# Solved Questions on Consumers and Producers Surplus and Deadweight Losses

1- The demand and cost functions of a monopolist are given by,
Qd=260-2P
MC=2Q+10
Find:
a) Consumers surplus,
b) Producers surplus,

Solution
a) Calculating equilibrium price and quantity,
Q=260-2P
Or, 2P=260-Q
Or, P=\frac{260}{2}-\frac{Q}{2}
Or, P=130-0.5Q

TR=P×Q
TR=(130-0.5Q)×Q
TR=130Q-0.5Q^2

MR=\frac{d(TR)}{dQ}
=frac{d(130Q-0.5Q^2)}{dQ}
=130-Q

For equilibrium,
MR=MC......(i)
Putting MR and MC functions in equation (i) we get,
130-Q=2Q+10
Or, 130-10=2Q+Q
Or, 120=3Q
Or,\frac{120}{3}=Q
\therefore\Q=40

Putting the value of Q in demand function, we get,
Q=260-2P
Or, 2P=260-Q
Or, 2P=260-40
Or, P=\frac{220}{2}
Or, P=110

Quantity demanded at zero price,
Q=260-2×0
Q=260

Price at zero quantity demanded,
Q=260-2P
Or, 0=260-2P
Or, 2P=260
Or, P=\frac{260}{2}
\therefore\P=130

#### Calculating (CS)

CS=\frac{1}{2}(BH)
Where,
B=base [(equilibrium quantity) - (zero)]
H=height [(Price at zero quantity) - (Equilibrium price)]
CS=\frac{1}{2}[(40-0)(130-110)]
=\frac{1}{2}(40×20)
=\frac{800}{2}
=400

#### Calculating (PS),

MR at Q=40,
MR=130-Q
=130-40
=90
MC at Q=0
MC=2Q+10
MC=0+10
=10

PS=\frac{1}{2}(40-0)(90-10)+(110-90)(40-0)
=\frac{1}{2}3200+800
=1600+800
=2400

Calculating DWL

P=MC
130-0.5Q=2Q+10
Or, 130-10=2Q+0.5
Or, 120=2.5Q
Or,\frac{120}{2.5}=Q
\therefore\Q=48

DWL=\frac{1}{2}[(110-90)(48-40)]
=\frac{1}{2}(20×8)
=\frac{160}{2}
=80

2-The demand and supply functions for a commodity are given as;
Qd=1100-100P
Qs=-100+50P
Suppose that a price ceiling is of Rs.4 is imposed on the commodity.
Now find out,
a) Consumers surplus (CS), Producers surplus (PS) and Total surplus (TS) before the imposition of the ceiling price.
b) Consumers surplus (CS), Producers surplus (PS) and Deadweight loss (DWL) after the imposition of the ceiling price.

Solution
a) For equilibrim,
Qd=Qs......(i)
Putting demand and supply functions in equation (i) we get,
1100-100P=-100+50P
Or, 1100+100=50P+100P
Or 1200=150P
Or, \frac{1200}{150}=P
therefore\P=8

Putting the value of P in demand function,
Qd=1100-100P
=1100-100×8

=1100-800

=300
Demand price at zero quantity,
Qd=1100-100P
100P=1100-Qd
100P=1100-0
P=\frac{1100}{100}
P=11

Supply price at zero quantity,
Qs=-100+50P
-50P=-100-Qs
50P=100+0
P=\frac{100}{50}
P=2

#### Calculating CS,

CS=\frac{1}{2}(BH)
Where,
B=300 (Equilibrium quantity - zero)
H=3 (demand price - Equilibrium price)
CS=\frac{1}{2}(300×3)
=\frac{900}{2}
=Rs.450

#### Calculating PS,

PS=\frac{1}{2}(BH)
Where,
B=300 (Equilibrium quantity-zero)
H=6  (Equilibrium price - supply price)
PS=\frac{1}{2}(300×6)
=\frac{1800}{2}
=900

#### Calculating TS

TS=CS+PS
=450+900
=1350

#### Imposition of price ceiling,

b) When a price ceiling P=Rs. 4 is imposed, then quantity supplied is,
Qs=-100+50P
=-100+50×4
=-100+200
=100
When Qs=100 then demand price is,
Qd=1100-100P
Or, 100=1100-100P
Or, 100P=1100-100
Or, P=\frac{1000}{100}
\therefore\P=10

