15.1 Elasticity of any function

Enter any function into the field to display the function graph. A moving point on the graph shows you the elasticity of the function at that point.

An elasticity $𝜖$ indicates how strongly a variable reacts to the change of another variable. All changes are indicated in %.
As an example to illustrate this, we use the price elasticity of demand or the substitution elasticity of a production function.
The price elasticity of demand is the percentage change in demand for a one percent change in price.

$$\text{Price elasticity of demand}=\frac{\text{percentage change in quantity}}{\text{percentage change in price}}$$

The substitution elasticity of a production function (with two factors of production) is the percentage increase of one factor of production, which is necessary to keep the produced quantity constant if the other factor of production is decreased by one percent.

$$\text{Elasticity of substitution of a production function}=\frac{\text{percentage quantity change factor x}}{\text{percentage quantity change factor y}}{|}_{F\left(x,y\right)=\mathit{const}.}$$

Although the elasticity is easy to understand in terms of content and mathematically clearly defined, the implementation of the most important economic examples (price elasticity of demand and substitution elasticity) deviates from the standard representation. In the case of the elasticity of substitution, not the original production function (since it is a function of two variables) but its isoquant is used. In the case of the price elasticity of demand, the sign is changed and the ordinate and abscissa are swapped. The presentation of the price elasticity is shown in the next graphic.

1. The elasticity is scale invariant .
   Thus, it does not matter in which unit the input and output quantity is measured. Since only percentage changes are included in the calculation of the elasticity, the scale is not important.
   Example: 1 % of 100.000 € are 1.000 €. If you use T € as a scale, you get 1% of 100T € are 1T €, which is the same.
2. The elasticity ${𝜖}_f$ of a function $f$ is linked to the derivative, but is not the same! While the derivative of a function relates the change of the output ($\mathit{\Delta y}$) to the change of the input ($\mathit{\Delta x}$), $f^{\prime }=\frac{\mathit{\Delta y}}{\mathit{\Delta x}}$, for the calculation of the elasticity one uses the relative changes($\frac{\mathit{\Delta y}}{y}$ and $\frac{\mathit{\Delta x}}{x}$), thus, ${𝜖}_f=\frac{\frac{\mathit{\Delta y}}{y}}{\frac{\mathit{\Delta x}}{x}}$. The relation between derivative and elasticity (the approximation by means of $\mathit{\Delta y}$ and $\mathit{\Delta x}$ is identical for both) results thus in ${𝜖}_f=f^{\prime }\cdot \frac{x}{y}$
   As a consequence, the elasticity of a linear demand is not constant, but low for a low price and high for a high price. Rather, the elasticity is the derivative on a double logarithmic scale (see below).
3. At a fixed point (x,p) the price elasticity of demand is higher the flatter (!) the demand curve is. The demand elasticity measures the strengthAt (c.p.) a higher price the demand elasticity is higher.
   At (c.p.) a higher quantity the demand elasticity is lower.
   Reason: The representation of demand as a function of price measures absolute values!

Here, we assume a linear or concave demand function, which implies a decreasing marginal utility with a non-increasing rate of purchase. This implies that the elasticity of demand increases with the price.
At ${𝜖}_P=1$ the maximum revenue is reached.
Descriptive explanation:
If starting from ${𝜖}_P=1$ the price is increased, then the quantity is reduced disproportionately, since with a higher price, ${𝜖}_P>1$ applies. A price increase by 1 reduces the quantity by more than 1. Thus, the sales revenue decreases.
If starting from ${𝜖}_P=1$ the price is reduced, then the quantity increases under-proportionately, since with a lower price, ${𝜖}_P<1$ applies. A price reduction by 1 increases the quantity by less than 1. Thus, the sales revenue decreases.
In both cases, the sales revenue decreases.
If the variable costs are negligible (e.g. museum, cinemas), then sales revenue maximization is equal to profit maximization and the enterprise behaves optimally, if it tries to maximize the revenue. If the demand curve or another function is plotted in a coordinate system with two logarithmic axes, the elasticity corresponds to the slope of the curve in this coordinate system. In other words, the elasticity of a function is obtained by differentiating the logarithm of the function with respect to the logarithm of the input variables.
In the derivation of the assertion, we use $\frac{d\log <!--\; FUNCTION\; APPLICATION\; -->\left(f\left(x\right)\right)}{\mathit{dx}}=\frac{f^{\prime }\left(x\right)}{f\left(x\right)}$ and $\frac{d\log <!--\; FUNCTION\; APPLICATION\; -->\left(x\right)}{\mathit{dx}}=\frac{1}{x}.$ Furthermore, it should be noted that extending this fraction by $\frac{\mathit{dx}}{\mathit{dx}}$ is formally not quite correct, but allows the reader an intuitively understandable and comprehensible derivation.
Derivation

$$\frac{d\log <!--\; FUNCTION\; APPLICATION\; -->\left(f\left(x\right)\right)}{d\log <!--\; FUNCTION\; APPLICATION\; -->\left(x\right)}=\frac{\frac{d\log <!--\; FUNCTION\; APPLICATION\; -->\left(f\left(x\right)\right)}{\mathit{dx}}}{\frac{d\log <!--\; FUNCTION\; APPLICATION\; -->\left(x\right)}{\mathit{dx}}}=\frac{\frac{f^{\prime }\left(x\right)}{f\left(x\right)}}{\frac{1}{x}}=f^{\prime }\left(x\right)\cdot \frac{x}{f\left(x\right)}$$