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47-419: There are some further constraints upon the sustaining of a Stackelberg equilibrium. The leader must know ex ante that the follower observes its action. The follower must have no means of committing to a future non-Stackelberg leader's action and the leader must know this. Indeed, if the 'follower' could commit to a Stackelberg leader action and the 'leader' knew this, the leader's best response would be to play

94-407: A Cournot equilibrium quantity, the follower will choose a deviant quantity that will hit the leader's profits. After all, the quantity chosen by the leader in equilibrium is only optimal if the follower also plays in equilibrium. The leader is, however, in no danger. Once the leader has chosen its equilibrium quantity, it would be irrational for the follower to deviate because it too would be hurt. Once

141-477: A Stackelberg follower action. Firms may engage in Stackelberg competition if one has some sort of advantage enabling it to move first. More generally, the leader must have commitment power. Moving observably first is the most obvious means of commitment: once the leader has made its move, it cannot undo it—it is committed to that action. Moving first may be possible if the leader was the incumbent monopoly of

188-708: A corresponding distinction of great theoretical importance must be drawn between two alternative methods of defining these quantities. Quantities defined in terms of measurements made at the end of the period in question are referred to as ex post ; quantities defined in terms of action planned at the beginning of the period in question are referred to as ex ante . ) Focusing attention on the relation between saving and investment, Myrdal argued that one may without any contradiction consider that, as they are made by separate agents, ex ante saving and investment decisions are not at parity in general while ex post saving and investment are recorded in bookkeeping balance exactly: There

235-480: A quantity that maximises its payoff, anticipating the predicted response of the follower. The follower actually observes this and in equilibrium picks the expected quantity as a response. To calculate the SPNE, the best response functions of the follower must first be calculated (calculation moves 'backwards' because of backward induction). The profit of firm 2 {\displaystyle 2} (the follower)

282-451: A standard tool in macroeconomics . Prices are quantities that directly refer to a point of time: they are determined at a point of time, after an ex ante adjustment process has taken place. As for the macroeconomic quantities, Myrdal proposed to refer to the point of time at which they are calculated. Gunnar Myrdal further explained that ex ante disparity and ex post balance are made consistent through price changes, which result from

329-486: A time period for which they are reckoned. But in order to be unambiguous they must also refer to a point of time at which they are calculated. (Gunnar Myrdal, Monetary Equilibrium, London : W. Hodge 1939: 46–7) Economist G. L. S. Shackle claimed the importance of Gunnar Myrdal´s analysis by which saving and investment are allowed to adjust ex ante to each other. However, the reference to ex ante and ex post analysis has become so usual in modern macroeconomics that

376-524: Is ex-post (actual) (or ex post ). Buying a lottery ticket loses you money ex ante (in expectation ), but if you win, it was the right decision ex post . The ex-ante (and ex-post ) reasoning in economic topics was introduced mainly by Swedish economist Gunnar Myrdal in his 1927–39 work on monetary theory, who described it in this way: An important distinction exists between prospective and retrospective methods of calculating economic quantities such as incomes, savings, and investments; and [...]

423-438: Is a New Latin phrase meaning "before the event". In economics , ex-ante or notional demand refers to the desire for goods and services that is not backed by the ability to pay for those goods and services. This is also termed as ' wants of people'. Ex-ante is used most commonly in the commercial world , where results of a particular action, or series of actions, are forecast (or intended). The opposite of ex-ante

470-431: Is because once leader has committed to an output and observed the followers it always wants to reduce its output ex-post. However its inability to do so is what allows it to receive higher profits than under Cournot. An extensive-form representation is often used to analyze the Stackelberg leader-follower model. Also referred to as a “ decision tree ”, the model shows the combination of outputs and payoffs both firms have in

517-480: Is clear (if marginal costs are assumed to be zero – i.e. cost is essentially ignored) that the leader has a significant advantage. Intuitively, if the leader was no better off than the follower, it would simply adopt a Cournot competition strategy. Plugging the follower's quantity q 2 {\displaystyle q_{2}} , back into the leader's best response function will not yield q 1 {\displaystyle q_{1}} . This

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564-417: Is in fact no contradiction at all between the statement of an exact bookkeeping balance ex post and the obvious inference that in a situation when saving is increasing without a corresponding increase of investment, or perhaps with an adverse movement in investment, there must be a tendency ex ante to a disparity. (Gunnar Myrdal, Monetary Equilibrium, London : W. Hodge 1939: 46) This analysis has become

