Economics: Graphing Marginal Revenue & Elasticity

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In this lesson, we're going to take a behind-the-scenes look at marginal revenue, and one of the things that we'll discover is the relationship between marginal revenue and a familiar concept, the elasticity of demand.
Let's start with our downward-sloping demand curve. Any firm that has market power is dealing with a downward-sloping demand curve. That is, they can either sell a larger quantity at a lower price or a smaller quantity at a higher price. A downward-sloping demand curve is what it means to have market power. If you as an individual firm, if you as a restaurant operator at the airport, have to be concerned about the quantity that you sell or you might depress your price, you've got market power and you're dealing with a downward-sloping demand curve. That means you have a special problem. That is, do you want higher prices and fewer customers or lower prices and more customers?
Let's look at how you sort that decision out. Let's start with a particular price quantity combination on our demand curve. Let's suppose that the price is initially a high price - c[0], maybe $10 per meal - and the quantity is initially a small quantity - we'll call this quantity q[0] - maybe it's twenty meals a day served. If the price is p[0] and the quantity is q[0], what will your total revenue be? Can you identify total revenue in this picture? The answer is: Total revenue is always price times quantity - it is the area of this rectangle.
What we're concerned with is how total revenue changes when the price and quantity change along the demand curve. Suppose you decide that you want to serve more customers at a lower price. If so, you may drop your price down to p[1], and the quantity that you are going to be selling will then increase accordingly. So let me put the new quantity in here. It will be q[1] at a price of p[1]. So the total revenue that you will be earning with the new lower price and larger quantity is the area of this rectangle here - below p[1] and up to q[1]. The change in price results in a change in quantity. A lower price means more customers served, and the total revenue is different in the two cases.
Let's look at how total revenue has changed, first geometrically and then intuitively. You'll notice that our two total revenue areas have in common this box in the corner, but our original situation involved this additional rectangle on top, which we lose when we drop the price. The new total revenue box adds this revenue in addition. We're going to call this little rectangle right here Area One. Area One is added when we lower the price and the quantity increases. The intuitive explanation of Area One is that when the price is lower, the firm is going to add new sales or new customers. Area One represents the additional revenue that comes when the firm lowers its price and adds additional sales. Area One is an increase in total revenue that results from a lower price and larger quantity. So let me write over here on the side the formula for Area One. So Area One, which I'll write as a rectangle one, this Area One is going to be equal to the new quantity minus the old quantity, or the change in quantity, multiplied by the new price. So Area One = P[1][ ]x (q[1] - q[0]).
The intuitive explanation for Area One is the additional revenue that comes from new sales at this new lower price. However, in order to get those additional customers, you had to reduce the price, and that resulted in a loss of revenue. This little box up here was lost when the firm went from p[0] to p[1]. Let's call this Area Two. Rectangle Two, or Area Two, then, is the lost revenue that is associated with lowering the price. Here's the intuition: See, you could have sold this many restaurant meals - say ten meals a day - you could have sold them at the higher price; but when you lowered the price, you gave up revenue that you would have earned on those meals anyway. You're now selling the same meals at a lower price, and that's a reduction in your revenue. Even though you added all of these new sales, your old sales now are less lucrative for you because you've cut the price from say $10 per meal down to $8 per meal. You gave up $2 on each of those meals that you would have sold anyway at the higher price. Area Two is the reduction in total revenue associated with reducing the price at which you're selling units that you would have sold anyway. So this area is going to be Area p[0] minus p[1] multiplied by the quantity - q[0] - that you would have sold at the higher price - the lost margin - the lost price increment - on all of these meals you would have sold anyway.
The marginal revenue that you earn whenever you drop your price from p[0] to p[1] and your quantity increases from q[0] to q[1], the marginal revenue is going to be the combination of Area One and Area Two. When the price falls, you gain Area One but you lose Area Two. Subtract Area Two from Area One, and you have the firm's marginal revenue.
Well, this raises a question, and that is is marginal revenue positive or negative? When you lower your price and increase your sales, are you adding to your total revenue or are you subtracting from it? The answer depends on the relative size of Area One and Area Two. If Area One is bigger, then your marginal revenue is greater than zero. If Area Two was bigger, then marginal revenue is negative. Look, what's going on here? When you lower your price, you're always going to be losing money on the units you could have sold at the higher price. The question is are you adding enough sales to compensate for that lost revenue? Are you adding enough sales so that Area One - the increment that comes from increasing your customer base - are you adding enough sales so that Area One, the extra revenue from adding more units sold, offsets the loss from giving up money that you would have earned on these units that you would have sold anyway?
Well, what are we talking about here? We're talking about how many extra sales you make when you lower your price a little bit. Does this concept sound familiar? We have a name for it. Do you remember what it is? The answer is we're talking about the elasticity of demand. Elasticity of demand refers to the responsiveness of quantity demanded to a change in the price of a product. If your demand is very elastic, a small change in price results in a big increase in your sales, and Area One is greater than Area Two. If, on the other hand, demand is inelastic, then the given change in price will add only a small increase in quantity demanded, and Area One will be smaller than Area Two. This is what happens when demand is inelastic. So if Area One is bigger than Area Two, then demand is elastic and a decrease in price results in an increase in total revenue. When demand is elastic, marginal revenue is positive. However, if demand is inelastic, Area One is smaller than Area Two, and in that case marginal revenue is negative. Selling extra units shrinks your total revenue.
What I'll do now is show you the relationship between Areas One and Two and the concept of elasticity. It will involve a few lines of math, but I think the payoff makes it worth it. Here I've reproduced the formulas for Area One and Area Two. Area One is the price times the change in quantity. Area Two is the change in price times the original quantity. What I want to do is convert this into a familiar measure, elasticity of demand, but in order to do that, I'm going to have to use a couple of math tricks. The first one is I want to talk about small changes in price and quantity. In the picture that we drew earlier, we had a large change in price and quantity just to make the diagram easier to read. But if we have a small change in price and quantity, we could write the difference between q[1] and q[0] as - the change in quantity. This difference - p[0] minus p[1] - we can write that as . The interesting thing about a small change is that the subscripts are no longer that important. We don't have a big difference like $10 for Price Zero and $8 for Price One. Instead, we have very small differences - $10 versus $9.99. So when the prices are that close, we don't have to be meticulous about keeping the subscripts, we can let them go because p[0]and p[1] are pretty close.
The next thing to notice is that whenever we write our deltas, we want to be careful that we have the new price minus the old one, or the new quantity minus the old one. Here I have the old price minus the new price, so when I write this as a change in price, I have to put a negative sign out in front of it to indicate that it's going the opposite direction from the change in quantity. Keeping these two things in mind, if you're careful in measuring the direction of your change and that the change is very, very small, we can write the ratio of Area One to Area Two in an interesting way. Let me go ahead and write the ratio of Area One to Area Two will be equal to the price times the change in quantity - that's Area One. Area Two, then, will be negative change in price multiplied by the quantity. In order to write the fraction in this way, I'm using my two assumptions - a small change so that Price Zero and Price One are very close and Quantity Zero and Quantity One are very close - and I'm using my assumption that delta represents a change with the new minus the old. So a negative here means we were subtracting the old from the new. By the way, we had to do that here for Area Two in order to get a positive area. If we subtracted p[1] minus p[0], we'd get a negative and it wouldn't be measuring the area, which is always positive.
Now let's see if we can simplify this expression. The first thing that I will do is I'll break the fraction into two parts and put the price terms in their own fraction multiplied by the quantity terms. So no big change yet, I just broke the fraction kind of into two separate parts. The next thing that I'm going to do is I'm going to use this price term, and instead of taking it this way, I'm going to write it as its own reciprocal. I'm going to write it as of this fraction. So what I get next is . So all I did between this step and this step was flip this fraction over and put it in the denominator of another fraction. I took the reciprocal of it. Well, now what I have is this: I have - that is, the percentage change in quantity demanded divided by the percentage change in price. One more thing - I've still got a negative sign that I have to put out in front.
Well, what have I got? What I've got is the percentage change in quantity divided by the percentage in price; and since these two usually move in opposite directions, with a negative sign out in front, I've got the absolute value of that fraction. Ah, hah! That's familiar. We now have a familiar measure of price responsiveness. This term is exactly the elasticity of demand. So if Area One is greater than Area Two, then the value of this fraction is greater than one. That is, the quantity changes a lot in response to a given change in price, and that's what we call elastic demand. If Area One is smaller than Area Two, then this fraction is less than one, and that's what we call inelastic demand. If Area One is equal to Area Two, this fraction is equal to one and we have unit elastic demand.
So, if a firm lowers the price if its product and it sells a lot of the goods, if Area One is greater than Area Two, that's because customers are relatively responsive to a change in price. Demand is elastic, and the marginal revenue is positive. That is, a reduction in price and an increase in quantity increase total revenue. If, on the other hand, a firm lowers its price and customers don't respond, Area One is small relative to Area Two. This elasticity measure is less than one, the demand is inelastic and marginal revenue is negative. Anytime demand is inelastic, marginal revenue is negative. Anytime demand is elastic, marginal revenue is positive.
This goes back to a lesson that we discussed earlier, and that is that an increase in quantity sold in a particular market will result in an increase in total revenue if the demand for the product is elastic. On the other hand, if the demand is inelastic, then a restriction of quantity and a raising of the price will increase the total revenue. I've gone to the trouble of all of this math just to make very clear to you that marginal revenue is intimately related to the concept of elasticity of demand.
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Graphing the Relationship between Marginal Revenue and Elasticity Page [1 of 3]