Transcript Oligopoly

MicroEconomics
Oligopoly
Students:
Ana Oliveira
Fernando Vendas
Miguel Carvalho
Paulo Lopes
Vanessa Figueiredo
Class Assigment – Microeconomics – “Oligopoly”
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Presentation Structure
• Introduction
• Competition Model
– Sequential Game
P. Lopes and F. Vendas
• Quantity Leadership
• Price Leadership
– Simultaneous Game
• Simultaneous Price setting
• Simultaneous Quantity Price setting
• Collude (corporate game)
• Resume
• Exercise
V. Figueiredo
M. Carvalho
A. Oliveira and F. Vendas
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Initial Framework
Market Structure
Pure competition
Small competitors
Pure Monopoly
One Large Firm
However
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Class Framework
Monopolistic Competition
Form : OLIGOPOLY
“ Strategic interaction that arise in an industry with small
number of firms.” – Varian, H. (1999 , 5th)
Many Different Behavior
Patterns of Behavior
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Study Framework
Restrict to the case of 2 firms
Duopoly
Simple to understand
Strategic interaction
Homogeneous
product
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Sequential Game
Quantity Leadership
In this case, one firm makes a choice before the other firm,
according Stackelberg model, thus, our study will start from this
model. Suppose, firm 1 (leader) and it chooses to produce a quantity
(y1) and firm 2 (follower) responds by choosing a quantity (y2). Each
firms knows that equilibrium price in the market depends on the total
output. So we use the inverse demand function p(Y) to indicate that
equilibrium, as function of industry output.
Y = y1 + y2
The leader has to consider the follower´s profit-maximization
problem, then we should think : What output should the leader
choose to max its profits ?
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Sequential Game
The follower's Problem
Assume that the follower wants to maximize its profits
max p(y1+y2)y2 – c2(y2)
y2
The follower's profit depends on the output choice of the leader, but
the leader´s output is predetermined and the follower simply views it
as a constant.The follower wants to choose an output level such that
marginal revenue (MR) equals marginal cost :
MR2 = p(y1+y2) +
∆p
∆y2
y2= MC2
When the follower increases its output, it increase its revenue by
selling more output at the market price, but it also pushes the price
down by ∆p, and this lowers its profits on all the units that were
previously sold at the higher price.
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Sequential Game
The follower's Problem
The profit max choice of the follower will depend on the choice made
by leader – the relationship is given by :
y2 = f2 (y1) - Reaction function
(Profit output of the follower as a function of the leader´s choice.)
How follower will react to the leaders choice of output
p(y1+y2) = a – b (y1+y2)
(consider cost (C) equal to 0)
So the profit function to firm 2 (follower) is :
∏2 (y1+y2) = ay2 – b y1y2 – by22
So, we use this form to draw the isoprofit lines (Fig.1)
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Sequential Game
The follower's Problem
Fig.1 - The isoprofit lines graffic
Monopolistics
This reaction curve gives the profit-maximizing output for the follower
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Sequential Game
There are lines depicting those combination of y1 and y2 that yield a
constant level of profit to firm 2. Isoprofit lines are comprised of all
points which satisfy equations
ay2 – b y1y2 – by22 = ∏2
Firm 2 will increase profits as we move to Isoprofit lines that are
further to the left. Firm 2 will make max possible profits when it's a
monopolist, thus, when firm 1 chooses to produce zero units of
output, as illustrated in fig 1.
This point will satisfy the usual sort of tangency condition (RF). To
understand it , we use :
MR2(y1,y2)= a – by1 – 2by2
(MR=MC ; MC=0)
So, we have reaction curve of firm 2
y2 = a-by1
2b
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Sequential Game
Leadership problem
It's action influence the output choice of the follower. This
relationship is given by f2= (y1) [y2= f2(y1) ]. As we made , in case
of the follower, the profit max problem for the leader is
max p(y1+y2)y1 – c1(y1)
y1
Note that the leader recognizes that when it chooses output y1, the
total output produced will be y1+ f2(y1) , its own output plus the
output of the follower, so he has the influence in output of the
follower. Let's see what happen :
f2 (y1) = y2 =
a-by1
2b
It is the reaction function as illustrated in the previous slide
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Sequential Game
Leadership problem
Since we assume MC=0 the leader´s profit are :
∏1 (y1+y2)= p(y1+y2)y1= ay1 – by12 –by1y2
But the ouput of the follower , y2 , will depend on the leader´s
choice via reaction function y2= f2 (y1). Simplifying all the calculus
and set the MC as zero and MR as (a /2) – by1 , we simple find :
y*
=
1
a
2b
In order to find the follower output we substitute y*1 into the the
reaction function:
y*
= a
2
4b
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Sequential Game
Leadership problem
This two equations give a total industry output
y*1
+
y*2
3a
= 4b
The Stackelberg solution can also be illustrated graphically using the
Isoprofit curves (Fig.2). Here we have illustrated the reaction curves
for both firms and the isoprofit curves for firm 1.