611-411: Is revenue minus cost. Revenue is the product of price and quantity and cost is given by the firm's cost structure, so profit is: Π 2 = P ( q 1 + q 2 ) ⋅ q 2 − C 2 ( q 2 ) {\displaystyle \Pi _{2}=P(q_{1}+q_{2})\cdot q_{2}-C_{2}(q_{2})} . The best response

658-414: Is the follower's quantity as a function of the leader's quantity, namely the function calculated above. The best response is to find the value of q 1 {\displaystyle q_{1}} that maximises Π 1 {\displaystyle \Pi _{1}} given q 2 ( q 1 ) {\displaystyle q_{2}(q_{1})} , i.e. given

705-402: Is the very thing that makes a Cournot strategy profile a Nash equilibrium in a Stackelberg game that prevents it from being subgame perfect. Consider a Stackelberg game (i.e. one which fulfills the requirements described above for sustaining a Stackelberg equilibrium) in which, for some reason, the leader believes that whatever action it takes, the follower will choose a Cournot quantity (perhaps

752-400: Is to find the value of q 2 {\displaystyle q_{2}} that maximises Π 2 {\displaystyle \Pi _{2}} given q 1 {\displaystyle q_{1}} , i.e. given the output of the leader (firm 1 {\displaystyle 1} ), the output that maximises the follower's profit is found. Hence,

799-444: Is very general. It assumes a generalised linear demand structure and imposes some restrictions on cost structures for simplicity's sake so the problem can be resolved. for ease of computation. The follower's profit is: The maximisation problem resolves to (from the general case): Consider the leader's problem: Substituting for q 2 ( q 1 ) {\displaystyle q_{2}(q_{1})} from

846-708: The General Theory do?", in J. Pheby (ed), New Directions in Post-keynesian Economics, Aldershot: Edward Elgar.) In European Union law, ex ante regulation is a type of regulation that is designed to prevent companies engaging in harmful conduct. It is usually used in sectors where there is a anti-competitive behavior. For example, such regulations are used in the telecommunications sector as well as in relation to data protection (the General Data Protection Regulation

893-709: The Stackelberg game. The image on the left depicts in extensive form a Stackelberg game. The payoffs are shown on the right. This example is fairly simple. There is a basic cost structure involving only marginal cost (there is no fixed cost ). The demand function is linear and price elasticity of demand is 1. However, it illustrates the leader's advantage. The follower wants to choose q 2 {\displaystyle q_{2}} to maximise its payoff q 2 × ( 5000 − q 1 − q 2 − c 2 ) {\displaystyle q_{2}\times (5000-q_{1}-q_{2}-c_{2})} . Taking

940-484: The behavior of economic agents, which is based on ex ante anticipations: For these anticipations determine the behaviour of the economic subjects and consequently those changes in the whole price system which during a period actually occur as a result of the actions of individuals. (Gunnar Myrdal,Monetary Equilibrium, London : W. Hodge 1939: 121) In context of ex-ante , the Swedish economist Myrdal also dealt with

987-583: The best response function of the follower (firm 2 {\displaystyle 2} ), the output that maximises the leader's profit is found. Hence, the maximum of Π 1 {\displaystyle \Pi _{1}} with respect to q 1 {\displaystyle q_{1}} is to be found. First, differentiate Π 1 {\displaystyle \Pi _{1}} with respect to q 1 {\displaystyle q_{1}} : Setting this to zero for maximisation: The following example

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1034-431: The best response of the leader were it that the follower would play Stackelberg if it (the leader) played Stackelberg. In this case, the best response of the leader would be to play Stackelberg. Hence, what makes this profile (or rather, these profiles) a Nash equilibrium (or rather, Nash equilibria) is the fact that the follower would play non-Stackelberg if the leader were to play Stackelberg. However, this very fact (that

1081-679: The best responses. Now the best response function of the leader is considered. This function is calculated by considering the follower's output as a function of the leader's output, as just computed. The profit of firm 1 {\displaystyle 1} (the leader) is Π 1 = P ( q 1 + q 2 ( q 1 ) ) ⋅ q 1 − C 1 ( q 1 ) {\displaystyle \Pi _{1}=P(q_{1}+q_{2}(q_{1}))\cdot q_{1}-C_{1}(q_{1})} , where q 2 ( q 1 ) {\displaystyle q_{2}(q_{1})}

