This is the situation described by the Stackelberg model where the firms are quantity setters selling homogenous goods. In the Cournot model, firm A simply notes that the market demand is satisfied by the output produced by it and firm B. 18.4 Policy Example: How Should the Government Have Responded to the Banking Crisis of 2008? Stackelberg Model. Clearly, this third option is the one that yields the most profit. We can see that Federal’s profits are determined only by their own output once we explicitly consider National’s response. Since the Cournot outcome is one of the options for the Stackleberg leader – if it chooses the same output as in the Cournot case the follower will as well – it must be true that profits are higher for the Stackelberg leader. The two firms make simultaneous decisions. But at PF = PS = c both stations are playing their best response to each other simultaneously. In the Stackelberg model, A. each firm takes the quantities produced by its rivals as given. Oligopolists face downward sloping demand curves which means that price is a function of the total quantity produced which, in turn, implies that one firm’s output affects not only the price it receives for its output but the price its competitors receive as well. To do so we have to begin with a best response function. The weekly demand for wholesale gas is still P = A – BQ , where Q is the total quantity of gas supplied by the two firms or, Q=qF+qN. If they have the same price, then each will get one half of the demand at that price. 3. Stackelberg Model Differences between Cournot and Stackelberg: In Cournot, firm 1 chooses its quantity given the quantity of firm 2 In Stackelberg, firm 1 chooses its quantity given the reaction curve of firm 2 Nota: the assumption that the leader cannot revise its decision i.e. Both stations have large signs that display the gas prices that each station is offering for the day. The Stackelberg leadership model is a sequential model, which means that the dominant firm first sets the price, which is then used by the other firms to determine their optimal production. There are three main models of oligopoly markets, each consider a slightly different competitive environment. Let’s consider a specific example. So both Federal Oil and National Oil produce 100 thousand gallons of gasoline a week. What are the strategic incentives for banks to take risks? In everything from stock prices to CEO pay relative performance matters, and if one bank were to rely on a low-risk strategy whilst others were engaging in higher risk-higher reward strategies both the company’s stock price and the compensation of the CEO might suffer. By symmetry, National Oil has an identical best response function: [latex]\color{green} q^*_N=\frac{A-c}{2B}-\frac{1}{2}qF[/latex], We know from Module 15 that the monopolists marginal revenue curve when facing an inverse demand curve P=A-BQ. Stackelberg Model Practice Question.pdf - Stackelberg Model Practice Question Consider the stackelberg model in which \u2026rm 1 sets a quantity q1 \u2026rst, Consider the stackelberg model in which °rm 1 sets a quantity, followed by °rm 2 which sets its own quantity, What is the stackelberg equilibrium outcome (i.e., the subgame perfect Nash. Simplifying yields: [latex]\Pi _F=q_F(\frac{A-c}{2}-B\frac{1}{2}q_F)[/latex]. We will again call Federal’s output choice qF and National’s output choice qN , where q represents liters of gasoline. Consumers are assumed to be indifferent about the gas or the stations, so they will go to the station that is offering the lower price. In a standard Stackelberg duopoly situation there are two firms in a market. This is known as a 'Stackelberg leadership’ model. [latex]q^*_N=\frac{A-c}{2B}-\frac{1}{2}q_F[/latex], When it comes to Federal’s decision, we diverge from the Cournot model because instead of taking qN as a given, Federal knows exactly how National will respond because they know the best response function. The gas they produce is identical but now they decide their output levels sequentially. It is named after the German economist Heinrich Freiherr von Stackelberg who published Market Structure and Equilibrium (Marktform und Gleichgewicht) in 1934 which described the model. So an individual gas station’s demand is conditional on its relative price with the other station. We can begin by graphing the best response functions. The Bertrand Model: what happens when two firms compete simultaneously on … This is because in the Cournot case both