数理经济学代写 EC301代写 经济学考试代考 经济学代写

EC301 Test

数理经济学代写 You have 90 minutes plus 15 minutes reading time. During the reading time, you may write on the question paper but not inside your answer booklet.

You have 90 minutes plus 15 minutes reading time. During the reading time, you may write on the question paper but not inside your answer booklet.

ANSWER TWO QUESTIONS

Question 1.

(a) Define separating Perfect Bayesian equilibrium (PBE) in this game. (5 marks)

(b) Compute all the separating PBE outcomes. For each equilibrium outcome, specify a set of posterior beliefs that, together with the action taken by each seller’s type (i.e., the low and high quality seller), constitute a PBE. (12 marks)

(c) Can you compare the separating PBE’s from the point of view of maximising the sum of buyers and the company’s utility? (5 marks)

(d) Define pooling PBE. (5 marks)

(e) Compute all the pooling PBE outcomes of the game. For each equilibrium outcome, specify a set of posterior beliefs that, together with the action taken by each seller’s type, constitute a PBE. (12 marks)

(f) Can you compare the pooling PBE’s from the point of view of maximising the sum of buyers and seller’s utility? (5 marks)

(g) Now suppose the company cannot advertise the new product. Which of the pooling and separating PBE’s with advertising are better for the company (i.e., give higher payoff) than the equilibrium when advertising is impossible? (6 marks)

Question 2. 数理经济学代写

A risk-neutral entrepreneur hires a project manager. The manager may select to work hard (H) or shirk (S). Shirking involves zero cost, while hard work costs the manager 10 utils (or units of utility). If the manager works hard, then the project yields either a revenue of 140 (thousand pounds) or 40, with equal probability. If the manager shirks then revenue is 40 with probability 1. The manager is risk averse; when her wage is w, her utility is √ 100w minus the cost of effort (which is zero if she shirks and 10 if she works hard). The manager’s outside option is to work for a different company where she would obtain a total payoff of 40. If indifferent between the outside option and the contract offered by the entrepreneur, assume the manager accepts the contract offered. The goal of the entrepreneur is to maximise his expected profit (revenue minus wage payments).

(a)

Suppose the entrepreneur observes the revenue realisation and whether or not the manager worked hard and can offer a contract with the manager’s wage that depends on the work effort and revenue realisation.

(a1) What is the contract the entrepreneur offers if he wants the manager to shirk? (5 marks)

(a2) What is the contract the entrepreneur offers if he wants the manager to work hard? (5 marks)

(a3) What is the optimal wage contract for the entrepreneur? Does the entrepreneur prefer to induce the agent to work hard or to shirk? What is the entrepreneur’s profit? (5 marks)

(b) 数理经济学代写

Now suppose the entrepreneur does not observe effort and can only offer a contract with wage that depends on the revenue realisation.

(b1) What is the contract the entrepreneur offers if he wants the manager to shirk? (5 marks)

(b2) What is the contract the entrepreneur offers if he wants the manager to work hard? In answering the question, describe the incentive and individual rationality constraint of the manager. Explain why both constraints have to hold as equalities. Write down the optimisation problem of the entrepreneur. (15 marks)

(b3) What is the optimal wage contract for the entrepreneur? Does the entrepreneur prefer to induce the manager to shirk or to work hard? What is the entrepreneur’s profit? (5 marks)

(c)

Due to a negative shock to the economy, the project’s revenue decreases. Now, if the manager works hard, then revenue is either 58 or 40 thousand pounds with equal probability, while if the manager shirks revenue is 40 with probability 1. Suppose the entrepreneur does not observe effort.

(c1) How does the negative shock to the economy affect the optimal wage contract when the entrepreneur wants to induce hard work and when he wants to induce shirking? What is the preferred wage contract for the entrepreneur? (10 marks)

Question 3. 数理经济学代写

Consider the following public goods game played between Andy and Bob. Each can provide the public good or not provide it, hoping to free ride on the other player. The cost of providing equals 5 to each player. There are two types of Andy:

Payoffs are given as follows:

• If neither player provides the good, both get utility of zero.
• If one player provides the good, then he gets utility equal to his value for that good minus his cost of providing it. This is independent of whether the other player provides the good or not.
• If one player doesn’t provide the good and the other player does provide it, then the non providing player has a free-ride and gets utility equal to his value for the good.

Players simultaneously decide whether to provide the good (action P) or not (action N).

(a) Assume a_H= 20, a_L= 3, b_H = 15, b_L= 12. Formulate this as a Bayesian game. (Players, Actions, Signals, States, Beliefs, Payoffs). (10 marks)

(b) Draw the game in extensive form, starting with a move by nature that determines the state. (5 marks)

(c) Write down Andy’s pure strategy set, mixed strategy set and behavioural strategy set, explaining the notation you are using. For an arbitrary behavioural strategy write the set of mixed strategies that are equivalent to it. (10 marks)

(d) Use iterative elimination of strictly dominated strategies to find the unique Bayesian Equilibrium of this game. (5 marks)

(e) Change the value of b_L to b_L= 8. Find all pure and mixed strategy Bayesian Equilibria of this game. (20 marks)

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