Chapter 6 Productivity

Joe Grice, Josh Martin, and Stuart Newman

What is productivity and why it is important?

What are the main productivity measures, and how do they relate to each other?

How productive is the UK, and how does it compare with other countries?

Which sectors and firms have the highest (and lowest) productivity?

What is the productivity puzzle, and what explains it?

Can we improve the way we measure productivity?

6.1 Introduction

Paul Krugman, an economist who won the Nobel prize, is regularly quoted as saying “Productivity isn’t everything, but in the long run, it’s almost everything.” What he meant by that is that productivity growth (or its absence) has a major influence on the quantity and quality of goods and services available to us and so, ultimately, our standard of living. If we are interested in raising people’s standard of living, we need to be equally interested in raising productivity. It is as simple as that.

We all have an intuitive sense of productivity. At a personal level, we usually know when we’ve been productive and when we have not. But what does it mean to scale that up to the whole economy? How would you measure it?

A key feature of any economy is the amount of output it is producing. At aggregate level, we often measure this as Gross Domestic Product (GDP). Essentially, it is the aggregate amount of goods and services that have been produced and that are available for consumption, or for investment which may lead to higher output in future.

factors of production
These are the inputs into the production process in order to produce output. These include the amount of labour and capital available. This covers tangible capital, such as machines and buildings, but also intangible capital such as the stock of human education.

At any time, the economy will have available to it a given set of factor endowments that can be used to generate this production. These are known as factors of production.

At individual firm level the amount and combination of factors employed can usually be varied; a firm producing drainpipes can produce more of them if it wishes by employing more people, or it can buy new machines so that its existing staff can produce more pipes.

At whole economy level, however, the factor endowments are more often fixed. Other things being equal, if one firm is employing more staff, it is likely to be attracting them from a different firm.

productivity
Productivity is the relationship between inputs and outputs in the economy – it captures how efficiently production inputs are being used to produce a given level of output, capturing how much output is generated per unit of input. The most recognised measure is labour productivity, which captures the amount of output per worker or per hour worked. Productivity plays a key part in determining a country’s long-term economic growth.

Aggregate factor endowments are not absolutely fixed. Working populations do change over time and capital can be increased by means of investment. But, over time, the greatest scope for increasing output is likely to come from using given factors of production – production inputs – more efficiently. Essentially, it is the degree of such efficiency that represents productivity.

The Solow-Swan growth model

The key model for how productivity growth happens was developed by Robert Solow in 1956. Trevor Swan also independently developed the growth model. The model focuses on exogenous growth by analysing changes in the workforce, the savings rate and the rate of technological progress.

The model demonstrates that output growth is linked to the growth in inputs (labour and capital) as well as new ideas and new technology. While labour is dependent on the population, the level of capital is driven by savings (as savings are equal to investment), so an economy can grow faster by saving more, leading to more capital per worker.

Importantly, the Solow-Swan model also demonstrates that technological advances and improvements in the productivity rate of labour and/or capital will produce higher rates of growth.

As we shall see, however, productivity growth has not been a positive story in recent years, and this is a matter for concern. Before we turn to these issues, we need to consider a little more carefully what productivity is and how it can be measured.

6.2 What is productivity?

At its simplest, productivity is the results you receive for the effort you put in: outputs from a certain level of inputs. Put another way, productivity is the rate of conversion of inputs to outputs.

From an economic viewpoint, an output is a good or service that an organisation produces, and an input is a resource that is used in the production of that good or service. For a firm, labour (workers) and capital (machines and so on) are the most important inputs.

6.2.1 The Cobb-Douglas production function

These elements were formalised by Charles Cobb and Paul Douglas. In a seminal article in the 1920s, they proposed a stylised algebraic representation of the production process:

\[Y = f(K,L)\]

On the face of it, this formula is pretty simple. It states that output (or production, represented by Y) is a function of the amount of capital (K) and labour (L) used in the production process. It is also quite general. But Cobb and Douglas went on to make their formula more specific.

\[Y = AK^{\propto}L^{1 - \propto}\]

This was intended to deal with two issues:

  1. The relative importance of labour and capital in producing output. This is captured in the parameter \(\propto\). If \(\propto\) was zero, then only labour would be relevant to producing output, and capital would drop out of the formula. At the other end of the scale, if \(\propto\) was one, then only capital would be relevant. Intermediate values of \(\propto\) represent situations between these two extremes. The idea was that empirical evidence could be used to estimate the actual value of \(\propto\) in a given economy. At a more granular level, some industries, such as oil and gas extraction, are heavily reliant on a high ratio of capital to labour, whereas others, such as many service industries, are more reliant on labour inputs. The former would have a relatively high value of \(\propto\) and the latter a lower one.
  2. The level of productivity. This is A in the equation. If we assume A is constant, then a firm can increase its level of output only by increasing its levels of capital and labour. As we noted above, however, at aggregate level at least, there will be limits to the extent that this is possible. So attention focuses on A. To what extent can output be increased by a higher level of A – in other words, supported by higher productivity?

6.2.2 What Cobb-Douglas tells us about productivity

The formula makes it obvious that, while there are a number of ways productivity might be improved, two of the most obvious ones will be:

You can read Cobb and Douglas’s original paper:

Cobb C W, Douglas P H (1928), ‘A theory of production’, American Economic Review, Volume 18, Number 1, Supplement, pages 139 to 165

Nearly a hundred years on, the Cobb-Douglas formulation has been found to be consistently a good way to capture and approximate reality. It is simple and easy to apply.

It has some inflexibilities though and, when studying productivity, you may see other production function formulations too. Examples include the Constant Elasticity of Substitution model (CES) and the Translog model.

6.3 Measuring productivity

Let’s consider a stylised example of a hypothetical bakery called Daily Bread. We will look at the various measures of productivity that can relate to its operations and how, by working through the other components of the Cobb-Douglas formula, we can arrive at estimates of A – productivity itself.

To illustrate the calculations, we will look at two ways in which Daily Bread could operate: one involving a single shift working and one with a double shift. We can then calculate what change in productivity is generated by a change between these two modes.

gross value added (GVA)
Gross Value Added (GVA) is the value generated by any unit engaged in the production of goods and services. It measures the contribution to the economy of each individual producer, industry or sector. Simplistically it is the value of the amount of goods and services that have been produced, less the cost of all inputs and raw materials that are directly attributable to that production.

