Profit and elasticity

The equilibrium price is not, in general, the one that maximises profit: they are two distinct objectives. And to measure how much demand “reacts” to price we introduce elasticity.

Observation — Producer's profit vs equilibrium

If the unit cost of production is $c_0=5$ € per box, the profit at equilibrium is $\pi^* = (p^* - c_0)\cdot q^* \approx (14{,}58 - 5)\cdot 4\,793 \approx 45\,950\ \text{€/month}.$ As the price varies, $\pi(p) = (p - c_0)\cdot D(p)$ is a parabola in $p$ whose vertex gives the price of maximum profit, in general different from the equilibrium price: maximising revenue and clearing the market are two distinct objectives. The analytical determination of the maximum is the classic optimisation problem that we will take up again when studying derivatives in Year 4.

Observation — Elasticity of demand

A measure of “how much demand responds to price” is the elasticity: $\varepsilon_D = -\frac{\Delta D / D}{\Delta p / p}.$ For a linear demand $D(p)=a-bp$: $\varepsilon_D = bp/(a-bp)$, increasing in $p$. When $\varepsilon_D>1$ demand is elastic, when $\varepsilon_D<1$ inelastic. The rigorous form with derivatives (logarithmic derivative) will be seen in Year 5.

Links

Topics: Functions and properties
Concepts: Elasticity · Market equilibrium
Functions: Parabola
Skills: Modelling