2.4 The theory of production, cost and the firm In the production process, firms turn inputs (or factors of production) into outputs (or products) via a process of transformation using the available technology. Inputs can be divided into three broad categories: labour (L), capital (K) and materials (M); and can further divided into subcategories such as: skilled and unskilled labour, fixed and flexible capital and so on. For example, a wind turbine manufacturer uses inputs which include the labour of its assembly workers, engineers and the entrepreneurial efforts of its managers; capital, such as its buildings, equipment, inventories and materials such as steel and fibreglass; and other running costs such as electricity and water.
It is worthwhile highlighting that in addition to commodities such as steel, petroleum, cement and so on, raw materials can be thought to include environmental inputs such as waste water and the atmosphere, especially as firms are being faced with paying for access to the environment as a pollution sink.
This relationship between inputs and outputs can be described by a production function – which relates the output Q for every specified combination of inputs: (2.4) Just as the theory of consumer choice describes the consumption decisions of the individual through the interaction of indifference curves and the budget line, the theory of production describes the behaviour of the firm by the interaction of the isoquant and isocost curves respectively (Figure 2.3).
For graphical simplicity, the following discussion will limit the number of inputs to two. Two examples will be developed using the framework in Figure 2.
3, one modelling the relationship between capital and labour, and a second modelling the relationship between capital and GHG emissions, exemplifying how ‘the environment’ may be considered a factor of production. An isoquant is a downward sloping convex (to the origin) curve which shows all the possible combinations of inputs (in our first case, labour and capital) that can yield the same output (Q). An important difference to the theory of consumer choice is that while indifference curves can only give ordinal rankings of preferred market baskets, isoquant curves have a quantitative measure associated with them – thus also give a cardinal ranking. A set of isoquants, or isoquant map, describes the firm’s production function.
In Figure 2.3 Q 50 gives all the combinations of labour and capital that yield 50 units of output a year. Q 100 lies above and to the right of Q 50 because it takes more of either labour or capital or both to obtain a higher level of output. Figure 2.3 The theory of production Note that labour and capital are flows, meaning a firm uses a certain amount of each factor over a period of time. Isoquants highlight the flexibility that firms have when making production decisions – for example, a shortage of cheap, low, skill labour may result in increased automation of the production process using capital, or the import of labour intensive components from low-wage countries. Thus isoquants are another way of describing the substitutability between factors of production.
The slope of the isoquant at any point measures the marginal rate of technical substitution (MRTS) – the ability of the firm to substitute one factor of production for another while maintaining constant output. This tells us that the productivity that any one input can have is limited. As a lot of labour is added in place of capital (say, as the firm moves from point b to point c in Figure 2.3) the marginal productivity of labour falls and the MRTS decreases. (2.5) The relative slopes between isoquants can also be used to demonstrate the law of diminishing returns. For example, in the short run, capital is often difficult to expand.
In Figure 2.3, as the firm moves from A to B to C it applies only more labour to achieve greater output. At each point the slope of the isoquant can be seen to flatten out, showing there are diminishing returns to labour. Conversely, if the firm were to expand output using only capital, moving from E to D to C, and as the MRTS increases the productivity of capital falls and that of labour rises, showing that there are diminishing returns to capital. Because adding one factor while holding the other constant eventually leads to lower and lower increments to output, the isoquant must become steeper, as more capital is added in place of labour, and flatter when labour is added in place of capital. The relative slopes of different isoquants also reflects the nature of the technology embodied in the production process.
For example, a relatively flat isoquant reflects a production process where the marginal productivity of capital is very high – to keep producing the same amount one less unit of capital requires a large amount of labour to replace it. For example, this might be the case in an industry using DNA sequencing technology, which requires large computers to process vast quantities of information with relatively little labour input. Conversely, a steeply sloping isoquant suggests a high marginal product of labour and a large investment in capital to replace relatively small amounts of labour to maintain production. This would be common in labour intensive industries such as some textiles or fruit and vegetable harvesting. In the last example, one factor of production was held constant while the other was increased to demonstrate the law of diminishing returns.
What if more than one input was allowed to vary? The measure of increased output associated with increases in some or all inputs is fundamental to understanding nature of the production process. There are three main cases: Increasing returns to scale are said to exist when a proportional doubling of inputs leads to more than a proportional doubling of outputs. For example, this could arise because the increased size of the firm allows greater specialisation of workers which boosts productivity and enables the use of larger, more sophisticated machinery. In these cases it is economically advantageous to have a small number of large firms supplying the market at a relatively low cost, than many small firms, at relatively high costs. These are typical characteristics of the electricity supply sector.
Constant returns to scale are said to exist when an increase in inputs leads to the same increase in outputs. In this case, the size of a firm’s output does not affect the productivity of its factors. With constant returns to scale, one plant using a particular production process can be easily replicated, so that two plants produce twice as much output. For example, a large company which supplies sandwiches might provide the same service per customer as a small company and use the same ratio of capital (store space) and labour (kitchen and service staff). Decreasing returns to scale exists when an increase in inputs leads to a less than proportional increase in outputs.
Decreasing returns arise in large scale operations when the difficulties of increased complexity associated with the management of a large operation begin to introduce inefficiencies leading to reductions in the productivity of both labour and capital. Just as consumption behaviour is not determined only by consumer preferences, but also be the budget line; production is not just determined by the isoquant line. Firms face costs when using factors of production, these are represented by the isocost line. This shows all possible combinations of labour and capital that can be purchased for a given total cost (C). (2.6) and (2.7) The formula for the isocost line is given by equation (6) and is determined solely by the relative price of labour (w) and capital (r).
