Can the law of returns be stated mathematically?

Economics Economics Questions
1

I have been rereading Human Action since the end of my exams recently, and was puzzling over Mises’ section on the Law of Returns. First of all, I feel this section is very neat, especially given the way he demonstrates how the law of returns derives from the quantitative definiteness of means using limiting arguments dependent upon the alteration of the ratio of produced goods to applied factors of production.

From reading the section, I feel it should not be a problem to further generalise the argument, and express it mathematically too, but then I thought this has probably been done before, especially by neoclassicals(though perhaps not as rigorously given their hatred of a priorism). Can anyone point me to a source expressing the law of returns mathematically? I feel given the requirements given by quantitative definiteness alone should help constrain and specify what classes of production curves, whether 3 dimensional(2 axes for the quantities of complimentary factors of production, and one for the produced good) or n dimensional are allowable even theoretically.

Aside of pure curiosity, I feel such an excercise could also provide some useful insights with regard to the problem of economic calculation. Indeed I am itching to tackle this puzzle myself once I get back to tackling the economic calculation problem again.

2

My initial thoughts. The relationship between producers goods and consumer goods is simply for notation purposes. I don’t really–yet–see the value of seperating the two within the many levels of a production fuction. Second, Mises is refering to the law of diminishing returns right? If so, here is some mathematical treatment of the proof http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD699154&Location=U2&doc=GetTRDoc.pdf which incorporates some set-theory and combinatorial analysis.

3

I had the same interest a while back.

In footnote 25 of chapter 1 of MESPM, Rothbard references Stigler:

The book is a little beyond my budget however. And I would only be interested in the proof. I’ve thought it through though and the reasoning seems pretty straightforward.

Imagine two factors of production (FOP), A and B. Since every production process requires complementary (multiple) FOP, if the ratio of A/B = 0 or B/A = 0 then the change in total phyiscal product (TPP) = 0. And if A/B → infinity then B/A → 0 (B/A approaches 0) and vice versa. Now because average physical product is measured as TPP/A (assumung A is the FOP is under consideration), APP → 0 when A → 0. Also, change in TPP → 0 when A → infinity since B/A → 0. Since the change in TPP → 0 as A → infinity, the APP → 0. Since at the boundaries of the APP curve, [A = 0, A = infinity], the APP is 0, and since APP > 0 between the boundaries of the interval given that TPP > 0 and A is finite, there must be some optimal/optima combinations which maximizes APP.

Those are my thoughts.

4

Talk about using a cannon to kill a misquito. It would take me hours to read through that.

5

Hi Stephen, your thoughts somewhat echo mine, though I haven’t used Rothbard’s terminology. Jeremy’s link is pretty involving, incorporating technology, which I didn’t think was necessary to just produce a proof or mathematical illustration of the law of returns. I’ve done some work on this myself now to satisfy my curiosity, I think it’s fair to say I’ve produced a mathematical way of expressing the law and generalising it using calculus, though Mises verbal proof of the initial proposition is more than enough, though it could be illustrated graphically.

I’m tempted to write it up in Latex and produce a pdf. I’d be more willing to if others would like to see what I’ve done.

6

I wouldn’t mind.

7

I’ve produced a "zeroth order" draft,(I don’t know how to attach files to posts, so please see the link to download the pdf) and not yet included the mathematical treatment but instead just stated the theorem and reproduced the proof in the file attached. There is a statement of Mises that has confused me however from pg 128 in his proof however, and I was wondering if anyone could enlightne me as to why it is true:

“On the other hand, an increase in the quantity of B available could not increase the output of D if the supply of C does not increase.”

I understand his earlier statements but I do not see how the above necessarily follows, unless one makes special assumptions. I have alluded to this in the final part of the attached pdf, and don’t think I can honestly progress until I’ve resolved my confusion. Does anyone have any ideas?

8

please let me know if you have difficulties downloading…

9

The variable’s exponents are < 1…

10

I’m sorry Esuric, could you elaborate on that? As far as I am aware I haven’t made use of exponents, if you are referring to p_{1}, I merely used the 1 as an index, to differentiate this from p, the number of consumer good D produced if c had not been increased by a factor x.

11

Abskebabs, your summary looks pretty good.

I’ve mulled over Jeremy’s link, which is a draft of this:

Shephard, Ronald. March 1970. Proof of the Law of Diminishing Returns. Zeitschrift für Nationalökonomie Journal of Economics 30(1-2):7-34.

