Answer to Question #135837 in Microeconomics for Mefi

Question

Given the demand function PD=27-Q²and supply function PS=2Q+3 Assuming perfect competition find the consumer surplus and producer surplus

Answer 1

At equilibrium

$PD=PS$

$(27-Q^2)=(2Q+3)$

take like terms on same side

$27-3=2Q+Q^2$

$24=2Q+Q^2$

$Q^2+2Q-24=0$

Write 2Q as difference.

$Q^2+6Q-4Q-24=0$

Factor Q and 4 out

$Q(Q+6)-4(Q+6)$

Factor (Q+6) out of expression

$(Q+6)(Q-4)$

Hence

$Q+6=0$

$Q-4=0$

Hence Q = 4 or -6

(Note point of equilibrium should be positive)

Q at equilibrium = 4

P at equilibrium

[27-4²] = [2(4)+3] = 11

The consumer surplus is,

$\int(27-Q^2)dQ - (11)(4)$

$[27Q - Q^3/3]- 44$

Since Q = 4

$[27(4)-4^3/3]-44$ = 42.67=$42.67

The producer surplus is

$(11)(4)-\int(2Q+3)dQ$

$44-[2Q^2/2+3Q)$

Since Q=4

$44-[2(4^2)/2+12]= 16$ =$16