# A Simple Guide to Measuring Multi-Factor Productivity

Office for National Statistics (ONS) and other statistical offices publish regular estimates of “multi-factor productivity’” or MFP, but what exactly is MFP? This guide aims to give the non-expert user an understanding of MFP by explaining the underlying concepts through straightforward stylised scenarios.

But first, what is productivity and why does it matter? To understand productivity, it helps to think of the economy as a system that converts inputs into outputs, as shown in Figure 1.

Multi-factor productivity (MFP) is a crucial economic measurement that accounts for the efficient use of multiple inputs like labor, capital, energy, materials etc. in generating output. While labor productivity measures output per hour worked, MFP provides a more comprehensive productivity measurement by considering how efficiently a firm combines labor, capital, intermediate inputs and technology.

In this article, we provide a simple, step-by-step guide to understanding the concept and measurement of multi-factor productivity.

## What is Productivity and Why Does it Matter?

At its core, productivity is a measure of efficiency – how well we convert inputs like labor, capital etc. into outputs like goods and services. It matters because higher productivity growth enables us to produce more output using the same amount of inputs. This frees up resources that can be utilized to increase consumption and living standards over time.

For a chocolate factory, key inputs would include:

• Labor – measured by number of workers or hours worked
• Capital – machinery, equipment, buildings etc.
• Materials – cocoa, sugar, milk powder etc.

The sole output is chocolate bars of different sizes. Productivity is the rate of conversion of these inputs into chocolate bar output.

## Starting with Labor Productivity

The simplest productivity measure is labor productivity i.e. output per labor hour. For a small bakery the inputs are skilled bakers unskilled sales assistants, capital like ovens and premises, and materials like flour, yeast etc.

The output is standard loaves of bread. Assume a baker works 8 hours a day, producing 600 loaves. The sales assistant works 7 hours.

• Daily output = 600 loaves
• Total hours worked = 8 + 7 = 15
• Labor productivity = Output/Hours
= 600/15 = 40 loaves per hour

If work hours fall, output and productivity decline. If a second baker is hired for a 7 hour evening shift, daily output rises to 1200 loaves. With a second sales assistant, total hours are 27 and labor productivity is 44.4 loaves per hour.

Simple labor productivity treats all labor hour inputs equally. But there are differences in skills and wages across worker types. Quality Adjusted Labor Input (QALI) accounts for variations in labor quality by weighting hours worked by relative wages.

For the bakery, assume the skilled baker earns £12 per hour while the assistant earns £8 per hour

• In single shift:
• Baker works 8 hours
• Assistant works 7 hours
• Baker share of wage bill = £96/(£96 + £56) = 63%
• In double shift:
• Baker works 14 hours
• Assistant works 13 hours
• Baker share of wage bill = £168/(£168 + £104) = 62%

Total hours rise by 58.8% from single to double shift. But the baker’s hours rise faster, so the quality-adjusted labor input rises slightly less, by 58.2%.

QALI reflects the decline in share of skilled (higher paid) labor.

## Accounting for Changes in Capital

Capital inputs like ovens and buildings provide ongoing services to production. Capital services are estimated from rental prices that reflect the productive value of capital assets.

Assume the bakery rents:

• Premises for £75 per day
• Oven for £13 per day plus £10 per hour of use

In double shift, premises rent is unchanged but oven rent rises with greater use. Total capital costs increase by 34% despite unchanged physical capital, reflecting higher capital services flow.

## Measuring Output through Value Added

Output is measured through value added (GVA) – the value of output less intermediate inputs like flour. With loaf price at £1 and flour cost £0.5 per loaf, daily GVA equals turnover less materials cost.

Double shift working raises GVA from £300 to £600. Part of this is down to increased capital and labor costs. The residual £120 is profit.

## Weighting Labor and Capital Cost Shares

The next step is weighting cost shares of inputs. On average over both scenarios, labor costs are 48% of GVA while capital takes the remaining 52%. This gives weights for aggregating QALI and capital services.

## Calculating Multi-Factor Productivity

Multi-factor productivity measures output growth unexplained by input growth.

• Output (GVA) rises by 69.3%
• Quality-adjusted labor input rises 58.2% (weighted by 48% labor cost share = 27.9%)
• Capital services rise 34% (weighted by 52% capital cost share = 17.7%)
• So MFP change = Output growth – Input growth
= 69.3% – 27.9% – 17.7% = 23.7%

The positive MFP reflects efficiency gains from greater capital utilization in double shift working.

## Key Takeaways

Measuring multi-factor productivity involves:

• Tracking output growth through value added
• Quality-adjusting labor input using relative wages
• Weighting growth in labor, capital inputs by cost shares
• Calculating productivity residual unexplained by input growth

MFP provides a comprehensive productivity metric capturing efficiency gains from innovation, economies of scale, better organization and technology.

Focusing on MFP alongside labor productivity gives a fuller view of the productivity trends driving economic growth and living standards. Understanding the breakdown of productivity changes into labor, capital and MFP components helps identify promising areas for policy focus.

## Labour and capital shares

So far, we have separately measured the changes in quality-adjusted labour input and capital services between the two scenarios, and we have now established a measure of output growth in terms of gross value added (GVA).

The next step is to measure the respective weights of labour and capital in production. To do this we will use the cost information in Table 5, as shown in Table 6.

Table 6: Labour and capital weights

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

Note that, echoing our published multi-factor productivity (MFP) estimates, we again take the average labour cost share over the two scenarios. Note also that we define the capital cost share as one minus the labour cost share, that is, we are treating any residual profit as a return on capital.

In our simple example, labour costs account for 48% of GVA averaged over the two scenarios, with capital accounting for the remaining 52%. For comparison, the typical labour share in the UK economy is around two-thirds.

## Capital inputs

So far, we have focused entirely on labour inputs. The next stage is to take account of capital inputs to the production process. Capital input includes anything that provides an ongoing use to output without being used up in the production process. In our bakery, capital inputs would include things such as the oven and the building of the bakery; as they can be used in the production process more than once, they are not simply “used up” in each production cycle. Inputs such as the ingredients are not capital inputs, as once flour is used to make a loaf of bread, the same flour cannot be used again to produce another loaf of bread.

How do we measure capital inputs? For our published multi-factor productivity (MFP) estimates we estimate capital services. Conceptually, these are flows of productive services, directly comparable to flows of labour services measured by hours worked (and QALI). 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, for our purposes, we can cut through this complexity by noting that this method essentially models the costs that firms would pay if they were to rent all their capital assets in competitive markets.

For our purposes, we will assume that our bakery rents its premises, oven and any other capital, and that the rents paid are fair reflections of the capital services provided by these assets.

Let’s assume that the bakery rents its premises for the equivalent of £75 per day. Assume further that the bakery rents an oven and associated equipment for the equivalent of £13 per day plus £10 for each hour that it is used (this is like leasing a photocopier and making payments based on the number of copies made). The different terms for the two types of capital reflect the fact that the life of premises is not materially affected by the intensity of its use, whereas equipment such as ovens will suffer wear and tear through use.

This information is summarised in Table 4, which follows the same format as Table 3. Note that unlike labour, we cannot distinguish a “quality” component for capital. As in Table 3, the column labelled “Average cost shares” shows the average share of each type of capital employed across the two scenarios. The change in capital services is then calculated as the change in costs of each type of capital weighted by its average cost share, and summed across both types.

Table 4 shows that a move from single- to double-shift working involves an increase of 34.0% in capital services employed in production. This 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.

Table 4: Introducing Capital Services

Asset types Charge unit Cost (£/charge unit) Cost (£/day) Cost shares (%) Average cost shares (%) Change in costs (%) Change in capital services (%)
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
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
Premises 43.4 0.0 0.0
Oven etc 56.6 60.0 34.0
Total 100.0 34.0 34.0

We should note that the example in Table 4 relies on rental charges fairly reflecting the value of each type of capital. In the real world, rental markets for capital assets are either thin or non-existent and most capital assets are owned directly by the firms that use them, albeit often financed by borrowings.

Moreover, even where rental markets exist, rental prices may include bundled labour services (such as a crane that is supplied with an operator) and margins for the rental organisation, and long-term rental terms will typically include an allowance for general inflation, which we are assuming away in our simplified scenarios.

## Operations Management: Single-Factor & Multi-Factor Productivity

What is multifactor productivity?

Learning about multifactor productivity can help you measure inputs, such as labor and capital resources, and outputs, such as revenues and products. In this article, we define multifactor productivity, discuss its benefits and importance, explain how to measure it and share an example calculation.

What is the difference between partial factor productivity and multifactor productivity?

Multifactor productivity Whereas the partial factor productivity formula uses one single input, the multifactor productivity formula is the ratio of total outputs to a subset of inputs. For example, an equation could measure the ratio of output to labor, materials, and capital.

How does multifactor productivity differ from labor productivity?

Multifactor productivity differs from labor productivity by comparing output not just to hours worked, but to a combination of inputs. What are these combined inputs? For any given industry, the combined inputs include labor, capital, energy, materials, and purchased services.

How is total factor productivity calculated?

TFP is calculated by dividing output by the weighted geometric average of labour and capital input, with the standard weighting of 0.7 for labour and 0.3 for capital. Total factor productivity is a measure of productive efficiency in that it measures how much output can be produced from a certain amount of inputs.