Question

If the supply function is $p = 4 - 5x + x^2$, then find the producer’s surplus when price is 18.

Hint:

To find producer's surplus, integrate the supply curve up to equilibrium quantity, subtract from total revenue.

Answer 1

Producer’s surplus = Revenue – Area under supply curve up to equilibrium quantity
First, solve $18 = 4 - 5x + x^2$
Rewriting: $x^2 - 5x + 4 - 18 = 0 \Rightarrow x^2 - 5x - 14 = 0$
Use quadratic formula: \[ x = \frac{5 \pm \sqrt{25 + 56}}{2} = \frac{5 \pm \sqrt{81}}{2} = \frac{5 \pm 9}{2} \]
So, $x = 7$ or $x = -2$ → Take $x = 7$
Revenue = $18 \times 7 = ₹ 126$
Now compute area under supply curve from 0 to 7:
\[ \int_0^7 (4 - 5x + x^2) dx = \left[4x - \frac{5x^2}{2} + \frac{x^3}{3}\right]_0^7 \]
Substitute: $4(7) - \frac{5.49}{2} + \frac{343}{3} = 28 - 122.5 + 114.33 = ₹ 45$ (approx)
Producer’s surplus = ₹ $126 - 45 = ₹ 81$