The key point here is that productivity is essentially a ratio of outputs (Y in our Cobb-Douglas formulation above) to inputs (\(K^{\propto}L^{1 - \propto}\)). The output side we can deal with quite quickly. What we are concerned with is the gross value added from the economy’s production processes, as discussed in Chapter 2 (see Sections 2.2.1 and 2.2.2).

Furthermore, since we are interested in real living standards, we are mainly interested in the volume measure of GVA (see Chapter 2), abstracting from movement in price levels. Measuring output is far from straightforward. But in our context of measuring productivity, there are no new issues more complex than those discussed in Chapter 2.

The new questions are how to measure the inputs, which are the factors of production such as labour and capital.

6.3.1 Measuring labour productivity

labour productivity
The total output produced relative to the amount of labour used to produce it, capturing the efficiency of the workforce. It reflects output per unit of labour input. There are several possible measures of labour, including the number of employees, the number of jobs or the number of hours worked.

Labour productivity is a commonly used measure for productivity as a whole. It can be thought of as productivity of labour without regard to what is happening to capital. In the short term at least, capital is likely to be pretty fixed, so this may not be an unreasonable assumption if we want to estimate Daily Bread’s productivity.

The bakery produces output: loaves of bread. For simplicity we assume that all Daily Bread’s loaves are the same size and quality. We can measure its output by counting the number of loaves it bakes in 24 hours.

Daily Bread uses inputs in the form of labour, capital (its baking oven and premises it operates in), and materials including flour, yeast and salt. We will assume the capital component is fixed, so Daily Bread can vary its output of loaves only by varying the amount of labour and materials used. The capacity of the oven is limited to 100 loaves per hour, and we might assume that a skilled baker can operate at the same rate, mixing and preparing the dough for the next 100 loaves while the previous batch is in the oven.

Suppose our baker works eight hours a day, and that the first hour each day is spent preparing the first batch of loaves for baking and the last hour for cleaning up, ordering materials and so on. This implies six hour-long baking cycles per day, and if we assume that Daily Bread’s skilled baker works at full capacity, output is 600 loaves per day.

Suppose also that Daily Bread employs a sales assistant to sell the output. The sales assistant works for seven hours a day, as there is no bread to sell in the first hour. (Note, that while the baker might physically produce loaves of bread, the sales assistant provides the vital service of marketing them to customers, and productivity in this example depends on selling bread as well as making it.)

In this scenario, output is 600 loaves per day, hours worked would be 15, so (daily) labour productivity would be 600 divided by 15, or 40 loaves per hour. We set this out in Figure 6.1.

  Labour input (hours) Output (loaves) Labour productivity (loaves per hour)
Bakers 8    
Sales Assistants 7    
Total 15 600 40.0

Figure 6.1 Single shift working at the Daily Bread bakery

Single shift working at the Daily Bread bakery

If productivity depends only on labour inputs, how might it change?

Imagine that the baker and the sales assistant each want to work fewer hours. The baker works seven hours, so the sales assistant works six hours each day.

Two hours per day are needed for preparation and cleaning up, so there are now five baking cycles. Output would be 500 loaves, hours worked would be 13, and labour productivity falls to 500 divided by 13. Daily Bread’s productivity would be 38.5 loaves per hour instead of 40.

Imagine the head baker goes on holiday, and a less skilled baker takes over. This baker can prepare only 90 loaves per hour, rather than 100. In the eight-hour shift, that implies 540 loaves a day, but both staff members work the same hours. So labour productivity would be 540 divided by 15. It falls to 36 loaves per hour while the head baker is on holiday.

If the CEO of Daily Bread wanted to improve productivity, one obvious way would be to ask the staff to work longer hours. But our baker doesn’t want to work more hours. And so the decision is made to institute a double shift.

In the new environment, the head baker works 7 hours per day. Daily Bread recruits another equally efficient baker who takes over at the end of his or her shift. If the second baker also works 7 hours per day, at full capacity of 100 loaves per hour, then daily output will increase to 1,200 loaves (12 baking cycles, as two hours per day would still be required for preparation and clean up).

Of course that means a second sales assistant, so that sales can take place over 13 hours rather than 7 hours. And so total hours worked would be 27, and labour productivity would be 1,200 divided by 27. Productivity jumps to 44.4 loaves per hour (Figure 6.2).

  Labour input (hours) Output (loaves) Labour productivity (loaves per hour)
Bakers 14    
Sales Assistants 13    
Total 27 1,200 44.4

Figure 6.2 Double shift working at the Daily Bread bakery

Double shift working at the Daily Bread bakery

This increase in labour productivity reflects increased efficiency in the use of labour and capital, for example, by reducing the proportion of down-time relative to baking cycles.

The change from single to double shift working has raised productivity from 40.0 to 44.4 loaves per hour worked. But how should we express this as a change? We use natural logarithmic changes, which are the same whether you are measuring an increase or a decrease, unlike arithmetic percentage changes.

How it’s done Logarithmic changes vs percentage changes

The usual way to measure changes is to calculate the percentage change. As it happens, 44.4 is exactly 11% higher than 40, so we might say the improvement in productivity has been 11%. But we could just as well turn the calculation round and consider the reduction in productivity that single shift working represents as compared with that of the double shift. The percentage reduction would be given by (100(1 – 40/44)) or 9.1%.

Are we to say that the change in productivity between the two modes of working is 11% or 9.1%, or something else?

As we can see, arithmetic percentages can lead to ambiguity and confusion. So let us consult the guidance published by Measuring Productivity. OECD Manual: Measurement of Aggregate and Industry-level Productivity Growth.

This is detailed, comprehensive and informative, if not necessarily the best holiday reading, and it suggests that productivity change should normally be expressed in logarithmic percentages rather than arithmetic ones.

If we consider the increase in double shift working over single shift, the logarithmic percentage change is given by

ln(44) – ln(40) = 9.5 log percent

or, the other way round, the reduction in productivity obtained by the single shift compared with the double shift is

ln(40) – ln(44) = –9.5 log percent

Either way round, the change in productivity is 9.5 log percent, with no ambiguity.

