We saw in class and in your reading that if the relationship between some variable \(y\) (say quantity sold) and another variable \(x\) (say price per unit) was described by the curve

\[ y = \alpha x^{\beta_{1}} \]

then \({\beta_{1}}\) was the elasticity, which is given by

\[ {\beta_{1}} = \frac{\partial y}{\partial x}\frac{x}{y} = \frac{\partial y/y}{\partial x/x} \]

At the end of class, I made the claim that \({\beta_{1}}\) is approximately the percentage change in y for a 1% change in \(x\). (That is if we are ignore the error terms; if we include them, \({\beta_{1}}\) will be approximately the percentage change of our predicted value of \(y\) for a 1% increase in \(x\)). But why is that? Let \(y_0\) be the price when \(x = x_0\) and \(y_1\) be the price when \(x\) increases by 1% (i.e. when \(x = 1.01x_0\)). The percentange change is given by

\[\begin{align} 100\frac{y_1 - y_0}{y_0} &= 100\frac{\alpha (1.01x)^{\beta_{1}} - \alpha x^{\beta_{1}}}{\alpha x^{\beta_{1}}}\\ &= 100(1.01^{\beta_{1}} - 1) \end{align}\]

It turns out that \(100(1.01^{\beta_{1}} - 1)\approx {\beta_{1}}\) if \({\beta_{1}}\) isn’t too far from zero:

curve(100*(1.01^x-1), from=-12, to=12, xlab=expression(beta), ylab=expression(100(1.01^beta-1)), ylim=c(-12, 12), xlim=c(-12, 12))
abline(0,1, lty=2)

Visually, it’s clear that the approximation is very good when \({\beta_{1}}\) is small. We can also plot the error in our approximation:

curve(100*(1.01^x-1) - x, from=-12, to=12, xlab=expression(beta), ylab=expression(100(1.01^beta-1) - beta), xlim=c(-12, 12))

As long as \({\beta_{1}}\) is relatively small, the error in our approximation is also small. In particular, if \({\beta_{1}}\) is less than about 5 in absolute value, our approximation will understate the true impact of a 1% increase in demand by no more than \(\approx 0.15\) percentage points.

(How did I know that \(100*(1.01^{\beta_{1}}-1)\approx {\beta_{1}}\) when \({\beta_{1}}\) is small? I took a Taylor expansion of \(100*(1.01^{\beta_{1}}-1)\) about \({\beta_{1}}=0\).)