Arrow-Debreu Equilibrium (**1**)
In principle, time/date can be incorporated in the state of nature. Consider an exchange economy with I agents and K goods:
Distinguish two dates: date 0 (ex ante), date **1** (ex post). There are S mutually exclusive state of nature.

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現象 プレイヤー プレイヤー プレイヤー プレイヤー 「協力」 「協力」 「協力」 「協力」 「裏切り」 「裏切り」 「裏切り」 「裏切り」
軍拡競争 国 軍縮 軍拡
国際貿易政策 国 関税引き下げ 税率据え置き
男女間**の**協力 カップル 相手に従う 相手に要求

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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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(a) If a consumer’**s** preference is complete and transitive, her demand behaviors always satisfy the weak axiom of revealed preference.
(b) Even if a firm’**s** technology shows increasing return to scale, the marginal product (with respect to some input) can be decreasing.

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

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Suppose % is a preference relation on X. Then, show the followings. (a) Re‡exive: For any x 2 X, x x.
(b) Transitive **1**: For any x; y; z 2 X, if x y and y z, then x z. (c) Transitive 2: For any x; y; z 2 X, if x y and y z, then x z. (d) Transitive 3:For any x; y; z 2 X, if x y and y % z, then x % z. where and are de…ned as follows:

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Ann and Bob are in an Italian restaurant, and the owner offers them a free 3- slice pizza under the following condition. Ann and Bob must simultaneously and independently announce how many slice(**s**) she/he would like: Let a and b be the amount of pizza requested by Ann and Bob, respectively (you can assume that a and b are integer numbers between **1** and 3). If a + b ≤ 3, then each player gets her/his requested demands (and the owner eats any leftover slices). If a + b > 3, then both players get nothing. Assume that each players payoff is equal to the number of slices of pizza; that is, the more the better.

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where x is a vector of choice variables, and a := (a **1** , ..., a m ) is a vector of
parameters ( パラメータ ) that may enter the objective function and constraint.
Suppose that for each vector a, the solution is unique and denoted by x(a).
◮ A maximum-value function, denoted by M (a), is defined as follows:

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“Soon after Nash ’s work, game-theoretic models began to be used in economic theory and political science,. and psychologists began studying how human subjects behave in experimental [r]

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with x = (y, z) where y is a scalar, z is an n-dimensional consumption vector, and V (·) is a real valued function. The consumption set X = R n +**1**
+ .
(a) Show that if V is concave, U is quasi-concave. (b) Show that if U is quasi-concave, V is concave. 5. Question 5 (4 points)

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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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with x = (y, z) where y is a scalar, z is an n-dimensional consumption vector, and V (·) is a real valued function. The consumption set X = R n+**1**
+ .
(a) Show that if V is concave, U is quasi-concave. (b) Show that if U is quasi-concave, V is concave. 5. Question 5 (4 points)

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2. Duopoly Game (20 points)
Consider a duopoly game in which two firms, denoted by Firm **1** and Firm 2, simultaneously and independently select their own prices, p **1** and p 2 , respectively.
The firms’ products are differentiated. After the prices are set, consumers demand A − p **1** + p 2

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Lotteries (**1**)
We consider preferences and choices over the set of “lotteries.” Let S be a set of consequences (prizes). We assume that S is a finite set and the number of its elements (= |S|) is S. A lottery p is a function that assigns a nonnegative number to

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4. Exchange Economy (12 points)
Consider the following exchange economies with two agents and two goods. Derive competitive equilibrium prices and allocations in each case.
(a) Two agents, **1** and 2, have the following indirect utility functions: v **1** (p **1** , p 2 , ω ) = ln ω − a ln p **1** − (**1** − a) ln p 2

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endowment of time is 2ω **1** units. There is no (initial) endowment of consumption
good. Each individual has a common utility function U (x) = ln x **1** + 2a ln x 2 . Sup- pose that only Ann owns the firm and its production function is y 2 = √z **1** , where
y 2 is the output of consumption good and z **1** is the input of (total) labor. Let the

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Explain.
(b) Show that any risk averse decision maker whose preference satisfies indepen- dence axiom must prefer L 2 to L 3 .
3. Question 3 (4 points) Suppose a monopolist with constant marginal costs prac- tices third-degree price discrimination. Group A’**s** elasticity of demand is ǫ A and

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