All the questions are in the file I uploaded
Economics 640: Macroeconomics Fall 2021 Professor Snudden Assignment 2 (100 points: Due Oct. 31st, 6 p.m.) All questions are good examples of exam questions. As with the first as- signment all graphs and tables should stand alone, be clear with correct titles, legends, axises, and notes. No one should be able to tell what software made the figures (No Stata blue!). Don’t lose marks due to carelessness in presen- tation. For the group component, students must work in a group of two to four people. You are not allowed to work on the group component of the assign- ment individually. Each student must hand in the individual component as a separate file. For the group component I don’t recommend dividing up the questions but instead recommend that everyone do all the questions and then come together to confirm answers. All solutions must be written in LaTeX. Assignments are to be submitted in one PDF file to MyLearnignSpace by the student with the last name that is alphabetically first before 6 p.m. on the Sunday that the assignment is due. Present the tables and graphs with concise write ups. Please only use Stata 16. Attach a copy of the Stata .do program file at the end. Good luck! 1 Group Questions: 1. (25 points) Estimate a measure of total factor productivity for the Cana- dian economy. (a) Visit Statistics Canada Table 36-10-0104-01 and download nominal (current price) quarterly data, 1961Q1 and 2021Q2, for four types of investment: Residential structures; Non-residential structures, ma- chinery and equipment; Intellectual property products; General gov- ernments gross fixed capital formation. i. Using these variables produce one graph that includes all these measures as a ratio of GDP from 1961Q1 and 2021Q2. ii. What is the ranking of each variable from the largest to the small- est share of GDP on average over the sample? iii. Comment on any major changes in the relative size of each type of investment. (b) Impute a quarterly measure of the private and public capital stock for Canada. Approximate the initial value of the capital stock by taking the quarterly value of investment in 1961Q1 and dividing through by a quarterly depreciation rate Ki1961Q1 = I i1961Q1 δi , where i ∈ {private, public}. Use general governments gross fixed cap- ital formation for the measure of Ipublic and total private investment from the national accounts for the measure of Iprivate. Then calculate the capital stock by using the perpetual inventory equation. Kit+1 = (1− δ)Kit + I it , t > 1961Q1 After doing so, to remove starting point bias, drop data prior to 1991Q1. i. Create two graphs from 1991Q1–2021Q2, one for public and the other for private investment. In each graph show three estimates of the capital to GDP ratio using δ ∈ [0.01, 0.0175, 0.025]. ii. What is an appropriate measure of δ for both private and public investment? Feel free to include references from the literature. iii. Which is larger, the pubic or private capital stock? (c) Calculate a measure of total labor, adjusted for hours worked. In our models, the measure of labor in the model is N s = h − l where h is normalized to one. Hence, we need aggregate labor but normalized 2 https://doi.org/10.25318/3610010401-eng by hours. This is one case where the data and the model are hard to match. One option is to we use total employment and adjust it for average hours worked by employed persons. Let hrsave be the index of average hours worked employee Visit Statistics Canada Table 36- 10-0206-01. Following real GDP normalize the index of hours to 2012q1. Nt = employmentt ∗ hrsavet /hrsave2012q1 Graph the log of real GDP, the log of capital stock (δ = 0.0125), and the log of your measure of total labor from 1991Q1–2021Q2. (d) Suppose the aggregate production function for Canada is given by: Yt = e ztAtK α t N 1−α t where At = µAt−1 with µ > 1. Take logs and write down an equation that you would use to measure the Solow residual. Assume α = 0.34. Calculate the Solow residual, take the quarter-over-quarter growth rate and graph the growth rate of the Solow residual. (e) Estimate an AR(1) process of the residuals using maximum likeli- hood. Report your estimates of µ, ρ and σ2� . Do these estimates make sense? (f) Suppose that in fact, the aggregate production function is given by Yt = e ztKα1t N α2 t O 1−α1−α2 t where Ot is the value of crude oil in Canadian dollars. What does this imply regarding the bias of the Solow residual in part d)? (g) Calculate the real value of Ot in Canadian dollars by using the esti- mate of petroleum consumption in Canada from the EIA, the price of WTI crude oil, the USD-CAD nominal exchange rate, and Cana- dian CPI from assignment 1. Convert to the same units of GDP, and calculate a new Solow residual. Assume 1 − α1 − α2 is the average of the value of Ot/Yt between 1991Q1–2021Q2 and adjust α1 and α2 accordingly. Graph the quarter-over-quarter growth rate of the new and old Solow residuals in one graph. (h) Re-estimate the AR(1) process of the residuals using maximum likeli- hood. Report your estimates of mu, ρ and σ2� . How do these compare to part e)? Bonus +1 grade