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)