Time is discrete and denoted = 1, 2, 3.... The economy is populated by a sequence of two-period-lived overlapping generations. As usual, denotes the number of people born in time and population grows at rate , = 1. The population of initial old is given by: 0. Individuals only care about their consumption when old: (1,, 2,+1) = 2,+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 units of output. The young also possess an investment technology, where units of invested at date yields () units at date + 1. > 0, < 0. There are only two types of government securities. One is issued by the central bank in the form of interest-bearing reserves1, and the other is issued by the Treasury in the form of interest bearing debt. Both securities are nominal denominated in dollars. Let denote the supply of reserves at date , and let denote the supply of bonds at date . The total public debt at date is denoted = + . Reserves and bonds yield gross nominal rates of return equal to and , respectively. The central bank is delegated control over {, , } where the variable := / represents the fraction of public debt monetized by the central bank. The fiscal authority is responsible for tax and spend decisions, which we denote and , respectively (also denominated in nominal terms). The primary deficit (surplus, if negative) is given by . The fiscal authority chooses the path of the primary deficit and rate at which to issue new debt, = /1. The consolidated nominal government budget constraint is given by: + 11 + 11 = + + As usual, := 1/ denote the price level. Denote by lower case variables the variable per old population: = /1, = /1 The Market clearing conditions imply that total demand for bonds and reserves (and total debt) equal supply: h = /. Also denote 1 := 11 + (1 1)1. 1. Show that the government BC is equivalent to: [3 points] = (1 1/) h (1) 2. What does the RHS of the equation above represents? How does the ratio 1/ impact the hability of the government to run primary deficits? [3 points] 3. Write down the young and old budget constraint: [3 points] From now on, = 0 so represents primary surplus per old person. For simplicity, we assume that represents a lump-sum tax (transfer, if negative) that is applied to the old only. Moreover, we assume that the initial stock of debt 0 +0 = 0 is in the hands of the initial old. A stationary government policy is {, , , , }, invariant through time. Assume that 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 return. Historically, the yield on bonds has been positive > 1 and the yield on reserves zero = 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: (2) (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). 4. Combine the young and old constraints into a single constraint [3 points] [HINT: solve the young BC for ] 5. Using the constraint above and the reserve requirement constraint (2), write down the Lagrangian and then take first-order conditions with respect to and [3 points] 6. When is the Reserve requirements 2 constraint binding? [3 points] Combining both FOCs, we may find the Fisher equation below2, which equates the real interest rate (marginal product of capital) to the inflation-adjusted rate of return on government debt. () = ((1 + ) )[(+1)/] (3) We may assume a stationary equilibrium from now on: 7. Find the inflation rate +1/ (or /+1). [3 points] In equilibrium, the old must pay taxes consistent with satisfying the government budget constraint (1). 8. Using the government BC (1) and the unique agent budget constraint you previously found, show that the equilibrium level of consumption must be: [3 points] = () + h. (4) Since it must be that = h we'll focus our attention to the Fisher equation that characterize a stationary equilibrium (3) and the inflation rate +1/, rewritten here as: ( h) = ((1 + ) ) (/+1) This equation is enough to answer the folling 2 questions. Assuming > (reserves are scarce),suppose that the central bank surprises individuals by suddenly raising its policy rate , while keeping all (, ) constant. 9. What happens to the level of real debt h and capital stock , and inflation +1/? How would your answer change with an increase of ? [19 points] Imagine, instead, that is market determined and that the monetary authority influences the interest rate through open market sales/purchases of government debt. That is to say, imagine that the central bank chooses instead of 10. Using = , h = , = h show that h = [/(+)]y. What are the effects of a permanent reduction in ? [19 points] Assume now that = = (excess reserves). The Fisher equation boils down to: ( h) = (/+1) This equation is enough to answer the following questions: 11. What are the effects in the level of real debt h and capital stock , and inflation of an increase in ? [19 points] 12. In this case, > . What are the effects of an open market operation represented by a change in ? [19 points] Can someone try to solve these problems not necessarily for solving all the parts Can someone just try to solve last four parts?Thank u