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: