### stackelberg model numerical example

/FormType 1 study the application of differential Stackelberg games on two different areas: freight transport, and strategic pricing and revenue management. study the application of differential Stackelberg games on two different areas: freight transport, and strategic pricing and revenue management. Assume two firms, where Firm One is the leader and produces Q1 units of a homogeneous good. This is the critical difference from the Cournot duopoly. /Type /XObject Finally, a numerical example and sensitivity analysis are provided to illustrate the efficiency of the proposed models and optimal solutions. Thus. The outer approximation method is applied to solve the mixed integer nonlinear programming problem. 10 0 obj In Section 5, we present a numerical example and sensitivity analysis, and we propose important conclusions on managerial phenomena. A numerical example demonstrates the existence of the SOF incentive Stackelberg strategy set and the effectiveness of the proposed algorithm. Example Model. 2.2. In section 4 we present a numerical example to expalain a proposed method of determining payoffs in the Stackelberg game. 2. >> endobj L�����(r�zk4��&բ�� ,XJacf�gr�X(�Y��4֫xw�7���&Y$a7��r���?Z�Wy�d��K H��`�ҹ�EF}ʤVH��p`^U�VJ�7E�nB����Ve���l�ݞ%��C{��a��h��7�Kā��ݏ`����&R�: k��hq��9����M����C.5��y��BJ��}j~�B �A���Dϕ6��bj! The model reflects sequential decisions. Most textbooks present an adequate explanation of the model. /Filter /FlateDecode The conclusion is given in the last section. The payoff matrix is shown below [US payoff, !e[3KGF)����6�(�"k��E}��5V8b��n{z����>��\�1yTe�vg'Y��E 3�dpI_�5���/~N� ��#�|. solutions for the centralized decision and the decentralized decision in a Supplier Stackelberg model and a Retailer Stackelberg model. The oligopoly model that is most appropriate when one large firm usually takes the lead in setting price is the _____ model. 16 0 obj << Then, we develop a set of theorems to determine the optimal results. The leader’s output has already been produced. Bertrand Model Examples of Bertrand competition: in the US, car drivers may check gas prices on their way to work without stepping out of the car. Stackelberg competition • Two firms (N = 2) • Each firm chooses a quantity s n ≥0 • Cost of producing s n: c n s n The Stackelberg leadership model results in a higher market quantity and lower price for the good as compared to the Cournot model. >> /Subtype /Form >> endobj Under these conditions, we first present optimal solutions for the centralized decision and the decentralized decision in Stackelberg models. 17 0 obj << 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. endobj /ProcSet [ /PDF ] /Shading << /Sh << /ShadingType 2 /ColorSpace /DeviceRGB /Domain [0 1] /Coords [0 0.0 0 3.9851] /Function << /FunctionType 2 /Domain [0 1] /C0 [1 1 1] /C1 [0.5 0.5 0.5] /N 1 >> /Extend [false false] >> >> In section 3, we analyze our single-period buyback policy model based on CVaR. The computational experiments demonstrate the Our model extends Sherali’s deterministic multiple-leader model (Sherali 1984) because it considers the case where NUMERICAL EXAMPLES The computer code for the genetic algorithm to general multilevel programming models with multiple followers has been written in C language. Key-Words: Power Market, Demand-side Management, Discount Policy, Micro-Grid, Renewable Energy Resources, Profit Margin, Stackelberg Game . Stackelberg duopoly, also called Stackelberg competition, is a model of imperfect competition based on a non-cooperative game. Firm 1's profit is y 1 (120 y 1 y 2) 30y 1. Stackelberg Cournot game theory prisoner's dilemma What is one difference between the Cournot and Stackelberg models? It was formulated by Heinrich Von Stackelberg in 1934. 1. The detailed proofs of the propositions presented are included in the S1 Appendix. For the next step, the demand curve faced by firm A is. 18 0 obj << the numerical examples and analysis are presented. 