A Switch Point Along The Same Wage Curve With Multiple Agricultural Commodities

Figure 1: Wage Curves Around An Anomalous Switch Points

This post presents another anomalous switch point. A switch point is anomalous in that it has properties that cannot hold for a switch point in a model of single production, with inputs of labor and circulating capital alone.

This example is one with multiple agricultural commodities and intensive and extensive rent. The technology and the endowments of land are the same as in this example.

Required net output, that is, final demand, varies. I start by postulating that final demand consists of 28 bushels wheat and 28 bushels rye. Under this assumption, Alpha, Beta, Epsilon, and Lambda are feasible techniques.

The cost-minimizing technique at a given rate of profits must be:

• Feasible.
• Have non-negative prices for all commodities produced under the technique, have a non-negative wage, and have non-negative rents on all scarce lands.
• Such that no process not operated under the technique obtains extra profits.

Epsilon is cost-minimizing up to a rate of profits of approximately 223.6 percent. A reswitching of the order of efficiency occurs over the range at which Epsilon is cost-minimizing. After the switch point, as illustrated in Figure 1, Alpha is cost-minimizing.

Figure 1 also illustrates a fake switch point at a rate of profits of approximately 219.0 percent. The wage curves for Alpha and Delta intersect at the fake switch point. The wage curve for Alpha is also the wage curves for Epsilon and Zeta. Likewise, the wage curve for Delta is also the wage curves for Eta and Theta. Epsilon is the unique cost-minimizing technique at and around this fake switch point. The prices of produced commodities (iron, wheat, and rye) differ, at the switch point, between the techniques for the two intersecting wage curves. In this sense, the fake switch point resembles the one in the example from Bidard and Klimovsky. The rent on type 2 land is positive under Epsilon, which would not be the case is this switch point was non-fake. Nor are the rents on type 1 land zero under Eta and Theta at this fake.

But consider again the switch point between Epsilon and Alpha. Under Alpha, only type 1 land is farmed, but only partially. Epsilon extends Alpha to produce wheat on type 2 land, to the extent of its endoment. The switch point lies along a single wage curve, which is anomalous.

Suppose that final demand was small enough that both Alpha and Gamma were feasible. For example, Alpha, Beta, Gamma, and Delta are the only feasible techniques when required net output consists of 10 bushels wheat and 10 bushels rye. Then Gamma is cost-minimizing from before the switch point, from approximately 176.8 percent. Alpha is cost-minimizing after the switch point. As with Epslion, under Gamma wheat is produced on type 2 land. But, unlike Epsilon, type 2 land is not farmed under Gamma to the extent of its endowment and the process in which wheat is produced on type 1 land is no longer operated. With this final demand, the example is one of reswitching between Gamma and Delta, at a lower rate of profits than shown.

Or suppose final demand consisted of 30 bushels wheat and 30 bushels rye. Then Beta, Epsilon, Iota, Kappa, and Lambda are feasible. Then this is a switch point between Epsilon and Iota. The same processes are operated under Epsilon and Iota, but which land is scarce varies. Iota is a technique in which landlords obtain intensive rent on type 1 land. The rent on type 1 land is negative under Iota before the switch point.

I have now found switch points:

• Along a single wage curve, with no intersecting wage curve; with variation in which processes are operated between the cost-minimizing techniques.
• In which wage curves intersect, but no variation occurs in which processes are operated between the cost-minimizing techniques.

The above switch point between Epsilon and Alpha combines these two phenomena, in some sense. I have also found fake switch points:

• In which wage curves intersect; the cost-minimizing technique does not vary.

These results suggest that concept of a switch point is not tightly tied to intersections of wage curves in models of joint production.