C.1 The Formula for Elasticity
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 \(\epsilon_{Y,X} = \frac{\%\Delta Y}{\%\Delta X}\) The formula (Note: This assumes a small percentage change, so that we don’t have to worry about whether to use the initial or final value as the basis for the percentage change. Of course, a change between $1.00$ and $1.01$ is only approximately a $1\%$ change…but we’ll be using calculus soon enough to determine the elasticity.) for a percentage change in some variable $X$ is \(\%\Delta X = 100\% \times \frac{\Delta x}{x}\) Plugging this expression into our formula for elasticity gives us \(\epsilon_{Y,X} = \frac{\%\Delta Y}{\%\Delta X} = \frac{100\% \times \frac{\Delta y}{y}}{100\% \times \frac{\Delta x}{x}}\) We can rewrite this as \(\epsilon_{Y,X} = \frac{\Delta y}{\Delta x} \times \frac{x}{y}\)
Note that this is related to but not equal to the absolute change in $Y$ per absolute change in $X$ given by $\Delta Y / \Delta X$. In particular, for the special case where $Y = X$, the elasticity and slope are the same thing. But if $Y > X$, then the same absolute change would represent smaller percentage change in $Y$ than it would in $X$; so the elasticity would be less than $\Delta Y/\Delta X$.
Example
Suppose the price of a good increased from $€1.00$ to $€1.01$, and this resulted in the quantity demanded decreasing from 50 million units to 48 million units. In this case we consider price to be the exogenous variable $X$ and quantity to be the endogenous variable $Y$; so using the formula above we have \(\begin{aligned} \Delta x &= €1.01 - €1.00 = +€0.01\\ x &= €1.00\\ \Delta y &= 48 \text{ million units} - 50 \text{ million units} = -2 \text{ million units}\\ y &= 50 \text{ million units} \end{aligned}\) We calculate the percentage change in price as \(\text{\% change in price} = 100\% \times \frac{\Delta x}{x} = 100\% \times \frac{+€0.01}{€1.00} = 1\%\) and the percent change in quantity as \(\text{\% change in quantity} = 100\% \times \frac{\Delta y}{y} = 100\% \times \frac{-2 \text{ million units}}{50 \text{ million units}} = -4\%\) Therefore we would say that a $1\%$ increase in the price led to a $4\%$ decrease in the quantity demanded, so the elasticity (written $\epsilon$) would be $-4$: \(\epsilon_{Q,P} = \frac{\text{\% change in quantity}}{\text{\% change in price}} = \frac{-4\%}{+1\%} = -4\) We could also just plug this directly into the formula to obtain \(\epsilon_{Y,X} = \frac{\Delta y}{\Delta x} \times \frac{x}{y} = \frac{-2 \text{ million units}}{+€0.01} \times \frac{€1.00}{50 \text{ million units}} = -4\) Note that in each case all the units cancel, leaving the measure of elasticity as a unitless measure.