Arrow’**s** Requirements of the SWF (1)
Unrestricted Domain (UD) The domain of f must include all possible combinations of individual preference relations on X.
Weak Pareto Principle (WP) For any pair of alternatives x and y in X, if xP i y for all i, then xP y.

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(e) The social welfare function introduced by Arrow is to derive social UTILITY by adding up individual utilities.
**2**. Externalities (25 points) Consider a one-consumer, one-firm economy (or equiv- alently an economy with many identical consumers and firms.) There are two private commodities. The firm also produces a level of pollution b. The produc- tion set of the firm is the convex set γ = {(y 1 , y **2** , b | G(y 1 , y **2** , b ) ≤ 0)}, where G

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A good is called normal (resp. inferior) if consumption of it increases (resp. declines) as income increases, holding prices constant.. Show the following claims.[r]

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(b) We will get (B; Z) in the following iterated elimination process: Step 1: We can erase X since X is strictly dominated by Z.
Step **2**: Given step 1, we can erase A since A is strictly dominated by B.
Step 3: Given steps 1 and **2**, we can erase Y since Y is strictly dominated by Z. (c) Any combinations of x and y that satisfy x + y = 100 are Nash equilibria. Clearly, there are 101 such equilibria, i.e., (0; 100)(1; 99):::(100; 0).

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(nw1) means student **s** prefers an empty slot at school c to her own assignment, and (nw**2**) and (nw3) mean that legal constraints are not violated when **s** is assigned the empty slot without changing other students’ assignments.
The second property is about no-envy, which is also widely used in the context of school choice. But due to the structure of controlled school choice, as in Definition 1, even when a student prefers a school to her own and there is a student with lower priority in the school, the envy is not justified if the student’**s** move violates the legal constraints. Definition **2** formally states the condition for a student to have justified envy.

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るい ひとみ ひとみ ひとみ ひとみ あい あい あい あい
1 位 位 位 位 ともき ともき ともき ともき ともき ともき ともき ともき だいき だいき だいき だいき **2** 位 位 位 位 こうき こうき こうき こうき こうき こうき こうき こうき ともき ともき ともき ともき 3 位 位 位 位 だいき だいき だいき だいき だいき だいき だいき だいき こうき こうき こうき こうき

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Solve the following problems in Snyder and Nicholson (11th):. 1.[r]

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Using this minimax theorem, answer the following questions.
(b) Show that Nash equilibria are interchangeable; if and are two Nash equilibria, then and are also Nash equilibria.
(c) Show that each player’**s** payo¤ is the same in every Nash equilibrium.

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Let w = (w 1 , w **2** , w 3 , w 4 ) ≫ 0 be factor prices and y be an (target) output.
(a) Does the production function exhibit increasing, constant or decreasing returns to scale? Explain.
(b) Calculate the conditional input demand function for factors 1 and **2**. (c) Suppose w 3 >

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すべて**の**プレーヤーに支配戦略が無いゲームでも解け る場合がある
「支配される戦略**の**逐次消去」（後述）
（お互い**の**行動に関する）「正しい予想**の**共有＋合理性」 によってナッシュ均衡は実現する！

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Proof of Pratt’**s** Theorem (1) Sketch of the Proof.
To establish (i) ⇔ (iii), it is enough to show that P is positively related to r. Let ε be a “small” random variable with expectation of zero, i.e., E(ε) = 0. The risk premium P (ε) (at initial wealth x) is defined by

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A bargaining situation is described by a tuple hX, D, % 1 , % **2** i: X is a set of possible agreements: a set of possible consequences that the two players can jointly achieve.
D ∈ X is the disagreement outcome: the event that occurs if the players fail to agree.

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(a) Derive firm 1’**s** payoff function and the best reply function.
(b) Solve the pure-strategy Nash equilibrium of this game. How much profit does each firm earn?
(c) Now suppose that firms decide prices sequentially: firm 1 sets its price p 1 first, and firm **2** chooses price only after observing firm 1’**s** price. Find the subgame perfect equilibrium of this game. How much profit does each firm earn?

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Three firms (1, **2** and 3) put three items on the market and can advertise these products either on morning (= M ) or evening TV (= E). A firm advertises exactly once per day. If more than one firm advertises at the same time, their profits become 0. If exactly one firm advertises in the morning, its profit is 1; if exactly one firm advertises in the evening, its profit is **2**. Firms must make their daily advertising decisions simultaneously.

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where u i (x, θ i ) is the money-equivalent value of alternative x ∈ X.
This assumes the case of private values in which player i’**s** payoff does not depend directly on other players’ types. If it does, then it is called common values case. The outcome (of the mechanism) is described by

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Solve the following problems in Snyder and Nicholson (11th):. 1.[r]

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(a) Find a Bayesian Nash equilibrium of the game in pure strategies in which each player i accepts an exchange if and only if the value v i does not exceed some
threshold θ i
(b) How would your answer to (a) change if the value of player i’**s** house to the other player j becomes 5

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Solve the following problems in Snyder and Nicholson (11th):. 1.[r]

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Solve the following problems in Snyder and Nicholson (11th):. 1.[r]

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Connection between UMP and EMP | UMP と EMP **の**関係
There is a strong link between the utility maximization problem (UMP, 効用最 大化問題 ) and the expenditure minimization problem (EMP, 支出最小化問題 ). Let us first consider the following practice question.

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