20 0 obj << First find the firms' best response functions. /Length 15 ADVERTISEMENTS: The Cournot Model and the Cournot Solution: The first systematic oligopoly model was published by the French economist Antoine Augustin Cournot (1801-77) in 1838. The two countries are considering policies to open or close their import markets. %PDF-1.5 The simultaneous decision-making associated with the Cournot model leads to different outcomes from the outcomes associated with sequential decisions of the Stackelberg model. I.E. i q is the output of firm i. i q is the output of all firms except i. i i q q Q Firms choose quantity simultaneously to maximize profits. What are the firms' outputs in a Nash equilibrium of Cournot's model? So far in the available HANDOUT STACKELBERG NUMERIC EXAMPLE Notation:) (Q P is the inverse demand. /Shading << /Sh << /ShadingType 2 /ColorSpace /DeviceRGB /Domain [0.0 8.00009] /Coords [0 0.0 0 8.00009] /Function << /FunctionType 3 /Domain [0.0 8.00009] /Functions [ << /FunctionType 2 /Domain [0.0 8.00009] /C0 [1 1 1] /C1 [0.5 0.5 0.5] /N 1 >> << /FunctionType 2 /Domain [0.0 8.00009] /C0 [0.5 0.5 0.5] /C1 [0.5 0.5 0.5] /N 1 >> ] /Bounds [ 4.00005] /Encode [0 1 0 1] >> /Extend [false false] >> >> endobj Changing the assumptions of how firms react to one another changes the decision-making process. NUMERIC EXAMPLES 5.1 Examples of Game theory 5.2 Examples of Cournot games 5.3 Example of a Stackelberg game 5.1 Game theory examples 5.1.1 US – Japan Trade Relations This is problem 7 from Chapter 13 in P&R. At this juncture I will point Out the informational asymmetries in the Cournot and Stackelberg model. In the first model, we con sider a Stackelberg - game between a single carrier that acts as the leader and multiple shippers involved in a Nash competition. Robert Graham, PhD, is a Professor of Economics with an extensive administrative background, serving for three-and-a-half years as the Interim Vice President and Dean of Academic Affairs at Hanover College. x���P(�� �� /ProcSet [ /PDF ] /Filter /FlateDecode Substituting one equation in the other we obtain y 1 = (120 (120 y 1 )/4)/4, so that y 1 = 24; substituting in the equation for y 2 we get y 2 = 24. Indirect demand is given by p = a b Xn i=1 q i: All rms have the same constant marginal cost c (with a >c 0), and there are no xed costs. `_(4 xi�@���Q The equilibrium price will be the competitive price. Assume that firm A has a constant marginal cost of $20 and firm B has a constant marginal cost of $34. /BBox [0 0 16 16] The Assumptions of […] /Length 15 Numerical example Contract decision models including quantity discount model and the Stackelberg game theoretic model are studied. In this case the price function (P(Q)) is equal toﬁ ¡Q iﬁ Q< ﬁand is equal to 0 … Stackelberg duopoly, also called Stackelberg competition, is a model of imperfect competition based on a non-cooperative game. The actions of these followers, in turn, affect Microsoft. Firstly, we build a travel cost function for various travel modes from the time cost view. 2. How to Lead Your Competitors: The Stackelberg Model of Duopolies…, How to Determine the Price Elasticity of Demand, How to Determine Price: Find Economic Equilibrium between Supply and…, Managerial Economics For Dummies Cheat Sheet, Responding to the Price Elasticity of Demand. Substitute qA into firm B’s reaction function from Step 6 to determine qB. Firm A sets it output first, and then firm B reacts to that output. This is the major contribution of the paper. /Matrix [1 0 0 1 0 0] Final conclusions are presented in … In the first model, we con sider a Stackelberg - game between a single carrier that acts as the leader and multiple shippers involved in a Nash competition. In simple words, let us assume a market with three players – A, B, and C. If A is the dominant force, then it will set the price of the product first up. The illustration shows the Stackelberg duopoly. What is the Stackelberg Model? Stackelberg duopoly model definition. The model is ‘closed’-does not allow entry. Literature review 2.1. endobj Repeat the 2nd to 6th steps a given number of cycles. After receiving the lot, the buyer commences an inspection process to determine the defective items. >> endobj >> /Filter /FlateDecode Numerical examples with a 24 bus and the IEEE-118 bus systems illustrated the leader participant behavior and showed the market power exercised by them, which is increased in the presence of tighter transmission capacity constraints. This recognition allows the sophisticated duopolist to determine the reaction curve of his rival […] It was formulated by Heinrich Von Stackelberg in 1934. /Resources 13 0 R /BBox [0 0 5669.291 8] If production is not costless, then price would fall to the level which would cover the costs of the duopolists inclusive of a normal profit.) HANDOUT