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1
GROWTH, ACCUMULATION AND CONVERGENCE
In this problem set we become familiar with the mechanics and the implications of the HarrodDomar and Solow growth models. You may (but do not have to) work in pairs. If you choose to
work in pairs, hand in a single copy of the answers, and make sure you put write in both your
names above. In this problem set, you will be asked to present several graphs. Place all graphs
in an appendix at the end of your homework. Please titles and make sure you clearly label the
axes of each graph. You may use this Word template to record your answers.
1. The Relationship between Income per Capita and Indicators of Welfare and
Empowerment
This question asks you to empirically explore the strength of the relationship between income per
capita and several direct indicators of welfare. To answer this question, download the Excel data
set “World Bank Data” on SmartSite. This file contains data from 2012 for 1561
countries and
comes from the World Bank’s country-level data sets . The variables are as follows:
y: GDP per capita in 2005 US Dollars.
Life Expectancy: Life expectancy at birth in years;
Sanitation: % of households with access to improved sanitation2
;
Immunization: % of children ages 12-23 months immunized against diphtheria, pertussis,
and tetanus
Parliament: % of parliamentary seats in a single or lower chamber held by women
Labor Ratio: Ratio of labor force participation rate of women to men.3
A. Generate the following three scatter plots with : 1) Life Expectancy versus y; 2) Sanitation
versus y and 3) Immunization versus y. In each scatter plot, put the first variable on the vertical
axis and y on the horizontal axis and add a trend-line through the data points.4
B. Discuss the relationship between income per capita and these three indicators of welfare. Do
you conclude that higher income per capita leads to better performance in terms of these
indicators? Is the relationship strong? What other features of the relationship seem interesting to
you?
1
The World Bank data set contains information on over 220 countries. I have included only those countries with
complete data for all of the variables in 2012.
2
The improved sanitation facilities include flush/pour flush (to piped sewer system, septic tank, pit latrine),
ventilated improved pit (VIP) latrine, pit latrine with slab, and composting toilet.
3
Labor force participation rate is the proportion of the population ages 15 and older that is economically active: all
people who supply labor for the production of goods and services during a specified period.
4 Use the chart option “insert trendline”.
2
C. Now let’s turn attention to an issue raised by students in the first lecture; namely does higher
income per capita seem to bring greater empowerment of women? To look at this question,
create similarly structured scatter plots for: 1) Parliament versus y and 2) Labor Ratio versus y.
Discuss your graphs. What features stand out to you? What do you conclude about the
relationship between these two measures of gender empowerment and the level of income per
capita?
2. Mechanics of the Harrod-Domar Growth Model
Now let’s walk through the mechanics of the H-D model for an imaginary country, which we’ll
call South. Let’s assume the following parameter values for South: The incremental capitaloutput ratio, v, is 1.25; capital depreciates at a rate of 2% per year (d = 0.02); the population
grows at 2% per year (n = 0.02); and the savings rate is 10% (s = 0.1).
A. In the initial year (t=0) the per-capita capital stock equals 200 and the population is 1.
Continuing under the assumption that the economy evolves according to the Harrod-Domar
model, fill in Table 2 below. (NOTE: Do not use the H-D growth equations yet. Instead, first
completely fill in columns A – G using the production function from the H-D model and the
equation for capital accumulation, then calculate the growth rates using the definition of growth
rate: g(t) = [Y(t+1)-Y(t)]/Y(t))
Table 2. Evolution of a Harrod-Domar Economy
(A)
Year
t
(B)
Population
L(t)
(C)
Total
Capital
Stock
K(t)
(D)
Per Capita
Capital
Stock
k(t)
(E)
Total
Income
Y(t)
(F)
Per Capita
Income
y(t)
(G)
Total
Savings
S(t)
(H)
Growth
Rate of
Total
Income
g(t)
(I)
Growth Rate
of Per Capita
Income
g
*
(t)
0 1.00 200.00 200.00
1
2 NA NA
B. Now let’s check your work. Write down the Harrod-Domar growth equations (one for
aggregate income and one for per-capita income). Use these equations to find the growth rate of
aggregate income, g, and the growth rate of per-capita income, g
*
for South. Recall that the
equation for the growth rate of per-capita income is approximate.