Measures of labour productivity often only go this far, calculating the productivity of the economy (or the bakery) by dividing the volume of output (number of loaves of bread) by the total amount of labour used (hours worked). This is the measure most often reported, partly because it is fairly simple.

multi-factor productivity (MFP)
Multi-factor productivity reflects the overall efficiency with which labour and capital inputs are used together in the production process. It is recorded in a growth accounting framework, in which the growth in MFP is measured as a residual – that is, the part of GDP growth that cannot be explained by changes in labour and capital inputs. It is also called total factor productivity or TFP.

But we can do better and get closer to a true measure of productivity, often known as multi-factor productivity (MFP).

6.3.2 Measuring labour quality

At Daily Bread the replacement baker could not produce as many loaves in an hour, because we counted output as the number of loaves produced. The difference in the quality of labour could be measured. But often differences in the labour input from different types of workers are not taken into account when we calculate labour productivity, even though it may be important.

This section introduces quality-adjusted labour input, which breaks down labour input into hours worked and labour quality.

quality-adjusted labour input (QALI)
An input into estimating multi-factor productivity (MFP). QALI accounts for changes in the composition (or “quality”) of the employed workforce as well as changes in hours worked. It weights hours worked by different types of workers by their relative income share, reflecting their contribution to economic production.

The measure known as quality-adjusted labour input (QALI) attempts to take into account differences in types of labour. Economic theory suggests that wages reflect the value that labour adds to production. For example, a skilled sheep shearer is likely to be able to shear more sheep in a given time than a general farm labourer tackling a sheep for the first time (and many more sheep than any of us could shear). And so, on a farm, employing an experienced sheep shearer is likely to produce more output (and thus more value added) in a given time than if you employed an economist to do the job. But, for farm work, the shearer’s hourly wages would be correspondingly higher as well.

A QALI measure accounts for these variations in labour composition or “quality” by weighting the hours worked of different worker types by their relative pay shares, that is, their shares of the total wage bill.

Back at Daily Bread, the head baker and the sales assistant are working together in a single shift, and two skilled bakers and two sales assistants sometimes work in a double shift. Let’s assume that the training and experience required of a baker is reflected in a going rate for bakers of £12 per hour, while the going rate for sales assistants is lower, at £8 per hour.

Figure 6.3 will incorporate this information.

Törnqvist index
A Törnqvist index is a form of index where the weight is constructed using an average of the relevant variable in the current and base period. As such, it is a symmetric index as it gives equal weight to both periods.

But we need to be more specific as to how we are going to incorporate these apparent quality differences. The international guidance recommends that doing so should be based on a Törnqvist index (see the Appendix 2, “Basic Statistical Skills” for a reminder of what this means).

Using this formula, the change in the quality-adjusted labour input between two periods is given as Qtt−1 where

Qtt−1 = Pi (Hours workedit / Hours workedit−1) (Weightit + Weightit−1 /2)

and

Hours workedit is the hours worked by the ith group in period t. (In our case, we have only two groups: bakers when i = 1 and sales assistants when i = 2)

Weightit is the share of the total pay bill in period t earned by the ith group.

If you look at this all at once, it is a formidable looking formula. But think about the process used to calculate it, and it is actually not quite as forbidding as it looks. For each group of workers:

Intuitively, this gives us what we want. The QALI rises if the hours worked by the higher wage workers (in our case, the bakers) increases faster than the total hours worked.

Once more, we can also simplify the calculation by working in natural logarithmic terms. If we work in terms of log Qtt−1, we can change the formula to become more straightforward. The log change in QALI then becomes the change in hours worked expressed as logarithms, weighted as before by the average income shares.

Applying the formula to the data (see Figure 6.3):

Scenario Worker types Labour input
(hours)
Pay
(£/hour)
Pay bill
(£/day)
Pay shares (%)
Single shift working Bakers 8 12 96 63.2
Sales Assistants 7 8 56 36.8
Total 15 152 100.0
Double shift working Bakers 14 12 168 61.8
Sales Assistants 13 8 104 38.2
Total 27 272 100.0

Figure 6.3a Introducing quality-adjusted labour input (QALI) at Daily Bread

Introducing quality-adjusted labour input (QALI) at Daily Bread

  Average pay shares (%) Change in hours (log%) Change in QALI (log%)
Bakers 62.5 56.0 35.0
Sales Assistants 37.5 61.9 23.2
Total 100 58.8 58.2

Figure 6.3b Change from single to double shift working at Daily Bread

Change from single to double shift working at Daily Bread

What does this tell us? If we made no distinction between hours supplied by the bakers and the sales assistants, we see total hours worked under the double shift system comes to 27 as against 15 with the single shift. So the increase in hours worked is log(27) less log(15) or 58.8 (log) percent. This is more than the increase we calculated in the QALI.

Another way of putting this is that the labour composition or “quality” has declined by 0.6 (log)% in the double shift regime.

The key to what is going on is to look at the composition of the workers in the two cases. In the single shift illustration, there were 8 high-skilled (high-cost) worker hours and 7 low-skilled (low-cost) ones. In the double shift illustration, there are 14 high-skilled and 13 low-skilled worker hours. The ratio of high- to low-skilled worker hours is reduced.

The double shift system has allowed relative economising on the high-cost labour input. That saving is a further factor contributing to the higher productivity that the double shift system allows, and that is what our analysis is picking up.

6.3.3 Measuring the contribution of capital inputs

capital input
Capital input includes anything that provides a means of producing output without being completely used up in the production process.

So far, we have focused entirely on labour inputs. The next stage is to take account of capital inputs to the production process.

At Daily Bread, it is easy to see the capital inputs. They include the oven and the building that the baker and the sales assistant work in.

Are the ingredients like flour capital inputs? No, because once flour is used to make a loaf of bread, the same flour cannot be used again to produce another loaf of bread. These are intermediate goods, as discussed in Chapter 2 on GDP.

capital services
These are flows of productive services from capital assets, rather than the capital stock of those assets. They are directly comparable to flows of labour services measured by QALI.

For productivity purposes, we measure capital inputs by estimating the services they provide. In principle, measurement of capital services requires lots of information on the accumulation of capital assets over time, as well as a number of assumptions on the lives of different types of assets, how the productive efficiency of assets changes over their lifetimes, and the nature of the returns on capital.

