Homework: Week 3

Group 4: Maxine Catchlove, Sam Ridsdale, Stephanie Schott, Tim Hawley

August 15th 2015

 

Task 1

 

‘A Contribution to the Empirics of Economic Growth’

by Mankiw, Romer and Weil

 

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 data source is ‘Real National Accounts’ constructed by Summers and Heston, which includes real income, government and private consumption, investment and population, and covers the period 1960-1985. The authors created three samples, the first consisting of 98 countries, the second 75 countries, and the third just the 22 OECD countries at that time, excluding Luxembourg. However they didn’t split the data up into these samples; some countries are included in all three samples.

The first sample excludes oil producers, because the bulk of their GDP comes from the resource extraction and therefore is not value added.

The second sample excludes countries who received a “D” grade for their data from Summers and Heston, and those who had a population of less than one million, since idiosyncratic factors may dominate the determination of their real income. The “D” grade was a result of the countries’ real income figures being based on little primary data.

The third sample consisted only of the 22 OECD countries with a population of more than one million (meaning that the ever-outlying Luxembourg is not included). Two advantages this sample has are that it has both good quality of data and small variation in omitted country-specific factors, however it is a relatively small sample size.

 

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

The assumption is that the rates of saving and population growth are independent of country-specific factors shifting the production function. Making this assumption means that they were able to estimate log income per capita at a given time with ordinary least squares.

They give three reasons for this assumption:

  1. This assumption is made in many models, not only in Solow model.

  2. Making this assumption allows the authors to assess theoretical work which has focused on the relationship between saving, population growth and income and has been used to criticise the Solow model.

  3. Because the model predicts the signs and the magnitude of the coefficients on saving and population growth, they can determine if there are important biases in the estimates obtained using Ordinary Least Squares.

The authors also make the standard assumptions of the original model; that the rates of saving, population growth and technological progress are exogenous; the two inputs, \(K\) and \(L\), are paid their marginal products; \(L\) and \(A\) grow exogenously at rates \(n\) and \(g\); and \(g\) and \(\delta\) are constant while \(A\) differs across countries

 

Why would population growth be correlated with the technology available to a country?

Technology improves agricultural productivity and creates a boom in available resources. This provides sustenance for a larger population. Technological progress in medicine reduces the death rates while also improving the birth rate, and increases lifespans. Subsequently, the potential for further technological progress increases, because there are more people available to research and create new technology, and the chances of discovering new technology increases with the number of people doing research.

 

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

The authors add human-capital accumulation to the model, thus making the production function:

 

\(Y{t} = K{t}^{\alpha}H{t}^{\beta}(A{t}L{t})^{1-\alpha-\beta}\)

 

where H is the stock of human-capital. They assume that one unit of consumption can be transformed costlessly into one unit of physical or human capital, and both depreciate at the same rate.

They also take into account human-capital investment in the form of education.

They use a proxy for the rate of H-K accumulation. It measures the percentage of the working-age population in secondary-school. They multiply this rate by the fraction of the working-age population that is of school-age (15-19). This new variable is imperfect. The age ranges in the two series are not the same, it does not include the input of teacher and ignores primary and higher education, taking into account only secondary school.

Making this inclusion helps because they found that it improves the performance of the Solow model. The human-capital measures entered significantly in the 3 samples. The log of the investment + the log of \((n + g + \delta)\) + the percentage of the population in secondary school explain 80% of the cross-country variation in income per capita in their first sample. Moreover, it reduces the size of the coefficient on physical capital investment and improves the fit of the regression compared to their first results (without human-capital).

The inclusion of human capital also helps because it explains results that appear anomalous from the vantage point of the textbook Solow model. This augmented model predicts a slower rate of convergence than the un-augmented model. Accumulation of physical K and population growth have greater impacts on income when accumulation of human K is taken into account and omitting the human-capital biases the coefficients on saving and population growth. The augmented model accounts for 80% of the cross-country variation in income

 

Does conditional convergence hold?

Yes, in the three samples the coefficient on the initial level of income was negative, which is evidence of convergence, and the inclusion of investment and population growth rates improved the fit of the regression. The measure of human capital lowered the coefficient on the initial level of income. It also improved the fit of the regression.

After controlling the variables that the Solow model says determine the steady state, there is substantial convergence in income per capita at the rate predicted.

 

Task 2

 

The following models use data mostly taken from Australian System of National Accounts, 2013-14 by the ABS, available here.

 

Part 1: Estimating the depreciation rate in Australia

 

To estimate the depreciation rate we have used ‘All sectors ; Gross fixed capital formation: Chain volume measures’ as a measure of investment from June 1960 until June 2014 and we have used the June 2014 value of ‘Non-financial - Produced assets: Volume measures’ as a measure of the current capital stock, which is $5,096.3 billion.

 

hist_inv <- read.csv("C:/Users/Maxine/Desktop/ECO3EGS/hist_inv.csv")

inv_flip <- hist_inv %>% arrange(n():1)

hist_inv <- hist_inv[[2]]

current_k <- 5096.3

 

We then created two functions.

