# Elasticity -- A Quantitative Approach

The first elasticity tutorial took a qualitative approach to elasticity. The idea conveyed was

Elasticity = Responsiveness

The elasticity of Q with respect to P is the responsiveness of Q to changes in P.

Q is the quantity demanded. P is the price per unit.

Elasticity has a quantitative meaning, too. Elasticity is a specific way of measuring responsiveness.

Suppose P changes, and Q changes as a result.
The elasticity of Q with respect to P is the relative change in Q divided by the corresponding relative change in P.

You can also say:
The elasticity of Q with respect to P is the percentage change in Q divided by the corresponding percentage change in P.

For example, if the price of something goes up by 1% and sales fall by 2%, the elasticity of quantity demanded with respect to price is -2%/1% = -2.
Don't neglect the minus sign. For demand, the elasticity number is negative, because price and quantity move in opposite directions.

You can write the elasticity formula this way:

```               (Change in Q) / (level of Q)
Elasticity = ---------------------------------
(Change in P) / (level of P)```
The numerator of this fraction is the change in Q relative to the level of Q.
The denominator of the fraction is the change in P relative to the level of P.

The elasticity concept can be applied to other things besides Quantity and Price. You can use the concept whenever changing one thing makes something else change. For example, some researchers have estimated the elasticity of population health status with respect to the population's medical care expenditure. This elasticity is the percentage difference in health status divided by the percentage difference in medical care expenditure. (This elasticity usually comes out small, by the way.)  In this example, health status goes in the place of Q and medical care spending goes in the place of P.  In the formula, you can consider Q and P as placeholders for anything that changes in response to changes in something else.

There's a shorter way to say "the elasticity of Q with respect to P." You can say "the P elasticity of Q," as in "the price elasticity of demand" or "the income elasticity of demand." Here's the formula again for the P elasticity of Q:

```               (Change in Q) / (level of Q)
Elasticity = ---------------------------------
(Change in P) / (level of P)```
Let's try to apply this. Suppose that a medical practice finds that when the price of a certain test is is \$10, the quantity demanded is 100 tests per month, while if the price is \$30, the quantity demanded is only 60 tests per month.
What's the price elasticity of demand? Let's take this one step at a time, because I have some points to make.

The first point is that, when you figure the changes in Q and P, you do both in the same direction of time. Let's call the \$10 price and 100 quantity the "before" situation, and the \$30 price and 60 quantity the "after" situation. To be consistent, you have to calculate all changes as "after" minus "before" or "before" minus "after." Let's calculate all changes as "after" minus "before."

What is the change in Q, "after" minus "before"? (The "after" quantity is 60. The "before" quantity is 100.)

Now for the "levels" in the elasticity formula.

What number do you think you should use for the level of Q in the numerator of the fraction?
100 60 I'm not sure.

(I'm leaving some space here, so you won't be influenced by the discussion that follows. This space will go away if you pick the answer I like.)

Here is the formula again.

```               (Change in Q) / (level of Q)
Elasticity = ---------------------------------
(Change in P) / (level of P)```

You have a choice of what level of Q should go into the formula. There is no one right answer. The phrase "level of Q" is ambiguous. You could use the "before" level of Q, which is 100 in our example. You could use the "after" level of Q, which is 60 in our example. You could use something in between.

Whichever you use, be consistent between the top of the fraction and the bottom. If you use the "before" level of Q in the top, use the "before" level of P in the bottom.

If the change in Q is small relative to the levels of Q, it doesn't matter much which Q you use for the level of Q. The elasticity comes out about the same regardless.

If you want a middle choice use the "arc elasticity." It uses values halfway between "before" and "after." Let's show that.

Here's the definition of arc elasticity:

```                   (Change in Q) / (average of Q's)
Arc Elasticity = ------------------------------------
(Change in P) / (average of P's)```
To get the average of the Q's, add them and divide by 2. The same goes for the average of the P's.

Let's use that in the calculation.

The data again are:
If the price is \$10, demanded quantity is 100 tests per month.
If the price is \$30, demanded quantity is 60 tests per month.

What is the average of the Q's?

(When you answer the above question correctly, you'll be able to answer this next question.)

... so the (Change in Q) / (average of Q's) is what?

Leaving some space until the above question is answered correctly.

