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Current time:0:00Total duration:7:11

Monopolist optimizing price: Total revenue

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

Video transcript

what I want to start thinking about in this video is given that we do have a monopoly on something and in this example in this video we're going to have a monopoly on oranges given that we have a monopoly on oranges and a demand curve for oranges in the market how do we maximize our profit and to answer that question we're going to think about our total revenue in different four different quantities and from that we'll get the marginal revenue for different quantities and then we can compare that to our marginal cost curve and that should give us a pretty good sense of where what quantity we should we should produce to optimize things so let's just let's just figure out total revenue first so obviously if we produce nothing if we produce zero quantity we'll have nothing to sell you know revenue is total revenues price times quantity your price is six but your quantity is zero so your total revenue is going to be zero if you produce nothing if you produce one unit and this over here is actually 1,000 pounds per day and we'll call a unit a thousand pounds per day if you produce one unit then your total revenue is one unit times five dollars per pound so it'll be five times actually 1,000 so it'll be five thousand dollars and you could also view it as the area as the area of this rectangle right over here you have the height is price and the width is quantity but we can plot that five times one if you produce one unit you're going to get five thousand dollars so this right over here is in thousands thousands of dollars and this right over here is in thousands of pounds thousands of pounds just to make sure that we're consistent with this right over here let's keep going so that was this point or when we produce one thousand pounds we get five thousand dollars if we produce two thousand pounds if we produce two thousand pounds now we're talking about our price is going to be four dollars or if we can say our price is four dollars we're gonna we can sell 2,000 pounds given this demand curve and our total revenue is going to be the area of this rectangle right over here height is price width is quantity four times two is eight so if I produce two thousand pounds then I will get a total revenue of eight thousand dollars so this is seven and a half eight is going to put us something right right about there and then we can keep going if I produce or if the price is $3 per pound I can sell 3000 pounds my total revenue is this rectangle right over here three times three is nine thousand dollars so if I produce three thousand pounds I can get a total revenue of nine thousand dollars so right right about there and let's keep going if I produce or if the price is $2,000 or is if the price is $2 per pound I can sell four thousand pounds my total revenue is two times four which is eight thousand dollars so if I produce four thousand pounds I can get a total revenue of eight thousand dollars it should be even with that one right over there just like that and then if I produce or if the price is $1,000 I'm looking for new colors if the price is $1 per pound I should say I can produce I can sell five thousand pounds my total revenue is going to be one times five or five thousand dollars so it's going to be even with this year so if I produce five thousand units I can get five thousand dollars of revenue and if I if the price is zero the market will demand six thousand pounds per day if it's free but I'm not gonna generate any revenue because I'm going to be giving it away for free so I will not be generating any revenue in this situation so our total revenue curve it looks like and if you've taken algebra you would recognize this as a as a as a downward facing parabola our total revenue our total revenue looks like this our total revenue it's easier for me to draw a curve with that dotted line our total revenue looks something like that and you can even solve it algebraically to show that it is this downward facing parabola the formula right over here of the demand curve its y-intercept is six so if I wanted to write price as a function of quantity we have price is equal to it's equal to 6 minus quantity or if you wanted to write it in the traditional slope-intercept form or MX plus B form and if that doesn't make any sense you might want to review some of our algebra playlists you could write it as is equal to negative Q plus 6 obviously these are the same exact thing you have a y-intercept of 6 you have a y-intercept of 6 and you have a negative 1 slope if you if you increase quantity by 1 you decrease price by 1 or another way to think about it if you decrease price by 1 you increase quantity by 1 so that's why you have a negative 1 slope so this is price as a function of quantity what is total revenue well total revenue total revenue is equal to price times quantity but we can write price as a function of quantity we did it just now this is what it is so we can rewrite it or we could even rewrite it like this we can rewrite the price part as so this is going to be equal to negative Q plus 6 times quantity times quantity and this is equal to total total revenue and then if you multiply this out you get total revenue is equal to Q times Q is negative Q squared plus 6 plus 6 Q so you might recognize this this is a this is clearly a quadratic since you have a negative out front before the before the second-degree term right over here before the Q squared it is a downward-opening parabola so it makes complete sense now I'm going to leave you there in this video because I'm trying to make an effort not to make my videos too long but in the next video what we're going to think about is what is the marginal revenue we get at each at for each of these quantities and just as a review just as a review marginal revenue marginal revenue is equal to change in total revenue change in total revenue divided by change in quantity or another way to think about it the marginal revenue at any one of these quantities is the slope is the slope of the line tangent to that point and you really have to do a little bit of calculus in order to actually calculate slopes of tangent lines but we'll approximate it with a little bit of algebra but we're essentially want to do is figure out the slope so if we wanted to figure out the marginal revenue when we're selling 1,000 pounds so exactly how much more total revenue do we get if we just barely increase if we just start selling another millions of pounds of oranges what's going to happen and so what we do is we're essentially trying to figure out the slope of the tangent line at any of at any at any point and you can see that because the change the change in total revenue is this change in total revenue is that and change in quantity and change in quantity is that there so we're trying to find the instantaneous slope at that point or you could think of it as the slope of the tangent line and we'll continue doing that in the next video