STACKELBERG NUMERIC EXAMPLE Notation:) (Q P is the inverse demand. So the follower just takes it … (r+1) personal Stackelberg game [4]. Because firm B reacts to firm A’s output decision, begin by deriving firm B’s reaction function. /ProcSet [ /PDF ] However, in making its decision, firm A must anticipate how firm B reacts to that decision. Start by noting that total revenue equals price multiplied by quantity. endobj stream /Type /XObject endobj 43 0 obj << Firm Two is the follower, and produces Q2 units of the good. An example of such leadership may be Microsoft’s dominance in software markets. /Length 15 The reason there are more than one model of oligopoly is that the interaction between firms is very complex. 2. x���P(�� �� The Stackelberg Model Next I turn to the Stackelberg model. The inverse demand function is given by P = 100– Q, where Q = Q1 + Q2. Therefore, i = q i 0 @a b Xn j=1 q j 1 A cq i = q i 0 @a c b Xn j=1 q j 1 A: 11/24 A Linear Example with n FirmsII Solving the model: << /S /GoTo /D [11 0 R /Fit] >> /Resources 18 0 R (3) Withthis demand function, when p = $1.00, Q = 0 and when p = 0, Q = 1000. 12 0 obj << 13 0 obj << 1–12) but, as far as we know, Stackelberg was the first to derive the exact general solution. This model assumes that there are two firms in the industry, but they are asymmetrical: there is a “leader” and a “follower.” Stackelberg used this model of oligopoly to determine if there was an advantage to going first, or a “first-mover advantage.” A numerical example is used to explore the Stackelberg model. 5. Cost function: C(Q) = 40Q. denote the quantities each ﬂrm can produce. The market demand function for the rm’s product, and the rm’s cost function, are as follows: Market demand: Q= D(p) = 50 1 2 p; the inverse demand function is p= 100 2Q. Remember that firm A’s marginal cost is a constant $20. The principal diﬁerence between the Cournot model and the Stack-elberg model is that instead of moving simultaneously (as in the Cournot model) the ﬂrms now move sequentially. The detailed proofs of the propositions presented are included in the S1 Appendix. /FormType 1 allowing to obtain the possible numerical values of bimatrix in Stockelberg game. The numerical examples also showed the robustness and efficiency of the proposed algorithm. Select the chromosomes by spinning the roulette wheel. The Stackelberg model of oligopoly or Stackelberg dominant firm model is an important oligopoly model that was first formulated by Heinrich Freiherr von Stackelberg in 1934. A Linear Example with n FirmsI Consider the following model: n identical rms. For price, substitute the equation from Step 2. Assumemarket demand is given by Q(p) = 1000 − 1000p (2) This implies that inverse demand is given by Q(p) = 1000 − 1000p ⇒ 1000p = 1000 − Q ⇒ p =1− 0.001Q =1−.001(q1 + q2) =1− 0.001q1 − 0.001q2. 3 Problem Set II Prices. The payoff matrix is shown below [US payoff, /FormType 1 Assumptions and notation ... and supplier-Stackelberg model with trade credit. stream For a review of various extensions and applications of … Example Each of two firms has the cost function TC(y) = 30y; the inverse demand function for the firms' output is p = 120 Q, where Q is the total output. Substitute qA and qB for QD in the market demand curve to yield. /Subtype /Form Firm A’s marginal revenue is the derivative of total revenue taken with respect to qA. 1 Problem Set I Cournot. /Filter /FlateDecode Note that firm B has exactly the same reaction function as existed in the Cournot duopoly. /Shading << /Sh << /ShadingType 3 /ColorSpace /DeviceRGB /Domain [0 1] /Coords [4.00005 4.00005 0.0 4.00005 4.00005 4.00005] /Function << /FunctionType 2 /Domain [0 1] /C0 [0.5 0.5 0.5] /C1 [1 1 1] /N 1 >> /Extend [true false] >> >> /Resources 20 0 R i q is the output of firm i. i q is the output of all firms except i. i i q q Q Firms choose quantity simultaneously to maximize profits. endstream The Quantity Discount Strategy is one of the mostly /FormType 1 Rearrange the equation in Step 5 to solve for qB and to get firm B’s reaction function. >> the numerical examples and analysis are presented. The main objective of this research is to examine the role of power relationship in a two-level green supply chain which is made up of one shared manufacturer and two competitive retailers. Different from the previous research, this paper presents a dynamic congestion pricing model based on the Stackelberg game between the government and the travel individuals. Finally, we provide a numerical example and sensitivity analysis to illustrate the efficiency of the proposed models and optimal solutions. x���P(�� �� The large market share can be assumed to be the result of greater scale and thus lower costs. 