That's the numerator of the Arc Elasticity fraction. Next, we calculate the denominator.

Repeating the data:

If the price is \$10, demanded quantity is 100 tests per month.
If the price changes to \$30, demanded quantity is 60 tests per month.

This is our arc elasticity fraction so far:

```                               -0.5
Arc Elasticity = ------------------------------------
(Change in P) / (average of P's)```

Tackling the denominator, what is the change in P, "after" minus "before"?

And the average of the P's?

So, (Change in P) / (average of P's) is:

The full elasticity fraction will show here when this question is answered correctly.

Last step: Based on the above fraction, what is the arc elasticity?

Good! That's how you calculate an arc elasticity.

### Elastic and Inelastic

The terms "elastic" and "inelastic" can be given a precise meaning in terms of the number that comes out of the elasticity fraction.  The divider between elastic and inelastic demand is -1.  (For elasticities where you expect a positive relationship between Q and P, such as for elasticities of supply, the divider is +1.)

If the demand elasticity is more negative than -1, the demand is elastic.
If the demand elasticity is between -1 and 0, the demand is inelastic.

I avoided saying "higher" and "lower" in the above definition, because economists say that the elasticity of demand is "high" when it's a big negative number. They say that the elasticity of demand is "low" when it's a small negative number, meaning close to 0. The elasticity of demand for oil is low. The elasticity of demand for cubic zirconia (imitation diamond) is high.

In our example, we got a price elasticity of demand of -0.5. Is this elastic or inelastic demand?
Elastic demand Inelastic demand

### When does raising your price bring in more money?

For demand, there's a relationship between the elasticity and what happens to total revenue from customers when you change your price:
 If demand is elastic (if the elasticity is more negative than -1) then if the price goes up,the total amount customers spend goes down. If demand is inelastic (if the elasticity is between -1 and 0) then if the price goes up,the total amount customers spend goes up. If you have unitary elasticity (if the elasticity of demand is exactly -1) then if the price goes up,the total amount customers spend stays the same.

Let's see how this works in our example: At a price of \$10, you sell 100 tests per month. What is the total revenue (price times quantity)?

When the price is \$30, you do 60 tests per month. What is that total revenue?

So, raising the price from \$10 to \$30 does what to total revenue?
Revenue goes up. Revenue goes down.

Our calculated elasticity was -0.5, which we called "inelastic" because its absolute value was less than 1. Is this consistent with the idea that revenue going up when the price goes up?
Yes No

This is how elasticity of demand relates to pricing strategy. If your demand is inelastic, how would you change your price to bring in more money?
Raise price if demand is inelastic.
Lower price if demand is inelastic.

Suppose that you were in charge of setting prices at a hospital. You offer trauma services in your emergency department. You also offer counseling for weight control in your education department. Each of those services has an incremental cost, mainly for personnel and supplies to provide the service. Your hospital also has overhead costs, like the construction loan for your building and your salary as an administrator. To get the revenue to cover the overhead costs, you will have to mark up the prices on some services to levels well above what is needed to cover just the incremental costs. Think about the price elasticity of demand for emergency department services and for education department services.
Which department's prices will you mark up by a lot?
Emergency department
Education department

We have said that demand is called "inelastic" if the price-elasticity is between 0 and -1. Suppose the demand is perfectly inelastic. No response of quantity to price differences. What is the elasticity number if demand is perfectly inelastic?
0 -1 −∞

What if demand were elastic? How would you change your price to bring in more money?
Raise price if demand is elastic.
Lower price if demand is elastic.

Elastic demand typically happens when you are a small player in a big market, and there's a going market price. Undercut that price, and you attract business. Whether you make a profit depends on your costs, but that's a subject for another tutorial.

The extreme form of elastic demand is perfectly elastic demand. Perfectly elastic demand means you can sell as much as you want at the going market price. People will buy a practically unlimited amount at the current price, with no need for price cuts to get people to take more.

What is the elasticity number if demand is perfectly elastic?
0 -1 −∞

Small sellers of commodities (standardized products) in big markets can have demand that is perfectly elastic. A small farmer, for example, may find that the market will take all he or she can produce at the going price.

That's all for now. Thanks for participating!
For more use of the elasticity of demand concept, see the tutorial on monopoly.