Rates of Change

If \(y\) changes from \(y_1 \rightarrow y_2\), the difference, \(\Delta y = y_2-y_1\)

We can express the relative difference, comparing the difference with the original value of \(y_1\) as: \[\text{relative change in y}=\frac{y_2-y_1}{y_1}=\frac{\Delta y}{y_1} \]

It’s most common to talk about the percentage change in \(y\) (\(\% \Delta y\)), also called the growth rate of \(y\), which is 100 times the relative change: \[\text{percentage change in y}=\% \Delta y = \frac{y_2-y_1}{y_1}=\frac{\Delta y}{y_1}*100\%\]

Natural logarithms \((\ln)\) are very helpful in approximating percentage changes from \(y_1\) to \(y_2\) because: \[100*(\ln(y_2)-\ln(y_1))=\% \Delta y = \text{percentage change in y}\]

Elasticity

Using logs and percentage changes helps us talk about elasticity, an extremely useful concept with vast applications all over economics. Elasticity measures the percentage change in one variable (\(y\)) as a response to a 1% change in another (\(x\)) at a particular value of \(x\) and \(y\). \[\epsilon_{yx} = \frac{\% \Delta y}{\% \Delta x}=\cfrac{(\frac{\Delta y}{y})}{(\frac{\Delta x}{x})} =\frac{\Delta y}{\Delta x}*\frac{x}{y}\]

  • Interpretation: A 1% change in \(x\) will lead to a \(\epsilon_{yx}\)% change in \(y\)

For example, the price elasticity of demand measures the percentage change in quantity demanded to a 1% change in price (at a particular price point), note here: \(x=P\) and \(y=q\): \[\epsilon_D = \frac{\%\Delta q}{\%\Delta p} = \cfrac{\frac{\Delta q}{q}}{\frac{\Delta p}{p}} =\frac{\Delta q}{\Delta p}*\frac{p}{q}\]

  • Note that \(\frac{\Delta q}{\Delta p}\) is \(\frac{1}{slope}\) of the demand curve (which is \(\frac{\Delta p}{\Delta q}\))
  • Note though we would technically multiply by \(\frac{100}{100}\) to get percentage change, this term obviously is just 1. Elasticity is unitless.
  • Note also that on a graph we usually express \(q\) as our independent variable and \(p\) as our dependent variable

Derivatives (Calculus)

Often, \(\Delta y\) refers to a very small change in \(y\), a marginal change in \(y\). A rate of change is the ratio of two changes, such as the change between \(x\) and \(y=f(x)\): \[\frac{\Delta f(x)}{\Delta x} = \frac{f(x + \Delta x)-f(x)}{\Delta x}\]

  • This measures how \(f(x)\) changes as \(x\) changes
  • If \(\Delta\) is infinitesimally small, then we have expressed the (first) derivative of \(f(x)\) with respect to \(x\), written variously as \(f'(x)\) or \(\frac{df(x)}{dx}\) \[\frac{d f(x)}{d x} = \lim_ {\Delta x \to 0} \frac{f(x + \Delta x)-f(x)}{\Delta x}\]

The derivative of a linear function \((y=ax+b)\) is a constant (i.e. the slope), \(a\) \[\frac{d f(x)}{d x} = a\]

The derivative of the first derivative is the second derivative of a function \(f(x)\) with respect to \(x\), denoted \(f''(x)\) or \(\frac{d^2f(x)}{dx^2}\)

  • The second derivative measures the curvature of a function
  • It used for proving when a function has reached a maximum or minimum, or is concave or convex (often used in #Nonlinear-Functions-&-Optimization)