Intermediate macro
Assignment #2
Due Monday September 30th 2pm AEST
Question 1 [4 points]
Convergence in two samples of economies. Go to the website containing the Penn World Table
(https://www.rug.nl/ggdc/productivity/pwt/) and collect data on real GDP (expenditure-side real GDP at chained PPPs (in mil. 2017 USD)) and population from the earliest year available (usually, but not always, 1950) to the latest year (usually, but not always, 2019) for Australia, Japan, South Korea, Italy, Taiwan, China, Brazil, Albania and South Africa. Then construct real GDP per person each year as the ratio of its real GDP per person to that of Australia for that year (so that this ratio will be equal to 1 for Australia for all years.)
(a) Plot these ratios for Japan, South Korea, Italy, and Taiwan over the period for which you have data. Does your data support the notion of convergence with Australia? Give a brief explanation to rationalize your finding. (2 points)
(b) Plot these ratios for China, Brazil, Albania, South Africa, and Australia. Does this data support the notion of convergence with Australia? Give a brief explanation to rationalize your finding. (2 points)
Question 2 [20 points]
Solow growth model with capital utilization. In this question, you will explore how changes in the capital utilization rate and savings rate affect an economy.
Consider a version of the Solow (neoclassical) growth model in which the total capital stock K may not be fully utilized. Denote the capital utilization rate as u with 0 ≤ u ≤ 1 so that the aggregate production function is Y = (uK)α (AN)1−α where as before, K is total physical capital stock and N is employment.
Further, assume that the depreciation rate is an increasing function of the utilization rate and given
by,
δ(u) = δuβ , where 0 < δ < 1,β > 0
Thus, the capital accumulation equation (in per effective worker terms) is,
(1 + gA + gN)(kt+1 − kt) = syt − (δ(u) + gA + gN )kt
where k ≡ K/AN is capital per effective worker, y ≡ Y/AN is output per effective worker
(1) Using the parameter values in Table 1, calculate the steady-state values of capital per effective worker k ≡ K/AN, output per effective worker y ≡ Y/AN and consumption per effective worker c ≡ C/AN. Also calculate the growth rates of output per worker and consumption per worker along the balanced growth path. (2 points)
|
α
β
δ
u
s
|
1/3
4
15%
75%
15%
|
gA
gN
|
2.5%
1%
|
Table 1: Benchmark Parameter Values – Solow model with capital utilization
(2) Using the parameter values in Table 1, calculate the golden rule savings rate that maximizes steady state consumption per effective worker. (1 point)
(3) Suppose that the economy is initially in steady-state. In year t = 0 the capital utilization rate increase from u = 0.75 to u = 0.80 (i.e., from 75% to 80%) while all other parameters have their benchmark values.
(i) Calculate the new steady-state levels of capital per effective worker, output per effective worker and consumption per effective worker. Does long-run capital per effective workerincrease? Explain. Does long-run output per effective worker increase? Explain. (3 points)
(ii) Also calculate the long run growth rates of output per worker and consumption per worker? Do the long-run growth rates of output per worker and consumption per worker increase? Explain. (1 point)
(iii) Using equations, describe how you can calculate the time-paths of capital per effective worker, output per effective worker and consumption per effective worker. Plot the time paths of capital per effective worker, output per effective worker and consumption per effective worker for 100 years (t = 0, 1, ...., 100). (2 points)
(iv) Describe and explain the dynamics of capital per effective worker and output per effective worker that you plotted in (iii). Discuss which variables move in the short-run and why. In your explanation, you should describe how effective investment and effective depreciation are impacted in response to an increase in capital utilization. You may use a diagram to aid in your explanation or alternatively describe how the effective investment and effective depreciation curves will be impacted following an increase in u. (4 points)
Note: You will probably want to use a spreadsheet program to implement these calcula- tions.
(4) Using the expression of steady state capital per effective worker, output per effective worker, and consumption per effective worker to derive how the steady states of each of these variables depend on the capital utilization rate. With all other parameters as in Table 1, plot how the steady state capital per effective worker, output per effective worker, and consumption per effective worker depends on the capital utilization rate for u = 0, 0.01, 0.02, 0.03, ..., 1. In your plot, the horizontal axis should be the capital utilization rate ranging from 0 to 1 and the vertical axis is the steady state capital per effective worker, output per effective worker, and consumption per effective worker. For each of these three variables, what is the capital utilization rate that maximizes each variable? (3 points)
(5) Suppose instead that at time t = 0, the savings rate decreases to 10% per year. With all other parameters as in Table 1 (in particular, with the capital utilization rate u back at the value in Table 1).
(i) Calculate the new steady-state levels of capital per effective worker, output per effective worker and consumption per effective worker. Does long-run consumption per effective worker increase? Explain. (2 points)
(ii) Calculate and plot the time-paths of (a) capital per effective worker, output per effective worker and consumption per effective worker for 100 years (t = 0, 1, ...., 100) and of (b) log output per worker and log consumption per worker. Describe and explain the short-run effect of the change in the saving rate on these variables. (2 points)
Note: To answer part (ii) of this question we need to know the level of productivity at year t = 0. Assume this initial level of productivity is A0 = 1. Here log means the natural logarithm, ln(·) = loge (·).
Deadline and format
This assignment contributes 12.5% toward your final grade. The assignment is due by 2pm on Monday September 30th. Your assignment should not exceed 1500 words in length. You should use relevant diagrams and/or algebra to reinforce your argument where appropriate. Citations, labels in diagrams, symbols in equations and numbers in tables will not count towards the word limit.
The assignment can be done in groups subject to the following rules: All members of the group will be given the same mark. No more than three students may make up a group. Students may choose to work and hand in an assignment on their own. No two groups may hand in the same assignment. All assignments will be electronically screened for plagiarism.
Assignments must be submitted in electronic format by 2pm on the due date. No extension will be considered. You can apply for special consideration if you cannot complete the assignment because of illness or other circumstances. In case that your application is successful, the weight of this assignment will be shifted to the final exam in the overall assessment. Moreover:
• You must keep a copy of your assignment.
• Plagiarism or other forms of academic dishonesty will result in disciplinary proceedings being brought against you.
• 10% of the available marks will be deducted for each full hour of delay after the due date for up to 10 hours.