microeconomics ev prof. davide vannoni. exercise session 4 monopoly and deadweight loss 1.exercise...

MICROECONOMICSEV

Prof. Davide Vannoni

Exercise session 4

1. Exercise on monopoly and deadweight lossmonopoly and deadweight loss

2. Exercise on natural monopoly

3. Exercise on oligopoly: simultaneous quantity quantity choiceschoices (Cournot) and sequential (Stackelberg)

4. Oligopoly: collusion and price warcollusion and price war

5. Oligopoly: price choicesprice choices simultaneous and sequential

D. Vannoni e M. Piacenza

Microeconomia C, A.A. 2007-2008 Esercitazione 4

3

Exercise n. 1

A monopoly maximizes the profit and bears a linear and increasing marginal cost. He produces 20 units, in correspondence of which the marginal cost (increasing) is equal to 40.

The demand function (linear) crosses the marginal costs in correspondence with a price equal to 60 and with a quantity equal to 30 units.

Determine:

- The inverse demand function and the marginal cost function (both represented by lines);

- The welfare deadweight loss.



4

Forma Generale funzione di Domanda (inversa)

: p = a – bq

Forma Generale Ricavo Marginale

: RM = a –2bq

Condizione equilibrio : RM = CM

analogo) (sistema 2

2120

120;2;1020

3060

4040

3060

20240

qCM

qp

abb

ba

ba

ba

ba



5

0 20 40

Demand

MC

MR

30 60 80 100 120

120

100

80

60

40

20

Deadweight loss:Deadweight loss:

20021

20304080.S.P

60p

80202120p

C

M



6

Exercise n.2

In lake Wobegon, Minnesota, live 10 families; for each family the demand for electricity is Qi = 50 – P. The production cost of electricity of Lake Wobegon Electric’s (LWE) is TC = 500 + Q.

a) If the regulatory authority of LWE wants to guarantee that in this market there is not a deadweight loss, which price would it impose to LWE? Which will be the production of electricity? Compute the consumer's surplus and the profit of LWE at such a price.



7

The market demand for electricity is the sum of the quantity demanded by the households.

Graphically, we sum horizontally the demand for each family . Algebraically, the total demand is:

10

1

10(50 ) 500 10 50 0.1M ii

Q Q P P P Q



8

In order to avoid a deadweight loss, the regulatory authority will impose to LWE a price equal to the marginal cost. Since the total cost curve is TC = 500 + Q, the marginal cost is:

MC = 1$

Equating price and MC and solving for quantity we get the quantity:

P = 50 – 0.1Q = CM = 1 Q = 490

The profit ofThe profit of LWE in correspondence of such a price is:

Π = TR – TC = 1∙490 – (500 + 490) = - 500$ (green area)

The consumer's surplus is:

CS = 0.5 ∙ (50 – 1) ∙ 490 = 12005 $ (equal to 1200.5 $ per household)



9

P

50

P = 50 – 0.1 Q

RM CMe = 500/Q +1

1 CM = 1 490 Q



10

b) If the regulatory authority wants to avoid that LWE is not losing money (-500$), what will be the lowest price to be charged? Find the production level, the consumer's surplus and the profit at this price. Is there a deadweight loss?

The lowest price to be imposed without implying losses for LWE is equal to the average cost of productionaverage cost of production:

AC = TC/Q = (500/Q) + 1

Imposing P = AC and solving for Q we get the production level:

P = 50 – 0.1Q = AC = (500/Q) + 1



11

Solving for Q we get the quadratic equation:

0.1Q2 – 49Q + 500 = 0

2 4

2

b b acQ

a

In our case:

2 10.449 49 (4)(0.1)(500)

479.62(0.1)

QQ

Q



12

Notice that:

a) If Q = 10.4 MR >MC

therefore the firm will increase profits if quantity increases;

b) In correspondence of the second solution, since the quantity (Q = 479.6) is lower than the quantity for which P = MC (Q = 490), the price will be higher (2.04) and the consumer's surplus lower.

The consumer's surplus is:

CS = 0.5 ∙ (50 – 2.04) ∙ 479.6 = 11500 $

Finally, the deadweight loss (in violet color) is given by:

P.S. = (2.04 - 1) ∙ (490 – 479.6) ∙ 0.5 = 5.4 $



13

P Q = 10.4 50 P = 50 – 0.1 Q RM CMe = 500/Q +1 2.04 1 CM = 1 479.6 Q



14

Exercise n. 3

Two firms manufacture the same product. Firm 1 has a marginal cost of 1 $ and firm 2 has a marginal cost of 2 $. Fixed costs are zero for both firms. The inverse demand is P = 6 - 0.01Q (Q = q1 + q2)

a) Compute the reaction functions and find the Cournot Cournot equilibriumequilibrium

b) Assume that firm 2 2 is a StackelbergStackelberg leader and find the new equilibrium



15

a) Profits of firm 1 and 2 are:

