Slutsky Equation (**1**)
When the price of a good declines, there are two conceptually separate reactions: The consumer is expected to substitute the relatively cheaper good for the now relatively more expensive good (= substitution effect), and to arrange her purchases of all goods due to the expansion of her effective income, i.e., the budget set (= income effect).

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Continuous
(a) Show that if % is represented by a linear utility function, i.e., u(x **1** ; x 2 ) = x **1** + x 2
with ; > 0, then % satis…es the above three properties.
(b) Find the preference relation that is **1**) Additive and Strictly monotone but not Continuous, and 2) Strictly monotone and Continuous but not Additive.

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(5) Suppose that this game is played finitely many times, say T (≥ 2) times. De- rive the subgame perfect Nash equilibrium of such a finitely repeated game. Assume that payoff of each player is sum of each period payoff.
(**6**) Now suppose that the game is played infinitely many times: payoff of each player is discounted sum of each period payoff with some discount factor δ ∈ (0, **1**). Assume specifically that A = 16, c = 8. Then, derive the condition under which the trigger strategy sustains the joint-profit maximizing prices you derived in (3) (as a subgame perfect Nash equilibrium).

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Introduction to Market Design and its Applications to School Choice.. Yosuke YASUDA.[r]

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(a) Derive all pure strategy Nash equilibria.
(b) Show that the following type of Nash equilibria does NOT exist: One firm chooses pure strategy M , and other two firms use mixed strategies.
(c) Derive a symmetric mixed strategy Nash equilibria. You may assume that each firm chooses M with probability p and E with probability **1** − p, then calculate an equilibrium probability, p.

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A function f (x) is homothetic if f (x) = g(h(x)) where g is a strictly increasing function and h is a function which is homogeneous of degree **1**. Suppose preferences can be represented by a homothetic utility function. Then, show the followings.
(a) The marginal rate of substitution between any two goods depends only on the ratio of the demands consumed. That is M RS ij is identical whenever x x j i takes

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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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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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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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(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 the decision maker’**s** preferences under uncertainty are described by the vNM utility function, u(x) = √ x.
(a) Is the decision maker risk-averse, risk-neutral, or risk-loving? Explain why. (b) Calculate the absolute risk aversion and the relative risk aversion, respectively.

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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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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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not have any Nash equilibrium, including mixed strategy equilibrium. 5. Question 5 (**6** points, Review)
A crime is observed by a group of n people. Each person would like the police to be informed but prefers that someone else make the phone call. They choose either “call” or “not” independently and simultaneously. A person receives 0 payoff if no one calls. If someone (including herself) makes a call, she receives v while making a call costs c. We assume v > c so that each person has an incentive to call if no one else will call.

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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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“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 strategy in dynamic games is a complete action plan which prescribes how the player will act in each possible.. contingencies in future..[r]

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