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程序代写案例-EC 502

时间：2021-02-19

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

about when Lt > L∗?

(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?

1

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

tradeoffs among the various options and come to your own choice.

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

include your team members’ names at the top of your writeup.

2

(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,

3

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?

4

学霸联盟

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

about when Lt > L∗?

(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?

1

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

tradeoffs among the various options and come to your own choice.

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

include your team members’ names at the top of your writeup.

2

(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,

3

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?

4

学霸联盟