It was followed by this:

Shephard, Ronald and Fare, Rolf. March 1974. The Law of Diminishing Returns. Zeitschrift für Nationalökonomie Journal of Economics 34(1-2):69-90.

I’ve read those two articles, and I must say, they do not solve the problem (they reject the Mises explanation), and among other things, introduce a very static and useless to Austrian theory concept (admitted even by John Hicks in Capital and Time, 3rd part–although Hicks still refused to accept Hayek’s point-input-point-output): the general production function.

Mises got his explanation probably from Edwin Cannan, who in turn generalized Edward West (here I go, from my notes):

Robert-Jacques Turgot in 1768 observed of land: “if it were once tilled the produce will be greater; tilling it a second, a third time, might not merely double and triple, but quadruple or decuple the produce, which will thus augment in a much larger proportion than the advances increase, and that up to a certain point, at which the produce will be as great as possible compared with the advances… if the advances be still increased, the produce will still increase, but less, and always less and less until….anaddition to the expenditure will add nothingwhatever to the produce” (Cannan 1892:54-55). This was the first inductive statement. Edward West in 1815, however, observed of prices of products of land as a factor of production that greater prices acted as an incentive for greater cultivation of land inferior to currently cultivated land and that lesser prices acted as an incentive for lesser cultivation of land inferior to currently cultivated land (Cannan 1892:65). This was the first deductive statement.

West understood diminishing returns must exist at the margin:—“if [the] same rich land would continue to yield the same proportionate return to the work of 20 and 30 and 100 as it did to that of 10 labourers, [then] the inferior land would never be cultivated at all. Thatthis diminution of the return of the soil to the additionalexpense bestowed on it takes place gradually may also beproved by the same reasoning.The gradations of the quality of the soil must be infinite” (West [1815] 1903:14).

Turgot had recognized initially increasing marginal returns as arising from fixed cost, which is capital good such as knowledge, technology, technique, and so on, spreading across an increasing quantity of product. Cannan continues, “Turgot’s law is just as true of manufactures as of agriculture. At any given time (or if the reader prefers, circumstances remaining unchanged) increase of labour up to a certain point is attended by increasing proportionate returns (called for short increasing returns) and beyond that point further increase of labour is attended by diminishing proportionate returns (called for short diminishing returns). Mankind cannot produce an unlimited amount of calico any more than an unlimited amount of wheat” (Cannan [1914] 1922:67-68). Cannan adds, “If population is not large enough to bring all industry up to this point, returns will be less than they might be, and the remedy is increase of population; if, on the other hand, population is so great that the point has been passed, returns are again less than they might be, and the remedy is decrease of population… [and] position of the point is perpetually being altered by the progress of knowledge” (Cannan [1914] 1922:69).

Cannan elaborated: “if returns did not diminish, the produce of any piece of land would be unlimited, ‘and this would have the same effect as an unlimited quantity of land convenient for cultivation’ ” (Cannan 1929:232 citing West [1815] 1903:38).

For instance, “if the price of corn were to fall….[then] some land would be withdrawn from cultivation, and the rent of that land which remained in cultivation would be lowered” (West [1815] 1903:40). For instance, “high prices [of products of land] are caused by the necessity of resorting to inferior soils, for the purpose of producing some part of the required supply. But if those inferior soils did not exist, or were left uncultivated, prices [of products of land] would be still higher, as the supply [of products of land] would be still scantier, until those high prices would call inferior soils into cultivation” (Longfield 1934:133).

Cannan argued: “if this is true of land, it is equally true of any other materials or instruments… if an ounce of coal could be made to give as much heat as an unlimited number of tons, if the best house could be made to accommodate comfortably the population of the world, and if the best locomotive could pull an unlimited number of trains in all required direction at once, [then] there would be no need of using more coal, houses, or locomotives… [then] they would have no value” (Cannan 1929:238).

Next, we know from Menger, if even one of several complementary higher-order goods is non-existent, then their complements also lose their goods-character, because they cannot be used to produce the lower-order good that their goods-character is contingent upon:—“even the largest quantity of land cannot be employed for the production of a quantity of grain, however small, unless we have at our disposal the (complementary) quantities of seed, labor services, etc., that are necessary for the production of this small quantity of grain” (Menger [1871] 2007:85). Wicksteed agreed, “Of course you cannot indefinitely increase a product in proportion to the increase of certain selected factors of production if you do not increase the other factors” (Wicksteed 1910:530).

