Application of Calculus (Commerce) • Topic 2 of 2

Marginal Functions & Elasticity of Demand

Derivatives turn the cost/revenue functions into marginal quantities and measure how sensitive demand is to price.

Marginal cost and revenue

  • Marginal cost $\text{MC}=\dfrac{dC}{dx}$ — the approximate cost of one extra unit.
  • Marginal revenue $\text{MR}=\dfrac{dR}{dx}$ — the approximate revenue from one extra unit.

Profit maximisation

Profit $P(x)=R(x)-C(x)$ is maximised where $P'(x)=0$, i.e.

$$\text{MR}=\text{MC},\qquad\text{with}\quad P''(x)<0.$$

Elasticity of demand

The price elasticity of demand measures the responsiveness of quantity to price:

$$E_d=\frac{p}{x}\cdot\frac{dx}{dp}.$$

Demand is called elastic if $|E_d|>1$, inelastic if $|E_d|<1$, and unit elastic if $|E_d|=1$.

Two figures: a total cost curve with a tangent whose slope equals marginal cost dC/dx, and a downward-sloping demand curve showing that quantity falls as price rises with the elasticity formula. Marginal Cost (slope of C) & Elasticity of Demand (a) Marginal cost = slope of tangent x (units) ₹ C(x) x₀ slope = dC/dx = MC (b) Demand: p falls as x rises x (quantity) p (₹) p = 12 − 0.8x high p, low x low p, high x Elasticity of demand: E_d = (p / x) × (dx / dp) | E_d | > 1 elastic   •   | E_d | < 1 inelastic   •   | E_d | = 1 unit elastic
Solved Examples
1
Worked Example
If $C(x)=x^2+4x+10$, find the marginal cost at $x=5$.
Solution

$\text{MC}=C'(x)=2x+4$. At $x=5$: $2(5)+4=14$ (in ₹ per unit).

Answer: $14$ (in ₹ per unit).

2
Worked Example
If $R(x)=20x-x^2$, find the marginal revenue at $x=3$.
Solution

$\text{MR}=R'(x)=20-2x$. At $x=3$: $20-6=14.$

Answer: $14.$

3
Worked Example
Profit $P(x)=-x^2+40x-100$. Find the profit-maximising output.
Solution

$P'(x)=-2x+40=0\Rightarrow x=20$. $P''=-2<0$, confirming a maximum. Output $=20$ units.

Answer: Output $=20$ units.

4
Worked Example
Demand $x=50-2p$. Find $E_d$ at $p=10$.
Solution

$\dfrac{dx}{dp}=-2$. At $p=10,\ x=50-20=30$. $E_d=\dfrac{p}{x}\dfrac{dx}{dp}=\dfrac{10}{30}\cdot(-2)=-\dfrac{2}{3}$. Since $|E_d|<1$, demand is inelastic.

Answer: $E_d=-\dfrac{2}{3}$; since $|E_d|<1$, demand is inelastic.

Key Points

  • Marginal cost $\text{MC}=C'(x)$; marginal revenue $\text{MR}=R'(x)$.
  • Profit is maximised where $\text{MR}=\text{MC}$ and $P''(x)<0$.
  • Elasticity $E_d=\dfrac{p}{x}\dfrac{dx}{dp}$.
  • $|E_d|>1$ elastic, $<1$ inelastic, $=1$ unit elastic.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.Marginal revenue is:
Explanation: MR is the derivative of revenue.
Q2.Profit is maximised when:
Explanation: Set $P'=0\Rightarrow \text{MR}=\text{MC}$ (with $P''<0$).
Q3.If $C(x)=x^2+4x+10$, $\text{MC}$ at $x=5$ is:
Explanation: $C'(5)=2(5)+4=14$.
Q4.Demand is inelastic when:
Explanation: Inelastic means $|E_d|<1$.
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