However, we can cut through this complexity by noting that the value of the capital services a firm uses is the same as the costs that firms would pay if they were to rent all their capital assets in competitive markets.

Taking this further, we can consider the two costs that an owner of capital would incur, that he or she needs to recover from the rental charged:

In a competitive rental market for capital, these costs will be reflected in the market rental rate that emerges. Let’s assume the market rent paid by our Daily Bread on its capital is a fair reflection of the capital services provided by these assets. For illustration, we assume that the costs to Daily Bread are:

The different rental payments for the two types of capital reflect the fact that you don’t pay a higher rent to your landlord if you use the building more because it doesn’t wear out, whereas equipment such as ovens will suffer wear and tear and have to be replaced sooner if they are used more intensively.

Figure 6.4a summarises these assumptions.

Asset types Charge unit Cost
(£/charge unit)
Cost (£/day) Cost shares (%)
Single shift working Premises Day 75 75 50.7
Oven etc. Day plus hours used 13 per day plus 10 per hour 73 49.3
Total 148 100.0
Double shift working Premises Day 75 75 36.1
Oven etc. Day plus hours used 13 per day plus 10 per hour 133 63.9
Total 208 100.0

Figure 6.4a Introducing capital services

Introducing capital services

Asset types Charge unit Cost
(£/charge unit)
Cost (£/day) Cost shares (%)
Single shift working Premises Day 75 75 50.7
Oven etc. Day plus hours used 13 per day plus 10 per hour 73 49.3
Total 148 100.0
Double shift working Premises Day 75 75 36.1
Oven etc. Day plus hours used 13 per day plus 10 per hour 133 63.9
Total 208 100.0

Figure 6.4aa A single shift

A single shift

The premises’ cost is £75 per day. The cost of the ovens, etc., is £13 plus £10 for each of the six hour-long bakes carried out under that shift arrangement, or £73.

Asset types Charge unit Cost
(£/charge unit)
Cost (£/day) Cost shares (%)
Single shift working Premises Day 75 75 50.7
Oven etc. Day plus hours used 13 per day plus 10 per hour 73 49.3
Total 148 100.0
Double shift working Premises Day 75 75 36.1
Oven etc. Day plus hours used 13 per day plus 10 per hour 133 63.9
Total 208 100.0

Figure 6.4ab A single shift

A single shift

So the total capital services used under this system is £148 per day.

Asset types Charge unit Cost
(£/charge unit)
Cost (£/day) Cost shares (%)
Single shift working Premises Day 75 75 50.7
Oven etc. Day plus hours used 13 per day plus 10 per hour 73 49.3
Total 148 100.0
Double shift working Premises Day 75 75 36.1
Oven etc. Day plus hours used 13 per day plus 10 per hour 133 63.9
Total 208 100.0

Figure 6.4ac A double shift

A double shift

The cost of the premises is the same at £75 a day. But there are now 12 bake cycles each day and so the cost of the ovens, etc., rises to £133.

Asset types Charge unit Cost
(£/charge unit)
Cost (£/day) Cost shares (%)
Single shift working Premises Day 75 75 50.7
Oven etc. Day plus hours used 13 per day plus 10 per hour 73 49.3
Total 148 100.0
Double shift working Premises Day 75 75 36.1
Oven etc. Day plus hours used 13 per day plus 10 per hour 133 63.9
Total 208 100.0

Figure 6.4ad A double shift

A double shift

The total capital services used with the double shift system come to £208.

We can now use this information to calculate the change in capital services between the two patterns of shift working. To be consistent with the information we calculated for quality adjusted labour input, we want this change to be in natural log percentage terms and we can apply the same formula for this purpose (Figure 6.4b):

Asset types Average cost shares (%) Change in costs (log %) Change in capital services (log %)
Premises 43.4 0.0 0.0
Oven etc. 56.6 60.0 34.0
Total 100.0 34.0 34.0

Figure 6.4b Change from single to double shift working

Change from single to double shift working

Asset types Average cost shares (%) Change in costs (log %) Change in capital services (log %)
Premises 43.4 0.0 0.0
Oven etc. 56.6 60.0 34.0
Total 100.0 34.0 34.0

Figure 6.4ba Premises

Premises

We can calculate the average cost share between the two cases as (50.7 + 36.1)/2 or 43.4%. But as it happens, in this particular case, this is academic because the change in costs for premises is zero. So the product is also zero.

Asset types Average cost shares (%) Change in costs (log %) Change in capital services (log %)
Premises 43.4 0.0 0.0
Oven etc. 56.6 60.0 34.0
Total 100.0 34.0 34.0

Figure 6.4bb Ovens

Ovens

The average cost share is 56.6%. This time, it does matter. The log change in costs is given by ln(133) – ln(73) which gives a change of 60 (log) percent. The change in capital services is given as the product of the average cost share and this change in cost. So the change in capital services in respect of ovens, etc., is 34.0 (log) percent.

Asset types Average cost shares (%) Change in costs (log %) Change in capital services (log %)
Premises 43.4 0.0 0.0
Oven etc. 56.6 60.0 34.0
Total 100.0 34.0 34.0

Figure 6.4bc Total

Total

The total change in capital services is the sum of those for premises and ovens, etc. This is also 34.0%, because there is no change stemming from use of premises.

At first sight, this increase in capital services in moving to a double shift system may seem counter-intuitive: after all, the amount of physical capital employed has not changed. But in moving to double shift working, the bakery is using its physical capital more intensively and therefore the flow of capital services has increased. The greater number of baking cycles will have increased the wear and tear on the ovens, and this is reflected in the higher capital cost that has to be borne.

We should note that the example in Figure 6.4 relies on rental charges fairly reflecting the value of each type of capital. In the real world, rental markets for capital assets are often non-existent and most capital assets are owned directly by the firms that use them, albeit often financed by borrowing.

Moreover, even where rental markets exist, rental prices may be unrepresentative, as they might include bundled labour services (such as an operator that is supplied with a crane) and profit margins for the rental organisation. Long-term rentals will typically include an allowance for general inflation, which we are not including in our simplified scenarios.

