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

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.