Cannan concluded, “A scientific law should be true at all times and places, and should not be….capable of being suddenly replaced by a contrary law… [W]e [don’t] say that a falling balloon is ‘subject to the law of gravity,’ but a rising balloon is ‘subject to another law, that of rising bodies,’ while a balloon which remains at the same level is ‘subject to the law of constant height’ ” (Cannan [1914] 1922:70-71).

12

@abskebabs I’m slightly confused

Therefore if c was increased by a factor x to give the value p1, then the following
inequalities would hold:

c + x = p1

p1 > p and p1c < pcx (so p1cx < pc , as expected).

Can we say

p < p1

and

c < cx ---------> so p1c < pcx

If the value of p/c does not have an optimal value(a global maximium) for finite c, then we can expect it to continually increase as we increase the quantiy of C.

Just becasue an optimal doesn’t exist of p/c doesn’t mean that all increases in c are proportional with increases in p. Therefore, isn’t is possible a function of c could resemble p^-c = f(c) where p^-c > 0

Thus, even though an increase in c, increases p, it is not proportional. Therefore, in the limit, c→∞, p→negative∞

Finally, since all future is uncertain wouldn’t this entire production function be a function of present time factors and thus every varable in the production fuction should be multiplated by the changing variables of time?

13

Thanks for your response Jeremy to clarify, when I said c was increased by a factor x, to avoid confusion I suppose I should have said multiplied by a factor x, in order that p1 instead of p of the consumer good is produced.

Also:

"p1 > p and p1c < pcx (so p1cx < pc , as expected).

Can we say

p < p1

and

c < cx ---------> so p1c < pcx"

What I actually stated was that one could still have increases in p to p1, when the factors are employed beyond their optimal ratio, and c is increased to cx, but that this would not be optimal with regard to the amount produced per producer good employed. Therefore we could have p1>p, and p1c<pcx(and so (p1/cx)<(p/c).)

Secondly, it is not completely obvious from the statements that p<p1 and c<cx that p1c<pcx, which is the reason we need to do more than just mathematical illustration to prove the proposition as Mises does. It might also be worht noting in addition, that there are also cases where p1<p, after c is multiplied by a factor x beyond its optimal value. One can imagine the case in which one is producing woolen shirts with dye in a factory. If the level of dye exceeds the required amount, then one does not have a suboptimally increased output, but zero output since one is left with useless products that don’t satisfy the specified requirements of the desired consumer good.

"If the value of p/c does not have an optimal value(a global maximium) for finite c, then we can expect it to continually increase as we increase the quantiy of C.

Just becasue an optimal doesn’t exist of p/c doesn’t mean that all increases in c are proportional with increases in p. Therefore, isn’t is possible a function of c could resemble p^-c = f(c) where p^-c > 0

Thus, even though an increase in c, increases p, it is not proportional. Therefore, in the limit, c→∞, p→negative∞"

That is correct, you’ve pointed out an assumption I employed that I haven’t yet made explicit(though I’ve mentioned a similiar point in my rough notes that I haven’t typed up yet). I did discount a lot of functions that I didn’t consider “realistic”, e.g. p/c=(constant/c^2) etc, since for such a function as we allow c–>0, p–>infinity etc, and we cannot identify a maximium with the methods of calculus, regardless of the fact that the “producer good” in such a process would be best employed at a quantity 0.

I think I’m finding it hard to visualise your counter example, and how p–>-infinity even though p/c does not attain a highest value. Am I correct in ascertaining that you’re talking about p=f(c)^(-1/c)? I didn’t claim that the increases in p would necessarily be proportional(they may well be more or less than) by increasing c if p/c had no maximium, though perhaps I should have been more explicit about this as well.

“Finally, since all future is uncertain wouldn’t this entire production function be a function of present time factors and thus every varable in the production fuction should be multiplated by the changing variables of time?”

Uncertainty is inescapably implied by action, though I think it is superfluous to a demonstration of the law of returns, since this theorem can be accurately described and established with or without considering uncertainty. However, uncertainty in the Knightian sense defies treatment via the methods of probability calculus, so I don’t think it could be adequately accounted for in a treatment aiming at mathematical illustration with the use of time factors. In a sense, if we knew how our combinations would change with time, including time in our treatment, this wouldn’t represent uncertainty, since we wouldn’t be uncertain about the changes, at least in a non-class probabilistic sense.

