2 Presentation Structure IntroductionCompetition ModelSequential GameQuantity LeadershipPrice LeadershipSimultaneous GameSimultaneous Price settingSimultaneous Quantity Price settingCollude (corporate game)ResumeExerciseP. Lopes and F. VendasV. FigueiredoM. CarvalhoOliveira and F. Vendas
3 Initial Framework Market Structure Pure competition Small competitors Pure MonopolyOne Large FirmHoweverComeçamos com a nossa apresentação com duas formas de estrutura de mercado :Concorrência pura (vários pequenos competidores)Monopólio puro (existe apenas uma grande empresa)Contudo...
4 Class Framework Monopolistic Competition Many Different Behavior Form : OLIGOPOLY“ Strategic interaction that arise in an industry with small number of firms.” – Varian, H. (1999 , 5th)Many Different BehaviorPatterns of BehaviorNo mercado existem estes dois extremos, pelo que há no mercado um grande número de competidores, mas não tantos que possamos considerar nula a influência no preço.Esta forma é conhecida por Oligopólio (Competição Monopolista).Para caracterizar esta forma será necessário conhecer os diferentes comportamentos e modelos.
5 Study Framework Restrict to the case of 2 firms Duopoly Simple to understandStrategic interactionHomogeneous productPara simplificar estudaremos o caso dos duopólios (2 firmas/empresas)O caso do duopólio permite captar vários aspectos importantes das empresas envolvidas em interacções estratégicas e com produtos homogeneos (idênticos)
6 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 + y2The leader has to consider the follower´s profit-maximization problem, then we should think : What output should the leader choose to max its profits ?Liderança de quantidadeNeste caso, uma empresa faz a escolha antes da outra, de acordo com o modelo de Stackelberg.O nosso estudo começa por aqui : suponhamos que a firma 1 é a lider e que escolhe produzir uma qty y1. A firma 2 responde com a escolha da qty y2.Ambas as firmas sabem que a preço de equilíbrio do mercado depende da qty total produzida.Então utilizamos a função inversa da procura para indicar o preço de equilíbrio como função da produção (output).Para decidir sobre a sua produção, o leader tem que considerar a maximização do lucro do follower.
7 Sequential Game The follower's Problem Assume that the follower wants to maximize its profitsmax 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.y2O problema do seguidorAssumimos que follower (firma2) quer maximizar os seus lucros, obviamente.Mas a produção do leader é predeterminada (já a concluío) e simplesmente o follower a ecncara com uma constate. (portanto, está traduzida na forma aqui mencionada)Contudo o follower quer escolher um nível de produção em que a receita marginal seja idêntica ao custo marginal. (formalula seguinte).Quando o follower aumenta a sua produção, aumenta a sua receita ao vender mais produto ao preço de mercado, mas também empurra o preço para baixo em ∆p, e isso diminui os seus lucros em todas as unidades previamente vendidas ao preço mais alto.∆p∆y2MR2 = p(y1+y2) y2= MC2
8 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 outputp(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 – by22So, we use this form to draw the isoprofit lines (Fig.1)É importante observar que a escolha do follower em maximizar os lucros, dependerá da escolha feita pelo líder, esta relação pode ser dada por : (formula) que se designa por função da reacção.Esta função , traduz o lucro(max) da produção do follower como função da escolha do lider. Uma vez que ela nos mostra como o follower reagirá à escolha de produção do líder.Após um simples calculo de derivação da curva de reacção (no caso simples da procura linear). Nesse caso a função de procura inversa assume a forma ....p(y1+y2)=a-b(y1+y2)Assume-se C=0 e obtemos a função do lucro do follower , que pode ser utilizada para desenhar as linhas de isolucro (fig1)
9 Sequential Game The follower's Problem Fig.1 - The isoprofit lines grafficMonopolisticsThis reaction curve gives the profit-maximizing output for the follower
10 MR2(y1,y2)= a – by1 – 2by2 (MR=MC ; MC=0) Sequential GameThere 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 equationsay2 – b y1y2 – by22 = ∏2Firm 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 2y2 =Essas linhas apresentam as combinações de y1 e y2 que proporcionam um nível constante de lucro ao follower. Isto é, as linhas de isolucro são compostas por todos os pontos (y1,y2) que satisfzem esta equação.Observem que os lucro do follower aumentará à medida que nos movermos em relação às linhas isolucro, mais à esquerda.Isso é verdadeiro porque se fixarmos a produção do follower num determinado nível , os lucros do follower aumentam à medida que a produção do lider diminui. O follower alcançará o max de lucro qd se tornar monopolista, ou seja, qd o líder escolher em não produzir.Portanto o follower vai tentar escolher o ponto mais à esquerda possível. Esse ponto satisfará a condição de tangência ( a inclinação da linha isolucro terá de ser vertical na escolha optima). O ponto dessa tangente descreve a curva de reacção do follower.Para verificarmos esse resultado de maneira algébrica, precisamos de uma expressão para a receita marginal em função do lucro do follower. Que é dada por esta, tendo em consideração estes presupostos MR=MC e MC=0.Depois de alguns cálculos, obtemos a curva de reação do follower, essa curva não é mais que a linha recta que vocês viram na gráfico 1.a-by12b
