# Competition with increasing returns to scale

When I criticize mainstream economics for neglecting increasing returns to scale there is one argument that is always brought up. In its most basic form the argument goes: an industry with increasing returns to scale is a natural monopoly. Since most companies are not monopolies, increasing returns to scale must not be common. I have never found this argument convincing.

A more sophisticated form of the above argument is a technique economists sometimes use to estimate the cost functions of firms . Many early attempts to measure returns to scale simply assumed that firms produce at the point where average costs are the lowest, and that all firms must face decreasing returns if they increase production beyond their current size. These early measurements inferred from the existence of small firms that increasing returns to scale must be rare.

Why is a firm that faces increasing returns to scale a natural monopoly according to economic theory? Suppose at a particular time several firms share a market equally. All have identical cost structures, all of which exhibit increasing returns to scale. The firms are all assumed to produce the same product. Then, suppose one firm (firm A) obtains a slightly larger market share in one period (due to randomness). Firm A now has lower costs than the others, which means it can charge a lower price. Since customers only care about product price, more will choose to purchase from firm A, compounding this firm’s advantage further. Eventually, the process will continue until all customers are buying from firm A. At this point no other firm is able to compete with firm A, since they cannot match the cost advantages firm A achieves due to its size. Competition is not possible and the market ends up dominated by a single firm with monopoly power.

Whenever I see the above argument used to dismiss increasing returns to scale I am reminded of a well known joke.

A policeman sees a drunk man searching for something under a streetlight and asks what the drunk has lost. He says he lost his keys and they both look under the streetlight together. After a few minutes the policeman asks if he is sure he lost them here, and the drunk replies, no, and that he lost them in the park. The policeman asks why he is searching here, and the drunk replies, “this is where the light is”.

The economists dismissing increasing returns to scale because they don’t understand how it does not always lead to monopolies are the equivalent of the drunk searching in the light. Both should know they ought to be looking elsewhere. The drunk will never find his keys in the light, and economists will not have success modelling the economy if they only focus on situations where modelling competition is easy.

The conclusion that increasing returns to scale leads to monopolies relies on a large number of assumptions about the way markets work. It makes no sense to assume increasing returns to scale are not present because you can’t explain competition using a highly simplified model when they exist.

In fact, there are ways in which economists can explain competition when firms face increasing returns. The most common way this is done is by assuming that consumers care about variety. A single firm will not dominate the market because consumers want to buy a wide variety of goods. The most common mathematical description of this scenario is known as Dixit-Stiglitz monopolistic competition.

In Dixit-Stiglitz competition models, consumer taste for variety is modelled through assuming a complicated set of preferences designed to give a simple answer. The preferences are such that

$\frac{dp}{dq}=-\frac{ p}{\sigma q}$

where $p$ is price $q$ is quantity and $\sigma>1$ is a parameter representing the importance of variety to consumers (lower sigma means consumers care more about variety). Deriving the above expression from consumer preferences requires that each firm is small enough to not be able to significantly affect the overall price level.

Firms are assumed to be profit maximizing, so, similar to the perfectly competitive case we obtain the following first order condition for a maximum

$\frac{d \pi}{dq}=0=p+q \frac{dp}{dq} -\frac{dc(q)}{dq}$

where $\pi$ is profit. Substituting in the expression for $\frac{dp}{dq}$ we obtained above and simplifying gives

$p=\frac{\sigma}{\sigma-1} \frac{dc(q)}{dq}$

In economic terms this states that price is a markup over marginal cost that depends on how much consumers care about variety. The more consumers care about variety, the higher the profit maximizing price.

Typically it is assumed that firms cost functions have the form $c(q)=aq+f$ where $a$ is a constant cost per unit and $f$ is a fixed cost. This cost function is chosen to simplify the math. Substituting this cost function into our expression above gives us an expression for the price charged by a firm,

$p=\frac{\sigma}{\sigma-1} a$

We then need to check to make sure that this point is actually a maximum, by checking that the second derivative of the profit function is less than zero, similar to the perfectly competitive case.

$\frac{d^2 \pi}{d^2 q}=\frac{d}{dq}\big(p(1-\frac{1}{\sigma})-a\big)=\frac{dp}{dq}(1-\frac{1}{\sigma})$

Substituting our expression for $\frac{dp}{dq}$ from above gives

$\frac{d^2 \pi}{d^2 q}=-\frac{p}{q\sigma}(1-\frac{1}{\sigma})$

Since $\sigma >1$ and $p, q$ are positive $\frac{d^2 \pi}{d^2 q}<0$ which means we have a maximum.

A different cost function without constant marginal costs could have been used instead and the results would be broadly similar.

At this point it is assumed that free entry drives the profits of firms to zero. Substituting our expression for price and cost into the profit equation we find that each firm is able to sell a quantity that is given by $q=\frac{f}{a} (\sigma-1)$. The general conclusion is that firms in Dixit-Stiglitz competition set price as a markup over marginal cost and sell a quantity which depends on their cost function and consumer preference for variety.

