Recent labor relations controversies and ongoing arguments about the minimum wage have raised questions as to how a supply chain should share the utility it produces.

If we ask the wrong question, however, we’ll get the wrong answer. “What is a fair share?” asks how a supply chain should divide a finite pie between its suppliers, workers, investors, and customers. “What is the optimum share?” asks how the outputs should be allocated to maximize the size of the pie so all can have more.

The Cobb-Douglas production function seeks to quantify the contributions of capital and labor to total productivity and output, which provides at least some basis for an optimum allocation. One of its key inputs is total factor productivity (TFP), which deserves far more attention than it now receives. Ever-increasing TFP is why all but the poorest Americans are more affluent than most people around the world.

Frederick Winslow Taylor (Principles of Scientific Management, Harper Brothers, 1911) made this clear long ago when he wrote, “... the one element more than any other which differentiates civilized from uncivilized countries—prosperous from poverty-stricken peoples—is that the average man in the one is five or six times as productive as the other.”

However, TFP is no longer increasing as rapidly as it should. It’s actually declining in many economic sectors.

“Productivity Now: A Call to Action for U.S. Manufacturers,” by Justin Rose, Henk van Duijnhoven, Christian Köpp, and Jonathan Van Wyck (Boston Consulting Group, 2016), warns, “Productivity growth—output growth that exceeds the growth of the workforce and capital employed—has been the lifeblood of the U.S. manufacturing sector for much of the recent past.... However, the past decade has seen the emergence of an alarming macroeconomic trend. The U.S. has struggled to improve productivity, which has increased at a tepid pace of 0.7% over the past 10 years.” The article adds that large productivity gains “will tip competitiveness from low-cost countries back to the United States.” Lack of productivity gains will, of course, do the opposite.

TFP and the Cobb-Douglas production function

The Cobb-Douglas production function relates a system’s total output to inputs of labor and capital. This is admittedly outside my area of expertise, and I don’t recall seeing it in any of my MBA macroeconomics classes. However, it does make sense, and one of the inputs is TFP. Total factor productivity consists simply of the outputs divided by the combined inputs.

TFP is therefore a direct measurement of an operation’s efficiency. Investopedia adds, “Total factor productivity (TFP), also known as Solow’s residual, is a measure of the share of production growth that cannot be explained by increases in inputs like capital and labor. It captures the impact of technical development, increases in production, and other unnoticed elements.”

We’ll see, in fact, that adding more capital and/or workers has diminishing marginal returns in the absence of greater efficiency. The definition also suggests that the “unnoticed elements” might be the most important elements in the entire proposition. When Henry Ford and contemporaries like Frank Gilbreth and Taylor took notice of them, American productivity increased enormously.

This perception is reinforced by Umit Gumusten (2024); emphasis is mine:

“At its very core, total factor productivity is generally supposed to be an indicator of the efficiency at which an economy or firm manages to use inputs, such as labor, capital, and technology, in producing goods and services.... TFP takes into consideration all aspects of production and reflects the competency to generate more output without the rise in inputs.”

It’s also very telling that the combined inputs consist of capital, labor, energy, materials, and services. Waste of any of these inputs reduces total factor productivity. This includes not just inefficient uses of time, materials, and energy, but also misguided purchases of expensive corporate headquarters and office buildings, a practice that Ford denounced long ago (My Life and Work, Doubleday, Page & Co., 1922).

To put this in perspective, here is what Ford’s office looked like in the Highland Park plant in 1913. The office of Ford’s production chief, Charles Sorensen, is occupied primarily by what looks like a planning or drafting table, a lot of mechanical parts, and at least two car wheels. The Spartans would have loved this work environment because it contained almost nothing not necessary to get the job done. Many modern executives would have refused to work in it.

The Cobb-Davis function (Allin Cottrell, Wake Forest University) is shown below in Equation 1; it relates total output to total factor productivity, capital, and labor.

