Productivity: The Only Sustainable Source of Growth

Productivity is output divided by the people producing it, or by the hours they worked. Productivity is called the only sustainable source of growth for an arithmetic reason rather than a rhetorical one. An economy can add workers only until it runs out of people, and it can raise the output each worker produces with no such limit. In a total, the two look identical.

Underneath that sits one line of arithmetic that cannot be argued with. Total output is the number of people producing multiplied by the amount each of them produces. The line is not a theory about how economies behave, it is a restatement of what a per worker figure means, and it holds in every economy in every year. The arithmetic buys a question that can be put to any growth rate at all: of the rise in the total, how much was more hands and how much was more from each hand? The two answers feel the same in a headline and they are nothing alike underneath.

Three things are already in hand: reading an output total, separating a real series from a nominal one, and working an employment growth rate from two head counts. None of that is repeated. Output per worker is computed from those same published counts, one of its two sources of growth has a ceiling and the other has none, and four things raise the one without a ceiling. A ratio can also be made to rise by taking people out of the bottom of it, and that is a job loss rather than a gain.

Two of the three Sankhya years below are pure, meaning one of the two ingredients of a growth rate is exactly nil. A pure year happens in a teaching example and almost never outside one, and it is the easiest place to see which of the two ingredients did the work.

What is productivity, and which denominator does the ratio divide by?

ProductivityHow much output comes out of a given amount of input. In most published use the input is labour, so the figure is output divided by workers or by hours worked. is a ratio, and a ratio is only ever as clear as the two counts making it up. The top is output, already familiar: a volume of production valued in money, with price effects taken out so that a rise means more was made. The bottom is where the interesting choice sits, and almost every confused argument about productivity turns out to be two people dividing by two different things.

The simplest bottom is a head count. Output per workerTotal output divided by the number of people at work. It says nothing about how long any of them worked, so two economies with the same figure can have very different working days. takes the whole output of an economy and divides it by everybody producing it. In year 1 of Sankhya, an invented country, output was Rs 16,00,000 crore and 8.00 crore people were at work. Do the division and the crore in the top cancels the crore in the bottom, leaving rupees per person: Rs 2,00,000/- of output for each person working, across the whole year. In year 2 output was Rs 16,80,000 crore and the same 8.00 crore people were at work. Divide again and each person produced Rs 2,10,000/-. That is a rise of Rs 10,000/- on Rs 2,00,000/-, or 5.00 per cent.

The other bottom is a count of hours, and it answers a different question. Output per hourTotal output divided by the total hours actually worked. It strips out how long the working day is, so it compares the productiveness of an hour rather than of a person. divides the same output by the hours actually worked rather than by the heads. Say the average Sankhya worker in year 2 put in 2,100 hours across the year, roughly a forty hour week. Then Rs 2,10,000/- of output per worker is Rs 100/- of output per hour. Now imagine a later year, invented purely to make the point, in which the working week shortens to about thirty six hours, so average hours fall 10.00 per cent to 1,890, and each hour becomes 5.00 per cent more productive at Rs 105/-. Multiply those out: output per worker is Rs 105/- times 1,890 hours, or Rs 1,98,450/-, down 5.50 per cent from Rs 2,10,000/-.

Output per hour rose 5.00 per cent and output per worker fell 5.50 per cent in the same invented year, with no disagreement between the two figures at all. The denominator has to be named before the number is allowed to mean anything. A cook at home makes the same point. Two hours at the stove produces four dishes, or two an hour. Cut the time to ninety minutes and the same cook produces three, still two an hour. Nothing about the cook has changed and the day now carries one dish fewer. Which of the two facts is meant has to be settled before anybody is praised or blamed.

If output can only rise two ways, how much of a given year came from each?

Turned around, the definition of output per worker gives the identity everything here runs on. If output per worker is output shared out over the workers, then output is the workers multiplied back up by what each one produces. Nothing has been assumed to get there, and no behaviour has been claimed. In rates, the same statement says that the factor by which output multiplies in a year equals the factor by which the workforce multiplies, times the factor by which output per worker multiplies. That is a decompositionSplitting one figure into the separate pieces that add or multiply up to it, so each piece can be traced to a cause rather than left inside a single number., and its purpose here is attribution: to say where a growth rate came from, not merely how large it was.

