where x 1 and x **2** are chosen independently by nature, and each of which is uniformly distributed between 0 and 1.
The bidders observe their own valuations before engaging in the auction. The seller and the rival do not observe a bidder’**s** valuation; they only know the distribution.

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(d) The perfect Bayesian equilibrium puts NO restriction on beliefs at information sets that are not reached in equilibrium.
(e) In the simple moral hazard problem we studied in class, the optimal wage (= **s**( )) is NOT necessarily increasing in outcome (= x).

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The main theorem shows that the condition that a schools’ priority profile ≻ C
has a common priority order for every type t ∈ T is sufficient for the existence of feasible assignments which are both fair and non-wasteful. This condition may be strong and hard to be satisfied when the classification of types is coarse. For instance, if the type set is {high income, low income} and there is a priority for students who live in each school’**s** walk zone, priority orders for high income students will differ across schools in general. However, this can be modified by making a finer type classification, {high income, low income} × {c 1 ’**s** walk zone, c **2** ’**s** walk zone,...}.

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

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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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Similarly, player **2** must be indi¤erent amongst choosing X and Y , which implies 4q + 6(1 q) = 7(1 q)
, 5q = 1 , q = 1=5.
Thus, the mixed-strategy equilibirum is that player 1 takes A with probability 1=5 (and B with probability 4=5) and player **2** takes X with probability 3=4 (and Y with probability 1=4).

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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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Players 1 (proposer) and **2** (receiver) are bargaining over how to split the ice-cream of size 1. In the first stage, player 1 proposes a share {x, 1 − x} to player **2** where x ∈ [0, 1] is player 1’**s** own share. Player **2** can decide whether accept the offer or reject it. If player **2** accepts, then the game finishes and players get their shares. If player **2** rejects, the game move to the second stage, in which the size of the ice-cream becomes δ(∈ (0, 1)) of the original size due to melting. In the second stage, by flipping a coin, the ice-cream is randomly assigned to one of the players. Suppose each player maximizes expected size of the ice-cream that she can get. Derive a subgame perfect Nash equilibrium of this game.

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

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Eco 601E: Advanced Microeconomics II (Fall, **2**nd, 2013)
Final Exam: January 28
1. Dynamic Game (24 points)
Consider the following two-person dynamic game. In the first period, game A is played; after observing each player’**s** actions, they play game B in the second period. Assume that the payoffs are simply the sum of the payoffs of two games (i.e., there is no discounting).

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(c) Confirm that by choosing the tax t appropriately, the socially optimal level of pollution is produced.
(d) Add a second firm with a different production function. Now the consumers observe a pollution level b = b 1 + b **2** . Show that the social optimum can still

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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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However, it is difficult to assess how reasonable some axioms are without having in mind a specific bargaining procedure. In particular, IIA and PAR are hard to defend in the abstract. Unless we can find a sensible strategic model that has an equilibrium corresponding to the Nash solution, the appeal of Nash’**s** axioms is in doubt.

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

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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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