# Rates of Change

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

- \(\Delta y\) is shorthand that means “change in \(y\)”, NOT \(\Delta * y\) (it’s not an entity itself)
- In calculus, the change in \(y\) is often written formally as \(dy\)

We can express the **relative difference**, comparing the difference with the original value of \(y_1\) as:

\[\text{relative change in y}=\frac{y_2-y_1}{y_1}=\frac{\Delta y}{y_1} \]

- e.g. if \(y_1=3\) and \(y_2=3.02\), then the relative change in \(y\) is:

\[\frac{y_2-y_1}{y_1}=\frac{3.02-3}{3}=0.0067\]

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:

\[\text{percentage change in y}=\% \Delta y = \frac{y_2-y_1}{y_1}=\frac{\Delta y}{y_1}*100\%\]

- e.g. if \(y_1=3\) and \(y_2=3.02\), then the percentage change in \(y\) is:

\[\frac{y_2-y_1}{y_1}*100=\frac{3.02-3}{3}*100=0.67\%\]

- Just move the decimal point over two digits to the right to get a percentage
- This is most common when we measure inflation, GDP growth rates, etc.

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)