Pricing and Downloaded Music

I was reading Slashdot the other day and inevitably there was an article regarding the RIAA. I remember some commenters saying how prices of music have to rise in order to make up lost money to the record companies and artists.

This is completely preposterous. The people making statements like this are either trying to start a flame war, or have no understanding of basic economics. I have two points to make here.

First, on demand. Suppose there are x albums out there that have had songs downloaded. According to these commenters (and also it seems, the record companies say the same), that means that there is a total of px dollars lost. This is completely false. Let's look at x, which is a number of albums. If these people had acquired the albums by paying for them, there would not have been x albums bought, since by the law of demand there would be someone, somewhere who wouldn't have thought the album was worth p. Some twelve year old whose parents refuse to buy them rap music, or somebody who did legitimately buy a bunch of CDs but decided to download some other ones they wanted.
On top of this, we can take into account the elasticity of demand. Albums are not necessary to survival. In fact, they can even be considered a luxury good. Therefore their price is elastic, which means that a change in price results in a much higher change in quantity demanded¹, so x albums acquired for free is actually much higher than the amount of albums that would be demanded at price p. In short, a lot of people who downloaded the music wouldn't have bought it anyway, so there isn't as much revenue lost as people say there is.

Now for the second point. These people are saying that record companies have to increase prices in order to make up for lost profits. Let's see why this is silly.
Ignore for the moment the people who downloaded the music, and think only of the legitimate buyers. Suppose you are a legitimate buyer. If the record company is raising the price of CDs, would you want to buy as many CDs? Probably not. You have other things to spend your money on. So therefore, the quantity demanded (and consequently, sold) q falls with an increase in price p.
Looking at this mathematically, revenue r is equal to the price of the good, times the number of the good sold, so pq. However, q is a function of p, so we can write q(p) = b - ap which is the general form of a linear demand curve (not all demand curves are linear, but all demand curves do slope downward and the result is the same regardless of whether it is linear or not). Now:

r = pq = p(b - ap) = pb - ap2

This resulting revenue function is a parabola opening downward, and it doesn't take a genius to know that there is a maximum value for revenue here. So increases in price actually cause revenue to fall!
Then take profits, which are equal to revenue minus cost. However, the cost for producing an additional CD is negligible compared to the price, so we can ignore it in terms of the function². So profits are essentially a reflection of revenue, and therefore we can say that the company, before we take into account the music downloading, will already be at the optimal point for gathering revenue. If they raise their prices, this will cause profits to fall due to the resulting loss of demand for higher priced items. We'll call this point the optimal production point.

Why do the record companies then sue people for downloading music? Why would they care? The reason being that file sharing online has negatively affected demand. What this does is reduces both the price at the optimal production point, and the quantity demanded. So both p and q drop, and revenue and profit drop accordingly. People no longer demand the good in the traditional sense, as they can just get it online for free. Suing people is an attempt to counteract this drop in demand to bring it back to what it was before.

In my opinion, the media companies will never stop music piracy. They should just bite the bullet and stick with their new optimal pricing point that the free market has given them.


¹ For those interested, the formula for the elasticity of a function is f' * x / f where f' is the derivative of f with respect to x, and both are functions of x. It's basically a ratio of percentage changes.
² A wise reader would note that there is a large fixed cost involved with making a CD, but in this case the company would want to make a large enough revenue to offset these costs. This doesn't affect the function, but rather limits the domain of p to be between two values. Also, we can assume that marginal cost for a CD (aka the cost to make a new CD) is more or less constant.