The weekly demand and supply functions for sport man 5 times

The weekly demand and supply functions for sport- man 5 times 7 tends are given by p = -0.1 x^2 - x + 40 and p = 0.1 x^2 + 2x + 20 respectively, where p is measured in dollars and x is measured in units of hundred. Find the equilibrium quantity and prices. Sketch the related graph and write a short interpretation.

Solution

Equlibrium is attained when Supply = Demand

-.1x² - x + 40 = .1x² + 2x + 20

or .2x² + 3x - 20 = 0 or 2x² + 30x - 200 = 0

that is, 2x² + 40x - 10x - 200 = 0 => 2x(x + 20) - 10(x+20) = 0

or (2x -10)(x + 20) = 0, x - 5 or -20

Since units sold cannot be negative, x = 5

P = .1x² + 2x + 20 = $32.5

Interpretation

For Prices >32.5, we will have lesser demand than equlibrium quantity. This is because, higher the price, lesser the demand. Demand curve will keep decreasing until we reach equlibrium. However, a higher price means more profit and thus more supply. SUpply curve shifts upwards till it intersects demand curve and we reach equlibrium.