Week 3 Homework - Group 12

Joshua Dawe, Jack Nguyen, Edward Southerington

August 17, 2015

PART 1

What is their data source? How do their samples differ? Which countries do they exclude? Do they include multiple observations of each country, or just a single observation?

The authors derive their data from the Real National Accounts data set compiled by Summers and Heston (1988). Summers and Heston use annual cross country data from the 1960-1985 period reflecting real income, government and private consumption, investment and population. The data set forms a time series, that is there are multiple observations for each country taken as an annual cross section of data.
Mankiw, Romer and Weil split the data set into three samples. The first sample includes data from 98 countries. These countries include all of the countries from the Summers and Heston set for which data is available, excluding those countries where oil production is the primary industry. The second sample includes data from 75 countries. The exclusion criteria for this sample is extended to exclude countries who data quality rating by Summers and Heston is D and also countries whose population was less than 1 million in 1960. The final sample only considers data from 22 countries, these countries are characterised by their membership of the OECD and each have populations over 1 million.

The authors make an identifying assumption when estimating the parameters of an un-augmented Solow model. What are their assumptions?

When estimating the parameters for the un-augmented Solow model, Mankiw, Romer and Weil assume that the depreciation /(delta/) and the rate of technical change g are constant across countries. That is, neither variable will differ significantly in a cross country analysis. However, it is important to note that the authors assume that the first time series data point of any country will be given by lnA (0) = /(alpha/) + /(epsilon/) where /(alpha/) is a constant and /(epsilon/) is country specific. Therefore, while the rate of technical change should be equivalent across countries, the nominal value of technical advantage will be relative to this starting point. Furthermore, the authors argue that savings rates s and population growth n are independent of the country specific factors which may shift the production i.e. /(epsilon/). This is important for the regression analysis of the data and its fit to the model which relies on this assumption.

Can you think of a good reason that population growth would be correlated with the technology available to a country?

The technology available to a country is partially determined by the value of the country specific shock variable /(epsilon/) i.e. ln A(0) = /(alpha/) + /(epsilon/). All future data points will be relative to this initial value. It seems possible that the income of a country at this initial time series point may be a determinant of the country specific technology available denoted by /(epsilon/). We know that population growth is positively correlated with income, therefore by transient logic, it may be also correlated with technology available.

How do the authors augment the model to include human capital? Does it help?

The authors alter the initial production function equation derived from the Solow model \[ Y (t) = K (t) ^(/(alpha/))(A(t)L(t))^(1 - /(alpha/)) 0 < /(alpha/) < 1. \] To the following version \[ Y (t) = K (t) ^(/(alpha/))H^(/(beta/))(A(t)L(t))^(1 - /(alpha/) - /(beta/)) 0 < /(alpha/) < 1. \] This transformation requires a number of assumptions including that a unit of either physical capital or a unit of human capital can be costlessly traded for a unit of consumption and that both human and physical capital depreciate at the same rate. The transformation also assumes that \[ alpha + beta < 1\].

Does conditional convergence hold?

Conditional convergence appears to hold in the period 1960-1985. The data on page 427 shows increasing levels of correlation as conditions are added to the model. The conditions set in the augmented model appear to result in the highest levels of convergence.

PART 2

Part 1

library(dplyr); library(readr)

## 
## Attaching package: 'dplyr'
## 
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## 
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

invHis <- read.csv("investment.csv")

invHis1 <- invHis %>% arrange(n():1)

depRate <- function(dep, inv, capital) {
  
  t <- length(inv)
  remCap <- sum(inv/((1+dep)^(0:(t-1))))
  
  difference <- abs(capital-remCap)
}

optim(0.07, depRate, lower = 0, upper = 0.2, inv = invHis1$Investment/1000, capital = 5096, method = "Brent")

## $par
## [1] 0.0411923
## 
## $value
## [1] 0.0001865847
## 
## $counts
## function gradient 
##       NA       NA 
## 
## $convergence
## [1] 0
## 
## $message
## NULL

We find that depreication is 4.1%

Part 2

From the paper “Labour’s share of growth in income and prosperity”, a visiting researcher paper by Dean Parham, the assumed labour share is 55%. Applying this to to GMI to find total capital and dividing capital by total of (COE, GOS and GMI)

GMI <- 130748
GMIcapShare <- GMI*.45
GOS <- 540302
COE <-755009

alpha <- (GMIcapShare+GOS)/(GMI+GOS+COE)

##s = average of GFC of past 5 years divide by average of GDP of past 5 years

gfc1 <- 365284
gfc2 <- 379094
gfc3 <- 422600
gfc4 <- 430844
gfc5 <- 424279

GFCaver <- (gfc1+gfc2+gfc3+gfc4+gfc5)/5

gdp1 <- 1397902
gdp2 <- 1430354
gdp3 <- 1483675
gdp4 <- 1520944
gdp5 <- 1559662

GDPaver <- (gdp1+gdp2+gdp3+gdp4+gdp5)/5

s <- GFCaver/GDPaver

n15 <- (12583.7-12304.4)/12304.4

g <- 0.016
d <- 0.041

k <- (s/(d+n15))^(1/(1-alpha))
y <- k^alpha
#output per worker
return(y)

## [1] 2.874384

#output per person
y1 <- y*0.65
return(y1)

## [1] 1.86835

Part 3

library(reshape2); library(dplyr); library(ggplot2)

s <- 0.25
delta <- 0.05
n <- 0.02
alpha <- 0.4
n1 <- n*0.97

k <- rep(NA, 200)
kp <- rep(NA, 200)
y <- rep(NA, 200)
yp <- rep(NA, 200)
k[1] <- 1
kp[1] <- 1

for (t in 2:200) {
  y[t] <- k[t-1]^alpha
  k[t] <- (1-delta-n)*k[t-1]+s*y[t]
  
  yp[t] <- kp[t-1]^alpha
  kp[t] <- (1-delta-n1)*k[t-1]+s*yp[t]
}

data.frame(t = 1:200, k, kp) %>% melt(id = "t") %>% 
  ggplot(aes(x = t, y = value, colour = variable)) +
  geom_line() +
  ylim(0, 12)

##output per worker
return(y[200])

## [1] 2.336242

#outpur per woker after polio epidemic
return(yp[200])

## [1] 2.336819

y1 <- y[200]*.65
yp1 <- yp[200]*.62
#output per person
return(y1)

## [1] 1.518558

#output per person after polio epidemic
return(yp1)

## [1] 1.448828

What happens to output per worker?

WIth the polio epidemic, the labour force will decrease but the capital does not change. This will result in a higher output per worker.

What happens to output per person?

Since participation rate would decrease by 3%, then the output per person would decrease as there is now less workers in comparison to the population, as shown with the calculations above.

What happens to the children?

There are no children in the workforce, so the data above would not change, but their quality of life would be different. If the children ever get a serious infection of polio, it could prevent them from ever joining the workforce, affecting the future workforce.