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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M P 1 : max
q≥0 p(q)q − c(q).
where p(q)q is a revenue and c(q) is a cost when the output is fixed to q. Let π(q) = p(q)q − c(q) denote the revenue function. Assume that the firm’**s** objective function π(q) is convex and differentiable. Then, the first order condition is:

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

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消去**の**プロセスが途中で（場合によっては最初から）止まる 理論的な結果（**の**予測）をあまり絞ることができない
ナッシュ均衡**の**概念**の**方がより“強い”
逐次消去**の**途中でナッシュ均衡においてとられる行動が消去

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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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JR Jehle and Reny, Advanced Microeconomic Theory, **3**rd.
→ The copies of related chapters will be distributed in class.
NS Nicholson and Snyder, Microeconomic Theory: Basic
Principles and Extensions, 11th (older versions would be OK). Symbols that we use in lectures

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Problem Set **2**: Due on May 14
Advanced Microeconomics I (Spring, 1st, 2013)
1. Question 1 (6 points)
(a) Suppose the utility function is continuous and strictly increasing. Then, show that the associated indirect utility function v(p, ω) is quasi-convex in (p, ω). (b) Show that the (minimum) expenditure function e(p, u) is concave in p.

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

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Problem Set **2**: Posted on November 18
Advanced Microeconomics I (Fall, 1st, 2013)
1. Question 1 (7 points)
A real-valued function f (x) is called homothetic if f (x) = g(h(x)) where g : R → R is a strictly increasing function and h is a real-valued function which is homo- geneous of degree 1. Suppose that preferences can be represented by a homothetic utility function. Then, prove the following statements.

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◮ with probability p, a consumer with wealth x will receive a
times of her current wealth x
◮ with probability 1 − p she will receive b times of x.
Thm Assume that the assumptions of Pratt’**s** Theorem holds. Then, for any proportional risk, the decision maker 1 is more risk

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

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Problem Set **2**: Due on May 10
Advanced Microeconomics I (Spring, 1st, 2012) 1. Question 1 (**2** points)
Suppose the production function f satisfies (i) f (0) = 0, (ii) increasing, (iii) con- tinuous, (iv) quasi-concave, and (v) constant returns to scale. Then, show that f must be a concave function of x.

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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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Substituting into p+q = **3**=4, we achieve q = 1=**2**. Since the game is symmetric, we can derive exactly the same result for Player 1’**s** mixed action as well. Therefore, we get the mixed-strategy Nash equilibrium: both players choose Rock, Paper and Scissors with probabilities 1=4; 1=**2**; 1=4 respectively.

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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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Problem Set **2**: Posted on November 4
Advanced Microeconomics I (Fall, 1st, 2014)
1. Question 1 (7 points)
A real-valued function f (x) is called homothetic if f (x) = g(h(x)) where g : R → R is a strictly increasing function and h is a real-valued function which is homo- geneous of degree 1. Suppose that preferences can be represented by a homothetic utility function. Then, prove the following statements.

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

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You and your n − 1 roommates (n ≧ **2**) each have five hours of free time that could be used to clean your apartment. You all dislike cleaning, but you all like having a clean apartment: each person i’**s** payoff is the total hours spent (by everyone) cleaning, minus a number c (> 0) times the hours spent individually cleaning. That is,

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