EV uses the current prices as the base and asks what income change at the current prices would be equivalent to the proposed change in terms of its impact on utility.
CV uses the new prices as the base and asks what income change would be necessary to compensate the consumer. That is, EV (resp. CV ) requires to keep a consumer’**s** utility constant before (resp. as a result of) a price change.

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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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Two neighboring homeowners, **1** and 2, simultaneously choose how many hours to spend maintaining a beautiful lawn (denoted by l **1** and l 2 ). Since the appearance of one’**s** property depends in part on the beauty of the surrounding neighborhood, homeowner’**s** benefit is increasing in the hours that neighbor spends on his own lawn. Suppose that **1**’**s** payoff is expressed by

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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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2. Duopoly Game (20 points)
Consider a duopoly game in which two firms, denoted by Firm **1** and Firm 2, simultaneously and independently select their own prices, p **1** and p 2 , respectively.
The firms’ products are differentiated. After the prices are set, consumers demand A − p **1** + p 2

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

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

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ここで Apple ’**s****の**行動は Apple が Google **の**行動をどう予想
するかによって決まる
Google **の**最適な戦略は Google が「 Apple が Google
**の**行動をどう予想するか」をどう予想するかによって 決まる

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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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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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6. Cost Minimization Problem
A …m can rent capital (K) at a rental price r and hire labor (L) at a wage w. To produce anything at all requires one unit of capital, i.e. r **1** = r is a …xed cost; this is sunk in the short run, but not sunk in the long run. If in a unit of time the …rm employs L units of labor, and rents K units of capital (in addition to the one unit needed as a …xed cost), its output Q is given by one of the following production function:

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(c) When an indirect utility function takes Gorman form, its original utility func- tion must be quasi-linear.
(d) A firm’**s** cost function is homogeneous of degree 0 in the input price vectors. (e) If an allocation is Pareto efficient, it must be in the core.

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4. Exchange Economy (12 points)
Consider the following exchange economies with two agents and two goods. Derive competitive equilibrium prices and allocations in each case.
(a) Two agents, **1** and 2, have the following indirect utility functions: v **1** (p **1** , p 2 , ω ) = ln ω − a ln p **1** − (**1** − a) ln p 2

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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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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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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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“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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