3
C. Use the approximate doubling time formula from lecture to calculate approximately how
many years it will take South to double:
-Aggregate Income: _______
-Per Capita Income: _______
For parts D - F you will need to create graphs in Excel. You might find it useful to set up a
spreadsheet with the same structure as Table 2 above.
D. Using the same parameter values, compute and graph South’s income per capita over the next
100 years. Title this graph “Figure 2D/E”.
What is South’s income per capita after 50 years? _____
What is South’s income per capita after 100 years? _______
E. Our second country is called North. North is exactly the same as South except that it starts
with a greater capital stock. In the initial year (t=0), North has a capital stock of 600. On the
same graph that you made in part D, graph North’s income per capita over the next 100 years.
What is North’s income per capita after 50 years? ______
What is North’s income per capita after 100 years? ______
Do North and South converge over time given their common savings rate according to
this model? Explain the reasons for convergence, or lack thereof, in this model.
F. Finally, suppose a newly elected, populist leader of South wants her country to catch up with
North. She is a brilliant orator and is confident that she can convince her people to consume less
and save more in order to fulfill her economic catch-up mission. She has hired you to tell her
what savings rate is necessary to catch up with North over a 50-year time horizon.
What is this savings rate? ________ (You can answer this either by trying different
savings rate in the spreadsheet you have been using for parts D and E or by figuring it out
with pencil and paper.)
Are you surprised by your answer? Do you think that Zimbabwe could really catch up
with the USA by following this savings plan? (BRIEFLY explain the rationale for your
answer to this question.)
3. Solow Model with No Technological Change
Let’s now modify the technology assumption used in the prior problem and assume that output in
all countries (South and North) is produced according to the following constant-returns-to-scale
production function that lies at the heart of the standard Solow growth model:
𝑌(𝑡) = 10[𝐾(𝑡)]
0.5[𝐿(𝑡)]
0.5
4
A. Express the production function in per capita terms (i.e., derive an expression for y as a
function of k).
Assume we have the same values as in problem two for the following exogenous parameters: s =
0.10; n = 0.02; d = .02.
B. What are the steady state levels of capital per worker and income per worker?
Assume also that our two countries start with the same initial conditions as in problem two;
namely, that L(0)=1 for both South and North and that K(0) = 200 in South and K(0) = 600 in
North.
C. On one graph plot the income per capita levels for the two countries over 200 years under
exogenous technological change. Title this graph “Figure 3A”.
D. On a separate graph, plot the income per-capita growth rates over the same time period for
both countries. Title this graph “Figure 3B”.
E. Does the model reach a steady state?
F. Does South converge to North according to this model when both countries have the same
savings rate?
G. How and why are the implications of this model for South different from those in the Harrod--
Domar model? Why can growth not be sustained?
H. Suppose now that the populist leader raises savings rate in South to the levels that you
identified in 2F above.
What happens in both short and long terms to per-capita income levels and growth rates
after the savings rate increases in South?
Do South and North still converge? Why or why not?
4. Solow Model with Exogenous Technological Progress
Now assume that there is an exogenous rate of technological progress of 2% per year. To
capture this, we need to slightly modify the production function as follows:
𝑌(𝑡) = 10[𝐾(𝑡)]
.5[𝑇(𝑡)𝐿(𝑡)]
.5
In the equation above, 𝑇(𝑡), is the state of technology and measures how “effective” our workers
are. To capture exogenous technological progress, assume that in our initial period, 𝑇(0) = 10
and that 𝑇(𝑡) grows exogenously by 2% every year for every country.
5
A. Assume that both North and South have a 10% savings rate and that initial capital stocks and
all other exogenous parameters are the same as in problem 3.
On one graph plot the per capita income levels for the two countries over 200 years under
exogenous technological change. Title this graph “Figure 4A”.
On a separate graph, plot the per-capita income growth rates over the same time period
for both countries. Title this graph “Figure 4B”.
B. Compare and contrast your findings in 4A with the results you obtained under the HarrodDomar model. Even though long run growth rates converge between South and North in both
the Harrod--Domar and this “Solow-with-exogenous-technological-change” model, why do percapita income levels converge in one model, but not in the other?

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