6.3.4 Calculating labour and capital shares

labour share
The labour share of income estimates the income received by labour in the generation of value added, which includes compensation of employees.
capital share
The capital share of income estimates the income received by capital in the generation of value added, which includes gross operating surplus.

So far, we have separately measured the changes in quality-adjusted labour input and capital services between the two scenarios. To calculate productivity, we need a combined measure of inputs, and we get that by weighting together labour and capital inputs. For this, we need the labour share and the capital share, which are usually taken to be the shares of the costs of production.

Before accounting for the labour and capital shares, we need to properly account for our bakery’s output. Earlier, we thought about output as the number of loaves of bread. In this section, and the next, we will need to go back to monetary values, so we will need to think about the price of a loaf.

Let’s assume that each loaf is sold for £1. So total revenue (gross output) from the 600 loaves in the single shift pattern is £600 a day, and from the 1,200 loaves in the double shift system, daily revenue is £1,200.

However, the bakery will also incur the cost of the ingredients to make the loaves – the intermediate inputs. Let’s assume these are 50p for every loaf. So half of the revenue is eaten up by intermediate costs, and gross value added (the value of the output added deducting intermediate costs) is half the revenue – £300 a day in the single shift system, and £600 with double shifts.

Gross value added generated by the business goes to three main places:

With a clear breakdown of the Daily Bread’s accounts, we can combine the various inputs. To do this, and thus calculate productivity, we need to calculate the labour and capital shares of the production costs and we need the Cobb-Douglas function again.

From the Cobb-Douglas production function, the capital share is \(\propto\) and the labour share is 1 minus \(\propto\). Since they must add to 1, we can just as easily find the labour share, and calculate the capital share as 1 minus that. This is easier, so we’ll do that.

Based on the selling price of our loaves, and the costs of intermediates, and the costs of the wages of the bakers and shop assistants from Figure 6.3, we have all we need. This calculation is summarised in Figure 6.5.

Scenario GVA (£/day) Labour costs (£/day) Labour costs (% of GVA) Average labour cost share (%) Average capital cost share (%)
Single shift working 300 152 50.7    
Double shift working 600 272 45.3    
Average       48.0 52.0

Figure 6.5 Labour and capital shares

Labour and capital shares

Scenario GVA (£/day) Labour costs (£/day) Labour costs (% of GVA) Average labour cost share (%) Average capital cost share (%)
Single shift working 300 152 50.7    
Double shift working 600 272 45.3    
Average       48.0 52.0

Figure 6.5a Gross value added

Gross value added

This is £300 a day in the single shift system, and £600 a day in the double shift system.

Scenario GVA (£/day) Labour costs (£/day) Labour costs (% of GVA) Average labour cost share (%) Average capital cost share (%)
Single shift working 300 152 50.7    
Double shift working 600 272 45.3    
Average       48.0 52.0

Figure 6.5b Labour costs

Labour costs

From Figure 6.3, we know the wage bill costs per day are £152 per day and £272 per day under the one and two shift modes of working respectively.

Scenario GVA (£/day) Labour costs (£/day) Labour costs (% of GVA) Average labour cost share (%) Average capital cost share (%)
Single shift working 300 152 50.7    
Double shift working 600 272 45.3    
Average       48.0 52.0

Figure 6.5c The labour share of output

The labour share of output

This is 50.7% in the single shift system, and 45.3% with double shifts.

Scenario GVA (£/day) Labour costs (£/day) Labour costs (% of GVA) Average labour cost share (%) Average capital cost share (%)
Single shift working 300 152 50.7    
Double shift working 600 272 45.3    
Average       48.0 52.0

Figure 6.5d The average labour share

The average labour share

This is therefore 48.0% and the capital share by implication is 52.0%. These calculations are important for the Törnqvist index formula noted above.

At Daily Bread the labour cost share is 48%, averaged over the two scenarios. The typical real-life labour share in the UK economy is around two-thirds.1

6.3.5 Bringing it all together

The measurements for labour and capital input noted above form part of a growth accounting framework that comes directly from the Cobb-Douglas production function. Each part of the function needs to be accounted for to determine its contribution to growth.

Revisiting the formula, we can see that so far, we’ve discussed output (Y), the labour (L) and capital (K) inputs and the weights (\(\propto\) and (1−\(\propto\))) we apply to each of them.

\[Y = AK^{\propto}L^{1 - \propto}\]

The remaining element is the A, which, as we have mentioned, is productivity, and which we want to quantify.

A is multi-factor productivity (MFP), as we defined it in section 6.3.1, and it measures the amount of output growth that cannot be accounted for by changes in inputs of quality-adjusted labour and capital.

Back at Daily Bread, we now have all we need to calculate the change in A when it moves from single to double shift working, shown in Figure 6.6.

Change in output (GVA) (log%) Weighted change in QALI (log%) Weighted change in capital services (log%) Change in MFP (%)
69.3 27.9 17.7 23.7
Change in QALI (log%) Labour cost share (%) Change in capital services (log%) Capital cost share (%)
58.2 48.0 34.0 52.0

Figure 6.6 Decomposition of change in output for Daily Bread

Decomposition of change in output for Daily Bread

Change in output (GVA) (log%) Weighted change in QALI (log%) Weighted change in capital services (log%) Change in MFP (%)
69.3 27.9 17.7 23.7
Change in QALI (log%) Labour cost share (%) Change in capital services (log%) Capital cost share (%)
58.2 48.0 34.0 52.0

Figure 6.6a The increase in output (measured by gross value Added)

The increase in output (measured by gross value Added)

In log percent terms, this is given by ln(600) less ln(300) or 69.3 log percent.

Change in output (GVA) (log%) Weighted change in QALI (log%) Weighted change in capital services (log%) Change in MFP (%)
69.3 27.9 17.7 23.7
Change in QALI (log%) Labour cost share (%) Change in capital services (log%) Capital cost share (%)
58.2 48.0 34.0 52.0

Figure 6.6b The (log) change in quality-adjusted labour input (QALI)

The (log) change in quality-adjusted labour input (QALI)

The figure in Figure 6.3 (58.2%) multiplied by the labour share in Figure 6.5 (48.0%) gives a weighted contribution of 27.9% to the change in output.

