7.3 THE COURNOT MODEl
In the previous section we concluded that, if irms' sales are limited by the output they produced beforehand, then in equilibrium, firms set prices such that total demand just clears total output.**s** This analysis can be taken one step back: What output levels should irms choose in the irst place? Suppose that output decisions are made simultaneously before prices are chosen. Based on this analysis, irms know that, for each pair of output choices (ql, q2), equilibrium prices will be Pl = P2 = P(ql + q2). This implies that irm i'**s** proit is given by fj =qi ( P(ql + q2) - c ) , assuming, as before, constant marginal cost, c. The **game** wherein irms simultaneously choose output levels is known as the Cournot model. 76 Speciically, suppose there are two firms in a market for a homogeneous product. Firms choose simultaneously the quantity they want to produce. The market price is then set at the level such that demand equals the total quantity produced by both irms.

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Consider a consumer problem. Suppose that a choice function x(p; !) satis…es Walras’**s** law and WA. Then, show that x(p; !) is homogeneous of degree zero. 6. Lagrange’**s** Method
You have two …nal exams upcoming, Mathematics (M) and Japanese (J), and have to decide how to allocate your time to study each subject. After eating, sleeping, exercising, and maintaining some human contact, you will have T hours each day in which to study for your exams. You have …gured out that your grade point average (G) from your two courses takes the form

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安田予想で未受賞**の**候補者たち
Robert Barro (1944-, マクロ、成長理論) → イチオシ！
Elhanan Helpman (1946-, 国際貿易、成長) → 誰ともらう**の**か？
Paul Milgrom (1948-, 組織**の**経済学、オークション) → 今年は厳しい… Ariel Rubinstein (1951-, ゲーム理論) → 今年は厳しそう…

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If the stage game has a unique NE, then for any T , the finitely repeated game has a unique SPNE: the NE of the stage game is played in every stage irrespective of the histor[r]

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3(a - e)/4, is greater than aggregate quantity in the Nash equilib- rium of the Cournot game, 2(a - e)/3, so the market-clearing price is lower in the Stackelberg game.. Thus, i[r]

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A tree starts with the initial node and ends at.. terminal nodes where payoffs are specified..[r]

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Review of Lecture **5**
Indifference property in mixed strategy NE.
If a player chooses more than one strategy with positive
probability, she must be indifferent among such pure strategies: choosing any of them generate same expected payoff.

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Both the Bertrand and Cournot models are particular cases of a more general model of oligopoly competition where firms choose prices and quantities (or capacities.). Ber[r]

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Q = K 1 =4
L 1 =8 Then, answer the following questions.
(a) In the short run, the …rm is committed to hire a …xed amount of capital K(+1), and can vary its output Q only by employing an appropriate amount of labor L . Derive the …rm’**s** short-run total, average, and marginal cost functions. (b) In the long run, the …rm can vary both capital and labor. Derive the …rm’**s**

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(**5**) Suppose that this **game** is played finitely many times, say T (≥ 2) times. De- rive the subgame perfect Nash equilibrium of such a finitely repeated **game**. Assume that payoff of each player is sum of each period payoff.
(6) Now suppose that the **game** is played infinitely many times: payoff of each player is discounted sum of each period payoff with some discount factor δ ∈ (0, 1). Assume specifically that A = 16, c = 8. Then, derive the condition under which the trigger strategy sustains the joint-profit maximizing prices you derived in (3) (as a subgame perfect Nash equilibrium).

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Exist exactly one for ANY exchange problem. Always Pareto efficient and individually rational[r]

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1. Course Description
This is an introductory course in **game** theory, which will provide you with mathematical tools for analyzing strategic situations ‐ your optimal decision depends on what other people will do. In particular, we will study central solution concepts in **game** theory such as Nash equilibrium, subgame perfect equilibrium, and Bayesian equilibrium. Game theory has been widely recognized as an important analytical tool in such fields as economics, management, political science, phycology and biology. To illustrate its analytical value, we will cover a variety of applications that include international relations, development, business competition, auctions, marriage market, and so forth. There is no prerequisite for this course, although some background on microeconomics and familiarity of probabilistic thinking would be helpful.

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(a) Show that there is no pure-strategy equilibrium in this **game**.
(b) Is there any strictly dominated strategy? If yes, describe which strategy is dominated by which strategy. If no, briefly explain the reason.
(c) Derive the mixed-strategy Nash equilibrium.

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Find (all) pure‐strategy Nash equilibrium if it exists. iii.[r]

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C) Now suppose that the rule of the **game** is modified as follows. If exchange occurs, each individual receives 3 times as much amount as the bill she will have. For example, if individual 1 receives $**5** and 2 receives $10 initially and both wish to exchange, then 1 will receive $30 (= $10 x 3) and 2 will receive $15 (= $**5** x 3). Nothing happens if they do not exchange. Then, does trade occur in a Bayesian Nash equilibrium? Explain.

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(c) Solve for the total saving S by all types who save and the total borrowing B.. by all types who borrow.[r]

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1. Rationality
Players can reach Nash equilibrium only by rational reasoning in some games, e.g., Prisoners’ dilemma.
However, rationality alone is often insufficient to lead to NE. (see Battle of the sexes, Chicken **game**, etc.) A correct belief about players’ future strategies

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If the stage game has a unique NE, then for any T , the finitely repeated game has a unique SPNE: the NE of the stage game is played in every stage irrespective of the histor[r]

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3(a - e)/4, is greater than aggregate quantity in the Nash equilib- rium of the Cournot game, 2(a - e)/3, so the market-clearing price is lower in the Stackelberg game.. Thus, i[r]

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