Macroeconomics 137
Problem set 4
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Question 1 Download the data for Consumption, Investment, GDP, from following sources for the period 1951-01-01 to 2020-01-01.
Monthly consumption expenditure: https://fred.stlouisfed.org/series/PCEPI
Quarterly investment expenditure: https://fred.stlouisfed.org/series/GPDIC1
Quarterly GDP: https://fred.stlouisfed.org/series/GDPC1
Quarterly Govt Expenditure: https://fred.stlouisfed.org/series/GCEC1
Using the EDIT GRAPH option in the FRED database, extract the quarterly series for Consumption.
Store all the series in one excel spreadsheet. Make sure that you are copying them for same time period.
a. Compute 8 quarter moving average for each of the data series. Plot the actual series and the con- structed moving average series.
b. Subtract the actual series from the moving average trend-line constructed in the previous part. Lets call this cycle for each of the series. Plot the cycle series for each data component listed above.
c. Calculate standard deviation of the cycle series constructed in part b. How volatile are investment, consumption, and govt spending relative to GDP.
d. Re-calculate the standard deviation by splitting the sample into pre 1984, and 1984—2008 sample. What do you observe?
Question 2
Solve Problem 13.4 in Kurlat textbook.
Question 3: from end of Chapter 8, Eggertsson
Consider this model
ˆ(ı)t =ˆ(r)t + Et πt+1
ˆ(m)t = ηy Y(ˆ)t - ηi ˆ(ı)t
ˆ(m)t = ˆ(m)t- 1 + ˆ(µ)t - πt
ˆ(µ)t = γˆ(µ)t- 1 + ϵt
ˆ(r)t = ρˆ(r)t- 1 + υt
where ϵt and υt are iid with zero mean and 0 < ρ,γ < 1. Here the notation is the same as in class (i is interest rate, m is real money balances, π is inflation, Y is output, r is and exogenous process for the real interest rate, and µ is the nominal growth rate of money). The variable Y(ˆ)t is an exogenous endowment, which is iid with zero mean.
i. Solve for inflation as a function of current and expected future values of ˆ(µ)t ˆ(r)t , and Y(ˆ)t. Show that there is a unique bounded solution. Hint: it is useful here to write the system in terms of a single forward looking equation ofˆ(m)t first and then use the third quation to solve for inflation, given the evolution already defined for ˆ(m)t .
ii. Solve for inflation as a function of only current and/or past values of ˆ(µ)t , ˆ(r)t , and Y(ˆ)t. Hint: use the last two equations and the fact that Et Y(ˆ)t+j = 0 for j > 0 to solve for the expectation of ˆ(µ)t and ˆ(r)t from your last answer.
iii. Show and impulse response function for πt and it to ϵt , i.e. to an increase in money growth.
iv. How well do you think your answer to iii fits with empirical evidence. Does this model imply any effect on output?
Question 4. Taylor Rule
Consider the following approximate economy. There is a Fisher equation
ˆ(r)t = ˆ(ı)t - (Et P(ˆ)t+1 - P(ˆ)t )
that says that real interest rate, the exogenous sequence {ˆ(r)t }, is equal to the nominal interest rate, ˆ(ı)t , minus expected inflation (all expressed as percentage deviation from steady state). The policy rule is either a price level rule ˆ(ı)t = ϕp P(ˆ)t or a Taylor rule ˆ(ı)t = ϕπ (P(ˆ)t -P(ˆ)t- 1 ). What is the ”Taylor principle” (i.e. the value of ϕπ that ensures a unique bounded solution for the price level), and what is the corresponding principle for the price level rule?
Question 5 Let’s imagine that the representative household lives for two periods only. They have to decide how much they are going to consume in period 1 and in period 2. Let’s assume that their preferences are described by the utility function:
log c1 + β log c2
The household takes as given its current and future income y1 and y2 , lump-sum taxes t1 and t2 and the interest rate and simply solves a consumer optimization problem:
max log c1 + ξ2 β log c2
{c1 ,c2 ,b2 }
s.t. (1)
c1 + b2 = y1 - t1 (2)
c2 = y2 - t2 + (1 + r)b2 (3)
b2 is a one-period real bond that gives real return r. There is no cash in this economy.
1. Set up Lagrangian problem and derive the consumption Euler equation.
2. Rearrange your Euler equation to write c2 as a function c1 and interest rate. Substitute this resulting equation into the household budget constraint to solve for consumption in period 1.
3. What is the marginal propensity to consume out of period one income. i.e. ∂y1/∂c1 . How does it depend on ξ2 ?
Firms hire workers to produce y1 in first period. Firm labor demand condition gives W1 /P1 = 1, i.e.
real wage in period 1 is one. We will assume that W1 = 1, nominal wages are fixed at one. We also assume that output in period 2 is fixed at y2 = 1. Furthermore, assume that all output must be consumed either by the government or by agents. In particular, we will assume that government spending is only done in the first period. y1 = c1 + g1 . As a result in the second period y2 = c2 .
4. What is the price level in period 1, i.e. P1 ?
5. Let’s define the natural interest rate to be real interest rate such that output in period one is equal to one. Write down an expression for the natural interest rate.
6. Write down a condition for ξ2 that has to be satisfied for the natural interest rate to be negative?