 

The first function will, given some depreciation rate, find the difference between the ABS estimate of current capital stock and an estimate calculated using historical investment:

k_diff_estimator <- function(depreciation, hist_inv, current_k){
  
  t <- length(hist_inv)
  
  k_hat <- sum(hist_inv/((1+depreciation)^(0:(t-1))))
  k_diff <- abs(current_k - k_hat)
  
  return(k_diff)  
  
}

 

The second function will also find this difference, but it will return an estimate of the depreciation rate, given the difference:

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

 

We can then use this function with R’s optim function:

optim(0.07, dep_estimator, lower = 0, upper = 0.2, series = inv_flip$Investment/1000, finalvalue = current_k, method = "Brent")
## $par
## [1] 0.04118641
## 
## $value
## [1] 0.0001000624
## 
## $counts
## function gradient 
##       NA       NA 
## 
## $convergence
## [1] 0
## 
## $message
## NULL

 

The above output tells us that the depreciation rate is approximately 4.1%.

 

We can then use this to find the difference between our estimate of capital stock and the ABS estimate:

depreciation <- 0.041

k_diff_estimator(depreciation, hist_inv/1000, current_k)
## [1] 2770.023

 

The estimated difference is approximately $2.8 billion.

 

Part 2: Balanced growth output in Australia

 

To find the steady state output per worker and per person for 2015 we have created a model with parameters calculated using data from the ‘National Income Account’. The model shows the change in the steady rate from 2009 to 2015.

 

To determine an average savings rate we used the ratio of Gross Fixed Capital to Gross Domestic Product from 2009 to 2014.

 

To determine the capital share of income, we assumed that it can be calculated by finding the ratio of Gross Operating Surplus to the difference of Gross National Income and Gross Mixed Income, which has remained constant at around 0.4 for the period.

 

Additionally assuming that the depreciation rate over the period was as calculated above and there was a technical growth rate of 1.6% gives us the following parameters:

 

Savings rate, \(s = 26.1\%\)

Capital Share of Income, \(\alpha = 40\%\)

Depreciation rate, \(\delta = 4.1\%\)

Technical Change, \(g = 1.6\%\)

 

The last parameter needed is the population growth rate, which we assumed to be given by the growth rate of the total number of the labour force for each period, as counted by the ABS. This gives us figures ranging from 1.2% to 2.2%

 

Finally, assuming the standard production function, \(Y_{t} = K_{t-1}^{\alpha}(A_{t}L_{t})^{1-\alpha}\) we can derive the steady state of capital per worker and output per worker to be given by these two equations:

 

(1)     \(k* = (\frac{s}{\delta + n + g})^{1/(1-\alpha)}\)

(2)     \(y* = A \times (k*^{\alpha})\)

 

Therefore we can calculate the steady state value by first, calculating the per worker capital and output levels for 2009 through to 2015 with the following model:

 

LF <- read.csv("C:/Users/Maxine/Desktop/ECO3EGS/LF.csv")

n <- LF[[3]]

s <- 0.274
a <- 0.4
d <- 0.041
g <- 0.016

y <- rep(NA, 7)
k <- rep(NA, 7)
A <- rep(NA, 7)

A[1] <- 1
k[1] <- (s/(n[1]+d+g))^(1/(1-a))
y[1] <- A[1]^(1-a)*k[1]^a

for (t in 2:7){

A[t] <- A[t-1]*exp(g*t)  
y[t] <- A[t]^(1-a)*k[t-1]^a
k[t] <- (1-d-n[t])*k[t-1]+s*y[t] 

}

out <- data.frame(y,k)

out
##          y        k
## 1 2.338646 8.363943
## 2 2.383981 8.538113
## 3 2.473950 8.734027
## 4 2.594232 8.975555
## 5 2.751653 9.230370
## 6 2.947625 9.509391
## 7 3.190290 9.781349

 

 

Then we can use the final values of y and k to calculate the steady state level of each variable:

 

k_S <- (s/(n[7]+d+g))^(1/(1-a))
y_S <- A[7]*(k_S^a)

y_S
## [1] 3.519729

 

The 2015 steady state level of output per worker is 3.519729.

 

Output per person can be calculated by multiplying output per worker by the 2015 participation rate, which the ABS puts at 65.1%:

\(3.190290 \times 0.651 = 2.076879\)

 

The steady state level of output per person is:

\(3.519729 \times 0.651 = 2.291344\)

Part 3: A labour force shock

 

The result of yupsters causing a polio epidemic in 2009 which permanently decreased the participation rate by 3% would change the model to:

 

k2 <- rep(NA, 7)
y2 <- rep(NA, 7)

n2 <- n*0.97

A[1] <- 1
k2[1] <- (s/(n2[1]+d+g))^(1/(1-a))
y2[1] <- A[1]^(1-a)*k[1]^a

for (t in 2:7){

A[t] <- A[t-1]*exp(g*t)  
y2[t] <- A[t]^(1-a)*k2[t-1]^a
k2[t] <- (1-d-n2[t])*k2[t-1]+s*y2[t] 

}

out2 <- data.frame(y,y2,k,k2)

out2
##          y       y2        k       k2
## 1 2.338646 2.338646 8.363943 8.472110
## 2 2.383981 2.396266 8.538113 8.647586
## 3 2.473950 2.486590 8.734027 8.844792
## 4 2.594232 2.607343 8.975555 9.087339
## 5 2.751653 2.765310 9.230370 9.343662
## 6 2.947625 2.962043 9.509391 9.624706
## 7 3.190290 3.205709 9.781349 9.900033

 

What happens to output per worker?

Output per worker increases after the labour shock, due to the capital stock remaining the same.

 

 

What happens to output per person?

Output per person decreases to:

\(3.205709 \times 0.621 = 1.990745\)

 

What happens to the children?

The children either contract polio, become orphans or suffer from a relatively lower standard of living. As a result, a town hall meeting is held: