# Elas, elas, that great city!

Let's say Starbucks is having a bad quarter, and wants to make more money. They decide that to do that, they will simply charge ten times as much per drink - for example, instead of charging about \$2 for a simple cup of coffee, they charge about \$20. Should you expect them to have ten times as much revenue?

No - people will buy less. They make buying decisions based, in part, on price. In economics this is measured as "the price elasticity of demand".

People will buy less if you raise prices for a couple of reasons:

The Substitution Effect

People like more than one thing. I like wine, and I like lemonade. Sometimes I strongly prefer wine, and sometimes I strongly prefer lemonade. But if wine cost \$1,000 a bottle, then I'd sometimes prefer buying lemonade to buying wine; wine's not that much better, even with a nice dinner that goes better with wine. This is called the substitution effect.

The Income Effect

Let's take an extreme case and consider someone who is an absolute Starbucks fanatic. They drink coffee every day, and would prefer spending \$20 on a Starbucks coffee to spending \$2 on some other coffee and \$18 on anything else. But even fanatically loyal customers aren't made of money. If someone's discretionary income after the necessities of life (pretending for a moment that coffee is not in this category) is \$500 a month, and coffee costs \$2, they can easily afford their Starbucks-a-day habit. If Starbucks coffee costs \$20, then they can only buy 25 coffees a month, not enough to sustain their daily consumption. So even with perfectly loyal customers, after a point Starbucks isn't making any extra revenue, though they may make the same money on fewer coffees. This is called the income effect.

How Big Is This Effect?

Economics uses the term "elasticity" to describe how responsive one quantity is to changes in another quantity. So for example, the price elasticity of demand describes how much the amount people buy changes, for a given change in price.

Since there's no intrinsic meaning to the units you use to measure something - 100 centimeters is the same as one meter is the same as three feet (approximately) is the same as one yard - elasticity is measured in relative quantities. A good approximation of this is percentages. If you raise the price by 10%, by what percentage does demand go down?

Interpreting Elasticity

If elasticity is less than 1, that means that changes in demand are less than enough to offset changes in price - so if you raise prices, revenues go up. For example, last month a Starbucks branch was selling coffee for \$2, and selling 1,000 coffees a day, for \$2,000 in revenue. This month it raises prices by 10%, to \$2.20 per coffee. If the price elasticity of demand is -0.5, that means that each 10% increase in price causes about a 5% decrease in the number of coffees bought, so people only buy 950 coffees, and this Starbucks gets \$2,090 in revenue.  This is good news if you are trying to make more money by raising prices.

On the other hand, if elasticity is greater than 1, that means that you can't generate more revenue by raising prices - people will buy more than enough to offset the revenue loss. In the example above, let's say that the elasticity of demand is -1.5. That means that a 10% increase in price causes about a 15% decrease in the number of coffees bought, so people only bought 850 coffees, for total revenue of \$1,870.

(Note: Raising prices can be profitable even with elasticity greater than 1, in cases where your per-unit costs are very high relative to the price. For example, if Starbucks were selling coffee for 1 cent, which is probably less than the cost of the materials, then it would benefit from raising prices at least until it was charging more than the cost of making an extra cup of coffee.)

Interpreting Elasticity

It turns out that it's easy to find websites telling people how to calculate elasticity, which is mainly useful for economics students doing their homework. But it's hard to find websites telling people how to use elasticity to make predictions. So I'll do that now. But to do that I need to define elasticity in a more mathematically precise way.

Let's say that you believe some good has a constant price elasticity of demand, so that the same percentage change in price always produces the same percentage change in quantity. Here, the "percentage" language is a little tricky. Let's look at the coffee example again, and say that economists have measured consumer behavior and determined that price elasticity is -0.5. If you hear that price has gone up by 10% to \$2.20, you might think that demand would fall by 5%, to 950 coffees a day. And if price only went up by 1%, to \$2.02, you might think that demand would fall by 0.5% to 995 coffees per day. So far, it's clear.

But let's say price goes up by 1%, ten times. How much does demand fall? One way to calculate this is to say that price increases to 101%^10=1.1046, about 10.46% higher than it was before, for a final price of \$2.21, so demand should decrease by 5.23% to about 948 coffees. Another way to calculate it is to say that demand falls about 0.5% each time, so the final demand is 99.5%^10=95.11% of the original quantity, about a 4.89% reduction to a total quantity of 951 coffees. Why are these numbers different?

These numbers are slight different because we're just using a linear approximation, which is good for small numbers, but gets worse when we use bigger numbers.

Once More, With Calculus

We can write the percentage definition of elasticity like this:

E=((Q2-Q1)/Q1)/((P2-P1)/P1). If we call the difference in prices DP and the difference in quantities DQ, we can say, E=(DQ/Q)/(DP/P).

As DP gets bigger, E gets harder to measure - depending on what differences you take, in what directions, you get answers that diverge more and more, like in the example above. But as DP gets closer to 0, E gets easier to measure, becomes more consistent, and eventually converges on a single value. Here's how we'd express that in calculus:

E=(dQ/Q)/(dP/P)

We can rewrite this as:

E*P^-1dP=Q^-1dQ

And then integrate both sides to get:

E*ln(P)=C+ln(Q), where C is an additive constant. The next trick is to take the exponential function of both sides, to get

exp[E*ln(P)]=exp[C+ln(Q)]

(exp[ln(P)])^E=exp[C]*exp[ln(Q)]

P^E=K*Q, where K is just a scaling constant. So if you want to calculate the ratio of quantities produced by a change in prices, you can take the equation for the first and second price-quantity pair:

P1^E=K*Q1

P2^E=K*Q2

And divide the second equation by the first to get:

P2^E/P1^E=K/K*Q2/Q1

(P2/P1)^E=Q2/Q1

quantity_ratio=price_ratio^E

So, for example, with elasticity of -0.5, if you double price, you get about 2^-0.5=.707 times the demand, a 29.3% reduction. This works on any scale you want, even for very big changes in price.

Provided, of course, that your elasticity figure is correct, and that your assumption that elasticity is constant is also correct. Which it may not be.