Task 1

The data source is from the Real National Accounts constructed by Robert Summers and Alan Heston. There are three different samples used – the first is the most comprehensive, and includes all countries for which data is available excluding countries for which oil is the primary production. The second sample excludes countries with a population under one million, as there is likely to be a higher degree of error. The third sample consists of the 22 OECD countries, excluding those with a population under a million. The disadvantage of the third sample is that it is small, consisting of only 22 countries as opposed to the 98 and 75 of the first and second samples respectively. From each sample they use just one observation.

The authors assume that the growth rate and depreciation rate are the same across countries. The growth rate refers fundamentally to the advancement of the knowledge which is the same across all countries, and the depreciation rate is assumed to be the same as there is no reason to believe otherwise, and no data to use to estimate rates across different countries if there were to differ. The other assumption made by the authors is that the saving rate and population growth rate are not effected by other, country-specific factors that may shift production.

Population growth would have a positive effect on technology – the more human capital a country has, the more ‘talent’ it is likely to have and subsequently more ideas will be produced. Ideas form the basis of technological change, so the higher a country’s population the more likely it is to experience faster technological growth.

The authors add human capital to the model as an endogenous variable, under the assumptions that any unit of consumption can be spent on either human capital or physical capital equally, and that human capital follows the same production function as physical capital. The addition of human capital to the model eradicates the high coefficients on population growth and investment, providing a more accurate result and improving the model’s overall performance.

Conditional convergence does hold for population growth, human and physical capital model.

Task 2

Part 1: Estimating the depreciation rate in Australia

Non-financial assets: Volume Measures June 2014: $5096.3 billion

\(K = \$5096.3b\)

library(readr); library(dplyr); library(ggplot2); library(reshape2)
inv <- read_csv("Investment.csv")

inv2 <- inv %>% arrange(n():1)

dep <- function(depreciation, series, finalvalue){
  objective <- finalvalue - sum(series/((1+depreciation)^(0:(length(series)-1))))
  abs(objective)
}

optim(0.07, dep, lower = 0, upper = 0.2, series = inv2$Investment/1000, finalvalue = 5096, method = "Brent")
## $par
## [1] 0.0411923
## 
## $value
## [1] 0.0001865847
## 
## $counts
## function gradient 
##       NA       NA 
## 
## $convergence
## [1] 0
## 
## $message
## NULL

\(\delta = 0.041\)

delta <- 0.041
K <- 5096.3
inv <- inv$Investment
kdiffest <- function(delta, inv, K){
  
  t <- length(inv)
  
  khat <- sum(inv/((1+delta)^(0:(t-1))))
  
  kdiff <- abs(K - khat)
  
  return(kdiff)  
  
}

Part 2: Balanced growth output in Australia

\(\alpha = 40\%\)

Gross mixed income = $130748 million

\(g = 1.6\%\)

\(s = 27.4\%\)

\(\kappa = \left(\frac{s}{n + \delta}\right)^\left(\frac{1}{1-\alpha}\right)\)

\(k[1] = \kappa\)

labour <- read_csv("LF.csv")
date <- labour$date
# Defining parameters
alpha <- .4 #After several hours discussion with Pardy
n <- labour$growth
delta <- 0.041
g <- 0.016
s <- 0.274
k <- rep(NA, 7)
y <- rep(NA, 7)
kappa <- (s/(n[1] + delta))^(1/(1-alpha))
k[1] <- kappa
y[1] <- k[1]^alpha
for (date in 2:7){
  y[date] <- k[date-1]^alpha
  k[date] <- (1-delta-n[date])*k[date-1]+s*y[date]
}
data.frame(date = date[1:7], k ) %>% melt(id = "date") %>% 
  ggplot(aes(x = date, y = value, colour = variable,group = 1)) +
  geom_line() +
  ylim(0, 18)
## Warning: Removed 6 rows containing missing values (geom_path).

graph tweet

Part 3: A labour force shock

s <- 0.25
delta <- 0.05
alpha <- 0.40
k <- rep(NA, 200)
k[1] <- 1
kfall <- rep(NA, 200)
kfall[1] <- 1
y <- rep(NA, 7)
yfall <- rep(NA, 200)
c <- delta + n
n <- 0.02
nfall <- n*0.97
y[1] <- k[1]^alpha
yfall[1] <- kfall[1]^alpha
for (t in 2:200) {
  y[t] <- k[t-1]^alpha 
  k[t]<- (1-delta-n)*k[t-1]+s*y[t]
  yfall[t] <- kfall[t-1]^alpha
  kfall[t] <- (1-delta-nfall)*kfall[t-1]+s*y[t]
}
data.frame(t = 1:200, k, y, kfall, yfall ) %>% melt(id = "t") %>% 
  ggplot(aes(x = t, y = value, colour = variable)) +
  geom_line() +
  ylim(1, 9)

  • What happens to output per worker?

Output per worker is constant. If the K/L ratio remains constant, output per worker is also constant, even after a permentant 3% decrease in the labour force.

  • What happens to output per person?

Output per person (y) reduces, but by less than 3%. This is because the Labour force is a smaller than the total population, therefore a 3% reduction in the labour force leads to a reduction in output but less than proportional. - Not graphed

  • What happens to the children?

R.I.P