# Orthogonal Time in Euclidean 3-space

## Abstract

This paper begins by asking a simple question: can a farmer own and fully utilize precisely five tractors and precisely six tractors at the same time? Of course not. He can own five or he can own six – but he cannot own five and six at the same. The answer to this simple question eventually led this author to Alfred Marshall’s historically-ignored, linguistically-depicted ‘cardboard model’ where my goal was to construct a picture based on his written words. More precisely, in this paper our overall goal is to convert Marshall’s (‘three-dimensional’) words into a three-dimensional picture so that the full import of his insight can be appreciated by all readers.

After a brief digression necessary to introduce the reader to the intricacies of Euclidean 3-space, plus a brief digression to illustrate the pictorial problem with extant theory, the paper turns to Marshall’s historically-ignored words. Specifically, it slowly constructs a visual depiction of Marshall’s ‘cardboard model’. Unfortunately (for all purveyors of extant economic theory), this visual depiction suddenly opens the door to all manner of Copernican heresy. For example, it suddenly becomes obvious that we can join the lowest points on a firm’s series of SRAC curves and thereby form its LRAC curve; it suddenly becomes obvious that the firm’s series of SRAC curves only appear to intersect because mainstream theory has naively forced our three-dimensional economic reality into a two-dimensional economic sketch; and it suddenly becomes obvious that a two-dimensional sketch is analytically useless because the ‘short run’ (SR) never turns into the ‘long run’ (LR) – no matter how long we wait.

This article is about one of the standard diagrams found in economics textbooks. This familiar diagram shows the relationship between short run average cost and long run average cost. The author argues in favour of an alternative three-dimensional diagram. The alternative diagram is not just visually different; it also presents a different relationship between short run and long run average costs.

The title and subtitle of the paper give the impression of it being difficult and esoteric, but in fact the writing is clear, and the author is sympathetic to non-technical readers. As a result, this article is accessible to a general audience. I appreciate how careful the author is in describing the diagrams. Some of the comments on economic theory, though, are quite vague and difficult to understand. For example, the second-last sentence on page 6 and the corresponding footnote 5 seem like incomplete thoughts and are hard to decipher.

The author proposes a new diagram that differs from the original in two ways. Firstly, the new diagram is three-dimensional. Secondly, the author objects to the “tangency solution” that we see in the standard diagram. Normally, the long run average cost curve is drawn as an envelope of the short run curves. In other words, the long run curve is tangent to the short run curves. The author argues that the long run curve should instead cross through each short run curve at its lowest point. This can be seen in figures 8 and 9.

The new diagram is partly based on a description given in a footnote by Alfred Marshall. That the new diagram is three-dimensional does seem to fit with Marshall’s description. The half-pipe surface implied in the new diagram is consistent with Marshall’s footnote, I think. However, Marshall does not make any reference in this footnote to a long run average cost curve. So, this aspect of the new diagram requires some separate justification.

I feel that that justification is currently missing from the paper. Also, I suspect that the author has an interpretation of the long run cost curve that is different from the standard interpretation. This may be because his background is in engineering rather than economics. The usual interpretation is as follows. In the short run the producer must work with a fixed amount of capital. In the long run, on the other hand, the producer can choose the amount of capital that he or she would like to use. Thus, a producer has greater flexibility in the long run. The producer in the long run can choose the best amount of capital for the quantity of output that he or she seeks to produce.

Under this standard interpretation, then, figure 8 can’t be right. In figure 8, the long run curve is sometimes above some of the short run curves. This would mean that, for some quantities, the average cost of production is higher in the long run than it is in the short run. This would imply that the producer does not have full flexibility to choose the best amount of capital in the long run.

To take an example, in the three-dimensional figure 9 we see that the long run curve meets the minimum point of the short run curve for k=5. However, as we can see from the alternative viewing angle in figure 8, the short run curve for k=6 is lower than the short run curve for k=5 for this quantity of output. So, if the producer wishes to produce this quantity of output then the producer would prefer to use 6 units of capital rather than 5.

The author may well have a perfectly coherent interpretation of the long run average cost curve that is just different from the usual meaning. However, I think that this interpretation needs to be explained and justified carefully in the paper. It may also be appropriate to give a different name to the curve to avoid confusion.

An additional minor point is that the way we usually draw the long run average cost curve depends on whether the amount of capital can vary continuously or comes in discrete amounts. The usual smooth long run curve corresponds to the continuous case. Here, we draw just a few short run curves, but we imagine that there are infinitely many more that are invisible. The smooth long run curve is tangent to these invisible short runs curves as well as to the visible ones. In the discrete case, there are no invisible short run curves; the ones that are drawn in the diagram represent the full range of options available to the producer in the long run. In this case, the long run curve has a scalloped shape as it coincides with whichever short run curve is lowest as we move along from left to right. I think it might be worth noting this in the paper.

The question addressed in this paper was already addressed and resolved in the sophisticated discussion by Paul Samuelson in his Foundations of Economic Analysis. See the pages for “Wong” in the index.