One of the most commonly-used concepts in economics iselasticity. It is used to measure how much one variables changes when another variable changes. For example, elasticity can measure the relationship between:

• price and quantity demanded
• income and quantity demanded
• tax rates and labor supply
• currency values and exports or imports

In all these cases, we are interested in the relative change in two variables and that's where calculus comes in: the derivative.

However, elasticity is most useful if measured without regard to the units involved. In other words, we'd like to measure the effect of change on demand for, say, paper whether we measure it by sheet, by ream or by pound.

To make our measurements independent of the units, elasticity measures percent change, or in calculus the derivative divided by the variable itself, sometimes called the "dot product." It looks like this:

y/yx/x

=yxxy

For elasticity the x variable is price and y variable is quantity, so elasticity =

QPPQ

In your textbook, you will see elasticity defined, using a midpoint formula as:

%changeiny%changeinx

=y1y2/y1+y2/2/x1x2/x1+x2/2

The midpoint formula is necessary because you'd get a different answer if you used y1 and x1 in the denominator versus using y2 and x2. The midpoint is compromise--but not as good as calculus in which the change is minimized so there is no concern about what goes in the denominator.

Here's an example of price elasticity of demand using calculus. If we know the demand curve has a function:

Q=100xp1+1

where P = $3 and Q = 50units

QP=100p2

Using the formula for elasticity

QPPQ

Elasticity =

100xp2xp/q

inserting numbers for p and q

100x32x3/50=

- 0.67

A typical elasticity, somewhat inelastic and occurring for such items as movies or shoes.

Often for convenience, economists leave off the minus sign for price elasticity of demand. However, we need to remember that quantity demanded always falls when price rises.

Now you try one for income elasticity.

If: Quantity cars demanded =Income2+3xincome+40

And Income = $1000;

What is the income elasticity of demand?

A) About 2, meaning that much more of the items is purchases as income rises. The item is income elastic.

B) About 0.5, meaning that somewhat more, but not proportionately as much of the item is purchased as income rises. The item is income inelastic.

C) About -2, meaning that less of the item is purchased as income rises. The item is a an inferior good.