In the short run, fixed costs include capital, K, whereas labour, L, is considered variable. Fixed costs are represented as a horizontal line and do not vary whatever level of production we achieve.

The two graphs show how the two phases pan out. In the first phase (I), variable costs (and therefore total costs, seeing as fixed costs are a constant) grow slower than growth at first, before reaching a point of inflection (II) and beginning to grow much faster than the output they are capable of generating.

This is related with returns to scale. In phase I, where the elasticity of scale is greater than 1, there are increasing returns to scale, while phase III corresponds to decreasing returns to scale. In point II, the elasticity of scale equals 1, which represents constant returns to scale.

If we translate this into average and marginal costs, an optimal level is reached along the stretch between which marginal costs are equal to average variable and fixed costs respectively. This coincides with the stretch just before costs begin to grow exponentially.

Average costs are those associated to one unit of production. Costs per unit grow quicker as production increases, so we find the arithmetic average as the sum of costs divided by the sum of production:

Av. fixed costs:		Av. variable costs:		Av. total costs:

Marginal costs are a very important concept in Economics because they show costs at a very specific point in time: they show the cost associated with producing one additional unit at any given production level.

They are the partial derivative of total or variable costs. This is an important point: because the partial derivative as respects quantity is zero for fixed costs (as these are independent of production levels), the partial derivative of variable and total costs is the same. Marginal costs, as any derivative, are tangent to total and variable cost curves at each point.

When analysing costs, the first thing to know is that there are fixed and variable costs:

• Fixed costs: fixed costs cannot be altered in short run. They mainly include things like rent, repayments on technology, etc…  which require time to change, and are often associated with indirect production costs (all that which is not a direct input). They do not depend on the level of production and cannot be adjusted accordingly.

• Variable costs: variable costs depend on the level of production, and can include things such as direct inputs (raw materials) and contracted labour hours (not fixed salaries).

As seen in the graph, fixed costs are represented by a straight horizontal line, independent of quantity. Variable costs follow a curve as they grow slower than production at first (phase I) before reaching a point of inflection (II) and beginning to grow exponentially (phase III). Total costs are simply the sum of fixed and variable costs, as can be seen on the graph.

The way in which fixed and variable costs affect production is related with returns to scale. In phase I, where the elasticity of scale is greater than 1, there are increasing returns to scale, while phase III corresponds to decreasing returns to scale. In point II, the elasticity of scale equals 1, which represents constant returns to scale.

Period analysis show the inter-temporal dimension of production theory. It was developed by Alfred Marshall in his “Principles of Economics”, 1890, and has remained practically unaltered since. It tries to explain how equilibrium is achieved and explains the adjustment processes to reach it, going from the short-run equilibrium to the long-run equilibrium. His method is able to classify forces with references to the length of time needed for the production process; and then confine in ceteris paribus those forces that are of less importance with regards to the specific time we are considering. Four periods can be distinguished: very short-run, short-run, long-run and very long-run.

The very short-run period is normally not studied as it is considered irrelevant. It is a period characterized by having all its production factors or inputs, fixed; thus implying there can be no change in their quantities. Costs are therefore considered to be fixed and logically there is inexistence of technological progress in this period.

The short-run period is a period in which some factors remain fixed while others may vary in their quantity. Capital elements, such as equipment, are considered fixed factors while labour is considered to be a variable factor.  During this period, cost analysis has a large dependence on marginal cost and average total costs. But, since not all factors can be increased, this will result in diminishing returns, which will imply that marginal costs will increase.  A firm should stop production until marginal costs reach average total costs in order to maximize profits. In this period technological progress is still non-existing.

The long-run period is formed by the succession of several short-run periods and it is considered to be like an envelope of many of these. As inputs can be increased, economies of scale can be achieved, so the cost of producing an extra unit of output will decrease, and thus reduce the average costs. Inputs and outputs should be increased until the point where the optimum size of the firm is achieved and from whereon an increase in size will only mean diseconomies of scale. All inputs may be varied but the basic technology of production still cannot be changed.

Finally, it is in the very long-run period when new production processes and new products come as a result of the existence of technological changes within this period, which allow for new ways of producing and different input requirements.

As in consumer’s theory (where consumption duality is analysed), the firm´s input decision has a dual nature. Finding the optimum levels of inputs, can not only be seen as a question of choosing the lowest isocost line tangent to the production isoquant (as seen when minimising cost), but also as a question of choosing the highest production isoquant tangent to a given isocost line (maximising production). In other words, having a cost function that sets a budget constraint, solving for the inputs allocation that gives the highest output.

The way to solve either problem is very similar: we look for the Lagrangian function and obtain first order conditions, then solve the system.

Video – Production duality:

Cost minimisation tries to answer the fundamental question of how to select inputs in order to produce a given output at a minimum cost.

A firm’s isocost line shows the cost of hiring factor inputs. This line gives us all possible combinations of inputs (here labour and capital) that can be purchased at a given cost.

Assuming that a certain amount of output wants to be achieved, we have several possible combinations to achieve it, but only one that minimises costs. The isocost line tangent to the isoquant, which represents the amount of output targeted, will reveal the input combination that results in the lowest cost, for that given output.

We can also use the method of Lagrangian systems to analytically solve a constrained minimisation problem. The first derivatives determine a system of equations that can be resolved by submitting our sought output to the restriction presented by the minimisation of costs:

Video – Cost minimisation:

Production maximisation must be seen as an optimisation problem regarding the production function, represented by isoquants, and a constraint regarding production costs, represented by an isocost line.

Producers are therefore faced with the following problem: faced with a set of possible production levels and a fixed budget, how to choose the level which maximises their production?

If we know the production function of a certain producer, and we know their budget, we have the two restrictions necessary to maximise their production. This can be done graphically, with the point where isocost and isoquant meet defining an optimum, as shown in the adjacent figure.

It can be also done mathematically, through a Lagrangian, where the first derivatives determine a system of equations that can be resolved by submitting our production function to the restriction presented by the budget:

Video – Production maximisation: