find producer surplus calculus

Question

Find producer's surplus at the market equilibrium point if supply function is

  is p=0.8x+3 and the demand function is p=70/(x+5)

thank you

Answer 1

Producer Surplus = about 13.52

set the prices equal from the supply and demand curves.

.8x + 3 = 70/(x+5)

(x+5)(.8x+3) = 70

.8x^2 + 7x + 15 = 70

.8x^2 + 7x -55 = 0

8x^2 +70x -550 = 0

(8x+110)(x-5) =0

x = 5

p=70/(5+5) = 7

producer surplus = PS = the integral of 70/(x+5) minus px = 70ln(x+5) evaluated from 0 to 5

minus 7(5) = 70(ln10-ln5) - 35 = 70ln2 - 35 = 70(.693) 48.52 - 35 = 13.52

PS = 13.52 = the area between the demand curve and the equilibrium price line = area between the hyperbolic curve p=70/(x+5) and the horizontal line p=7, which is the same as the area between the demand curve and the x axis all minus the rectangular area between p=7 and p=0 (the x axis) from x=0 to x=5