00:01
So here we're given a demand function and a cost function and we're told to maximize revenue, right? now that's a very unusual choice, right? it's not, i'm going to assume that we're thinking, are we thinking about maximizing revenue or maximizing net revenue, which is what we call profits? i would say the argument here is to maximize net revenue.
00:36
Because if there's no reason for a monopolist to maximize revenue, that is irrational.
00:41
What the monopolist wants to do is to maximize net revenue profits.
00:47
For example, they gave us a cost function.
00:52
If the goal is simply to maximize revenue, there's no point in giving us a cost function.
00:57
We don't need the cost function.
00:59
So i'm going to assume that this entire question is referring to net revenue, right? so net revenue is equal to price times quantity minus costs.
01:16
Now, if you want to do this for revenue, you would just set for revenue, it would just be price times quantity.
01:23
Maybe i'll do both.
01:25
So net revenue here would be if we sub in to make everything a function of key, q, we get a gorgeous net revenue function.
01:35
For revenue, we just get the revenue part.
01:40
I am now going to differentiate with respect to q, right? and this says i want to find the maximum, right? these things, if we think about the top of a function, we know that the top of a function has a slope equal to zero, right? so i want to differentiate these things to find where the slope is equal to zero.
02:00
So i get 500 minus 4q minus 4q is equal to 0, which gives me 500 is equal to 8q, right? if i differentiate here, i get 500 minus 4q is equal to 0.
02:18
500 equals to 4q, q is equal to 125, q is equal to 62 .5.
02:27
To get the price, we plug back into the demand curve, right? the demand curve tells us what the price is...