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Lagrange’s Method | ラグランジュ（の未定乗数）法 (1) There are two approaches to solve this type of optimization problems with equality constraints: substitution ( 代入 ) and Lagrange’s method. Lagrange’s method is a powerful way to solve constrained optimization problems, which essentially translates them into unconstrained problems.

1. Course Description    This  is  an  advanced  course  in  microeconomics,  emphasizing  the  applications  of  mathematical tools and models to the study of individual economic decisions and their  aggregate  consequences.  We  begin  with  a  parsimonious  set  of  hypotheses  about  human behavior and the ways in which individual choices interact, and then examine  the  implications  for  markets.  This  entails  treatments  and  applications  of  consumer  theory and theory of the firm, under the ideal conditions implied by our hypotheses.   

1. Course Description    This  is  an  advanced  course  in  microeconomics,  emphasizing  the  applications  of  mathematical tools and models to the study of individual economic decisions and their  aggregate  consequences.  We  begin  with  a  parsimonious  set  of  hypotheses  about  human behavior and the ways in which individual choices interact, and then examine  the  implications  for  markets.  This  entails  treatments  and  applications  of  consumer  theory and theory of the firm, under the ideal conditions implied by our hypotheses.   

1. Course Description    This  is  an  advanced  course  in  microeconomics,  emphasizing  the  applications  of  mathematical tools and models to the study of individual economic decisions and their  aggregate  consequences.  We  begin  with  a  parsimonious  set  of  hypotheses  about  human behavior and the ways in which individual choices interact, and then examine  the  implications  for  markets.  This  entails  treatments  and  applications  of  consumer  theory and theory of the firm, under the ideal conditions implied by our hypotheses.   

This  is  an  advanced  course  in  microeconomics,  emphasizing  the  applications  of  mathematical tools and models to the study of individual economic decisions and their  aggregate [r]

1. Course Description    This  is  an  advanced  course  in  microeconomics,  succeeding  to  Advanced  Microeconomics I (ECO601E) in which we study individual economic decisions and their  aggregate consequences under ideal situations. In this course, we extend our previous  analyses  to  incorporate  imperfectly  competitive  market  structures,  dynamic  market  competitions,  and  incomplete  information.  To  this  end,  we  study  game  theory,  a  collection  of  mathematical  tools  for  analyzing  strategically  interdependent  situations.  Course grade will be determined by combining grades on three homework assignments  (45%)  and  a  final  exam  (55%).  Each  problem  set  will  be  distributed  in  class.  You  are  encouraged to form study groups, but must write up solutions independently. 

(a) Derive all pure strategy Nash equilibria. (b) Show that the following type of Nash equilibria does NOT exist: One firm chooses pure strategy M , and other two firms use mixed strategies. (c) Derive a symmetric mixed strategy Nash equilibria. You may assume that each firm chooses M with probability p and E with probability 1 − p, then calculate an equilibrium probability, p.

A function f (x) is homothetic if f (x) = g(h(x)) where g is a strictly increasing function and h is a function which is homogeneous of degree 1. Suppose preferences can be represented by a homothetic utility function. Then, show the followings. (a) The marginal rate of substitution between any two goods depends only on the ratio of the demands consumed. That is M RS ij is identical whenever x x j i takes

3. Auction (14 points) Suppose that a seller auctions one object to two buyers, = 1, 2. The buyers submit bids simultaneously, and the buyer with higher bid receives the object. The loser pays nothing while the winner pays the average of the two bids b + b

Introduction to Market Design and its Applications to School Choice.. Yosuke YASUDA.[r]

(a) Show that the above data satisfy the Weak Axiom of revealed preference. (b) Show that this consumer’s behavior cannot be fully rationalized. Hint: Assume there is some preference relation % that fully rationalizes the above data, and verify that % fails to satisfy transitivity.

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

(a) Show that if u(x 1 , x 2 ) and v(x 1 , x 2 ) are both homogeneous of degree r, then s (x 1 , x 2 ) := u(x 1 , x 2 ) + v(x 1 , x 2 ) is also homogeneous of degree r. (b) Show that if u(x 1 , x 2 ) and v(x 1 , x 2 ) are quasi-concave, then m(x 1 , x 2 ) := min{u(x 1 , x 2 ), v(x 1 , x 2 )} is also quasi-concave.

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Google の最適な戦略は Google が「 Apple が Google の行動をどう予想するか」をどう予想するかによって 決まる

Consider the following exchange economies with two agents and two goods. Derive competitive equilibrium prices (price ratio) and allocations in each case. (a) Two agents, a and b, have the following indirect utility functions: v a (p 1 , p 2 , ω) = ln ω − α ln p 1 − (1 − a) ln p 2

4. Exchange Economy (12 points) Consider the following exchange economies with two agents and two goods. Derive competitive equilibrium prices and allocations in each case. (a) Two agents, 1 and 2, have the following indirect utility functions: v 1 (p 1 , p 2 , ω ) = ln ω − a ln p 1 − (1 − a) ln p 2

endowment of time is 2ω 1 units. There is no (initial) endowment of consumption good. Each individual has a common utility function U (x) = ln x 1 + 2a ln x 2 . Sup- pose that only Ann owns the firm and its production function is y 2 = √z 1 , where y 2 is the output of consumption good and z 1 is the input of (total) labor. Let the

(a) Suppose % is represented by utility function u(·). Then, u(·) is quasi-concave IF AND ONLY IF % is convex. (b) Marshallian demand function is ALWAYS weakly decreasing in its own price. (c) Lagrange’s method ALWAYS derives optimal solutions for any optimization

(b) If consumer’s choice satis…es the weak axiom of revealed preferences, we can always construct a utility function which is consistent with such choice behav- iour. (c) If a consumer problem has a solution, then it must be unique whenever the consumer’s preference relation is convex.

vNM Utility Function (1) Note the function U is a utility function representing the preferences on L(S) while v is a utility function defined over S, which is the building block for the construction of U (p). We refer to v as a vNM (Von Neumann-Morgenstern) utility function.