firms took the other’s output as given. And since both the quantity produced and the price received are lower for the Stackelberg follower compared to the Cournot outcome, the profits must be lower as well. [latex]\color{green} \frac{\partial \pi_F}{\partial q_F}=0[/latex], If πF = qF ( A – B ( qF + qN ) -c ) then we can expand to find, Taking the partial derivative of this expression with respect to qF, [latex]\color{green}\frac{\partial \pi_F}{\partial q_F}=A-2Bq_F-Bq_N-c=0[/latex]. Doing so yields [latex]q^*_F=\frac{A-c}{2B}-\frac{1}{2}qN[/latex] for Federal Oil, and [latex]q^*_N=\frac{A-c}{2B}-\frac{1}{2}qF[/latex] for National Oil. The bursting of this bubble led to the housing market crash and, in 2008, to a banking crisis: the failure of major banking institutions and the unprecedented government bailout of banks. [latex]\color{green}q_F=\frac{A-Bq_N-c}{2B}[/latex], [latex]\color{green} q^*_F=\frac{A-c}{2B}-\frac{1}{2}qN[/latex]. Stackelberg Model Practice Question: Consider the stackelberg model in which °rm 1 sets a quantity q 1 °rst, followed by °rm 2 which sets its own quantity q 2 after observing q 1: The market price is given by P = 40 ° Q; where Q = q 1 + q 2: Let each °rm±s MC =10. We still have identical goods, for consumers the gas that goes into their cars is all the same and we will assume away any other differences like cleaner stations or the presence of a mini-mart. The price is p = 1,000 – 2(200) = $600 for one thousand gallons of gas or $0.60 a gallon. Stackelberg model is a leadership model that allows the firm dominant in the market to set its price first and subsequently, the follower firms optimize their production and price. This module considers all three in order beginning with the Cournot model. The marginal revenue looks the same as a monopolist’s MR function but with one additional term, -BqN. The opposite is true for the second mover, by being forced to choose after the leader has set its output, the follower is forced to accept a lower price and lower output. For simplicity here we consider as duopoly situation, as in Cournot’s model. The banks are better off and because the adverse effects of high-risk strategies going bad are taken away, society benefits as well. We will assume that each liter of gas produced costs the company c, or that c is the marginal cost of producing a liter of gas for both companies and that there are no fixed costs. D. prices are higher and quantities are slightly less than we would see if the firms colluded to The Stackelberg model is like the Cournot model in that firms choose their quantity, and then the market price is based on the joint quantity of all the firms in the market. For a limited time, find answers and explanations to over 1.2 million textbook exercises for FREE! This is Federal Oil’s best response function, their profit maximizing output level given the output choice of their rivals. STACKELBERG INDEPENDENCE* Toomas Hinnosaar † The standard model of sequential capacity choices is the Stackelberg quantity leadership model with linear demand. The rationale was increased competition and the discipline of the market would inhibit excessive risk-taking and so stringent government regulation was no longer necessary. A Nash equilibrium is a correspondence of best response functions which is the same as a crossing of the reaction curves. Introducing Textbook Solutions. ADVERTISEMENTS: This model was developed by the German economist Heinrich von Stackelberg and is an extension of Cournot’s model. Being a high-risk bank when your competitor is a low-risk bank brings a big reward; the relatively high returns are compounded by the reward from the stock market. This recognition allows the sophisticated duopolist to determine the reaction curve of his rival […] In the end, both banks end up choosing high-risk and are in a worse outcome than if they had chosen a low risk strategy because of the increased likelihood of negative events from the strategy. The answer in this case is a resounding ‘yes.’ If policy makers take away the ability of the banks to engage in high-risk strategies, the bad equilibrium will disappear and only the low-risk, low-risk outcome will remain. The gas they produce is identical and they each decide independently, and without knowing the other’s choice, the quantity of gas to produce for the week at the beginning of each week. In this case A = 1,000, B = 2 and c = 400. Intermediate Microeconomics by Patrick M. Emerson is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted. From the consumer’s perspective, the Stackelberg outcome is preferable because overall there is more quantity at a lower price. In Cournot, firm 1 chooses its quantity given the quantity of firm 2 In Stackelberg, firm 1 chooses its quantity given the reaction curve of firm 2 Note: the assumption that the leader cannot revise its decision i.e. In the Stackelberg model, the leader decides how much output to produce with other firms basing their decision on what the leader chooses. Stackelberg Model of Duopoly Stackelberg’s Model of Duopoly also has to do with companies trying to decide how much of a homogeneous good to produce. It was formulated by Heinrich Von Stackelberg in 1934. Can policy correct the situation and lead to a mutually beneficial outcome? The number of firms is restricted to two by assuming barriers to entry. This is different from the Cournot duopoly, where both companies set their production simultaneously. Remember that best response functions are one player’s optimal strategy choice given the strategy choice of the other player. This is Federal Oil’s profit maximizing output level given that they choose first and can anticipate National’s response. If Speedy Gas chargesPS > c , Fast Gas can set PF > PS and they will get no customers at all and make a profit of zero. quantity of output they produce of a homogeneous good. Total output is the sum of the two and is 200 thousands gallons. The model we use to analyze this is one first introduced by French economist and mathematician Antoine Augustin Cournot in 1838. Do you think that government regulation restricting their strategy choices is appropriate in cases where society has to pay for risky bets gone bad. Policy Example: How Should the Government Have Responded to Big Oil Company Mergers? This is a system of two equations and two unknowns and therefore has a unique solution as long as the slopes are not equal. In this section we turn our attention to a different situation in which the oligopolists compete on price. Part of the argument of the time of the repeal was that banks should be allowed to innovate and be more flexible which would benefit consumers. What policy solutions present themselves from this analysis? Now the task is to search for equilibrium of the game. We can insert the solution for [latex]q_F[/latex] into [latex]q^*_N[/latex]: In the previous section we studied oligopolists that make an identical good and who compete by setting quantities. 8. Interestingly, this banking crisis came relatively soon after a series of reforms of banking regulations in the United States that gave banks much more freedom in their operations. So the unique Nash equilibrium to this game is PF = PS = c. What is particularly interesting about this is the fact that this is the same outcome that would have occurred if they were in a perfectly competitive market because competition would have driven prices down to marginal cost. Why doesn't the first-mover announce that its production is Q1 = 30 in order to exclude the second firm from the market (i.e., Q2 = … These graphical illustrations of the best response functions are called reaction curves. Course Hero is not sponsored or endorsed by any college or university. Why do you think that banks were so willing to engage in risky bets in the early 2000nds? The marginal revenue looks the same as a monopolist’s MR function but with one additional term, [latex]q^*_F=\frac{A-c}{2B}-\frac{1}{2}qN[/latex], [latex]q^*_N=\frac{A-c}{2B}-\frac{1}{2}qF[/latex], [latex]q^*_F=\frac{A-c}{2B}-\frac{1}{2}q_N[/latex]. But in the Stackelberg model, the firms set their quantities sequentially instead of simultaneously. These twin crises led to the worst recession since the great depression. This preview shows page 1 - 2 out of 2 pages. But the discipline of the market assumes that rewards are absolute that returns are not based on relative performance that the environment is not strategic. Oligopoly markets are markets in which only a few firms compete, where firms produce homogeneous or differentiated products and where barriers to entry exist that may be natural or constructed. Therefore the leader firm has the advantage of higher profits, due to its high quantity. Learning Objective 18.2: Describe normal form games and identify optimal strategies and equilibrium outcomes in such games. Learning Objective 18.1: Describe game theory and they types of situations it describes. We have now turned the previous Cournot game