# トップPDF Lec1 9 最近の更新履歴 yyasuda's website ### Lec1 9 最近の更新履歴 yyasuda's website

A vector z in this space is interpreted as a production combination; positive components in z are interpreted as outputs and negative components as inputs. (Note that the firm’s profit can be expressed as pz = Pk i=1 p i z i where p ∈ R k ++ is a vector of prices.)

14 さらに読み込む ### PS1 最近の更新履歴 yyasuda's website

(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. ### PracticeF 最近の更新履歴 yyasuda's website

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 ### Final 最近の更新履歴 yyasuda's website

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 ### en 最近の更新履歴 yyasuda's website

Reconsidered” American Economics Review, Vol.101: 399-410.    Abdulkadiroglu, Che and Yasuda (forthcoming), “Expanding ‘Choice’ in School Choice” American Economic Journal: Microeconomics.    Gale and Shapley (1962), “College Admissions and the Stability of Marriage” American Mathematical Monthly, Vol.69: 9-15.

84 さらに読み込む ### Midterm 最近の更新履歴 yyasuda's website

(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. ### PQ1 最近の更新履歴 yyasuda's website

Solve the following problems in Snyder and Nicholson (11th):. 1.[r] ### PS1 最近の更新履歴 yyasuda's website

(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. ### Lec1 最近の更新履歴 yyasuda's website

 【戦略】 個々プレイヤーがとることできる行動  【利得】 起こり得る行動組み合わせに応じた満足度、効用 Q: ゲーム解（予測）はどうやって与えられる？ A: 実はノイマン達は一般的な解を生み出せなかった…

22 さらに読み込む ### Micro1 最近の更新履歴 yyasuda's website

More on Roy’s Identity | もっとロア恒等式 Roy’s identity says that the consumer’s Marshallian demand for good i is simply the ratio of the partial derivatives of indirect utility with respect to p i and ω after a sign change.

34 さらに読み込む ### Midterm1 10 最近の更新履歴 yyasuda's website

2. Revealed Preference (10 points) Consider the following choice problem. There are 4 feasible elements, and we denote the set of all elements as X = fa; b; c; dg. Suppose individual choice behaviors are described by two di¤erent choice functions, f 1 and f 2 . ### Final1 13 最近の更新履歴 yyasuda's website

(a) If a consumer’s preference satisfies completeness and transitivity, her prefer- ence can be ALWAYS represented by some utility function. (b) It is POSSIBLE that an expenditure function is a convex function of prices. (c) If the utility function is quasi-linear, the compensating variation is ALWAYS ### Final1 10 最近の更新履歴 yyasuda's website

where ; > 0. Let w 1 ; w 2 > 0 be the prices for inputs x 1 and x 2 respectively. Then, answer the following questions. (a) Sketch the isoquant for this technology. Hint: Isoquant is the combination of inputs that achieves a certain given level of output. (corresponds to “indi¤erence curve” in consumer theory.) ### Final1 11 最近の更新履歴 yyasuda's website

(d) If the relative risk aversion of some risk averse decision maker is independent of her wealth, then her absolute risk aversion MUST be decreasing in wealth.. (e) The competitive equi[r] ### Final1 12 最近の更新履歴 yyasuda's website

Suppose that the decision maker’s preferences under uncertainty are described by the vNM utility function, u(x) = √ x. (a) Is the decision maker risk-averse, risk-neutral, or risk-loving? Explain why. (b) Calculate the absolute risk aversion and the relative risk aversion, respectively. ### Midterm1 14 最近の更新履歴 yyasuda's website

(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 ### PracticeM 最近の更新履歴 yyasuda's website

(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. ### Final1 14 最近の更新履歴 yyasuda's website

is increasing in x 1 , the marginal product of x 2 must be negative. (c) Let (x, p) be a competitive equilibrium. Suppose u i (y i ) > u i (x i ) for some bundle y i . Then show that p · y i > p · x i . Does this depend on whether utility ### Lec2 1 最近の更新履歴 yyasuda's website

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.

15 さらに読み込む ### PS2 1 最近の更新履歴 yyasuda's website

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