Question

The demand function P=4500-15Q2 and the supply function P = 5Q2 + 2500. Determine the following:

Consumer surplus.

Producer surplus.

Answer 1

At equilibrium,

$4,500-15Q^2=5Q^2+2,500$

$4,500-2,500=5Q^2+15Q^2$

$2,000=20Q^2$

$\frac {2,000}{20} =Q^2$

$\sqrt100=\sqrt Q^2$

$Q=10$

Price is calculated below

$5(10)^2+2,500$

$500+2,500= 3,000$

Consumer surplus is calculated as

$∫_0^q​d(Q)dQ−P\times Q$

$CS=∫_0 ^{10} 4,500−15Q^2dQ−{3,000\times 10}$

$=[4,500 Q−\frac{15Q^3}{3}​]−30,000$

Since Q= 10;

$=[45,000−5,000]−30,000$

$=\$ 10,000$

producer surplus is calculated as;

$P\times Q−∫_0^q s(Q)dQ.$

$PS={ 3,000\times 10}−∫_0^{10}5Q^2 +2,500 dQ$

$=30,000−(\frac{15Q^3}{3}+2,500Q)$

Since Q =10

$PS=30,000−{5,000−2,500}$

$=\$ 0$