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Arrieta Guinoza, Anghela
Industrial Organization /Group EUS
PROBLEM SET 1
PROBLEM 1: A set of symmetric firms has the following total cost function:
,
Where is the quantity produced by each individual firm.
The inverse demand function is the following,
,
Where is the total quantity produced in the market, the sum of all quantities produced by all entrants.
(1) The quantity of the minimum efficient scale.
MCi (qi) = AVCi (qi) 30000-600q+3q^2 = 30000-300q+q^2 3q^2 -q^2 = -300q+600q 2q^2 -300q = 0
(2) The lowest price that any firm trying to maximize profits and avoid losses can offer.
MCi (150) = AVCi (150) 30000-600(150) +3(150)^2 = 30000-300(150) + (150)^2 7500 = 7500
MCi ( qi) = ∂TC/∂q = 30000 - 600q+3q^2 AVCi (qi) = TC/q = 30000-300q+q^2
qMES = 150
Lowest Price that any firm can offer 7500
150
7500
MCi (qi) AVCi (qi)
P
Q qEME
(3) The equilibrium price in perfect competition.
To maximize profits, the price should be equal to the marginal cost.
P (Q) = MC (Q) Pe (Q) = 30000-600q+3q^2 Pe (q) = 30000-600(150) +3(150)^2
(4) The total quantity traded in the perfect competition equilibrium.
SLR = DEMAND
7500=10000-0.1Q
0.1Q = 10000-
(5) The profits of each and every firm in the perfect competition equilibrium.
In perfect competition (in the long run), there are no profits.
PROFITS = TR-TC PROFITS= pq – 30000q-300q^2 +q^3 PROFITS= (7500150) – (30000150-300150^2 +150^3 )
Pe (Q) = 7500
Qe=
Reservation Price: (When Q=0) P (Q) =10000-0.1Q P= 10000
SLR (AVC=MC)
Qe=
Pe=75 00
PROFITS= 0
P
Q
(9) The producer surplus in perfect competition equilibrium.
PS = peqe-MCq PS = 0
The supply curve is flat; this means that the entire surplus is in the consumer side and not on the producer side. That’s why in this case producer surplus is equal to 0.
(10) Welfare in perfect competition equilibrium.
W = CS+PS W = 31.250.000+
(11) The consumer surplus in the monopoly equilibrium.
CS= (10000-Pemon)/ CS=12500(10000-8750)/
(12) The producer surplus in the monopoly equilibrium.
PS = (Pmon-MC)Qmon PS = (8750-12500)
(13) Welfare in the monopoly equilibrium.
W=CS+PS W= 7.812.500+15.625.
W= 31.250.
CS=7.812.
MR D
MC
Pmon = 8750
Ppc = 750 0
Qmon = 12 500 Qpc = 2^5000
P
Q
PS=15.625.
W=23.437.
(14) Net welfare loosed caused by monopolization.
Total Welfare in perfect competition model = 31.250. Total Welfare in monopoly = 23.437. So, there is a welfare loosed of 7.812.500 caused by monopolization.
PROBLEM 2: The monopoly operating firm providing the sanitary water for a city has the following total cost function.
TC (Q) =0.05+0.06Q
Where Q is the litres of water provided per household and per hour
The inverse demand function is the following
P (Q) =0.12-0.01Q
It shows the Euros that each household is willing to pay per hour of water provided by the monopoly water operating firm.
Compute the following results in equilibrium in the case of a free monopoly, and the optimum in the case of efficiently regulated monopoly:
(1) The quantity of litres of water per household /hour provided by the monopoly operating water firm in the case of no regulation.
Max profits = TR(q)-TC(q) First order condition: ∂TR (q)/ ∂q = ∂TC (q)/ ∂q MR=MC
TR=P(q)q TR= (0.12-0.01q)q TR=0.12q-0.01q^2
TC = 0.05+0.06Q
MR=MC
0.12-0.02q = 0. 0.12-0.06=0.02q
MR = 0.12-0.02q
MC = 0.
qemon= 3
(6) The profits per household/hour of service that would obtain the monopoly operating firm in the case of first-best efficient regulation.
Profits=TR-TC Profits=pq-(0.05+0.06q) Profits=0.066-(0.05+0.06(6))
(7) The quantity of litres of water per household/hour provided by the monopoly operating water firm in the case of second-best efficient regulation.
Second-best efficient regulation = maximize welfare without losses
MU(q)=AVC(q) 0.12-0.01q= (0.05+0.06q)/q 0.12q-0.01q^2 =0.05+0.06q -0.01q^2 +0.06q-0.05=
(8) The price per litre provided that the monopoly water operating firm will offer in the case of second-best efficient regulation.
P* 2 =0.12-0.01q P* 2 =0.12-0.01(5)
(9) The profits per household/hour of service that would obtain the monopoly operating firm in the case of second-best efficient regulation.
Profits=TR-TC Profits=pq-(0.05+0.06q) Profits=0.075-(0.05+0.06(5))
Profits=-0.05 € per household / hour (LOSSES)
q* 2 1
P* 2 =0.07 €
Profits=- 0 € per household / hour
AVC
D
Q
P
P* 2 =0.07 €
P* 1 =0.06 €
q* 2 = 1 q* 2 = 5 q* 1 = 6