To understand the graffic, firm 2 is behaving as a follower, which
means that it will choose an output along its reaction curve , f2(y1).
Thus, firm 1 wants to choose an output combination on the reaction
curve that gives it the highest possible profits.
But, it means, picking that point on the reaction curve that touches
the lowest isoprofit line (as illustrated). It follows by the usual logic
of maximization that the reaction curve must be tangent to the
isoprofit curve at this point.
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Sequential Game
Leadership problem
Fig.2 - Isoprofit curves (Stackelberg equilibrium)
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Price Leadership
Instead of setting quantity, the leader may instead set
the price, in this case the leader must forecast the
follower behaviour.
•
What is the follower problem?
•
Profit Maximization
•
In equilibrium the follower must
always set the same price as the
leader.
•
If one firm charged a lower
price…
•
Suppose that the leader has a price:
•
“p”
•
In this model the follower
takes the price as being
outside of is control since it
was already set by the leader.
•
max(y2) py2 – c2(y2)
•
This determines the supply
curve to the follower S(p);
•
The follower takes this price and
wants to maximize profits:
•
The follower wants to choose an
output level where the price equals to
the marginal cost.
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Price Leadership
•
What is the leader problem?
•
It realizes if it sets a price “p”
the follower will supply S(p)
•
The amount of output that the leader
will sell will be…
•
R(p) = D(p) – S(p)
•
Supose that the leader has a a
constant marginal cost of production:
•
“c”
•
Then the profits that achieves for any
price “p” are given by:
•
•
∏1(p)=(p-c)[D(p)– S(p)]=
=(p-c)R(p)
•
In order to maximize the profits the
leader wants to chose a price and a
output combination...
•
Where the marginal revenue
equals the marginal cost.
(Residual demand curve facing the leader)
However, the marginal revenue should be the marginal revenue for
the residual demand curve (the curve that actually measures how
much output it will be able to sell at a each given price).
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Price Leadership
Graphical illustration
The marginal revenue curve associated will have the same vertical
intercept and be twice the step.
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Price Leadership
Algebraic example 1/2
• Inverse Demand Curve:
• D(p) = a - bp
• Follower cost function:
• C2(y2) = y22/2
• Leader cost function:
• C1(y1) = cy1
• The follower wants to
operate where price is
equal to marginal cost:
• MC2(y2) = y2
• Setting price
marginal cost
• p=y2
equal
to
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Price Leadership
Algebraic example 2/2
•
Solving for the supply curve:
•
y2=S(p)=p
•
The demand curve facing leader
(residual demand curve) is:
•
•
R(p) = D(p)-S(p)=
=a-bp-p=a-(b+1)p
•
Solving for p as function of the
leader’s output y1:
•
p=a/(b+1) – y1/(b+1)
•
This is the inverse demand
function facing the leader.
•
MR1 = a/(b+1) – 2y1/(b+1)
•
•
MR1=a/(b+1)–2y1/(b+1)=
=c=MC1
•
Setting marginal revenue equal
to marginal cost:
•
Solving for the leader’s profit
maximization output:
•
y1*=(a-c(b+1))/2
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Comparing Price Leadership and
Quantity Leadership
We’ve seen how to calculate the equilibrium price and
output in case of quantity leadership and price
leadership. Each model determines a different
equilibrium price and output combination.
Quantity leadership
Capacity choice

Quantity Leader
Price leadership
Price setting

Price and supply decision
“We have to look at how the firms actually make their decisions in
order to choose the most appropriate model”
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Simultaneous Game
Simultaneous Quantity Setting
Leader – follower model is necessarily asymmetric.
• Cournot Model
Each firm has to forecast the other firm´s output choice.
Given its forecasts, each firm then chooses a profit-maximizing
output for itself.
Each firm finds its beliefs about the other firm to be confirmed.
