(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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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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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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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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(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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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) 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

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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]

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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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(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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