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very helpful for students gives an understanding of economics
Typology: Study notes
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Genc AUS
Consumption at the golden rule of capital level is given by largest distance between the LRSS output depreciation (and investment) function as shown in the following figure: At this level, we obtain the following representation of consumption:
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¿ Scenario: Too much capital
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does not have a tendency to move toward the Golden Rule steady state. Achieving
Remember: Policymakers can affect the national saving rate:
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d c ¿ d k
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Step-by-Step Transition Dynamics Step 1: Reduction in Savings Rate (s ↓)
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Step 2: Capital Dynamics
Step 3: Output and Investment Decline
Step 4: Consumption Dynamics
R Program to Visualize the Transition You may use the following program to generate the transition dynamics. I also provide the output of this program in the form of an attached Excel file.
library(ggplot2)
f <- function(k, alpha = 0.3) { return(k^alpha) }
delta <- 0. s_initial <- 0. s_final <- 0. alpha <- 0. k_gold <- (alpha / delta)^(1 / (1 - alpha))
time <- seq(0, 50, by = 1) k <- numeric(length(time)) y <- numeric(length(time)) i <- numeric(length(time)) c <- numeric(length(time))
k[1] <- k_gold * 1.
for (t in 2:length(time)) { s <- ifelse(t < 5, s_initial, s_final) k[t] <- k[t-1] + s * f(k[t-1], alpha) - delta * k[t-1] y[t] <- f(k[t], alpha) i[t] <- s * y[t] c[t] <- y[t] - i[t] }
df <- data.frame( Time = time, Capital = k, Output = y, Investment = i, Consumption = c )
ggplot(df, aes(x = Time)) + geom_line(aes(y = Capital, color = "Capital")) + geom_line(aes(y = Output, color = "Output")) + geom_line(aes(y = Investment, color = "Investment")) + geom_line(aes(y = Consumption, color = "Consumption")) + labs(title = "Transition to Golden Rule Capital", y = "Per Worker Values", color = "Variable") + theme_minimal()
