# Mathematics of a price war

In my last post I looked at two companies competing over a natural monopoly. We saw that once they started competing it would not then make sense for either company to drop out. This is why price wars tend to continue once they have started.

Here is the maths behind the problem:

To keep things simple let’s assume that the two companies are only going to compete for two years.

They can drop out before they even enter the market, after a year or after two years.

Let’s assume that the second company drops out at the start with probability x and after the first year with probability y. This leaves the probability of him dropping out after two years as 1-x-y.

We will say that the cost to each company of competing is represented by ‘c’ each year and the profit they can make in a monopoly is ‘p’.

Let’s work out the profit that the first company will make at the end of each year.

It profit at the start is zero because they haven’t yet entered the market.

After the end of the first year it has either incurred a cost c if the other company hasn’t dropped out, which has a probability of 1-x, or it has a profit of p if the other company dropped out at the start, which has a probability of x.

In other words its expected profit after a year = xp – c(1-x)

After two years its expected profit = 2xp + (p-c)y – 2c(1-x-y)

This is because 2p is two years’ worth of profit and he gets this if the other company dropped out at the start, which happens with probability x. This gives us the 2xp bit.

If the other company drops out after a year, which happens with a probability y, then he will make a profit p in the second year, but have incurred a cost c in the first year. Overall he makes (p-c) with a probability y. This gives us the (p-c)y part of the equation.

The last bit of the equation is the two years of cost (2c) multiplied by the probability that the other company waits until the final year to drop out (1-x-y)

We now know the expected profit for the first company at the end of each year based on the probability of second company dropping out at each point. The second company will set the probabilities so that first company’s profits are the same whatever strategy it chooses.

This means that the expected profit of dropping out at the start must equal the expected profit of dropping out at the end of year 1, so xp – c(1-x) = 0

The other two equations must be equal as well so:
xp – c(1-x) = 2xp + (p-c)y – 2c(1-x-y)

This is only true if y=0.

y is the probability that the second company drops out after a year. y is 0 so there is no chance that the company will drop out half way through the game. The game is symmetrical so the same applies to the first company. They either drop out at the start or keep going to the end.

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