point if you are the first to find the links to the Canadian government source of data on crude oil consumption in Canada and the USD-CAD nominal exchange rate. 3 https://www150.statcan.gc.ca/t1/tbl1/en/tv.action?pid=3610020601 https://www150.statcan.gc.ca/t1/tbl1/en/tv.action?pid=3610020601 https://www.eia.gov/international/data/world https://www.eia.gov/dnav/pet/pet_pri_spt_s1_m.htm https://www.eia.gov/dnav/pet/pet_pri_spt_s1_m.htm http://knoema.com/OECDKSTEI2018/key-short-term-economic-indicators-monthly-update 2. (25 points) Consider the closed economy Solow model without a govern- ment. Savings is a constant s fraction of output, s = St Yt . The aggregate production function is given by Yt = zK α t N 1−α t where Kt is the aggregate capital stock, Nt denotes the work force that grows at rate n, and z denotes total factor productivity. Capital evolves according to Kt+1 = (1− δ)Kt + It where δ > 0. (a) Rewrite the production function in terms of output per worker. (b) Solve for the per capita dynamic equations for capital, kt, consump- tion ct, and output yt. (c) Solve for the steady state quantities k, y, i, and c. (d) How does the steady state quantity of y change if there is a change in δ? (e) Suppose that α = 0.3, s = 0.15, δ = 0.012, n = 0.005 derive the values of the steady state quantities k, y, i, and c. (f) Now suppose the depreciation rate is time varying, δt, δt = δ̄ · et where δ̄ = 0.012 and represents the average depreciation rate and et+1 = ρet + �t where �t ∼ iid(0, 1) and ρ = 0.8. Simulate the model in Stata. Is the model stable? (g) What is the value of the steady state quantities k, y, i, and c? Is it the same as calculated in part (e)? (h) Produce an IRFs that reports the responce of s, c, y, i to a one stan- dard deviation shock to et. Could this shock solely explain macroeco- nomic business cycles? Could this shock help explain part of macroe- conomic business cycle? (i) Structural macroeconomic models are suppose to be based on core fundamental parameters some of which govern the behaviour of house- holds and firms. There are four such parameters in the model. Are all of them a candidate to be shocked to explain business cycle move- ment? 4 Individual Questions: 1. (5 points) Consider the closed economy Solow model without a govern- ment. Savings is a constant s fraction of output, s = St Yt . The aggregate production function is given by Yt = zKt; where Kt is the aggregate capi- tal stock, Nt denotes the work force that grows at rate n, and z denotes to- tal factor productivity. Capital evolves according to Kt+1 = (1−δ)Kt+It where δ > 0. (a) Rewrite the production function in terms of output per worker. (b) Solve for the dynamic equation for capital per worker as a function of capital per worker and exogenous parameters. (c) Show that the economy is on a balanced growth path. (d) Derive three exact expressions for how the gross growth rate of con- sumption per worker changes to an increase in either z, s, or δ. 2. (10 points) Consider the closed economy Solow model without a govern- ment. Savings is a constant s fraction of output, s = St Yt . The aggregate production function is given by Yt = ztK α t N 1−α t where Kt is the aggregate capital stock, Nt denotes the work force that grows at rate n, and zt denotes total factor productivity. Capital evolves according to Kt+1 = (1 − δ)Kt + It where δ > 0. Suppose also that zt = (1 + µ) t, that is TFP grows exogenously at rate µ. (a) Rewrite the production function in terms of effective unit of labor. (b) Solve for the dynamic equations for capital, kt, consumption ct, and output yt in terms of effective unit of labor (in effective unit of labor). (c) Derive an expression for the growth rate of capital per capital and define it γ. (d) Show that the economy in per capital terms is on a balanced growth path. What is γ on the balanced growth path? (e) Solve for the steady state quantities k, y, and c in terms of effective unit of labor. (f) Derive the golden rule for consumption in terms of effective unit of labor. 3. (10 points) Consider the two-period model. The consumer’s preferences over current and future consumption (ct and ct+1) is given by U(c1) + βU(c2) where U(c) = ln(c). 5 Households receive income only in the first period equal to y1. Households can save s for the second period and receive interest rate r. (a) Set up the households optimization decision and solve for the marginal rate of substitution. (b) Solve for the optimal levels of current consumption, future consump- tion, and saving. (c) The government decides people aren’t saving as much as they should. A mandatory savings regime is implemented where the government takes τ from household in the first period and gives back (1 + r)τ in the second period. Set up the households optimization decision and solve for the marginal rate of substitution. (d) Solve for the new optimal levels of current consumption, future con- sumption, and saving. (e) How have the optimal levels of current consumption, future consump- tion, and saving changed? 4. (25 points) Consider the basic set-up of the Solow model considered in the second group question of this assignment