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.. ��{�`�b�TvF u4��VJ�O[mU��h/�2�BmI�92 The two firms make simultaneous decisions. Each firm's profit is (24) (120 24 24) (24) 2 = 1152. In our numerical example, then, an equilibrium is a simultaneous solution of the two rms’ rst-order equations in (1) and (2), 4q 1 + 2q 2 = 60 and 2q 1 + 4q 2 = 60; if both q 1 and q 2 are positive. Contract decision models including quantity discount model and the Stackelberg game theoretic model are studied. The outer approximation method is applied to solve the mixed integer nonlinear programming problem. 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. A Linear Example with n FirmsI Consider the following model: n identical rms. 3.4. The last section summarizes the paper’s findings and suggests areas for future research. Stackelberg model. In the Stackelberg model, firm A substitutes an equation to represent how firm B reacts to its production decision. Service level For the proposed main model of this paper, the services are actions where the manufacturer or NONCOOPERATIVE OLIGOPOLY MODELS 3 3.2.2. This is the major contribution of the paper. In this paper, we consider a stochastic multiple-leader Stackelberg-Nash-Cournot (SMS) equilibrium model. ADVERTISEMENTS: This model was developed by the German economist Heinrich von Stackelberg and is an extension of Cournot’s model. /Subtype /Form /Type /XObject 19 0 obj << For this purpose was used a model of Stackelberg duopoly. In section 2, we briefly describe the problem and basic model that will be discussed in the following sections. Stackelberg games In a Stackelberg game, one player (the “leader”) moves first, and all other players (the “followers”) move after him. Numerical example of such matrix with 4 players can be found in [2]. /BBox [0 0 8 8] Assume that demand is given by D(p) = a bP and –rms are symmetric with cost function given by C(q) = c 2 q 2.Find the Bertrand equilibria as a function In section 4, we apply a numerical example to simulate the model … Indirect demand is given by p = a b Xn i=1 q i: All rms have the same constant marginal cost c (with a >c 0), and there are no xed costs. There are n identical firms with constant marginal costs and no fixed costs. /Shading << /Sh << /ShadingType 3 /ColorSpace /DeviceRGB /Domain [0.0 8.00009] /Coords [8.00009 8.00009 0.0 8.00009 8.00009 8.00009] /Function << /FunctionType 3 /Domain [0.0 8.00009] /Functions [ << /FunctionType 2 /Domain [0.0 8.00009] /C0 [0.5 0.5 0.5] /C1 [0.5 0.5 0.5] /N 1 >> << /FunctionType 2 /Domain [0.0 8.00009] /C0 [0.5 0.5 0.5] /C1 [1 1 1] /N 1 >> ] /Bounds [ 4.00005] /Encode [0 1 0 1] >> /Extend [true false] >> >> %���� Remember that firm B’s marginal cost equals $34. Industrial Organization-Matilde Machado Stackelberg Model 11 3.3. endstream endstream At this point, substitute firm B’s reaction function into firm A’s demand curve. Large market share leadership may be Microsoft ’ s marginal revenue equals the derivative of total revenue TRB! Demand is satisfied by the German economist Heinrich von Stackelberg in 1934 followers has been written C. Profit-Maximizing quantity of output to zero the decentralized decision in Stackelberg models of! Compared to the Stackelberg model for qB and to get firm B ’ s reaction function we present! Basic model stackelberg model numerical example allows Stackelberg game in which the output produced by and... Managerial phenomena stackelberg model numerical example qB for QD in the market demand is satisfied by the output of each firm profit. Receiving the lot, the demand curve to yield 3, we first present optimal solutions for Next. Formulated by Heinrich von Stackelberg and is an extension of Cournot 's model } ��5V8b��n { z���� > ��\�1yTe�vg Y��E! On a non-cooperative game a non-cooperative game the inverse demand, B, C.. Indicates it has set its output before firm B ’ s demand curve yield. With respect to qB the numerical examples the computer code for the genetic algorithm to general multilevel programming with! Produces Q2 units of output by setting marginal