Multi-product firms are firms that produce multiple goods, and therefore have to deal with allocating inputs more properly in order to attain higher production levels. This is a greater problem than the one single-product firms face, the production maximisation problem, since multiproduct firms must allocate their factors not only to produce one good, but multiple goods.

Indeed, production analysis can be extended to include multiproduct firms, as it can be derived from F. Y. Edgeworth‘s “Mathematical Psychics”, 1881­­. In this case, a company produces two goods, X and Y, using the same two factors, L and K, and subject to the following restrictions:

In this first figure, which is known as an Edgeworth box, if we take A as a starting point, we can see that it is far from an optimum, as we could increase our production of X whilst maintaining production of Y constant by moving from A to B, and maintain production of X constant whilst increasing production of Y by moving to point C. These optimum points pertain to the contract curve, along which production is efficient. Along all these points, the marginal rate of technical substitution for both products is equal: MRTS_x=MRTS_Y.

From this Edgeworth box and, more precisely, the contract curve that takes all efficient production levels into account, we can derive the production possibility frontier (PPF), shown in the figurebelow. This is just another way to look at the same problem: how to better allocate our resources in order to obtain the maximum output possible. However, by using the PPF, we concentrate more on the amounts being produced, rather than the inputs being used.

Of course, what is missing here to complete the analysis would be the cost of both inputs and how much is required to produce each type of good. In this case, by factoring in the selling price of each, we would have a closed optimisation problem that we would be able to resolve analytically, determining an efficient production function that could optimise the ratio of production of one good to another.

The production possibility frontier (PPF) represents the quantity of output that can be obtained for a certain quantity of inputs using a given technology. Depending on the technology, the PPF will have a certain shape.

As you can see on the adjacent figure, this PPF (blue curve) slopes downwards. This slope, which equals the marginal rate of transformation between X and Y, shows us how, in order to increase the output X, the quantity of Y must decrease. In fact, the marginal rate of transformation measures the tradeoff of producing more X in terms of Y.

This frontier determines the maximum output (of both X and Y) that can be obtained given the technology. Production at point A will produce more quantity of Y and less of X than production at point B. However, both are technically efficient, since they maximize the output. For example, production at point C is technically inefficient because, at any point on the PPF, more combined output is produced using given the technology. Also, point D is unattainable given the technology, being this is the reason why it is outside the PPF.

The PPF can be derived from the contract curve on an Edgeworth box. In this box, we see the quantity of inputs (K, L) being used in the production of each good (X,Y). In fact, we can see how, for each quantity of each product, the quantity of each input can change. The isoquants (green curve for X, red for Y) determine how much a certain input has to increase in order to compensate the decrease in the other input, maintaining the quantity of output produced unaltered. The slope of these curves is given by the marginal rate of technical substitution of each output.

The points where the isoquants of different outputs combination intersect, which are Pareto-optimal, allow us to draw the contract curve, from which the PPF can be derived. Since the technology is given, only one PPF can be derived from the contract curve (as opposed to the case of the utility possibility frontier).

Video – Production possibility frontier:

Production in the very long run differs from long run production in that there may be changes in technology. There are three main types of technological advances:

• New technology allows for completely new products
• New technology allows us to produce equal quantities with less inputs, displacing our isoquants
• New technology allows us to produce greater quantities with equal inputs, displacing our production function

These technological advances may displace our ratio of capital to labour one way or the other. Let’s see it graphically first:

-the first technological change shown in the adjacent figure is called Hicks-neutral, which translates into an unchanged ratio of capital’s marginal product to labour’s marginal product. John Hicks devised it in his book “The Theory of Wages”, 1932. Analytically: Y = AF(K,L)

-the second technological change is called Harrod-neutral or labour-augmenting increases efficiency of labour. Roy Harrodformulated it in his book “Towards a Dynamic Economics”, 1948. Analytically: Y = F(K,AL)

-the third and last technological change shown is called Solow-neutral or capital-augmenting , which increases the efficiency of capital. It was devised by Robert Solow in his article “Technical Change and the Aggregate Production Function”, 1957. Analytically: Y = F(AK,L)

Finally, we must also take into account the possibility of endogenous growth derived from endogenous technical changes. This can be easily plotted by using a Cobb-Douglas function, such as:­­

The law of returns to scale explains how output behaves in response to a proportional and simultaneous variation of inputs. Increasing all inputs by equal proportions and at the same time, will increase the scale of production. Returns to scale differ from one case to another because of the technology used or the goods being produce. Therefore, it is closely related to economies of scale happening within the business’ production process.

