As part of our introduction to the theory of the firm, we first consider the nature of production of different goods and services in the short and long run.

The concept of a production function

The production function is a mathematical expression which relates the quantity of factor inputs to the quantity of outputs that result. We make use of three measures of production / productivity.

• Total product is simply the total output that is generated from the factors of production employed by a business. In most manufacturing industries such as motor vehicles, freezers and DVD players, it is straightforward to measure the volume of production from labour and capital inputs that are used. But in many service or knowledge-based industries, where much of the output is “intangible” or perhaps weightless we find it harder to measure productivity
• Average product is the total output divided by the number of units of the variable factor of production employed (e.g. output per worker employed or output per unit of capital employed)
• Marginal product is the change in total product when an additional unit of the variable factor of production is employed. For example marginal product would measure the change in output that comes from increasing the employment of labour by one person, or by adding one more machine to the production process in the short run.

The Short Run Production Function

The short run is defined in economics as a period of time where at least one factor of production is assumed to be in fixed supply i.e. it cannot be changed. We normally assume that the quantity of capital inputs (e.g. plant and machinery) is fixed and that production can be altered by suppliers through changing the demand for variable inputs such as labour, components, raw materials and energy inputs. Often the amount of land available for production is also fixed.

The time periods used in textbook economics are somewhat arbitrary because they differ from industry to industry. The short run for the electricity generation industry or the telecommunications sector varies from that appropriate for newspaper and magazine publishing and small-scale production of foodstuffs and beverages. Much depends on the time scale that permits a business to alter all of the inputs that it can bring to production.

In the short run, the law of diminishing returns states that as we add more units of a variable input (i.e. labour or raw materials) to fixed amounts of land and capital, the change in total output will at first rise and then fall.  Diminishing returns to labour occurs when marginal product of labour starts to fall. This means that total output will still be rising – but increasing at a decreasing rate as more workers are employed. As we shall see in the following numerical example, eventually a decline in marginal product leads to a fall in average product.

What happens to marginal product is linked directly to the productivity of each extra worker employed. At low levels of labour input, the fixed factors of production - land and capital, tend to be under-utilised which means that each additional worker will have plenty of capital to use and, as a result, marginal product may rise. Beyond a certain point however, the fixed factors of production become scarcer and new workers will not have as much capital to work with so that the capital input becomes diluted among a larger workforce.
As a result, the marginal productivity of each worker tends to fall – this is known as the principle of diminishing returns.   

An example of the concept of diminishing returns is shown below. We assume that there is a fixed supply of capital (e.g. 20 units) available in the production process to which extra units of labour are added from one person through to eleven.

• Initially the marginal product of labour is rising.
• It peaks when the sixth worked is employed when the marginal product is 29.
• Marginal product then starts to fall. Total output is still increasing as we add more labour, but at a slower rate. At this point the short run production demonstrates diminishing returns.

The Law of Diminishing Returns
Capital Input	Labour Input	Total Output	Marginal Product	Average Product of Labour
20	1	5		5
20	2	16	11	8
20	3	30	14	10
20	4	56	26	14
20	5	85	28	17
20	6	114	29	19
20	7	140	26	20
20	8	160	20	20
20	9	171	11	19
20	10	180	9	18
20	11	187	7	17

 

Average product will continue to rise as long as the marginal product is greater than the average – for example when the seventh worker is added the marginal gain in output is 26 and this drags the average up from 19 to 20 units. Once marginal product is below the average as it is with the ninth worker employed (where marginal product is only 11) then the average will decline.

This marginal-average relationship is important to understanding the nature of short run cost curves. It is worth going through this again to make sure that you understand it.

Criticisms of the Law of Diminishing Returns

How realistic is this notion of diminishing returns? Surely ambitious and successful businesses do what they can to avoid such a problem emerging.

It is now widely recognised that the effects of globalisation, and in particular the ability of trans-national corporations to source their factor inputs from more than one country and engage in rapid transfers of business technology and other information, makes the concept of diminishing returns less relevant in the real world of business. You may have read about the expansion of “out-sourcing” as a means for a business to cut their costs and make their production processes as flexible as possible.
In many industries as a business expands, it is more likely to experience increasing returns. After all, why should a multinational business spend huge sums on expensive research and development and investment in capital machinery if a business cannot extract increasing returns and lower unit costs of production from these extra inputs?

Long run production - returns to scale

In the long run, all factors of production are variable. How the output of a business responds to a change in factor inputs is called returns to scale.

• Increasing returns to scale occur when the % change in output > % change in inputs
• Decreasing returns to scale occur when the % change in output < % change in inputs
• Constant returns to scale occur when the % change in output = % change in inputs
•  

A numerical example of long run returns to scale
Units of Capital	Units of Labour	Total Output	% Change in Inputs	% Change in Output	Returns to Scale
20	150	3000
40	300	7500	100	150	Increasing
60	450	12000	50	60	Increasing
80	600	16000	33	33	Constant
100	750	18000	25	13	Decreasing

 

In the example above, we increase the inputs of capital and labour by the same proportion each time. We then compare the % change in output that comes from a given % change in inputs.

• In our example when we double the factor inputs from (150L + 20K) to (300L + 40K) then the percentage change in output is 150% - there are increasing returns to scale.
• In contrast, when the scale of production is changed from (600L + 80K0 to (750L + 100K) then the percentage change in output (13%) is less than the change in inputs (25%) implying a situation of decreasing returns to scale.

As we shall see a later, the nature of the returns to scale affects the shape of a business’s long run average cost curve.

The effect of an increase in labour productivity at all levels of employment

Productivity may have been increased through the effects of technological change; improved incentives; better management or the effects of work-related training which boosts the skills of the employed labour force.

The length of time required for the long run varies from sector to sector. In the nuclear power industry for example, it can take many years to commission new nuclear power plant and capacity. This is something the UK government has to consider as it reviews our future sources of energy.