Y = total output; A = TFP; K = capital, the value of plant, equipment, infrastructure, and so on; N = labor, e.g., in hours; Alpha = output elasticity of capital, and 1 minus alpha (sometimes depicted as beta) is the output elasticity of labor. Alpha is between 0 and 1 in all cases, which means that, in the absence of higher TFP, it’s futile or even worse to try to increase output by adding capital or labor.

Remember that TFP is the portion not explained by direct measurements of capital and labor. If, for example, a motion efficiency improvement doubles output without the need for more labor, this doubles TFP. This suggests that TFP is our best friend. The Investopedia reference adds a very encouraging note; the Chinese Dragon isn’t the manufacturing superpower it would like us to believe: “Viewed through the lens of total factor productivity, China has managed to become an economic superpower through its sheer size rather than through gains in productivity.”

This is why we should have never sent manufacturing jobs over there. We can easily get them back if we diligently use the productivity improvement methods we inherited from Ford, Gilbreth, and Taylor, which were subsequently improved upon by Taiichi Ohno and others.

Marginal product of the factors

It’s useful to look at the marginal product of the factors, which shows the incremental effects of adding capital or labor. The marginal product of capital (MPK, per the Cottrell reference) is:

However, if we remember the original function back in Equation 1, we can substitute as follows to get Equation 2:

Similarly, we can derive the marginal product of labor (MPL) as shown below to get Equation 3:

The effect of output elasticity is particularly noteworthy. If the output elasticity of labor is 0.4 and labor increases is 10%, output will increase 4%. The key takeaway is that throwing more human and/or physical resources at a problem is unlikely to be cost effective. So, we must look instead for ways to use what we have more effectively. Ford realized this more than 100 years ago as shown in My Life and Work (emphasis is mine):

“The farmer makes too complex an affair out of his daily work.... Power is utilized to the least possible degree. Not only is everything done by hand, but seldom is a thought given to logical arrangement. A farmer doing his chores will walk up and down a rickety ladder a dozen times. He’ll carry water for years instead of putting in a few lengths of pipe. His whole idea, when there is extra work to do, is to hire extra men.”

When might the elasticity of labor approach 1? In the musical Fiddler on the Roof, the tailor, Motel, saves enough money to buy a used sewing machine. The machine will allow him to do a lot more work, and better quality work in the bargain. Its introduction increases his total factor productivity several-fold. But because he can operate only one machine, purchase of another will not increase output. So the elasticity of capital is zero.

Suppose, however, he has a brother who also is a tailor and is willing to work all night and sleep during the day. Addition of another shift doubles output, so the elasticity of labor is close to 1. Alternatively, if the brother is willing to work only during the day, addition of another machine will almost double the output, which then makes the elasticity of capital close to 1—but only because there is an additional worker to operate it.

Another key takeaway from Fiddler on the Roof is Motel’s eagerness to get a machine that will make him more productive so he can earn more money while charging less for his work. Compare this to the International Longshoremen’s Association’s recent demands that automation be kept out of their workplace to “protect jobs”—and, long ago, the resistance of Luddites to automated weaving machinery.

Total earnings by factor

Cottrell then derives the total earnings of each factor by multiplying them by their quantities, as shown below in Equations 4 and 5. We must bear in mind that, even though TFP (A) is not visible in these equations, output Y is directly proportional to it as shown by Equation 1.

This looks like it could provide at least a theoretical model for a fair allocation of revenues between investors and labor. If the enterprise is capital-intensive, as it is in the semiconductor, automotive, and pharmaceutical sectors, then investors are certainly entitled to most of the income. Were it not for their contribution, the enterprise and the jobs wouldn’t even exist. If there’s unmet demand for the product, though, reinvestment is better than payment of dividends. If the enterprise is labor-intensive, as it still is in poorly designed construction jobs and farm jobs, then the workers are entitled to a commensurate share.