1 = (6 - 0.01·q1 - 0.01·q2)·q1 - q1

2 = (6 - 0.01·q2 - 0.01·q1)·q2 - 2·q2

Maximizing profits:

d/dq1 = ’1 = 6 - 0.02·q1 - 0.01·q2 - 1 = 0

from which qq1 1 = 250 - ½ q= 250 - ½ q22 (inverting: q2 = 500 - 2q1)

d/dq2 = ’2= 6 - 0.02·q2 - 0.01·q1 - 2 = 0

from which qq2 2 = 200 - ½ q= 200 - ½ q11



16

The equilibrium point is where the reaction functions intersect:

500 - 2q1= 200 - ½ q1

q1 = 300/1.5 = 200; q2 = 500 - 2·200 = 100

p = 6 - 2 -1 = 3

1 = 600 - 200 = 400 and 2 = 300 – 200 = 100



17

Q2

Q1

Reaction function firm 1

500

250

Reaction functionfirm 2

200

400

100

200

Cournot Equilibrium



18

b) In the Stackelberg model, firm 2 (leader)Stackelberg model, firm 2 (leader) includes in its profit function the reaction function of firm 1:

2 = (6 - 0.01·q2 - 0.01·qq11)·q2 - 2·q2 = = [6 - 0.01·q2 - 0.01·(250 - ½ q250 - ½ q22)]·q2 - 2·q2

’2 = 6 - 0.01·q2 - 2.5 - 2 = 0

from which q2 = 1.5/0.01 = 150 and q1 = 250 – 75 = 175

p = 6 - 1.5 - 1.75 = 2.75

1 = (2.75 - 1)·175 = 306 and 2 = (2.75 - 2)·150 = 112.5

It exists a so called first mover advantage



19

Q2

Q1

Reaction functionFirm 1

500

250

Reaction functionFirm 2

200

400

100

200

Cournot Equilibrium

Stackelberg Equilibrium

175

150



20

Exercise n. 4

In an oligopoly there are 4 firms manufacturing an homogenous good with the following marginal costs c1=10, c2=10, c3=12, c4=11. The firms start to collude and, after a recession, adopt price strategies à la Bertrand.

a) Which will be the equilibrium prevailing after the price war?

b) Is it possible to find equilibrium quantities and prices?

c) If firm 1 and firm 2 are colluding, which will be the market price?



21

a) At the end of the price war, only firm 1 and 2 will remain (lowest marginal costs);

a) The equilibrium price will be equal to 10, while the quantity in undetermined. A usual assumption is to divide it equally among firms.

a) A collusive agreement will allow firm 1 and 2 to fix a price equal (or slightly lower than) 11, i.e. firm 4's marginal cost.



22

Exercise n. 5

Two firms compete using price as strategic variable. The demand functions are:

Q1 = 20 – P1 + P2

Q2 = 20 + P1 – P2

Marginal costs are equal to zero.



23

a) Suppose that the two firms are simultaneously chosing the price. Find the Nash equilibrium. Which will be the price, quantities and profits?

In order to find the Nash equilibrium we compute for each firm the reaction function with respect to the price of the other firm, then we solve the system of two equations with two unknown variables.

With marginal costs equal to zero, the profits are:

1 = P1Q1 = P1(20 - P1 + P2) = 20P1 - P12 + P2P1

’1 = MR1 = 20 - 2P1 + P2



24

Al the profit maximizing price MR1 = 0, from which we get the reaction function of firm 1:

P1 = (20 + P2)/2

Symmetrically for firm 2:2: P2 = (20 + P1)/2.Substituting one reaction function into the other:

P1 = [20 + (20 + P1)/2]/2 = 15 + P1/4 P1 = 20 $

Due to symmetry, P2 = 20 $

Q1 = 20 , Q2 = 20 and 1 = 2 = 400



25

P2 Equilibrio con scelte simultanee funzione di reazione impresa 1 funzione di reazione impresa 2 20 10 20 P1



26

If firm 1 chooses the price before firm 2, it can take into account the reaction function of firm 2. The profit of firm 1 is:

1 = P1Q1 = P1[20 - P1 + (20 + P1)/2]

To determine the price which maximizes the profit:

1’ = 20 - 2P1 + 10 + P1

Imposing 1’= 0 P1 = 30 $

b) Suppose now that firm 1 chooses as first the price, followed by firm 2. Which will the the equilibrium prices, quantities and profits?



27

Substituting in the reaction function of firm 2, we get P2 = (20 + 30)/2 = 25 $.

In correspondence of such prices we get:

Q1 = 20 – 30 + 25 = 15Q2 = 20 + 30 - 25 = 25

Profits are therefore:1 = 30·15 = 450 $ 2 = 25·25 = 625 $

If firm 1 fixes its price acting as the first mover, firm 2 can exploit this and gain more profits (second mover advantage)



28

P2 Equilibrio con scelte simultanee funzione di reazione impresa 1 funzione di reazione impresa 2 25 20 Equilibrio con impresa 1 leader 10 20 30 P1



29

c) In this example of price competition with differentiated products, which situation will you prefer? (i) two firms choosing simultaneously the price; (ii) you are the first choosing the price; (iii) your competitor is the first mover choosing the price

Looking at the profits attainable in the different cases, the ranking should be (iii), followed by (ii) and (i).

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