25. /2 peints My Noes Ask Your Teacher The quantity demanded q (in units of a hu

Question

25. /2 peints My Noes Ask Your Teacher The quantity demanded q (in units of a hundred) of the Mikado miniature cameras per week is related to the unit price p (in dollars) by p- -0202100 and the quantity q (in units of a hundred) that the supplier is willing to make available in the market is related to the unit price p (in dollars) by p-01o2 20+50 If the market price is set at the equilibrium price, find the consumers' surplus and the producers' surplus. (Round your answers to the nearest dollar) consumer's aurplus producer's surplus

Explanation / Answer

Given

p = - 0.2 q^2 + 100

p = 0.1 q^2 + 2q + 50

First, find the equilibrium quantity q. How?

Equate the two equations, then solve for q:

- 0.2 q^2 + 100 = 0.1 q^2 + 2q + 50

0.3 q^2 + 2 q - 50 = 0

multiply with 10

3 q^2 + 20 q - 500 = 0

3 q^2 + 50 q - 30 q - 500 = 0

3 q^2 - 30 q + 50 q - 500 = 0

3q ( q - 10 ) + 50 ( q - 10) = 0

( q - 10 ) ( 3q + 50 ) = 0

q - 10 = 0 , 3q + 50 = 0

q = 10 , 3 q = -50 ==> q = -50/3

q has to be positive

so q = 10

Now, solve for the equilibrium price by substituting the value of q to the demand equation:

p = - 0.2 (10)^2 + 100

p = - 0.2( 100 ) + 100

p = - 20 + 100

equilibrium price p = 80

Substituting the value of q to the supply equation should result in the same equilibrium price.

p = 0.1 (10)^2 + 2(10) + 50

p = 0.1(100) + 20 + 50

p = 10 + 20 + 50

equilibriumr price p = 80

Consumer surplus:

=>Int (-0.2q^2+100) dq on [0,10] - 10*80

=> (-0.2/3) q^3 + 100 q on [0,10] - 10*80

=> 933.333333 - 800

=> 133.33

Producer surplus:

=> 10*80 - Int (0.1 q^2 + 2 q + 50)dq on [0,10]

apply power int (x^n) dx = x^(n+1) / (n+1)

=> 800 - (( 0.1/3) q^3 + q^2 + 50 q)) on [0,10]

=> 800 - (33.33333 + 100 + 500)

=> 166.67

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