(b) Revenue equivalence theorem claims that the equilibrium bidding strategy under the …rst-price auction is ALWAYS identical to the one under the second- price auction.
(c) EVERY perfect Bayesian equilibrium is a weak perfect Bayesian equilibrium. 2. Dynamic Game (14 points)
(c) If a player randomizes pure strategies X and Y in a (mixed strategy) Nash equilibrium, she MUST be indi¤erent between choosing X and Y .
2. Monopoly (10 points)
Suppose a monopoly …rm operates in two di¤erent markets, A and B. Inverse demand for each market is given as follows.
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
justifiably claim an empty slot under ν k .
End: The algorithm ends when there is no student who is unassigned or whose assigned school is not the most preferable one among the schools for which she is in the non-rejected status. Then the tentative assignments become final. Obviously, this algorithm may not end in finite steps because of complex reproposal steps. However, owing to these reproposal steps, we allow students to repropose to schools at which they might justifiably claim an empty slot or justifiably envy some student if they did not repropose to. Therefore, though this algorithm may not work well in general, under some conditions it succeeds in producing an assignment and reducing justified envy and claims. First, we will show that CDAAR ends in finite steps when ≻ C has a common priority order for every type and an ordering
Problem Set 2: Due on May 10
Advanced Microeconomics I (Spring, 1st, 2012) 1. Question 1 (2 points)
Suppose the production function f satisfies (i) f (0) = 0, (ii) increasing, (iii) con- tinuous, (iv) quasi-concave, and (v) constant returns to scale. Then, show that f must be a concave function of x.
How to Measure Welfare Change | 厚生の変化をどうはかるか？
When the economic environment or market outcome changes, a consumer may be made better off ( 改善 ) or worse off ( 悪化 ). Economists often want to measure how consumers are affected by these changes, and have developed several tools for the assessment of welfare ( 厚生 ).
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.
Two players, 1 and 2, each own a house. Each player i values her own house at v i
and this is private information. The value of player i’s house to the other player j(6= i) is 3
2 v i . The values v i are drawn independently from the interval [0, 1] with uniform distribution. Suppose players announce simultaneously whether they want to exchange (E) their house of not (N). If both players agree to an exchange, the exchange takes place. Otherwise no exchange occurs.
(c) Any finite game has at least one Nash equilibrium in pure strategies. 2. Expected Utility (16 points)
Suppose that an individual can either exert effort or not. Her initial wealth is $100 and the cost of exerting effort is c. Her probability of facing a loss $75 (that is, her wealth becomes $25) is 1
However, it is difficult to assess how reasonable some axioms are without having in mind a specific bargaining procedure. In particular, IIA and PAR are hard to defend in the abstract. Unless we can find a sensible strategic model that has an equilibrium corresponding to the Nash solution, the appeal of Nash’s axioms is in doubt.
where u i (x, θ i ) is the money-equivalent value of alternative x ∈ X.
This assumes the case of private values in which player i’s payoff does not depend directly on other players’ types. If it does, then it is called common values case. The outcome (of the mechanism) is described by
Proof of Pratt’s Theorem (1) Sketch of the Proof.
To establish (i) ⇔ (iii), it is enough to show that P is positively related to r. Let ε be a “small” random variable with expectation of zero, i.e., E(ε) = 0. The risk premium P (ε) (at initial wealth x) is defined by
Problem Set 2: Due on May 14
Advanced Microeconomics I (Spring, 1st, 2013)
1. Question 1 (6 points)
(a) Suppose the utility function is continuous and strictly increasing. Then, show that the associated indirect utility function v(p, ω) is quasi-convex in (p, ω). (b) Show that the (minimum) expenditure function e(p, u) is concave in p.