# トップPDF PS1 最近の更新履歴 yyasuda's website ### PS1 最近の更新履歴 yyasuda's website

with x = (y, z) where y is a scalar, z is an n-dimensional consumption vector, and V (·) is a real valued function. The consumption set X = R n+1 + . (a) Show that if V is concave, U is quasi-concave. (b) Show that if U is quasi-concave, V is concave. 5. Question 5 (4 points) ### PS1 最近の更新履歴 yyasuda's website

with x = (y, z) where y is a scalar, z is an n-dimensional consumption vector, and V (·) is a real valued function. The consumption set X = R n+1 + . (a) Show that if V is concave, U is quasi-concave. (b) Show that if U is quasi-concave, V is concave. 5. Question 5 (4 points) ### 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. ### PS1 最近の更新履歴 yyasuda's website

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

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

long-run total, average, and marginal cost functions. 7. Expected Utility Suppose that an individual can either exert e¤ort or not. The cost of e¤ort is c. Her initial wealth is 100. Her probability of facing a loss 75 (that is, her wealth becomes 25) is 1 ### Final1 13 最近の更新履歴 yyasuda's website

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 ### 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 ### Lec2 1 最近の更新履歴 yyasuda's website

St Petersburg Paradox (1) The most primitive way to evaluate a lottery is to calculate its mathematical expectation, i.e., E[p] = P s∈S p(s)s. Daniel Bernoulli first doubt this approach in the 18th century when he examined the famous St. Pertersburg paradox.

15 さらに読み込む ### Final1 14 最近の更新履歴 yyasuda's website

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 ### 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 ### 最近の更新履歴 yyasuda's website

 政府（官僚組織、政治家）はどのように行動するか？  政治経済学 政治経済学 政治経済学 政治経済学  私企業中でなにが起こっているか？  組織経済学、企業統治（コーポレート・ガバナンス） 組織経済学、企業統治（コーポレート・ガバナンス） 組織経済学、企業統治（コーポレート・ガバナンス） 組織経済学、企業統治（コーポレート・ガバナンス）

70 さらに読み込む ### Final1 12 最近の更新履歴 yyasuda's website

i (p, u) denote the Hicksian demand function of good i and e(p, u) denote the expenditure function. Then, state the Shephard’s lemma. (c) Using envelope theorem, derive either (a) Roy’s identity, or (b) Shephard’s lemma. You can assume that the first order conditions guarantee the optimal solution, i.e., ignore the second order conditions. ### Lec1 14 最近の更新履歴 yyasuda's website

Overlapping Generations Model (2) Proof. Suppose that each member of generation t + 1 transfers one unit of its endowment to generation t. Now generation 1 is better off since it receives 3 unit of consumption in its lifetime. None of the other generations are worse off.

14 さらに読み込む ### Lec1 11 最近の更新履歴 yyasuda's website

Pareto Efficiency (1) A situation is called Pareto efficient if there is no way to make someone better off without making someone else worse off. That is, there is no way to make all agents better off. To put it differently, each agent is as well off as possible, given the utilities of the other agents.

15 さらに読み込む ### Lec1 12 最近の更新履歴 yyasuda's website

Second Welfare Theorem (1) Theorem 12 Consider an exchange economy with P i∈I e i ≫ 0, and assume that utility function u i is continuous, strongly increasing, and strictly quasiconcave for all i ∈ I. Then, any Pareto efficient allocation x is a competitive equilibrium allocation when endowments are redistributed to be equal to x.

15 さらに読み込む ### Lec1 13 最近の更新履歴 yyasuda's website

Second Welfare Theorem Theorem 9 Suppose the conditions stated in the existence theorem are satisfied. Let (x ∗ , y ∗ ) be a feasible Pareto efficient allocation. Then, there are income transfers, T 1 , ..., T I , satisfying P i∈I T i = 0, and a price vector p such that for all j ∈ J and for all i ∈ I.

13 さらに読み込む ### EX2 1 最近の更新履歴 yyasuda's website

Solve the following problems in Snyder and Nicholson (11th):. 1.[r] ### Final1 08 最近の更新履歴 yyasuda's website

(! x ; ! y ) = (1; 1) (a) Assume there are only two individuals in this economy. Then, draw the Edgworth-box and show the contract curve. Find a general equilibrium (equilibrium price and allocation) if it exists. If there is no equilibrium, explain the reason.