# 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

- 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 = KHL

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.)

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(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

- 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.

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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 - 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?

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(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 - 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

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