Sankhya year 2 is the first of the two pure cases. Output went from Rs 16,00,000 crore to Rs 16,80,000 crore, a rise of 5.00 per cent. The people producing numbered 8.00 crore when the year opened and 8.00 crore when it closed, so the workforce factor is exactly one. Output per worker went from Rs 2,00,000/- to Rs 2,10,000/-, a factor of 1.05. Multiply: one times 1.05 is 1.05, the 5.00 per cent the year began with. There were no additional people anywhere in the arithmetic to credit it to, so every single point of Sankhya year 2 growth was the same people producing more each.

Sankhya year 3 is the mirror image. Output went from Rs 16,80,000 crore to Rs 17,47,200 crore, a rise of 4.00 per cent. The number of people at work went from 8.00 crore to 8.32 crore, a rise of 4.00 per cent, a factor of 1.04. Output divided by workers in each year gives Rs 16,80,000 crore over 8.00 crore, or Rs 2,10,000/-, and Rs 17,47,200 crore over 8.32 crore, again Rs 2,10,000/-. The productivity factor is exactly one. Multiply: 1.04 times one is 1.04. Every point of Sankhya year 3 growth was additional people, and the amount each person produced did not move by a single rupee.

Those two years are pure cases, and pure cases do not happen. In a real year both sources move, usually by unequal amounts, sometimes in opposite directions, and the total on its own says nothing about the split. The reason to work the pure cases first is that they calibrate the eye: once 5.00 per cent that was entirely output per worker and 4.00 per cent that was entirely additional people have been seen, a mixed year stops looking like a single fact and starts looking like a sum nobody has done yet.

Take a year that did not happen, built on the same starting point as Sankhya year 2. Suppose the workforce had risen 2.00 per cent, from 8.00 crore to 8.16 crore, and output per worker had risen 3.00 per cent, from Rs 2,00,000/- to Rs 2,06,000/-. Multiply the two counts back together: 8.16 crore people at Rs 2,06,000/- each is Rs 16,80,960 crore of output, or growth of 5.06 per cent. Set that beside the year that did happen. Sankhya year 2 produced Rs 16,80,000 crore and grew 5.00 per cent. The two totals sit Rs 960 crore apart on an economy above sixteen lakh crore, close enough that any headline rounds them together, and underneath one of them the workforce did not move at all while under the other sixteen lakh people started working.

The 0.06 points between 5.06 and the 5.00 that comes from adding 2.00 and 3.00 is the cross term, the same creature that separates nominal growth from real growth plus inflation. Here it is the productivity gain applied to the additional workers as well as to the ones already there. The cross term stays negligible while both rates are small and stops being negligible as soon as they are not. Multiply the factors instead of adding the rates and nobody ever has to decide when to start caring.

All three Sankhya years and the mixture, in one place

The whole spine sits on one table, with every rate worked from the two levels beside it rather than carried over from anywhere. Down the last column the argument is already visible before any of it is argued.

Two things in that table are worth pausing on. Year 3 has the larger workforce and the higher output, and it is the year in which nobody got any better at anything. Year 2 has the smaller output total and it is the year in which every producer improved. A reader looking only at the output row would rank year 3 above year 2, and a reader looking at the per worker row would say year 3 was the year Sankhya stood still. Both readers are looking at correctly computed figures from the same table.

Why does growth from additional people run out, and growth from output per worker not?

Side by side, the two sources invite a question that is arithmetic rather than economics: how much of each is left? People are countable and finite, so for additional people the answer is a number that can be looked up. For output per worker there is no answer at all. Nothing anywhere in the arithmetic supplies one.

Work it on Sankhya. In year 3, 8.32 crore people were at work, out of a working age population of 13.20 crore. The labour force, meaning everyone working or looking for work, was 8.80 crore, so participationHow much of the working age population turns up in the labour force, meaning people either at work or looking for it. It can never pass one hundred per cent, since everybody counted is already inside the population underneath. was 66.67 per cent and those actually at work were 63.03 per cent of the working age population. Now suppose something impossible: every single person of working age goes to work, nobody studies, nobody is ill, nobody cares for a child or a parent at home and nobody retires. Employment goes from 8.32 crore to 13.20 crore. That is a rise of 58.65 per cent. Not per year. In total, once, for ever, at that working age population.

