# Elasticity Definitions

Our study of comparative statics is essentially a study of cause and effect: how a change in an **exogenous variable** affects some **endogenous variable**. An “exogenous variable” is an underlying parameter of the model; it’s called “exogenous” because it is determined “outside” the model. An “endogenous variable” is determined *inside* the model: that is, it is the result of whatever process the model is trying to describe.

Elasticity measures *how responsive* an endogenous variable is to changes in an exogenous variable. For example, we may be interested in how responsive the quantity of a good demanded by a consumer is to a change in the price of that good, or in how responsive a firm’s hiring practices might be to an increase in the minimum wage.

First, let’s note that elasticity is a *rate*: it measures the percentage change in an endogenous (dependent) variable *per* percentage change in an exogenous (independent) variable.

Intuitively, we might think of elasticity as answering the question: “If some independent variable $X$ increases by $1\%$, what is the resulting percentage change in the dependent variable $Y$?” Mathematically, we may write this as \(\epsilon_{Y,X} = \frac{\%\Delta y}{\%\Delta x}\)

## Degrees of Elasticity

There are two important aspects of any measure of elasticity: its sign, and its magnitude.

The sign of an elasticity indicates whether the two variables are *positively related* or *negatively related*. For example, we generally assume that an increase in the price of a good results in a decrease in the quantity demanded of that good; therefore demand elasticity of the sort we looked at above will generally be negative. However, if we think about how an increase in the price of a good affects the demand for *some other* good, the sign becomes important: depending on whether the two goods are complements or substitutes, it might be negative or positive.

The magnitude of elasticity — specifically, whether it is greater or less than one — describes whether a change in the exogenous variable results in a *proportional*, *less than proportional*, or *greater than proportional* change in the endogenous variable. That is, we can characterize elasticity as follows:

**Perfectly inelastic**($|\epsilon| = 0$ or $|\% \Delta Y| = 0$): the endogenous variable*does not change*when the exogenous variable increases**Inelastic**($|\epsilon| < 1$, or $|\% \Delta Y| < |\% \Delta X|$): the percentage change in the endogenous variable is*less than*the percentage change in the exogenous variable.**Unit elastic**($ |\epsilon| = 1$, or $|\% \Delta Y| = |\% \Delta X|$): the percentage change in the endogenous variable is*the same as*the percentage change in the exogenous variable**Elastic**($|\epsilon| > 1$, or $|\% \Delta Y| > |\% \Delta X|$): the percentage change in the endogenous variable is*greater than*the percentage change in the exogenous variable**Perfectly elastic**($|\epsilon| = \infty$, or $|\% \Delta X| = 0$): any change in the exogenous variable would cause the exogenous variable to change “*infinitely*” in percentage terms (usually this means going from a positive number to zero or vice versa)

## Example: Elasticity along a Linear Demand Curve

A demand curve shows the quantity demanded at various different prices. As such, price ($P$) is the exogenous variable, and quantity ($Q$) is the endogenous variable.

If we consider a linear demand curve — one where each dollar increase in price results in the same reduction in quantity demanded — it might be tempting to think that the elasticity is constant. However, a given change in price represents a very different *percent* change in price, depending on the price level: if the original price is €100, a €2 increase represents only a 2% change, while if the original price is €20, that same €2 increase represents a 10% change!

In the following graph, try dragging the point up and down along the demand curve to see how the *percent* change in price and quantity, and therefore the elasticity, changes along the curve: