No isoquant here, sorry baby

My editorial

I am changing the topic, slightly. In my last update I was discussing about the first takeaway from my research in 2017, namely about collective intelligence. This time, I switch to the second one: the mathematical construct of the type ‘Y = Kµ*L1-µ*A’, probably the best known for its application in the Cobb – Douglas production function (Cobb, Douglas 1928[1]). The best known, yes, although not the only known. The so-called ‘new economic geography’ (see: Krugman 1991[2];  Krugman 1998[3]) uses the same logical frame for slightly different a purpose. Anyway, the thing I want to discuss is the issue of perfect substitution. There is an assumption, regarding the Cobb – Douglas production function, that as long as I use the ‘Y = Kµ*L1-µ*A’ framework, I have to assume perfect substitution between factors K and L on the right side. I even had that remark from my audience, as I was presenting that paper of mine, ‘Settlement by Energy – Can Renewable Energies Sustain Our Civilisation?’ , at a conference. The remark went more or less as: “If you apply the mathematics of Cobb – Douglas production function to the combination of food and non-edible energy, you must imply perfect substitution between these two? Do you?”. My immediate answer was “No, I don’t. We cannot replace food with electricity and vice versa”. Still, my pondered answer (i.e. the answer that I could articulate if the scientist in question, who asked his question, was still there, and which I cannot articulate in this context because this scientist is not in front of me anymore) is different: “Firstly, the assumption of perfect substitution is a false necessity in the Cobb-Douglas production function. Secondly, substitution between food and energy can go all the way from a nearly perfect one, through imperfect one, down to no substitution at all”.

Good. I have addressed my pondered answer to the thin air in front of me, which stared back at me, blankly. I need to develop an argumentation. I start from the beginning. Professor Charles W. Cobb, and professor Paul H. Douglas have never claimed perfect substitution between labour and capital. Really. You can read their seminal paper in any direction you want, you will not find such an assumption. Quite the contrary, the very idea of defining labour and capital as two separate aggregates, factors of aggregate output, gives like the shade of a suspicion that they didn’t really treated them as mutual perfect substitutes. Thus, the idea of perfect substitution comes from those, who used to interpret the writings of Cobb and Douglas. How could it come up to the surface of intellectual prowess of economic sciences? Well, some people say that if I have an equation like ‘Y = Kµ*L1-µ*A’, with a condition µ < 1, any decrease in one factor, accompanied with an exactly corresponding decrease in the second factor, must produce the same output on the left side. With a given µ, if I take 10% out of my capital and add 10% to my labour (e.g. if I sell my house and move to the countryside, and grow my own cattle, and my own carrots), my output should stay rock solid, without flinching even by an inch. The reasoning seems almost perfect. The ‘almost’ comes from the fact that it is false. Yes, baby: this is bullshit. I am developing on that. In Tables 1 and 2, below, I present a simulated set of data: 31 one consecutive periods in time, starting with labour and capital being supplied in equal amounts, 500 units each. Then, I increase the supply of capital each year by one unit, and I correspondingly decrease the supply of labour. ‘Correspondingly’ means that each year, I take off the supply of labour the very same proportion, which one unit of capital adds to its base. I compute the model output with the Y = Kµ*L1-µ formula. In Table 1, I give to my µ the value of µ = 0,75, thus roughly what you can find out today if you apply the original methodology by Charles Cobb and Paul Douglas to the present-day data. In Table 2, I make my µ equal to µ = 0.25, thus exactly as Cobb and Douglas posited it regarding the first two decades of the 20th century. As you can see, when I make capital dominant, with µ = 0.75, its incremental increase is bound to produce incremental increase in the aggregate output, even in the presence of the corresponding decrease in the supply of labour. If, on the other hand, I make labour the dominant factor, and I posit µ = 0.25, incremental increase in capital, accompanied by exactly proportional a decrease in labour, just has to produce decreasing an output.

Thus, no isoquant here, sorry baby. Still, there is one special case when that assumption of perfect substitution holds. This is when µ = 0.5, or, when I attribute to both of my factors the same relative importance in making the aggregate output.

