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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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.

（一般に）ナッシュ均衡は複数存在する場合がある
プレイヤー全員にとってあるナッシュ均衡よりも別**の**ナッシュ
均衡**の**方が望ましい場合もある
良い均衡（Mac, Mac）ではなく悪い均衡（ Win , Win ）が選 ばれてしまう危険性がある

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

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]

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.

◮ 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]

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.

5. Mixed Strategy (20 points)
Consider a patent race game in which a “weak” firm is given an endowment of 4 and a “strong” firm is given an endowment of 5, and any integral amount of the endowment could be invested in a project. That is, the weak firm has five pure strategies (invest 0, 1, **2**, **3** or 4) and the strong firm has six (0, 1, **2**, **3**, 4 or 5). The winner of the patent race receives the return of 10. Both players are instructed that whichever player invests the most will win the race and if there is a tie, both lose: neither gets the return of 10.

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.

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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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.

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]

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.

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

elimination of strictly dominated strategies can never be selected (with positive probability) in a mixed-strategy Nash equilibrium.[r]

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