cournot model of non collusive oligopoly

An oligopoly market structure is characterized by barriers to entry and a few firms. The two firms are assumed to produce a homogeneous product. Firm 1s profit- maximising output is thus a decreasing function of how much it thinks firm 2 will produce. Alternatively, it has been assumed that each maximises his profit on the assumption that his rivals price remains unchanged. It depends on whether the product is homogeneous or differentiated, whether there is a dominant firm, whether firms compete based on output or price, etc. It is possible to generalize the Cournot model by considering a situation in which there are many firms. The small firms meet the entire demand P1C at the price OP1. The expression for National is symmetric: Note that we have now described a game complete with players, Federal and National; strategies, [latex]q_F[/latex] and [latex]q_N[/latex]; and payoffs, [latex]\pi_F[/latex] and [latex]\pi_N[/latex]. The total combined revenue for the two firms is then: $TR=P \times Q=(300-Q) \times Q =300Q-Q^2$. Should this fact be regarded as tacit collusion? An oligopoly (from Greek , oligos "few" and , polein "to sell") is a market structure in which a market or industry is dominated by a small number of large sellers or producers. With these assumptions in place, we can express Federals profit function: [latex]\pi_F=P \times q_Fc \times q_F = q_F (P-c)[/latex], Substituting the inverse demand curve, we arrive at the expression, Substituting [latex]Q=q_A+q_B[/latex] yields. In a game of a finite number of repetitions of a Cournot-type model of an industry, if firms are satisfied to get close to (but not necessarily achieve) their optimal responses to other firms' sequential strategies, then in the resulting noncooperative "equilibria" of the sequential market game, (1) if the lifetime of the industry is large compared to the . 24.1. There are two common models that describe the monopolistic competition in an oligopoly: Cournot and Bertrand Competition. Firm 1 maximises 1 with respect to q1, treating q2 as a parameter and firm 2 maximising 2 with respect to q2, treating q1 as a parameter. Empirical evidence suggests that higher costs force a further price rise above the kink. As a result of high price charged by the firm, buyers will shift to products of other sellers who have kept their prices at the old level. Economic models of perfect competition and of monopoly only need to include individual behavior that takes the environment as given. The essence of the Cournot model is that each duopolist treats the output level of its competitor as fixed and then decides how much to produce. The Stackelberg model considers quantity-setting firms with an identical product that make output decisions simultaneously. Lets return to the example of two oil companies: Federal Gas and National Gas. Firstly, it does not explain how the ruling price is determined. Kreps and Scheinkmans (1983) 4 showed that a more realistic model in which firms first choose capacity and then compete in prices has the same solution as Cournot. Ferreira (2104) 9 shows that this is not the case. These cookies ensure basic functionalities and security features of the website, anonymously. Plagiarism Prevention 5. In a collusive oligopoly, the competition is being eliminated by a few firms through a formal agreement. That doesn't look so intimidating, does it? The Cournot Model The Cournot model is an economic model in which the industry that makes homogenous goods products . Mathematically, this intersection is found by simultaneously solving, [latex]q^*_F=\frac{A-c}{2B}-\frac{1}{2}q_N[/latex] and The model has been successful to help understanding international trade under more realistic assumptions. Nash equilibrium is an equilibrium in which no firm has an incentive to deviate from its strategy. These outputs would inevitably add up to 150 and thus maximize joint profits. Under the circumstance, some of these firms act together or collude with each other to reap maximum advantage. Oligopolies: News networks Fox, CNN, and MSNBC. In a Cournot equilibrium, neither firm will find it profitable to change its output once it is able to discover the choice actually made by its rival. The Cournot model does not say anything about the dynamics of the adjustment process, i.e., whether the firms adjust their output until the Cournot equilibrium is reached. He then became the private secretary of a field marshal who required assistance in writing his . That is, raising the price either above or lowering it below the marginal cost would be worse for the firm. A Nash equilibrium is a correspondence of best response functions, which is the same as a crossing of the reaction curves. These cookies help provide information on metrics the number of visitors, bounce rate, traffic source, etc. Cournots adjustment process is somewhat unrealistic. Functional