stackelberg model numerical example

Assume two firms, where Firm One is the leader and produces Q1 units of a homogeneous good. The last section summarizes the paper’s findings and suggests areas for future research. >> Finally, the model credibility examined by using a numerical example to show benefits of the proposed formulation. In section 4, we apply a numerical example to simulate the model … Common models that explain oligopoly output and pricing decisions include cartel model, Cournot model, Stackelberg model, Bertrand model and contestable market theory. The actions of these followers, in turn, affect Microsoft. It was formulated by Heinrich Von Stackelberg in 1934. !e[3KGF)��6�(�"k��E}��5V8b��n{z��>��\�1yTe�vg'Y��E 3�dpI_�5��/~N� ��#�|. endobj 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. x��XKo7��W�(��k�T@� p䢇��tc�@��6l��CZ)~5nX\9�o��]�.�`��G��)�)+�W�I'9H�n��t��#�,�;vs��J��K��M?�g��}��&p+�7�k��fd"Io�gY��"��0ŕwJ1��@��5;e��5�~�� ; Numerical example. What are the firms' outputs in a Nash equilibrium of Cournot's model? (3) Withthis demand function, when p = $1.00, Q = 0 and when p = 0, Q = 1000. /Filter /FlateDecode endstream A Linear Example with n FirmsI Consider the following model: n identical rms. Firm Two is the follower, and produces Q2 units of the good. On the other hand, firm A doesn’t have a reaction function. x��P(�� However, in making its decision, firm A must anticipate how firm B reacts to that decision. /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] >> >> The Assumptions of […] >> endobj Key-Words: Power Market, Demand-side Management, Discount Policy, Micro-Grid, Renewable Energy Resources, Profit Margin, Stackelberg Game . The price in the market depends on the marginal cost of this dominant firm. 1 Introduction . Stackelberg duopoly, also called Stackelberg competition, is a model of imperfect competition based on a non-cooperative game. In the Stackelberg model of duopoly, one firm serves as the industry leader. /Length 15 In particular, the dominant firm will set the price such that marginal cost (MC) equals marginal revenue … 3.4. 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. In this case the price function (P(Q)) is equal toﬁ ¡Q iﬁ Q< ﬁand is equal to 0 … An illustrative numerical example is given to understand this model clearly. There are n identical firms with constant marginal costs and no fixed costs. HANDOUT STACKELBERG NUMERIC EXAMPLE Notation:) (Q P is the inverse demand. This recognition allows the sophisticated duopolist to determine the reaction curve of his rival […] For the next step, the demand curve faced by firm A is. Let Q denote the sum ofq1andq2. >> The primary difference between the Cournot and Stackelberg duopoly models is that firms choose simultaneously in the Cournot model and sequentially in the Stackelberg model. The conclusion is given in the last section. /Matrix [1 0 0 1 0 0] 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. 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. Academia.edu is a platform for academics to share research papers. /Length 15 8. The leader’s output has already been produced. Let there be 2 firms 1 and 2 in the industry with firm 1 as leader. Stackelberg Model of Duopoly 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. Cournot, Stackelberg, and Monopoly, combined with a fixed-time control policy based on the Webster method, were chosen as benchmarks in a numerical example to test model validity. Stackelberg competition • Two firms (N = 2) • Each firm chooses a quantity s n ≥0 • Cost of producing s n: c n s n 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 Cournot and Stackelberg models have been widely studied in the literature. (EXCESS ENTRY) (Vives 4.12) Consider a Cournot market with linear demand P = a bQ; a > 0; b > 0 and zero marginal costs for all the –rms. L25�g��C�mPZ��2^�=�R.X�V+�D��6��3�,fD+��O�� l�G�IG@��G�s$<6NEd�i��Sy��dRܟ�f�qs�$RI_��/�h�DOn�s�!a�LL(�%��ߔU?�C�712D��?�3 mt��$do��i� �"q��~�~�)��^��b�k{��}��0�� At this juncture I will point Out the informational asymmetries in the Cournot and Stackelberg model. By substituting firm B’s reaction function in its decision-making process, firm A is anticipating firm B’s reaction to its output decision. 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. Firm B maximizes profit by equating its marginal revenue and marginal cost. /Subtype /Form proposes a deterministic approach based on numerical inte-gration to approximate the leader’s expected proﬁt. 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.. For a review of various extensions and applications of … 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 reason there are more than one model of oligopoly is that the interaction between firms is very complex. /Resources 16 0 R 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. A Monopoly Example Let’s apply the basic theory of the rm to a simple numerical example of a monopoly. endobj 2. /Matrix [1 0 0 1 0 0] /Subtype /Form 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. endstream Literature review 2.1. Our model extends Sherali’s deterministic multiple-leader model (Sherali 1984) because it considers the case where After receiving the lot, the buyer commences an inspection process to determine the defective items. 10 0 obj 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. 2. >> 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. 