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Problem Set 6: Due on July 30 Advanced Microeconomics II (Spring, 2nd, 2012) 1. Question 1 (7 points) Suppose a government auctions one block of radio spectrum to two risk neutral mobile phone companies, i = 1, 2. The companies submit bids simultaneously, and the company with higher bid receives a spectrum block. The loser pays nothing while the winner pays a weighted average of the two bids:

  A strategy in dynamic games is a complete action plan which prescribes how the player will act in each possible.. contingencies in future..[r]

   If the stage game has a unique NE, then for any T , the finitely repeated game has a unique SPNE: the NE of the stage game is played in every stage irrespective of the histor[r]

   Both the Bertrand and Cournot models are particular cases of a more general model of oligopoly competition where firms choose prices and quantities (or capacities.).   Ber[r]

elimination of strictly dominated strategies can never be selected (with positive probability) in a mixed-strategy Nash equilibrium.[r]

  A strategy in dynamic games is a complete action plan which prescribes how the player will act in each possible.. contingencies in future..[r]

  A tree starts with the initial node and ends at.. terminal nodes where payoffs are specified..[r]

3(a - e)/4, is greater than aggregate quantity in the Nash equilib- rium of the Cournot game, 2(a - e)/3, so the market-clearing price is lower in the Stackelberg game.. Thus, i[r]

  Paul Romer (1955-, 内生的成長理論) → 学界から消えた！？   Ben Bernanke (1953-, マクロ、金融) → FRB議長を辞めたのは好材料？   Douglas Diamond (1953-, 銀行取付) → 金融は無い？   清滝信宏 (1955-, マクロ、金融) → まだ早い

  Exist exactly one for ANY exchange problem.   Always Pareto efficient and individually rational[r]

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!!

Consider a consumer problem. Suppose that a choice function x(p; !) satis…es Walras’s law and WA. Then, show that x(p; !) is homogeneous of degree zero. 6. Lagrange’s Method You have two …nal exams upcoming, Mathematics (M) and Japanese (J), and have to decide how to allocate your time to study each subject. After eating, sleeping, exercising, and maintaining some human contact, you will have T hours each day in which to study for your exams. You have …gured out that your grade point average (G) from your two courses takes the form

payoff) while M gives 1 irrespective of player 1’s strategy.   Therefore, M is eliminated by mixing L and R .   After eliminating M , we can further eliminate D (step 2) and L (step 3), eventually picks up ( U , R ) as a unique outcome.

3(a - e)/4, is greater than aggregate quantity in the Nash equilib- rium of the Cournot game, 2(a - e)/3, so the market-clearing price is lower in the Stackelberg game.. Thus, i[r]

Prisoners’ Dilemma: Analysis
   ( Silent , Silent ) looks mutually beneficial outcomes, though    Playing Confess is optimal regardless of other player’s choice!    Acting optimally ( Confess , Confess ) rends up realizing!!

6. Cost Minimization Problem A …m can rent capital (K) at a rental price r and hire labor (L) at a wage w. To produce anything at all requires one unit of capital, i.e. r 1 = r is a …xed cost; this is sunk in the short run, but not sunk in the long run. If in a unit of time the …rm employs L units of labor, and rents K units of capital (in addition to the one unit needed as a …xed cost), its output Q is given by one of the following production function:

   If the stage game has a unique NE, then for any T , the finitely repeated game has a unique SPNE: the NE of the stage game is played in every stage irrespective of the histor[r]

(a) If an agent is risk averse, her risk premium is ALWAYS positive. (b) When every player has a (strictly) dominant strategy, the strategy profile that consists of each player’s dominant strategy MUST be a Nash equilibrium. (c) If there are two Nash equilibria in pure-strategy, they can ALWAYS be Pareto

(c) Solve for the total saving S by all types who save and the total borrowing B.. by all types who borrow.[r]

    5. Bayesian Nash Equilibrium (12 points)  There are three different bills, $5, $10, and $20. Two individuals randomly receive one  bill each. The (ex ante) probability of an individual receiving each bill is therefore 1/3.  Each  individual  knows  only  her  own  bill,  and  is  simultaneously  given  the  option  of  exchanging her bill for the other individual’s bill. The bills will be exchanged if and only  if  both  individuals  wish  to  do  so;  otherwise  no  exchange  occurs.  That  is,  each  individuals can choose either exchange (E) or not (N), and exchange occurs only when  both  choose  E.  We  assume  that  individuals’  objective  is  to  maximize  their  expected  monetary payoff ($).