# Factors of Production

In the textbook we analyse the labour market and consider a firm's decision to hire employees. However there are other factors of production such as capital and land. Firms must decide what combination of these factors to employ. This decision is considered below.

## Combining factors of production

When combining resources managers will seek to find the optimal combination of factors of production.

This means they will hire them up to the point where:
marginal product of labour  =   marginal product of capital   =  marginal product of land
price of labour                          price of capital                         price of land

This is known as the equi-marginal condition. This means the last employee is as productive per £ as the last piece of land and the last machine. If this was not the case the firm would want to reallocate its resources. For example, if the marginal product of labour per £ was higher than the marginal product of capital per £ the firm would want to become more labour intensive and less capital intensive. As more labour was employed the marginal product would fall (according to the law of diminishing returns). This should continue until the equi marginal condition is fulfilled.

The equi marginal condition can be shown using isoquant and isocost analysis.

## Isoquants

An isoquant shows all the combinations of factors of production that generate the same total output.

An isoquant is downward sloping because if more of one factor is used then less of the other factor has to be used to produce the same overall output.

If the law of diminishing returns applies then an isoquant is convex to the origin. As more units of one factor are used then the extra output produced by this factor will diminish. This means that successively less of the other factor has to be given up to maintain the same overall total output. The gradient of the isoquant shows the marginal rate of  technical substitution between the factors of production.
The slope of the isoquant is given by:
Marginal Product of Labour
Marginal Product of Capital

For example, if the extra output from another unit of labour is twice that of the capital this means two units of capital must be sacrificed for another unit of labour if total output is to be maintained. The gradient of the isoquant will be:
-2  =  -2
1

The further an isoquant is from the origin the higher the level of total output as more factors of production are being used and therefore the higher the output will be.

## Isocosts

For any given level of output a firm will want to minimise costs. To do this it considers the isocost. An isocost shows the maximum combination of factors that can be hired for a given level of spending. For example if the budget is £100 and the price of labour is £10 an hour and the price of capital is £20 then the maximum affordable combinations are shown below.

A change in the budget with the same relative prices leads to a parallel shift in the isocost.

The slope of the isocost is shown by the relative prices of the factors of production. If the price of capital fell to £5 the combinations are shown by a new isocost.

The slope of isocost is given by:
Price of Labour
Price of Capital

## The least cost combination

The aim of a firm is to minimise costs for a given level of output. To do this it will want to be on the isocost nearest to the origin but tangent to the required isoquant. A firm could produce at F but it is cheaper to produce the same output at G. In this case the cost minimisation combination of factors to produce 200 units is L1 labour and K1 capital.
The least cost combination therefore occurs where the gradient of the isoquant equals the gradient of the isocost. i.e. the rate of substitution between the factors of production equals the relative prices of the factors of production.

This means:
Marginal product of labour  =  Price of labour
Marginal product of capital     Price of capital

Rearranging we get:
Marginal product of labour  =  Marginal product of capital
Price of labour                        Price of capital

i.e. the extra output per £ of labour equals the extra output per £ of capital. If this were not the case the firm would reallocate resources e.g. if the marginal product per £ of labour was higher than the marginal product per £ of capital the firm would employ more labour and less capital. Employing more labour would lead to a lower marginal product (due to the law of diminishing returns) until the marginal products per £ were equal.

## Changes in the relative prices

If the price of capital falls this changes the slope of the isocost. The firm will now cost minimise using L2 of labour and K2 capital.

## Returns to a factor and returns to scale

Isoquant analysis can also be used to analyse the returns to a factor and returns to scale.

In the diagram above the amount of capital is kept constant at K1. To produce 100 units of output L1 units of labour are needed. When the amount of labour is doubled to L2 output only increases by 90 to 190; the extra labour has added less than before because of diminishing returns to a factor. If the same amount of labour is added again to get to L3 output increases to 260 i.e. by 70 units. Once again there have been diminishing returns to a factor.

In this diagram we see increasing returns to scale. Initially K1 capital and L1 labour can be used to produce 100 units of output. If the capital and labour input are doubled (and assuming these are the only factors of production) output rises to 300 i.e. it more than doubles. If the amount of  capital and labour are again doubled to K4L4 amount rises to 800 units. Again it more than doubles. A doubling of all factors of production leads to an increase in output that is more than twice as much. There are increasing returns to scale.