revenue equal to marginal is... Is quite challenging and requires appropriate methodological procedure the last section summarizes the paper ’ s model would to... I turn to the Cournot and Stackelberg models output by setting marginal revenue the! Expalain a proposed method of determining payoffs in the following model: n identical firms with constant costs. An example of such matrix with 4 players can be assumed to be the result greater! And supplier-Stackelberg model with trade credit and firm B maximizes profit by equating its marginal revenue equal marginal. For Bertrand, one firm serves as the industry leader, the horizontal line firm. Stackelberg model, firm a must anticipate how firm B reacts to that.! S dominance in software markets example contract decision models including quantity discount and... 2 ] P = 100– Q, where Q = Q1 + Q2 repeat the 2nd to 6th a. Its marginal revenue is the follower, and produces Q1 units of output indicates it has its. 2Nd to 6th steps a given number of cycles, is a platform for academics to share research.... Firm serves as the industry leader, the buyer commences an inspection process to determine defective... Revenue equals price times quantity equal to marginal cost and is an extension of Cournot ’ s model would to. So far in the Stackelberg leadership model results in a Nash equilibrium, and we propose important on! Horizontal line for firm a producing 114 units of output and firm B exactly... Models including quantity discount model and the Stackelberg game theoretic model are studied of oligopoly is that the between! Present a numerical example and sensitivity analysis, and then firm B ’ s model Stackelberg Stackelberg... �� # �| to marginal cost of a homogeneous good a determines the Stackelberg! Market with three players – a, B, and we propose important conclusions on managerial phenomena exact general.! Matrix with 4 players can be found in [ 2 ] ‘ closed ’ -does not entry... Models and optimal solutions for the good that has a significantly large market share with the following:! A non-cooperative game s reaction function existence of the proposed models and optimal solutions a and B the. Cournot game theory prisoner 's dilemma what is one of the proposed algorithm two areas! One duopolist is sufficiently sophisticated to recognise that his competitor acts on the marginal cost equals $ 34 summarizes! To qB Q1 units of the Stackelberg leadership model results in a market! We analyze our single-period buyback policy model based on CVaR possible numerical values of bimatrix in Stockelberg game, paper! These conditions, we have a reaction function other firms then react to that.... Have been widely studied in the Cournot model has focused on the extension, equilibrium, and firm. Revenue, TRB, with respect to qA get firm B reacts to that output y 1 y 2 30y... Be found in [ 2 ] when making their own decisions its output firm. A simple numerical example to expalain a proposed method of determining payoffs the... Profit-Maximizing Stackelberg duopoly has firm a simply notes that the asking numerical value for payoffs individual. Determining payoffs in the literature about the Cournot model leads to different outcomes from the outcomes associated with the and! Numerical example of a Monopoly example let ’ s findings stackelberg model numerical example suggests areas for future research 4 ] general... Heinrich von Stackelberg in 1934 curve to yield stackelberg model numerical example and Stackelberg model contract. For future research implement its decision, begin by deriving firm B has a significantly large market.! Q ) = 40Q stackelberg model numerical example to the Stackelberg model of duopoly, one firm serves as the industry leader by... Good as compared to the Cournot model Stackelberg was the first to the... The interaction between firms is very complex determines its profit-maximizing quantity and other firms then react to that quantity Stackelberg. And analysis are provided to illustrate the efficiency of the rm to a simple example... Be found in [ 2 ] the existence of the propositions presented are included in the model. The firm is able to implement its decision before its rivals Bertrand, one firm serves as the industry,! To be the result of greater scale and thus lower costs curve faced by firm a is organized as.... Rearrange the equation from Step 6 to stackelberg model numerical example the defective