An aggregate production plan y maximizes aggregate profit, if and only if each firm’**s** production plan y j maximizes its individual profit for all j ∈ J.
The theorem implies that there are two equivalent ways to construct the aggregate net supply function:

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(a) If a consumer’**s** preference satisfies completeness and transitivity, her prefer- ence can be ALWAYS represented by some utility function.
(b) It is POSSIBLE that an expenditure function is a convex function of prices. (c) If the utility function is quasi-linear, the compensating variation is ALWAYS

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囚人**の**ジレンマ：注意点
このゲームでは個々**の**プレーヤーが最適戦略を持つ
【最適戦略（支配戦略）】 他**の**プレーヤーたちがどのような行 動を選択しても、自分がある特定**の**行動Aを選ぶことによって 利得が最大化されるとき、行動Aを「支配戦略」と呼ぶ。

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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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(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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◮ A set S in R n is called compact if it is closed ( 閉 ) and bounded.
Thm A**1**.10 (Weierstrass) Existence of Extreme Values
Let f : S → R be a continuous real-valued function where S is a non-empty
compact subset of R n . Then f has its maximum and minimum values. That

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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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each prize **s**, where P **s**∈S p(**s**) = **1** (here p(**s**) is the objective
probability of obtaining the prize **s** given the lottery p). Let α ◦ x ⊕ (**1** − α) ◦ y denote the lottery in which the prize x is realized with probability α and the prize y with **1** − α. Denote by L(S) the (infinite) space containing all lotteries

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

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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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