Change in output (GVA) (log%) Weighted change in QALI (log%) Weighted change in capital services (log%) Change in MFP (%)
69.3 27.9 17.7 23.7
Change in QALI (log%) Labour cost share (%) Change in capital services (log%) Capital cost share (%)
58.2 48.0 34.0 52.0

Figure 6.6c The (log) change in capital services

The (log) change in capital services

The figure in Figure 6.4 (34.0%) multiplied by the capital share in Figure 6.5 (52.0%) gives a weighted contribution of 17.7%.

Change in output (GVA) (log%) Weighted change in QALI (log%) Weighted change in capital services (log%) Change in MFP (%)
69.3 27.9 17.7 23.7
Change in QALI (log%) Labour cost share (%) Change in capital services (log%) Capital cost share (%)
58.2 48.0 34.0 52.0

Figure 6.6d The change in productivity between our two scenarios

The change in productivity between our two scenarios

This is the residual change in output that cannot be explained by the change in inputs. Subtracting the contributions due to changes in labour and capital inputs from the change in output leaves a residual of 23.7 log percent that cannot be attributed to changes in inputs and therefore represents a change in MFP.

The figure of 23.7% is the productivity improvement that moving from single to double shift at Daily Bread provides.

How it’s done A further decomposition of change in output per hour

As we saw earlier, if we divide output by hours worked, we arrive at a simple measure of labour productivity. Figure 6.7 shows how this can similarly be decomposed into three components:

  • A component reflecting the change in labour composition per hour worked (change in QALI – change in hours worked)
  • A component reflecting the change in capital services per hour worked, or capital deepening (change in capital services – change in hours worked)
  • A residual MFP component.
Change in output per hour (log%) Weighted change in labour composition (log%) Weighted change in capital services per hour worked (log%) Change in MFP (log%)
10.5 −0.3 −12.9 23.7
Change in labour composition (log%) Labour cost share (log%) Change in capital services per hour worked (log%) Capital cost share (log%)
6.0 48.0 −24.8 52.0

Figure 6.7 Decomposition of change in output per hour

Decomposition of change in output per hour

Note that while capital services increase by 34.0% between the two scenarios, capital services per hour fell by 24.8% (the difference between these numbers is, of course, the growth in hours worked of 58.8%). So, the double shift scenario is less capital intensive than the single shift scenario.

The change in MFP (23.7%) is identical whether we decompose output growth or growth of output per hour. The fact that it is positive suggests that the single shift scenario is inefficient: the bakery can become more efficient by using its fixed capital (that is, its premises and the fixed element of its baking equipment) more efficiently.

6.4 UK productivity: A set of paradoxes

In the introduction, we argued that what is happening to productivity is of primary importance not least as the overriding medium-term influence on people’s standard of living. Having looked at the various productivity measures and how they relate to each other, and how in practice they can be measured, it is now time to consider what the experience has been in the UK.

In the context of the UK, the discussion can be ordered around some paradoxes.

  1. Low UK productivity. Why has the UK had apparently poor productivity as compared with similar countries?

For a long time, the UK “productivity puzzle” referred to this question. But now we have two more:

  1. A productivity slowdown. Since the 2008/09 financial crisis, UK productivity growth has been a fraction of that in previous decades.
  2. Persistent differences in productivity between firms. This occurs even within the same sector and appears to be large. So apparently low-productivity firms are not learning by observing high-productivity firms.

We will look at each of these paradoxes in turn.

6.4.1 Paradox 1: International comparisons

ONS Resource

The ONS publishes periodic comparisons of productivity achieved in the major economies, based as far as possible on OECD methodologies and data. For example, see International comparisons of UK productivity (ICP), first estimates, Statistical bulletins.

Figure 6.8 is taken from an ONS productivity release. This chart shows UK productivity in 2015 and 2016 as 100 and the value for the other G7 countries relative to it. Productivity in Canada is a little lower than in the UK, while Japan was about 10% lower. But in all the other G7 countries, productivity was higher than in the UK – in the case of Germany more than a third higher. Taking an average of other G7 countries weighted by their economy’s size, UK productivity was around 18% lower than these economic competitors.

Figure 6.8 Gross domestic product per hour worked: G7 countries

Gross domestic product per hour worked: G7 countries

There is very little change in these relative positions between the two years. In fact, this stability goes back much further, as shown in Figure 6.9. With a few gyrations from year to year, the overall picture shows little change over the last two decades.

Figure 6.9 GDP per hour worked relative to the UK

GDP per hour worked relative to the UK

(1995 to 2016, UK = 100 in all years)

As against the rest of the G7 as a whole, the UK has become, if anything, marginally less productive relatively. This failure of the UK to improve its comparative productivity position comes in the face of continuing government programmes intended to address the issue.

Policies to boost productivity growth

Successive governments since the 1960s have recognised the productivity problem and have adopted policies and programmes designed to improve matters. The Department for Business, Energy, and Industrial Strategy (BEIS) as it is now called, formerly the Department for Business, Innovation and Skills, has been the home of many of these. Policies have generally aimed to increase capital investment by businesses, increase business innovation (through research and development (R&D) and science), or make finance more available for small and medium sized enterprises (SMEs). There have been many policies in recent decades, including the following:

The current policy framework is set out in The UK’s industrial strategy published in 2017.

These policy frameworks were no doubt constructed in accordance with the best evidence and appear to have targeted important issues. But all we can see is what actually happened. We do not have the counterfactual figures on what would have been the outcome in the absence of such policies. The position might have looked much worse. All the same, it seems clear that over a long period, successive policies have failed to improve the UK’s comparative productivity performance.

Diagnosing the causes of this persistent weakness is problematic, because the UK scores well when we look at most of the likely reasons for it:

One well-regarded comparative assessment of educational attainment in different countries is provided by the OECD’s Programme for International Student Assessment (PISA), which has run every three years for over two decades. It assesses 15-year-old students in mathematics, science, and reading, in a consistent way across countries. In the 2018 exercise, the UK ranked in the mid-teens of 79 countries across the board – a small improvement on the previous rankings.

In this light, it is hard to discern a clear explanation of the persistent low UK productivity relative to most other countries. To make matters worse, it has now been joined by the second of the trio of puzzles: the sharp deceleration of UK productivity growth since 2007.

6.4.2 Paradox 2: The collapse in productivity growth

Until around 2007, UK productivity was growing at a steady rate of around 2% a year. As we have seen, this turned out not to be sufficiently fast to close the gap in productivity performance between the UK and other G7 countries. But it did nevertheless support a long-term increase in real wages and living standards. Moreover, it was a well-established trend: productivity had been growing at this rate or a little higher for the previous six decades.

But from 2007 onwards, there was a sharp break in behaviour. Productivity growth slowed to much lower rates. In fact, by the end of 2019, productivity was estimated to be only 3% higher than it had been at the end of 2007. The implied average growth of only 0.25% a year is clearly much lower than what had preceded it. The suddenness of the break is demonstrated clearly in Figure 6.10.

Figure 6.10 Productivity, UK, January to March 1997 to October to December 2020

Productivity, UK, January to March 1997 to October to December 2020

The yellow line shows the actual path estimated for productivity, with the break in growth rates clearly visible. The red dotted line projects the levels of productivity that would have been achieved, had growth been sustained at the levels of the previous 60 years. As a matter of straightforward arithmetic, we can then calculate the “productivity gap” – the shortfall in productivity as a result of this mysterious change in behaviour. By the end of 2019, this came to some 23%. By equally simple arithmetic, we can then derive an estimate of “missing output”, how much more GDP or national income there would have been had productivity continued to grow as before. This equates to approximately £350 billion, which is clearly a very large amount of money: to put it in context, it would represent enough missing national income to finance the National Health Service in its entirety each year more than twice over.

What caused this abrupt break in productivity growth?

Many theories have been advanced to explain this break, but none has reached consensus – it is likely that there is no single explanatory factor, but rather a combination of things that have come together. Based on Figure 6.10, it is clear that explanations must be sufficiently large to explain the considerable break in the time series; come into effect around 2007 (or at least increase notably from this point onwards); be ongoing or long-lasting; and be apparent in many developed countries, but especially pronounced in the UK.

Some theories that were popular shortly after the 2007/08 economic downturn have since fallen out of favour:

Robert Gordon summarises his argument in relation to the US in this book:

Gordon R (2016), The rise and fall of American growth, Princeton University Press

Other explanations, based on the slowdown in investment, are more credible:

ONS Resource

Read more about the various explanations for the productivity puzzle in the ONS article on Productivity measurement, and see the further reading section for more accounts of the arguments.

Macroeconomic explanations have not so far been underpinned by extensive empirical testing that might serve to confirm or refute them. But these macro-based theories may explain the sharp break in behaviour since the financial crisis, because a change in microeconomic factors alone are an implausible explanation. We cannot argue that skills or management practices changed suddenly for the worse in 2008.

Examples of research arguing that the failure to invest in innovation is driving low investment:

Carlin W, Soskice D (2018), ‘Stagnant productivity and low unemployment: Stuck in a Keynesian equilibrium’, Oxford Review of Economic Policy 34

Oulton N (2019), ‘The UK (and Western) productivity puzzle: Does Arthur Lewis hold the key?’, International Productivity Monitor, Voume.36, pages 110 to 141

6.4.3 Paradox 3: The behaviour of disaggregated productivity

We have focused on experience at the level of the whole economy. But it is instructive to look at what has been happening at lower levels of aggregation – by sector or by individual firm, for example.

ONS Resource

In the UK, the main source of information on individual business is the Annual Business Survey (ABS) carried out each year by the ONS of around 62,000 respondents. The ABS covers the non-financial business sector and relates to approximately two-thirds of the UK economy. It provides information on businesses’ turnover and intermediate purchases to give a good approximation to their gross value added. This approximation is called aGVA.

In addition, the Interdepartmental Business Register (IDBR) can be used to derive employment information about the firms responding to the ABS. The Business Register Employment Survey and HMRC records are both used to provide this employment information.

The Interdepartmental Business Register (IDBR) is a comprehensive list of UK businesses – around 2.7 million businesses in all sectors of the economy – used by government for statistical purposes, and it provides the sampling frame for business surveys such as the ABS. The two main sources of input are value added tax (VAT) and pay as you earn (PAYE) records from HMRC, with additional information coming from Companies House, Dun and Bradstreet and ONS business surveys. The main characteristics that are covered by the IDBR are:

  • employment (various size bands)
  • geography
  • industry
  • domestic or foreign ownership.

Together, these sources allow the construction of reasonable productivity estimates by individual firm, computed as aGVA divided by the employment headcount.

In 2020, the ONS published an article ‘Firm-level labour productivity measures from the Annual Business Survey, Great Britain: 1998 to 2018’. This article contains a wealth of interesting analysis, with the main results summarised in Figure 6.11.

Classification Sub-classification GVA (£000)
Industry Accommodation and food services 18.0
Education, health and social care, and other services 19.0
Administrative services 20.0
Retail 25.0
Transport and storage 41.0
Construction 43.5
Real estate 47.0
Professional, scientific and technical services 48.0
Manufacturing 61.0
Information and communication 74.5
Firm size
(by employment band)
1 to 9 employees 21.5
10 to 49 29.0
50 to 99 34.0
100 to 249 36.0
250 to 99 36.5
1,000 and over 27.5
Age of firm 2 years or younger 24.5
3 to 5 years 28.0
6 to 10 years 25.5
11 to 20 years 29.5
21 years and older 31.5
By ownership of firm Domestically owned 25.5
Foreign owned (EU) 40.5
Foreign owned (Non-EU) 42.5

Figure 6.11 Average (median) aGVA per worker by firm characteristics in 2018, £000s

Average (median) aGVA per worker by firm characteristics in 2018, £000s

There is a clear difference in productivity levels between various sectors of the economy and an interesting relationship between productivity and firm size.

For the most part, there is a positive relationship between size and productivity. The productivity of the median firm with between 250 and 999 employed was some 70% higher than those of the smallest firms. The exception to this trend is the lower productivity of firms that employ more than 1,000 as compared with the 250 to 999 band. But this is likely to be explained by the fact that this group has a high representation of businesses in sectors where productivity is typically lower than average, such as the retail industry.

One possible explanation of this positive relationship may be that larger firms are able to exploit economies of scale, reducing the proportional impact of their overheads and other fixed costs. At least as plausible, however, is the likely correlation between firm size and age. Some smaller firms may be relatively new firms that are still in the process of discovering and exploiting the efficiencies that are open to them. Some confirmation of this possibility is provided by looking at the relationship between firm age and average productivity.

