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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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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The main theorem shows that the condition that a schools’ priority profile ≻ C
has a common priority order for every type t ∈ T is sufficient for the existence of feasible assignments which are both fair and non-wasteful. This condition may be strong and hard to be satisfied when the classification of types is coarse. For instance, if the type set is {high income, low income} and there is a priority for students who live in each school’**s** walk zone, priority orders for high income students will differ across schools in general. However, this can be modified by making a finer type classification, {high income, low income} × {c **1** ’**s** walk zone, c **2** ’**s** walk zone,...}.

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(c) There are two pure-strategy Nash equilibria: (A; X) and (B; Y ).
(d) Let p be a probability that player **2** chooses X and q be a probability that player **1** chooses A. Since player **1** must be indi¤erent amongst choosing A and B, we obtain
**2**p = p + 3(**1** p) , 4p = 3 , p = 3=4.

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るい ひとみ ひとみ ひとみ ひとみ あい あい あい あい
**1** 位 位 位 位 ともき ともき ともき ともき ともき ともき ともき ともき だいき だいき だいき だいき **2** 位 位 位 位 こうき こうき こうき こうき こうき こうき こうき こうき ともき ともき ともき ともき 3 位 位 位 位 だいき だいき だいき だいき だいき だいき だいき だいき こうき こうき こうき こうき

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Prisoners’ Dilemma: Analysis (3)
(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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for all **s** i ∈ S i , which is identical to Nash equilibrium condition. To establish
uniqueness, assume on the contrary that there is another Nash equilibrium **s** ∗∗ 6= **s** ∗ . Pick player j with **s** ∗∗
j 6= **s** ∗ j . Since **s** ∗∗ j is a Nash equilibrium strategy,

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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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St Petersburg Paradox (**1**)
The most primitive way to evaluate a lottery is to calculate its
mathematical expectation, i.e., E[p] = P **s**∈S p(**s**)**s**.
Daniel Bernoulli first doubt this approach in the 18th century when he examined the famous St. Pertersburg paradox.

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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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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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Combination of dominant strategies is Nash equilibrium. There are many games where no dominant strategy exists[r]

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A good is called normal (resp. inferior) if consumption of it increases (resp. declines) as income increases, holding prices constant.. Show the following claims.[r]

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Let w = (w **1** , w **2** , w 3 , w 4 ) ≫ 0 be factor prices and y be an (target) output.
(a) Does the production function exhibit increasing, constant or decreasing returns to scale? Explain.
(b) Calculate the conditional input demand function for factors **1** and **2**. (c) Suppose w 3 >

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Using this minimax theorem, answer the following questions.
(b) Show that Nash equilibria are interchangeable; if and are two Nash equilibria, then and are also Nash equilibria.
(c) Show that each player’**s** payo¤ is the same in every Nash equilibrium.

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u **2** (x, y **2** ) = **2** ln x + y **2** .
u 3 (x, y 3 ) = 3 ln x + y 3 .
(a) Assume that the public good is purchased, privately and that person 3 is the first to go to the market and buy the public good. Assume he does not act strategically; he ignores persons **1** and **2** when he buys x, and thinks only of his own utility maximization problem. What is the outcome? How much of the public good does person 3 buy? How much do persons **1** and **2** buy? (b) Use the Samuelson optimality condition to find the Pareto optimal quantity

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