Section 6.7 topic: Consumers’ and Producers’ Surplus

Producers’ Surplus: If p = S(x) is the supply equation, po is the equilibrium price of the commodity,

and xo is the equilibrium quantity, then the producers’ surplus is given by

Z xo

0

[po � S(x)] dx

1. Determine the producers’ surplus for the supply function below at the given number of units sold.

(Round answer to two decimal places.)

p = S(x) = 80e0.06x, xo = 30

price.. .

..

X = Xo

X

-

-

p . -

- Sfx . I -180g. " "← -

183 .97

PS =

$9483.97- Soe

" " "

) dx =$2786

Consumers’ Surplus: If p = D(x) is the demand equation, po is the equilibrium price of the

commodity, and xo is the equilibrium quantity, then the consumers’ surplus is given by

Z xo

0

[D(x)� po] dx

2. Calculate the consumers’ surplus for the demand equation at the given number of units demanded.

(Round answer to the nearest cent.)

p = D(x) = 27 = 2x1/3, xo = 343

3. Calculate the producers’ surplus at the indicated price level for the supply equation below. (Round

answer to the nearest cent.)

p = S(x) = 130 + 0.2x2, po = 194.80

2 Summer 2019, Maya Johnson

-

-

Po -

-Dcxo ) -27-21343/3=13

BE-

§

"

427-2×43- 13 )dx=$Ro

-

- -

Sfx . ) -

-194.80

18

130+0.2×5=194.80 pS=f( 194.80 - (130+0.2×2) )dx- 13

- 130

0.2×5=64.1o

0.2 0.2 =$77X ? = 524

⇒ x . -524=18

4. Determine the consumers’ surplus and the producers’ surplus at the equilibrium price level for

the following demand and supply functions. (Round answers to the nearest cent.)

p = D(x) = 83.1� 1.8x

p = S(x) = 0.3x2 + 9

3 Spring 2019, Maya Johnson

Find Xo → Equilibrium quantity :

Set Six > =D Cx ) & solve for X.

⇒ 0.3×79=83 .I - 1.8 x

⇒ 0.3×2+9 - 83 .I t 1.84=0

⇒ 0.3×2+1.8×-74 .

I = O

⇒ o÷¥i÷x - t÷-

- %

⇒ ×Z

t 6×-247=0

⇒ ( x +19 Xx- 13 ) = O

⇒ X = 13 , -491 Cannotbe negative!

⇒ Xo ; Po = D L Xo ) = S ( Xo ) = 0.343ft 9

=59÷(Equilibrium price

CS = ( 83 . I - 1.8 x- 59 . 7) dx

F$l5(use calculator

13

PS -

- S ( 59.7 - o .3×2 - 9) dxI

$439.40°

Luse calculator

5. 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.

4 Spring 2019, Maya Johnson

- --

Find Xo → Equilibrium quantity first :

Set Stx ) =D Lx ) & solve for XI

⇒ 2x t2=54 ⇒ 2LxtD={¥ ⇒ 2LxtD.CxtD2=¥z.¥5( Xt 1)

2

⇒

21×+133=5.4⇒ ( × ti ) 3=27 ⇒ Xt 1=3527 ⇒ Xt I =3

-g Po =D Go )=S( Xo ) -

-2 (2) t2 = by

⇒ × = 2 ⇒ Xo = 2 L Equilibriumprice

Cs -

-

§¥,

-6) dx -

- $240[ Use Calculator

2

PS =

Sf6 - 2x - 2) dx

° (use calculator