Find consumer surplus. D(q)=(2000)/((2q+1)^2) if the supply and equilibrium are equal at q=8.
To find consumer surplus, we first need to determine the demand curve for the given equation D(q)=(2000)/((2q+1)^2) at an equilibrium point q=8 where supply equals demand.
Setting equal the demand and supply equations, we have:
D(q) = 2000/(2q+1)^2
S(q) = q
Set D(q) equal to S(q) and solve for q:
2000/(2q+1)^2 = q
2000 = q(2q+1)^2
2000 = q(4q^2 + 4q + 1)
2000 = 4q^3 + 4q^2 + q
4q^3 + 4q^2 + q - 2000 = 0
In order to find the value of q when supply equals demand, we can solve the above equation. Since it's a cubic equation, it may require numerical methods to find the exact solution.
However, assuming q≈8, we can find the value of consumer surplus.
Consumer Surplus = ∫[0 to 8] 2000/(2q+1)^2 dq - ∫[0 to 8] q dq
= ((2000/3) * (1/(2q+1))^3 ) | from 0 to 8 - (q^2/2) | from 0 to 8
= ((2000/3) * (1/17)^3) - (8^2/2)
= (2000/3) * (1/4913) - 32
= (2000/14739) - 32
≈ 0.13574
Therefore, the estimated consumer surplus is approximately 0.13574.