Compute the cyclical components 1600 for the log of each

Question

Assignment 1 - Exercise

Exercises for Hands on HP filtering

Preparation before working with this exercise: Follow the instructions in the pdf file "MAKRO_II_Introduction to HP Filtering" that you will find in the folder OPGAVER on the Macro II Absalon page. It tells you how to download and save to a folder of your choice (called Makro II, say) an add-in for Excel that allows you to use the Hodrick-Prescott (HP) filter and how to connect it to Excel on your computer. It also tells you how to work with this tool in practice.

In OPGAVER you will also find two Excel files named "Exc for Chap 13_1_GDP data_US_UK_SV_DK" and "Exc for Chap 13_2 data_Denmark" which contain the relevant data for Exercise 1 and Excercise 2 below, respectively. Save these files to the relevant folder on your computer (Makro II, say), copy the files and do your operations in the copied files.

Exercise 1 - Output gaps on annual basis in different countries

The Excel sheet "Exc for Chap 13_1_GDP data_US_UK_SV_DK" contains annual GDP volumes for USA, UK, Sweden and Denmark in 2011 US dollars (chained PPP's) for 1950-2014. The source is Penn World Table.

1. For each country: Create the transformed series, the natural log of GDP, and find the trend of this series by HP-filtering with λ = 100. In a diagram plot the actual series for the log of GDP as well as its trend value against years for the entire period 1950-2014. Comment on an intuitive basis on how well you think that the HP filter finds the trend of the series. Comment particularly on this aspect for the end points, that is, the first and last 4-5 years.

2. For each country: Compute for each year the 'output gap' in percent as ln(GDP) minus the trend value for ln(GDP) times 100 and plot output gap against years. You may make four separate plots or one with all four output gap series. Comment on these aspects: Do the output fluctuations overall (qualitatively) look similar across countries? Do business cycles seem to become smaller or larger over time? And how strongly do the business cycles seem to be synchronized between the countries? For these questions you may want to disregard the first and last 4-5 years.

Exercise 2 - Volatility of GDP, private consumption and private investment over the business cycle in Denmark

The Excel sheet "Exc for Chap 13_2 data Denmark" contains quarterly data for the volumes of GDP, private consumption and private investment for Denmark in 2010 prices for the period 1966,1 (first quarter of 1966) to 2015,4. The source is OECD Economic Outlook, 98 database.

1. Apply the HP filter with λ = 1600 to the series for the (natural) log of GDP, consumption, and investment to obtain the trends for each of these three series. Plot the actual data as well as the trends against time (quarters), inspect the graphs and comment on an how well the HP filter seems to find the trend of the series. Comment on end point problems. Try changing λ to 100 and do the same. Intuitively, from inspection of the plots, does this substantially affect how well the HP filter captures the trend of quarterly data?

2. Compute the cyclical components (λ = 1600) for the log of each of the series, that is, the log of the actual value minus the trend of this. Also compute the cyclical components in percent (actual ln(variable) minus trend ln(variable) times 100); the latter are the "output gap" and the "consumption gap" and "investment gap" in percent, respectively.

3. Plot these gaps against time in the same diagram disregarding the first and last 16 quarters. Which of the variables can be seen to be more volatile, as measured by the size of the fluctuations around trend? Is this in line with the evidence presented in Chapter 13?

Let c_t be the cyclical component of ln(GDP) in period t, and x_t the cyclical component of the log of different series (GDP, consumption, investment).

4 Disregarding again the first and last 16 quarters, compute the contemporaneous coefficients of correlation, ρ(x_t, c_t),for x_t being the cyclical component of the log of consumption and investment, respectively. Then compute the lagged and leaded coefficients of correlation, ρ(x_t-n, c_t) for n =2, 1, -1, -2 and for x_t being the cyclical component of the log of GDP, consumption and investment, respectively (you may have to disregard a few more quarters at the ends). Use this to set up a table like Table 13.3 for Denmark for the involved variables. Compare to the findings for Denmark in Table 13.3b. Is the overall picture confirmed? Is any of consumption or investment a leading or lagging variable to GDP?

Assignment 2 - Exercise

In the main text of chapter 16 we considered the following utility maximization problem

max_{C_1,C_2} U = u (C₁) + 1/(1 + φ) u (C₂) (1)

s.t. C₁ + 1/(1 + r) C₂ = (Y₁^L - T₁ + 1(1 + r) (Y₂^L - T₂) + V₁) (2)

and showed that when u (C) = σ/(σ-1)C^(σ-1)/σ the solution for consumption in period 1 is

C₁ = θ(Y₁^L - T₁ + 1(1 + r)(Y₂^L - T₂) + V₁), θ ≡ 1/(1 + (1 + r)^σ-1(1 + φ)^-σ) (3)

1. Show that in case of a temporary change in taxes, specifically a change in taxes in period 1, the effect on consumption in period 1 is

∂C₁/∂T₁ = -θ (4)

Comment on the size of this derivative.

