Orthogonal Time in Euclidean 3-space

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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.

Posted for comments on 20 Oct 2017, 1:50 pm.

Comments (4)

  • Conal Duddy says:

    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.

    • Richard Everett Planck says:

      Dr. Ellerman, I also thank you for your comment. I didn’t realize that the question addressed in my paper had already been addressed by Paul Samuelson in his ‘Foundations of Economic Analysis’. Quite honestly, I have resisted reading his ‘Foundations’ because I was worried that it might confuse me. Basically, I wanted to follow my own line of thinking to its conclusion without serious distractions. I’m now adequately finished with most of my work so I can now turn to a reading of his classic text. Encouraged by your comment, I just placed an on-line order for his book and I intend to start reading it as soon as it arrives. Again, I thank you for your comment.

  • David P Ellerman says:

    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.

    • Richard Everett Planck says:

      Dr. Duddy, I thank you for your comments. And I appreciate your recognition that I attempted to translate ‘engineering language’ into ‘economic language’ by using ‘ordinary English’. However, my effort obviously back-fired because I failed to specifically mention/describe/discuss that my use of ‘short run’ vs ‘long run is totally different from the accepted economics distinction. I shall now offer an analogy as an attempt to clarify the difference.

      Let us envision a young man who wants to be a commercial fisherman. He buys a brand-new eight-meter oar-powered fishing boat (somewhat like the old-fashioned whalers). He also buys nets and hires several men to help him row and fish. They set out to sea the next day and he discovers that his decision regarding the amount of his human and physical capital was perfect. So, over ‘the long haul’ he changes nothing. For 20 years, as long as he makes no changes in the amount of his human capital or the amount of his physical capital, this is still the ‘short run’ according to my definition. Certainly, he is permitted to vary the day-to-day utilization of his capital (quit early because fishing is poor on some particular day or work ‘overtime’ because fishing is exceptionally good on some other day) but he is still operating on his original SRAC according to my definition.

      Now let us envision the same young man who starts off with precisely the same amount of human and physical capital but we let his results be different. Specifically, he is forced to quit fishing at 10:00 AM because his boat is already overloaded with fish. So we permit him to return to port and sell his fish so he can immediately go to the boat dealer, turn in his original boat (thus incurring a modest financial loss), buy a (larger) brand-new ten-meter fishing boat and hire a few more men to help with the rowing and the fishing. This (planned) change’ in the amount of his human and physical capital is how I distinguish between the ‘long run’ and the ‘short’ run. Thus, my distinguishment has nothing to do with calendar or clock because, in the first situation, my ‘short run’ was 20 years but in the second situation, my ‘long run’ was more like 20 hours.

      Regarding your comment about the new diagram being three-dimensional…. Yes, it is. First, I recognized its shape immediately because of the three-dimensional pVt sketches that I had studied in numerous thermo classes. Second, I also immediately realized its full potential: Marshall had only scratched the tip of the iceberg. One ‘nicety’ of his three-dimensional scheme is that it invalidates the appropriateness of the ‘ceteris paribus’ assumption because it is simply not possible to ‘hold everything else constant’ while we vary one ‘piece’ and see how it affects a second ‘piece’ of the puzzle. Thus a three-dimensional sketch does seem to be the minimum number of dimensions (Chiang, 1984) that might provide usable answers.

      Your comment about the tangency solution is easily addressed (but your critique is very well founded and will be discussed in a bit more detail below). First, I illustrated my LR curve where my LR curve intersected each of the SR curves at their lowest point. I define this ‘lowest point’ as the ‘design output level’ (DOL). [This is the label that I use to give a name to a particular state of affairs but it is not the same as in ‘mainstream’ economics where “… the firm will plan an optimal-size plant for production…” ] Specifically, it’s that unique level of output where all factors of production are working at 100% of the design capacity (no more, no less). But in real life of course, the average business does not operate at its DOL; according to US government data, most businesses seem to operate at roughly 80-85% of capacity (depending on whether the economy is in a ‘recessionary’ phase or in an ‘expansionary’ phase). So – in an (apparently less-than-successful) effort to retain pedagogical simplicity – I decided to illustrate my basic contention by drawing my LRAC curve as intersecting the lowest point on each of a series of SRAC curves.

      Perhaps Figure 8 and Figure 9 now make more sense. Let us assume that our farmer makes what (according to my meaning) is a ‘long run’ decision and buys one additional tractor so as to reduce the expected cost per bushel of harvesting his corn (Fig. 9). Understand that our farmer is now on his k = 6 SRAC. But our farmer is helpless before the vagaries of nature so what happens if his fields yield only ½ of the expected crop? The answer is that his cost/bushel on his new (lower) SRAC could, in fact, be higher than if he had not made what I call a ‘long run’ decision to buy the extra tractor. Thus it is possible, according to my distinction between the ‘short run’ and the ‘long run, to have portions of the new SRAC (i.e., SRAC2, Fig. 8) be higher than portions of the old SRAC (i.e., SRAC1) even though, in the normally-expected range of output, the new SRAC is lower than the old SRAC. And, when the firm’s LRAC and its possible SRACs are ‘collapsed’ into a two-dimensional sketch (Fig. 8), portions of the the firm’s LRAC will appear to be (illogically) higher than portions of some of its SRACs. This is unfortunate because mainstream theory rules out this possibility even though it probably happens rather often in real life. It’s quite easy to envision an over-eager farmer who made (what I call) a ‘long run’ decision to buy an additional tractor and then discoved, too late, that it was not really needed. Unfortunately, he cannot ‘un-buy’ the tractor which means that this year’s ‘long run’ costs per bushel are now higher than last year’s ‘short run’ costs per bushel.

      In summary, there are two requirements for my distinguishment between the ‘short run’ and the ‘long run’: (1) ‘short run’ means (a) temporary and (b) no ‘cost-reversal’ penalty fees whereas (2) ‘long run’ means (a) expected permanency and (b) ‘penalty fees’ for reversal. Put even more simply, in my ‘short run’ the farmer is permitted to move only to the right or to the left on his current SRAC because my definition of ‘short run’ analytically constrains him to stay on that SRAC whereas, in my ‘long run’, the farmer is analytically permitted (and has decided) to ‘jump’ to a different (and hopefully sustainably ‘lower’) SRAC.

      Finally, by way of closure and apology, I have been using my definition of ‘long run’ for nearly three decades so, in this paper, I completely forgot that I was using an existing economic term but had given it an entirely new meaning. What do the kids say these days… “Sorry. My bad.”