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

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

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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,발음듣기

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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 a line.발음듣기

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So if the demand curve looks like that, this is price.발음듣기

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This is quantity. That is demand right over there.발음듣기

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I'm going to show that the marginal revenue curve has twice the slope.발음듣기

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It is twice as steep as this.발음듣기

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It's really twice the negative slope.발음듣기

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So let me just write price as a function of quantity.발음듣기

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We know we get price. Since this is a line, we can essentially write it in our traditional slope y-intercept form.발음듣기

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In algebra class, you would write if this was y and this was x, you would write y = mx+b where m is the slope and b is the y-intercept.발음듣기

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I'll do something very similar, but instead of y and x we have P and Q.발음듣기

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So, if P = m x Q where m is the slope plus ... plus the P-intercept + b.발음듣기

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So this right over here is b and if you were to ...발음듣기

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If you were to take your, If you were to 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.발음듣기

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That is your slope, change in P / change in Q.발음듣기

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Now what is going to be our total revenue?발음듣기

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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.발음듣기

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So if this is total revenue, total revenue as a function of quantity.발음듣기

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Well, total revenue is just price x quantity.발음듣기

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Total revenue is just price x quantity.발음듣기

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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.발음듣기

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So we get total revenue is equal to and I'll write it all in blue.발음듣기

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We have mQ + b and then we're going to multiply that x Q.발음듣기

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We're going to multiply that x Q or we get total revenue = mQ^2. mQ^2 + b x Q And this is a parabola and it's actually going to be a downward sloping parabola because m is going to be negative.발음듣기

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This is downward sloping. m has a negative slope, so m is negative.발음듣기

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So we know ... We know that M < 0 over here.발음듣기

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That's one of the assumptions we'll make.발음듣기

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If m < 0, this is going to be a downward opening parabola.발음듣기

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Total revenue will look something, total revenue will look something like that.발음듣기

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That is our total revenue. Now, the marginal revenue as a function of quantity is just the derivative and this is the calculus part.발음듣기

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It's the slope of the tangent line at any given point and that is what the derivative is.발음듣기

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It's the slope of the tangent line at any point as a function as a function of quantity.발음듣기

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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.발음듣기

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So we essentially just have to take the derivative of this with respect to Q.발음듣기

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So we get D, TR/DQ.발음듣기

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So how much does total revenue change with a very, very small change in quantity, infinitely small, infinitesimal change in quantity and this comes straight out of calculus. m is a constant. Q^2, the derivative of Q^2 with respect to Q is 2Q. So it's going to be 2Q x the constant.발음듣기

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So it's going to be 2m ... 2mQ.발음듣기

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And then b is a constant.발음듣기

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We're assuming it's given. b is a constant.발음듣기

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The derivative of bQ with respect to Q is just going to be b.발음듣기

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It's just going to be b.발음듣기

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And so right over here, this is our marginal revenue curve. or I should say our marginal line. It is 2mQ + b.발음듣기

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So notice, it has the same y-intercept as our demand curve so definitely starts right over there, but it has twice the slope.발음듣기

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The slope of our demand curve is m.발음듣기

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The slope of our marginal revenue curve is 2m, is 2m and this is a negative slope, so this will be twice as negative.발음듣기

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So it will look something like this.발음듣기

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It will look something like this, just like that.발음듣기

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So no matter what your demand curve is, 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, it's twice the negative slope which is kind of ... what's going to be generally true.발음듣기

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Anyway, if you understood that, great.발음듣기

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You now feel good that this is always the case for a linear demand curve like this.발음듣기

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If you did not understand it, don't worry.발음듣기

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You can proceed with the economics playlist.발음듣기

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