Combination of dominant strategies is Nash equilibrium. There are many games where no dominant strategy exists[r]

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Outcome of JRMP May Violate Weak Stability
The same example as before: There are two hospitals h **1** , h 2 in one
region with regional cap 10.
Each hospital has a capacity of 10 and a target capacity of 5. There are 10 doctors, d **1** , . . . , d 10 such that

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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

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

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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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

(a) If a consumer’**s** preference satisfies completeness and transitivity, her prefer- ence can be ALWAYS represented by some utility function.
(b) It is POSSIBLE that an expenditure function is a convex function of prices. (c) If the utility function is quasi-linear, the compensating variation is ALWAYS

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**

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

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

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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

for i = **1**, · · · , n.
We now have to show that p ∗ is a competitive equilibrium. Using Warlas’ law, we can show that z i (p ∗ ) = 0 for all i. It is common to show the existence of equilibrium by applying a version of fixed-point theorems in Economics.

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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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An Arrow-Debreu security is a contract that agrees to pay one unit of a numeraire (a currency or a commodity) if a particular state of nature occurs and pays zero in all the other states. Suppose there exist no market for contingent commodities, but Arrow-Debreu securities for all state **s** ∈ S are exchanged at t = 0.

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

(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**

Three Firms (**1**, 2 and 3) put three items on the market and can advertise these products either on morning (= M ) or evening TV (= E). A firm advertises exactly once per day. If more than one firm advertises at the same time, their profits become 0. If exactly one firm advertises in the morning, its profit is **1**; if exactly one firm advertises in the evening, its profit is 2. Firms must make their daily advertising decisions simultaneously.

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?

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.)