Provide detailed explanations to. the attached questions.

Problem 4 The current yield curve for default-free zero-coupon bonds is as follows: Maturity (years) YTM 10% WN N 11% 12% a. What are the implied one-year forward rates for years 2 and 3? b. What would be the shape of the yield curve according to the Expectations hypothesis? c. If you believe in the liquidity preference theory, what would be your guess as to the upper limit on the short-term interest rate expected for the third year?Question 2 It is intuitive to think that the presence of more agents in the economy "shrinks" its core, since there are more coalitions that can object a given allocation. You will understand in this question why this is indeed the case.' Fix a standard, two-person exchange economy & = ((u', w' ), (u', w?)). Define its replica as the four-person exchange economy 82 = ((u', w'), (ul, w ), (u', w'), (u', wa)). where (u', w3) = (u', wi ) and (u', wi ) = (uz, w? ). 1. Argue that if (p, x', x?) is a competitive equilibrium for &, then (p, x', x3, x3, x* ) with x' = x' and x* = x', is an equilibrium for &?. 2. Argue that if both utility functions are strictly quasi-concave, and (p, x', x', x3, x* ) is a competitive equilibrium for &', then, x' = x3 and x? = x*. 3. Argue that if both utility functions are strictly quasi-concave, and (x], x3, x3, x* ) is in the core of &', then, x' = x3 and x? = x*. 4. Argue that if both utility functions are monotone and strictly quasi-concave, and (p, x', x?) is a competitive equilibrium for &, then (x], x], x', x?) is in the core of &?. 5. Suppose that u'(x) = u'(x) =x'x. wi = (1,0) and w2 = (0, 1). Argue that allocation ((0, 0), (1, 1) ) is in the core of &, yet allocation ((0, 0), (1, 1), (0, 0), (1, 1)) is not in the core of &?. 6. Use these results to argue, informally, that the replication of agents does not affect the set of equilibrium allocations of the economy but shrinks its core.' We are using the term "shrink" loosely, since the presence of more agents changes the dimension of the allocation space, so comparing the sizes of the cores will require some refinement of the argument. In the limit, one can show that replication ad infinitum reduces the core to just the set of equilibrium allocations.4. Consider the following moral hazard problem with limited liability. Suppose a risk neatml restaurant owner wants to hire a riskneutral waitress to serve the customers in his restaurant. The number of customers served is stochastic; it can be either (j = 100 people or q = 50 people. When the waitress exerts effort, then in = 3/4 is the probability that the dumber will be q and 1/4 is the probability that the number will be q. When the waitress does not exert effort, then 7T0 = 1/4 is the probability that thenumber will be a and 3/4 is the probability that the number will be q. Exerting effort is costly to the individual and suppose this cost is equal to if) = 2. Silppose the owner receives a payoff from serving of the customers by the payoff function S(q) = :32. Suppose the contract the owner proposes consists of payments {(13.9} to the individual, depending on whether the serving number is high or not. (a) Suppose that the owner can observe if the waitress exerts effort or not. Set the optimization problem and nd the optimal transfers when the owner induces effort. Show that the owner prefers inducing effort over not inducing effort. (b) Now suppose that the owner cannot observe if the waitress exerts effort or not. Set the optimization problem and nd the optimal transfers when the owner induces effort. What is the cost of inducing effort? (c) Suppose again the owner cannot observe if the waitress exerts effort. Now suppose also that the landlord cannot impose a transfer less than I, where I = 2. Set the optimization problem and nd the optimal transfers when the owner induces effort. What is the expected limited liability rent of the waitress? (d) Suppose once again the owner cannot observe if the waitress exerts effort. Now suppose also that the owner cannot impose a transfer less than l, where l = 0 this time. Set the optimization problem and nd the optimal transfers when the owner induces effort. What is the expected limited liability rent of the waitress? (e) Now suppose that the owner makes the waitress a shareholder of the prots by using a simple linear sharing rule. Set the optimization problem and nd the optimal linear sharing rule. 3. To slow down transmissions of the viral disea 'lockdown" hence ordering business to shut and the citizens to adhere to limiting physical distancing, but leaving the timing of the lockdown to be decided by the two regional governments of Krona. Assume that both regions have roughly the same population size of 10 million citizens. Region 1 entered lockdown at the beginning of month 3, region 2 entered lockdown at the beginning of month 2. See Next Page The following table shows the cumulative infections and fatalities due to the viral disease over time and the time of the lockdown in each region at the end of each month. Month 1 Month 2 Month 3 Month 4 cases fatalities cases fatalities cases fatalities cases fatalities Region 1 100 3 5000 300 9000 540 15000 900 Region 2 100 5 2200 130 4500 270 5500 330 a) The minister for the economy argues that after both regions entered lockdown, in month 3 the number of cases in region 2 more than doubled, whereas the growth rate is lower in region 1 arguing that the lockdown was not effective in preventing the spread of the disease. Do you agree? Can you provide a better estimate using your knowledge of empirical methods introduced in the module? [word limit 250 words; 8 marks]. b) What kind of data would you prefer to have to estimate the full impact of the lockdowns on infections and fatalities? Explain how this would help to improve your estimate from the previous question. [word limit 200 words; 4 marks]. c) In response to the lockdown the economy in region 1 contracted by $1.1bn, and $1.114bn in region 2. What can we learn from these figures about the statistical value of life? [word limit 100 words; 3 marks]. d) Suppose a vaccine against the disease was available at the beginning of month 1 (and assume it works immediately by protecting the citizens from the disease). 60% of population require the vaccine to stop the disease from spreading. The vaccine is available for $20. Should the government introduce the vaccine? State any assumptions you implicitly make underlying your advice and briefly discuss the limitation of your advice. [word limit 250 words; 5 marks]