The assignment is marked out of 25 points. The weight for each part is indicated following the question text.

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 1950 to the latest year (2019) for the United States, Germany, France, South Korea, Taiwan, Brazil, India, China, and South Africa.

Then construct real GDP per person each year as the ratio of its real GDP per person to that of the United States for that year (so that this ratio will be equal to 1 for the United States for all years.)

(a) Plot these ratios for Germany, France, South Korea, and Taiwan over the period for which you have data. Does your data support the notion of convergence with the United States? Give a brief explanation to rationalize your fifinding. (2 points)

(b) Plot these ratios for Brazil, India, China and South Africa. Does this data support the notion of convergence with the United States? Give a brief explanation to rationalize your fifinding.(2 points)

Question 2 [14 points]

Solow growth model. In this question, you will explore how changes in the saving rate, the rate of technological progress and the rate of population growth affffect an economy’s growth.

Consider the Solow (neoclassical) growth model with aggregate production function Y =Kα(AN)1α.

Each period lasts a year.

(1) Using the parameter values in Table 1, calculate the steady-state values of capital per effffective worker k K/AN, output per effffective worker y Y/AN and consumption per effffective worker c C/AN. Calculate the golden rule saving rate. Also calculate the growth rates of output per worker and consumption per worker along the balanced growth path. (2 points)

(2) Suppose that the economy is initially in steady-state. In year t = 0 the saving rate increases from s = 0.15 to s = 0.20 (i.e., from 15% to 20%) while all other parameters have their benchmark values.

(i) Calculate the new steady-state levels of capital per effffective worker, output per effffective worker and consumption per effffective worker. Does long-run consumption per effffective worker increase? 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. (3 points)

(ii) Calculate and plot the time-paths of (a) capital per effffective worker, output per effffective worker and consumption per effffective 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 effffect of the change in the saving rate on these variables. (3 points)

Note: Here log means the natural logarithm, ln(·) = loge(·). 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. You will probably want to use a spreadsheet program to implement these calculations.

(3) Suppose instead that at time t = 0, the rate of technological progress increases to 2.0% per year. With all other parameters as in Table 1 (in particular, with the saving rate s back at its benchmark value of 15%), calculate and plot the time-path of log output per worker for 100 years after the change in the rate of the technological progress (for t = 1, 2, …, 100), again assuming A0 = 1. Compare the time-path of log output per worker from the increase in the saving rate in part (2) to the time path with the increase in the rate of technological progress.

How many years pass before output per worker surpasses the level that would be obtained from the increase in the saving rate? What does this suggest about the relative importance of level versus growth rate effffects? Explain. (3 points)

(3) There is increasing concern about recent declines in the rate of population growth observed across a number of developed economies. For example, in Japan average annual growth rate of the population in the last decade has been 0.1%.

We will now explore the implications of a negative population growth rate and contrast its implications with those resulting from changes in savings rate and rate of technological progress.

Suppose that at time t = 0, the rate of population growth declines to 2.0% while all other parameters are as in Table 1 (in particular, the saving rate s is back at its benchmark value of 15%, and the the rate of technological progress at 1.5%.). Calculate and plot the timepath of log output per worker for 100 years after the decline in population growth rate (for t = 1, 2, …, 100), again assuming A0 = 1. Compare the time-path of log output per worker from the increase in the saving rate in part (2) and the increase in rate of technological progress in part (3) to the time path with the decline in the rate of population growth. How many years pass before output per worker surpasses the level that would be obtained from the increase in the saving rate? In light of your results, is there cause for concern if population growth rates would continue to decline? Explain. (you may assume that N0 = 1 to facilitate your explanation). (3 points)

Question 3 [6 points]

Transition Dynamics of Physical and Human Capital. You will explore the dynamics of human capital and physical capital using the human capital accumulation model. The aggregate production function is Y = AKαH1α . The values of parameters are shown as in Table 2. Each period lasts a year.

(1) Suppose an economy starts from K0 = 2 and H0 = 1. Calculate and plot the time paths of log human capital, log physical capital and log output for 100 years (t = 0, 1, 2, …, 100). Describe the dynamics of human capital and physical capital. What are the growth rates of human capital, physical capital and output in the long run? Has the ratio of human capital to physical capital converged to a steady state? If so, what is the steady state ratio of human capital to physical capital? (3 points)

(2) Now suppose another economy also starts from K0 = 2 and H0 = 1 but their production process also utilizes land, Z = 1, as a fifixed input so that their aggregate production function is Y = AKαH1αβZβ where α = 1/3 as before, and β = 1/10. Calculate and plot the time paths of log human capital, log physical capital and log output for 100 years (t = 0, 1, 2, …, 100) assuming all other parameter values as in Table 2. Describe the dynamics of human capital and physical capital. In comparison with your answers in part (1), how do the dynamics of human capital and physical capital change? What are the growth rates of human capital, physical capital and output in the long run? Do the ratios of human capital to physical capital converge to a steady state? If so, what is the steady state ratio of human capital to physical capital?

Explain the difffferences between the two economies. (3 points)