Eco 601E: Advanced Microeconomics II (Fall, 2nd, 2013)
Final Exam: January 28
1. Dynamic Game (24 points)
Consider the following two-person dynamic game. In the first period, game A is played; after observing each player’s actions, they play game B in the second period. Assume that the payoffs are simply the sum of the payoffs of two games (i.e., there is no discounting).
This lecture is mostly based on Chapter 14 “Mechanism Design” of Tadelis (2013).
There are many economic and political situations in which some central authority wishes to implement a decision that depends on the private information of a set of players. The theory of mechanism design is the study of what kinds of mechanisms such a central authority (or mechanism designer) can devise in order to reveal some or all of the private information from the group of players who interact each other.
(b) Let p be a probability that player 2 would choose Rock, and q be a probability that she chooses Paper. Note that her probability of choosing Scissors is written as 1 p q. Under mixed strategy Nash equilibrium, player 1 must be indi¤erent amongst choosing Rock, Paper and Scissors, which implies that these three actions must give him the same expected payo¤s. Let u R ; u P ; u S be his expected payo¤s by selecting
(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.
Using this minimax theorem, answer the following questions.
(b) Show that Nash equilibria are interchangeable; if and are two Nash equilibria, then and are also Nash equilibria.
(c) Show that each player’s payo¤ is the same in every Nash equilibrium.
Let w = (w 1 , w 2 , w 3 , w 4 ) ≫ 0 be factor prices and y be an (target) output.
(a) Does the production function exhibit increasing, constant or decreasing returns to scale? Explain.
(b) Calculate the conditional input demand function for factors 1 and 2. (c) Suppose w 3 >
However, it is difficult to assess how reasonable some axioms are without having in mind a specific bargaining procedure. In particular, IIA and PAR are hard to defend in the abstract. Unless we can find a sensible strategic model that has an equilibrium corresponding to the Nash solution, the appeal of Nash’s axioms is in doubt.
(d) 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.
(e) 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 1 = c 2 = 8. Then, derive the
Hint: Your answers in (a) – (c) may change depending on the value of θ.
4. Duopoly (20 points)
Consider a duopoly game in which two firms, denoted by firm 1 and firm 2, simul- taneously and independently select their own price, p 1 and p 2 . The firms’ products are differentiated. After the prices are set, consumers demand 24 − p i +
Proof of Pratt’s Theorem (1) Sketch of the Proof.
To establish (i) ⇔ (iii), it is enough to show that P is positively related to r. Let ε be a “small” random variable with expectation of zero, i.e., E(ε) = 0. The risk premium P (ε) (at initial wealth x) is defined by
(a) Find a Bayesian Nash equilibrium of the game in pure strategies in which each player i accepts an exchange if and only if the value v i does not exceed some
threshold θ i
(b) How would your answer to (a) change if the value of player i’s house to the other player j becomes 5
Edgeworth Box | エッジワース・ボックス
The most useful example of an exchange economy is one in which there are two people and two goods. This economy’s set of allocations can be illustrated in an Edgeworth box ( エッジワース・ボックス ) diagram.
How do people actually play this game? When the proposer makes an ofer below 20 percent, the recipient rejects it about half of the time. Higher ofers are also rejected, but with lower frequency. The threat of rejection results in larger ofers than in the dictator game, and recipients enjoy significantly higher payoffs on average, even though some offers are rejected. In an ultimatum game experiment that was otherwise identical to the dictator game experiment discussed in the previous section (the same prize, subject pool, and sample size), 71 percent of proposers divided the $10 prize equally; none proposed keeping it all for themselves. Of the rest, 4 percent ofered $1, 4 percent ofered $2, 17 percent offered $4, and 4 percent ofered $6 36 As with the dictator game, specific results have varied a bit from experiment to experiment. Even so, virtually every study con irms that many subjects reject very low offers, and the threat of rejection produces larger offers.