11 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 ismax 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 slidey1Já vimos como o follower escolherá a sua produção, dada a escolha do lider. Agora vamos verificar o problema do lado do lider, que passa também por maximizar os seus lucros.O lider tem conhecimento que as suas acções influenciam a escolha de produção do follower. Essa relação é resumida pela função da reacção. Como vimos no caso do follower , o lider também tem um problema de maximização do lucro, que é dad pela forma....Obeservem que o lider reconhece que qd escolhe em produzir y1, a produção total será y1+f2(y1), isto é, a sua própria produção mais a produção do follower, então qd pensa em variar a sua produção, tem que considerar a influência que exerce sobre o follower (firm 2). Algebricamente falando, vamos ver : (curva de oferta linear era igual à função de recção, como ilustrado no slide anterior)a-by12b
12 Sequential Game Leadership problem Since we assume MC=0 the leader´s profit are :∏1 (y1+y2)= p(y1+y2)y1= ay1 – by12 –by1y2But 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:Assumimos que os CM=0, os lucros do lider serão dados por esta formaMas a produção do follower y2, dependerá da escolha do lider através da função de reacção. Ao simplificarmos a expressão a receita marginal da função será (a/2)/by1, se igualarmos isso ao custo marginal , que no exemplo é zero (0) e resolvermos em função de y1, obtemos y1(*)=a/2b a função.Para encontrar a produção do follower basta substituir y1(*) na função de reacção obtida anteriormente.a2by*1a4by*2
13 Sequential Game Leadership 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.3a4by*1y*2Estas duas equações dá a produção total destas duas empresas.O modelo de stackelberg pode ser visto graficamente na figura 2 com o uso das curvas de isolucro. Nela são ilustradas as curvas de reacção de ambas as empresas e as curvas isolucro do lider.Para entender este gráfico, a empresa 2 comporta-se com follower, o que siginifica que escolherá uma produção sobre a sua curva de reacção. Portanto, o lider quer escolher uma combinação de produção que lhe forneça os maiores lucros possíveis.Mas escolher esse ponto, siginifica escolher o ponto da curva de reacção que toca a curva isolucro mais baixa, conforme demonstra a figura 2 , isto é, que a curva de reacção tem que ser tangente à curva da isolucro nesse ponto.
14 Sequential Game Leadership problem Fig.2 - Isoprofit curves (Stackelberg equilibrium)
15 Price LeadershipInstead 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?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.Profit MaximizationIf 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);
16 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).
17 Price Leadership Graphical illustration The marginal revenue curve associated will have the same vertical intercept and be twice the step.
18 Price Leadership Algebraic example 1/2 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 costD(p) = a - bpC2(y2) = y22/2C1(y1) = cy1MC2(y2) = y2p=y2
19 Price Leadership Algebraic example 2/2 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:y2=S(p)=pR(p) = D(p)-S(p)==a-bp-p=a-(b+1)pp=a/(b+1) – y1/(b+1)MR1 = a/(b+1) – 2y1/(b+1)MR1=a/(b+1)–2y1/(b+1)==c=MC1y1*=(a-c(b+1))/2
20 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 leadershipPrice settingPrice and supply decisionQuantity leadershipCapacity choiceQuantity Leader“We have to look at how the firms actually make their decisions in order to choose the most appropriate model”
21 Simultaneous Game Simultaneous Quantity Setting Leader – follower model is necessarily asymmetric.Cournot ModelEach 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.
22 The profit-maximization problem of firm 1 is them Simultaneous Quantity SettingAssuming:Firm 1decides to produce y1 units of output, and believes that firm will produced y2eTotal output produced will be Y = y1 + y2eOutput will yield a market price of p(Y) = p( y1 + y2e )The profit-maximization problem of firm 1 is themmax p(y1 + y2e ) y1 – c(y1)y1For 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.
23 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*y1* = ƒ1(y2* )y2* = ƒ2(y1* )CournotequilibriumSo the output choices (y1*, y2*) satisfy
24 Cournot EquilibriumEach 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.Cournot EquilibriumReaction curve for firm 1Reaction curve for firm 2y1y2Figure - Cournot EquilibriumIs the point at which the reaction curves cross.
25 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 fig27.4Firm 2 can reason the same way, so firm 2 choice next period will be:Y2t+1=ƒ2(y1t)Firm 1 choice in period t +1 will be:Y1t+1=ƒ1(y2t)These two equations describe how each firm adjusts its output in the face of the other firm´s choice
26 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.
