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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

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

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,...}.

(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 2p = p + 3(1 p) , 4p = 3 , p = 3=4.

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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!!

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,

(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).

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)

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

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

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)

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

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)

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

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]

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 >

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.

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