cookies help to perform certain functionalities like sharing the content of the website on social media platforms, collect feedbacks, and other third-party features. as well. This way a system of n equations is formed, and its solution is the Cournot equilibrium. [latex]\frac{\partial \pi_F}{\partial q_F}[/latex], [latex]\frac{A-c}{2B}-\frac{1}{2}[/latex], and they will get no customers at all and make a profit of zero. [latex]=(1,000 2Q)q_F[/latex] These cookies help provide information on metrics the number of visitors, bounce rate, traffic source, etc. In this paper, we model the learning behavior of firms in repeated Cournot oligopoly games using Q -learning. These cookies will be stored in your browser only with your consent. Interestingly, the solution to the Cournot model is the same as the more general Nash equilibrium concept introduced by John Nash in 1949 and the one used to solve for equilibrium in non-cooperative games in chapter 17. The most popular models are price (Bertrand) and quantity (Cournot) competition. If you were to maximize your personal benefit, you would probably use this time to do something productive. Third, the total output is larger in the Stackelberg outcome than in the Cournot outcome. On their face, these decisions seem to make little sense. Imagine that the Lucky Firm goes through the same exercise and finds its reaction function. Another area of application of the model is the analysis of futures markets. B. There are several assumptions in Cournot's model: What's the difference between Cournot and Bertrand? Unfortunately, economic theory does not suggest which assumptions to use. The Cournot model is a one-period method in which each firm has to forecast the other firm's output choice. This action-reaction pattern continues, since firms . And by symmetry, we know that the two optimal quantities are the same: [latex](\frac{A-c}{3B}, \frac{A-c}{3B})[/latex]. TOS4. Intermediate Microeconomics by Patrick M. Emerson is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted. This functional relation between the expected output of firm 2 and the optimal output choice of firm 1 can be expressed as: This functional relation is simply the reaction function, which gives firm 1s optimal choice as a function of its beliefs about the firm 2s choice. quantities) and prices are determined through market clearing. Although digital newspapers and websites have experienced tremendous growth in popularity in recent years, most people still consume news on televisions. The understanding, after all these results, is that the Cournot model can be seen as a reduced form of a more complicated description of an oligopolistic market. But opting out of some of these cookies may affect your browsing experience. By symmetry, we know that National Gas has the same best response function: Solving for the Nash equilibrium, we get the following: [latex]q^*_N=150-\frac{q_F}{2}[/latex] The cookie is used to store the user consent for the cookies in the category "Other. The model of capacity choice plus price competition is no longer equivalent to the Cournot in the presence of futures markets. The two firms are assumed to produce a homogeneous product. The model of Bertrand needed more complications, like the introduction of sufficiently high heterogeneity in the produced goods, to show similar properties. Terms of Service 7. Thus, the dominant firm has nothing to sell in the market. The cookie is set by the GDPR Cookie Consent plugin and is used to store whether or not user has consented to the use of cookies. An example of dominant firm price leadership is shown in Fig. Content Guidelines 2. The model is based on two assumptions. This fact explains stickiness of prices. If new competitors want to enter, they have to spend millions on new factories and other infrastructure. This work by Mapping Ignorance is licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0, 2023 Mapping Ignorance Equilibrium is achieved when MC curve passes through the discontinuous portion of the MR curve. Interested in learning about what this model predicts? In this section, we turn our attention to a different situation in which the oligopolists compete on price. We know that the second movers best response is the same as in section 18.1, and the solution to the profit optimization problem above yields the following best response function for Federal Gas: substituting this into Nationals best response function and solving the following: [latex]q^*_N=\frac{A-c}{2B}-\frac{1}{2}\left [ \frac{A-c}{2B} \right ][/latex], [latex]q^*_N=\frac{A-c}{2B}-\left [\frac{A-c}{4B} \right][/latex], ([latex]q^*_F[/latex], [latex]q^*_F[/latex]). Stackelberg's Duopoly Model are: There are four common non- collusive oligopoly models. A collusion curve shows all the possible output combinations the colluding firms can produce to maximize joint profits. It is rational for each firm to assume that