2. Solve the following below: 2a .Find the equilibrium quantity. A. 82.3775 B. 6

Question

2. Solve the following below:

2a .Find the equilibrium quantity.

A. 82.3775

B. 6.594

C. 12.4928

D. 13.9049

2b. Find the equillibrium price.

A. 82.3775

B. 6.594

C. 12.4928

D. 13.9049

2c. Find the producer's surplus.

A. 82.3775

B. 6.594

C. 12.4928

D. 13.9049

The supply function for a product is givern by p= 0.05x2 +0.2x+9 where is the unit price in dollars and x is the number available for sale each week, measured in thousands of units. The demand function for the product is given by p--0.022-04x+16, where x is the number of units demanded in thousands, and p is the unit

Explanation / Answer

We have given Supply function S(x)=p=0.05x2+0.2x+9 where p is unit price in dollars

and Demand function D(x)=p=-0.02x2-0.4x+16 where p is unit price in dollars

to find the equilibrium quantity and the equillibrium price we set S(x)= D(x)

0.05x2+0.2x+9=-0.02x2-0.4x+16

0.07x2+0.6x-7=0

by comparing ax^2+bx+c=0 we get a=0.07,b=0.6,c=-7 and x=[-b+/-sqrt(b^2-4ac)]/2a

x=[-0.6+/-sqrt((0.6)^2+(4*0.07*7))]/(2*0.07)

x=[-0.6+sqrt(0.36+(1.96))]/(0.14),x=[-0.6-sqrt(0.36+(1.96))]/(0.14)

x=[-0.6+sqrt(2.32)]/(0.14),x=[-0.6-sqrt(2.32)]/(0.14)

x=6.59396157981,x=-15.1653901512

the positive solution gives the equilibrium point (qe,pe)

plug x=6.59396157981 into p=0.05x2+0.2x+9=0.05(6.59396157981)2+0.2*(6.59396157981)+9=12.4928087818

the positive solution gives the equilibrium point (xe,pe)=(6.594,12.4928)

2a) the equilibrium quantity xe=6.594

2b) the equillibrium price pe=12.4928

2c) the producer's surplus is xepe-integration of (0 to xe)(S(x))dx

=(6.594*12.4928)-integration of (0 to 6.594)(0.05x2+0.2x+9)dx

=82.3775232-[(0.05)x3/3+(0.2)x2/2+9x] from 0 to 6.594

=82.3775232-[((0.05)(6.594)3/3+(0.2)(6.594)2/2+9*(6.594))-((0.05)(0)3/3+(0.2)(0)2/2+9*0)]

=82.3775232-[((0.05)(6.594)3/3+(0.2)(6.594)2/2+9*(6.594))]

=13.9048957236

the producer's surplus is 13.9049

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