27 Many firms in Cournot Equilibrium More than two firms involved in a Cournot equilibriumEach 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 outputThe marginal revenue equals marginal cost condition for firm isUsing the definition of elasticity of aggregate demand curve and letting si=yi/Y be firm i´s share of total market outputLike the expression for the monopolist, except (si)
28 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 facesIf 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.
29 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 equilibriumThe only equilibrium is the competitive equilibrium
30 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 BeersOrganization of the Petroleum Exporting Countries (OPEC)Port Wine Institute (IVP)
31 Collusion Profit-maximization when colluding maxy1, y2 p(y1, y2)[y1+y2] – c1(y1) – c2(y2)The optimality quantity is given byp(y1*, y2*) + (∆p/∆Y)[y1* +y2* ] = MC1 (y1* )p(y1*, y2*) + (∆p/∆Y)[y1* +y2* ] = MC2 (y2* )From there we may conclude that in equilibriumMC1 (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.
32 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.
33 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 ) = 0Which rearranging gives∆ π 1/ ∆y1 = p( y*1 , y*2 ) + (∆p/∆y) y*1 – MC1 (y*1 ) = - (∆p/∆y) y*2Following∆ π 1 / ∆y1 > 0So that are always incentives for firm 1 individually to cheat firm 2 if it thinks that firm 2 will stick to the agreement.
34 Collusion Game Theory – Brief example PRISONER’S DILEMMAEach 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.Prisoner BConfessDon’tconfessPrisoner A5201Firm BKeep PricesLower pricesFirm A1001014050
35 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.
36 Collusion How to maintain a Cartel? (1) Monitor others participants behavior“Beat any price” strategyThreat 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.”
37 Collusion How to maintain a Cartel? (2) Punish disrespects to the dealtit-for-tat - “I’ll do this time what you did last time”Πm – monopoly profitsΠd – one time profitΠc – Cournout profitPresent value of cartel behaviour - Πm + (Πm/r)Present value of cheating - Πd + (Πc/r)Πd > Πm > Πcr < (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.RegulationGovernment RegulationExamplesInstituto do Vinho do Porto
38 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 modelThe oligopoly behavior depends on the characteristics of the market
39 Sequential, Simultaneous or Cooperative game Resume 2Questions:- What if they change the price?What if they change amount produced?What if they introduced a new product?Sequential, Simultaneous or Cooperative gameExample: Television broadcasting in Portugal RTP, SIC, TVI
40 Resume 3 Stackelberg Model – Quantity Leadership A firm (leader) decides its own production before the others – dominant firm or natural leaderThe others firms (followers) decide after they know the leader’s decisionWhen the leader chooses an output, it will take into account how the follower will respondExample: Computer firm, IBM
41 Resume 4 Price Leadership A firm (leader) sets the price and the others choose how much they will produce at that priceWhen the leader chooses a price, it will take into account how the follower will respondExample: McDonalds
42 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 confirmedEach firm believes that a change in its output will not lead to followers to change their productionsEach firm has a small market share, that implies that price will be very close to the marginal price – nearly competitiveExample: Banking business
43 Resume 6 Bertrand Competition – Simultaneous Price Setting Each firm chooses its price based on that it expects the price of the other firms will beCompetitive equilibriumExample: Pump Gas
44 (clients, governments…) Resume 7CollusionGroup of firms that jointly collude to set prices and quantities that maximize the sum of their profitsBehave like a single monopolistTypically unstableProblem: temptation to cheat to make higher profits (may break the cartel)Firms need a way to detect and punish cheatingPunish Strategies(clients, governments…)Example: Cartel
45 Comparing Oligopoly models... Assuming demand function P=a-by and marginal cost = 0ModelsFirm 1QuantityQ1Firm 2Q2TotalQ1+Q2MarketPriceprofit∏1∏2∏1+∏2Colludea/4ba/2ba/2a2/8ba2/4bCournota/3b2a/3ba/3a2/9b2a2/9bBertranda/bStackelberg3a/4ba/4a2/16b3a2/16bEvidences...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;
46 Exercise Stackelberg model 1/2 The Demand Curve is:Marginal cost for Leader and Follower:P = 10 - Q= 2Questions:What will be the equilibrium price for both?What will be the equilibrium quantity for both?
47 Exercise Stackelberg model 2/2 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:MR2=P(Q1+Q2)+(∆P/∆Q2)*Q2MR2 = 10-Q1-2Q2MR = MC = 2R2(Q1) = Q2* == 4-(Q1/2)P1=10 – Q1– 4 + (Q1/2) == 6 - (Q1/2)MR1 = 6-Q1AndMR1 = MC = 2Anwsers:What will be the equilibrium price for both: = 4What will be the equilibrium quantity for both? Q1 = 4; Q2=2
48 MicroEconomincs Oligopoly Bibliografy:Intermediate Microeconomics- Varian, H.Price Theory and Apllications- Landsburg, S.