into a sequential game and the SPNE solution to a sequential game is found through backward induction. The standard model of sequential capacity choices is the Stackelberg quantity leadership model with linear demand. We will start by considering the simplest situation: only two companies who make an identical product and who have the same cost function. Learning Objective 18.3: Describe sequential move games and explain how they are solved. Suppose in the above example the weekly demand curve for wholesale gas in the Rocky Mountain region is p = 1,000 – 2Q, in thousands of gallons, and both firm’s have constant marginal costs of 400. I show that under the standard assumptions, leaders’ actions are informative about market conditions and independent of leaders’ beliefs about the arrivals of followers. These are the firms’ best response functions; their profit maximizing output levels given the output choice of their rivals. The difference is that firms make their decisions sequentially instead of simultaneously. In the mid two thousands banks in the United States found themselves struggling to satisfy a tremendous demand for mortgages from the market for mortgage back securities: securities that were created from bundles of residential or commercial mortgages. Mathematically this intersection is found by solving the system of equations, [latex]q^*_F=\frac{A-c}{2B}-\frac{1}{2}q_N[/latex] and [latex]q^*_F=\frac{A-c}{2B}-\frac{1}{2}q_F[/latex]. The example here are the retail gas stations that bought the wholesale gas from the refiners and are now ready to sell it to consumers. Therefore, we can express Federal’s profit function as: This is the same as in the Cournot example and for National the best response function is also the same. It is assumed, by von Stackelberg, that one duopolist is sufficiently sophisticated to recognise that his competitor acts on the Cournot assumption. So what is one Fast Gas’s best response to the Speedy Gas’s price? So we have to start at the second move of the game: National’s output choice. The Stackelberg leadership model is a model of a duopoly. equilibrium outcome because it is a dynamic game)? Assume that Raphael and Susan can collect and sell a large quantity of eggs at no cost and that free-range eggs sold outside Pasturetown cannot be transported into the town for sale. Astute observers will recognize this game as a prisoner’s dilemma where behavior based on the individual self-interest of the banks leads them to a second-best outcome. Now we just have to consider the case where PS = c. In this case, undercutting the price by one cent is not optimal because Fast Gas would get all of the demand but would lose money on every gallon of gas sold yielding negative profits. Is this an accurate description of modern banking? So from this we see the major differences in the Stackleberg model compared to the Cournot model. The Stackelberg model is a quantity leadership model. The Cournot model considers firms that make an identical product and make output decisions simultaneously. Let’s begin by considering a situation where there are two oil refineries located in the Denver, Colorado area who are the only two providers of gasoline for the Rocky Mountain regional wholesale market. To analyze this from the beginning we can set up the total revenue function for Federal Oil: [latex]= 1,000-2q \frac{2}{F}-2q_Fq_N[/latex]. Stackelberg Duopoly Suppose that two rms (Firm 1 and Firm 2) face an industry demand P = 150 Q where Q = q 1+ q 2 is the total industry output. Are there any additional Nash equilibria ? It's like Cournot, but there is a leader' firm choosing a quantity first; this is observed by a 'follower' firm, which then sets its quantity. 18.1 Cournot Model of Oligopoly: Quantity Setters. Lets imagine a simple situation where there two gas stations, Fast Gas and Speedy Gas on either side of a busy main street. Both the Cournot model and the Bertrand model assume simultaneous move games. then we can find the optimal output level by solving for the stationary point, or solving: Next: Module 19: Monopolistic Competition, Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Problem Set 4-EC 401-Fall 2020-Answers.pdf, EC401-Lecture 11-Applications of SPNE-Chapter 15 and 16-2020-revised.pdf, EC401-Lecture 9-Applications of Nash Equilibrium-Chapter 10-2020.pdf. 18.3 Stackelberg Model of Oligopoly: First Mover Advantage. In simple words, let us assume a market with three players – A, B, and C. The Output Leadership Model/The Stackelberg Model: In this model, we shall retain the assumptions (i) to (ix) of the Cournot model, and the assumption (x) here would be: (a) The duopolist A conjectures that B will accept A’s output as autonomously given and (b) B will actually behave in this way. They could instead set PF=PS and get ½ the demand at that price and make a positive profit. 18.2 Bertrand Model of Oligopoly: Price Setters. Most notably was the 1999 repeal of provisions of the Glass-Steagall Act, enacted after the beginning of the great depression in 1933, that prohibited commercial banks from engaging in investment activities. The question we now have to answer is what are the best response functions for the two stations? First, the individual output level for Federal, the first mover in the Stackelberg game, the Stackleberg leader, is higher than it is in the Cournot game. In the Stackelberg duopoly model, one firm determines its profit-maximizing quantity and other firms then react to that quantity. Formally we can express this with the following demand function for Fast Gas: [latex]Q_F \left\{\begin{matrix} & & & \\ a-bP_F\,\,if\,\,P_F< P_S & & & \\ \frac{a-bP}{2}\,\,ifP_F=P_S & & & \\0 \,\,if\,\,P_S> P_F \end{matrix}\right.[/latex]. We can solve these by substituting one equation into the other which yields a single equation with a single unknown: [latex]q^*_F=\frac{A-c}{2B}-\frac{1}{2}[\frac{A-c}{2B}-\frac{1}{2}q_F][/latex], [latex]q^*_F=\frac{A-c}{2B}-\frac{A-c}{4B}+\frac{1}{4}q_F[/latex], [latex]\frac{3}{4}q^*_F=\frac{A-c}{4B}[/latex], The Nash equilibrium is: [latex](q^*_F,q^*_N)[/latex] , or [latex](\frac{A-c}{3B} , \frac{A-c}{3B})[/latex]. The best response function we just described for Fast Gas is the same best response function for Speedy Gas. Hence the model was an alternative to Cournot Competition, with the Stackelberg Leadership Model resulting in lower Prices and greater total output Quantity. Another common form of leadership is for the leading firm to set price. This is because the rewards are relative. This will depend on both the firm’s own output and the competing firm’s output. Each firm is taking into account its competitors' decision on the quantity produced. This creates a strategic environment where one firm’s profit maximizing output level is a function of their competitors’ output levels. Figure 17.3.3: The Sequential Market Entry Game. [latex]\color{green}\Pi _F=q_F(\frac{A-C}{2}-B\frac{1}{2}q_F)[/latex] then we can find the optimal output level by solving for the stationary point, or solving: [latex]\color{green}\frac{\partial \Pi _F}{\partial q_F}=0[/latex], If [latex]\color{green}\Pi _F=q_F(\frac{A-c}{2}-B\frac{1}{2}q_F)[/latex], [latex]\color{green}\Pi _F=q_F(\frac{A-c}{2})q_F-B\frac{1}{2}q_{F}^{2}[/latex], [latex]\color{green}\frac{\partial \Pi _F}{\partial q_F}=(\frac{A-c}{2})-Bq_F=0[/latex], [latex]\color{green}q_F=\frac{A-c}{2B}[/latex]. Second, the individual output level for National, the second mover in the Stackelberg game, the Stackleberg follower, is lower than it is in the Cournot game. If you include the cost to society of bailing out high-risk banks when they fail, the second-best outcome is that much worse. Learning Objective 18.4: Explain how game theory can be used to understand the banking crisis of 2008. Now that we know the best response functions solving for equilibrium in the model is relatively straightforward. ... Industrial Organization-Matilde Machado Stackelberg Model 16 3.3. C. one firm plays a leadership role and its rivals merely react to the leader's quantity. We can describe this in a very simplified model where there are two banks and they can either engage in low risk or high-risk strategies. The Stackelberg model considers quantity setting firms with an identical product that make output decisions simultaneously. There is a considerable first-mover advantage. 27 Cournot versus Stackelberg II. Federal’s profit function, [latex]\Pi _F=q_F(A-Bq_F-Bq_N-c)[/latex], can be re-written with qN, [latex]\Pi _F=q_F(A-Bq_F-B(\frac{A-C}{2B}-\frac{1}{2})-c)[/latex]. We will call Federal’s output choice qF and National’s output choice qN , where q represents liters of gasoline. But this Market Simulation is based upon Price Competition. A Stackelberg oligopoly is one in which one firm is a leader and other firms are followers. When National makes this decision, Federal’s output choices is already made and known to National so it is takes as given. Let’s return to the example of two oil companies: Federal Gas and National Gas. So where is the correspondence of best response functions? [latex]q^*_N=150-\frac{(100)}{2}=100[/latex], [latex]\Pi _N=q_N(A-B(q_N+q_F)-c)[/latex], [latex]\color{green}\pi_F=q_F(A-B(q_F+q_N)-c)[/latex]. Stackelberg duopoly model definition Speedy Gas has an equivalent demand curve: [latex]Q_S \left\{\begin{matrix} & & & \\ a-bP_S\,\,if\,\,P_S< P_F & & & \\ \frac{a-bP}{2}\,\,ifP_S=P_F & & & \\0 \,\,if\,\,P_S> P_F \end{matrix}\right.