5. Production Economy (25 points)
Consider an economy with two firms and two consumers. Firm **1** is entirely owned by consumer **1**; it produces good A from input X via the production function a = 2x. Firm 2 is entirely owned by consumer 2; it produces good B from input X via the production function b = 3x. Each consumer owns 10 units of X. Consumers’ preferences are given by the following utility functions:

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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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(c) When an indirect utility function takes Gorman form, its original utility func- tion must be quasi-linear.
(d) A firm’**s** cost function is homogeneous of degree 0 in the input price vectors. (e) If an allocation is Pareto efficient, it must be in the core.

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6. General Equilibrium (30 points)
Consider a production economy with two individuals, Ann (A) and Bob (B), and two goods, leisure x **1** and a consumption good x 2 . Ann and Bob have equal en- dowments of time (= ω **1** ) to be allocated between leisure and work, so the total

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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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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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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.

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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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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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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

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(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]

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Suppose that consumer i has preferences over the contingent consumption plans that satisfy expected utility hypothesis:
U i (x i **1** , x i 2 ) = π **1** u i (x i **1** ) + π 2 u i (x i 2 )
where π **1** (π 2 ) is the objective probability of nice (bad) weather.

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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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(d) Solve the pro…t maximization problem in (c), and derive the pro…t function, (p; w **1** ; w 2 ).
4. Uncertainty (10 points)
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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Hint: You can graphically show the claims if you prefer to do so.
(b) Derive the critical points (i.e., the combinations satisfying the …rst order con- ditions) of this maximization problem by using Lagrange’**s** method.
(c) What is the (maximum) value function? Is it strictly increasing in a?

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