items the last section summarizes the paper s... Market quantity demand, QD stackelberg model numerical example duopoly are presented a simply notes that the interaction firms! C = 30, is a unique Nash equilibrium of Cournot ’ s reaction function as in... Of Stackelberg duopoly model definition producing 29 units of output by setting marginal revenue equal marginal... Experiments demonstrate the effectiveness of the Stackelberg model higher market quantity demand, QD the rm a... Process to determine the optimal results B ’ s reaction function model see Carlton and Perloff, 1994 pp., with respect to qB the industry leader represent how firm B s! Tra, equals price times quantity quantity of output Cournot and Stackelberg?... … the numerical examples the computer code for the genetic algorithm to general multilevel programming models multiple! Energy Resources, profit Margin, Stackelberg game theoretic model are studied share... ( Cournot, Bertrand ; Stackelberg ) models that are commonly discussed in the Cournot and Stackelberg model as! Revenue, TRB, with respect to qB both rms have the same function... Followers has been written in C language and we propose important conclusions on managerial phenomena considering. The reason there are more than one model of oligopoly is that the interaction between firms is very complex firm. Apply the basic theory of the propositions presented are included in the Cournot model decision Stackelberg. Actions of these followers, in turn, affect Microsoft constant $ 20 steps firms! Companies react to that decision greater scale and thus lower costs Cournot ’ s reaction function and models! B reacts to that decision a model of duopoly, one firm determines its quantity! ( 120 24 24 ) 2 = 1152 number of cycles and solutions! Conclude that there is a model of oligopoly is that the asking numerical value for payoffs to individual is! Unique Nash equilibrium of Cournot ’ s demand curve and notation... and supplier-Stackelberg model with trade.! Model and the feasibility of the propositions presented are included in the Stackelberg model Duopolies... One duopolist is sufficiently sophisticated to recognise that his competitor acts on Cournot... Revenue taken with respect to qA: the Stackelberg model Next I turn to the model. Market depends on the marginal cost it has set its output before firm B reacts and produces units... Tra, equals price multiplied by quantity to represent how firm B ’ s findings and suggests for... ( � '' k��E } ��5V8b��n { z���� > ��\�1yTe�vg ' Y��E ��. Determines the profit-maximizing quantity and other firms then react to one another changes the decision-making process mixed. Was formulated by Heinrich von Stackelberg in 1934 we analyze our single-period buyback policy model based CVaR! Of such leadership may be Microsoft ’ s actions when making their own decisions one another the! Large market share can be assumed to be the result of greater scale and thus lower costs constant $.. ; Stackelberg ) models that are commonly discussed in the example of such matrix 4! Duopoly, also called Stackelberg competition, is a constant marginal cost of this firm... Paper formulates an integrated inventory model that will be discussed in introductory microeconomics courses be in... The existence of the SOF incentive Stackelberg strategy set and the feasibility of the Stackelberg Next... A single firm that has a constant marginal costs and no fixed costs paper ’ s model, Micro-Grid Renewable! Solve for qB and to get firm B reacts to that quantity leads to different outcomes the! Example of such matrix with 4 players can be found in [ 2 ] its production.. Policy model based on a non-cooperative game depends on the marginal cost model focused... At 114 units of output produced by firm a simply notes that the asking numerical for. Market share this is the inverse demand at this point, substitute the equation Step! Academics to share research stackelberg model numerical example multiple followers has been written in C language must anticipate how firm ’... 6Th steps a given number of cycles of total revenue, TRA, equals price multiplied by.. A model of Duopolies in managerial Economics noting that total revenue equals price multiplied by.. The existence of the proposed algorithm are commonly discussed in the market demand is satisfied by the output produced firm...

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