Mathematical Economics | 经济代写 | econ代写 | math代写 | 计量经济代写 – Mathematical Economics

Mathematical Economics

Mathematical Economics | 经济代写 | econ代写 | math代写 – 这是经济方面的代写,结合math相关的内容,计量经济, 是比较有代表性的计量经济等代写方向

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Summer Examinations 2021/

Mathematical Economics 1A

Time Allowed: 2 Hours

Read all instructions carefully – and read through the entire paper at least once before you start entering your answers.

Answer TWO questions ONLY.

Approved pocket calculators are allowed.

You should not submit answers to more than the required number of questions. If you do, we will mark the questions in the order that they appear, up to the required number of questions in each section.

1. Consider the following game.

Mum Fink
Mum 4 , 4 0 , x
Fink x, 0 2 , 2
(a) Suppose x = 5. What are the Nash equilibria of the normal form game above?
(12 marks)
(b) For each value of x (0 , ), what are the pure-strategy Nash equilibria of the normal
form game above? (13 marks)
Now, suppose that this game might be played twice instead of once (with no
discounting). In fact, the number of times the game is played depends on the actions of
the players. More specifically, if both players play M um in the first period, then the
game is played a second time. If either plays F ink in the first period, then the game is
not played a second time. The game is never played a third time.


(Question 1 continued overleaf)


(c) Take x = 5. What are the pure-strategy subgame-perfect Nash equilibria of this game?
(13 marks)
(d) For which x (0 , )do all pure-strategy subgame-perfect Nash equilibria feature both
players playing F ink in the first period? (12 marks)

2. There are three types of workers in the world, I, II, and III; each type has probability 1 / 3. The output of a type I worker is 100 , the output of a type II is 80 , and the output of a type III is 40. Employers observe only the workers level of education e , not her type t , and pay a wage w equal to the expected output given the beliefs about the workers type (I, II or III) induced by her education. The workers utility is the wage she gets minus the cost of the education. The cost of education for a type I is 5 per year, the cost to a type II is 10 per year, and the cost to a type III is 20 per year. The worker can choose her years of education freely from [0 , )- that is, shes not constrained to integer amounts.

(a) Find a pooling PBE in pure strategies. Specify beliefs. (17 marks)
(b) Find the least years of education for each type for which there is a pure fully separating
PBE (one in which each type gets a different number of years of education). If no fully
separating equilibria exist, show this. (17 marks)
(c) For what levels of education are there pooling PBEs? For which of these do the
associated equilibria survive the Cho-Kreps Intuitive Criterion? (16 marks)

3. Butch and Marcellus can each choose to contribute or not contribute to the picnic. They choose simultaneously. If at least one person contributes, both will enjoy the picnic, which is worth v > 0 to each of them; v is common knowledge. If neither contributes, they both get 0. The cost of contributing, however, differs among players and is their type. The common prior holds that players costs of contributing cB and cM are drawn from independent uniform distributions on[0 , 1].

(a) Suppose Marcellus strategy is to contribute if cM<  c  M , where c  M< 1 is some cutoff
point. For a given cB , what is Butchs expected payoff when he contributes, and what is
it when he doesnt? Find Butchs best response to Marcellus cutoff strategy.
(15 marks)
(b) Find a BNE in which Butch and Marcellus use symmetric cutoff strategies. (20 marks)
(c) For what values of v are there asymmetric BNEs? (15 marks)