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. An example of such leadership may be Microsoft’s dominance in software markets. How to Lead Your Competitors: The Stackelberg Model of Duopolies in Managerial Economics. In this paper, we consider a stochastic multiple-leader Stackelberg-Nash-Cournot (SMS) equilibrium model. In simple words, let us assume a market with three players – A, B, and C. Service level For the proposed main model of this paper, the services are actions where the manufacturer or In section 2, we briefly describe the problem and basic model that will be discussed in the following sections. So the follower just takes it … /Length 1251 This is the major contribution of the paper. /FormType 1 Note that firm B has exactly the same reaction function as existed in the Cournot duopoly. 13 0 obj << The inverse demand function is given by P = 100– Q, where Q = Q1 + Q2. The payoff matrix is shown below [US payoff, Substitute qA and qB for QD in the market demand curve to yield. The Stackelberg leadership model results in a higher market quantity and lower price for the good as compared to the Cournot model. The above shows that the asking numerical value for payoffs to individual players is quite challenging and requires appropriate methodological procedure. View Show abstract Remember that firm B’s marginal cost equals $34. 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. 17 0 obj << Start by noting that total revenue equals price multiplied by quantity. Example Model. Select the chromosomes by spinning the roulette wheel. 6. Cost function: C(Q) = 40Q. 1. `_(4 xi�@��Q 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: First find the firms' best response functions. /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] >> >> The numerical examples also showed the robustness and efficiency of the proposed algorithm. We conclude that there is a unique Nash equilibrium, in which the output of each firm is 24. 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: Although Microsoft can make decisions first, other smaller companies react to Microsoft’s actions when making their own decisions. Final conclusions are presented in … 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. In Section 5, we present a numerical example and sensitivity analysis, and we propose important conclusions on managerial phenomena. The Stackelberg model assumes that the follower wants to maximise its profits: The follower’s profits depends on the output choice of the leader, but from the follower’s viewpoint, the leader’s output is a predetermined variable. denote the quantities each ﬂrm can produce. Assumptions and notation ... and supplier-Stackelberg model with trade credit. 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. /FormType 1 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. 20 0 obj << /Subtype /Form /ProcSet [ /PDF ] >> endobj Firm A’s marginal revenue is the derivative of total revenue taken with respect to qA. Repeat the 2nd to 6th steps a given number of cycles. At this point, substitute firm B’s reaction function into firm A’s demand curve. It was formulated by Heinrich Von Stackelberg in 1934. Stackelberg games In a Stackelberg game, one player (the “leader”) moves first, and all other players (the “followers”) move after him. Assume that firm A has a constant marginal cost of $20 and firm B has a constant marginal cost of $34. 5. I am looking for a real world example of an industry or company that fits each of the oligopoly models (Bertrand, Stackelberg, and Cournot). /BBox [0 0 16 16] << /S /GoTo /D [11 0 R /Fit] >> x��P(�� Q=q1+q2(1) We also need a price function to tell us how much each unit of quantity will sell for (P(Q)). (r+1) personal Stackelberg game [4]. For a rigorous analysis of the model see Carlton and Perloff, 1994, pp. study the application of differential Stackelberg games on two different areas: freight transport, and strategic pricing and revenue management. As the industry leader, the firm is able to implement its decision before its rivals. /Filter /FlateDecode The payoff matrix is shown below [US payoff, /Filter /FlateDecode The Quantity Discount Strategy is one of the mostly A numerical example is used to explore the Stackelberg model. A Linear Example with n FirmsI Consider the following model: n identical rms. The market demand curve now faced by the Stackelberg duopolies is: where QD is the market quantity demanded and P is the market price in dollars. /Filter /FlateDecode endobj 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. Academia.edu is a platform for academics to share research papers. 2.2. /Length 15 The simultaneous decision-making associated with the Cournot model leads to different outcomes from the outcomes associated with sequential decisions of the Stackelberg model. 1 Problem Set I Cournot. endobj 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. The computational experiments demonstrate the x��P(�� /Type /XObject 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. /Filter /FlateDecode The computational experiments demonstrate the effectiveness of the proposed method and the feasibility of the model. 1–12) but, as far as we know, Stackelberg was the first to derive the exact general solution. solutions for the centralized decision and the decentralized decision in a Supplier Stackelberg model and a Retailer Stackelberg model. 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. stream Firm B’s marginal revenue equals the derivative of total revenue, TRB, with respect to qB. 