its competitors output remains fixed only when the two firms are choosing their output levels only once because then their output levels cannot be changed. These cookies will be stored in your browser only with your consent. [latex]A-2Bq_F-Bq_N[/latex][latex]=[/latex][latex]c[/latex]. There are ample examples of oligopoly. Such dynamic adjustment is explained by other models. For instance, it explains how otherwise wasteful subsidies could be beneficial to promote exports (Brander, 85) 1, or how dumping practices could actually be socially beneficial (Brander and Krugman, 1983) 2. Solving this by steps results in the following: [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] Image Guidelines 4. This website uses cookies to improve your experience while you navigate through the website. Functional cookies help to perform certain functionalities like sharing the content of the website on social media platforms, collect feedbacks, and other third-party features. Oligopolies often result from the desire to maximize profits, which can lead to collusion between companies. This duopolistic example shows that the firms marginal revenue curves include one extra term: [latex]MR_F(q_F)=A-2Bq_F-Bq_N[/latex] and [latex]MR_N(q_N)=A-2Bq_N-Bq_F[/latex], The profit maximizing rule tells us that to find the profit maximizing output, we must set the marginal revenue to the marginal cost and solve. But this seems to be a totally unrealistic assumption for a homogeneous product. If firm 1 decides to produce q1 units of output, it expects that the total output produced will be Q = q1 + q2e and industry output will yield a market price of p (Q) = p (q1 + q2e). The weekly demand for wholesale gas is still [latex]P = ABQ[/latex], where [latex]Q[/latex] is the total quantity of gas supplied by the two firms, or, We have now turned the previous Cournot game into a sequential game, and the [latex]SPNE[/latex] solution to a sequential game is found through backward induction. Thus if there are a large number of firms, none can exert much influence on the market price. Then, in 2001, Chevron acquired Texaco for $38.7 billion. Iso-profit curves are lines showing those combinations of two competitors products q1 and q2 which yield a constant level of profit to firm 2. Privacy Policy 9. A success story [], Your email address will not be published.Required fields are marked *. The resulting equilibrium is a Nash equilibrium in quantities, called a Cournot (Nash) equilibrium. This was the essence of the earliest model of oligopoly developed by Cournot as far back as 1838: each firm acts independently on the assumption that its decision will not provoke any response from rivals; this is not generally accepted nowadays as providing a useful framework in which to analyse contemporary oligopoly behaviour. Key Differences Between Collusive Oligopoly and Non-Collusive Oligopoly. Time is precious, so why waste it? Uncertainty about the interaction of rival firms makes specification of a single model of oligopoly impossible. The policy implications of this analysis are presented in Section 3. An iso-profit curve for firm 1 is the locus of points indicating different levels of output of firm 1 and its rival firm 2, which yield to firm 1 the same level of profit, as shown in Fig. Let's look at an example of a Cournot model with equations and graphs! A Cournot Nash equilibrium describes a Nash equilibrium in a Cournot model of oligopoly, in which firms choose how much to sell (i.e. In actual practice, the analysis of price leadership is complicated, particularly when new firms enter the industry and try to become the leader or dominant. When a firm purchases assets of another firm, acquisition takes place. [latex]C = 400[/latex]. Viewed through the lens of the models of oligopoly studied in this chapter, the FTCs decision to demand a divestment in oil refining and wholesale gas operations but mostly allow the retail side to consolidate makes sense. 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. Cournot saw a colluding duopoly acting akin to a monopoly in terms of price and quantity setting. Between these two extreme cases one finds the realm of the oligopolies, where firms cannot take prices or own demand as given, since they depend on their own choices. It is a stable equilibrium, provided that A's reaction curve is steeper than B's reaction curve. For instance, an oligopoly with say four firms will establish a price above the competitive one. Have you read up to this point? Cournot's Duopoly Model: In 1838, A French economist, Augustin Cournot has developed a model on oligopoly. Figure 3 shows the collusion curve in yellow, which has some very important insights. (This condition is satisfied by the assumption we made that the highest points of successive isoprofit curves of A lie to the left of one another, while the highest points of B's isoprofit curves lie to the right of each