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Simultaneous Quantity Setting
Assuming:
Firm 1decides to produce y1 units of output, and believes that firm will
produced y2e
Total output produced will be Y = y1 + y2e
Output will yield a market price of p(Y) = p( y1 + y2e )
The profit-maximization problem of firm 1 is them
max
p(
y1 + y2e ) y1 – c(y1)
y
1
For any given belief about the output of firm 2 (y2e ), there will be
some optimal choice of output for firm 1 (y1).
y1 = f1(y2e )
This reaction function gives one firm´s optimal choice as
a function of its beliefs about the other firm´s choice.
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Simultaneous Quantity Setting
Similarly, we can write: y2 = ƒ2(y1e ) Which gives firm2´s optimal
choice of output for a given expectation about firm 1´s output, y1e.
• Each firm is choosing its output level assuming that the other
firm´s output will be at y1e or y2e.
• For arbitrary values of y1e and y2e this won´t happen - in general
firm 1´s optimal level of output, y1, will be different from what
firm 2 expects the output to be, y1e.
Seek an output combination (y1*, y2*)
•Optimal output level for firm1 (assuming firm 2 produces y2*) is y1*
•Optimal output level for firm2 (assuming firm 1 produces y1*) is y2*
So the output choices (y1*, y2*) satisfy
y1* = ƒ1(y2* )
y2* = ƒ2(y1* )
Class Assigment – Microeconomics – “Oligopoly”
Cournot
equilibrium
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Cournot Equilibrium
• Each firm is maximizing its profits, given its beliefs about the other firm´s
output choice.
• The beliefs that optimally chooses to produce the amount of output that the
other firm expects it to produce are confirmed in equilibrium.
• In a Cournot equilibrium neither firm will find it profitable to change its
output once it discovers the choice actually made by the other firm.
y2
Reaction curve for
firm 1
Cournot
Equilibrium
Reaction curve
for firm 2
Figure - Cournot Equilibrium
y1
Is the point at which the reaction
curves cross.
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Adjustment to Equilibrium
At time t the firm are producing
outputs (y1t, y2t), not necessarily
Grafico livro pag 480 fig27.4
equilibrium outputs.
If firm 1 expects that firm 2 is
going to continue to keep its output
at y2t, then next period firm 1
would want to choose the profit–
maximizing output given that
expectation, namely ƒ1(y2t).
Firm 1 choice in period t +1 will be:
Firm 2 can reason the same way,
so firm 2 choice next period will be:
Y1t+1=ƒ1(y2t)
Y2t+1=ƒ2(y1t)
These two equations describe how each firm adjusts its output in
the face of the other firm´s choice
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Adjustment to Equilibrium
The Cournot equilibrium is a stable equilibrium when the
adjustment process converges to the Cournot equilibrium.
Some difficulties of of this adjustment process:
Each firm is assuming that the other´s output will be fixed from
one period to the next, but as it turns out, both firms keep
changing their output.
Only in equilibrium is one firm´s output expectation about the
other firm´s output choice actually satisfied.
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Many firms in Cournot Equilibrium
More than two firms involved in a Cournot equilibrium
Each firm has an expectation about the output choices of the other firms in the
industry and seek to describe the equilibrium output.
Suppose that are n firms:
Total industry output
y  y1  ... yn
The marginal revenue equals marginal cost condition for firm is
p(Y ) 
p
yi  MC ( yi )
Y
Using the definition of elasticity of aggregate demand curve and letting
si=yi/Y be firm i´s share of total market output


1
p ( y ) 1 
  MC ( yi )

(
Y
)
/
s

i 

Like the expression for the monopolist, except (si)
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Many firms in Cournot Equilibrium
Think of Є(Y)/si as being the elasticity of the demand curve facing the firm:
< market share of the firm

> elastic the demand curve it faces
If its market share is 1  Demand curve facing the firm is the market
demand curve  Condition just reduces to that of the monopolist.
If its market is a very small part of a large market  market share is
effectively 0  Demand curve facing the firm is effectively flat condition
reduces to that of the pure competitor: price equals marginal cost.
If there are a large number of firms, then each firm´s influence on the
market price is negligible, and the Cournot equilibrium is effectively the same
as pure competition.
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Simultaneous Price Setting
Cournot Model described firms were choosing their quantities and letting
the market determine the price.
Firms setting their prices and letting the market determine the quantity
sold  Bertrand competition.
What does a Bertrand equilibrium look like?
