Homework: Week 2
Group 4: Maxine Catchlove, Sam Ridsdale, Stephanie Schott, Tim Hawley
Part 1: Solow in Solowâs words
What is Solowâs point?
Solow’s point is to provide a mechanism for an aggregate production function that allows for the segregation of shifts in the production function from movements along the function. This was done in order to determine whether variations in output were either determined by technical change or the availability of capital per worker. This was done by building on older models with the use of an additional time series, the share of labour or property in total income and an additional assumption that factors are paid their marginal products (according to Solow this could be extended to monopolistic factor markets as well).
What does Solow mean by neutrality? What is Hicks Neutrality? What is Harrod Neutrality?
Neutral technical change refers to shifts in the production function that increase or decrease output attainable from given inputs while maintaining the same marginal rate of substitution. Solow’s neutrality then implies a production function of Y= F {K(t),L}, a production function is neutral when F is independent labour and capital.
Hicks neutrality refers to a shift that does not affect the balance of capital and labour in the production function. For example, in a production function Y=A*F(K,L), only A changes.
Harrod neutrality refers to the increase in efficiency of all factors proportionally for a given capital output ratio, more specifically Harrod focused on the increased efficiency of labour brought on by technological progress spurring economic growth.
Is his model different to the one we talked about in class?
Solow’s model uses the derivative of capital given output times the change in capital to determine growth over time. The model introduced to us previously considered growth in output to only be a product of technical change. Solow’s model is more complicated but perhaps more fitting, particularly when looking at pre and post-war data!
What data does he use to test his hypothesis?
Solow uses American data dating from 1909 to 1949, although he states that real net national product is the most accurate data for production functions he must settle for GNP data from that period. He also expresses the benefit of annual flow of capital services but again must settle for the less utopian estimate of capital stock in existence. Solow sources labour force figures from Douglas and the Economic Almanac. He uses a time series data of real private non-farm GNP and employed capital per man hour from Kendrick. Solow also uses Goldsmith estimates of 1909-1949 (eliminating government, agriculture and consumer durables). He creates measures of capital utilization by reducing Goldsmith figures by the unemployment rate and the share-of capital series built on a number of sources and assumptions.
Does Solow find a constant rate of technological progress in the United States?
No,The data in chart two fluctuates throughout the period of the end of World War 1 and the stock market crash of 1929. Interestingly the rate of technical progress is quite high throughout WW2 and drops after 1945. The overall upward shift of technical progress is around one per cent for the first half of the data and increases to two per cent for the second half that averages out at 1.5% over the data. Solow attributes 87.5% of the increase in output per man hour to technical change and 12.5% to increased use of capital.
Does Solow find technological progress to be neutral?
On average technical change during the 1909-1949 period was neutral but with some outliers undoubtedly caused by the economic climate of war. This is most striking during World War 2 in 1943-1945 and in the years immediately following the war.
What does chart 4a show?
Chart 4 demonstrates persistent but non-violent diminishing returns of capital given a level of technical progress. Chart 4a is a linear regression function used to apply to the data in chart 4 to create a curve using parametric forms.
What is âsaturationâ?
Solow refers to ‘capital saturation’ when analysing the production function. The kind of capital saturation he is referring to is a situation in which the marginal product of capital falls to zero, meaning that depreciation of capital is greater than production. Solow uses gross marginal product as an indicator of capital saturation. Saturation would occur if the gross marginal product fell below .03-.05 which is the marginal rate of depreciation.
Part 2: Simulating shocks in a Solow model
Production Function:
\(Y_{t} = K_{t-1}^{\alpha}L_{t}^{1-\alpha}\)
Parameters:
Savings rate, \(s = 25\%\)
Depreciation rate, \(\delta = 5\%\)
Population growth, \(n = 2\%\)
Capital Share of Income, \(\alpha = 40\%\)
Variables:
Capital per worker, \(k\)
Output per worker, \(y\)
Deriving the steady state values for each variable:
(1) \(sk^{\alpha} - (\delta + n)k = 0\)
(2) \((\delta + n)k = sk^{\alpha}\)
(3) \(k^{1-\alpha} = \frac{s}{\delta + n}\)
(4) \(k* = (\frac{s}{\delta + n})^{1/(1-\alpha)}\)
Therefore,
\(k* = (\frac{0.25}{0.05 + 0.02})^{1/(1-0.4)}\)
\(k* = 8.344556914\)
(5) \(y = \frac{Y}{L}\)
(6) \(y = \frac{K^{\alpha}L^{1-\alpha}}{L}\)
(7) \(y = \frac{K^{\alpha}}{L^{\alpha}}\)
(8) \(y = k^{\alpha}\)
Therefore,
\(y* = (k*)^{\alpha}\)
\(y* = 8.344556914^{0.4}\)
\(y* = 2.336475936\)
Model 1: Initial value of capital per worker = 1
s <- 0.25
d <- 0.05
n <- 0.02
a <- 0.4
k <- rep(NA,200)
y <- rep(NA,200)
k[1] <- 1
y[1] <- k[1]^a
for (t in 2:200){
y[t] <- k[t-1]^a
k[t] <- s*y[t]+(1-d-n)*k[t-1]
}
plot.ts(k)
plot.ts(y)
We observe convergence to the steady state values of k and y calculated earlier.