In all cases, however, efficiency and productivity improvements that increase the output Y will deliver more for workers and investors alike. Ford made this clear in My Life and Work: “It is utterly foolish for Capital or for Labour to think of themselves as groups. They are partners. When they pull and haul against each other—they simply injure the organization in which they are partners and from which both draw support. It ought to be the employer’s ambition, as leader, to pay better wages than any similar line of business, and it ought to be the workman’s ambition to make this possible.”

Customers also must get a share

If we double TFP, this doesn’t mean investors and workers can get twice as much money as they did previously. Unless demand for the product is totally inelastic, i.e., independent of price, we must reduce the price to sell twice as much output. The only way Ford, and then his competitors, could sell millions of automobiles was to get the price down to where people could afford them. Price elasticity is the marginal change of sales volume (Y in our case) divided by the marginal change in the price as shown by Equation 6.

In this case, a possible formula for fair compensation of workers and investors, as derived from equations 4 and 5, is as follows, where P is the price as mandated by Equation 6 or, if necessary, a more complex formula that relates sales to price.

Wages (and dividends) must be supported by productivity

The Bureau of Labor Statistics provides a spreadsheet of TFP by major industry (download here), with an index of 100 representing the status in 2017. It’s much easier to use if you select “Total Factor Productivity” from the drop-down list in Column D, and excludes everything else. That for “Motor vehicles, bodies and trailers, and parts” is 98.146, i.e., down from what it was in 2017.

This suggests that the astronomical wage concessions the United Automobile Workers “won” from General Motors, Stellantis, and Ford in 2023 are built entirely on an illusion and a foundation of sand. The UAW cites wage increases between 33% and 160%, faster progression to highest pay, and other gains. These gains are simply not supportable in the absence of higher TFP. If the automakers raise prices to cover the higher wages, people will buy fewer new cars—and this could even result in layoffs. This is according to the laws of economics, as opposed to what management wants, the union wants, or what you or I want.

The TFP figures are depressing in general because many show declines, or only small increases, from 2017. One encouraging standout is “Support activities for mining” at 164.20, relative to 2017. This reinforces the need to improve efficiency across the board if we are to see the kind of genuine economic growth that is needed to, for example, roll back the federal deficit. Meanwhile, Gina Potthoff Kacik (Chicago Booth Review, 2023) warns, “U.S. Construction Has a Productivity Problem.”

She writes, “A construction worker in 2020 actually produced less than a construction worker in 1970, they calculate, reinforcing an observation made by The Economist, among others.”

Some of the reasons are obvious in a fraction of a second to anybody who knows what to look for. I recently saw some roofers working on a nearby house. They were carrying bundles of shingles up a ladder, rather than using conveyors that are made for this express purpose. While nothing in this article constitutes engineering advice, carrying heavy loads up ladders may violate OSHA regulations as well as waste labor on a nonvalue-adding activity. If the workers had fall protection, it was invisible (like the emperor’s proverbial new clothes).

Ford, on the other hand, achieved unprecedented business results by making automobiles affordable to those who built them. This involved a combination of higher wages and lower prices. In his preface to Horace Lucien Arnold and Fay Leone Faurote’s Ford Methods and the Ford Shops (The Engineering Magazine, 1915), Charles Buxton Going writes:

“Ford’s success has startled the country, almost the world, financially, industrially, mechanically. It exhibits in higher degree than most persons would have thought possible the seemingly contradictory requirements of true efficiency, which are: constant increase of quality, great increase of pay to the workers, repeated reduction in cost to the consumer. And with these appears, as at once cause and effect, an absolutely incredible enlargement of output reaching something like one hundred fold in less than 10 years, and an enormous profit to the manufacturer.”

The workers got unprecedented wages, and customers got lower prices, but this was in the context of a hundredfold increase in output over the course of 10 years. Productivity gains of this nature aren’t forthcoming from today’s automakers or most other industries.

However, the good news is that proven off-the-shelf productivity improvement techniques have been available for well over 100 years, and they have subsequently evolved through, as but one example, the Toyota Production System. The only real obstacle to realization of substantial TFP growth seems to consist of failure to use these methods.