(c) If a player randomizes pure strategies X and Y in a (mixed strategy) Nash equilibrium, she MUST be indi¤erent between choosing X and Y .
**2**. Monopoly (**10** points)
Suppose a monopoly …rm operates in two di¤erent markets, A and B. Inverse demand for each market is given as follows.

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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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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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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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X
c∈C
max n 0, q τ(ˆ c **s**) − |ν l τ(ˆ **s**) (c) \ {ˆ **s**}| o
holds for any step l in the cycle, at any school c which ˆ **s** is admitted, q τ(ˆ c **s**) = |ν l τ(ˆ **s**) (c)| holds for any step l in the cycle. Hence, ˆ **s**’**s** rejected status for any school which ˆ **s** once proposed to cannot change to the non-rejected status by reproposal conditions (i) or (iii). Moreover, since a student **s** such that **s** ∈ S τ(ˆ **s**) and f (ˆ **s**) < f (**s**) cannot be assigned to a school which ˆ **s** prefers to her own assignment, reproposal condition (ii) does not apply to ˆ **s**. Therefore, ˆ **s** is always assigned to the same school in the cycle. Now we can separate the set of students who are always unfree because they do not change their assignments in the cycle. With the set of students who are always free in the cycle, only the reproposal condition (iii) could apply and it is when there was a reproposal before step t ′ . But a reproposal based on (iii) gives

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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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How to Measure Welfare Change | 厚生**の**変化をどうはかるか？
When the economic environment or market outcome changes, a consumer may be made better off ( 改善 ) or worse off ( 悪化 ). Economists often want to measure how consumers are affected by these changes, and have developed several tools for the assessment of welfare ( 厚生 ).

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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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4. Question 4 (5 points)
Consider a game of election with asymmetric information among voters. Whether candidate A or candidate B is elected depends on the votes of two citizens (denoted by 1 and **2**). The economy may be in one of two states, α and β. The citizens agree that candidate A is best if the state is α and candidate B is best if the state is β. The payoff for each citizen is symmetric and given as follows: 1 if the best candidate wins, 0 if the other candidate wins, and 1/**2** if the candidates tie. Suppose that citizen 1 is informed of the true state, whereas citizen **2** believes it is α with probability 0.9 and β with probability 0.1. Each citizen may either vote for candidate A, vote for candidate B, or not vote.

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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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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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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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Open Set and Closed Set (**2**)
Boundary and interior
◮ A point x is called a boundary point of a set S in R n
if every ε-ball centered at x contains points in S as well as points not in S. The set of all boundary points of a set S is called boundary, and is denoted ∂S .

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

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