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## Some Thoughts On Pricing, Part III

What happens when competition decides they will become organized and the price of a product is exactly 10 pesos?  If I am bound by the price too due to politics (perhaps the government itself sets the price because it has such powers) or some other factor, and I have to price myself at 10 pesos, then the rational consumer is faced with identical products at identical prices to choose from.  Perhaps he will then choose at random.  If F1 is my only competitor, he will choose me at the shops half the time.  If instead there are Fn competing companies, I will be chosen perhaps $\frac{1}{n+1}$ of the time.  My expected profit in such a situation is easily calculated as:

$E_p(10) = \frac{1}{n+1} \cdot (9) + \frac{n}{n+1} \cdot (-1) = constant$

Having established such, let us assume that I am not bound by any politics.  Then it is only obvious that I would want to price at 9.99, since this virtually guarantees that the rational consumer choosing as by the C1 axiom will pick me over any other product: I am guaranteed in effect selling 100 percent of my product, and furthermore at maximal expected profit, since selling at anything less than 9.99 would mean obtaining less for product I am sure to give away.  In terms of expected profit, we can make

$E_p(x) = x - 1$

if I price at less than ten $x < 10$, or

$E_p(x) = -1$

if I price at more than ten $x > 10$, for any amount of competition against me (does not depend on n, since everyone is pricing at 10).  Maximal expected profit is at a price of 9.99 in these cases.

I like to consider this particular example the limiting case in which the Gaussian distribution is tightly wound around 10 pesos. The tighter the certainty around ten pesos, the closer I am to the above distribution (except the definition at 10 pesos, which we reasoned in a different way).  This is because the probability of selling my product at less than 10 pesos is essentially 1, where selling at a price above 10 pesos is essentially zero.  Perhaps this can be more easily seen upon inspection of the following graphs.

First, I have graphed what happens as the certainty of F1's pricing becomes tighter and tighter (one other competing firm).

In this next graph, I have shown what happens as the certainty of F1...F5's pricing becomes tighter.

For 40 competing firms, this is what happens.

All these graphs show that indeed the limiting distribution will not depend on the number of companies competing against me as they converge or stack upon a single price quote.  The more in agreement companies are about what the price should be, the less their ability to sell (oppositely for me) and the more of the pie I can take, and my expected profit per product will balloon to an absolute maximum of 8.99 pesos (9.99 revenue - 1 cost).

For a company thinking this way we have differences of pennies across comparable products, much perhaps as Elisa remarks in her comments about pricing in Switzerland.

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