How long does 58.65 per cent last? Grow employment at year 3's own rate of 4.00 per cent a year and follow it: after eleven years the workforce is 12.81 crore, still inside the ceiling, and after twelve years the arithmetic wants 13.32 crore, more people than exist of working age. The entire stock of growth available from additional people, measured against a ceiling no economy has ever come close to touching, is used up inside twelve years at an ordinary hiring rate. After that, employment can grow only as fast as the working age population itself grows, and that count was settled by births roughly two decades earlier, so nothing decided in the current year moves it.

Now run the other source at exactly the same 4.00 per cent for the same twelve years. Output per worker goes from Rs 2,10,000/- to Rs 3,36,217/-, a multiple of 1.601, and there is no line anywhere in that calculation that says stop. Carry it to twenty five years and the multiple is 2.6658, and still nothing says stop. A source with a ceiling that arrives on a schedule nobody controls, set against a source that has no ceiling in the arithmetic at all, is the whole content of the word sustainable.

Two honest qualifications follow. The argument is stronger without overclaiming. The first is that no economy reaches full participation, and none should be expected to: the real limit binds long before 13.20 crore, and that binding makes the ceiling nearer than the calculation above, not further. The second is that having no ceiling in the arithmetic is not the same as being easy: no count of people stops output per worker from rising, and nothing at all promises that it will. Four things actually raise it, and after the fact very little of any rise can be explained.

What actually raises output per worker, and how much of a rise can anyone explain?

Four things, and they are worth naming with an everyday version attached, because the abstract list is forgettable and the concrete one is not.

The first is more or better equipment for each person to work with, and the name for it is capital deepeningMore equipment, tools, machinery or buildings behind each worker. Deepening, because the amount of capital per person gets deeper rather than merely bigger in total.. A tailor with one machine and a tailor with three are not equally productive, and neither is a delivery rider on a cycle and one on a scooter. The second is skills: the same rider who knows every lane in the neighbourhood delivers more in an hour than one reading a map. The third is the organisation of the work, and it is the one people forget. Two identical kitchens with identical equipment produce different numbers of meals if one of them has thought about where the chopping happens and where the plates are stacked. The fourth is technology, meaning a genuinely better way of doing the thing: a payment takes three seconds instead of a trip to a branch.

Now the honest part. The four overlap so heavily that separating them is not merely difficult, it is often impossible in principle. A new machine raises output only because somebody was trained to run it, and the training was worth doing only because the machine arrived, so which of the two gets the credit? The usual answer is to measure what can be measured, count the equipment and count the years of schooling, and then subtract. Suppose output per worker rose 5.00 points in a Sankhya year, and an exercise of that kind attributes 1.50 points to equipment and 0.80 points to skills. The two counted pieces come to 2.30 points. The remaining 2.70 points, or 54 per cent of the whole rise, is whatever is left.

The leftover is a residualWhatever is left after every measurable piece has been subtracted from a total. It is arrived at by subtraction, so it contains both the real thing being sought and every mistake made measuring the rest., and it has a name of its own, total factor productivity, sometimes called the Solow residual after Robert Solow, who set out the growth accounting it falls out of. The name makes it sound like a discovery, and it is closer to a confession: it is defined as the part nobody could account for, so every measurement error in the pieces that were counted ends up inside it too. That is not an argument for ignoring it. The residual is an argument for reading the sentence carefully whenever somebody says that better technology explained most of a productivity gain. The usual evidence for that sentence is that nothing else did.

Is working longer the same thing as being more productive?

No, and mixing the two is the commonest misreading of the word anywhere outside a statistics office. Watch a tailor for two days. On the first day the tailor works eight hours and finishes four shirts, or half a shirt an hour. On the second day the tailor stays two hours later and finishes five shirts. Output is up 25.00 per cent and it took 25.00 per cent more time, so the rate is still half a shirt an hour. Nothing about the tailor changed. The day got longer.

