# Economics Applications of Derivatives

##### 585 words — categories: calculus

Some interesting things you can do with derivatives in the context of economics are to:

- Find the
**price elasticity of a demand function**. - Find the maximum of a total-revenue function.
- Characterize demand in terms of elasticity.

Retailers and manufacturers often need to know how a small change in price will affect the demand of a product. If a small increase in price produces no change in demand, then the price increase is a reasonable decision to make. However, if the price increase leads to a (large) drop in demand, then the increase is likely ill advised. To measure the *sensitivity* of demand to a price change, economists calculate the *elasticity of demand*.

Suppose that \(q\) represents the quantity of goods purchased and \(x\) is the price per unit of the goods. These quantities are related by the demand function

\[q = D(x)\]

If there’s a change \(\Delta x\) in the price per unit, the percent change in price is given by

\[\frac{\Delta x}{x} = \frac{\Delta x \cdot 100}{x} \%\] A change in the price produces a change in quantity sold \(\Delta q\) given by

\[\frac{\Delta q}{q} = \frac{\Delta q \cdot 100}{q} \%\] The ratio of the percent change in quantity to the percent change in price is

\[\frac{\Delta q / q}{\Delta x / x}\] The limit as \(\Delta x\) approaches 0 gives us the elasticity of demand \(E\).

\[E(x) = - \frac{x \cdot D^\prime(x)}{D(x)}\] The negative sign is because the price \(x\), and the demand \(D(x)\), are both nonnegative. Because the demand \(D(x)\) is normally decreasing, its derivative \(D^\prime(x)\) is usually negative. By inserting a negative sign in the definition, economists make \(E(x)\) nonnegative and easier to work with.

## DVD Rentals

Suppose that a DVD rental business in the early 2000s found that demand for rentals of its DVDs is given by

\[q = D(x) = 120 - 20x\] where \(q\) is the number of DVDs rented per day at \(x\) dollars per rental. Find:

- The quantity demanded when the price is $2 per rental.
- The elasticity as a funciton \(x\).
- The elasticity at \(x = 2\) and at \(x = 4\). Interpret the meaning of these values.
- The value of \(x\) for which \(E(x) = 1\). Interpret the meaning of this price.
- The total revenue function, \(R(x) = x \cdot D(x)\).
- The price \(x\) at which total revenue is a maximum.

```
from sympy import *
from sympy.abc import x
demand = 120 - 20*x
q = lambdify(x, demand, "numpy")
q(2)
```

`## 80`

```
elasticity = simplify((x * diff(demand, x)) / demand)
elasticity
```

`## x/(x - 6)`

```
E = lambdify(x, elasticity, "numpy")
E(2) * -1
```

`## 0.5`

`E(4) * -1`

`## 2.0`

An elasticity less then 1 means that a small percentage increase in price will cause an even smaller percentage decrease in the quantity sold. An elasticity greater than 1 means that an increase in price will cause a larger percentage decrease in quantity sold (in this case by a factor of 2).

`solve((x / (6-x)) - 1, x)`

`## [3]`

```
revenue = x * demand
revenue
```

`## x*(120 - 20*x)`

```
fprime = diff(revenue, x)
fprime
```

`## 120 - 40*x`

`solve(fprime, x)`

`## [3]`

`fprime.diff(x) < 0`

`## True`

## Theorems for Elasticity of Demand

- Total revenue is increasing at those \(x\)-values for which \(E(x) < 1\). The demand is
*inelastic*. - Total revenue is decreasing at those \(x\)-values for which \(E(x) > 1\). The demand is
*elastic*. - Total revenue is maximized at the value(s) of \(x\) for which \(E(x) = 1\). The demand has
*unit elasticity*.