Kindly help me to solve the questions as attached below.
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].A. factory produces racing lakes. The rst stage is to assemble the bile: on the factoryr oor, which tak an exponentially distributed length of time with mean 1|} hours. Once amenibled, the bike is then immediately hrspected. o If the bike passes inspection {which happens at a rate of 1 per hour}. then the bike is shipped to its new owner and does not return to the factory oor or for inspection. a If it Idem not pass inspection [which happens at a rate of Ill per hour] then it is start baclt to the factory oor to be reassembled. The distribution of time it takes to reassernbie is the same as for the original assembly. This transfer between assembly and inspection wili continue indenitely until the bike passes inspection. Let I; be the status of a bike at time t1 and assume that {list :3 } is a continuous time Markov chain with state space, S, given by: S = {'being assembled or reassembled', 'being inspected', Ishipped"'}. [a] Draw a statesoace diagram for the process {Int 2 ill}. [b] Reiahel the states for notational convenience: 1 is \"being assembled or reassembled', 2 is 'being inspectod', and 3 is 'shipped'. Using ofh] notation where necessary, compute the following for VERY small It: ii] Pixa - 2Win: - 1}; [ii] P{X;. = SIXD = 1} [the answer to this is not zero]; [iii] P'IIXHH = III: = 2}: (iv) PUD. = 3|.Xn = 3). {c} 1Write down the transition matrix of the jump chain of the process {Int 3 D}. {d} Does the jump chain of the process {Xi 3 D} have an equilibrium distribution? Justify your answer. If it does have an equilibrium distribution, nd it. If it doesn't, find all invariant distributions instead. {e} What is the probability that a bilte currently being inspected will pass [and so be shipped to its new miter)? {f} How many times, on average, will the biloe be inspected before eventually being shipped? Ex 17.3: Fish and Coconuts, Part II Let's decentralize the model from exercise 5.2. Now, let's think of "Chuck" as two firms and one consumer. Each firm (i.e., the "fish firm" that produces good 1 and the "coconut firm" that produces good 2) has the production function f(L) = VI Both firms take the price of their good as given. The labor supply is L = 100, supplied inelastically. For simplicity, let's fix p2 = 1, and let p1 = p; so throughout this problem, we'll be interested in the equilibrium price ratio p = p1/P2. (a) Assuming the two firms take the price of their good as given, find their equilibrium profit-maximizing combinations of outputs, Y, and Y2, as a function of p. You may do this in one of two ways: . Easy way: Use the fact that the firms will produce at the point along the PPF that sets MRT = p. Remember that you found the equation of the PPF and an expression for the MRT in exercise 5.2! . Hard/thorough way: Find the firms' individual supply functions, S1(p, w) and S2(w) (since p2 = 1). Find their labor demands, and set labor demand equal to the total labor supply. Solve for the equilibrium wage rate as a function of p - that is, w*(p). Finally, plug w* (p) back into the supply functions to get Yr (p) = Si(p, w*(p)) and Y,(p) = S2(w* (p)). (b) Find the total monetary value of the fish and coconuts produced given p1 = p and py = 1: that is, evaluate M(p) = pYi(p) + Y2(p) given the functions you derived in part (a). (c) Now suppose the consumer has preferences u($1, 22) = 45 Inc1 + 1012. Find her optimal quantity of fish, X], as a function of p. (Note: because this is a quasilinear utility function, it won't be dependent on her income!) Set this equal to the equilibrium quantity of fish you found in part (a), Y; and solve for the equilibrium price ratio, p*. Note: this is a nasty quadratic equation, use a calculator or Wolfram alpha to solve! Confirm that at that price ratio, the quantity of fish and coconuts supplied (and therefore demanded) is the same as you found in question 5.2(b).Problem 4 [32 points]: The table below shows the response variable y to be explained by explanatory variables x and u: Observation i 1 2 3 4 5 6 7 8 9 10 Response Variable y 22 44 18 22 3 35 34 20 8 5 Explanatory Variable x 3 9 5 3 7 8 4 Explanatory Variable u 2 5 3 8 4 -2 5 5 (a) [6 points] Suppose a simple linear regression is performed based on EV x only. That is, Vi = a, tax, te,, i=1,2,..., 10 Using the "Im" function in R, obtain the OLS estimates a,, a, and R2. (b) [6 points] Suppose a simple linear regression is performed based on EV u only. That is, Vi = Yo + yu; +e,, i=1,2,...,10 Using the "Im" function in R, obtain the OLS estimates yo, 7, and R2. (c) [13 points] Suppose a multiple linear regression is used to explain y by both x and u. That is, Vi = Bot Bix; + Bu, te,, i=1,2, ...,10 Based on matrix operations in R (i.e. A%*%B, t(A), solve(A) on Ch3 page 32), (i) show that OLS estimates B =(Bo, B,, B2)' = (4.225, 3.96875, 0.33333)', (ii) compute the values of SYY, RSS, o' and Var(B). (iii) compute the value of R', and show that it's equal to the sum of R2 from part (a) and (b). [Note: The relationship is true only if vectors x and u are orthogonal. That is, I'm = Corr(x,u) = _ _(x, -x)(u, -1) =0] (d) [7 points] Consider a new data point with (x, u) = (1, 1). What is the best point estimator for the response, and a 95% prediction interval for the response