Now change one thing instead. Give the tailor a better machine, or a layout where the cloth is already cut and stacked when the tailor sits down, and let the day stay at eight hours. Six shirts come out instead of four. That is three quarters of a shirt an hour against half a shirt an hour, a rise of 50.00 per cent, and the tailor went home at the same time as before. Productivity is about what a given amount of effort produces, so it is the exact opposite of a demand for more effort. A call for higher productivity that arrives as a call for longer hours has quietly become a different demand.

The choice of denominator between workers and hours is not a technicality either. Measure the first day against the second on a per worker basis and the tailor looks 25.00 per cent better. Measure the same two days per hour and the tailor is exactly where the tailor started. Both figures are correctly computed and only one of them is about productiveness.

Where does a productivity gain actually land?

Put the Sankhya year 2 gain into rupees first, because the abstract version of this question is unanswerable and the concrete version is at least clear. Each person at work went from producing Rs 2,00,000/- in the year to producing Rs 2,10,000/-, so the gain is Rs 10,000/- a head. Multiply by the 8.00 crore people at work and the gain is Rs 80,000 crore, exactly the difference between Rs 16,00,000 crore and Rs 16,80,000 crore. Nothing is hiding: the whole of the extra output is sitting there, and the only remaining question is who ends up holding it.

There are three places it can go and any mixture of them is possible. The gain can go into pay, so the people producing take home more. It can go into prices, so buyers pay less for the same thing and keep the difference without ever knowing a productivity gain happened. Or it can stay with the producer as a wider margin. A household meets all three without naming any of them: a raise, a cheaper phone plan for the same data, and a shop that quietly makes more per sale than it did last year.

The split is decided by bargaining power, by how much competition the producer faces, and whatever else was going on in that year, including things that have nothing to do with productivity at all. A producer in a crowded market tends to find the gain competed into the price whether it wanted that or not. A producer with few rivals keeps more of it. Workers who can move easily to another employer capture more of it than workers who cannot. Which of the three ought to receive it is a distributional questionA question about how something already produced gets shared out among the people and businesses with a claim on it, rather than about how much of it was produced., and the arithmetic above stops at the producing.

One warning before leaving this block. Because the gain can land in a price rather than in a pay packet, a person can be more productive every year and feel none of it in their own income, and that experience is not evidence that the productivity figure was wrong. The mismatch is evidence that the productivity figure and the pay packet are two different measurements, and knowing which of the two is in hand is the whole skill.

How does a lender read two borrowers whose revenue rose by the same 5.00 per cent?

Move the decomposition into a lending file and it stops being an economics exercise and starts being a credit question. Take two invented garment units inside Sankhya, each producing Rs 4,00,00,000/- of revenue last year with 200 people, or Rs 2,00,000/- of revenue for each person. Both come to a lender this year showing revenue of Rs 4,20,00,000/-, a rise of exactly 5.00 per cent. On the revenue line they are indistinguishable.

Underneath, they are not. The first unit hired: it now runs 210 people, so revenue per person is still Rs 2,00,000/- and every rupee of the rise came from additional hands. The second unit did not hire: it still runs 200 people, so revenue per person rose to Rs 2,10,000/- and every rupee of the rise came from each person producing more. At a wage of Rs 1,20,000/- a year, the first unit wage bill went from Rs 2,40,00,000/- to Rs 2,52,00,000/-. The second unit wage bill did not move.

Now let the demand that produced the rise go away, and let revenue fall back to Rs 4,00,00,000/-. The second unit is back where it started with the same 200 people and the same wage bill. The first unit is back at the old revenue carrying Rs 12,00,000/- a year of additional wages, and it will keep carrying them until it decides to let people go. Letting people go is slow, costly and sometimes not possible at all. Growth bought with additional workers arrives with a cost that stays after the revenue leaves, and growth from output per worker does not. A lender who reads only the revenue line has read the least informative row in the file. The household version is the same shape: a household that raised its income by sending a second earner out to work has a different balance sheet from one that raised the same income through a promotion, because the first one also acquired a commute, a lunch cost and a person who now cannot do the things they used to do at home.