other.) In this case the demand curve facing the firm is the market demand curve. [latex]/frac{3}{4}q^*_F=25[/latex] Being a leader in the industry, the dominant firms supply curve is represented by the MCL curve. In other words, every price will be matched by an equivalent price cut. Clearly, this third option is the one that yields the most profit. Since, cournot competition is an economic model in which rival companies offering an identi View the full answer Transcribed image text: Under non-collusive oligopoly, firms seeking to maximise profits must make assumptions about their rivals. It turns out that when firms compete in an oligopoly, a similar model of behavior known as the Cournot model applies. We can insert the solution for [latex]q_F[/latex] into [latex]q^*_N[/latex]: [latex]q^*_N=150-\frac{(100)}{2}=100[/latex]. Check out our articles that might satisfy your hunger for learning: The Cournot model in economics is a model of oligopoly where firms producing homogeneous products compete in quantities. One important characteristic of an oligopoly market is interdependence among sellers. We don't need to go over all these calculations for the Lucky Firm as we know that its reaction function is symmetric and is: We know that the Cournot equilibrium occurs when the two functions intersect. The Cournot model is based on the following assumptions: (i) There are only two non-collusive firms, i.e., there exists the simplest example of oligopoly, viz., duopoly. The difference between Cournot and Bertrand models is that in the Cournot model, firms compete in quantities. Augustin Cournot came up with the model of oligopoly in 1838. Oligopoly is a fascinating market structure due to interaction and interdependency between oligopolistic firms. Our mission is to provide an online platform to help students to discuss anything and everything about Economics. In cournot model, firms make assumptions about their rival's choice of output. Both stations have large signs that display the gas prices that each station is offering for the day. This chapter considers all three in order, beginning with the Cournot model. We can now plot our Cournot equilibrium on a diagram! Answer: The question is ill-poised. The Happy Firm and the Lucky Firm happen to produce the same quantities, but this doesn't have to be the case. Other works have challenged this view, but the discussion in the present article is not about this ongoing research, the issue is rather whether the Cournot model can still be viewed as a reduced form of the more realistic scenarios in the presence of futures markets. Each firm exercises its most profitable strategy, given its competitor's choice. The amount of profit that the firms were making jointly before cooperation was: By colluding, they can enjoy higher profits of: $\pi_1+\pi_2=P \times Q = (300-150) \times 150 = 22,500$. Cournot Oligopoly - January 1989. The next example is still more complex. Kemplerer and Meyer (1989) 6, on their part, showed that when firms in an oligopoly compete in supply functions (like electricity auction markets, where firms propose both a quantity and a price to sell) the result is again the Cournot behavior, provided there is some uncertainty in the demand. He is a member of ARP-Sociedad para el Avance del Pensamiento Crtico (Society for the advancement of critical thinking). Firms also want to put their resources to use in the best way given the other party's action! So the equilibrium condition is the same as that of a monopolist, i.e., MR = MC, where MR = p(Q) [1 1/|e(Q)|]. In this case, the Federal Trade Commission (FTC) was the agency that provided oversight, and in the end, they approved the merger with the following condition: they had to sell their stake in two massive oil refineries. The original model was presented in a simple way by assuming that two firms (called duopolists) have identical products and identical costs. Now firm 1 produces q1t+1 firm 2 will react by producing q2t+1. The two curves have the same form because the two firms in our example are the same. [latex]q^*_F=\frac{A-c}{3B}=\frac{1,000 400}{(3)(2)}=\frac{600}{6}=100[/latex]. This is because in the Cournot case, both firms took the others output as given. Total output is the sum of the two and is 200,000 gallons. Sweezy uses kinked demand curve to describe price rigidity in oligopoly market structure. This sort of competition leads to an inefficient equilibrium. Each reaction curve shows the relationship between a firms profit-maximising output and the amount it thinks its competitor will produce. When National makes this decision, Federals output choices are already made and known to National, so it is taken as given. These cookies track visitors across websites and collect information to provide customized ads. The history of his model is the history of a success. Uploader Agreement. Allaz and Vila (1993) 8 showed that in this simple form, a