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Simultaneous Price setting
Simultaneous Quantity Price setting
Collude (corporate game)
P. Lopes and F. Vendas
One Large Firm
Form : OLIGOPOLY
“ Strategic interaction that arise in an industry with small number of firms.” – Varian, H. (1999 , 5th)
Many Different Behavior
Patterns of Behavior
Restrict to the case of 2 firms
Simple to understand
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 ?
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.
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)
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
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
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:
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.
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.
Instead of setting quantity, the leader may instead set the price, in this case the leader must forecast the follower behaviour.
(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).
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
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:
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.
“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.
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 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.
y1* = ƒ1(y2* )
y2* = ƒ2(y1* )
So the output choices (y1*, y2*) satisfy
Reaction curve for firm 1
Reaction curve for firm 2
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
maxy1, y2 p(y1, y2)[y1+y2] – c1(y1) – c2(y2)
p(y1*, y2*) + (∆p/∆Y)[y1* +y2* ] = MC1 (y1* )
p(y1*, y2*) + (∆p/∆Y)[y1* +y2* ] = MC2 (y2* )
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.
∆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
∆π1/ ∆y1 =p( y*1 , y*2 ) + (∆p/∆y) y*1 – MC1 (y*1 ) = - (∆p/∆y) y*2
∆π1 / ∆y1 > 0
Πm – monopoly profits
Πd – one time profit
Πc – Cournout profit
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.
Sequential, Simultaneous or Cooperative game
Example: Television broadcasting in Portugal RTP, SIC, TVI
Stackelberg Model – Quantity Leadership
Example: Computer firm, IBM
Cournot Model – Simultaneous Quantity Setting
Example: Banking business
Bertrand Competition – Simultaneous Price Setting
Problem: temptation to cheat to make higher profits (may break the cartel)
Firms need a way to detect and punish cheating
The Demand Curve is:
Marginal cost for Leader and Follower:
The Marginal Revenue Curve is:
The Firm 2 Reaction Function:
Replacing in the Firms’1 demand function:
The Marginal Revenue for firm 1 is:
Intermediate Microeconomics- Varian, H.
Price Theory and Apllications- Landsburg, S.