for i = **1**, · · · , n.
We now have to show that p ∗ is a competitive equilibrium. Using Warlas’ law, we can show that z i (p ∗ ) = 0 for all i. It is common to show the existence of equilibrium by applying a version of fixed-point theorems in Economics.

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i (p, u) denote the Hicksian demand function of good i and e(p, u) denote
the expenditure function. Then, state the Shephard’**s** lemma.
(c) Using envelope theorem, derive either (a) Roy’**s** identity, or (b) Shephard’**s** lemma. You can assume that the first order conditions guarantee the optimal solution, i.e., ignore the second order conditions.

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いよいよゲーム理論**の**中身を見ていこう！
まずは**1**時点**の**（静学的な）ゲームを分析
各プレイヤーは独立かつ同時に戦略を決定
相手**の**決定を知らずに自分**の**戦略を決めるような状況 決定**の**タイミングは文字通り“同時”である必要は無い！

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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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(a) Show that the above data satisfy the Weak Axiom of revealed preference. (b) Show that this consumer’**s** behavior cannot be fully rationalized.
Hint: Assume there is some preference relation % that fully rationalizes the above data, and verify that % fails to satisfy transitivity.

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

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2 α ◦ x ⊕ (**1** − α) ◦ y ∼ (**1** − α) ◦ y ⊕ α ◦ x: The consumer does
not care about the order in which the lottery is described.
3 β ◦ (α ◦ x ⊕ (**1** − α) ◦ y) ⊕ (**1** − β) ◦ y ∼ (βα) ◦ x ⊕ (**1** − βα) ◦ y:
A consumer’**s** perception of a lottery depends only on the net probabilities of receiving the various prizes.

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6. General Equilibrium (30 points)
Consider a production economy with two individuals, Ann (A) and Bob (B), and two goods, leisure x **1** and a consumption good x 2 . Ann and Bob have equal en- dowments of time (= ω **1** ) to be allocated between leisure and work, so the total

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(a) Suppose % is represented by utility function u(·). Then, u(·) is quasi-concave IF AND ONLY IF % is convex.
(b) Marshallian demand function is ALWAYS weakly decreasing in its own price. (c) Lagrange’**s** method ALWAYS derives optimal solutions for any optimization

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is increasing in x **1** , the marginal
product of x 2 must be negative.
(c) Let (x, p) be a competitive equilibrium. Suppose u i (y i ) > u i (x i ) for some
bundle y i . Then show that p · y i > p · x i . Does this depend on whether utility

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Consider the case that M ≻ m. By I and C, there must be a single number v(**s**) ∈ [0, **1**] such that
v(**s**) ◦ M ⊕ (**1** − v(**s**)) ◦ m ∼ [**s**]
where [**s**] is a certain lottery with prize **s**, i.e., [**s**] = **1** ◦ **s**. In particular, v(M ) = **1** and v(m) = 0. I implies that

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