side effect of the futures market is that firms behave more competitively (it works as if firms have to compete twice). We also share information about your use of our site with our social media, advertising and analytics partners who may combine it with other information that youve provided to them or that theyve collected from your use of their services. Common models that explain oligopoly output and pricing decisions include cartel model, Cournot model, Stackelberg model, Bertrand model and contestable market theory. Augustin Cournot came up with the model of oligopoly in 1838. Profits of firm 2 will increase as it moves to iso-profit curves what are further and further to the left. The basic model and main findings are developed in Section 2. The lower supply and higher prices do material harm to consumers, however, and it is for this reason that the FTC stepped in and demanded that the merged company sell off its interest in two big refining operations. The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional". The reaction curve of firm 2 is the locus of points of highest profits that firm 2 can attain, given the level of output of its rival. If they have the same price, then each will get one-half of the demand at that price. We can then plug the value of Q2 into the equation for Q1 (1) to get: $Q_1=150-\frac{1}{2}\times(150-\frac{1}{2}Q_1)$$Q_1=150-75+\frac{1}{4}Q_1$. While the group holds a great deal. Cournot illustrated his model with the example of two firms each owning a spring of mineral water which is produced at zero marginal cost. Research, however, never ends, and the Cournot model faces new challenges. Merger and acquisition take place because the management comes to a conclusion that a consolidated firm is powerful than the sum of individual firms. [latex]\pi_F=q_F(A-B(q_F+q_N)-c)[/latex]. b) each firm chooses simultaneously and non-cooperatively its own product's price to maximize its own profit. can be re-written, replacing [latex]q_N[/latex] with the best response function: [latex]\Pi _F=q_F(A-Bq_F-B(\frac{A-C}{2B}-\frac{1}{2})-c)[/latex], If the profit function is [latex]\Pi_F[/latex][latex]=[/latex][latex]q_F([/latex][latex]\frac{A-C}{2}-[/latex][latex]B[/latex][latex]\frac{1}{2}[/latex][latex]q_F)[/latex], then we can find the optimal output level by solving for the stationary point, or solving, [latex]\frac{\partial \Pi _F}{\partial q_F}[/latex][latex]=[/latex][latex]_0[/latex], If [latex]\Pi_F[/latex][latex]=[/latex][latex]q_F([/latex][latex]\frac{A-c}{2}-[/latex][latex]B[/latex][latex]\frac{1}{2}[/latex][latex]q_F)[/latex], then we can expand to find, [latex]\Pi_F[/latex][latex]=[/latex][latex]q_F([/latex][latex]\frac{A-c}{2}[/latex][latex])q_F[/latex][latex]-B[/latex][latex]\frac{1}{2}[/latex][latex]q_{F}^{2}[/latex], Taking the partial derivative of this expression with respect to [latex]q_F[/latex], we get, [latex]\frac{\partial \Pi _F}{\partial q_F}[/latex][latex]=([/latex][latex]\frac{A-c}{2}[/latex][latex])[/latex][latex]-[/latex][latex]Bq_F=[/latex][latex]_0[/latex], [latex]q_F=[/latex][latex]\frac{A-c}{2B}[/latex]. He studied Economics at the University of the Basque Country and obtained his PhD at Northwestern University. Non-collusive Oligopoly 217 p 0 o' X Figure 9.1 Firm B reacts on the Cournot assumption, and will produce one-half of the unsupplied section of the market, i.e. Sulit dimasuki perusahaan baru (Terdapat hambatan masuk yang cukup kuat bagi perusahaan di luar pasar untuk masuk dalam pasar) 4. If on Monday a firm can sign a contract with some buyers to deliver a quantity at an agreed price, that agreement constitutes a futures contract. This is why modern economists generalize the presentation of the Cournot model by using the reaction curves approach. However, this model leads to a paradox: a market with only two firms is enough to achieve perfect competition, a fact not easily found in the real economy. This is common, since collusion is illegal and price wars are costly. Therefore, they produce the same quantities: What is interesting to see is something called a collision curve. Learning Objective 18.3: Describe the different outcomes when oligopolist firms choose quantities sequentially. The history of his model is the history of a success. To better explain this phenomenon, we have presented the nine best examples of oligopoly in different industries. 2 The model To investigate the problem of interest, we employ a Cournot model with complementary Advertisement cookies are used to provide visitors with relevant ads and marketing campaigns. ISSN 2529-8992 In a Non Collusive Cournot Duopoly model with two firms, zero costs and linear demand curve. This means the price is lower because the demand curve is downward sloping. Rearrange to find Q1:$2Q_1=300-Q_2$$Q_1=150-\frac{1}{2}Q_2$ (1)We found the reaction function for the Happy Firm! In such a situation the Cournot model will generate sub-optimal outcome. Prohibited Content 3. 