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]
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]
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]
December 4, 2010, “Soukairou Hall ( 想海樓ホール )” 1 st floor ,
GRIPS, National Graduate Institute for Policy Studies, Tokyo
Organized by GRIPS, Hitotsubashi University Global COE program “Innovation in the Japanese Corporation-Education and Research Center for Empirical Management Studies” and Osaka University and Kyoto University Global COE program “Human Behavior and Socioeconomic Dynamics”
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
Q = K 1 =4
L 1 =8 Then, answer the following questions.
(a) In the short run, the …rm is committed to hire a …xed amount of capital K(+1), and can vary its output Q only by employing an appropriate amount of labor L . Derive the …rm’s short-run total, average, and marginal cost functions. (b) In the long run, the …rm can vary both capital and labor. Derive the …rm’s
(1) Write the payoff functions π 1 and π 2 (as a function of p 1 and p 2 ).
(2) Derive the best response function for each player. (3) Find the pure-strategy Nash equilibrium of this game.
(4) Derive the prices (p 1 , p 2 ) that maximize joint-profit, i.e., π 1 + π 2 .
Two neighboring homeowners, 1 and 2, simultaneously choose how many hours to spend maintaining a beautiful lawn (denoted by l 1 and l 2 ). Since the appearance of one’s property depends in part on the beauty of the surrounding neighborhood, homeowner’s benefit is increasing in the hours that neighbor spends on his own lawn. Suppose that 1’s payoff is expressed by
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.
R n + := {(x 1 , ..., x n )|x i ≥ 0, i = 1, ..., n} ⊂ R n .
For any x, y ∈ X, x % y means x is at least as preferred as y. Consumption set contains all conceivable alternatives.
A budget set is a set of feasible consumption bundles, represented as B(p, ω) = {x ∈ X|px ≤ ω}, where p is an n-dimensional positive vector interpreted as prices, and ω is a positive number interpreted as the consumer’s wealth.
(a) The intersection of any pair of open sets is an open set.
(b) The union of any (possibly infinite) collection of open sets is open.
(c) The intersection of any (possibly infinite) collection of closed sets is closed. (You can use (b) and De Morgan’s Law without proofs.)
Problem Set 2: Posted on November 4
Advanced Microeconomics I (Fall, 1st, 2014)
1. Question 1 (7 points)
A real-valued function f (x) is called homothetic if f (x) = g(h(x)) where g : R → R is a strictly increasing function and h is a real-valued function which is homo- geneous of degree 1. Suppose that preferences can be represented by a homothetic utility function. Then, prove the following statements.