Pareto Optimal Allocations

NEWCASTLE UNIVERSITY _______________________ RESIT ASSESSMENT PERIOD 2020/2021 _______________________ ECO3001 Advanced Microeconomics Time allowed - 24 hours Your completed work must be submitted by 09:30am on 01/09 /2021 You may submit your work at any point during the 24 - hour period of the exam. (IMPORTANT NOTE: Time in the UK is currently GMT+1hr, you may check what time it is in the UK at the following link. https://www.thetimezoneconverte r.com/ ) It is expected that this paper will take 3 hours to complete. To ensure browser compatibility with Canvas for downloading and uploading exams we recommend using Google Chrome You are not expected to, and it is not advisable to, work on this assessment constantly for the 24 - hour period. The assessment has been designed to be worked on for a much shorter period of time in line with the length of a standard exam sitting. 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Where appropriate the bibliography is not included in the word count. You should write the total number of words at the top of your submission, if you are scanning in handwritten notes you should count the average number of words per line and multiply by the ave rage number of lines per page to give a rough approximation d) Submissions must be in the form of a Microsoft Word file . Do NOT submit in .pages formats from Apple products . Any handwritten answers/diagrams/graphs etc should be scanned into the document usin g Microsoft Lens. Please note - Turnitin can only accept one document for upload. If you are scanning anything in you must combine them into one file prior to upload. If you are scanning in any documents you must ensure that you have at least 20 typed wo 20 times) within the document you submit This is to ensure Turnitin/Canvas will allow you to upload the document. You must ensure you check your file before submitting. 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If you have any issues accessing your assessment or University IT systems you can contact IT support on +44 (0)191 208 5999 at any time, or by email at [email protected] (monitored 08:00 - 17:00 GMT+1hr, Monday to Friday). * Remember, if you encounter IT difficulties that significantly affects your ability to complete this exam or when uploading your answer(s) you ma y submit a PEC detailing these. [ECO3001 ] NEWCASTLE UNIVERSITY _______________________ RESIT EXAM 2020/21 _______________________ ADVANCED MI CROECONOMICS Candidates should answer ALL questions . [Please turn over] [ECO3001] 2 1. [20 marks] Consider an economy with two consumers (Consumer 1 and Consumer 2) and two goods ( x and y ). Preferences are given by the utility functions and The initial endowments are , , and , and a ssume that the price of good y is given and it is . (a) Illustrate and analyse this pure exchange economy. Find the set of Pareto optimal allocations. (b) Find and characterise the competitive equilibrium of this economy. (c) What does the Second Fundamental Theorem of Welfare Economics say? Without calculating any transfers, illustrate it in your Edgeworth Box. 2. (A ) [1 0 marks] Solve the following sequential - move game: (B) [1 0 marks] Prove that the number of non - empty coalitions that can form with N players is , then consider the following 3 - person coalitional g ame described by its characteristic function: v (1) = 7; v (2) = 0; v (3) = 12; v (1,2) = 23; v (2,3) = 24; v (1, 3) = 27; v (1,2,3) = 35. Check if the coalitional game is super - additive and essential, then normalise the payoffs. Explain the concept of core, check for its existence in the game above, then find and illustrate the core allocations. [ECO3001] 3 3. (A) [10 marks] Find the equilibrium of the one - shot game below. 1/2 C D C 4, 4 - 3, 9 D 5, - 4 - 1, - 2 State the Folk Theorem and depict the feasible and individually rational payoffs that could be achieved if the above stage game was repeated infinitely. Construct explain. (B) [10 marks] Consider the following 3 - person coalitional game, described by the values (worth) of each possible non - empty coalition: v (1) = 5; v (2) = 1; v (3) = 4 v(1,2) = 11; v (2,3) = 9; v (1,3) = 24 v (1,2,3) = 66 Calculate the Shapley value for each person, assuming that all scenarios under which the grand coalition forms are equally likely. Interpret your results. 4. [2 0 marks] Consider a simple model of screening with education, where firms offer menus of contracts conting ent on education levels. There are two t ypes of students: high ability (H), with productivity 4, and low ability (L), with productivity 1 . Assume q is the proportion of L students in the population. Acquiring education y is costly and the cost functions a re c(y) = y for the L students, and c(y) = y/2 for the H students. (a) Discuss in detail, present diagrammatical ly and demonstrate that there cannot exist a pooling equilibrium. (b) Using an appropriate diagram, consider the possibility of a separating equilibrium and derive the conditions for its existence. 5. [ 2 0 marks] Explain how the male marriage premium can be an equilibrium outcome in a simple model of inter - linked frictional labour and marriage markets. Consider the following scenar io: Wages are uniformly distributed between 2 and 12 . The male reservation wage function is piecewise linear, with the pure labour market r eservation wage being equal to 3 . Calculate the male marriage premium if sing le men need at least a wage of 8 to be a ccepted by women, but they are only prepared to match this up to a wage of 5 . Assume that in equilibrium there are 14 bachelors and 24 eligible single men. END OF PAPER