Argmax, An Example

Profit Maximization

Suppose a company wishes to produce a quantity, q, of computers that will maximize profits. Suppose that the price of computers is $100. Also assume that the total cost to the firm increases with the quantity produces according to the equation given below.

So,
Revenue = 100 q.
Cost = 100 + 5q²

Profits(q) = Revenue - Cost = 100q - (100+5q²).

We want to then find the quantity that maximizes profit.

q* = argmax (Profits(q)).

To find the maximum, we can take the first derivative and set it equal to zero. This will find the top of the profit curve (see graph below).

d(Profits)
----------- = 100 - 10q.
d q

Setting this equal to zero gives,

100 - 10q* = 0, or
q* = 10.

Profits at q=10 are then 100 x 10 - (100 + 5 x 10²) = 400.

We can use shorthand q* = argmax [Profit(q)] = argmax [100q - (100 + 5q²)] = 10.