Table 1 – Simulation of the Y = Kµ*L1-µ*A production function, with µ = 0,75

Period Capital Labour Output
1 500 500 500
2 501 499,001996 500,4997502
3 502 498,0079681 500,999002
4 503 497,0178926 501,4977567
5 504 496,031746 501,9960159
6 505 495,049505 502,4937811
7 506 494,0711462 502,9910536
8 507 493,0966469 503,487835
9 508 492,1259843 503,9841267
10 509 491,1591356 504,4799302
11 510 490,1960784 504,9752469
12 511 489,2367906 505,4700782
13 512 488,28125 505,9644256
14 513 487,3294347 506,4582905
15 514 486,381323 506,9516742
16 515 485,4368932 507,4445783
17 516 484,496124 507,937004
18 517 483,5589942 508,4289528
19 518 482,6254826 508,920426
20 519 481,6955684 509,4114251
21 520 480,7692308 509,9019514
22 521 479,8464491 510,3920062
23 522 478,9272031 510,881591
24 523 478,0114723 511,370707
25 524 477,0992366 511,8593557
26 525 476,1904762 512,3475383
27 526 475,2851711 512,8352562
28 527 474,3833017 513,3225107
29 528 473,4848485 513,8093031
30 529 472,5897921 514,2956348
31 530 471,6981132 514,781507

Table 2 – Simulation of the Y = Kµ*L1-µ*A production function, with µ = 0,25

Period Capital Labour Output
1 500 500 500
2 501 499,001996 499,5007488
3 502 498,0079681 499,00299
4 503 497,0178926 498,5067164
5 504 496,031746 498,0119206
6 505 495,049505 497,5185951
7 506 494,0711462 497,0267328
8 507 493,0966469 496,5363264
9 508 492,1259843 496,0473688
10 509 491,1591356 495,5598529
11 510 490,1960784 495,0737715
12 511 489,2367906 494,5891177
13 512 488,28125 494,1058844
14 513 487,3294347 493,6240648
15 514 486,381323 493,143652
16 515 485,4368932 492,6646391
17 516 484,496124 492,1870193
18 517 483,5589942 491,710786
19 518 482,6254826 491,2359324
20 519 481,6955684 490,7624519
21 520 480,7692308 490,2903378
22 521 479,8464491 489,8195837
23 522 478,9272031 489,3501829
24 523 478,0114723 488,8821291
25 524 477,0992366 488,4154157
26 525 476,1904762 487,9500365
27 526 475,2851711 487,485985
28 527 474,3833017 487,0232549
29 528 473,4848485 486,5618401
30 529 472,5897921 486,1017342
31 530 471,6981132 485,6429312

That piece of maths I present above is the mathematical part of my answer to that scientist who is not sitting in front of me anymore. Now, the existential part, namely about the actual substitution between food and non-edible energy. If I live with no technology at all, even without an ox to pull my cart, so if I live the life of a hunter gatherer, I need a lot of food to stay healthy and fit for hunting. A cautious estimation leads to some 7000 – 8000 kilocalories a day. If I switch from hunting to farming, and I progressively buttress my existence with technologies, like starting with a horse and going all the way up to a super harvester, my alimentary requirement will fall progressively, probably down to some 3000 kcal a day. If I make one more step and become a city boy, who I am now, actually, I can drive my alimentary necessity down to about 1800 kilocalories a day, although this is without sport. If I do sport, and if I am serious about, I will need more. The point is that at the highest alimentary intake, when I have no technology to replace my muscles, any such technology can make my alimentary requirement drop a little. This is nearly perfect a substitution. As I am surrounded by more and more technologies, each new one produces a decreasing marginal decrease in my need for food. This is imperfect substitution. It goes down to a point of no substitution at all, when the only conceivable next step is Matrix and me connected to some tubes sucking energy from my body, to feed those lazy computers.

By the way, if I were a mean and clever artificial intelligence, I would tap into energy created by human society rather that by the human body. The most energy-intensive activity we do, as humans, is living in a big city. This is where mean and clever artificial intelligences can find the greatest amount of joules to free ride on them. This artificial intelligence from Matrix, the one who preyed directly on human metabolism, was not really the sharpest knife in the drawer.

[1] Charles W. Cobb, Paul H. Douglas, 1928, A Theory of Production, The American Economic Review, Volume 18, Issue 1, Supplement, Papers and Proceedings of the Fortieth Annual Meeting of the American Economic Association (March 1928), pp. 139 – 165

[2] Krugman, P., 1991, Increasing Returns and Economic Geography, The Journal of Political Economy, Volume 99, Issue 3 (Jun. 1991), pp. 483 – 499

[3] Krugman, P., 1998, What’s New About The New Economic Geography?, Oxford Review of Economic Policy, vol. 14, no. 2, pp. 7 – 17


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