Question

Find the consumer's surplus at $P = 5$ for the following demand functions:

(a) P=25-2Q,

(b) P= 10/sqrt {Q} 

Answer 1

a)

$P=25-2Q$

but $P=5$

$5=25-2Q$

$2Q=20$

$Q=10$

Consumer surplus $=\intop^Q_0D(Q)dQ-PQ$

$\intop^{10}_0(25-2Q)dQ-5(10)$

$=[25Q-Q^2]_0^{10}-50$

$=250-100-50$

$=100$

b)

$P=\frac{10}{\sqrt{Q}}$

$P=5$

$5=\frac{10}{\sqrt{Q}}$

$\sqrt{Q}=2$

$Q=4$

Consumer surplus $=\intop^4_0\frac{10}{\sqrt{Q}}dQ-20$

$=10[2\sqrt{Q}]^4_0-20$

$=20(\sqrt{4}-0)-20$

$=40-20$

$=20$