## Sunday, November 4th, Profit Maximization with Inequalities in Math

All firms make money. Different pricing leads to different profits. How do they determine which prices to use?

Starting with a few assumptions. Firstly, we are going to use the traditional demand model, and assume that firms have 0 production costs. That is, their profit is equivalent to their revenue. Secondly, we are assuming that the number of customers at each price level can be approximated with a linear model.

Listing out a few prices and quantities, we see that the profit, which is equal to price times quantity, generally increases, then decreases.

Since the model is linear, we can write the equation aP+bQ=C, where a,b, and C are constants. Also, we want to maximize the quantity Profit=PQ.

To do this, we can substitute Q=(C-aP)/b, and let Profit=P(C-aP)/b. Now Profit is a function related to P only, so we can try to maximize it.

Instead of doing Calculus, which you would normally do in a high school math class, we can use something much simpler and useful, or the AM-GM inequality.

The inequality states that the Arithmetic mean of 2 positive real numbers (a+b)/2 is always greater than or equal to the Geometric mean of the 2 numbers, sqrt(ab).

This can simply be proved by squaring both sides, giving us 1/4(a^2)-1/2(ab)+1/4(b^2) is greater than or equal to 0, which is true because the left hand side can be written as a perfect square ((a-b)/2)^2.

Using the AM-GM inequality, we can now find a general formula for maximizing profit. We know that Profit=P(C-aP)/b, or b*Profit=P(C-aP).

To make the equation more usable for AM-GM, we can multiply a on both sides, giving us ab*Profit=aP(C-aP).

Directly applying AM-GM on the right hand side, we get that aP(C-aP)≤((aP+C-aP)/2)^2=C^2/4. Thus, ab*Profit≤C^2/4, and Profit≤C^2/4ab.

Note that the equality takes place when a=b, or P=C/2a in this case.

This can be used to solve for the optimal pricing for a given demand curve, for example, the following chart:

P | Q |

2 | 10 |

4 | 9 |

6 | 8 |

8 | 7 |

10 | 6 |

12 | 5 |

14 | 4 |

16 | 3 |

Here, P+2Q=22 is the equation, we can either do the work that provided us with the general formula again, or we can simply use our conclusion that Profit≤(22^2)/4*1*2=60.5, which occurs when price=11.

Note that although 11 is not listed on the chart, we assumed that it is a possible price to set, because the domain of the function is the real numbers such that both P and Q are positive.

With this in mind, we can increase the difficulty in a few ways:

One thing to do would be adding in the cost function, which can also be modeled as a linear increasing function. However this does not make things too different because instead of calculating the Profit as price times quantity, we can use Per-unit profit times quantity, with Per-unit profit=Price-cost. This would again give us a relationship between price and profit in the form of the formula.

Another thing to do would be deriving those without directly using the formula, or using them on equations not in the form of the formula. This should be simple with a bit of algebra tricks, and the specifics are left as an exercise to the reader.

To summarize, the AM-GM inequality is one of the most useful equations when it comes to maximizing or minimizing values, and it can be applied nicely if there is a way to turn the value you want to maximize or minimize into a function of one variable.

If you feel interested and want a bit of extra challenge, you can explore around the AM-GM inequality for more than 2 variables, or the more complete version of QM-AM-GM-HM inequality. Even better, you can try to do some maximization problem with calculus and then AM-GM, and compare how much time it takes for both.

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