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:

Isocost lines show combinations of productive inputs which cost the same amount. They are the same concept as budget restrictions when looking at consumer behaviour. Mathematically, they can be expressed as:

rK + wL = C

Where r is the cost of capital and w is the cost of labour. Generally, we think of r as the interest rate the financial markets offer, as capital requires investment. Even if the capital can be paid for using a company’s own resources, r is still equivalent to the opportunity cost of having the money tied up in investments rather than in liquid assets which offer a return (r) by lending it to the markets. The cost of labour (w) is the salary paid to employees per unit of time.

Isocosts are usually represented graphically together with isoquant lines (which are combinations of productive inputs which produce a fixed quantity of outputs). The two have a tangency point, which determines the optimal production (where production is maximised or cost minimised).

Video – Isocosts:

A production function shows how much can be produced with a certain set of resources. Generally, when looking at production, we assume there are two factors involved in production: capital (K) and labour (L), as this allows us graphical representations of isoquants. However, any analysis made with 2 factors can mathematically be extended to n factors.

Therefore, a production function can be expressed as q = f(K,L), which simply means that q (quantity) is a function of the amount of capital and labour invested. In the adjacent figure, q_x is function of only one factor, labour, and it can be graphically represented as shown (green).

It is well worth introducing here another concept: marginal productivity, which is how much more quantity we could produce by adding one unit more of a factor. As is logical, this will depend on how we are employing the factors we already have. The marginal product is the partial derivative of the production function with respect to the factor we are examining:

Marginal productivity decreases with each additional unit, as it can be seen in the above figure (cyan). At a certain point, the more workers we have, for example, the more each additional worker will be redundant if we do not invest in other necessary factors. This is the same as saying that the second derivative is negative. At that point, A, production is as efficient as possible.

Video – Production function:

The economic region of production shows the combinations of factors at a certain cost that make economic sense. Areas outside the economic region of production mean that at least one of the inputs has negative marginal productivity. This region is marked by what are called ridge lines, which are simply the boundaries beyond which one of the two factors is being overused. Therefore, outside the economic region of production, there is clear inefficiency, and the company would be better off using less of one of the two factors, bringing costs down whilst maintaining equal production output. Graphically:

Video – Economic region of production:

The marginal rate of technical substitution (MRTS) can be defined as, keeping constant the total output, how much input 1 have to decrease if input 2 increases by one extra unit. In other words, it shows the relation between inputs, and the trade-offs amongst them, without changing the level of total output. When using common inputs such as capital (K) and labour (L), the MRTS can be obtained using the following formula:

An isoquant shows the different combinations of K and L that produce a certain amount of a good or service. Mathematically, an isoquant shows:

f (K,L) = q₀

Graphically, the shape of an isoquant will depend on the type of good or service we are looking at. The shape of isoquants is also in close relation with the terms marginal rate of technical substitution (MRTS) and returns to scale.

The first example of isoquant map showed in the adjacent graph is the most common representation. It shows four convex isoquants (green), showing each curve what amount of capital K the producer can stop applying when increasing the amount of labour L, while maintaining the quantity of output produced constant. This relation gives us the MRTS between these inputs, which is the slope of the curve in each of its points.

Our second example is an isoquant map with four parallel lines (cyan). This is the case for inputs which are perfect substitutes, since the lines are parallel and MRTS = 1, that is the slope has an angle of 45º with each axis. It can also be the case for inputs that are perfect substitutes but in different proportions. In that case, the slope will be different and the MRTS can be defined as a fraction, such as 1/2 ,1/3 , and so on. For perfect substitutes, the MRTS will remain constant.

Our third example shows an isoquants map with four isoquants (red) that represent perfect complementary inputs. This is, there will not be an increase on the amount produced unless both inputs increase in the required proportion. The best example of complementary inputs are shovels and diggers, since the amount of holes will not increase when there are extra shovels without diggers. Notice that the elbows are collinear, and the line crossing them defines the proportion in which each input needs to increase in order to have an increase in the production. In this case the horizontal fragment of each isoquant has a MRTS = 0 and the vertical fractions a MRTS = ∞.

Isoclines are lines which ‘join up’ the different production regions. Having defined  and decided the optimal levels of K and L we need to produce the different quantities, the line that passes through these optimal levels is an isocline (cyan). In other words, it is the line that joins points where the MRTS of each isoquant is constant:

Video – Isoquants: