A Quick Lesson on Demand Elasticity and Revenue!

(All graphs shown below were drawn by me, using GIMP's grid and straight line tools. Yay, me.)

In a conversation I was having elsewhere, some issues relating to demand elasticity and revenue came up, and it occurred to me that it would likely be much easier to actually show the relevant graphs than to just keep trying to explain things with plain text. Hence, this post. Now, let's start by pointing out a few things. Some of these will probably be obvious, but it's good to make everything clear, to begin with, so bear with me.

1. When I say "demand," I really mean "effective demand" - that is, not how much of an item we want but how much of an item we actually can and will pay for. Obviously, if we were just talking about desire without follow through, none of this would matter at all, in real life.

2. "Quantity of demand" simply means the number of units of an item demanded. For instance, if, at a certain price, two hundred cartons of cigarettes are bought, the quantity of cigarettes demanded is 200 cartons.

3. When price goes up, quantity of demand goes down, in most cases. There are some exceptions - one of which we'll briefly show below - but this generally holds true.

Now, when we want to put this effect down on paper, we can do it with a graph. The graph below, for instance, has two axes, the vertical axis showing the price of some good and the horizontal axis showing the quantity of the good demanded. The black line sloping downwards between the two axes is the "demand curve," and each point on the curve corresponds to a certain combination of price and quantity demanded.

For instance, if we look at point A on the curve, we see that, when the price of the good is $20, 10 units are demanded. When price falls to $10 at points B, the quantity demanded rises to 20 units. (These numbers are merely given for the sake of illustration. We'll get into exactly how much a change in price changes quantity demanded, below.)

Now, let's consider the slope of the demand curve. The slope of a line, as we all know from math class, is equal to the change in the value graphed along the y-axis divided by the change in the value graphed along the x-axis between two points on the line. So, for the demand curve, the slope of the line is the change in price divided by the change in quantity demanded, between two points on the curve. We call the slope, in this case, the "elasticity of demand." The elasticity of demand tells us, roughly, how much the quantity demanded will change if the price increases or decreases.

(The definition of elasticity given above is actually somewhat simplified. In reality, since the numerical values of price and quantity of demand will almost always be presented on different scales, elasticity is really the percent change in the total price divided by the percent change in the quantity of demand. However, for ease of reference, I've used identical scales for both price and quantity of demand for all the relevant examples, below, so this distinction doesn't matter, for our purposes. Keep in mind, however, if you encounter graphs with differing scales.)

To illustrate, let's look at a special, ideal case, which we call "perfectly inelastic demand." For a perfectly inelastic demand curve, the slope of the line is nonexistent. That means that, in practice, the quantity of demand is exactly the same, no matter what price the product sells for. In the graph below, for instance, consumers demand ten units of the good at both point A and point B, even though the price at A is $10 higher.

This is also a great chance for us to talk about total revenue. Now, obviously, how much money someone makes from selling a product is equal to the number of units he sells times the price that he charges. For instance, at point A, above, we're selling 10 units for $20 each, so we're making $200 in total revenue. At point B, we're only making $100 in revenue, because price is lower.

This is reflected in the size of the rectangles formed from the red lines drawn between the points and the axes. In fact, the area of the rectangle colored in orange is equal to the revenue we'd get at point B, while the area of the orange rectangle plus the area of the green rectangle equals the revenue we'd get at point A. The rectangle for A is larger than the rectangle for B because we'd make more revenue at A than at B.

In most cases, though, we're not dealing with perfect inelasticity. Instead, we have either "relatively inelastic" or "relatively elastic" demand. "Relatively elastic demand," shown on the graph immediately below, means that the change in quantity of demand between two points is greater than the change in price - or, in mathematical terms, that the absolute value of the slope of the line is less than 1.

Let's consider the revenue we can make off of a relatively elastic product, at different prices. As you can see just by eyeballing the rectangles sketched in red on the graph above, we make more revenue at point B, when price is lower, than at point A, when the price is higher. (If you don't trust your eyes, here, quick multiplication will confirm this.) So, it's in our best interest, if we're the seller, for the price to be lower - we make more money, that way.

