Question

Suppose a profit-maximizing monopolist has total cost and marginal cost as follow. TC =8Q + 10 and MC = 8. It faces the demand curve P=20-1/5Q. 

1. What is the equilibrium price and output? 

2. What is the total profit? 

3. Calculate the consumer surplus, producer surplus, and deadweight loss if the firm acts as a monopolist. Illustrate your answer with a diagram. 

4. Calculate the consumer surplus, producer surplus, and deadweight loss if the firm acts as a perfectly price-discriminated monopolist. Illustrate your answer with a diagram.

Answer 1

P	Q	TC	MC	TR	MR	Profit
20	0	10	8	0		-10
19	5	50	8	95	19	45
18	10	90	8	180	17	90
17	15	130	8	255	15	125
16	20	170	8	320	13	150
15	25	210	8	375	11	165
14	30	250	8	420	9	170
13	35	290	8	455	7	165
12	40	330	8	480	5	150
11	45	370	8	495	3	125
10	50	410	8	500	1	90
9	55	450	8	495	-1	45
8	60	490	8	480	-3	-10
7	65	530	8	455	-5	-75
6	70	570	8	420	-7	-150
5	75	610	8	375	-9	-235
4	80	650	8	320	-11	-330
3	85	690	8	255	-13	-435
2	90	730	8	180	-15	-550
1	95	770	8	95	-17	-675
0	100	810	8	0	-19	-810

a) Equilibrium price is 14 and quantity is 30

b) Total profit is 170

c) PS = 30*(14-8) = 30*6=180
CS=0.5*30*(20-14)=0.5*30*6 = 90
DWL=0.5*(60-30)*(20-14) = 0.5*30*6=90

d) CS=0
PS=180+90+90=360
DWL=0

Answer 2

Answer 3

Solution:

For the monopolist, TC = 8Q + 10, thus, MC = 8

Inverse demand curve: P = 20 - 0.2*Q

a) Profit maximizing condition for the monopolist is where marginal cost equals the marginal revenue.

Finding marginal revenue:

Total revenue = price*quantity = P*Q

TR = (20 - 0.2Q)*Q = 20Q - 0.2*Q²

Marginal revenue, MR = = 20 - 0.2*2*Q = 20 - 0.4*Q

With the profit maximizing condition of MC = MR, we have

8 = 20 - 0.4*Q*

0.4*Q* = 20 - 8

Q* = 12/0.4 = 30 units

Equilibrium price, P* = 20 - 0.2*30 = $14

So, equilibrium price = $14, equilibrium quantity = 30 units

b) Total profit = TR - TC

TR = 20*30 - 0.2(30)² = 600 - 180 = $420

TC = 8*30 + 10 = $250

Profit = 420 - 250 = $170

c) Consumer surplus (area of a triangle) = (1/2)*(equilibrium quantity)*(maximum willingness to pay at 0 quantity - equilibrium price)

maximum willingness to pay at 0 quantity (or the vertical intercept), P(Q=0) = 20 - 0.2*0 = 20

CS = (1/2)*(30)*(20 - 14) = $90

Producer surplus = Profits + fixed cost = 170 + 10 = $180

So, total welfare under monopoly = CS + PS = 90 + 180 = $270

Deadweight loss = total maximal welfare (possible under perfect competition) - total welfare under monopoly

Total welfare under perfect competition: Under perfect competition, marginal cost pricing is followed, that is the equilibrium price charged is same as the marginal cost.

So, equilibrium price = MC = $8, then finding Q

8 = 20 - 0.2*Q

Q = 12/0.2 = 60

So, producer surplus = (8*60 - 8*60 - 10) + 10 = 0

Consumer surplus = (1/2)*(60)*(20 - 8) = $360

So, total welfare now = CS + PS = 360 + 0 = $360

Then, deadweight loss = 360 - 270 = $90

Following is the required diagram:

d) Under a perfectly price-discriminated monopolist, the monopolist charges each consumer his/her maximum willingness to pay. The monopolist charges such price, until the marginal cost is reached, beyond which he doesn't produce any more quantity and charges no lower price, as otherwise the loss will be incurred.

Thus, by following perfect price discrimination, no deadweight loss is generated, and entire consumer surplus is converted ito producer surplus. So, now

Consumer surplus = 0 (since, each consumer is charged their maximum willingness to pay. and thus they pay the highest price they can pay, and are left with no gain or surplus).

Producer surplus = maximum welfare possible (as that under the perfect competition) = area of the big triangle = (1/2)*(Q (where P = MC))*(Maximum price (vertical intercept) - equilibrium price (=Marginal cost))

PS = (1/2)*(60)*(20 - 8) = $360

Total welfare under perfect price discrimination = 0 + 360 = $360

Deadweight loss = Maximal welfare - total welfare under perfect price discrimination

DWL = 360 - 360 = $0

Intuition behind DWL = 0 in this case: Since, monopolist follow such pricing strategy till price fall as low as marginal cost, and complete efficiency is reached, distribution is not fair though, as consumer lose their entire surplus. So, total welfare as that under perfect competition is attained, only difference being that entire consumer surplus is converted into producer surplus.

Refer the below diagram for this:

Answer 4

Ans 3.

(a) Total revenue = Price × Quantity

= (20 - (1/5)Q) × Q

Total revenue = 20Q - (1/5)Q^2

Marginal revenue = 20 - 2/5Q

Monopoly operares where,

MR = MC

20 - 2/5Q = 8

12 = 2/5Q

Q = 5/2(12)

Q = 30 Units

Price = 20 - (1/5)30

Price = 14 units

(b) Total Profit = Total revenue - Total cost

= (20(30) - 1/5(30)^2) - (8(30) + 10)

= 600 - 180 - 240 - 10

= 170

Profit = 170 units

(c) To find the dead weight loss, we need to find the Perfect competition equilibrium output and price

Perfect competition occurs where,   

Marginal cost = Price

8 = 20 - (1/5)Q

1/5Q = 20 - 8

1/5Q = 12

Q = 60 units

Price = MC = 8 units

Below is the diagram which would illustrate the consumer , producer surplus, deadweight loss

Consumer surplus = Blue color region

= 1/2 (20-14) × 30

Consumer surplus = 90 units

Producer surplus = pink color region

Producer surplus =  (14-8) × 30 = 180 units

Deadweight loss = yellow color region

= 1/2(14-8) × (60-30)

Deadweight loss = 1/2(6)×(30)

Deadweight loss = 90 units

(d) In case of perfectly price discriminating nonopolist , all of the consumer surplus is extracted and goes to the producer.The producer charges the maximum price that a consumer is willing to pay for each unit of output.

The outcome is same as that of perfect competition ,i.e.

Marginal cost = Price

There is no consumer surplus as all the consuner surplus goes to the producer.There is not even the deadweight loss as the producer keeps selling the output until the price that a consumer is willing to pay equals the marginal cost

The below diagram would illustrate the same

The yellow color region depicts the Producer surplus and it equals to the area:

1/2*(20-8)*60

= 1/2(12)*(60)

Producer surplus = 360 units.

  

  

  

  

Answer 5

TC= 8Q+10

MC=8

P=20-1/5Q

a) P=MC

8=20-1/5Q

1/5Q=12

Q=60 (Equilibrium quantity)

P= 8(Equilibrium price)

b)Total profit= Total Revenue -Total Cost

                   = 8*60 - 8*60+10

                   =480-490

                   = -10

LOSS OF 10

C)