This is a continuation of Question 5 in Problem Set 3. We ended with specifying a condition for ξ2 that has to be satisfied for natural interest rate to be negative there. Assume that agents suddenly become patient
Question 1 This is a continuation of Question 5 in Problem Set 3. We ended with specifying a condition for ξ2 that has to be satisfied for natural interest rate to be negative there. Assume that agents suddenly become patient i.e. ξ2 increases. In particular, we will assume that 1 ξ2 β = 0.96.
1. What is natural rate of interest when ξ21β = 0.96?
2. Assume g1 = 0. If there exists a zero lower bound constraint on nominal interest rate, what is the output in period 1?
3. Assume g1 = 0. Derive the relationship between output in period 1,y1 , and price level in second period, P2 , for a general price level (not necessarily equal to one)? Is it increasing or decreasing? Also provide a brief intuitive explanation for your answer. Let’s consider a government that spends g1 , and g2 in periods 1 and 2. It raises lump-sum taxed t1 and t2 in respective periods. And it has access to one period borrowing through issuance of government bond b2 , that it issues in period 1.
4. Derive the government’s lifetime budget constraint.
5. Now assume that g1 = 0.01, and g2 = 0. If the government can only use taxes to finance its expenditure, i..e no borrowing is allowed, what sequence of taxes satisfies the governments budget constraint.
6. In the same problem as above, if the government is not allowed to raise taxes in period 1, i.e. t1 = 0. What sequence of borrowing and period 2 taxes satisfies government’s budget constraint. Now we will calculate the effect of fiscal expenditure during a liquidity trap in our model. Assume that g1 = 0.01, g2 = 0 and ξ21β = 0.96.
7. What is the output during a liquidity trap in this scenario?
8. Does it matter if the government borrows in first period to finance g1 = 0.01 or if it imposes a tax of 0.01 in the first period.
9. What happens to consumption c1 when g1 = 0.01 relative to the case when g1 = 0. 10. Intuitively explain how the government is able to generate higher output by spending g1 .
Question 2 Let’s solve for the model we saw in Eggertsson-Krugman (2012) paper. There are two representative agents in this economy, each one receiving an endowment 12 Y per period. Each agent maximizes ∞ X t Et β (i) log Ct t=0 s.t. 1 Dt (i) = (1 + rt−1 )Dt−1 (i) − Y + Ct (i) 2 1 with i = s or b and β(s) > β(b). Each agent needs to satisfy the debt limit. (1 + rt )Dt (i) ≤ Dhigh > 0 Further we assume that borrowers are constrained in that the debt limit is binding for them. Thus, high Dt (b) = D1+rt . Finally, we assume that all production is consumed so that Yt = Ct (s) + Ct (b)
1. Provide an expression for consumption of borrowers and savers in the steady state. What is the real interest rate in the steady state? Suppose that you are in steady state. Now suppose there is an unexpected reduction in Dhigh from D to Dlow and that the indebted agents need to satisfy this new debt limit in one period. We divide the time-periods into “short-run” S and “long-run” L. high
2. Provide an expression for consumption of borrowers in the long-run and short-run. (note steady state consumption will change with the new debt limit).
3. Solve for consumption of savers in the short-run and long-run using the savers’ budget constraint and their Euler equation.
4. Provide an expression for what happens to the real interest rate in the short-run?
5. Write down the condition that guarantees that the real interest rate (natural rate) is negative in the short-run in this economy? t Assume that there exists a nominal bond in net zero supply, which pays a nominal return it . If Πt = PPt−1 is the inflation rate in period t, then a no-arbitrage condition between holding a nominal bond and a real bond implies the following relation (Fisher equation): 1 + rt = (1 + it ) Pt Pt+1
6. Assume that PL = P ∗ i..e price in the long-run is fixed at P ∗ . If there exists a zero lower bound constraint on the short-term nominal interest rate, what would have to happen to prices in the shortrun for existence of equlibrium. Next, we endogenize production. Instead of savers and borrowers getting endowment yt , they both works and recieve profits from owning firms. Eventually, they both get paid equal share of total output in the economy. So the budget constraint that we wrote earlier are unchanged. In the short-run, firms hire workers to produce YS . The production function of the firm is YS = Lα S. They pay nominal wages WS . As you saw in intermediate macro, firm’s labor demand is given by equality of dYS S = αLSα−1 = W marginal production of labor to real wage: dL PS . Use the production function to substitute S 1 out labor LS for output YS as LS = YSα . And substitute this back into firm’s demand for labor to get: α−1 αYS α = WS PS Set α = 0.5 and WS = 1.
7. Simplify the labor demand condition using values of α and WS , to write an expression for PS as function of YS and α. 2 Under rigid nominal wages, short-run prices cannot adjust to guarantee an equilibrium at full-employment as you solved for in part 6 above. Assume that the natural rate in the short-run is negative. This implies that short-term nominal interest rate is set at iS = 0. Furthermore, suppose that the central bank sets PL = PS . Assume YL = 1.
8. Use your calculations for saver’s consumption in the short-run and the long-run (from part -3) and saver’s consumption Euler equation to write an expression for YS as function β, Dlow and Dhigh Let government impose a one-time small tax TS > 0 on savers in the short-run. This tax income is transferred to the borrower in the same period. Thus budget constraint of the borrower becomes: CS (b) = Dlow 1 YS + − Dhigh + TS 2 1 + rS
9. How does your expression in part 8 change with this tax. Provide some intuition for your mathematical derivation.
Question 3
1. Go to https://www.treasury.gov/resource-center/data-chart-center/interest-rates/pages/ textview.aspx?data=yield. Plot the graph of the yield curve for the US Treasuries as of Friday May 21, 2021. (you can make an excel graph and attach that graph with your answers).
2. Does the observed curve suggest that interest rates will rise or decline?
3. Compare the yield curve from a similar date last year (Friday May 22, 2020). What were the markets predicting on May 22, 2020 as to what would happen to interest rates over time. (you can make an excel graph and attach that graph with your answers).
4. Go to https://fred.stlouisfed.org. Find the data series “T10Y2Y ”. it is the spread between the 10-Year Treasury Constant Maturity and the 2-Year Treasury Constant Maturity rates. Document the dates when this spread has turned negative. Do recessions typically follow these dates? In what direction does Fed typically change the monetary policy rate following these dates? (you can look at the monetary policy rate by finding the data series titled FEDFUNDS on the same website) 3
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