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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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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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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) 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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(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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Arrow-Debreu Equilibrium (**1**)
In principle, time/date can be incorporated in the state of nature. Consider an exchange economy with I agents and K goods:
Distinguish two dates: date 0 (ex ante), date **1** (ex post). There are S mutually exclusive state of nature.

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

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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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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 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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(a) If a consumer’**s** preference is complete and transitive, her demand behaviors always satisfy the weak axiom of revealed preference.
(b) Even if a firm’**s** technology shows increasing return to scale, the marginal product (with respect to some input) can be decreasing.

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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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