Assuming that firms are selling identical products  Bertrand
equilibrium is the competitive equilibrium, where price equals marginal^
cots.
Consider that both firms are selling output at some price > marginal cost.
Cutting its price by an arbitrarily small amount
customers from firm 2.
firm 1 can steal all of the
Firm 2 can reason the same way!
Any price higher than marginal cost cannot be an equilibrium
The only equilibrium is the competitive equilibrium
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Collusion
Key Findings
•Companies collude so as to jointly set the price or quantity of a certain good.
This way it is possible to maximize total industry profits.
•The output produced by multiple firms that are colluding will be equal to the
one produced by one firm that has a monopoly.
•When firms get together and attempt to set prices and outputs so as to
maximize total industry profits, they are known as a Cartel.
•A cartel will typically be unstable in the sense that each firm will be tempted
to sell more than its agreed upon output if it believes that the other firms will
stick to what was agreed.
EXAMPLES OF COLLUSION:
•
•
•
De Beers
Organization of the Petroleum Exporting Countries (OPEC)
Port Wine Institute (IVP)
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Collusion
Profit-maximization when colluding
maxy1, y2 p(y1, y2)[y1+y2] – c1(y1) – c2(y2)
•The optimality quantity is given by
p(y1*, y2*) + (∆p/∆Y)[y1* +y2* ] = MC1 (y1* )
p(y1*, y2*) + (∆p/∆Y)[y1* +y2* ] = MC2 (y2* )
•From there we may conclude that in equilibrium
MC1 (y1* ) = MC2 (y2* )
If one firm has a cost advantage, so that it’s marginal cost curve always lies
bellow that of the other firm, then it will necessarily produce more output in
the equilibrium in the cartel solution.
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Collusion
Incentives not to respect the deal (1)
•The profit-maximizing point is D
but if firm 1 assumes that firm 2
will stick with the deal, it will
have incentives to produce G
because it will produce more and
will therefore produce more
revenue.
•Worse, if firm 1 thinks that firm
2 isn’t going to stick with the
deal, it will want to start to
produce G as fast as possible so
as to gain the maximum profits it
can.
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Collusion
Incentives not to respect the deal (2)
•Algebraically
∆ π1/ ∆y1 = p( y*1 + y*2) + (∆p/ ∆y) Y*1 – MC1(y*1)
p( y*1 , y*2 ) + (∆p/∆y) y*1 + (∆p/∆y) y*2 – MC1 (y*1 ) = 0
•Which rearranging gives
∆ π 1/ ∆y1 = p( y*1 , y*2 ) + (∆p/∆y) y*1 – MC1 (y*1 ) = - (∆p/∆y) y*2
•Following
∆ π 1 / ∆y1 > 0
•So that are always incentives for firm 1 individually to cheat firm 2 if it thinks
that firm 2 will stick to the agreement.
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Collusion
Game Theory – Brief example
PRISONER’S DILEMMA
•Each prisoner is in a different cell
and may assume that the other one is
not going to talk.
Prisoner B
Confess
Confess
Prisoner A
Don’t
confess
Don’t
confess
5
5
20
0
0
20
•The dominant strategy
example is to confess.
in
this
•But if both stay silent they will only
get 1 year each.
1
Firm B
1
Keep
Prices
Firm A
Keep
Prices
Lower
prices
Class Assigment – Microeconomics – “Oligopoly”
Lower
prices
100
100
140
10
10
140
50
50
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Collusion
Example of failed collusion
• OPEC has tried and succeeded to maintain a cartel for the oil market.
However they had some drawbacks, like in 1986 when Saudi Arabia dropped
the price from $28 to $10 for barrel.
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Collusion
How to maintain a Cartel? (1)
Monitor others participants behavior
• “Beat any price” strategy
Threat participants to respect the deal
•“If you stay at the production level that maximizes joint industry profits, fine.
But if i discover that you are cheating by producing more than this amount, i
will punish you by producing the Cournot level of output forever.”
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Collusion
How to maintain a Cartel? (2)
Punish disrespects to the deal
• tit-for-tat - “I’ll do this time what you did last time”
Πm – monopoly profits
Πd – one time profit
Πc – Cournout profit
•Present value of cartel behaviour - Πm + (Πm/r)
•Present value of cheating - Πd + (Πc/r)
•Πd > Πm > Πc
r < (Πm - Πc) / (Πd - Πm)
As long as the prospect of future punishment is high enough, it will
pay the firms to stick to their quotas.
