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,

c) Deadweight loss,

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`