1128-403: The equilibrium and chose some non-optimal quantity it would not only hurt itself, but it could also hurt the leader. If the follower chose a much larger quantity than its best response, the market price would lower and the leader's profits would be stung, perhaps below Cournot level profits. In this case, the follower could announce to the leader before the game starts that unless the leader chooses

1175-422: The firms (so the leader has no market advantage other than first move) and in particular c 1 = c 2 = 1000 {\displaystyle c_{1}=c_{2}=1000} . The leader would produce 2000 and the follower would produce 1000. This would give the leader a profit (payoff) of two million and the follower a profit of one million. Simply by moving first, the leader has accrued twice

1222-717: The first order derivative and equating it to zero (for maximisation) yields q 2 = 5000 − q 1 − c 2 2 {\displaystyle q_{2}={\frac {5000-q_{1}-c_{2}}{2}}} as the maximum value of q 2 {\displaystyle q_{2}} . The leader wants to choose q 1 {\displaystyle q_{1}} to maximise its payoff q 1 × ( 5000 − q 1 − q 2 − c 1 ) {\displaystyle q_{1}\times (5000-q_{1}-q_{2}-c_{1})} . However, in equilibrium, it knows

1269-418: The follower to punish in the next period so that the leader chooses Cournot quantities thereafter. The Stackelberg and Cournot models are similar because in both competition is on quantity. However, as seen, the first move gives the leader in Stackelberg a crucial advantage. There is also the important assumption of perfect information in the Stackelberg game: the follower must observe the quantity chosen by

1316-543: The follower will choose q 2 {\displaystyle q_{2}} as above. So in fact the leader wants to maximise its payoff q 1 × ( 5000 − q 1 − 5000 − q 1 − c 2 2 − c 1 ) {\displaystyle q_{1}\times (5000-q_{1}-{\frac {5000-q_{1}-c_{2}}{2}}-c_{1})} (by substituting q 2 {\displaystyle q_{2}} for

1363-433: The follower would play non-Stackelberg if the leader were to play Stackelberg) means that this profile is not a Nash equilibrium of the subgame starting when the leader has already played Stackelberg (a subgame off the equilibrium path). If the leader has already played Stackelberg, the best response of the follower is to play Stackelberg (and therefore it is the only action that yields a Nash equilibrium in this subgame). Hence

1410-565: The follower's best response function). By differentiation, the maximum payoff is given by q 1 = 5000 − 2 c 1 + c 2 2 {\displaystyle q_{1}={\frac {5000-2c_{1}+c_{2}}{2}}} . Feeding this into the follower's best response function yields q 2 = 5000 + 2 c 1 − 3 c 2 4 {\displaystyle q_{2}={\frac {5000+2c_{1}-3c_{2}}{4}}} . Suppose marginal costs were equal for

1457-656: The follower's problem: The maximisation problem resolves to (from the general case): Now solving for q 1 {\displaystyle q_{1}} yields q 1 ∗ {\displaystyle q_{1}^{*}} , the leader's optimal action: This is the leader's best response to the reaction of the follower in equilibrium. The follower's actual can now be found by feeding this into its reaction function calculated earlier: The Nash equilibria are all ( q 1 ∗ , q 2 ∗ ) {\displaystyle (q_{1}^{*},q_{2}^{*})} . It

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1504-403: The follower. Suppose firm i {\displaystyle i} has the cost structure C i ( q i ) {\displaystyle C_{i}(q_{i})} . The model is solved by backward induction . The leader considers what the best response of the follower is, i.e. how it will respond once it has observed the quantity of the leader. The leader then picks

1551-420: The game (the imperfection of knowledge) that results in neither player ( ceteris paribus ) being at a disadvantage. As mentioned, imperfect information in a leadership game reduces to Cournot competition. However, some Cournot strategy profiles are sustained as Nash equilibria but can be eliminated as incredible threats (as described above) by applying the solution concept of subgame perfection . Indeed, it

1598-418: The industry and the follower is a new entrant. Holding excess capacity is another means of commitment. The Stackelberg model can be solved to find the subgame perfect Nash equilibrium or equilibria (SPNE), i.e. the strategy profile that serves best each player, given the strategies of the other player and that entails every player playing in a Nash equilibrium in every subgame . In very general terms, let

1645-417: The leader also had a large cost structure advantage, perhaps due to a better production function ). There may also be cases where the follower actually enjoys higher profits than the leader, but only because it, say, has much lower costs. This behaviour consistently work on duopoly markets even if the firms are asymmetrical. If, after the leader had selected its equilibrium quantity, the follower deviated from