The relation by which output increases compared to the increase in inputs is the return to scale. We can measure the elasticity of these returns to scale in the following way:

That is, the sum of the partial derivatives of production with regards to each factors multiplied by the proportion each input makes up of the whole.

Depending in the proportion by which output increases compared to inputs, there are three different kinds of returns to scale:

-Increasing returns to scale (µ>1): when total output increases more than proportionately (green);

-Constant returns to scale (µ=1): when total output increases proportionately (blue);

-Decreasing returns to scale (µ<1): when total output increases less than proportionately (red).

When dealing with Cobb-Douglas functions, we can also determine which returns of scale are present, since α+β=µ.

When dealing with long run production, the main change from short run production is that we can vary the levels of fixed inputs we use (capital, K), as well as variable inputs (labour, L). Our levels of production will be determined by our returns to scale. It’s worth introducing here the concept homogenous functions. A function is considered homogenous if, when we have a multiplier, λ:

That is, we can reduce a production function to its common multiples multiplied by the original function. This is important to returns to scale because it will determine by how much variations in the levels of the input factors we use will affect the total level of production. Also, an homothetic production function is a function whose marginal rate of technical substitution is homogeneous of degree zero.

All this becomes very important to get the balance right between levels of capital, levels of labour, and total production. We can measure the elasticity of these returns to scale in the following way: 

That is, the sum of the partial derivatives of production with regards to each factors multiplied by the proportion each input makes up of the whole. Graphically: 

If µ is greater than one, we say that we have positive returns to scale (green). The more we produce, the less our costs will be per unit. If it is between one and zero, we have constant returns to scale (blue), and if it is negative, we will have negative returns to scale (red). When dealing with Cobb-Douglas functions, we can also determine which returns of scale are present, since α+β=µ.

All this will determine the average size of a business in the sector. Positive returns to scale, the most common scenario, will naturally attract a concentration of very large businesses, as is often the case in industrial, capital intensive sectors. Getting it right is crucial for another reason: in the long term, market equilibrium will determine that the total production of a certain good will be such to eliminate economic profits.

The short run is considered the period of time where fixed costs are still fixed, which basically means that, if you have a factory, you have to make do with it because you can neither sell it, nor make it bigger, nor rent half of it: you are stuck with it for the time being. Capital is also considered fixed, meaning that, in the short run, all you can play around with are your variable costs, being labour the most commonly used variable cost.

If we look at the adjacent graphs, we can see how marginal productivity (cyan, second graph) drops with each added unit of labour, under the law of diminishing returns. The two diagrams allow us to see three clear phases, characterised by the point of inflection that marks the law of diminishing returns. In the first diagram, we see how production increases exponentially with each added unit of labour, up to a point where added units begin to offer less and less return. In the second diagram, we introduce average productivity too; to define even further at what level we should produce.

In phase I, average and marginal productivity increase with each added unit, corresponding with the first diagram up to the point of inflection. Phase I ends in the extensive margin (or technical optimum), being this the line that splits phases I and II. Phase II is where we should ideally be. Average productivity drops with each added unit, but marginal productivity is still positive. The line that separates phases II and III is called the intensive margin (or technical maximum), from which there will be too much variable input for each level of fixed input. When marginal productivity becomes negative, we enter phase III, and we should consider reducing labour. We can also see that, at optimum level, marginal and average productivity coincide.

This all relates to the productive elasticity of our product, which is the difference in quantity that occurs as the result of a change in the level of productive inputs. Obviously, in phase I, a change of one unit in labour input will produce a much larger return in production levels than itself: ε_L>1. In phase two, 1> ε_L>0, and in phase III, 0> ε_L.

Algebraically, the elasticity of production can be defined as:

It must be noted that the elasticity of production for a given input (here labour, L) is the first half of the elasticity of scale, which in turn gives us the returns to scale.

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:

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: