Material bottlenecks

Part I: The definition of the village
Part II: Value and cost
Part III: The plow
Part IV: Material bottlenecks

Suppose that it takes a certain amount of wood to make a plow. And suppose further that wood is somewhat scarce in our village. The wood supply, then, is a material bottleneck to the manufacture of plows. In addition to the 90 hours it takes to make a plow, additional time is required to gather the wood. To get enough wood for one plow, someone has to walk for a day (~10 hours) to the forest, spend a day collecting wood, and then walk back to the village. Thus we must add the time it takes to collect the wood to the cost of a plow, making the cost of a plow ~120 hours (90 + 30).

Before the invention of the plow, locally available wood was sufficient for life-support needs (heating and cooking); the time spent collecting wood was trivial, so wood per se had little cost. Now, however, wood has value — almost as much value as a plow — so wood itself becomes a commodity. We see again that a material bottleneck just translates to a labor bottleneck: the labor necessary to procure (and possibly prepare) the required material.

Since the value of the plow is 900 pounds of food, it is still economically efficient to take 120 hours to make a plow: Since it is now economically efficient to gather wood in addition to making plows, the plow makes gathering wood a valuable activity, indeed just as valuable (up to the point where the villagers have enough wood to make plows) as making plows or growing food.

Any material bottleneck will show this behavior: Any material bottleneck makes supplying sufficient material just as valuable as making the plow.

Math: The upper limit on the labor equivalent is 1.1 pounds of food per hour — because food can be produced only by actually using a plow, not just by making one) — and the lower limit is 1.0 pounds per hour.

If it requires fewer than 818 (900 / 1.1) hours to make a plow, then the net efficiency [difference in productivity / (cost in hours * labor equivalent)] is greater than 1 for a labor equivalent of 1.1. If the cost is between 818 and 900, then the net efficiency is greater than 1 only if the labor equivalent is 1.0. Alternatively, if a plow costs between 818 and 900 hours, then the labor equivalent is equal to the value that will make the net efficiency of producing and using a plow to be exactly one.

In other words, if the cost of a plow is less than 818 hours, then the labor equivalent is 1.1 pounds of food per hour and price of the plow is the number of hours * 1.1 pounds of food. If the cost of the plow is between 818 and 900 hours, the price of the plow is 900 pounds, and the labor equivalent is 900 / cost. If the cost of a plow is greater than 900 hours, it doesn't make sense to make plows.

[The above is a first-order approximation; it doesn't fully take into account the cost of making a plow. With second-order effects, the price of a plow will approach 900 pounds of food up to 900 hours of labor, with a bend in the price curve near the 818 hour cost point. The difference is negligible at relatively low costs; the first-order approximation simplifies the algebra and gives us a close enough picture to examine the philosophical basis of economics.]