Solow-Swan model, economic growth and stata coding

Winter 2019
PP 57300
*Instructor: Jina
Homework 1
Due: Wednesday 23rd January, 2019
Directions: Please submit a single PDF write-up with your answers to questions, any required figures,
and the code / output of your coding language of choice (e.g., a Stata “log file”). Every answer in which
you need to use data should be replicable directly from your code (i.e., don’t read values from the data
browser – write a command!).
Human captial in the Solow-Swan model

1. Consider the following quote from EasterlyâAZs Elusive Quest for Growth (2001): “When there is
   a high level of human capital relative to physical capital, the return to investing in physical capital
   will be high and thus growth will be higher until physical and human capital are brought back into
   balance” (page 76-77). This question will ask you to analyze this statement in the context of the
   human capital-augmented Solow-Swan model:
   Y = KHL

where, for simplicity, we assume (a) no technological progress and (b) no population growth.
(a) What does it mean (in words) for any production function to exhibit constant returns to scale?
(b) Show that if  +  +
= 1, this production function exhibits constant returns to scale. (Hint:
multiply inputs by some constant.)
(c) Show that
y = kh
where lower-case letters indicate per capita values, i.e., y = Y/L, etc.
(d) What is the marginal product of capital per worker (MPk)? Note that the general definition
is:
MPk = dy
dk
(e) How does the MPk depend on the level of human capital h? That is, is the MPk higher when
h is higher? (You can show this formally by taking a cross-derivative, but you can also just
work it out by looking at the MPk equation.) Does this support Easterly’s claim that “when
there is a high level of human capital relative to physical capital, the return to investing in
physical capital will be high”?
(f) Physical capital is accumulated in the standard way:

K = SY − K
where we suppress the t-subscript on K. Show that the growth rate of k is:

k
k
= sk−1h − 
(g) Assume that human capital is exogenously given (i.e., it is from outside the model). and it
is initially h0. Find the steady state of capital, k, of this economy. (Remember that at k:

k = 0.)
1
(h) Now say that we exogenously increase human capital to h1 such that h1 > h0. In the short
term (i.e., keeping the level of physical capital fixed) what happens to output now that human
capital is h1?
(i) Will the level of physical capital per worker increase or decrease?
(j) Over time, what will happen to the growth rate of physical capital? Give your answer in words,
and draw a sketch of the growth rate versus time if you want.
Convergence and selection on outcomes

1. For us to observe convergence in real data, we would require that, in the long-run, initially poorer
   countries have had faster growth rates than initially richer countries. This would mean they would
   catch-up over time. Yet, constructing measures of GDP in the past is incredible difficult. Even if
   we can do it, we can often only do so for the countries that are rich today, because they often have
   better data storage, archives, etc. For example, we have a lot of data on Sweden going back many
   generations, but much less on Cambodia. Therefore, if we can only examine convergence using data
   from countries that are richer today, this might have some effect on our conclusions.
   3000
   3500
   4000
   4500
   5000
   5500
   1880 1900 1920 1940 1960 1980 2000 2020
   Year
   Randomly generated GDPpc
   Figure 1: Data generate by the “homework1 convergence.do” file
   Download the file “homework1 convergence.do”. This file will do a very simple simulation, and
   randomly generate GDP per capita in 1900 for 100 countries, as well as random long-run growth
   rates. This is what you see in figure 1. It then uses these rates to generate GDP per capita in
2. You don’t need to fully understand the first parts of the code, but you will need to run it to
   generate your data. The second part of the code (starting at “Plotting figures / data explorations”)
   requires you to write commands to plot figures as follows, and interpret them:
   (a) Plot a scatter of GDP per capita growth rates versus GDP per capita level in 1900
   (b) Overlay a line plot on this. What is the relationship between GDP per capita in 1900 and
   long-run GDP per capita growth rates? Do you conclude that there is convergence?
   2
   (c) Now let’s think about data quality. Let’s say we only have data for 1900 from countries that are
   above median wealth in 2000. Plot the same figure as before, but now only for those countries
   of above median wealth in 2000. What does the relationship between 1900 GDPpc and GDPpc
   growth in this new sample? Do you conclude there is convergence?
   (d) Finally, plot the same for the richest quartile (i.e., the countries above the 75th percentile of
   wealth) in 2000. What is the relationship now?
   (e) Explain briefly (one or two sentences) what this simulation implies. That is, what might we
   conclude or fail to conclude about the world and why?
   Many studies suffered from this data problem. Pritchett (1997) was novel in that he estimated a
   minimal level of GDP in the poorer countries. The conclusion of this simulation should be that you
   should not only be concerned about the data that you see, but also about the data that you don’t
   see.
   World Development Indicators data
3. The world development indicators (WDI) data is one of the most useful datasets for understanding
   development status around the world. You can find the databank resource here, which will allow
   you to download data. To familiarize yourself with this resource, you should navigate through the
   download portal and download data for all countries on a few indicators to answer the questions
   below. Note the distinction between countries and aggregates. Please write code in stata to
   answer each question. Once you have downloaded the data to your working directory, a useful
   command to start is “import”!
   (a) The WDI contains data on income levels around the world. Download GDP per capita data
   for 2015. Import it into stata and write a command to list the 10 highest and 10 lowest income
   countries.
   (b) In the Solow-Swan model, we saw that savings drove growth, and also that savings was equal
   to investment. One measure of investment in capital is gross capital formation. Download and
   plot a scatter plot of savings (as a percent of GDP) versus gross capital formation (as a percent
   of GDP) for 2015. Does it look like countries that save more invest more in capital?
   (c) In the Solow-Swan model, we see that steady states were determined in part by savings rates.
   Plot the same data of savings as a percent of GDP versus log GDP per capita in 2015. Do
   countries with higher incomes save more as a percent of GDP?
   (d) (Optional) Now download data for the aggregate high, upper middle, lower middle, and low
   income on growth rates through time. Plot the time-series from 1960-present of the annual
   GDP per capita growth rates in each group on a single figure. The command “reshape” might
   be helpfull
   3