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 is calculated as Revenue minus the area under the supply curve up to the equilibrium quantity.
First, solve the equation $18 = 4 - 5x + x^2$.
Rearranging the equation yields: $x^2 - 5x + 4 - 18 = 0 \Rightarrow x^2 - 5x - 14 = 0$.
Applying the quadratic formula:
\[x = \frac{5 \pm \sqrt{25 + 56}}{2} = \frac{5 \pm \sqrt{81}}{2} = \frac{5 \pm 9}{2}\]
This gives two possible values for x: $x = 7$ or $x = -2$. We select the positive value, $x = 7$.
Revenue is calculated as $18 \times 7 = ₹ 126$.
Next, calculate the area under the supply curve from 0 to 7:
\[\int_0^7 (4 - 5x + x^2) dx = \left[4x - \frac{5x^2}{2} + \frac{x^3}{3}\right]_0^7\]
Substituting the limits:
$4(7) - \frac{5(49)}{2} + \frac{343}{3} = 28 - 122.5 + 114.33 = ₹ 45$ (approximately).
Producer's surplus is therefore ₹ $126 - 45 = ₹ 81$.