Solow Growth Model with Technological Change

Luis Alberto Alaniz Castillo
April 27th of 2018

Generating the data

  • The savings rate 20 percent of income.
  • Capital share in output 40 percent.
  • Depreciation rate of capital 10 percent percent per year.
  • Population growth rate 1.8 percent per year.
  • Technological progress growth rate 3.0 percent per year.
s <- 0.2  # Savings rate
alpha <- 0.4  # Capital share in output
delta <- 0.1  # Depreciation rate
n <- 0.018  # population growth rate
a <- 0.03  # Technological progress growth rate

k0 <- seq(0, 6, length.out = 501)
t <- seq(0, 100, length.out = 501)
k0 <- matrix(k0, 501, 1)
t <- matrix(t, 501, 1)

y0 <- k0^alpha
c0 <- (1 - s) * y0

k <- matrix(0L, nrow = length(k0), ncol = length(t))

for (i in 1:501) {
    for (j in 1:501) {
        k[i, j] <- ((k0[i]^(1 - alpha) - (s/(delta + n + a))) * (exp(-(delta + 
            n + a) * (1 - alpha) * t[j])) + s/(delta + n + a))^(1/(1 - alpha))
    }
}

y <- k^alpha

ss <- (s/(delta + n + a))^(1/(1 - alpha))
a1 <- a * 100

Equations and and steady state of the model with technological change.

This is the equation for the transitional dynamics of the capital stock which can be solved analytically: \[ \dot{k} = sk^{\alpha}-(\delta+n+x)k \] This is the equation for the equilibrium path of the capital stock and the analytical solution of the equation above: \[ k = (((k_{0}^{(1-\alpha)}-\frac{s}{(\delta+n+x)})e^{-(1-\alpha)(\delta+n+x)t})+\frac{s}{(\delta+n+x)})^{\frac{1}{(1-\alpha)}} \]

This is the equation for the equilibrium path of output stock : \[ y = k^{\alpha} \]

The steady state of the model is 1.7 at that point all per capita variables grow at the rate 3.

Graph of the evolution of the capital stock through time for an initial capital stock.

M <- mesh(t, k0)
surf3D(x = M$x, y = M$y, z = k, colvar = k, plot = TRUE)

plot of chunk capital stock

Graph of the evolution of output through time for an initial output.

M <- mesh(t, y0)
surf3D(x = M$x, y = M$y, z = y, colvar = y, plot = TRUE)

plot of chunk output