Solow Model

Part 1: Solow in Solow’s words

  • What is Solow’s point? Solow write an aggregate production function which considers the relationship between output (Q), capital (K), labour (L) and technical change [A(t)]. Solow contends that the aggregate production function has the capacity to allow us to separate the variables, which impact the level of per capita output, to form the overall level of production. Namely, distinguishing between the impact of technological change such as an increase in human capital, infrastructure or its efficiency and the impact of differing levels of capital per worker. Solow points out the increase on capital to boost up investment could just lead to growth in output in short run. The most significant driving force to sustain long-term growth in output is technical change. Q=A(t).F(K,L)

  • What does Solow mean by neutrality? What is Hicks Neutrality? What is Harrod Neutrality? “Solow neutral” means the relative input shares “LFL/K.Fk” remain the same for a given labour output ratio. Solow neutrality implies the production function can be written as: Y = F [KT (t), L] (capital-augmenting form) Where F/F is independent of K and L, shifts in the production function are neutral. “Hicks neutral” means the ratio of marginal products remain the same for a given capital labour ratio. Hicks neutrality implies the production function can be written as: Y = T (t)F (K, L) “Harrod neutral” means the relative input shares K.Fk/LFL remain the same for a given capital output ratio Y = F [K, LT (t)] (labor-augmenting form) Is his model different to the one we talked about in class? The model is quite similar to the one we talked in class, but with some key distinctions. In class, we assume that the change in the degree of technical progress could be equal to growth in output, simplifying the theory making it easier to understand. That is ˙A/A= g. In Solow’s reading, he concluded that growth equals to change in technical progress plus “the derivative of capital over output function” times “the change in capital”. That is g=˙A/A+w.˙k/k. This version is more complicated, but provides perhaps a more realistic picture, more compatible with real world experience. What data does he use to test his hypothesis? Solow used non-farm GNP of US 1909-1949, the capital stock data 1909-1949 and hold the labour and capital constant to test the hypothesis.

  • Does Solow Find a constant rate of technological progress in the United States? The rate of technical progress was almost constant but not quite, the average rate of progress from 1909-29 is 0.9 per cent per year. However the average rate of progress from 1930-49 then rises to 2.25 per cent per year. This may be partially attributed to changes in the economic conditions caused by the cause by the Second World War and post-war economic recovery.

  • Does Solow find technological progress to be neutral? The technological progress is usually neutral but there was still some outliers. Form 1943-49, the technological progress appeared to not following the trend. Solow thoughts WWII and post-war development has some effect on the data. However the technological progress is appeared neutral to a large extent. Technological progress was affected by capital and investment.

  • What does chart 4a show? Chart 4 shows the relationship between the output per worker divided by the rate of technical progress and that divided by capital per worker. Showing a positive correlation between these two variables. Chart 4a tests the equation of the Cobb-Douglas function.

  • What is “saturation”? Saturation means the gross marginal product of capital falls to depreciation rate and lacks capital to increase investment. Leading to zero net marginal productivity. In a saturation state the economy and GNP would be slowing down.

Reference http://www.econ.boun.edu.tr/hatipoglu/ec308/lecture3.pdf

Question 2: Simulating shocks in a Solow model

\[ Y_{t} = K_{t-1}^{\alpha}L_{t}^{1-\alpha} \] \(s = 25\%\)

\(\delta = 5\%\)

\(n = 2\%\)

\(\alpha = 40\%\)

\[k* = \left(\frac{s}{\delta + n}\right)^\frac{1}{1 - \alpha}\] \[k* = \left(\frac{.25}{0.05 + 0.02}\right)^\frac{1}{1 - .40}\] \[k* = 8.3445569137\]

\[y* = k^\alpha\] \[y* = 8.3445569137^{0.40}\] \[y* = 2.3364759358\]

s <- 0.25
delta <- 0.05
n <- 0.02
alpha <- 0.40
k <- rep(NA, 200)
K <- rep(NA, 200)
k[1] <- 1
y <- rep(NA, 200)
c <- delta + n
for (t in 2:200) {
  y[t] <- k[t-1]^alpha 
  k[t]<- (1-delta-n)*k[t-1]+s*y[t]
}
plot.ts(k)

data.frame(t = 1:200, k[1], k,s*k^alpha,c*k ) %>% melt(id = "t") %>% 
  ggplot(aes(x = t, y = value, colour = variable)) +
  geom_line() +
  ylim(0, 12)

k levels out at steady state (8.3445569137)

