A fast-food outlet finds that the demand equation for its new side dish is given

Question

A fast-food outlet finds that the demand equation for its new side dish is given by

p = D(x) = 54 /(x + 1)2

where p is the price (in cents) per serving and x is the number of servings that can be sold per hour at this price. At the same time, the franchise is prepared to supply x servings per hour at a price of p cents based on the supply function given by

p = S(x) = 2x + 2

Find the equilibrium price, the consumers' surplus CS and the producers' surplus PS at this price level.

equilibrium price po = ____¢

consumers' surplus CS = ____¢

producers' surplus PS =____ ¢

Explanation / Answer

54/(x + 1)2  = 2x + 2

27/(x + 1)2  = x + 1

(x + 1)3=27

x+1 =3

x=2

equilibrium price po =54 /(2 + 1)2 =6 ¢

consumers' surplus=[0 to 2] D(x) - po dx

=[0 to 2] 54/(x + 1)2 - 6 dx

=[0 to 2] -54/(x + 1) - 6x

=[ -54/(2+ 1)   - 6*2]-[ -54/(0+ 1)   - 6*0]

=(-18-12)-(-54 -0)

=24¢

producers' surplus=[0 to 2] po -S(x) dx

=[0 to 2] 6 - ( 2x + 2) dx

=[0 to 2] 4 - 2x dx

=[0 to 2] 4x - x2

=4*2 - 22 -0+0

=4 ¢

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