00:01
So here we're talking about monopoly.
00:02
We know a demand curve, which is p equals 596 minus 4q.
00:08
We know the marginal cost, which is equal to 60.
00:12
And we want to try to maximize profit.
00:16
So the monopolist wants to set marginal revenue equals to marginal cost, but what the heck is marginal revenue? well, revenue is equal to price times quantity, but we have an expression for price.
00:27
So we can sub in and get an expression for revenue.
00:33
Now, marginal revenue is the derivative of revenue with respect to quantity, which is equal to 596 minus 8q, right, if i just take that derivative.
00:46
So graphically, what does this look like? quantity, price, our demand curve starts at 596 and sloped down.
00:56
The marginal revenue curve also starts at 596, but slopes downward at exactly at double the rate.
01:04
And then we have a marginal cost of 60.
01:07
So if i equate these two, i get the monopolist's optimal profit, which is going to be 596 minus 8q is equal to 60.
01:17
That means 536 is equal to 8q.
01:21
And i really hope they've given us an integer here.
01:24
I'm going to slam that in my calculator.
01:28
67.
01:29
What an interesting number.
01:32
67 is the monopolist's optimal quantity, right, by equating marginal revenue and marginal cost.
01:37
Now, the price is right off the demand curve.
01:40
If we produce 67 units, the demand curve tells us what the resulting price in the market is going to be.
01:47
So price will equal to 596 minus 4 times 67.
01:52
You again plug that into your calculator and this gives me a price of 328 right so 328 is the actual price now the profit has a interpretation right in this graph this blue square here is the profit right the profit is equal to price minus marginal cost times q, right? price minus marginal cost is how much you're making on every single unit.
02:29
Q, you're making that times q number of units.
02:32
So profit here is going to equal to 328 minus 60.
02:40
That's the profit margin times 67 units, which is, again, a question for the calculator.
02:46
But the calculator is happy to provide us with the answer.
02:51
And that yields a fairly large profit of 17 ,956...