Model 2: Initial value of capital per worker = steady state value
SS <- (s/(n+d))^(1/(1-a))
k[1] <- SS
y[1] <- k[1]^a
for (t in 2:200){
y[t] <- k[t-1]^a
k[t] <- s*y[t]+(1-d-n)*k[t-1]
}
plot.ts(k)
plot.ts(y)
Since the initial value of k was set at the steady state value we observe no change in k or y from period 1 to 200.
Model 3: Inital value of capital per worker = 1, savings rate changes from 25% to 30% in period 2
k1 <- rep(NA,200)
y1 <- rep(NA,200)
s1 <- rep(0.3,200)
s1[1] <- 0.25
k1[1] <- 1
y1[1] <- k1[1]^a
for (t in 2:200){
y1[t] <- k1[t-1]^a
k1[t] <- s1[t-1]*y1[t]+(1-d-n)*k1[t-1]
}data.frame(t = 1:200, k1, k) %>% melt(id = "t") %>%
ggplot(aes(x = t, y = value, colour = variable)) +
geom_line() +
ylim(0, 12.5)
data.frame(t = 1:200, y1, y) %>% melt(id = "t") %>%
ggplot(aes(x = t, y = value, colour = variable)) +
geom_line() +
ylim(0, 3)
We observe that an increase in the savings rate increases the steady state values of k and y. This is due to a higher level of investment.
Note: Variables k and y in the two graphs above are the values calculated in Model 1 where the initial value of capital per worker was 1, not the steady state value.
Part 3: Introducing Labour-Augmenting Technical Change
Model 1 (with Labour-Augmenting Technical Change): Initial value of capital per worker = 1
g <- 0.015
A <- rep(NA,200)
A[1] <- 1
k[1] <- 1
y[1] <- k[1]^a*A[1]^(1-a)
for (t in 2:200){
A[t] <- A[t-1]+A[t-1]*g
y[t] <- A[t]*(s/(d+n+g))^(a/(1-a))
k[t] <- s*y[t]+(1-d-n)*k[t-1]
}
plot.ts(k)
plot.ts(y)
Model 2 (with Labour-Augmenting Technical Change): Initial value of capital per worker = steady state value
SS <- (s/(n+d))^(1/(1-a))
k[1] <- SS
y[1] <- k[1]^a*A[1]^(1-a)
for (t in 2:200){
A[t] <- A[t-1]+A[t-1]*g
y[t] <- A[t]*(s/(d+n+g))^(a/(1-a))
k[t] <- s*y[t]+(1-d-n)*k[t-1]
}
plot.ts(k)
plot.ts(y)
Model 3 (with Labour-Augmenting Technical Change): Initial value of capital per worker = 1, savings rate change
k1 <- rep(NA,200)
y1 <- rep(NA,200)
s1 <- rep(0.3,200)
s1[1] <- 0.25
k1[1] <- 1
y1[1] <- k1[1]^a*A[1]^(1-a)
for (t in 2:200){
A[t] <- A[t-1]+A[t-1]*g
y1[t] <- A[t]*(s1[t-1]/(d+n+g))^(a/(1-a))
k1[t] <- s1[t-1]*y1[t]+(1-d-n)*k1[t-1]
}data.frame(t = 1:200, k1, k) %>% melt(id = "t") %>%
ggplot(aes(x = t, y = value, colour = variable)) +
geom_line() +
ylim(0, 165)
data.frame(t = 1:200, y1, y) %>% melt(id = "t") %>%
ggplot(aes(x = t, y = value, colour = variable)) +
geom_line() +
ylim(0, 50)