2. Suppose instead that the change in taxes is permanent, i.e. that both T₁ and T₂ are changed by the same amount, dT₁ = dT₂. Show that in this case

∂C₁/∂T₁ = -θ(1 + 1/(1 + r)), when dT₁ = dT₂ (5)

Compare the size of this effect with the one found in question 1 and explain the difference. Show that when r = φ we get ∂C₁/∂T₁ = -1 and explain this intuitively.

Until now we have looked at changes in taxes without taking into account that government has a budget constraint which should hold.

3. Argue that when government has a two-period time horizon beyond which it cannot have any debt (or assets), government must obey the following intertemporal budget constraint

D₁ + G₁ + 1/(1 + r)G₂ = T₁ + 1/(1 + r)T₂ (6)

where D₁ is government debt at the beginning of period 1 and G₁ and G₂ are government purchases in period 1 and 2 respectively.

4. Suppose government announces that it will change taxes in period 1 and that consumers realize that if the government IBC is to hold, this will require a change in period 2 taxes also (we assume that G₁ and G₂ are unchanged). Show that in this case

∂C₁/∂T₁ = 0 (7)

and explain this result intuitively.

The implication of the result in question 4 is that in any given period it will have no effect on private consumption whether government chooses to finance its expenditure by taxes or by borrowing as long as the intertemporal budget constraint holds. This is known as Ricardian Equivalence between taxes and debt.

5. Give reasons why Ricardian Equivalence is likely not to hold in practice.

Assignment 3 - Exercise

Consider the following model of a closed economy

y_t = y¯ + α₁ (g_t - g¯) - α₂ (r_t - r¯) + v_t, α₁ > 0, α₂ > 0 (A.1)

r_t ≡ i_t - π_t+1^e (A.2)

i_t = r¯ + π_t+1^e + h (π_t - π^∗) , h > 0 (A.3)

g_t = g¯ - k (y_t - y¯) , k > 0 (A.4)

π_t = π_t+1^e + γ (y_t - y¯) + s_t, γ > 0 (A.5)

π_t^e = π_t-1 (A.6)

With variables being as normal and unless otherwise stated, both s_t and v_t are zero.

1. Explain equations (A.1) - (A.6).

Equations (A.1) - (A.6) may be combined to yield the following AD and SRAS curves:

AD : y_t = y¯ - α (π_t - π^∗) + z_t (A.7)

SRAS : π_t = π_t-1 + γ (y_t - y¯) + s_t (A.8)

where

α ≡ α₂h/(1 + α₁k), zt ≡ 1/(1 + α₁k)v_t (A.9)

2. Show graphically and formally that the long run equilibrium is stable, i.e. that over time y_t and π_t will converge towards their long run values, but that deviations from the long run equilibrium will show persistence. Explain the economic mechanisms behind this.

Hint: For the graphical part of this and the next question, assume that the economy starts outside its long-run equilibrium, more specifically, in a situation such as the one in Figure 18.3 in the textbook, where output is low, and inflation is high.

Hint: With respect to the formal part: de?ne yˆ_t ≡ y_t - y¯ and πˆ_t ≡ π_t - π^∗ and show, using equations (A.7) and (A.8), that (when z_t = s_t = 0)

πˆ_t = βπˆ_t-1 and yˆ_t = βyˆ_t-1 where β ≡ 1/(1 + αγ) (A.10)

Suppose that the formation of expectations in equation (A.6) is replaced by

π_t^e = φπ_t-1^e + (1 - φ)π_t-1, 0 < φ < 1 (A.11)

3. Explain and show graphically that this will not change the long run equilibrium compared with question 2 and that the long run equilibrium is still stable, but that convergence to long run equilibrium will be slower than in the model with static expectations.

For the rest of this problem we now revert to the case of static expectations given in equation (A.6).

4. Assume that for some time the economy has been in a long run equilibrium when, in a single period, there is a negative supply shock, s_t > 0. Analyse the effects of this until the economy is back in long run equilibrium. Explain how the values of the parameters h and k affect y_t and π_t in the short run, i.e. the period of the shock. Comment and briefly compare with the case of a negative demand shock, v_t < 0.

Suppose that due to imperfections in the goods- and/or labour market the natural level of output, y¯, is less than the efficient level of output, y^∗, specifically

y^∗ = y¯ + ω, ω > 0 (A.12)

Due to this, fiscal policy is now changed so that it aims at stabilizing output around its efficient level. Specifically equation (A.4) is replaced by

g_t = g¯ - k (y_t - y^∗), k > 0 (A.13)

It is assumed that before the change of policy, the economy is in a long run equilibrium.

5. Analyse graphically and in economic terms the effects following this change of policy. Show that in the new long equilibrium run there will be no effect on real output but that inflation will increase to

π_LR = π^∗ + (α₁k/α₂h)ω (A.14)

Explain why the parameters, α₁, α₂, k and h affect π_LR the way they do.

Note - In the Assignment 1, it's only exercise 2.

Attachment:- Assignment Files.rar