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
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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.
D. Vannoni e M. Piacenza
Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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.
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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)
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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P
50
P = 50 – 0.1 Q
RM CMe = 500/Q +1
1 CM = 1 490 Q
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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)
Q
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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 $
D. Vannoni e M. Piacenza
Microeconomia C, A.A. 2007-2008 Esercitazione 4
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P Q = 10.4 50 P = 50 – 0.1 Q RM CMe = 500/Q +1 2.04 1 CM = 1 479.6 Q
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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
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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
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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Q2
Q1
Reaction function firm 1
500
250
Reaction functionfirm 2
200
400
100
200
Cournot Equilibrium
D. Vannoni e M. Piacenza
Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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
D. Vannoni e M. Piacenza
Microeconomia C, A.A. 2007-2008 Esercitazione 4
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Q2
Q1
Reaction functionFirm 1
500
250
Reaction functionFirm 2
200
400
100
200
Cournot Equilibrium
Stackelberg Equilibrium
175
150
D. Vannoni e M. Piacenza
Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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?
D. Vannoni e M. Piacenza
Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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.
D. Vannoni e M. Piacenza
Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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.
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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P2 Equilibrio con scelte simultanee funzione di reazione impresa 1 funzione di reazione impresa 2 20 10 20 P1
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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?
D. Vannoni e M. Piacenza
Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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)
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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
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Microeconomia C, A.A. 2007-2008 Esercitazione 4
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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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