Replacing perfect competition with Dixit-Stiglitz monopolistic competition explained several puzzles in the field of international trade. Models of trade based on comparative advantage predict that trade should be highest between dissimilar countries, while in reality trade is often highest between countries with very similar characteristics. Paul Krugman won his Nobel prize for applying monopolistic competition to models of international trade. Models based on monopolistic competition instead explain how trade could occur between countries with similar characteristics. Similar ideas have been used in economic geography to explain why people tend to congregate in cities. Despite the success of models incorporating increasing returns this way in several subfields of economics, incorporating increasing returns this way is still a relatively uncommon within the field. Monopolistic competition is used in many models, but those models typically incorporate increasing costs.

Despite the success of monopolistic competition, monopolistic competition is not really an ideal explanation of how competition works under increasing returns. In reality, firms often produce more than one variety of product. Since firms can produce more than one variety, arguments similar to the arguments against decreasing returns to scale discussed in a previous post apply. Firms ought to be able to decentralise the production of different varieties and achieve at worst the same efficiency as two smaller firms. In other words firms ought to have at worst constant returns to scope, ie it should never be less efficient for the same firm to produce multiple varieties than for two firms to produce the varieties separately.

I tend to think the success of Dixit-Stiglitz competition in explaining various economic phenomena is because it is the only tractable way economists have found to analyze increasing returns in their models. Dixit-Stiglitz competition does not need to be a perfect theory of competition in the presence of increasing returns to scale to explain its success in international trade theory. The fact that firms in Dixit-Stiglitz competition tend to set price as a markup over marginal cost also reflect the empirical reality of how firms set prices.

How do we reconcile the existence of increasing returns with the fact that most industries are not dominated by a single firm? I think the most likely explanation is simply that economic processes take time. I suspect the long run equilibrium for most industries is for a single firm to control the entire market. In fact, the equilibrium for the entire economy might be to have a single firm producing everything. However, this does not happen immediately. Consumers have a tendency to stick with the products they know and change infrequently, even if another product has a lower price. Companies might not have access to the technology to begin creating a different product, and firms might have difficulty scaling up production enough to supply the entire market even if they have a small cost advantage. Additionally, various outside factors could affect the market. For example, governments sometimes break up companies that get too large, disrupting what could be a natural process towards larger firms. New technologies could enable new entrants to compete with existing players and disrupt what would otherwise be a natural monopoly.

Finally, firms aren’t necessarily going to let other firms dominate the market if they can help it. Firms know the benefits of having monopoly power, and will fight to obtain it. So if there are initially a large number of firms in a market one firm will not dominate the market due to having a small cost advantage. Rather, firms will attempt to compete as best they can to achieve market dominance. Sometimes this will require selling at a loss, but if a firm does so for too long they may have difficulty obtaining financing to continue to do so.

The above description closely matches the Post-Keynesian theory of the firm, as portrayed, for example, in Marc Lavoie’s book Post-Keynesian Economics: New Foundations. Post-Keynesians believe that the primary objective of firms is growth, not profit maximisation. The connection between this idea and increasing returns is obvious: if firms’ costs are decreasing then it makes sense that both their market power and their profits will increase as they grow. Additionally, if firms fail to grow they will not be competitive with firms that do, and hence growth is a prerequisite for firm survival.

Post-Keynesians believe that firms seek profits as a means to fund expansion. If a firm demonstrates profitability it is likely going to be able to borrow more and sell more equities, thus funding its further expansion. There is a tradeoff between gaining more customers through charging the absolute lowest price possible, and obtaining profits to fund further expansion.

The story I told above about competition in the presence of increasing returns would no doubt benefit from being made more precise. However, for many applications, particularly in macroeconomics, it is probably not necessary to model competitive forces exactly. Assuming that prices are constrained by competitive pressures to be some level of markup is probably enough to gain understanding of the effect of returns to scale in many contexts. For example, assuming Dixit-Stiglitz competition with increasing returns resolved long-standing puzzles in the theory of international trade. Similar benefits could likely be gained by following a similar approach in other areas of economics.

## 3 thoughts on “Competition with increasing returns to scale”

1. Benedict says:

I tend to think of managerial incompetence as the explanation for the question you posed:

“How do we reconcile the existence of increasing returns with the fact that most industries are not dominated by a single firm?”

Most would agree that managerial incompetence is a big reason why some firms underperform. I think most would also agree that as firms get bigger the task of managing them becomes more complex, and it becomes harder to find the talent necessary to manage them effectively. However, I don’t think it is the cost overhead of management that limits returns to scale. Rather, I think the real challenge for firms is that it is hard to tell at the point if hiring whether a candidate is competent, especially when some people basically spend all of their time trying to disguise their own incompetence. Gavin Belson from the show Silicon Valley is a good example of a CEO who bases all his decisions on protecting his reputation, even as he runs his company into the ground. Therefore, although with decent management firms will experience increasing returns to scale, they have to survive the constant and somewhat random threat of abject incompetence rising to the top of their managerial hierarchy as they grow.

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