rate of profits of zero.

2.0 Technology

Consider the technology illustrated in Table 1. The managers of firms know of three processes of production. These processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. Two processes are known for producing corn, and their coefficients of production are specified in the last two columns in the table. Each process is assumed to require a year to complete and uses up all of its commodity inputs. (The coefficients were found by first creating an example with two wage curves tangent at a switch point. Selected coefficients were then varied to move the switch point to the wage axis. A binary search improved the approximation. Octave code was useful.)

Table 1: The Technology

This technology presents a problem of the choice of technique. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Similarly, the Beta technique consists of the iron-producing process and the corn-producing process labeled Beta.

3.0 Quantity Flows

Quantity flows can be analyzed independently of prices. Suppose the economy is in a self-replacing state, with a net output consisting only of corn. Table 2 displays (approximate) quantity flows for the Alpha technique, when the net output consists of a bushel of corn. The last row shows gross outputs, for each industry. The entries in the three previous rows are found by scaling the coefficients of production in Table 1 for Alpha by these gross outputs. The row for iron shows that each year, the sum 0.02848 + 0.3480 = 0.6328 tons are used as inputs in the iron and corn industries. These inputs are replaced at the end of the year by the output of the iron industry, with no surplus iron left over. Similarly, the output of the corn industry replaces the inputs of corn for the two industries, leaving a net output of one bushel corn.

Table 2: Quantity Flows for Alpha Technique

Table 3 shows corresponding quantity flows for the Beta technique. As above, the net output is one bushel corn. These tables allow one to calculate, for each technique, the labor aggregated over all industries per net unit output of the corn industry. Likewise, one can find the aggregate physical quantities of capital goods per net unit output of corn.

Table 3: Quantity Flows for Beta Technique

4.0 Prices

The choice of technique is analyzed based on prices of production and cost-minimization. Labor is assumed to be advanced, and wages are paid out of the surplus product at the end of the year. Corn is taken as the numeraire. Figure 1 graphs the wage-rate of profits for the two techniques. The cost-minimizing technique, at a given rate of profits, maximizes the wage. That is, the cost-minimizing techniques form the outer envelope, also known as, the wage frontier, from the wage curves. The Beta technique is cost-minimizing at all feasible rates of profits. At the switch point, the Alpha technique is also cost-minimizing. Furthermore, at the switch point, any linear combination of the techniques is cost-minimizing.

In calculating wage curves, one can also find prices for each rate of profits. Table 4 shows certain aggregates, as obtained from Tables 2 and 3 and the price of iron at the switch point.

Table 4: Aggregates at the Switch Point
Net Output1 Bushel Corn
Labor1.18036 Person-Years
Physical Capital0.06328 Tons
0.39217 Bushels
0.00525 Tons,
0.90330 Bushels
Financial Capital0.94957 Bushels

A certain sort of indeterminancy arises in the example. For a given quantity of corn produced net, the ratio of labor employed in corn production to labor employed in iron production varies at the switch point from approximately 17.7 to 223.7. A change in technique leaves total employment unchanged, given net output, even as it alters the allocation of labor between industries. It is also the case that, at the switch point, a change in technique, given net output, leaves the total value of capital unchanged, while, once again, altering its allocation between industries.

For non-fluke switch points, aggregate employment and the aggregate value of capital, per unit net output, vary with the technique. If the technique that is cost minimizing at an infinitesimally greater rate of profits than associated with the switch point has a greater value of capital per net output at the switch point, the real Wicksell effect is positive. If that technique has a smaller value of capital per net output, still using the prices at the switch point to value capital goods, is negative. (Edwin Burmeister argues that a negative real Wicksell effect is the appropriate formalization of the neoclassical idea of capital-deepening.) The fluke switch point presented here has a zero real Wicksell effect.

The indeterminacy at the switch point is related to both fluke properties of the switch point. Net output per worker, for a given technique, is shown by the intersection of the wage curve for the technique with the wage axis. Since both curves intersect the wage axis at the same point, they produce the same net output per worker. Thus, both techniques result in the same overall employment, per bushel corn produced net.

The wage curve also shows the value of capital per worker. For a given technique and rate of profits, the numeraire value of capital per person-year is the absolute value of the slope of the secant connecting the point on the wage curve specified by the rate of profits and the intercept with the wage axis. In the limit, when the rate of profits is zero, the value of capital per person-year is the absolute value of the slope of the tangent. The tangency of the wage curves at the switch point on the wage axis implies that both techniques have the same value of capital per person-year.

Update (10 Sept. 2017): Fixed transcription error in coefficients of production.


Sturai said...

I would ask a question without thinking to much about it. Do you need n≤3 commodities in order to have your bifurcations? Or could you reproduce them with n=2 and link them with the analysis on the r1=sr2?

Robert Vienneau said...

As I understand it, the bifurcations I have been exploring can exist in two commodity models and in models with any higher number of commodities. My idea is that for bigger models, one can tell a story of how technical progress changes the wage-curve frontier as formed by many such bifurcations.

Part of my point is to draw interesting graphs, even if I am not clear in explaining them.

I also want to illustrate that one can say something, without utility maximization or supply and demand, in open models.