00:02
A monopolist maximizes their profit when marginal revenue is equal to marginal cost.
00:11
Let's start by finding an expression for revenue.
00:13
Revenue is the price of each unit of the product, which is 200 minus q, multiplied by the number of units sold, which is q, and the cost is given to be 40 times q.
00:33
So marginal revenue is equal to the derivative of revenue with respect to the number of units sold, which is equal to, since this is 200q minus q squared, we get 200 minus 2q, okay, and then marginal cost is the derivative of cost with respect to q, which is just 40.
01:03
So we want to find marginal revenue and marginal cost when q is 90.
01:10
So marginal revenue at q equals 90 is equal to 200 minus 2 times 90, which is 200 minus 180, which is 20.
01:24
I see that 20 is not equal to marginal cost, which is 40.
01:31
Okay, so the monopolist is not maximizing profit.
01:36
Okay, so this number is static.
01:41
Okay, the value of marginal cost is fixed, so we need to adjust marginal revenue until it's equal to marginal cost, which is 40.
01:49
So we want to find when marginal revenue is equal to 40, and that will tell us how to maximize the profit.
01:56
Okay, so let's set 40 equal to marginal revenue, which is 200 minus 2q, and now we'll solve for q...