00:01
Consider the following demand and supply functions based by a monopolist.
00:10
Find the price and quantity that maximizes profits for the monopolist.
00:18
So we're going to determine the total revenue.
00:27
Revenue is equal to price times quantity.
00:36
So substitute in the demand function for price and that's our total revenue from the demand curve.
00:47
So now we're going to distribute this.
00:53
We end up with 20q minus 2q squared.
00:58
Then we're going to find the marginal revenue.
01:04
In order to do that, find the derivative of total revenue.
01:24
At equilibrium, marginal revenue is equal to marginal cost.
01:29
So we take our marginal revenue, set it equal to our marginal cost, which is 10.
01:59
Now solve for q.
02:22
So q is equal to 2 .5 at equilibrium.
02:32
Put the value of q into the demand function to find the price.
03:09
This is the equilibrium price.
03:15
So basically, the main thing to remember is that we need to set marginal revenue equal to marginal cost and then we can find the profit maximizing price and quantity.
03:52
So we're going to find the consumer surplus, producer surplus, and the deadweight loss.
04:01
So if you would prefer, you can draw the graph first.
04:20
So our marginal cost curve is going to be horizontal because it's just a constant equal to 10.
04:30
And then we have our demand curve and our marginal revenue curve is going to be like this.
04:41
So where these two curves cross, we can use that to determine the deadweight loss.
04:57
So you see that there is a point where marginal cost and demand cross and that combined with the point where marginal revenue and marginal cost cross gives us the base of this triangle.
05:13
Then we have to draw a line up to the demand curve from this point.
05:23
So you can see from this that we are going to need the quantity for the point where marginal cost and the demand curve cross and we also need this quantity which we already calculated because that is where marginal revenue and marginal cost cross.
05:45
So we can go ahead and just label that and we're going to calculate the other one.
05:49
And then at this quantity, we need to figure out the corresponding price on the demand curve which we did calculate.
06:02
So we know that this one is going to be 15.
06:09
So that would be the deadweight loss.
06:11
Now the producer surplus is going to be this area here.
06:21
So you could see that this ends up being a rectangle and we know this part already which is 2 .5 and we also know this part.
06:33
So we actually can get that now...