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.