Regulation
• Government Regulation
• Examples
• Instituto do Vinho do Porto
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Resume 1
• Few firms
• Homogeneous or different products
• Strategic interactions (the decisions of one
firm influence the results of the others)
• It is not possible to describe the oligopoly
behavior in just one model
• The oligopoly behavior depends on the
characteristics of the market
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Resume 2
Questions:
- What if they change the price?
- What if they change amount produced?
- What if they introduced a new product?
Sequential,
Simultaneous or
Cooperative game
Example: Television broadcasting in Portugal
RTP, SIC, TVI
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Resume 3
Stackelberg Model – Quantity Leadership
• A firm (leader) decides its own production before
the others – dominant firm or natural leader
• The others firms (followers) decide after they
know the leader’s decision
• When the leader chooses an output, it will take
into account how the follower will respond
Example: Computer firm, IBM
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Resume 4
Price Leadership
• A firm (leader) sets the price and the others
choose how much they will produce at that
price
• When the leader chooses a price, it will take into
account how the follower will respond
Example: McDonalds
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Resume 5
Cournot Model – Simultaneous Quantity Setting
•
It is supposed that both firms make their output choices
simultaneously and the expectations about the other firm’s
choices are confirmed
•
Each firm believes that a change in its output will not lead
to followers to change their productions
•
Each firm has a small market share, that implies that price
will be very close to the marginal price – nearly competitive
Example: Banking business
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Resume 6
Bertrand Competition – Simultaneous Price Setting
• Each firm chooses its price based on that it
expects the price of the other firms will be
• Competitive equilibrium
Example: Pump Gas
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Resume 7
Collusion
•
Group of firms that jointly collude to set prices and
quantities that maximize the sum of their profits
•
Behave like a single monopolist
•
Typically unstable
Problem: temptation to cheat
to make higher profits (may
break the cartel)
Firms need a way to
detect and punish
cheating
Punish Strategies
Example: Cartel
(clients,
governments…)
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Comparing Oligopoly models...
Comparing Oligopoly models...
Assuming demand function P=a-by and marginal cost = 0
Firm 1
Quantity
Q1
Firm 2
Quantity
Q2
Total
Quantity
Q1+Q2
Market
Price
Firm 1
profit
Firm 2
profit
∏2
∏1+∏2
Collude
a/4b
a/4b
a/2b
a/2
a2/8b
a2/8b
a2/4b
Cournot
a/3b
a/3b
2a/3b
a/3
a2/9b
a2/9b
2a2/9b
Bertrand
a/2b
a/2b
a/b
0
0
0
0
Stackelberg
a/2b
a/4b
3a/4b
a/4
a2/8b
a2/16b
3a2/16b
Models
∏1
Total
profit
Evidences...
•
The Firm 1 profit in the Stackelberg Model.
•
From Stackelberg Model to Bertrand Model.
•
•
In the model Stackelberg the total output is bigger than in Cournot model;
In Shared Monopoly model: smallest output and highest price;
•
In Bertrand model: highest output and smallest price;
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Exercise
Stackelberg model 1/2
•
The Demand Curve is:
• P = 10 - Q
•
Marginal cost for Leader and
Follower:
• =2
• Questions:
• What will be the equilibrium price for both?
• What will be the equilibrium quantity for both?
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Exercise
Stackelberg model 2/2
•
The Marginal Revenue Curve is:
•
•
MR2=P(Q1+Q2)+(∆P/∆Q2)*Q2
MR2 = 10-Q1-2Q2
•
Marginal cost:
•
MR = MC = 2
•
The Firm 2 Reaction Function:
•
•
R2(Q1) = Q2* =
= 4-(Q1/2)
•
Replacing in the Firms’1 demand
function:
•
•
P1=10 – Q1– 4 + (Q1/2) =
= 6 - (Q1/2)
•
MR1 = 6-Q1
•
MR1 = MC = 2
•
The Marginal Revenue for firm 1 is:
•
•
•
•
And
Anwsers:
What will be the equilibrium price for both: = 4
What will be the equilibrium quantity for both? Q1 = 4; Q2=2
Class Assigment – Microeconomics – “Oligopoly”
31/10/2003
47
MicroEconomincs
Oligopoly
Bibliografy:
Intermediate Microeconomics- Varian, H.
Price Theory and Apllications- Landsburg, S.
Class Assigment – Microeconomics – “Oligopoly”
31/10/2003
48