# The importance of competition

In an oligopoly firms produce a homogeneous product so they are directly competing against each other. In perfect competition the same holds with the main difference being that the number of firms is practically infinite. In some sense perfect competition is thus a special case of oligopoly.

Using this reasoning, we can see that as we vary the number of firms in a market of a homogeneous good, we will have a bunch of cases that are in-between an oligopoly with few firms and perfect competition. To determine the benefits of competition, we can ask ourselves how certain variables (such as price or welfare) change as we vary the number of firms.

In addition, in this post we will see why the assumption of perfect competition in many economic models is a reasonable one.

Let us thus consider a standard oligopoly model where the number of firms is F. Assume that the inverse demand function takes the form

$P^d(q) = p^{max} - mq.$

Firms are identical and they have constant marginal costs

$c(q_i) = \mu q_i \implies MC_i = \mu.$

Firm i faces the following inverse demand curve:

$p^{max} - mFq^{OL} = p^{max} - (F - 1)mq^{OL} - mq_i.$

A little explanation on this one: since firms are identical they all produce the same quantity, which we label q^OL. There are F firms, so the total quantity on the market is Fq^OL. Substituting this into the inverse demand function we get the left side. Then we can seperate Fq^OL into (F – 1)q^OL – q^OL. That is we treat one firm’s (firm i’s) demand seperately. The demand firm i specifically faces can be rewritten as q_i. Doing this we get the right side.

By definition, the inverse demand curve above thus gives the price at which firm i will be able to sell its products on the market. To calculate the revenue of firm i just simply multiply the inverse demand by the quantity firm i sells

$R_i = p^{max}q_i - (F - 1)mq^{OL}q_i - mq_i^2.$

Now we know that a profit maximizing firm will equate marginal revenue with marginal cost so

$p^{max} - (F - 1)mq^{OL} - 2mq_i = \mu.$

Using the fact that all firms are identical and that thus q_i (the amount firm i produces) is the same as the production of any other firm (q^OL), we can let q_i = q^OL above and then solve for q^OL to obtain

$q^{OL} = \frac{p^{max} - \mu}{(F+1)m}.$

This is the amount that gets produced by each firm. Clearly, if F goes to infinity this expression goes to zero meaning that no one firm will produce a large amount of goods in perfect competition. Something anyone who ever took a principles of micro course knows. Now plug the total production Fq^OL into the inverse demand function, do the algebra and rearrange to get

$p^{OL} = \frac{1}{F+1} p^{max} + \frac{F}{F+1} \mu.$

This is the price that gets charged by firms in this market. When we express the equation as above, we can clearly see that the price is a weighted average of p^max, the maximum price at which anything can be sold, and of mu, the marginal cost of firms. The weights relate to the number of firms. Clearly as F goes to infinity, the first weight goes to 0 while the second one goes to 1. This means that in perfect competition price will equal marginal cost. Again, something that anyone who ever took principles of micro knows.

It is worth taking a look at a numerical example with a less extreme case though. Assume that p^max is $80 and that marginal cost (mu) is$20. Then with three firms on the market we will have a price of

$p^{OL} = \frac{1}{4} \80 + \frac{3}{4} \20 = \35.$

Increasing the number of firms to just 10 will give us a price of

$p^{OL} = \frac{1}{11} \80 + \frac{10}{11} \20 = \25.5,$

which is a 27% price reduction and is quite close to the perfect competition price of \$20. But this doesn’t show the real power of competition. In reality, more competition (i.e. increasing F) will not only push prices down, but this will also let people consume more, so quantity will also change. Only when we look at these two effects together, can we really appreciate competition.

Consider the following graph, where p* and q* represent the price and quantity in perfect competition:

Note that because the price went up (because of the lack of competition) from p* to p^OL, the quantity consumed dropped from q* to Fq^OL. All these units would have been consumed had there been more competition. The value of this lost consumption is represented by the grey shaded triangle. This is what we call the deadweight loss, a term which should again ideally be familiar from principles of micro to anyone. It basically represents the value of the goods that could not be consumed because of the higher prices. Higher prices which in this case are caused by the lack of competition.

The deadweight loss is a good comprehensive measure of how much the lack of competition costs to society. It is better than just looking at price and quantity alone. So let us see how deadweight loss changes as we vary the number of firms. First note that the deadweight loss is just the area of the triangle in the picture, which is

$DWL(F) = \frac{1}{2} (p^{OL} - p^*)(Fq^{OL} - q^*)$

$DWL(F) = \frac{1}{2m} (\frac{p^{max} - \mu}{F + 1})^2.$

Here we make use of the fact that p* = mu, that is in perfect competition price will just equal the marginal cost. Furthermore, q* is calculated using the formula for q^OL, multiplying it by F and letting F go to infinity. If we do this, we can clearly see that q* will converge to (p^max – mu) / m.

Now let us examine the expression for deadweight loss. What’s remarkable is not that DWL decreases as the number of firms increases, but that it decreases with the square of the number of firms. When we went from 3 firms to 10, price decreased by 27%. But look what happens to DWL (letting m = 0.5):

$DWL(F) = (\frac{80-20}{3 + 1})^2 = \225$

$DWL(F) = (\frac{80-20}{10 + 1})^2 = \30$

We increased the number of firms 10 / 3 = 3.33-fold, this increased the denominator 11 / 4 = 2.75-fold. But the denominator gets squared, so altogether DWL decreased 2.75^2 = 225/30 = 7.5-fold. This is an 87% decrease in DWL. Why the big difference compared to a measly 27% decrease in price? Because DWL takes changes in quantity demanded (which is a result of the drop in price) into account as well.

Experimenting with different values of F we get:

This shows that for all practical purposes if a market has at least about 10 firms, it gets very close to perfect competition. This has two important implications:

1. It shows that competition has very strong potential effects. Increasing competition should therefore be a priority in markets where goods are roughly homogeneous and the number of competitors is less than 10 (with the exception of some special cases like network industries). This can dramatically improve welfare. But also note that if a market already has about 10 or more firms, promoting further competition will yield little additional welfare.
2. This result is also very encouraging from a modeling point of view. Economists often assume in their models that markets are perfectly competitive. Many times outsiders / beginners (such as MBA students or perhaps undergrad econ students) who study economics and/or hear about economic research complain that this assumption is “stupid” and/or unrealistic. The result above, however, shows that the assumption of perfect competition is quite reasonable once we have 10 or more firms in a market with homogeneous products.

Therefore, we have established a very strong result above. And I hope that by now you appreciate the benefits of competition even more.