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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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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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) Any finite game has at least one Nash equilibrium in pure strategies. **2**. Expected Utility (16 points)
Suppose that an individual can either exert effort or not. Her initial wealth is $100 **and** the cost of exerting effort is c. Her probability of facing a loss $75 (that is, her wealth becomes $25) is 1

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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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Axiomatic Approach (**2**)
PAR (Pareto Efficiency) Suppose hU, di is a bargaining problem with v, v ′ ∈ U **and** v ′
i > v i for i = 1, **2**. Then f (U, d) 6= v. The axioms SYM **and** PAR restrict the behavior of the solution on single bargaining problems, while INV **and** IIA require the solution to exhibit some consistency across bargaining problems.

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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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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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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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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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(a) Derive each partner’**s** payo¤ function.
(b) Derive each partner’**s** best reply function **and** graphically draw them in a …gure. (Taking m in the horizontal axis **and** n in the vertical axis.)
(c) Is this game strategic complementarity, strategic substitution, or neither of them? Explain why.

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Suppose a government auctions one block of radio spectrum to two risk neutral mobile phone companies, i = 1; **2**. The companies submit bids simultaneously, **and** the company with higher bid receives a spectrum block. The loser pays nothing while the winner pays a weighted average of the two bids:

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Suppose a seller auctions one object to two risk neutral buyers, = 1; **2**. The buyers submit bids simultaneously, **and** the buyer with higher bid receives the object. The loser pays nothing while the winner pays a weighted average of the two bids b+b **2** 0
where b is the winner’**s** bid, b 0 is the loser’**s** bid. Assume that the valuation of the

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(d) Now suppose that the …rms interact inde…nitely through time. They discount future pro…ts at a discount factor . For what value of is there an equilibrium where …rms follow the “trigger strategies”, i.e., they produce cartel output as long as the other …rm has always produced cartel output **and** otherwise they produce Cournot Nash output?

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合理的な豚：分析
子豚には最適戦略（支配戦略）が存在する！ 大豚**の**行動によらず「 待つ 」**の**が常に最適
子豚が合理的ならば絶対にスイッチを押さない 子豚**の**「 スイッチを押す 」は可能性から消去される

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Dual Problem - Theory | 双対問題 - 理論 (1)
Applying the duality idea to the consumer problem, we can establish the close relationship between the indirect utility **and** expenditure functions, **and** between the Marshallian **and** Hicksian demand functions.

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Problem Set **3**: Due on July 17
Advanced Microeconomics II (Spring, **2**nd, 2013)
1. Question 1 (4 points) Consider the following labor market signaling game. There are two types of worker. Type 1 worker has a marginal value product of 1 **and** type **2** worker has a marginal value product of **2**. The cost of signal z for type 1 is C 1 (z) = z **and** for type **2** is C **2** (z) = (1 − c)z. The worker is type 1 with probability

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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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