Subadditivity is an important concept because it is often used to justify imperfect competition, the classic nemesis of neo-classicists. The only real way to justify less than perfect competition is the kind of sector which lends itself to natural monopolies. Natural monopolies are those where barriers to entry are so high that it is worth making the investment only once, and where one producer will operate more efficiently than two. Traditionally, natural monopolies are state-run, generally in energy or communication sectors, although current trends are leaning towards privatisation.

When a large investment is required in order to begin production, these costs must naturally be passed on to the consumer. What differentiates natural monopolies is that generally, once the investment is made, marginal costs are infinitely small. In these cases, if there is only one company in the market, prices for consumers may actually be lower than if there are rivals, because of the way average costs drop with additional users.

So what exactly is the difference between subadditivity and economies of scale? Subadditivity is a more generic concept which encompasses economies of scale. We say subadditivity exists when:

That is, when the costs function (including initial investment) for the sum of all units at a given level of production is actually smaller than the sum of the various cost functions (given multiple companies) at the same given level. If we look at the diagram, we can see the traditional long run costs functions for two companies. We only consider economies of scale when additional units provide a lower average cost, but costs per unit begin to rise after a certain level. Subadditivity simply means that it is cheaper to produce the same production level when only company one is producing, and that the same production level is more expensive when company two joins the market.

 

This is why subadditivity is considered a necessary but insufficient conditionfor a natural monopoly to be considered optimal, whereas if economies of scale exist, this is a sufficient but not necessary condition for a natural monopoly to be sustainable. The cyan demand curve, within the economy of scale region, indicates a viable natural monopoly. The red line would indicate a non sustainable natural monopoly. The black line indicates the frontier of a profit generating monopoly (which would attract competitors) and the green line indicates a viable duopoly.

In the long run, no cost is fixed. We can determine our production level and adjust plant sizes, investment in capital and labour accordingly. As we can see in the diagrams below, this gives us unlimited options. Depending on the scale we choose to implement, each level of production will be associated to new, short run cost curves. When we exhaust the infrastructure these provide us, we can upgrade to a new production level and so forth. The actual long run cost curve is made up of all of these individual scenarios, built up year after year.

If we look at average costs, the curve these draw is also the build up of the individual short run curves. These form a U shape, as we can see in the diagram. When average cost is decreasing with each additional investment, we are enjoying economies of scale, but not yet working at maximum efficiency. We reach this maximum efficiency at the minimum point, before the average cost per unit begins growing again.

Marginal costs only really make sense in the long run for each individual level of production. At optimal levels of production for each level, they intersect the long run average cost curve where this meets the short run average cost curve. This is what determines our optimal level of production. In summary, at our optimal level of production:

AC_SR = AC_LR = MC_SR = MC_LR

which is known as Le Châtelier’s principle.

The other important lesson to take away from this is the fact that, when we reach the optimal point where efficiency is at its best, average and marginal costs are the same, as happened in the short run (see detailed graph below these lines). At this point, the slope of the tangent line, that is, the derivative, of the short run total cost (TC_SR) is equal to zero, which is consequent with the fact that we are operating at maximum efficiency. What we can also glean from the graph is the fact that TC_SR and TC_LR are tangent where short run average cost (AC_SR) and AC_LR also meet, which is not necessarily an optimum. In fact, the only time where these junctions do suppose an optimum is in case 2.

This economic phenomenon occurs when increasing output is translated into a decline of the firm´s average cost of production. Alfred Marshall was the first economist to distinguish economies of scale depending on their origin:

-Internal economies of scale occur when something inside the firm makes the average cost of production lower:

• Purchasing: buying raw materials in large quantities or big bulks brings better prices and discounts which allow average cost to be reduced.
• Specialisation: covering all areas of management and work can be very difficult, however when operating on a large scale, workers can specialize in the activities in which they are best and more productive.
• Flexibility:  a more efficient production process can be achieved by managers when scale means that different arrangement of the inputs can be made. Better capital can be purchased by firms that have more money, increasing the quality of their products and/or reducing cost of production.

-External economies of scale occur when something outside the business, but inside its industry, makes average cost of production lower. As the industry of the firm grows, more support may come from the suppliers’ side: more variety, more quality, more quantity, etc. Government may give extra aid if it considers that the industry has some special interest.

On the other hand, diseconomies of scale may appear due to the excessive scale of a firm. Diseconomies of scale occur when average cost per unit increases due to excessive size of production.

The main reasons for diseconomies of scale to appear are:

• Demotivation issues: in firms with many workers, standing out becomes harder. Each individual may find himself as meaningless, and therefore cause him to stop working as hard as he would with fewer workers.
• Communication: the communication process of larger firms becomes slower as information has to pass through more and more layers. Information can even become distorted either accidentally or deliberately. The longer it takes managers to make decisions, and these to be known by workers, the less efficient the firm becomes.

Different levels of production have different cost levels. It is generally assumed that higher levels of production help drive down cost (by reducing fixed costs per unit, by lowering the price of raw materials and intermediate goods through increased bargaining power…). However, this is not necessarily the case when an increase in production leads to a jump in fixed costs that cannot be fully absorbed by the increase in production (purchasing a new, larger factory or a heavy investment in infra utilised machinery). To find out whether increasing production will increase efficiency (whether the increase in production will be greater than the increase in cost), we can measure elasticity.  

This is the same as measuring the result of dividing the % change in output by the % change in input. When considering the diagram seen in the short run cost analysis entry, along phase one, when efficiency is increasing with production, μ > 1. When we reach the point of inflection, μ = 1, and when we begin to have diminishing returns to scale, output begins to grow slower than inputs, and μ < 1, which is in phase III.

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.

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.

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.