Integration/Calculus help?

Question

Alright so since you guys helped me oh so well with my first problem, why not try again? So heres my question. Roger is breeding tropical fish to sell to hobbyists. He finds that his supply curve looks like y1=0.06x^2+5 , where x is the number of fish, in hundreds, sold at y1 dollars each. Articles in his trade newsletters lead him to believe that the demand curve is y2=0.2x^2 - 4.8x + 31.

a) The equilibrium point is very close to seven fish. Calculate the producer surplus if Roger sells at y1.

b) What is the consumer surplus?

In my economics class this would have been simple to solve...but in calculus we have to use integration, and that is new for me this year, and my textbook is hard to understand. So if you could dumb it down into some steps for me it would be greatly appreciated. Thanks a lot!

for a)...at y1=7

Answer 1

for a)...at y1=7

It looks like that the surplus is based on integrating for x (area under the curve), from the equilibrium to your point. Thus, the first step is to find the actual equlibrium where the two formulae are equal so that you can determine the market price.

0.06x^2+5 = 0.2x^2 - 4.8x + 31
0 = .14x^2 - 4.8x + 26

Use the Quadratic formula:
x = (-b +/- (b^2 - 4ac)^1/2)/2a
x = (4.8 +/- (23.04 - 14.56)^1/2)/.28
x = 17.14 +/- 10.40
x = 6.74

Thus, the equlibrium quantity is 6.74 fish.
The market price is 0.06(6.74)^2+5 = $7.73

From the website I found (see links):

Producers’ Surplus:
Let p(x) be the unit price as a function of x, the supply quantity.
Let p be the market unit price.
Let x be the supply quantity such that p(x) = p.
The producers’ surplus is the difference between the total market price for x units and what producers are willing to receive for x units. That is:
PS = ∫(from x to 0) (p - p(x)) dx

Substitute, integrate, and solve:
PS = ∫(7, 0) ($7.73 - (0.06x^2+5)) dx
PS = ∫(7, 0) -0.06x^2 + 2.73

Use the Integration Power Rule: ∫ax^n = (ax^(n+1)) / (n+1)
PS = -0.02x^3 + 2.73x | (7, 0)

Now, plug in both values above after the |, and subtract the second from the first.
PS = (-0.02(7)^3 + 2.73(7) - (-0.02(0)^3 + 12.73(0))
PS = -6.86 + 19.11 = $12.25

Consumer's Surplus (from same site):
Let p(x) be the unit price as a function of x, the demand quantity (notice the change from supply to demand :)
Let p be the market unit price.
Let x be the demand quantity such that p(x) = p.
The consumers’ surplus is the difference between what consumers are willing to pay for x units and the total market price for x units. That is:
CS = ∫(from x to 0) (p - p(x)) dx

Substitute, integrate, and solve:
CS = ∫(7,0) ($7.73 - (0.2x^2 - 4.8x + 31)) dx
CS = ∫(7,0) (-0.2x^2 + 4.8x - 23.27) dx

CS = -.06667x^3 + 2.4x^2 - 23.27x | (7,0)
CS = -.06667(7)^3 + 2.4(7)^2 - 23.27(7) - (-.06667(0)^3 - 2.4(0)^2 + 38.73(0))

CS = -22.867 + 117.6 - 162.89
CS = -$68.16