Q.A firm's supply function is $Q_s = 5P$ (a straight line through the origin). Find the price elasticity of supply when price rises from ₹10 to ₹12, and explain why a linear supply curve through the origin always gives the same elasticity.

A straight-line supply curve passing through the origin has a special property: its elasticity is the same, and equal to 1, at every point.

Step 1 — Quantities at the two prices.

At $P=10$: $Q_s = 5(10) = 50$. At $P=12$: $Q_s = 5(12) = 60$.

Step 2 — Percentage changes.

Step 3 — Elasticity.

Why this always happens for a line through the origin: for $Q_s = 5P$, the slope $\frac{dQ_s}{dP}=5$ is constant, and $\frac{P}{Q_s} = \frac{P}{5P} = \frac{1}{5}$ at EVERY point. So point elasticity $E_s = \frac{dQ_s}{dP}\times\frac{P}{Q_s} = 5 \times \frac{1}{5} = 1$ regardless of which price is chosen.

Cross-check with the geometric-method table (Section 7): since $Q_s = 5P$ passes through the origin (when $P=0$, $Q_s=0$), the intercept-based rule directly gives $E_s = 1$ at every point — matching the calculation above without any further arithmetic.

Wrongly assuming elasticity must be computed afresh at every different price on a linear-through-origin supply curve — the whole point of this special case is that it does NOT change, so once the line is confirmed to pass through the origin, $E_s=1$ can be stated immediately.