Find the consumer and producer surpluses for this product.

Question

The demand and supply functions for a product are modeled by Demand: p = -0.2x+8 and Supply: p = 0.1x + 2 Where x is the number of units in millions. Demand: p = -0.2x+8 and Supply: p = 0.1x + 2 Where x is the number of units in millions. Find the consumer and producer surpluses for this product.

Answer 1

Demand: p=-.2x+8

Supply: p= .1x +2

set them equal, solve for x, then for p

-.2x + 8 = .1x +2

multiply by 10

- 2x +80 = x + 20

- 2x-x =20-80 = -60

x = -60/-3

x=20 million

p = .1(20)+2 = 4

p= -,2(20)+8 =-4+8= 4

equilibrium price = $4

it may help to graph the demand and supply lines

demand curve has y or p intercept = 8, and x intercept = 40

supply curve has y or p intercept = 2

plot price, p, on the y axis

they intersect at the point (20,4)

Area of a triangle = height times base, divided by 2 = A=hb/2

consumer surplus is the area of the right triangle formed by the demand line, the price line and the p axis

Area = CS =hb/2 = (8-4)(20)/2 = 40 million dollars

producer surplus is the area of the right triangle formed by the supply line, the price line and the p axis

Area = PS = hb/2 = (4-2))20/2 = 20 million dollars

assuming the price unit is in dollars (or replace with pounds, pesos, rubles, yen, crypto currency, or whatever)

or if you want, you could you use integral calculus to find the areas for consumer and producer surplus, but using geometry is easier to see. If you do use calculus, the geometry of right triangles will be a way to check your calculus solutions

CS = the integral of -.2x+8 - 4 evaluated from x =0 to x=20

-.2x^2/2 +8x -4x = .1x^2 +4x

-.1(20)^2 +4(20)

=-.1(400)+80

= -40+80 = 40 million dollars of consumer surplus

PS = the integral of 4 - (.1x +2) or 2-.1x, evaluated from x=0 to x=20

2x -.1x^2/2

2(20) - .1(20^2)/2

= 40-.1(400)/2

= 40 -.1(200)

= 40 -20

= 20 million dollars of producer surplus