Open Set and Closed Set (**2**)
Boundary and interior
A point x is called a boundary point of a set S in R n
if every ε-ball centered at x contains points in S as well as points not in S. The set of all boundary points of a set S is called boundary, and is denoted ∂S .

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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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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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Substituting into p+q = 3=4, we achieve q = **1**=**2**. Since the game is symmetric, we can derive exactly the same result for Player **1**’**s** mixed action as well. Therefore, we get the mixed-strategy Nash equilibrium: both players choose Rock, Paper and Scissors with probabilities **1**=4; **1**=**2**; **1**=4 respectively.

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that a plea bargain is allowed):
If both confess, each receives 3 years imprisonment.
If neither confesses, both receive **1** year.
If one confesses and the other one does not, the former will be set free immediately ( 0 payoff) and

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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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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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e z . The prices of the three goods are given by (p, q, **1**) and the consumer’**s** wealth is given by ω.
(a) Formulate the utility maximization problem of this consumer.
(b) Note that this consumer’**s** preference can be expressed in the form of U (x, y, z) = V (x, y) + z. Derive V (x, 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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or
u i ( i ; i ) u i (**s** i ; i ) for all **s** i **2** S i . (**2**)
7. Mixed strategies: Application
A crime is observed by a group of n people. Each person would like the police to be informed but prefers that someone else make the phone call. They choose either “call” or “not” independently and simultaneously. A person receives 0 payo¤ if no

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

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