This occurs when \(( \delta +n)\times\kappa = s\times \kappa^\alpha\)

  • Now initialise k1 to be equal to the steady state value that you derived above.

\[\kappa = \left(\frac{s}{n + \delta}\right)^\left(\frac{1}{1-\alpha}\right)\] \[k[1] = \kappa\] \[k[1] = 8.3445569137\]

s <- 0.25
delta <- 0.05
n <- 0.02
alpha <- 0.40
k <- rep(NA, 200)
K <- rep(NA, 200)
kappa <- (s/(n + delta))^(1/(1-alpha))
k[1] <- kappa
y <- rep(NA, 200)
for (t in 2:200) {
  y[t] <- k[t-1]^alpha 
  k[t]<- (1-delta-n)*k[t-1]+s*y[t] 
}


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

k starts at, and maintains steady state level

  • Change in savings rate \(s[1] = 25\%\) \(s[2:200] = 30\%\)
s <- rep(NA, 200)
s[1] <- 0.25
s[2:200] <- 0.3
delta <- 0.05
n <- 0.02
alpha <- 0.40
k <- rep(NA, 200)
K <- rep(NA, 200)
k[1] <- 1
y <- rep(NA, 200)
y[1] <- k[1]^alpha
for (t in 2:200) {
  y[t] <- k[t-1]^alpha 
  k[t]<- (1-delta-n)*k[t-1]+s[t]*y[t] 
}
yMap = data.frame(y, y[1])
plot.ts(yMap)

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

Labour Augmenting Technological Change

\(A = 1.5\%\)

A <- rep(NA, 200)
A[1] <- 1
g <- 0.015
s <- 0.25
delta <- 0.05
n <- 0.02
alpha <- 0.40
k <- rep(NA, 200)
K <- rep(NA, 200)
k[1] <- 1
y <- rep(NA, 200)
for (t in 2:200) {
  y[t]<- k[t-1]^alpha*A[t-1]^(1-alpha) 
  A[t] <- A[t-1]+A[t-1]*g 
  k[t]<- (1-delta-n)*k[t-1]+s*y[t] 
}
plot.ts(k)

data.frame(t = 1:200, k[1], k) %>% melt(id = "t") %>% 
  ggplot(aes(x = t, y = value, colour = variable)) +
  geom_line() +
  ylim(0, 120)

  • Now initialise k1 to be equal to the steady state value that you derived above.

\[\kappa = \left(\frac{s}{n + \delta}\right)^\left(\frac{1}{1-\alpha}\right)\] \[k[1] = \kappa\] \[k[1] = 8.3445569137\]

A <- rep(NA, 200)
A[1] <- 1
g <- 0.015
s <- 0.25
delta <- 0.05
n <- 0.02
alpha <- 0.40
k <- rep(NA, 200)
K <- rep(NA, 200)
kappa <- (s/(n + delta))^(1/(1-alpha))
k[1] <- kappa
y <- rep(NA, 200)
for (t in 2:200) {
 y[t]<- k[t-1]^alpha*A[t-1]^(1-alpha) 
  A[t] <- A[t-1]+A[t-1]*g 
  k[t]<- (1-delta-n)*k[t-1]+s*y[t] 
}


data.frame(t = 1:200, k[1], k) %>% melt(id = "t") %>% 
  ggplot(aes(x = t, y = value, colour = variable)) +
  geom_line() +
  ylim(0, 120)

  • Change in savings rate \(s[1] = 25\%\) \(s[2:200] = 30\%\)
A <- rep(NA, 200)
A[1] <- 1
g <- 0.015
s <- rep(NA, 200)
s[1] <- 0.25
s[2:200] <- 0.3
delta <- 0.05
n <- 0.02
alpha <- 0.40
k <- rep(NA, 200)
K <- rep(NA, 200)
k[1] <- 1
y <- rep(NA, 200)
y[1] <- k[1]^alpha
for (t in 2:200) {
  y[t]<- k[t-1]^alpha*A[t-1]^(1-alpha) 
  A[t] <- A[t-1]+A[t-1]*g 
  k[t]<- (1-delta-n)*k[t-1]+s[t]*y[t]  
}
yMap = data.frame(y, y[1])
plot.ts(yMap)

data.frame(t = 1:200, k[1], k) %>% melt(id = "t") %>% 
  ggplot(aes(x = t, y = value, colour = variable)) +
  geom_line() +
  ylim(0, 160)