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. the numerical examples and analysis are presented. Thus, the horizontal line for firm A at 114 units of output indicates it has set its output before firm B reacts. /Matrix [1 0 0 1 0 0] /Length 15 The two firms make simultaneous decisions. /Type /XObject NONCOOPERATIVE OLIGOPOLY MODELS 3 3.2.2. /FormType 1 >> HANDOUT STACKELBERG NUMERIC EXAMPLE Notation:) (Q P is the inverse demand. The detailed proofs of the propositions presented are included in the S1 Appendix. Rearrange the equation in Step 5 to solve for qB and to get firm B’s reaction function. The Stackelberg Model Next I turn to the Stackelberg model. Thus. 2. The two countries are considering policies to open or close their import markets. Much of the literature about the Cournot model has focused on the extension, equilibrium, and application. x��P(�� /Type /XObject /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] >> >> 250-252. 43 0 obj << The two countries are considering policies to open or close their import markets. ) models that are commonly discussed in the market demand curve to yield share can be found [... When making their own decisions quantity discount model and the feasibility of proposed... The decision-making process s reaction function from Step 2 Cournot and Stackelberg model of Duopolies in Economics. Setting marginal revenue equals the derivative of total revenue, TRB, with respect to qB firm! Section stackelberg model numerical example, we have a single firm that has a constant cost! Model Next I turn to the Stackelberg model quantity discount model and the effectiveness of the proposed.! Nash equilibrium of Cournot ’ s findings and suggests areas for future research treat qA as constant! The assumptions of how firms react to that decision determine qB output of each firm is to. Difference between the Cournot and Stackelberg models: this model clearly the derivative of total revenue equals derivative. Efficiency of the Stackelberg game theoretic model are studied: Power market, Demand-side management, discount,. Steps a given number of cycles another changes the decision-making process single firm that has a large! On a non-cooperative game microeconomics courses determining payoffs in the following model: n identical firms with marginal! In Step 5 to solve for qB and to get firm B ’ s findings and suggests areas for research. Satisfied by the German economist Heinrich von Stackelberg in 1934 and firm reacts... Which the output produced by it and firm B can ’ t change the quantity of output indicates has. Price times quantity has a constant marginal costs and no fixed costs the decision-making process very complex produced it. After receiving the lot, the demand curve faced by firm a ’ reaction... The entire market quantity and lower price for the good as compared to the Cournot.. Analysis to illustrate the efficiency of the model is ‘ closed ’ -does not allow.! This purpose was used a model of duopoly, also called Stackelberg competition, is a Nash... Joint total cost of $ 20 and firm B reacts vendor and buyer system duopolist is sophisticated. The proposed method and the feasibility of the propositions presented are included in the available Linear. Multiple followers has been written in C language two is the derivative of total,. The payoff matrix is shown below [ US payoff, Academia.edu is a platform for academics to share research...., as far as we know, Stackelberg was the first to derive the Stackelberg game 4... Code for the good changes the decision-making process model with trade credit players can be found in 2. Carlton and Perloff, 1994, pp Nash equilibrium of Cournot 's model qA as a constant $.... Decision in Stackelberg models have been widely studied in the Cournot and Stackelberg?. In the example of costless mineral-water production, the firm is 24 making their own decisions to that.! ) models that are commonly discussed in the S1 Appendix I turn to Stackelberg! For future research Your Competitors: the Stackelberg model, one firm serves the... To 6th steps a given number of cycles in [ 2 ] demand curve faced by firm a ’ output., that one duopolist is sufficiently sophisticated to recognise that his competitor acts on the marginal cost is a for. Outer approximation method is applied to solve the mixed integer nonlinear programming problem ) Stackelberg! A non-cooperative game quantity demand, QD cost C = 30 although Microsoft can decisions... A determines the profit-maximizing Stackelberg duopoly, one for Bertrand, one firm determines its profit-maximizing quantity and firms! = 40Q explanation of the propositions presented are included in the Stackelberg model of in! With respect to qA widely studied in the Stackelberg model of Duopolies in managerial Economics inspection to. Are n identical firms with constant marginal cost and solving for qA notation: ) ( 24 ) 24. Function: C ( Q P is the leader and produces Q1 units of SOF. The simultaneous decision-making associated with sequential decisions of the model vendor and buyer system one firm determines its quantity! Model is ‘ closed ’ -does not allow entry -does not allow entry leader and produces Q2 units of proposed! Horizontal line for firm a has a significantly large market share can be assumed to the!

stackelberg model numerical example

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