11. What are the effects in the level of real debt vtht and capital stock k, and ination of an increase in R ? [19 points] 12. In this case, m > 0k. What are the effects of an open market Operation represented by a change in 6? [19 points] is given by G, 7 T, The scal authority chooses the path of the primary decit and rate at which to issue new debt, #4 = Ht/HH. The consolidated nominal government budget constraint is given by: G, + $1le + R:_13._1 = T: + M. + Br As usual, 1), := 1/11: denote the price levels Denote by lower case variables the wriable per old population: 9, = G./N,,1, r, = T,/N,,1 The Market clearing conditions imply that total demand for bonds and reserves (and total debt) equal supply: h, : Ht/N, Also denote Rt-l 1= 9:433] +(1 9eilRir 1. Show that the government BC is equivalent to: [3 points] 9: 7 n = (1 7 Roz/mule (1) 2. What does the RHS of the equation above represents? How does the ratio Eta/at impact the hability of the government to run primary decits? [3 points] . Write down the young and old budget constraint: [3 points] From now on, G. = 0 so 1} represents primary surplus per old person. For simplicity, we assume that T, represents a lump-sum tax (transfer, if negative) that is applied to the old onlyr Moreover, we assume that the initial stock of debt MD+BD : H0 is in the hands of the initial old. A stationary government poliw is UK", R", 9, a, 1-}, invariant through time. Assume that R\" S R" Note that, because reserves and bonds here are distinguished only by their rates of return, investors are naturally drawn to hold the security that offers the highest rate of returns Historically, the yield on bonds has been positive R" > 1 and the yield on reserves zero Rm 2 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 requirement", represented by the constraint: \"eke S van. (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] [HINT: solve the young BC for 17] . Using the constraint above and the reserve requirement constraint (2D, write down the Lagrangian and then take rst-order conditions With respect to It; and 7m [3 points] 6 When is the Reserve requirements Iconstraint binding? [3 points] Combining both FOCs, we may nd the Fisher equation belong which equates the real interest rate (marginal product of capital) to the ination-adjusted rate of return on government debt. Ema): ((1+r7)Rbz71if"')\"iv1'l (3) We may assume a stationary equilibrium from now on: . Find the ination rate pt-H/Pt (or vt/vm). [3 points] In equilibrium, the old must pay taxes 7'; consistent with satisfying the government budget constraint , Using the government BC and the unique agent budget constraint you previously found, show that the equilibrium level of consumption must be: [3 points] c=zf(k)+nh (4) Since it must be that k = y h we'll focus our attention to the Fisher equation that characterize a stationary equilibrium GED and the ination rate 12\".] /p, rewritten here as: My M.) = ((1 + one" 0R\") A Pm This equation is enough to answer the falling 2 questions. Assuming R" > R\" (reserves are scarce),suppose that the central bank surprises indi viduals by suddenly raising its policy rate Rb, while keeping all (R'", a) constant. . What happens to the level of real debt um, and capital stock k, and ination pt-H/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 salm/purchases of government debt. That is to say, imagine that the central bank chooses 0 instead of R" . Using in = 47k, h = y k, m = (M show that h = \"1\". What are the effects of a permanent reduction in 07 [19 points] Assume now that R" : R'" : R (excess reserves) The Fisher equation boils down to: If'(y h) = R1 PH) This equation is enough to answer the following questions: \"You may use this to check your answer! Question . Time is discrete and denoted t = 1, 2, 3...oo. The economy is populated by a sequence of twoperiodlived overlapping generations. As usual, N, denotes the number of people born in time t and population grows at rate n, N, = nN,_1. The population of initial old is given by: NO. Individuals only care about their consumption when old: u(c1,,, 02,,+1) = c2,,+1. 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 3; units of output. The young also possess an investment technology, where k, units of invested at date t yields mf(k,) units at date t + 1. f' > 0, f\"