Question . Time is discrete and denoted t = 1, 2, 3...oo. The economy is populated by a sequence of two-period-lived overlapping generations. As usual, Na denotes the number of people born in time t and population grows at rate 72., N, = nNt_1. The population of initial old is given by: No. Individuals only care about their consumption when old: u(c1,g,02,+1) 2 cu\". Given this utility function, we know that agents will save all of their income. By assuming this type of utility, we are abstracting from the consumption-savings decision problem, however, we will focus on the portfolio-choice problem. The young are endowed with :9 units of output. The young also possess an investment technology, where kg units of invested at date if yields asf(kt) units at date t+ 1. f' > 0, f\" 1 and the yield on reserves zero Rm = 1. For this return structure, the demand for reserves would fall to zero in our model. To generate a demand for reserves when they are dominated in rate of return, we assume that investors structure their wealth portfolios in a manner that respects a \"reserve requiremen \Utkt S Utmt (2) a E (0, 1) is a parameter that may be interpreted as either a legislated minimum reserve requirement (investors are required to hold a minimum amount of cash against their private sector investments). . Combine the young and old constraints into a single constraint [3 points] [HINTz solve the young BC for 1):] . Using the constraint above and the reserve requirement constraint Gib, write down the Lagrangian and then take rstorder conditions with respect to Is: and mi [3 points] Page 2 . When is the Reserve requirements @ constraint binding? [3 points] Combining both FOCs, we may nd the Fisher equation belowE which equates the real interest rate (marginal product of capital) to the inationadjusted rate of return on government debt. :1: mat) = ((1 + 0)Rb aRm) \"\"1 (3) \"t We may assume a stationary equilibrium from now on: . Find the ination rate pt+1/pt (or v: /vt+1). [3 points] In equilibrium, the old must pay taxes 1",: consistent with satisfying the government budget constraint Gib. 8. 10. - '_r Using the government BC @D and the unique agent budget constraint you previously found, show that the equilibrium level of consumption must be: [3 points] 6 = rk) + nh (4) Since it must be that k = y it we'll focus our attention to the Fisher equation that characterize a stationary equilibrium @D and the ination rate pt+1/pt, rewritten here as: $f'(y 11th:): ((1+ 0)Rb aRm) pt Pt+1 This equation is enough to answer the folling 2 questions. Assuming Rb > Rm (reserves are scarce),suppose that the central bank surprises indi viduals by suddenly raising its policy rate Rb, while keeping all (Rm, ,u) constant. What happens to the level of real debt 11th,; and capital stock 16, and ination pt+1/pt? How would your answer change with an increase of R\"? [19 points] Imagine, instead, that Rb is market determined and that the monetary authority inu ences the interest rate through open market sales/ purchases of government debt. That is to say, imagine that the central bank chooses 9 instead of Rh Using m = 019, h = y k, m = 9h show that h = fey. What are the effects of a permanent reduction in 0? [19 points] Assume now that Rb = Rm = R (excess reserves). The Fisher equation boils down to: My h) = Bi Pt+1 This equation is enough to answer the following questions: 2You may use this to check your answer! 11. 12. What are the effects in the level of real debt vtht and capital stock is, and ination of an increase in R ? [19 points] In this case, m > 014:. What are the effects of an open market operation represented by a change in 6? [19 points]