EC 502: Problem Set 2
Due: Feb 24th (start of class)
1. Malthusian Pessimism
Imagine a world hundreds of years ago in a nation called Subsistia. There is no capital input
in the economy. Subsistian output is generated by using only labor and land as inputs in
a Cobb-Douglas production function: Yt = Qαt (AtLt)1−α with 0 < α < 1. Here Qt is the
quantity of land, Lt is the size of the population – assumed to be equal to the size of the
labor force – and At is the level of technology. Time is discrete. The amount of land in
Subsistia is fixed, with Qt = Q¯ > 0 for all periods t. The technology level is also fixed with
At = A¯ > 0 for all periods t. Population growth is endogenous, and the population size Lt
grows faster for higher levels of income per capita, e.g. because higher living standards lead
to better healthcare or consumption above subsistence levels. In particular, the growth rate
of the population n(·) is an increasing function of per capita income satisfying the formula
Lt+1 =
[
1 + n
(
Yt
Lt
)]
Lt,
with n(x) = γ(x− y∗) and γ, y∗ > 0.
(a) Sketch income per worker YtLt as a function of Lt. Is
Yt
Lt
increasing, decreasing, or
constant as a function of Lt?
(b) Based on your results from part (a), sketch the population growth rate n
(
Yt
Lt
)
as a
function of Lt. Is n increasing, decreasing, or constant as a function of Lt?
(c) Let L∗ be the population size such that n( YtLt ) = 0 when Lt = L
∗, i.e. let L∗ be
the steady state population size at which there is zero change in population. Write a
formula for L∗ in terms of Q¯, A¯, α, y∗, and/or γ.
(d) When Lt < L∗, will the population grow, decline, or stay the same in period t+ 1? How
(e) Assume that in the long run Lt → L∗, which is generally true in this model. Write a
formula for the long term value of output per worker YtLt in terms of Q¯, A¯, α, y
∗, and/or
γ. Does the long-term level of output per worker depend on the technology level?
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2. Jumping into Cross-Country Data
This question is an empirical exercise using cross-country data. Completion of this exercise
will require you to manipulate data, compute descriptive statistics, use OLS regression, and
plot data. The dataset is posted on Blackboard in the file cross country data.csv in CSV
format. This dataset is drawn from the Penn World Tables version 8.1. An observation
consists of the following information for a given country and year in the sample:
• Country: the name of the country
• Year: the year of the observation
• SavingsRate: the ratio of investment to GDP
• RealGDP: the value of gross domestic product (millions of 2005 US dollars)
• LaborForce: the size of the labor force (millions of people)
• HumanCapital: an index of human capital per person based on years of schooling and
Mincerian returns to schooling
• PhysicalCapital: the capital stock (millions of 2005 US dollars)
To complete this assignment, use any data analysis software or language you wish. De-
ciding which tool to use, and learning how to use that tool independently, is part of the
assignment. Several options include:
• The statistical programming language R – my suggestion. R is open source, i.e. free.
R is available at the link here. R is the most common statistical analysis tool for data
scientists and statisticians in private industry and academics. All of the tools – data ma-
nipuation, OLS regression, scatterplots, etc... – needed for this assignment are readily
available within R. R is easier to use together with the additional software package
RStudio, available for free here. RStudio functions as what is known as an integrated
development environment for R, which is a fancy way of saying that by using RStudio
you can easily see and manipulate the data and results from your R code. You might
be interested in the tutorial Econometrics in R, available here.
• The econometrics programming language Stata. Stata is popular with economists.
Stata can be bought at a discounted price for students at the link here. All of the
tools needed for this assignment are also readily available within Stata. You might be
interested in the Stata tutorial available at the link here.
• Microsoft Excel. Using the data analysis, manipulation, and plotting tools in Excel,
it is technically possible to complete this assignment. Doing so would involve much
more time-consuming manual work than one of the statistical programming language
options. However, Excel is conceptually simpler. You should feel free to weigh the
Important Note: To receive credit for completing this question you must turn in a writeup of
your answers which states the requested information in each part in summarized, easily
readable form. You will receive no credit for raw output from a statistical package. Although
statistical software can easily be used to churn out pages and pages of results, it is your
responsibility – not the TA’s – to extract answers from that output! If you choose to work in a
group – of up to five students – remember to turn in a separate writeup of your analysis and
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(a) Descriptive Statistics
For each nation in the sample, compute the following values:
• GDP per worker in 1985 and 2005, YL 1985 and YL 2005
• the average savings rate from 1985 to 2005, s
• the average growth rate of the labor force from 1985 to 2005, n. To compute
the average growth rate of a variable X from year t to year t + T , use the log
approximation formula
1
T
log
(
Xt+T
Xt
)
.
• the average growth rate of GDP per worker from 1985 to 2005, gY/L. Use the log
approximation formula above. Remember, economists use log to refer to natural
logarithms!
Report the following information:
• the number of countries in the sample of data
• the mean – across countries – of log (YL 1985), log (YL 2005), s, n, and gY/L
• the standard deviation – across countries – of log (YL 1985), log (YL 2005), s, n, and
gY/L
(b) Unconditional Convergence
Produce and report a scatterplot of YL 1985 (on the horizontal axis, in logs) and gY/L (on
the vertical axis, using exactly the formula from above). Then, estimate the following
OLS regression
gY/L = β0 + β1 log
(
Y
L 1985
)
+ ε.
Report the value of the estimated coefficient βˆ1. Does the estimated value of βˆ1 sug-
gest that living standards in nations which are initially poorer catch up, fall further be-
hind, or remain the same relative to wealthier nations?
(c) MRW Revisited
Following the analysis in MRW, estimate the following OLS regression
log
(
Y
L 2005
)
= β0 + β1 log(s) + β2 log(n+ g + δ) + ε.
To implement this regression, you should assume that g + δ = 0.05 is constant across
nations. Report the following information:
• The estimated value βˆ1. Does this value suggest that living standards in nations
which save more are higher, lower, or the same as in nations which save less?
• The value of α, the capital elasticity of output in the Solow model with physical
capital only, is implied by the estimate βˆ1. Is this implied value of α consistent
with a labor share of around 2/3?
• The R2 of the regression. What fraction of the variation in living standards in 2005
is explained by the variables in this regression?
(d) HJ Levels Accounting
Following the analysis in HJ, assume that α = 13 . Then, for each nation i in the dataset,
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compute the implied value of productivity A2005,i in the year 2005 using the following
decomposition
Y
L 2005,i
=
(
K
Y 2005,i
) α
1−α
(
H
L 2005,i
)
A2005,i.
Above, the subscript 2005, i refers to the value for country i in the year 2005 of the
following quantities:
• YL 2005,i: GDP per worker
• KY 2005,i: capital intensity, the ratio of the capital stock to GDP
• HL 2005,i: human capital per person, i.e. the raw index from the dataset
• A2005,i: implied productivity or technology
Report the mean and standard deviation – across countries – of log(A2005,i). Produce
and report a scatterplot of A2005,i (on the horizontal axis, in logs) and YL 2005,i (on the
vertical axis, in logs). Then, estimate the following OLS regression
log
(
Y
L 2005
)
= β0 + β1 log(A2005,i) + ε.
Report the value of the estimated coefficient βˆ1. Does the estimated value of βˆ1 sug-
gest that living standards are higher or lower in nations with higher productivity? Report
the R2 of the regression. What fraction of the variation in living standards in 2005 is
explained by productivity?
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