Rates of Change

If $y$ changes from $y_1\to y_2$, the difference, $\mathrm{\Delta }y=y_2-y_1$

• $\mathrm{\Delta }y$ is shorthand that means “change in $y$”, NOT $\mathrm{\Delta }\ast y$ (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: $$\text{relative change in y}=\frac{y_2-y_1}{y_1}=\frac{\mathrm{\Delta }y}{y_1}$$

• 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 $y$ ($\mathrm{\%}\mathrm{\Delta }y$), also called the growth rate of $y$, which is 100 times the relative change: $$\text{percentage change in y}=\mathrm{\%}\mathrm{\Delta }y=\frac{y_2-y_1}{y_1}=\frac{\mathrm{\Delta }y}{y_1}\ast 100\mathrm{\%}$$

• e.g. if $y_1=3$ and $y_2=3.02$, then the percentage change in $y$ is: $$\frac{y_2-y_1}{y_1}\ast 100=\frac{3.02-3}{3}\ast 100=0.67\mathrm{\%}$$
  • 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: $$100\ast (\ln (y_2)-\ln (y_1))=\mathrm{\%}\mathrm{\Delta }y=\text{percentage change in y}$$

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 $y$. $${ϵ}_{yx}=\frac{\mathrm{\%}\mathrm{\Delta }y}{\mathrm{\%}\mathrm{\Delta }x}=\frac{(\frac{\mathrm{\Delta }y}{y})}{(\frac{\mathrm{\Delta }x}{x})}=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}\ast \frac{x}{y}$$

• Interpretation: A 1% change in $x$ will lead to a ${ϵ}_{yx}$% 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 $y=q$: $${ϵ}_D=\frac{\mathrm{\%}\mathrm{\Delta }q}{\mathrm{\%}\mathrm{\Delta }p}=\frac{\frac{\mathrm{\Delta }q}{q}}{\frac{\mathrm{\Delta }p}{p}}=\frac{\mathrm{\Delta }q}{\mathrm{\Delta }p}\ast \frac{p}{q}$$

• Note that $\frac{\mathrm{\Delta }q}{\mathrm{\Delta }p}$ is $\frac{1}{slope}$ of the demand curve (which is $\frac{\mathrm{\Delta }p}{\mathrm{\Delta }q}$)
• 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, $\mathrm{\Delta }y$ 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 $y=f(x)$: $$\frac{\mathrm{\Delta }f(x)}{\mathrm{\Delta }x}=\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}$$

• This measures how $f(x)$ changes as $x$ changes
• If $\mathrm{\Delta }$ is infinitesimally small, then we have expressed the (first) derivative of $f(x)$ with respect to $x$, written variously as $f^{\prime }(x)$ or $\frac{df(x)}{dx}$ $$\frac{df(x)}{dx}=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}$$

The derivative of a linear function $(y=ax+b)$ is a constant (i.e. the slope), $a$ $$\frac{df(x)}{dx}=a$$

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

• 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)