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MicroEconomics Oligopoly. Students: Ana Oliveira Fernando Vendas Miguel Carvalho Paulo Lopes Vanessa Figueiredo. Introduction Competition Model Sequential Game Quantity Leadership Price Leadership Simultaneous Game Simultaneous Price setting

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MicroEconomics Oligopoly

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  • Students:

    • Ana Oliveira

    • Fernando Vendas

    • Miguel Carvalho

    • Paulo Lopes

    • Vanessa Figueiredo


Competition Model

Sequential Game

Quantity Leadership

Price Leadership

Simultaneous Game

Simultaneous Price setting

Simultaneous Quantity Price setting

Collude (corporate game)



P. Lopes and F. Vendas

V. Figueiredo

M. Carvalho

  • Oliveira and F. Vendas

Presentation Structure

Market Structure

Pure competition

Small competitors

Pure Monopoly

One Large Firm


Initial Framework

Monopolistic Competition


“ Strategic interaction that arise in an industry with small number of firms.” – Varian, H. (1999 , 5th)

Many Different Behavior

Patterns of Behavior

Class Framework

Restrict to the case of 2 firms


Simple to understand

Strategic interaction

Homogeneous product

Study Framework

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 ?

Sequential GameQuantity Leadership

Assume that the follower wants to maximize its profits

max p(y1+y2)y2 – c2(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 :

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.



Sequential GameThe follower's Problem


MR2 = p(y1+y2) + y2= MC2

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)

Sequential GameThe follower's Problem

Sequential GameThe follower's Problem

Fig.1 - The isoprofit lines graffic


This reaction curve gives the profit-maximizing output for the follower

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 =



Sequential Game

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)

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=

It is the reaction function as illustrated in the previous slide



Sequential GameLeadership 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 :


In order to find the follower output we substitute y*1 into the the reaction function:






Sequential GameLeadership problem



This two equations give a total industry output

+ =

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.



Sequential GameLeadership problem



Sequential GameLeadership problem

Fig.2- Isoprofit curves (Stackelberg equilibrium)

What is the follower problem?

In equilibrium the follower must always set the same price as the leader.

Suppose that the leader has a price:

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.

Price Leadership

Instead of setting quantity, the leader may instead set the price, in this case the leader must forecast the follower behaviour.

  • Profit Maximization

  • If one firm charged a lower 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);

Price Leadership

  • What is the leader problem?

  • The amount of output that the leader will sell will be…

  • Supose that the leader has a a constant marginal cost of production:

  • Then the profits that achieves for any price “p” are given by:

  • In order to maximize the profits the leader wants to chose a price and a output combination...

  • It realizes if it sets a price “p” the follower will supply S(p)

  • R(p) = D(p) – S(p)

    (Residual demand curve facing the leader)

  • “c”

  • ∏1(p)=(p-c)[D(p)– S(p)]=

  • =(p-c)R(p)

  • Where the marginal revenue equals the marginal cost.

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).

Price LeadershipGraphical illustration

The marginal revenue curve associated will have the same vertical intercept and be twice the step.

Inverse Demand Curve:

Follower cost function:

Leader cost function:

The follower wants to operate where price is equal to marginal cost:

Setting price equal to marginal cost

Price LeadershipAlgebraic example 1/2

  • D(p) = a - bp

  • C2(y2) = y22/2

  • C1(y1) = cy1

  • MC2(y2) = y2

  • p=y2

Solving for the supply curve:

The demand curve facing leader (residual demand curve) is:

Solving for p as function of the leader’s output y1:

This is the inverse demand function facing the leader.

Setting marginal revenue equal to marginal cost:

Solving for the leader’s profit maximization output:

Price LeadershipAlgebraic example 2/2

  • y2=S(p)=p

  • R(p) = D(p)-S(p)=

  • =a-bp-p=a-(b+1)p

  • p=a/(b+1) – y1/(b+1)

  • MR1 = a/(b+1) – 2y1/(b+1)

  • MR1=a/(b+1)–2y1/(b+1)=

  • =c=MC1

  • y1*=(a-c(b+1))/2

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.

  • Price leadership

  • Price setting

  • Price and supply decision

Quantity leadership

Capacity choice

Quantity Leader

“We have to look at how the firms actually make their decisions in order to choose the most appropriate model”

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.

Simultaneous Game

Simultaneous Quantity Setting


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)


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.

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 y1eor y2e.

  • For arbitrary values of y1eand y2ethis 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*) isy1*

  • Optimal output level for firm2 (assuming firm 1 produces y1*) isy2*

y1* = ƒ1(y2* )

y2* = ƒ2(y1* )



So the output choices (y1*, y2*) satisfy

Reaction curve for firm 1


Cournot Equilibrium

Reaction curve for firm 2


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.

Figure - Cournot Equilibrium

Is the point at which the reaction curves cross.

Adjustment to Equilibrium

At time t the firm are producing outputs (y1t, y2t), not necessarily 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).

Grafico livro pag 480 fig27.4

Firm 2 can reason the same way, so firm 2 choice next period will be:


Firm 1 choice in period t +1 will be:


These two equations describe how each firm adjusts its output in the face of the other firm´s choice

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.

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

The marginal revenue equals marginal cost condition for firm is

Using the definition of elasticity of aggregate demand curve and letting si=yi/Y be firm i´s share of total market output

Like the expression for the monopolist, except (si)

Many firms in Cournot Equilibrium

Think of Є(Y)/sias 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.

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 firm 1 can steal all of the customers from firm 2.

Firm 2 can reason the same way!

Any price higher than marginal cost cannot be an equilibrium

The only equilibrium is the competitive equilibrium

CollusionKey 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.


    • De Beers

    • Organization of the Petroleum Exporting Countries (OPEC)

    • Port Wine Institute (IVP)

CollusionProfit-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.

CollusionIncentives 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.

CollusionIncentives 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

∆π1/ ∆y1 =p( y*1 , y*2 ) + (∆p/∆y) y*1 – MC1 (y*1 ) = - (∆p/∆y) y*2

  • Following

∆π1 / ∆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.

CollusionGame Theory – Brief example


  • Each prisoner is in a different cell and may assume that the other one is not going to talk.

  • The dominant strategy in this example is to confess.

  • But if both stay silent they will only get 1 year each.

CollusionExample 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.

CollusionHow 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.”

CollusionHow 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

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

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

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

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

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

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: PumpGas

Resume 7


  • 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

(clients, governments…)

Example: Cartel

Comparing Oligopoly models...

  • 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;

The Demand Curve is:

Marginal cost for Leader and Follower:

ExerciseStackelberg model 1/2

  • P = 10 - Q

  • = 2

  • Questions:

  • What will be the equilibrium price for both?

  • What will be the equilibrium quantity for both?

The Marginal Revenue Curve is:

Marginal cost:

The Firm 2 Reaction Function:

Replacing in the Firms’1 demand function:

The Marginal Revenue for firm 1 is:

ExerciseStackelberg model 2/2

  • MR2=P(Q1+Q2)+(∆P/∆Q2)*Q2

  • MR2 = 10-Q1-2Q2

  • MR = MC = 2

  • R2(Q1) = Q2* =

  • = 4-(Q1/2)

  • P1=10 – Q1– 4 + (Q1/2) =

  • = 6 - (Q1/2)

  • MR1 = 6-Q1

  • And

  • MR1 = MC = 2

  • Anwsers:

  • What will be the equilibrium price for both: = 4

  • What will be the equilibrium quantity for both? Q1 = 4; Q2=2



Intermediate Microeconomics- Varian, H.

Price Theory and Apllications- Landsburg, S.

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