3. I am indifferent, or x is indifferent ( 無差別である ) to y: x ∼ y
Note that we implicitly assume that the elements in X are all comparable, and ignore the intensity of preferences.
A legal answer to the questionnaire P can be formulated as a function f which assigns to any pair (x, y) of distinct elements in X exactly one of the three values: x ≻ y, y ≻ x or x ∼ y. That is,
Hint: A separating equilibrium means that Amy takes different strategies in S and N , while she chooses the same strategy in a pooling equilibrium. Your answer in (c) might depend on the value p.
3. Question 3 (4 points) Let F and G be distribution functions with support [0, ω], and let f and g be the corresponding probability density functions. The hazard rate of F is the function λ : [0, ω) → R + defined by
A set S in R n is called compact if it is closed and bounded.
Theorem 3 (Weierstrass, Existence of Extreme Values)
Let f : S → R be a continuous real-valued function where S is a non-empty compact subset of R n . Then f has its maximum and minimum values. That is, there exists vectors x and x such that
M P 1 : max
q≥0 p(q)q − c(q).
where p(q)q is a revenue and c(q) is a cost when the output is fixed to q. Let π(q) = p(q)q − c(q) denote the revenue function. Assume that the firm’s objective function π(q) is convex and differentiable. Then, the first order condition is:
(c) There are two pure-strategy Nash equilibria: (A; X) and (B; Y ).
(d) Let p be a probability that player 2 chooses X and q be a probability that player 1 chooses A. Since player 1 must be indi¤erent amongst choosing A and B, we obtain
2p = p + 3(1 p) , 4p = 3 , p = 3=4.
(nw1) means student s prefers an empty slot at school c to her own assignment, and (nw2) and (nw3) mean that legal constraints are not violated when s is assigned the empty slot without changing other students’ assignments.
The second property is about no-envy, which is also widely used in the context of school choice. But due to the structure of controlled school choice, as in Definition 1, even when a student prefers a school to her own and there is a student with lower priority in the school, the envy is not justified if the student’s move violates the legal constraints. Definition 2 formally states the condition for a student to have justified envy.
(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).
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
Q = K 1 =4
L 1 =8 Then, answer the following questions.
(a) In the short run, the …rm is committed to hire a …xed amount of capital K(+1), and can vary its output Q only by employing an appropriate amount of labor L . Derive the …rm’s short-run total, average, and marginal cost functions. (b) In the long run, the …rm can vary both capital and labor. Derive the …rm’s
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.
f (x, a) s.t. g(x, a) = 0.
where x is a vector of choice variables, and a := (a 1 , ..., a m ) is a
vector of parameters that may enter the objective function, the constraint, or both. Suppose that for each vector a, the solution is unique and denoted by x(a).