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

53문장 0% 태국어 번역 0명 참여 출처 : 칸아카데미
0 영상보기
영어 to
태국어
2단 보기 1단 보기
번역 0%

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

It is twice as steep as this.발음듣기

다른 번역문 등록

It's really twice the negative slope.발음듣기

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

Now what is going to be our total revenue?발음듣기

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

Well, total revenue is just price x quantity.발음듣기

다른 번역문 등록

Total revenue is just price x quantity.발음듣기

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

So we get D, TR/DQ.발음듣기

다른 번역문 등록

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

다른 번역문 등록

So it's going to be 2m ... 2mQ.발음듣기

다른 번역문 등록

And then b is a constant.발음듣기

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

It's just going to be b.발음듣기

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

The slope of our demand curve is m.발음듣기

다른 번역문 등록

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

다른 번역문 등록

So it will look something like this.발음듣기

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

Anyway, if you understood that, great.발음듣기

다른 번역문 등록

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

다른 번역문 등록

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

다른 번역문 등록

You can proceed with the economics playlist.발음듣기

다른 번역문 등록

칸아카데미 더보기더 보기

전체보기
Top