# Numerical Simulations of Competition in Quantities

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1 Numerical Simulations of Competition in Quantities Devon Gorry Department of Economics & Finance Utah State University John Gilbert Department of Economics & Finance Utah State University Abstract We present a series of numerical simulation models built in Excel that can be used to explore the properties of various models of strategic competition in quantities and their economic implications. The resources incorporate both tabular and graphical data presentation formats and are built in such a way that they provide instant or live feedback on the consequences of changes in the economic system. We discuss the theory behind the models, how they can be implemented as numerical simulations in Excel, and ways in which the simulations can be used to enhance student understanding of the material. JEL: A22, C63, D43 Keywords: Excel, numerical models, competition in quantities 1 Introduction Models of strategic competition in quantities are a mainstay of the undergraduate curriculum at multiple levels. The Cournot model of oligopoly is frequently taught in both introductory and intermediate microeconomics, and its various extensions form a core component of most courses in industrial organization. Moreover, models of competition in quantities are also commonly used in the international trade classroom when covering new trade theory models of the reciprocal dumping variety. The models provide important insights into core concepts from game theory and into the Correspondence address: Devon Gorry, 3565 Old Main Hill, Logan, Utah Tel:

2 implications of variations in market structure for economic outcomes. Hence, it is important that students develop a solid understanding of the properties of these models. There are a number of ways to teach models of strategic competition in quantities like the Cournot model. At the principles level a graphical approach is likely to be adopted, while in intermediate and senior-level industrial organization classes, simple models of competition in quantities are often used to introduce more general concepts from game theory and are typically solved algebraically for simple functional forms. A useful analog to these approaches is the use of numerical examples, which are a good mechanism for developing economic intuition. Numerical simulation makes abstract results more compelling for students, allows faster and more general demonstration of model properties, and can help connect models based on a common foundation. There are a wide range of examples in the literature, including models from microeconomics principles (Mixon and Tohamy, 2000), international trade (Gilbert and Oladi, 2011), macroeconomics (Findlay, 2014) and financial economics (Guo and Gilbert, 2014). While there are many platforms on which numerical simulations can be built, Excel has a number of advantages for pedagogical purposes. 1 It allows for both tabular and graphical presentations of information, can be used to solve quite complex problems, and is widely available on many computer platforms. It is a also a program with which most students are already familiar, and it is widely used outside of the classroom. In this paper we describe a series of numerical simulation models built in Excel that can be used to support instruction on models of strategic competition in quantities. As in Gilbert and Guo (2014), the models are built in such a way that they respond instantly to changes, have a consistent presentation across various extensions, and gradually build complexity onto a common foundation. The paper is organized as follows. In the next section we set out the structure of the basic model and its solution, followed by a review of the geometry of the problem. We then turn to the question of how to implement the model as a numerical simulation in Excel. Next, we discuss how the resources we have built can be obtained and used, and we provide details on various extensions. Throughout we provide examples of exercises that can be used to explore the implications of differences in market structure for various economic variables of interest. We conclude with a few notes on how the approach we have taken in this paper could be extended to other topics. 1 Excel is useful in other contexts too. Wight (1999) discusses some general benefits of using spreadsheets beyond numerical simulation, while Briand and Hill (2013) consider the use of Excel for econometrics. 2

3 2 The Basic Cournot Duopoly We begin by sketching the basic Cournot duopoly model and its solution. We follow a standard, textbook approach and assume: (1) there are two firms (no entry); (2) firms have constant marginal and average costs, denoted c i ; (3) the firms are producing a homogeneous product; (4) the overall market demand function is linear, with the inverse demand being specified as P (Q) = a bq where a, b > 0 and a > c i ; and (5) the firms must choose their outputs simultaneously. 2 The Cournot output represents a Nash equilibrium in a game where firms simultaneously choose output. Thus, the equilibrium is a list of price and output levels for each firm such that neither firm can increase their profits by changing their output given that the other firm does not change their output. Formally, the Cournot-Nash equilibrium is a triplet {p c, q1 c, qc 2 } such that: Given q 2 = q c 2 ; qc 1 solves max q 1 π 1 (q 1, q c 2 ) Given q 1 = q c 1 ; qc 2 solves max q 2 π 2 (q c 1, q 2) p c = a b(q c 1 + qc 2 ) We begin by solving firm 1 s maximization problem, whereby firm 1 chooses it s profit maximizing quantity and takes firm 2 s quantity as given. This yields firm 1 s best response function, R 1 (q c 2 ). max q 1 (a b(q 1 + q c 2))q 1 c 1 q 1 The first order condition for a maximum is: 3 a 2bq 1 bq c 2 c 1 = 0 and the explicit best response function is: R 1 (q c 2) = a c 1 2b 1 2 qc 2 2 A more complete treatment of the model, and those that follow, can be found in typical industrial organization texts, such as Carlton and Perloff (2005). 3 It is easy to see that the second-order condition is also satisfied. 3

4 Using symmetry, firm 2 s maximization will yield a similar best response function, R 2 (q c 1 ). R 2 (q c 1) = a c 2 2b 1 2 qc 1 Equilibrium occurs at the intersection of these two best response functions. By substituting firm 2 s best response function into firm 1 s, we solve for q c 1 : q c 1 = a 2c 1 + c 2 3b and similarly for q c 2, q c 2 = a 2c 2 + c 1 3b Market quantity is given by Q c = q c 1 + qc 2, while the price is pc = a bq c. We can then substitute back for the firm s profits, consumer surplus, or any other economic variable of interest. 3 The Geometry of the Solution The solution to the model can be illustrated in several ways. In Figure 1, we show the profit maximizing output choice of firm 1 for any given level of output of firm 2. The line labeled D is the market demand curve. Subtracting the quantity produced by firm 2 (q 2 ) from the market demand curve yields the residual demand curve for firm 1, labeled RD 1. The line labeled c 1 is the firm 1 s marginal (and average) cost. The firm maximizes profit by producing at the quantity where the marginal revenue arising from the residual demand curve (labeled MR 1 ) is equal to the marginal cost, at point a. Drawing a vertical line up to the residual demand curve (at b) yields the market price. Since the distance bd is equal to q 2, the market clears. A similar diagram can be constructed for firm 2. The equilibrium outcome for both firms is described in Figure 2. The best response functions describe the profit maximizing output choices of each firm as a function of the output choices of the rival. The best response of firm 1 is labeled R 1, while that of firm 2 is labeled R 2. The Nash equilibrium is where the two best responses coincide, i.e., point a. The best response functions have the property that profit is maximized for firm 1 at point b, and for firm 2 at point c. These are the monopoly levels of output. Profit is minimized for firm 1 at point 4

5 P P b d a c 1 q 1 MR 1 RD 1 q 2 D Q Figure 1: The Decision of Firm 1 d, and for firm 2 at point e, which correspond to where price equals marginal cost, or the competitive level of output. Conceptually, we can envisage two sets of concave isoprofit curves, with higher profits to each firm the lower the rival s output. The loci of the peaks of the isoprofits are the best response functions. Hence, at the Cournot-Nash equilibrium, the isoprofit curves are orthogonal. Since they are concave, it must be possible to find a Pareto improvement over the Nash equilibrium. That is, if both firms could agree to reduce output, they could both enjoy higher profits. The solution to the joint profit maximization problem in fact lies at any point along the dotted line between c and b, which is the locus of tangencies of the isoprofits of each firm. 4 The point labeled f is one such solution, with each firm producing exactly half the monopoly level of output. Point g represents the equilibrium that would be obtained if the firms move sequentially (with firm 1 deciding on its output level first). We discuss this case further below. 4 Assuming symmetric marginal costs. If marginal costs differ, the joint profit maximizing solution is at the end point of the best response function of the lower cost firm. 5

6 q 2 e c q 2 a f g q 1 b R 1 d R 2 q 1 Figure 2: Equilibrium Outcome 4 Building an Excel Version There are a number of ways to build a simulation model in Excel. In many cases, the SOLVER add-in is used. This feature can solve optimization problems directly, but requires that the add-in be correctly installed and set up. It also requires that the optimization routine be launched, either manually or via a macro, before a change can be observed. Hence, it adds a degree of complexity to the model-building and is best used when the problem involves significant nonlinearities and/or when obtaining closed form solutions is difficult. In the case at hand, the use of linear demand and cost functions makes solving the problem for a closed form solution relatively simple, so we can implement the solution directly in the Excel sheets cells. This has the advantage of avoiding potential difficulties with SOLVER and also makes the entire model live in the sense that it responds instantly to any changes in parameters. The process is quite straightforward. We begin by designating cells to hold values for exogenous parameters. In Figure 1 these are cells D5, D6, D10 and D18. Next we allocate cells to contain the values of the endogenous variables, blocking them into logical groups (cells D11 through D14, D19 6