It describes the combinations of labour and capital which can be combined at the same cost. For example, if the wage rate was £10 per unit and interest rate £5 per unit, then a firm could replace one unit of labour with two units of capital with no change in total cost. It is the interaction between the isocost and isoquant curves which provides us with a description of the firm’s optimal production level and mix of inputs. In Figure 2.2, suppose a firm has decided it wishes to produce at Q100. Isocost line nm gives the total cost of the factors of production to yield this amount and intersects the isoquant at points b and c, each with their respective combinations of labour and capital.
However, the same amount of output can be produced at a lower cost along isocost line rs at point a. If we can assume the rational behaviour of the firm is to minimise costs for any given level of output in order to maximise profits, it follows that the firm will use the combination of inputs where the slopes of isoquant and isocost curves are just equal. At this point, the production of an additional unit of output costs the same, regardless of which input is used. It is worthwhile to briefly question how realistic is the assumption of profit maximisation? For small firms, which are managed by their owners, profit is likely to be a major objective, however other ojectives such as provision of a particular nonprofit service or lifestyle for the proprietor maybe also important. In larger firms, where managers have little contact with the owners (such as stockholders) there is likely to be even greater deviations from profit maximisation.
For example, managers may be more concerned with revenue maximisation in order to expand growth, firm size and prestige; they may also seek to maximise dividend payouts to shareholders or short-run profit (perhaps to earn a large bonus or to take a larger proportion of revenues in salary) at the expense of longer-term profit which seeks to maximise the value of the stream of profits over time. Thus the profit maximisation assumption has several potential serious weaknesses – and as noted by Alan Greenspan following the 2008 collapse of the banking system: “those of us who have looked to the self-interest of lending institutions are in a state of shocked disbelief”. The weakness of this central assumption – what Greenspan terms a “pillar of competitive markets” means that care must be taken when assessing what exactly markets are maximising and regulations put in place to support market abuse. In theory, firms or managers that do not place profit maximisation at the heart of their business are unlikely to survive and will become either take-over targets, or sacked by their boards respectively. However, in practice because markets are characterised by the diffuse nature of share ownership and weak corporate governance, problems regarding the goals and management of the company can go undetected creating systemic problems for the stability of markets (Kay, 2012).
This model can now be used to predict the effect of relative price changes among the factors of production on output. Changes in factor prices can occur for many reasons, be it movements in commodity prices, such as for oil; changes in interest rates; collective union agreements pushing wages up; or improved technologies which bring down the cost of inputs, such as in computing. For our example, instead of using labour as factor 2 we will use the atmosphere’s properties to absorb GHG pollution as a factor of production. This assumes that property rights to the environment can be adequately defined. In this case an energy production firm is required to face the costs it imposes on others through its pollution.
This is the case in example 2 of Figure 2.3, where the imposition of a price on GHG emissions is shown by the inwards rotation of the isocost curve from zx to zy. When the carbon price is increased for every tonne of GHG emissions the firm must pay the government an environmental charge. It is no long possible to produce Q50 at the same cost and the isocost line rotates inwards to reflect the higher price for factor 2. The shift in the point of profit maximisation from point e to g can be separated out into an income effect and a substitution effect. The substitution effect, shown diagrammatically as the shift from point e to h and gives the decline in the quantity of emissions (from f2 e to f2 h ), and the increase in the capital (from f1 e to f1 h – a cleaner production technology, such as carbon capture and storage, for example), required to maintain output at the same level.
The income effect (from point h to g) reflects that the firm now has less money to spend on inputs as it has to pay the government the carbon fee, and represents the fall in use of the environment (from f2 h to f2 g ) and fall in the use of capital (from f1 h to f1 g ) associated with having less money available to it. There are a couple of insights we can draw from this – the first is that the more the factors of production are substitutable, the easier the firm can deal with its GHG pollution without using the atmosphere as a waste sink and the more effective the fee will be in reducing pollution. Second, the greater the degree of substitution, the more easily the firm can avoid the effluent fee. 2.5 Technology and the production function To model more clearly the role of technology on the production process we can set out a production function. The production function describes the optimal combination of inputs and outputs for any given technology, across all levels of output (i.e.
points e and a, and so on, for each feasible isocost and isoquant curve). Technology in this context is understood to be a given state of knowledge about how to transform inputs into outputs. In Figure 2.4, a standard production function is shown where the increased use of inputs begins with increasing returns to scale (over the range a to b), transitions into constant returns (around point b), then into decreasing returns (from b to d) and then finally the situation of negative returns to scale is shown for output beyond point d. As technology becomes more advanced and is absorbed by the firm, the form of the production function changes as the firm can obtain more output for any given set of inputs. For example, improvements in nanotechnology may allow a producer of photovoltaic cells to supply a greater number of solar arrays each month for any given combination of labour, capital and raw materials.
This is shown in Figure 2.3 as the shift from Q tech1 to Q tech2 . In Figure 2.4 (a) the shape of the average and marginal product curves are shown, which are closely related. When the marginal product is greater than the average product, the average product is increasing, and when it is less than the average product, they both are falling. When the marginal product crosses the axis, output is maximised at point d and d’ respectively. This happens because an additional unit of input adds so much to the complexity of management, that it actually slows down the production process.
Note that even though the law of diminishing returns to scale still applies as we move along Q tech2 , it sets in at a higher rate of output and input. This is an important insight, which explains why, although there are diminishing returns to both labour and capital, output has been able to expand preventing the economy from falling into the so-called Malthusian trap. Next Page – Ch 2: The Cost of Production Previous Page – Ch 2: Equilibrium – the Basic Neoclassical Model and Extensions