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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Continuous
(a) Show that if % is represented by a linear utility function, i.e., u(x **1** ; x 2 ) = x **1** + x 2
with ; > 0, then % satis…es the above three properties.
(b) Find the preference relation that is **1**) Additive and Strictly monotone but not Continuous, and 2) Strictly monotone and Continuous but not Additive.

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

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(a) Derive all pure strategy Nash equilibria.
(b) Show that the following type of Nash equilibria does NOT exist: One firm chooses pure strategy M , and other two firms use mixed strategies.
(c) Derive a symmetric mixed strategy Nash equilibria. You may assume that each firm chooses M with probability p and E with probability **1** − p, then calculate an equilibrium probability, p.

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u i (**s** ′ i , **s** − i ) > u i (**s** i , **s** − i ) for all **s** − i ∈ S − i .
A strategy **s** ′
i is a weakly dominant strategy if playing **s** ′ i is
optimal for any combination of other players’ strategies: u i (**s** ′ i , **s** −i ) ≥ u i (**s** i , **s** −i ) for all **s** ∈ S and

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

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(a) Show that if u(x **1** , x 2 ) and v(x **1** , x 2 ) are both homogeneous of degree r, then
**s** (x **1** , x 2 ) := u(x **1** , x 2 ) + v(x **1** , x 2 ) is also homogeneous of degree r.
(b) Show that if u(x **1** , x 2 ) and v(x **1** , x 2 ) are quasi-concave, then m(x **1** , x 2 ) :=
min{u(x **1** , x 2 ), v(x **1** , x 2 )} is also quasi-concave.

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【戦略】 個々**の**プレイヤーがとること**の**できる行動
【利得】 起こり得る行動**の**組み合わせに応じた満足度、効用
Q: ゲーム**の**解（予測）はどうやって与えられる？
A: 実はノイマン達は一般的な解を生み出せなかった…

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More on Roy’**s** Identity | もっとロア**の**恒等式
Roy’**s** identity says that the consumer’**s** Marshallian demand for good i is
simply the ratio of the partial derivatives of indirect utility with respect to p i
and ω after a sign change.

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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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(b) Does this production function display increasing, constant, or decreasing re- turns to scale? Explain why.
(c) Formulate the cost minimization problem (you may denote a target output level by y). Then, solve it and derive the (minimum) cost function, c(w **1** , w 2 , y).

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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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problem with equality constraints. 2. Consumer Theory (30 points)
A consumer gets utility from 2 sources: drinking (measured in liters x) and time spent on the phone (measured in hours y). Each liter of drink costs $**4** and each hour on the phone costs $**4**. She has a total of $120 available for spending. Her utility function is given by:

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vNM Utility Function (**1**)
Note the function U is a utility function representing the preferences on L(S) while v is a utility function defined over S, which is the building block for the construction of U (p). We refer to v as a vNM (Von Neumann-Morgenstern) utility function.

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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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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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long-run total, average, and marginal cost functions.
7. Expected Utility
Suppose that an individual can either exert e¤ort or not. The cost of e¤ort is c. Her initial wealth is 100. Her probability of facing a loss 75 (that is, her wealth becomes 25) is **1**

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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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