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Anghela Arrieta Guinoza
Industrial Organization-EUS
PROBLEM SET 3
Two firms producing tablets are competing producing a very similar product but each
attracts consumers with different preferences. Some prefer tablets pads, the other prefer
tablets nexis.
We assume that the only product differentiation between the two tablets is horizontal and
that firms are competing a la Bertrand in prices.
The residual demand for each firm variety is the following:
and.
We assume that production of tablets has constant and equal marginal and average costs
(150 euros) as follows: ( and ).
[Warning: use the point to delimit the separation between units and decimal].
When these two firms compete according to Bertrand model with product differentiation,
calculate:
The demand functions (inverse) allow us to extract the demand of each firm :
P 1 =1000-2(Q 1 +1/4Q 2 )
P 1 =1000-2Q 1 -1/2Q 2
2Q 1 =1000-1/2Q 2 -P 1
Q 1 =500-1/4Q 2 - 0.5P 1
P 2 =1000-2(Q 2 +1/4Q 1 )
P 2 =1000-2Q 2 -1/2Q 1
2Q 2 =1000-1/2Q 1 -P 2
Q 1 =500-1/4Q 1 - 0.5P 2
First of all we need to take into account that a horizontal differentiation have certain
characteristics. Some consumers prefer one product variety, and others want another.
The differentiation is by consumers preferences (in localization).
In a Bertrand Model with product differentiation each firm has a different inverse
demand. Also MC and AV are constant and symmetric and the profits for each firm are:
i ( P 1 , P 2 ) pi c Q i P 1 , P 2
Each firm maximize its profits function independently in function of his control
variable, his price. For example the firm 1 will maximize its profits: (C.O.P) Max π1 (P 1 , P 2 ) = (P 1 -150) Q 1 (P 1 , P 2 )
- Q 1 =500 – 1/4(500-1/4Q 1 -0.5P 2 )-0.5P - Q 1 =500-125+1/16Q 1 +1/8P 2 -0.5P - Q 1 -1/16Q 1 =375+1/8P 2 -0.5P - 0.9375Q 1 =375+1/8P 2 -0.5P - Q 1 =400+2/15P 2 - 8/15P
- Q 2 =500 – 1/4(500-1/4Q 2 -0.5P 1 )-0.5P - Q 2 =500-125+1/16Q 2 +1/8P 1 -0.5P - Q 2 -1/16Q 2 =375+1/8P 1 -0.5P - 0.9375Q 2 =375+1/8P 1 -0.5P - Q 2 =400+2/15P 1 - 8/15P
- ∂ π1/ P 1 = 1(400+2/15P 2 -8/15P 1 ) + (P 1 -150)-8/15 = Max π1 (P 1 , P 2 ) = (P 1 -150)*(400+2/15P 2 -8/15P 1 ) - 400+2/15P 2 -8/15P 1 + (-8/15P 1 +80) = - -16/15P 1 =-400-2/15P 2 - - -16/15P 1 =-480-2/15P - 16/15P 1 =480-2/15P - P 1 =450+1/8P
The equilibrium quantity for the firm 2 will be the same as quantity for the firm 1 (because they have same MC and symmetric prices). Q 2 =194.
3. The total quantity of tablets sold in this equilibrium.
QTotal = Q1+Q2 = 194.29+194.29 = 388.
4. The profits of each firm in this equilibrium.
π1 (P1=514.28, P2=514.28) = (P 1 -150)Q 1 π1 (P1=514.28, P2=514.28) = (514.28- 150 )194.29 = 70777.07 PROFIT FIRM 1
π2 (P1=514.28, P2=514.28) = (P 2 -150)Q 2 π2 (P1=514.28, P2=514.28) = (514.28- 150 )194.29 = 70777.07PROFIT FIRM 2
GRAPHIC:
P 1
P 2 P1*=f (P2)
P2*=f (P1)
P1e=514.
Q 1 e=514.
P
Q
MC
MR RD
Question 2:
A country decides to liberalize entry into the broadband service market by allowing firms to enter using their
own networks or leased networks. In the country there is enough room for at most 10 firms competing to offer
their services using the available copper and cable lines.
We assume that this is a market with product differentiation where firms compete in quantities a la Cournot.
Each entering firm offers a single variety of a subscription contract with a fixed monthly price for unlimited
data transmission services.
The willingness to pay in euros per year for each variety of services that we name ( ) is a function that
depends on the production of each firm ( ) in thousands of subscribers, the number of varieties in the
market ( ), and the production of each competitors that enter in the market and that we name ( )
in millions of subscribers:
All the companies, both old and new, have to support monthly fixed costs of financing the infrastructure
without alternative uses (fixed and sunk costs) and monthly marginal cost of operating the service, as we see
in the following symmetric cost function:.
[Warning: use the point to delimit the separation between units and decimal].
If firms compete a la Cournot with product differentiation, calculate:
After using the excel template, the results are the following:
5. The optimum quantity of subscribers (in thousands) that each firm should attend to maximize the social welfare.
Now we will have to look for the point where the social welfare is maximized. This point is where there are only 2 firms in the market.
Optimum quantity of subscribers for each firm= 206.
6. The total optimum quantity of subscribers (in thousands).
Total quantity of subscribers = 412.12 (2 firms with 206.06 subscribers each one).
7. The optimum price of the annual subscription.
Optimum price = 459.
8. The number of firms in the social optimum.
As I’ve mentioned before, 2 firms maximize social welfare.
Optimal number of firms = 2
9. The price of annual subscription that maximize the consumer’s surplus.
The point where the consumer surplus is maximized is where there are 2 firms in the market.
Price that maximize the consumer surplus = 459.
10. Number of firms that maximize the consumer’s surplus.
The point where the consumer surplus is maximized is where there are 2 firms in the market.
Number of firms that maximize the consumer’s surplus = 2
GRAPHIC:
a=
a-bөQ
P1 (Q1)=(a-bөQ2)-bQ 1
P1 (Q1)=a-bQ 1
MC
Pcoure=177.
P1(Q1,Q2)=a-b(Q1+өQ2)
In this case:
a= b=1. c= f= ө=0.
