assignment seven
the oligopoly model (II): competition in prices
the federal funds market ………….1
sustainable cartels ………….4
spring
2016
microeconomi
the analytics of
cs
constrained optimal
microeconomics
assignment 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
► Demands are different from what we studied so far:
Q1 = 96 – 2P1 – 2P2
Q2 = 96 – 2P2 – 2P1
The difference is now that the two products are no longer
substitutes but rather complements:
● the two products have to be used together
● this implies that an increase in price of one product will
decrease the demand for both products
mergers of complements
► However the derivation of the reaction functions follows
the same “procedure”:
● write the profit functions for each firm
● maximize each profit function with respect to own price
for each firm
● this will give you the reaction function
● in fact we can use directly the previous result since now
the demands are
Q1 = a1 – b1P1 – d1P2
Q2 = a2 – b2P2 – d2P1
● there are no switchers in this case (the market size for
both products goes up/down simultaneously
● which implies that the reaction functions have the same
form as before but:
● we replace d1 with – d1 and d2 with – d2
P1 = 0.5∙a1/b1 + 0.5∙MC1 – 0.5∙d1∙P2/b1
P2 = 0.5∙a2/b2 + 0.5∙MC2 – 0.5∙d2∙P1/b2
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assignment 7
page | 1
microeconomics
assignment 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
mergers of complements
P2
54
► Using the general reaction functions
P1 = 0.5∙a1/b1 + 0.5∙MC1 – 0.5∙d1∙P2/b1
reaction function
firm 1
P2 = 0.5∙a2/b2 + 0.5∙MC2 – 0.5∙d2∙P1/b2
we get for our demand functions:
27
P1 = 0.5∙96/2 + 0.5∙6 – 0.5∙2∙P2/2
18
P2 = 0.5∙96/2 + 0.5∙6 – 0.5∙2∙P1/2
reaction function
firm 2
18
27
P1 = 27 – P2/2
P2 = 27 – P1/2
54 P1
► When the products are complements the
reaction function for the Bertrand model are
downward slopping.
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that is
► This is a system of two equations with two unknowns.
Since the system is symmetric we’ll solve for P1 = P2:
P = 27 – P/2 with P1 = P2 = 18
assignment 7
page | 2
microeconomics
assignment 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
mergers of complements
► Using the general reaction functions
Q1 = a1 – b1P1 – d1P2
Q2 = a2 – b2P2 – d2P1
we get the cross elasticities as:
firm 1: e1,2 = (∆Q1/∆P2)∙P2/Q1 = – d1∙P2/Q1
firm 2: e2,1 = (∆Q2/∆P1)∙P1/Q2 = – d2∙P1/Q2
For prices P1 = 20 and P2 = 10 we get:
Q1 = 96 – 2P1 – 2P2 = 96 – 2∙20 – 2∙10 = 36
Q2 = 96 – 2P2 – 2P1 = 96 – 2∙10 – 2∙20 = 36
For elasticities we get
firm 1: e1,2 = – d1∙P2/Q1 = – 2∙10/36 = – 0.55
firm 2: e2,1 = – d2∙P1/Q2 = – 2∙20/36 = – 1.11
 2016 Kellogg School of Management
assignment 7
page | 3
microeconomics
assignment 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
mergers of complements
changes in equilibrium (Bertrand - complements)
P2
► General reaction functions
P1 = 0.5∙a1/b1 + 0.5∙MC1 – 0.5∙d1∙P2/b1
P2 = 0.5∙a2/b2 + 0.5∙MC2 – 0.5∙d2∙P1/b2
54
If the marginal cost for Firm 2 decreases from the reaction function
we see immediately that this implies:
- price P2 decreases for each price P1
reaction function
firm 1
27
(0)
- reaction function of Firm 2 shifts left
As a result the price for Firm 2 decreases and price for Firm 1
increases.
reaction function
firm 2
(1)
27
54 P1
merger analysis
► The profit function of the resulting company is simply the sum of the individual profits:
 = 1 + 2 = (P1Q1) + (P2Q2) – TC(Q1,Q2) =
= P1(96 – 2P1 – 2P2) + P2(96 – 2P2 – 2P1) – [100 – 6(96 – 2P1 – 2P2) – 6(96 – 2P2 – 2P1)] =
= – 2(P1 + P2 – 30)2 + constant
► The above profit is maximized for any pair of prices (P1,P2) such that P1 + P2 = 30. Symmetric solution P1 = P2 = 15.
 2016 Kellogg School of Management
assignment 7
page | 4
microeconomics
assignment 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
limited capacities
unconstrained solution
► The “negotiation” process described in the problem, in which both parts can revisit their offers until agreement is reached, points
to a standard (simultaneous) Cournot solution:
 contractor one
residual demand
marginal revenue
profit maximization
reaction function
P1 = (6,300 – L2) – L1
MR1 = (6,300 – L2) – 2L1
MR1 = MC1 gives (6,300 – L2) – 2L1 = 6,000
L1 = 150 – L2/2
 contractor two
residual demand
marginal revenue
profit maximization
reaction function
P2 = (6,300 – L1) – L2
MR2 = (6,300 – L1) – 2L2
MR2 = MC2 gives (6,300 – L1) – 2L2 = 6,000
L2 = 150 – L1/2
 Cournot equations
L1 = 150 – L2/2
L2 = 150 – L1/2
L1
► Thus both contractors will settle for 100 levels (L1 = L2 = 100).
This is shown in the diagram as the intersection of the two reaction
functions.
300 reaction function
contractor 1
150
Cournot
Solution
100
reaction function
contractor 2
100 150
 2016 Kellogg School of Management
assignment 7
300
L2
page | 5
microeconomics
assignment 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
limited capacities
constrained solution
► The constraint solution is based on the same reactions functions with additional restrictions on the maximum allowed number of
levels:
 Cournot equations
L1 = 150 – L2/2 OR L1  50
L2 = 150 – L1/2 OR L2  50
► Thus both contractors will settle for 50 levels (L1 = L2 = 50).
This is shown in the diagram as the intersection of the two
“constrained” reaction functions.
L1
300
constrained
reaction function
contractor 2
constrained
Cournot
Solution
150
constrained
reaction function
contractor 1
50
50
 2016 Kellogg School of Management
assignment 7
150
300 L2
page | 6