'We assume that the products that the firms produce are homogeneous. Its output is a function of firms 2 output level so q1 = f1 (q2) just as q2 = f2(q1). Each firm is assuming that the others output will remain fixed from one period to the next, but both firms keep changing their output levels. When it comes to Federals decision, we diverge from the Cournot model because instead of taking [latex]q_N[/latex] as a given, Federal knows exactly how National will respond because they know the best response function. Firm 2 will make the maximum amount of profit when it is a monopolist, i.e., when firm 1 decides to produce zero unit of output. So in a situation where competition is based on price and the good is relatively homogeneous, as few as two firms can drive the market to an efficient outcome. (a) Cournot model (c) Bertrand model 57. In this sense, it is not a theory of pricing. positive.(Neary,2010). 24.3 we start with output (q1t, q2t) which are not equilibrium outputs. Non-Collusive Oligopoly There is competition with the companies in non-collusive Bertrand oligopoly c. Cournot oligopoly d. Monopoly; What is the main punishment differences in Cournot and Bertrand Models? In other words, in oligopolistic industries price is more stable than costs. We can obtain a firm's reaction curve using the MR R = MC R condition. Firm A produces half the total market demand to maximise revenue. But the model has certain limitations. The model we use to analyze this is one first introduced by French economist and mathematician Antoine Augustin Cournot in 1838. In the Cournot model each firm chooses its output level assuming1 that the other firms output will be q1e or q2e. There is no single model of profit-maximizing oligopoly behavior that corresponds to economists' models of perfect competition, monopoly, and monopolistic competition. A Cournot equilibrium is stable, and there is no incentive for the two firms to engage in price wars. Which one of the following ig not a non-collusive oligopoly model ? Here we assume that each firm has an expectation about the output choices of the other firms. A non-collusive oligopoly refers to a market situation where the firms compete with each other rather than cooperating. Jos Luis Ferreira is an Associate Professor at the Economics Department in Universidad Carlos III de Madrid. This is a system of two equations and two unknowns and therefore has a unique solution as long as the slopes are not equal. The Bertrand model shows us that it takes very little competition to result in highly competitive pricing, so a merger that might reduce the number of competing gas station brands by one is unlikely to have much of a material effect on prices and therefore will be unlikely to harm consumers. 24.3 (point E). [latex]q^*_F=100[/latex]. Now the task is to search for the equilibrium of the game. Structure 10.0 Objectives 10.1 Introduction 10.2 Non-Collusive Oligopoly 10.2.1 Cournot Model of Duopoly 10.2.2 Bertrand Model of Duopoly 10.2.3 Edgeworth Model 10.2.4 Chamberlin's Oligopoly Model 10.2.5 Kinked Demand Curve: Sweezy Model 10.2.6 Stackelberg Model 10.3 Let Us Sum Up 10.4 Key Words 10.5 Some Useful Books 10.6 Answer or Hints to Check Your . So what is Fast Gass best response to Speedy Gass price? On the other hand, retail gas is more akin to the Bertrand model, where a bunch of retailers are selling a homogenous good but are competing mostly on price. Share Your Word File For each possible level of firm 1s output, firm 2 wants to choose its own output in order to make its profits as large as possible. The question that arises now is: how do oligopoly firms remove uncertainty? In fact, firms enter into pricing agreements with each other instead of adopting competition or price war with each other. Cournot model shows that firms can operate as a cartel and gain higher profits if they join rather than compete against one another. Oligopolists face downward-sloping demand curves, which means that price is a function of the total quantity produced, which, in turn, implies that one firms output affects not only the price it receives for its output but the price its competitors receive as well. Necessary cookies are absolutely essential for the website to function properly. The cookie is used to store the user consent for the cookies in the category "Performance". For each choice of output by firm 1 (q1), firm 2 chooses the output level q2 = f2(q1) associated with the iso-profit curve farthest to the left. The definition of Cournot's model in economics is that it is a model of oligopoly where firms producing homogeneous products compete in quantities. Fast Gas could instead set, [latex]\frac{\partial \Pi _F}{\partial q_F}[/latex]. To analyze this from the beginning, we can set up the total revenue function for Federal Gas: [latex]TR(q_F)=pq_F[/latex]
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