A clear feature of the table is the lower productivity of young firms. Broadly, firms having been in existence for less than three years have productivity around a fifth lower than those of older firms. There is not much change, however, once firms have been in existence for more than three years. So if there is a “learning period” for new firms, this may be the extent of it.

At face value, ownership highlights striking differences. Firms with ownership outside the EU have productivity on average 93% higher than that of domestically owned firms. Firms with ownership within the EU similarly have productivity around 61% higher than domestically owned enterprises. In part, this may reflect the fact that foreign owned firms have greater access to world class know-how and technologies. As discussed in Chapter 9, and as noted earlier in this chapter, one of the clear advantages of countries receiving foreign direct investment is precisely this access to productive know-how. It may be this factor that is showing up in the productivity figures.

At the same time, care needs to be taken in interpreting ownership data. Young and small firms are more likely than larger or older enterprises to be domestically owned. As we have seen, these are likely to have lower productivity in any case. Furthermore, some reverse causation may be in operation. Overseas investors may be more prepared to invest in, and seek ownership of, high productivity firms than they would in respect of lower productivity ones. In that case, foreign ownership would be the consequence of firms having relatively high productivity, rather than the causal influence.

These analyses of the stories behind the aggregate figures for productivity are of great interest. But perhaps the most interesting feature relates to the distribution of productivity performance between different firms, which reveals what some describe as the third “productivity puzzle”. Figure 6.12 gives one such demonstration of UK experience.

Figure 6.12 Labour productivity by business in selected years: Kernel densities

Labour productivity by business in selected years: Kernel densities

ONS, Firm-level labour productivity measures from the Annual Business Survey, Great Britain: 1998 to 2018 using Annual Business Survey (ABS) and Inter-Departmental Business Register (IDBR) data

The horizontal axis shows the range of aGVA per worker achieved by firms included in the ABS. The “kernel densities” represent the proportion of employees whose firms achieve that level of productivity. By definition, the sum of all these kernel densities comes to one. The dotted lines represent the mean productivity achieved by the average employee. The kernel densities distribution is shown for selected years between 1998 and 2018.

There are several striking features revealed by the chart.

Taken together, these findings characterise the third productivity puzzle: the remarkable spread of productivity across different businesses, even fairly similar ones, and the lack of convergence over time.

This raises some important issues of policy relevance:

6.5 Improving productivity measurement

Given its importance in relation to people’s standard of living, productivity will remain a key issue for the UK economy for the foreseeable future. The future development of productivity statistics will be guided by the need for a better understanding of what drives productivity and the design of policies to improve the outcome.

6.5.1 Double deflation

double deflation
Double deflation is considered the best approach to producing volume estimates of gross value added (GVA). For every industry, the current price estimate of its output is deflated by a price index for output and the current price estimate of its inputs is deflated by an input price index. That is, outputs are deflated using deflators for the product produced, while the goods and services that are used as inputs to the production are deflated using relevant specific product deflators.

Since the numerator for productivity calculation is gross value added, the research agenda will be closely linked to the agenda for measuring the economy’s output in changing modern circumstances. But, in one sense, there is nothing additional from a productivity point of view here, over and above the issues discussed in the chapters of this book on GDP and hard to measure sectors. However, there is one issue in the context of improved measurement of GDP that is particularly important for measuring productivity, namely double deflation.

In November 2020, the ONS published early research findings into the production of alternative estimates of GDP in a new framework, including experimental estimates of double-deflated industry-level gross value added (GVA). Estimates were produced for the period 1997 to 2018 in the UK. These impacts reflect broader improvements to how the balancing of current price and volume estimates of GDP in the supply and use tables (SUTs) framework has been undertaken for these experimental estimates. That is, these estimates do not simply reflect the removal of single deflation bias.

In measuring GDP by the output route, ONS has used turnover as a proxy for value added and used a single deflator to obtain volume measures. It would clearly be preferable, as reflected in the international guidance, for the intermediate inputs into production processes to be deflated by an appropriate price index and for the gross outputs to be separately deflated by its own appropriate deflator. The volume of GVA under such double deflation is then the difference between the deflated output and the deflated inputs.

Arguably, this is important at whole economy level only in the short term. When available, the expenditure measure of GDP is to be preferred to the output measure, and the expenditure measure is reasonably well deflated already. But at sub-aggregate level, double deflation is of much more importance. Obtaining accurate estimates of the value added for particular sectors and individual firms requires the volumes of the gross output they produce and of the inputs they use to be measured accurately. Without double deflation, this is unlikely to be achieved. Going back to our example of the bakery, there is no reason to suppose that movements in the prices of loaves will be the same as those in the price of the flour and other intermediate inputs that are used in the production of the bread.

6.5.2 Missing capitals

Traditionally, economic analysis has concentrated on physical capital. At Daily Bread we regarded capital as the premises and ovens that are used to bake the bread. But there are other forms of capital which are equally relevant to economic production:

6.6 Summary

We began this chapter by setting out what productivity is, but also stressing its importance beyond being an economic ratio of technical interest only. On the contrary, as Paul Krugman reiterated, productivity – or the lack of it – is the crucial determinant of people’s material well-being.

Having considered various key measures of productivity and how they relate to one another, we also considered UK experience in recent years. Bluntly, it is far from reassuring. UK productivity remains low by comparison with most other comparable countries. At the same time, we have seen a sharp break from the growth trend of previous decades, with productivity essentially being on a flat plateau since 2007. This has consequences for our standard of living as well, for example, in the way it affects the government’s ability to finance growing demands on the NHS and other public services.

Understanding why productivity has evolved as it has is probably described as, at best, a work in progress. We have a clear picture of what has been happening and some pointers as to what may be significant determinants. But, to date, no fully convincing explanation has emerged.

In this context, compilation of high-quality estimates and measures of productivity growth will not, on its own, provide the fully convincing explanations that are needed. But it is also the case that without such quantitative evidence being available, progress in understanding and addressing these issues will be severely hampered. Accordingly, accurate measurement of productivity will remain a central agenda for the foreseeable future.

6.7 Further reading

Notes