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)

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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)

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(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.

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(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.)

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(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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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

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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**

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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

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(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

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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.

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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

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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

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

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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.

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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.

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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.

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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.

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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.

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

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(! 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.

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