"Relatively inelastic" demand is just the opposite - it's when price changes more than quantity of demand, between two points on the curve. In mathematical terms, that means that the absolute value of the slope of the line is greater than 1. The graph below displays a relatively inelastic demand curve.

Finally, let's consider how revenue works, when we have relatively inelastic demand. It's easy to see that the rectangle for point A is much larger than the rectangle for point B - hence, we make more revenue when price is higher than when price is lower, and it's in our best interest to increase price rather than to decrease it.

A FINAL CAVEAT: the demand curves we've presented above are somewhat simplified. In most cases, demand curves will not be lines but actual curves. This means that there will be a part of the curve where demand is relatively inelastic, part of the curve where demand is neither relatively inelastic nor relatively elastic, and part of the curse where demand is relatively elastic. The real difference between elasticity of demand curves is how much of the curve falls into a certain type of elasticity. Hence, when we say a good has relatively inelastic demand, we usually mean, in reality, that the price range for which the good has relatively inelastic demand is larger than the price range for which the good has relatively elastic demand.

But, in general, if you're just talking in simple terms, it's fine to use the line graphs shown above, in order to avoid a headache.

On Hating America

I think what most astonishes me about the over-heated rhetoric employed against Obama, of late, is the extent to which it ignores the masses of people who support him.

By any objective standard, one must acknowledge, Obama is ahead in the race for the presidency, at the moment. All the national tracking polls have him comfortably ahead, and the various, electoral projections have him either exceedingly close to or past the magic number of electoral votes, while Senator McCain languishes behind. Now, unless one assumes that all of these polls, which, in a broad sense, agree with one another, are all wrong, the obvious conclusion is that more people presently support Senator Obama than support Senator McCain.

Now, from the right, we're hearing a lot of talk about Obama's supposedly radical connections. We hear that he's super-liberal, that he's a Muslim, that he's a socialist, etc. Most vocally, though, we've heard the claim that Senator Obama is a terrorist or a terrorist sympathizer or a friend to terrorists. Moreover, the assertion from the right is that these claims are obviously true. No one is saying that if we dug in deep into Obama's past that we might find out that he has terrorist ties - they're saying that we already know for a fact that he does.

No one seems willing to carry these assertions to their logical conclusion, however, even though that conclusion, if true, would be quite shocking and serious. If you believe that it is public knowledge that Obama is a friend to terrorists and you acknowledge that the polls show he is widely ahead in the race, you must believe that all of his supporters either are also friends of terrorists or do not care if the person they vote for is a friend of terrorists. There are no other options. If we Obama supporters could easily learn what the extreme McCain supporters supposedly know, we would have to be either violently anti-American or completely disconnected not to do so and then promptly turn against him.

Why does no one say this explicitly, on the right? Well, because it's obviously untrue. It's absurd to imagine that the majority of people plugged into the American presidential race hate America and support terrorism or just don't care. There is no rational underpinning for such a belief, and it would be laughed out of the halls of public discourse if it ever grew brave enough to give voice to itself. After all, this argument essentially amounts to accusing the majority of likely, American voters of hating America.

There is, of course, one alternative path these extreme McCain-types can take. They can claim that the information about Obama's terrorist sympathies have been hidden and distorted by the media. Under that claim, we honestly don't know whether or not Obama <3's terrorists, because the facts have been kept away from us. The logical problem here is even more obvious - if we don't know the facts, it's impossible to conclude that Obama is a terrorist sympathizer, just as it is impossible to conclude that he is not one. (The complete lack of evidence of any such cover-up is another problem, but I'm really only aiming to point out the internal flaws in these arguments.)

In short, those extreme conservatives accusing Senator Obama of being anti-American are not only being intellectually dishonest, but they are also expressing either an incredibly deep cynicism or (more likely, I think) complete contempt for their fellow citizens.