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Optional calculus proof to show that MR has twice slope of demand

Video transcript

for those of you who are curious and have a little bit of a background in calculus I thought I would do a very optional and when I say it's optional you don't have to understand this in order to progress with the economics playlist but a very optional proof showing you that in general the slope of the marginal revenue curve for a monopolist is twice the slope of the demand curve assuming that the demand curve is aligned so the demand curve looks like that this is price this is quantity that is demand right over there I'm going to show that the marginal revenue curve has twice the slope it is twice as steep as this and it's really twice the negative slope so let me just write price as a function of quantity we know we get price since this is a line we can essentially write it in our traditional slope y-intercept form and in algebra class you would write if this was Y and this was X you would write y is equal to MX plus b where m is the slope and B is the y-intercept I'll do something very similar but instead of Y and X we have P and Q so if P is equal to M times Q where m is the slope plus plus the p-intercept plus B so this right over here is B and if you were to if you were to take your if you were to take take your change in P if you were to take your change in P and divide it by your change in Q your change in Q you would get M that is your slope change in P divided by change in Q now what is going to be our total revenue and this is kind of we're just kind of almost doing what we've done in the last few videos but we're doing it in general terms so this is total revenue total revenue as a function of quantity well total revenue is just price times quantity total revenue is just price times quantity we've already written we've already written price as a function of quantity right over here so we could take that and substitute it and substitute it in right over there so we get total revenue is equal to and I'll write it all in blue we have M Q plus B and then we're going to multiply that times Q we're going to multiply that times Q or we get total revenue is equal to M Q squared M Q squared plus B times Q and this is a parabola and it's actually going to be a downward sloping parabola because M is going to be negative this is downward sloping M is it has a negative slope so M is negative so we know we know that M is less than zero over here that's one of the assumptions we'll make if M is less than zero this is going to be a downward-opening parabola total revenue will look something totally revolutionize just the derivative and this is the calculus part it's the slope of the tangent line at any given point and that is what the derivative is it's the slope of the tangent line at any point as a function as a function of quantity so you give me a quantity I will tell you what the slope of the tangent line of the total revenue function is at that point so we essentially just have to take the derivative of this with respect to Q so we get D total revenue over DQ so how much does total revenue change with a very very small change in quantity and infinitely small infinitesimal change in quantity and this comes straight out of calculus M is a constant Q squared the derivative of Q squared with respect to Q is 2q so it's going to be 2 Q times the constant so it's going to be 2 m 2m Q and then B is a constant width we're assuming it's given B is a constant the derivative of BQ with respect to Q is just going to be B it's just going to be B and so right over here this is our marginal revenue curve or I used to say our marginal revenue line it is 2m Q plus B so notice it has the same y-intercept as our demand curve so it definitely starts right over there but it has twice the slope the slope of our demand curve is M the slope of our marginal revenue curve is 2 M is 2 m and this is a negative slope so this will be twice as negative so it will look something like this we'll look something like this just like that so no matter what your demand curve is if it's a few sumit sad out if you assume it's a line like this the marginal revenue curve will be a line with twice the slope and in this case is twice the negative slope which is kind of what's going to be generally true anyway if you understood that great you now feel good that this is always the case for a linear demand curve like this if you did not understand it don't worry you can proceed with the economics playlist you don't need that know calculus for for this for this playlist