Prisoners’ Dilemma: Analysis (3)
(Silent, Silent) looks mutually beneficial outcomes, though
Playing Confess is optimal regardless of other player’**s** choice!
Acting optimally ( Confess , Confess ) rends up realizing!!

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“Soon after Nash ’s work, game-theoretic models began to be used in economic theory and political science,. and psychologists began studying how human subjects behave in experimental [r]

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◮ A lottery p is a function that assigns a nonnegative number to
each prize **s**, where P **s**∈S p(**s**) = **1** (here p(**s**) is the objective
probability of obtaining the prize **s** given the lottery p).
◮ Let α ◦ x ⊕ (**1** − α) ◦ y denote the lottery in which the prize x

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ここで Apple ’**s****の**行動は Apple が Google **の**行動をどう予想
するかによって決まる
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

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A consumer has a utility function u(x, y, z) = min{x, y}+z. The prices of the three goods are given by (p x , p y , p z ) and the consumer’**s** wealth is given by ω.
(a) Note that the utility function u can be written in the form of U (V (x, y), z). Derive the functions V (x, y) and U (V, z).

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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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(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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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

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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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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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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.)

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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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