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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More on Roy’**s** Identity | もっとロア**の**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.

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Ann and Bob are in an Italian restaurant, and the owner offers them a free 3- slice pizza under the following condition. Ann and Bob must simultaneously and independently announce how many slice(**s**) she/he would like: Let a and b be the amount of pizza requested by Ann and Bob, respectively (you can assume that a and b are integer numbers between **1** and 3). If a + b ≤ 3, then each player gets her/his requested demands (and the owner eats any leftover slices). If a + b > 3, then both players get nothing. Assume that each players payoff is equal to the number of slices of pizza; that is, the more the better.

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

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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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【戦略】 個々**の**プレイヤーがとること**の**できる行動
【利得】 起こり得る行動**の**組み合わせに応じた満足度、効用
Q: ゲーム**の**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…

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

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

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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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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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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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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 is realized with probability α and the prize y with **1** − α. Denote by L(S) the (infinite) space containing all lotteries

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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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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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