Question

If the supply and demand function are given p=20e^0.4Q and p=100e^-0.2 respectively.find the equilibrium price and quantity,and calculate the consumers and producers surplus.

Answer 1

At equilibrium price, quantity demanded (Qd) = quantity supplied (Qs)

Therefore,

$20e^0.4Q = 100e^-0.2$

introducing logs;

$log 20e0.4Q = log 100e-0.2$

$0.4Q log 20e = -0.2 log 100e$

$0.4Q log 20e/log 20e = -0.2 log 100e/log 20e$

$0.4Q = -0.2 log 100e/log 20e$

$0.4Q = -0.2 x 1.537e$

$Q= -0.2x1.537e/0.4$

Q = 0.7685e

Consumer Surplus.

$q∗∫0d(q)dq−p∗q∗$

$=20e^0.4(-0.7685)$

$=20e^2.022$

=427.25

Producer Surplus.

$p∗q∗−q∗∫0s(q)dq$

$=100e^(-0.2)$

= 0.398e x 427.25

=427.25 + 170.05

= 597.30