Also, has anybody got any ideas how to tackle my previous question? I have some ideas, but I’d like to hear other people’s thoughts.

EDIT: I’m confused by the one star rating(as I am in general) that’s been given to this thread. Is this just a subject people consider frustrating and pointless?

14

I’ve made some more progress, and even “introduced” a new concept that I am calling “rescaling”, which I feel is used implicitly and very subtly by Mises, though needs to be made explicity I feel, since it underlies a lot of the reasoning surrounding the law of returns. It basically is the idea that if you maximise p/c w.r.t. c at a particular value of c in optimal combination with b at a certain ratio, can be reproduced at a completely different scale, with a different value of of c, as long as the ratio in which both means are combined is kept constant at the optimal level.

Or in plain English, if I know I get the most efficient returns using 7 workers in for every 3 machine parts used to build a car, I can increase output multiplying my factors by 3, so I have 21 workers with 9 machine parts and still be employing them to achieve the highest possible returns, and optimal utilisation of my complimentary factors of production. Though simple, I feel the theoretical power of this simple idea is actually quite potent, in helping us understand economic action in different contexts.

Comments and incisive criticisms are of course welcome! I’ve been wondering if I should turn this into a full blown paper, do you think most journals(including Austrian ones) would consider this “too elementary?”

15

Are you looking for a mathematical statement of the law of diminishing returns?

I.e. if I have a number of homogeneous X’s, then each additional X I can obtain is subjectively less valuable?

16

I believe that is rather a correct way of stating the law of marginal utillity, but yes, a mathematical way of illustrating the law of returns was initially what I enquired about, and have endeavoured to produce myself.

17

I’ve just been reading Rothbard’s derivation of the law of returns, and I must say it is astonishingly clear and elegant. It kind of rips the piss out of some of the clumsy stuff I’ve been fumbling about with in my own, and he avoids the statement Mises makes which confused the hell out of me.

18

In footnote 25 of chapter 1 of MESPM, Rothbard references Stigler:

[25]For algebraic proof, see George J. Stigler, The Theory of Price (New York: Macmillan & Co., 1946), pp. 44–45.

This is actually with regard to proving a that marginal product reaches a maximum value before average product. I had a thought about this and it seems to me this does not require a more involved proof than can be done with a knowledge of elementary caclulus as you can see:

d/da(p/a)=1/a(dp/da)-(p/a^2)

so therefore,

dp/da=a*d/da(p/a)+p/a

and its not too hard to see that dp/da(marginal product) may reach a maximum before p/a(average product). Eat your heart out George Stigler!

19

Wondering about the same topics I ended up in this old post, but better late than never. I guess you can think about it as follows: suppose you have 1 of B and 2 of C and p/c is always increasing, say p(b=1,c=1)=1 and p(b=1,c=2)=4 so when b=1 you have p/1=1/1=1 and p/2=4/2=2. Then if you add one more B you have to choose between keeping it wasted or moving one of your two Cs to take advantage of the new unit of B. But in this second case total product will be p(1,1)+p(1,1)=p/1+p/1=1+1=2, while in case both Cs were still working in complement with the first B you would have p(1,2)=2p/2=22=4 > 2. So there is no point in adding more Bs; instead, there is a point in concentrating all Cs in the smallest portion of B, because p/c will be larger. That is, B can’t be scarce, which amounts to say that B can’t be a factor of production or, in other words, a means to an end, as was previously supposed. HIH.

20

Or in plain English, if I know I get the most efficient returns using 7 workers in for every 3 machine parts used to build a car, I can increase output multiplying my factors by 3, so I have 21 workers with 9 machine parts and still be employing them to achieve the highest possible returns, and optimal utilisation of my complimentary factors of production.

I would say the conclusion assumes there is no interaction whatsoever between separate 3 groups of 7 workers and 3 machine parts - they are basically operating as 3 separate businesses. In real world, that is probably never the case - the groups share various facilities and services, which most probably skews the optimum point this or that way.

Also, there is an assumption that the business is small enough not to affect the market by its scaling.

Also, since I am writing anyway - how does the theory of diminishing returns address the fact that some resource may be complimentary to itself? I mean, there may be goals attainable by use of 2 X, which are not even partially attainable by use of 1 X, and at the same time very valuable? Two planks may allow crossing a chasm, while one may be only good enough to build a fire.