#### Calculation of CS after the imposition of price ceiling,

CS=NQ[NDP-CP+\frac{1}{2}(DP-NDP)]
Where,
CS=? Consumer surplus,
NQ=100, New quantity,
NDP=10, New demand price at zero quantity,
CP=4, Ceiling price,
DP=11, Initial demand price at zero quantity,

CS=100[10-4+\frac{1}{2}(11-10)]
=100[6+0.5]
=100×6.5
=650

#### Calculating PS after the imposition of ceiling price,

PS=\frac{1}{2}(BH)
Where,
B=100, New quantity,
H=2, (Ceiling price) - (Initial supply price at zero quantity)
PS=\frac{1}{2}(100×2)
=\frac{200}{2}
=100

#### Calculation of DWL,

WDL=\frac{1}{2}(BH)
Where,
B=IEQ-NEQ (Initial equilibrium quantity - New equilibrium quantity)
H=NDP-CP (New demand price - Ceiling price)

DWL=\frac{1}{2}[(300-100)(10-4)]
=\frac{1}{2}(200×6)
=\frac{1200}(2)
=600

3- Suppose the demand and supply functions are given by
Qd=100-0.5P
Qs=-50+P
Find:
a) Equilibrium price and quantity,
b) Consumers surplus, Producers surplus and total surplus,
c) if the government imposes price floor at Rs. 120, then find, consumers surplus, producers surplus, total surplus and deadweight loss.

#### Solution:

a) For market equilibrium,
Qd=Qs......(i)

Putting the demand and supply functions in equation (i), we get;
100-0.5P=-50+P
Or, 100+50=P+0.5P
Or, 150=1.5P
Or, \frac{150}{1.5}=P
\therefore\P=100

Putting the value of P in demand function, we get;
Qd=100-0.5×100
=100-50
=50
Hence equilibrium price= Rs. 100 & quantity = 50 units.

#### b) Calculating consumers surplus (CS)

Now let us convert demand function into price function as;
Qd=100-0.5P
0.5P=100-Qd
P=200-2Qd
CS= Area of  triangle
CS=\frac{1}{2}(BH)
where B =base
H = height
B = (equilibrium quantity)-(Zero)
=50-0
=50
H = (price at zero quantity demanded)-(equilibrium price)
=200-100
=100
CS=\frac{1}{2}50×100
=2500

#### Calculation of producers surplus (PS)

Now let us convert supply function into price function;
Qs=-50+P
-P=-50-Qs
P=50+Qs

PS= Area of Triangle
=\frac{1}{2}(BH)
Here,
B= (equilibrium quantity)-(zero)
=50-0
=50
H= (equilibrium price)-(price at zero quantity supplied)
=100-50
=50
PS=\frac{1}{2}(50×50)
=1250

#### Calculation of total surplus (TS)

TS=CS+PS
=2500+1250
=3750

#### C) If the government imposes price floor Rs. 120, then quantity demanded is;

P=200-2Qd
Putting the floor price in the above demand function to find the equilibrium quantity we get,
120=200-2Qd
Or, 2Qd=200-120
Or, 2Qd=80
Or, Qd=\frac{80}{2}
\therefore\Qd=40

Putting the value of Qd in the supply function to find out the supply price,
P=50+Qs
P=50+40
P=90

#### Calculation of CS after imposition of floor price,

CS= Area of  triangle
CS=\frac{1}{2}(BH)
where B =base
H = height
B = (equilibrium quantity)-(Zero)
=40-0
=40
H = (price at zero quantity demanded)-(floor price)
=200-120
=80
CS=\frac{1}{2}(40×80)
=1600

#### Calculation of PS after imposition of floor price;

PS=NQ[(FP-NSP)+\frac{1}{2}(NSP-SP)]
Where;
PS=? Producers surplus,
NQ=40 New quantity,
FP=120 Floor price,
NSP=90 New supply price,
SP=50 Supply price,

Putting the values in PS function we get,
PS=40[(120-90)+\frac{1}{2}(90-50)]
=40[30+20]
=40×50
=2000

#### Calculation of TS after the imposition of price floor,

TS=CS+PS
=1600+2000
=3600

#### Calculation of DWL

DWL=\frac{1}{2}(BH)
B= (Initial quantity)-(new quantity)
=50-40
=10
H= (Floor price)-(New supply price)
=120-90
=30
DWL=\frac{1}{2}(10×30)
=150