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
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.
1 Introduction
In the United States, affirmative action is an important method to create a diverse environment for students in schools or workers in firms. In the context of school choice, controlled school choice programs which try to balance the racial or socioe- conomic status at schools as well as to expand students’ choices are implemented. There are many examples of school boards that implemented controlled school choice programs such as Boston Public Schools, Educational Option in New York City high school match, Miami-Dade County Public Schools, or Chicago Public Schools.
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
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,
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]
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).
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.
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)
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)
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
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)
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 >