7 through D22 and D26 through D29). For consistency across the various sheets, we have adopted the coloring convention that white cells can be changed and those in gray are derived from the model. Figure 3: Excel Layout To fill in the endogenous variables, we enter the solutions as formulas involving the exogenous variable cells. We begin with the solutions for q1 c and qc 2. In cells D11 and D19, respectively, we enter the following formulas: (D5-2*D10 + D18)/(3*D6) (D5-2*D18 + D10)/(3*D6) These are just the solutions derived in Section 2 above expressed in terms of the appropriate cells. From there on, the remaining entries are straightforward: Total output in cell D26 is D11+D19. Price in cell D27 is D5-D6*D26. Revenue for firms 1 and 2 in cells D12 and D20 are D11*D27 and D19*D27, respectively. 7

8 Total costs for firms 1 and 2 in cells D13 and D21 are D11*D10 and D19*D18, respectively. Profits for firms 1 and 2 in cells D14 and D22 are D12-D13 and D20-D21, respectively. Consumer surplus in cell D28 is defined as (D5-D27)*D26/2. Total surplus in cell D29 is defined as D14+D22+D28. Note that it is not necessary to obtain closed form solutions for all of the endogenous variables. We can express the solutions either in terms of exogenous factors directly, or in terms of endogenous factors which have already been expressed in closed form. In effect, Excel will do the necessary substitutions for us. Figure 4: Excel Graphs To complement the numerical solutions, we present graphs, as shown in Figure 4. The two stacked graphs on the left of the figure present the residual demands faced by each firm, while on the right we have the best response functions. These depictions are much the same as those shown in Section 3 and are typical of textbook treatments. However, they are also live and will move directly as the parameters of the model are changed. 5 5 The graphs themselves are just connected scatterplots. The data underlying them, which is based on the visible cells, is hidden in the sheet to the right of the model. 8

10 way of developing an understanding of how the models work. Some possible starting points for the Cournot model might be (with learning objectives in italics): 1. Allow the marginal costs to differ across firms. How does the equilibrium output of the low-cost firm compare to the output of the high-cost firm? Which firm has higher profits? We find that the lower cost firm will produce more and make higher profits in equilibrium than the higher cost firm. 2. How does the profit of the two firms depend on the slope of the market demand curve? Can you explain the pattern you observe? The slope of the demand curve relates to the elasticity of demand, which in turn is a measure of the firm s collective market power. The flatter the curve, the less the market power, and consequently the lower the price and profit. 3. Could the cost of one firm be so high that it would choose to produce nothing? if so, how high would the cost have to be? How much would the other firm produce in this case? How would their profit compare to the case where both firms produced? If the cost of one firm is high enough (equal to the price intercept of the residual demand at their rival s optimal output) they will cease to produce. The other firm will respond by fully exploiting its monopoly power and will have greater profit. 4. Imagine that one firm received a subsidy, which lowers its marginal cost, potentially below zero. Is it possible to lower marginal cost so much that the rival chooses to exit the market? It is possible, if the subsidy is enough to drive the market price down to the marginal cost of the rival. In this case the price is at the competitive outcome. 5. Do consumers win or lose in the above scenarios? Consumer surplus depends on the market price, so they lose in scenario 3 and gain in scenario 4. 6 Extensions Following the approach outlined above for the Cournot duopoly case, we built several variations to allow students to see how different market structures affect the equilibrium outcome and how these market structures relate. With each variation, we solve the model and enter it into Excel exactly as outlined above for the Cournot case, taking care to maintain consistency in presentation across the models. Because of the similarity in the basic approach, we omit a detailed discussion and focus on a 10

11 brief description of the model extensions and the exercises that can help students learn key features of the extensions and how the models relate to one another. 6.1 Differentiated Goods The differentiated goods model extends the simple Cournot duopoly case by allowing the firms to face related but different inverse demand curves, while retaining the assumption of simultaneous moves. The model is useful for showing how product differentiation relates to market power. The inverse linear demands are given as follows: p 1 (q 1, q 2 ) = α 1 β 1 q 1 β 2 q 2 p 2 (q 1, q 2 ) = α 2 β 2 q 1 β 1 q 2 Here, β 2 1 β2 2 so that the price of a good responds more to the quantity of that good than to the quantity of the related good. The solution procedure is essentially the same as that set out in Section 2. Some exercises that can be conducted with this model include: 1. What happens to the prices, quantities, and profits of each firm as the slope parameter on the rival good approaches zero? When the β 2 parameters are zero, the goods are unrelated. In this case we have two monopolists. 2. What happens to the slope of the best response function of each firm when the rival slope decreases? Explain why the slope of the response function changes. It becomes steeper/flatter. This is telling us that the firm s do not respond to one another. 3. Set the demand rival slope equal to the demand slope. Compare your results to the Cournot duopoly model with the same demand slope and costs. Do you get the outcome you expected? With β 1 = β 2, the goods are perfect substitutes, i.e., homogeneous. In this case the model coincides with the standard Cournot model. 6.2 Many Firms The many firm case mirrors the Cournot duopoly case, but allows the user to choose the number of firms in the market, n, where n 1. We have set the problem up such that firm 1 may have a cost structure that differs from all the other firms, in order to maintain consistency with the other 11

12 Excel sheets. Since all firms enter simultaneously, the problem is not substantially different than the one discussed in section 2. This sheet is particularly useful for considering the relationship between Cournot, monopoly and competitive market outcomes. Some useful exercises include: 1. If you set number of other firms equal to zero, how does the outcome compare to that of a monopoly? With only one firm, the outcome will be monopoly. Cournot oligopoly converges to monopoly as the number of firms approaches If you set number of other firms equal to one, how does the outcome compare to the Cournot duopoly with the same demand and cost parameters? This case will simply replicate the Cournot duopoly. 3. What happens to the total output, price, and total surplus as the number of firms increases? As the number of firms increases, output rises, price falls and total surplus rises. The price should get closer and closer to marginal cost. The Cournot model converges to the competitive outcome. 6.3 Stackelberg Duopoly The Stackelberg duopoly returns to the homogenous good case and two firm assumptions of the Cournot duopoly. The model differs from Cournot by allowing firms to act sequentially instead of simultaneously. Firm 1 is the leader and chooses its quantity first while firm 2 is the follower and chooses its quantity after observing firm 1 s choice. This is a classic model for depicting first-mover advantage. The game is solved via backward induction. Firm 2 will solve the same problem as in the Cournot case, which will yield the same best response function. Firm 1 can predict firm 2 s best response function and will take this into account when choosing how much quantity to produce in the first period. Mathematically, firm 1 maximizes its profit subject to the market conditions, its own cost, and firm 2 s output, assuming that firm 2 will maximize its own profit in response to firm 1 s choice. 7 Some useful exercises with this model include: 1. Compare the total equilibrium output and price obtained in the Stackelberg Model to the Cournot Model (keeping costs and demand parameters constant). Why do the results differ? The total output is actually higher in Stackelberg than in Cournot, and the price correspondingly lower. 7 The outcome is depicted in Figure 2 as point g, the best isoprofit of firm 1 that can be reached while staying on firm 2 s best response. 12

13 2. Set costs of firm 1 equal to the costs of firm 2. Compare the first mover s (firm 1) production and profit to second mover s production and profit. Who is better off? How do the results compare to the Cournot Duopoly? Firm 1 produces more output and attains a higher profit - there is a first-mover advantage in this model. 3. In the Cournot duopoly model, suppose that firm 1 received a subsidy from the government that reduced its marginal cost, possibly below zero. Is it possible for a subsidy to generate the Stackelberg output levels? How much profit would firm 1 make if so? How much would they make net of the subsidy payments? This exercise reveals the logic underlying strategic trade policy (if we imagine the firms being in separate countries). A subsidy that induces firm 1 to produce the Stackelberg output level induces firm 2 to reduce its output similarly. The subsidy is equivalent to granting a first-mover advantage, and profit net of subsidy payments is maximized. 6.4 Stackelberg with Many Firms Next we combine the Stackelberg model with the many firms model. In this case, firm 1 moves first, and then all the other firms choose quantity simultaneously in the follower round after observing firm 1 s choice. Again, the model is solved via backwards induction. With a large number of firms in the second stage, we effectively have a short-run variant of the dominant firm with a competitive fringe model. 1. What happens to the profits of the first mover as the number of second movers increases? It declines. This is the key result from this model, the first mover advantage diminishes when the number of rivals is large (and disappears when the second stage is competitive.) 2. Is there some point where the number of rival firms would not rise further? If we imagine that firms can choose to be in the second stage or not then yes, eventually the price would fall below the marginal cost of the second mover firms, and entry would not make sense. We explore this idea further with the next model. 3. What happens to the output of the first mover as the number of second movers increases? It remains constant. The lesson is that being a first mover can lead to dominant market share, even if it doesn t always lead to a profit advantage. 4. Assume that the number of firms in the second stage must be such that profit for the typical 13

14 second mover is just above zero. If the marginal cost of the first mover falls, what must happen to the number of second movers? If the marginal cost of the first mover is lower, it can profitably produce more. For given demand conditions, this leaves less market share for the second movers. Hence, their number must decline unless they (temporarily) accept losses. 6.5 Monopolistic Competition In our final extension, we show how we can use the basic Cournot setup to derive a model of monopolistic competition. We return to a simultaneous move game with many firms, but the number of firms is no longer fixed. Instead, firms face both marginal and fixed costs and the number of firms that exist in the market is determined endogenously. The number of firms is decided by an entry condition, with firms entering until zero profit holds. Again for consistency across the models, we allow firm 1 to have (potentially) different costs than the remaining firms What is the profit of firm 1 if costs are the same as all other firms? All firms make zero profit in equilibrium in this case. 2. How does the profit of firm 1 change if the marginal cost falls? If the fixed cost falls? If firm 1 has lower costs (marginal or fixed) it will make higher profits, although only the marginal costs affect output. 3. What happens to the number of firms as fixed costs increase? The number of firms falls, some must exit in order to keep price at average cost. 4. How many firms do you think would enter if the fixed cost of production was equal to the profit firm s made in the Cournot duopoly (assuming symmetry)? The optimal number of firms would be two. The market outcome would replicate Cournot, with the exception of producer profits. 4. What would happen if fixed costs were very close to zero? The market outcome would replicate perfect competition. This can be verified by comparing to the Cournot model with many firms. 8 As it is set up, the costs of firm 1 should be no higher than the costs of the remaining firms. Also note that non-integer firm numbers are allowed. 14

15 7 Concluding Comments Numerical simulation is a very useful tool for reinforcing students economic intuition on model behavior and for emphasizing the linkages between various economic models. The common foundation of the models can sometimes be lost when covering many seemingly different structures. Hence, much can be learned by experimenting with model parameters and seeing how the results of different models converge, and numerical simulation models are well-suited to this task. Industrial organization, with its almost limitless variety of models of market structure, is a particularly useful area of application. In this paper we have outlined a set of new resources that can be used to aid in teaching basic models of strategic competition in quantities and extensions, along with suggestions for how they can be used. The models are all built in Excel, use a consistent interface, and are freely available for download. We presented a total of six models in this paper. There are of course many other possible variations. We could, for example, set up a model with differentiated goods and many firms or differentiated goods with sequential entry. The techniques that we have applied in building these models can be easily adapted to related problems, and indeed to practically any economic theory taught at the undergraduate level. For industrial organization in particular, it would be useful to develop similar models of strategic competition in prices. Starting with the differentiated goods model described in this paper, it would be possible to build a model of Bertrand duopoly, and perhaps extend it to a model of price leadership, for example. 15

16 References Briand, G. and R.C. Hill (2013) Teaching Basic Econometric Concepts using Monte Carlo Simulations in Excel International Review of Economics Education 12: Carlton, D. and J. Perloff (2005) Modern Industrial Organization (4th Edition), Addison-Wesley. Findley, T.S. (2014) Using MS Excel to Solve and Simulate the LC/PI Model of Consumption and Saving International Review of Economics Education, forthcoming. Gilbert, J. and R. Oladi (2011) Excel Models for International Trade Theory and Policy Journal of Economic Education 42(1):95. Guo, N. and J. Gilbert (2014) Demystifying Financial Markets for Saving and Insurance with Numerical Models Journal of Economic Education 45(1):78. Mixon, J.W. and S. Tohamy (2000) Using Microsoft Excel in Principles of Economics Computers in Higher Education Review 14(2). Wight, J. B. (1999) Using Electronic Data Tools in Writing Assignments Journal of Economic Education 30:

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