Problem Set **4**: Due on June 21
Advanced Microeconomics II (Spring, 2nd, 2012)
1. Question 1 (6 points)
A monopolist faces two kinds of consumers: students and non-students. The demand function of students is q = x(100 − 2p). The demand function of non-students is given as q = y(100 − p). Assume that marginal cost of production is zero.

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安田予想で未受賞**の**候補者たち
Robert Barro (1944-, マクロ、成長理論) → イチオシ！
Elhanan Helpman (1946-, 国際貿易、成長) → 誰ともらう**の**か？
Paul Milgrom (1948-, 組織**の**経済学、オークション) → 今年は厳しい… Ariel Rubinstein (1951-, ゲーム理論) → 今年は厳しそう…

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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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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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Both the Bertrand and Cournot models are particular cases of a more general model of oligopoly competition where firms choose prices and quantities (or capacities.). Ber[r]

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

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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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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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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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Prisoners’ Dilemma: Analysis
( Silent , Silent ) looks mutually beneficial outcomes, though
Playing Confess is optimal regardless of other player’**s** choice! Acting optimally ( Confess , Confess ) rends up realizing!!

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payoff) while M gives 1 irrespective of player 1’**s** strategy.
Therefore, M is eliminated by mixing L and R .
After eliminating M , we can further eliminate D (step 2) and L
(step 3), eventually picks up ( U , R ) as a unique outcome.

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Find (all) pure‐strategy Nash equilibrium if it exists. iii.[r]

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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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