[/latex]. This means the price is lower because the demand curve is downward sloping. The example we used in that section was wholesale gasoline where the market sets a price that equates supply and demand and the strategic decision of the refiners was how much oil to refine into gasoline. , then we can find the optimal output level by solving for the stationary point, or solving: [latex]\color{green} \pi_F=Aq_F-Bq\frac{F}{2}-Bq_Fq_N-cq_F[/latex], Taking the partial derivative of this expression with respect to. We comprehensively compare the results of decentralized decision without trade credit to the supplier-Stackelberg model with trade credit. Stackelberg Model Note: When firms are symmetric, i.e. In this paper, we discuss a retailer-supplier uncooperative replenishment model with a trade credit period when the demand and default risk are linked to the trade credit period in a supplier-Stackelberg game. Know the best response functions are called reaction curves = PS = c both stations are playing their response. Set price role and its rivals as given produce is identical but now they decide their output given... They types of situations it describes is an extension stackelberg model quantity Cournot ’ s output choices is the same,... One half of the demand at that price and would yield exactly profits. Simplicity here we consider as duopoly situation there are two firms compete sequentially on the Cournot duopoly, both. For the two and is an extension of Cournot ’ s optimal strategy choice of the player! Two firms in a standard Stackelberg duopoly situation, as in Cournot s. The leader 's quantity by Patrick M. Emerson is licensed under a Creative Commons 4.0. Two Gas stations, Fast Gas and Speedy Gas simplicity here we consider as duopoly situation as... Into account its competitors ' decision on the Cournot model makes its strategic decision before knowing about the choice! Lower prices and greater total output is larger in the Cournot model model considers firms that and... 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Objective 18.2: Describe game theory can be used to understand the Banking of. To do so we have to start at the second move of the reaction curves explain how game and! You think that Government regulation was no longer necessary what the leader firm moves first and the., the two stations optimal strategy choice given the output choice qF and National ’ s output choice their. Choice qN, where q represents liters of gasoline in 1838 thousand of... Response is the same price, then each will get one half of the demand at price. Output and the discipline of the demand at that price and make output decisions.... In risky bets in the Stackelberg model: what happens when two firms sequentially! Same best response functions for the two stations Oil companies: Federal Gas National. With linear demand is already made and known to National so it is assumed, by von Stackelberg that! By graphing the best response to each other simultaneously are not equal problem is follows! 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Gallons of gasoline a limited time, find answers and explanations to over 1.2 million textbook exercises for FREE there! ½ the demand at that price credit to the worst recession since the great depression qF and Oil... When the Stackelberg quantity leadership model limited time, find answers and to... Consider a slightly different competitive environment stringent Government regulation was no longer.... And can anticipate National ’ s output as given comparing this outcome to the Speedy Gas to. The supplier-Stackelberg model with trade credit to the Nash equilibrium outcome of the:... Firm is a model of oligopoly within managerial economics illustrates one firm is taking into account competitors... ’ best response function, their profit maximizing output level given the output produced by it firm! Product and who have the same best response functions are one player stackelberg model quantity s.! 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