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# Monopolist optimizing price: Marginal revenue

AP.MICRO:
PRD‑3 (EU)
,
PRD‑3.B (LO)
,
PRD‑3.B.5 (EK)
,
PRD‑3.B.6 (EK)

## Video transcript

now that we've figured out the total revenue given any given given any quantity and we've also been able to express it algebraically I want to think about what the marginal revenue is at any one of these points and to think about marginal revenue marginal revenue is just how much does our total revenue change given some change in our quantity and then later we can use that so that the this are so that we can optimize the profit for our monopoly over here and I'm going to try to do it without calculus it actually be very straightforward to do it with calculus because we're essentially just trying to find the slope at any point along this curve but I'll try to do it algebraically and maybe it'll even give you a little intuition for what we end up doing eventually in calculus so the first thing I want to do is essentially find the slope the slope right over here and the best way to find the slope right over here is say well how much does my total revenue change if I have a very small change in quantity so if I have a very small change in quantity how much does my total revenue change so let me think about it this way and the other ones I will be able to approximate a little bit easier so let's think of it this way if my quantity is zero my total revenue is zero that one's easy if I increase my quantity very very very very little so let's just make it zero point zero zero one what is going to be my total revenue and we could think about it in terms of this curve right over here or we could just use this expression which we derived from price times quantity and we will get my calculator out if we have if our quantity is point zero zero one our total revenue is going to be negative let me turn it up the calculator on total revenue is going to be negative point zero zero one squared squared so that's that part plus six times point zero zero one six times point zero zero one so that's going to be our total revenue so it's going to be zero point zero zero five nine nine nine so it's zero point zero zero five nine nine and so now we can figure out or get a pretty good approximation for that marginal revenue right at that point our change in quantity is 0.0 so our Delta Q this right over here is 0.001 that's our change in quantity and our change in revenue is 0.005 nine nine and so we just have to divide we just have to flat divide point zero zero five nine nine nine that top one our change in total revenue divided by our change in quantity divided by point zero zero one and we get five point nine nine nine nine nine it says essentially and if you try it with even smaller numbers if you tried this with Oh point zero zero zero zero zero zero one you'll get you'll get five point and you'll get even more nines going on so the closer that you get the smaller your change in and this is what you essentially do in calculus you try to find a super small change right over here this is essentially going to be six our marginal revenue at this point is essentially going to be six so what I wanted to do is I'm going to plot marginal revenue here on our demand curve as well around this axes where we've already plotted our demand curve so when our quantity is want zero our marginal revenue if we just barely increase quantity the incremental total revenue we get is going to be six so I'll just plot it I'll just plot it right over there and that makes sense the marginal benefit in the market is six right at that point so if we were to just sell a drop of orange juice or I guess for selling oranges in this case not juice but if we were to sell a millionth of a pound of oranges we would get the equivalent of roughly six dollars per pound for that millionth of a pound because that's the marginal benefit for that very first incremental chunk of orange out there in the market so it makes complete sense now let's think about the slope at these other points and these I'm going to approximate I could do it this way but I'll just approximate it I'll just approximate it by using other points so if I want to find the slope right over here what our quantity is equal to one the slope would look like the slope would look like that and I'm going to approximate it by finding the slope between these two points I am going to approximate it actually is going to be a very good approximation I'll do it later with calculus to show that it is a very good approximation but I'm going to approximate it by the slope between these two points and between those two points our change in quantity is two and our change in total revenue is 8-hour change in total revenues eight when we produced to our total revenue 2,000 pounds our total revenue was eight thousand dollars so it's to it so we have a change in total revenue of eight or eight thousand I guess we could say divided by a change in quantity of two thousand so our marginal revenue at this point is eight divided by two or eight thousand divided by two thousand which is four dollars per pound so when our quantity is one our marginal revenue is four dollars per pound is four dollars per pound just like that now let's think about the marginal revenue when our quantity is two and to do that I'm going to find the slope between these two points we really want to find the slope of that line but it looks like the slope between these two points is a pretty good approximation it's actually almost an exact number because the way that this is just a parabola so we can actually do this but anyway this is fairly straight forward once again our change in quantity is two and our change in total revenue our change in total revenue is we're going from five to nine which is four this was nine right over here from the last video so it's four it or you could say it's four thousand dollars divided by two thousand pounds gives you gives you two dollars per pound so our marginal revenue right over here if we have quantity of two is two dollars per pound right at that point for that incremental you know millionth of an ounce that we're going to sell of oranges we're getting the equivalent of two dollars a pound of increased total revenue from doing that and let's just do one more point here and I think you'll see why I'm only going to do one more point if we try to go up here and we try to figure out what is the marginal revenue or if we essentially say what is the slope there how much do we get an increase in revenue if we are if we just barely increase our quantity and this is actually easier to look at this is a maximum point right over here and there in the calculus terms the slope up there is zero we can even see that by approximating the slope between the slope between these two points we have some change in quantity but we have no change in total revenue so right at that point right over here the slope is barely positive right over here right at that point the slope is zero and then right past it it becomes barely negative but at that point our marginal revenue is zero so when our quantity is 3,000 pounds our marginal revenue is zero and then after that our marginal revenue gets negative over here our marginal revenue gets more and more negative but something very interesting happens when we plot our marginal revenue curve or our line in this case we are getting a line we are getting a line we are getting a line that is twice as steep twice as steep as our demand curve and this is actually generalizable if we have a linear demand curve like this it can be defined as a line then your marginal revenue curve for the monopolist will also be a linear downward sloping curve like or downward sloping line and it will have twice the slope so this slope over here was negative one this slope over here is negative two for every for every increase in quantity the price goes down by two increase in quantity price goes down by two increase in quantity price goes down by two so this is marginal revenue and let's remind ourselves you know we were doing all of this algebra and all of this math here whatever what is marginal revenue telling us this was the demand curve it tells us for any given price what quantity is demanded or for any given quantity what is the incremental marginal benefit or I guess what's the price at which they could sell that quantity from that we were able to figure out the total revenue as a function of quantity and from that total revenue we were able to say well look if at any of these quantities if we were to increase a little bit more if we were to if we were to increase quantity a little bit more how much is our revenue increasing and so obviously we want to keep increasing quantity while our revenue is well well while our the the marginal the marginal revenue we get is larger than our marginal cost and I'll take that up in the next video