Section 6.7 topic: Consumers’ and Producers’ Surplus...
Transcript of Section 6.7 topic: Consumers’ and Producers’ Surplus...
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