December **4**, 2010, “Soukairou Hall ( 想海樓ホール )” 1 st floor ,
GRIPS, National Graduate Institute for Policy Studies, Tokyo
Organized by GRIPS, Hitotsubashi University Global COE program “Innovation in the Japanese Corporation-Education and Research Center for Empirical Management Studies” and Osaka University and Kyoto University Global COE program “Human Behavior and Socioeconomic Dynamics”

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Consider a consumer problem. Suppose that a choice function x(p; !) satis…es Walras’**s** law and WA. Then, show that x(p; !) is homogeneous of degree zero. 6. Lagrange’**s** Method
You have two …nal exams upcoming, Mathematics (M) and Japanese (J), and have to decide how to allocate your time to study each subject. After eating, sleeping, exercising, and maintaining some human contact, you will have T hours each day in which to study for your exams. You have …gured out that your grade point average (G) from your two courses takes the form

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3(a - e)/4, is greater than aggregate quantity in the Nash equilib- rium of the Cournot game, 2(a - e)/3, so the market-clearing price is lower in the Stackelberg game.. Thus, i[r]

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If the stage game has a unique NE, then for any T , the finitely repeated game has a unique SPNE: the NE of the stage game is played in every stage irrespective of the histor[r]

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Exist exactly one for ANY exchange problem. Always Pareto efficient and individually rational[r]

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Paul Romer (1955-, 内生的成長理論) → 学界から消えた！？
Ben Bernanke (1953-, マクロ、金融) → FRB議長を辞めた**の**は好材料？ Douglas Diamond (1953-, 銀行取付) → 金融は無い？
清滝信宏 (1955-, マクロ、金融) → まだ早い

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A strategy in dynamic games is a complete action plan which prescribes how the player will act in each possible.. contingencies in future..[r]

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Q = K 1 =**4**
L 1 =8 Then, answer the following questions.
(a) In the short run, the …rm is committed to hire a …xed amount of capital K(+1), and can vary its output Q only by employing an appropriate amount of labor L . Derive the …rm’**s** short-run total, average, and marginal cost functions. (b) In the long run, the …rm can vary both capital and labor. Derive the …rm’**s**

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(1) Write the payoff functions π 1 and π 2 (as a function of p 1 and p 2 ).
(2) Derive the best response function for each player. (3) Find the pure-strategy Nash equilibrium of this game.
(**4**) Derive the prices (p 1 , p 2 ) that maximize joint-profit, i.e., π 1 + π 2 .

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Introduction to Market Design and its Applications to School Choice.. Yosuke YASUDA.[r]

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Two neighboring homeowners, 1 and 2, simultaneously choose how many hours to spend maintaining a beautiful lawn (denoted by l 1 and l 2 ). Since the appearance of one’**s** property depends in part on the beauty of the surrounding neighborhood, homeowner’**s** benefit is increasing in the hours that neighbor spends on his own lawn. Suppose that 1’**s** payoff is expressed by

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R n + := {(x 1 , ..., x n )|x i ≥ 0, i = 1, ..., n} ⊂ R n .
For any x, y ∈ X, x % y means x is at least as preferred as y. Consumption set contains all conceivable alternatives.
A budget set is a set of feasible consumption bundles, represented as B(p, ω) = {x ∈ X|px ≤ ω}, where p is an n-dimensional positive vector interpreted as prices, and ω is a positive number interpreted as the consumer’**s** wealth.

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A tree starts with the initial node and ends at.. terminal nodes where payoffs are specified..[r]

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

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If the two pennies match, then player 2 wins player 1’**s** penny; if the pennies do not match, then 1 wins 2’**s** penny.
Although the existence of Nash equilibrium is not guaranteed, the natural extension of strategies, mixed strategies, will almost always assure the existence of equilibrium.

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5. Bayesian Nash Equilibrium (12 points)
There are three different bills, $5, $10, and $20. Two individuals randomly receive one bill each. The (ex ante) probability of an individual receiving each bill is therefore 1/3. Each individual knows only her own bill, and is simultaneously given the option of exchanging her bill for the other individual’**s** bill. The bills will be exchanged if and only if both individuals wish to do so; otherwise no exchange occurs. That is, each individuals can choose either exchange (E) or not (N), and exchange occurs only when both choose E. We assume that individuals’ objective is to maximize their expected monetary payoff ($).

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(a) The intersection of any pair of open sets is an open set.
(b) The union of any (possibly infinite) collection of open sets is open.
(c) The intersection of any (possibly infinite) collection of closed sets is closed. (You can use (b) and De Morgan’**s** Law without proofs.)

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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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Combination of dominant strategies is Nash equilibrium. There are many games where no dominant strategy exists[r]

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

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