Monopolist optimizing price: Marginal revenue발음듣기
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Monopolist optimizing price: Marginal revenue
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Now that we figured out the total revenue 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.발음듣기
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To think about marginal revenue, marginal revenue is just how much does our total revenue change, given some change in our quantity.발음듣기
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It actually would 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 will even give you a little intuition for what we end up doing eventually in calculus.발음듣기
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The best way to find the slope right over here is say how much does my total revenue change if I have a very small change in quantity?발음듣기
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If I increase my quantity very, very, very, very little, so let's just make it 0.001, what is going to be my total revenue?발음듣기
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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, I'll get my calculator out, if our quantity is .001, our total revenue is going to be negative ...발음듣기
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Total revenue is going to be -.001², squared, so that's that part, plus 6 times .001, 6 times .001.발음듣기
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Now we can figure out or get a pretty good approximation for that marginal revenue right at that point.발음듣기
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That's our change in quantity, and our change in revenue is 0.00599, and so we just have to divide.발음듣기
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We just have to divide .005999, that top one, our change in total revenue divided by our change in quantity, divided by .001.발음듣기
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If you try it with even smaller numbers, if you tried this with .00000001, you'll get 5-point, and you'll get even more 9s going on.발음듣기
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The closer that you get, the smaller your change in, and this is what you essentially do in calculus.발음듣기
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What I want to do is I'm going to plot marginal revenue here on our demand curve as well or on this axis where we've already plotted our demand curve.발음듣기
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When our quantity is 0, our marginal revenue, if we just barely increase quantity, the incremental total revenue we get is going to be 6.발음듣기
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If we were to just sell a drop of orange juice or I guess we're 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 $6 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.발음듣기
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If I want to find the slope right over here, when our quantity is equal to 1, the slope would look like, the slope would look like that.발음듣기
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Between those two points, our change in quantity is 2, and our change in total revenue is 8.발음듣기
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So we have a change in total revenue of 8, or 8,000, I guess we could say, divided by a change in quantity of 2,000, so our marginal revenue at this point is 8 divided by 2, or 8,000 divided by 2,000, which is $4 per pound.발음듣기
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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.발음듣기
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It's actually almost an exact number because of the way that this is just a parabola, so we can actually do this.발음듣기
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Once again, our change in quantity is 2, and our change in total revenue, our change in total revenue is, we're going from 5 to 9, which is 4.발음듣기
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Right at that point, for that incremental millionth of an ounce that we're going to sell them oranges, we're getting the equivalent of $2 a pound of increased total revenue from doing that.발음듣기
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Let's just do one more point here, and I think you'll see why I'm only going to do one more point.발음듣기
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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 just barely increase our quantity, and this is actually easier to look at.발음듣기
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We have some change in quantity, but we have no change in total revenue, so right at that point.발음듣기
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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.발음듣기
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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 or downward-sloping line, and it will have twice the slope.발음듣기
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For every increase in quantity, the price goes down by 2; increase in quantity, price goes down by 2; increase in quantity, price goes down by 2.발음듣기
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Let's remind our self, we've been doing all of this algebra and all of this math here, what is marginal revenue telling us?발음듣기
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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.발음듣기
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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 increase quantity a little bit more, how much is our revenue increasing?발음듣기
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Obviously, we want to keep increasing quantity while our revenue is ... while the marginal revenue we get is larger than our marginal cost.발음듣기
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