1692-427: The leader believes that the follower is irrational). If the leader played a Stackelberg action, (it believes) that the follower will play Cournot. Hence it is non-optimal for the leader to play Stackelberg. In fact, its best response (by the definition of Cournot equilibrium) is to play Cournot quantity. Once it has done this, the best response of the follower is to play Cournot. Consider the following strategy profiles:

1739-435: The leader has chosen, the follower is better off by playing on the equilibrium path. Hence, such a threat by the follower would not be credible. However, in an (indefinitely) repeated Stackelberg game, the follower might adopt a punishment strategy where it threatens to punish the leader in the next period unless it chooses a non-optimal strategy in the current period. This threat may be credible because it could be rational for

1786-444: The leader plays Cournot; the follower plays Cournot if the leader plays Cournot and the follower plays Stackelberg if the leader plays Stackelberg and if the leader plays something else, the follower plays an arbitrary strategy (hence this actually describes several profiles). This profile is a Nash equilibrium. As argued above, on the equilibrium path play is a best response to a best response. However, playing Cournot would not have been

1833-400: The leader's move because it is irrational for the follower not to observe if it can once the leader has moved. If it can observe, it will so that it can make the optimal decision. Any threat by the follower claiming that it will not observe even if it can is as uncredible as those above. This is an example of too much information hurting a player. In Cournot competition, it is the simultaneity of

1880-402: The leader, otherwise the game reduces to Cournot. With imperfect information, the threats described above can be credible. If the follower cannot observe the leader's move, it is no longer irrational for the follower to choose, say, a Cournot level of quantity (in fact, that is the equilibrium action). However, it must be that there is imperfect information and the follower is unable to observe

1927-508: The maximum of Π 2 {\displaystyle \Pi _{2}} with respect to q 2 {\displaystyle q_{2}} is to be found. First differentiate Π 2 {\displaystyle \Pi _{2}} with respect to q 2 {\displaystyle q_{2}} : Setting this to zero for maximisation: The values of q 2 {\displaystyle q_{2}} that satisfy this equation are

Stackelberg competition - Misplaced Pages Continue

1974-451: The position of John Maynard Keynes to not include it in his work was currently considered as an oddity, if not a mistake. As Shackle put it: Myrdalian ex ante language would have saved the General Theory from describing the flow of investment and the flow of saving as identically, tautologically equal, and within the same discourse, treating their equality as a condition which may, or not, be fulfilled. (Shackle, G.L.S. (1989) "What did

2021-445: The price function for the (duopoly) industry be P {\displaystyle P} ; price is simply a function of total (industry) output, so is P ( q 1 + q 2 ) {\displaystyle P(q_{1}+q_{2})} where the subscript 1 {\displaystyle _{1}} represents the leader and 2 {\displaystyle _{2}} represents

2068-434: The profit of the follower. However, Cournot profits here are 1.78 million apiece (strictly, ( 16 / 9 ) 10 6 {\displaystyle (16/9)10^{6}} apiece), so the leader has not gained much, but the follower has lost. However, this is example-specific. There may be cases where a Stackelberg leader has huge gains beyond Cournot profit that approach monopoly profits (for example, if

2115-441: The question of the unit of time, which he proposed to solve by reducing the actual time-dimension of macroeconomic variables such as income, saving and investment to a point of time: Some of these quantities refer directly to a point of time. That is true of "capital value" as also of such quantities as demand and supply prices. Other terms – as e.g. "income", "revenue", "return", "expenses", "savings", "investments" – imply, however,

2162-577: The security domain. In this context, the defender (leader) designs a strategy to protect a resource, such that the resource remains safe irrespective of the strategy adopted by the attacker (follower). Stackelberg differential games are also used to model supply chains and marketing channels . Other applications of Stackelberg games include heterogeneous networks , genetic privacy , robotics , autonomous driving , electrical grids , and integrated energy systems . Ex ante The term ex-ante (sometimes written ex ante or exante )

2209-401: The strategy profile – which is Cournot – is not subgame perfect. In comparison with other oligopoly models, The Stackelberg concept has been extended to dynamic Stackelberg games. With the addition of time as a dimension, phenomena not found in static games were discovered, such as violation of the principle of optimality by the leader. In recent years, Stackelberg games have been applied in

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