Part I | Solow in Solow’s Words

Read Robert Solow’s (1957) Technical Change and the Aggregate Production Function and answer the following questions, accordingly.

What is Solow’s point?

Solow (1957) examines the effects of input growth and technological progress on output growth, using US data from 1909-1949. Solow aims to segregate the effects of technical change and capital intensity on output, assuming constant returns to scale and perfect competition. He estimates \(\frac{1}{8}\) of the doubling of output per worker, from the period 1909-1949, is attributable to increases in capital intensity and \(\frac{7}{8}\) is attributable to technical change. Thus technological development is required to sustain real output growth.

What does Solow mean by neutrality? What is Hicks Neutrality? What is Harrod Neutrality?

Solow (1957) defines technological progress to be neutral when relative input shares (MRS) remain the same for a given labour-output ratio. This form of progress is capital augmenting as it acts to increase output in the same way as an increase in the stock of capital. A “Solow neutral” production function can be written as:

\[Y(t) = F(A(t)K(t),L(t))\]

Hicks (1932) defines technological innovation as neutral” when the ratio of marginal products remain the same for a given capital labour ratio. A “Hicks neutral” production function can be written as:

\[Y(t) = A(t)F(K(t),L(t))\]

Harrod (1932) defines technological innovation to be neutral if relative input shares are unchanged for a given capital-output ratio. Considered labour augmenting as A acts to increase output in the same way as an increase in the stock of labour. A “Harrod neutral” production function can be written as:

\[Y(t) = F(K(t), A(t)L(t))\]

Is his model different to the one we talked about in class?

Solow’s (1957) modelling is concerned with the capital requirements for economic growth, hence, it is different from other models talked about in class. Note, Robert Solow’s attention to the role of technological progress in real economic growth as well as the limitations of dimishing returns on capital over time (cf. the Financial Gap Model).

What data does he use to test his hypothesis?

Solow uses US non-farm GNP, capital stock, and labour force employment statistics, for the period 1909 - 1949, to test his hypothesis. He further refines the capital stock data by reducing the amount of capital by the fraction of the labour force unemployed each year, in an attempt to account for idle capital. He concedes, however, that his assumption that labour and capital both suffer the same unemployment percentages is wrong, but is a step closer to the true amounts.

Does Solow find a constant rate of technological progress in the United States?

Solow (1957, p. 316) argues ‘the over-all result for the whole 40 years is an average upward shift of about 1.5 per cent per year’ and goes on to claim (as stated above) ‘over the 40 year period output per man hour approximately doubled’. The source of this economic growth - \(\frac{1}{8}\) is attributable to increases in capital investment levels and \(\frac{7}{8}\) is attributable to technical progress.

Does Solow find technological progress to be neutral?

Solow (1957, p. 320, emphasis added), using U.S. data between 1909 to 1949, argues ‘[technological] change during that period was neutral on average’. Notable events within this data set include: World War I (1914-18), The Great Depression (1929-39) and World War II (1939-45).

What does Chart 4a show?

Note, there is no ‘Chart 4a’ in Solow’s (1957) article However, ‘Chart 4’ illustrates the scatter of (Y-axis) output per worker \(q\) divided by technical change \(A\) and (X-axis) capital per worker \(k\). This relationship can be denoted by \(\frac{q}{A}=k\). Chart 4, according to Solow (1957, p. 318), shows ‘persistent but not violent diminishing returns’ of capital per worker. Whereas ‘Equation 4a’ (p. 318), i.e., \(q=\alpha+\beta k\), denotes a linear regression for the variable q used to fit the scattered data within Chart 4.

What is “saturation”?

According to Solow (1957), ‘capital-saturation’ or simply ‘saturation’ is an economic environment where the marginal product of capital approachs zero resulting in limited (if any) growth in real output for additional invested capital. That is, without technological advancement (or another kind of econonomic shock) capital-saturation will place downward pressure on interest rates and upward pressure on consumption.

Part II | Simulating Shocks within a Solow Model

1 - Initialise the exogenous parameters for the baseline model: The savibngs rate is 25%, depreciation is 5%, population growth rate is 2%, capital share of income is 40% and GDP comes from a Cobb-Douglas production function $Y_{t}=K_{t-1}^{} L_{t}^{1-}. This is a closed economy (savings = investment).

2 - Derive the steady state capital per worker and output per worker analytically.

3 - Initialise two variables, capital per worker k and output per worker y; these should be 200 periods long. (Hint: use rep(NA, 200))

4 - Initialise k1=1 and simulate the model forwards (updating capital and output) 200 periods. Does it converge?

5 - Now initialise k1 to be equal to the steady state value that you derived above. Simulate again. Any difference?

6 - Now initialise three new series called k1, y1, and s1, where the first two are capital and output per worker, and s1 is a series of savings rates. The first element of s1 is 0.25 (the same savings rate as above). But from the second period onwards, the savings rate is 0.3. Simulate this model.

7 - Generate a time-series plot that contains y and y1. (Hint: create a data frame that contains y and y1, and use plot.ts(yourdataframe))

N.B. (for a bonus mark) re-do all the above, introducing labour-augmenting technical change that grows at a rate of 1.5% per year.

Cobb-Douglas Production Function: \[Y = {K^\alpha}_{t-1} {L^\alpha}_t\] \[k = \frac{K}{L}\] \[y = \frac{Y}{L}\]

Exogenous Paramaters: \[\alpha = 0.4\] \[\delta = 0.05\] \[s = 0.25\] \[n = 0.02\]

# let k[1] <- 1
alpha <- 0.4
delta <- 0.05
n <- 0.02
s <- 0.25
k <- rep(NA,200)
y <- rep(NA,200)
k[1] <- 1
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, main="Captial Per Worker Over Time if k[1] = 1", xlab='time (t)', ylab='capital per worker (k)', ylim=c(0,10))

plot.ts(y, main="Output Per Worker Over Time if k[1] = 1", xlab='time (t)', ylab='output per worker (y)', ylim=c(0,10), col='red')

The two Graphs, see above, indicates capital investment does converge to a ‘steady state’ over time.

\[k^{*} = \left( \frac {s}{\delta+n} \right)^{\frac {1}{1-\alpha}}\] \[k^{*} = \left(\frac {0.25}{0.05 + 0.02}\right)^{\frac{1}{1-0.4}}\] \[k^* = 8.3445569137464\]

\[y^* = (k^*)^\alpha\] \[y^* = 8.3445569137464^{0.4}\] \[y^* = 2.33647593584899\]

# let k[1] <- (s/(n + delta))^(1/(1-alpha))
alpha <- 0.4
delta <- 0.05
n <- 0.02
s <- 0.25
k <- rep(NA,200)
y <- rep(NA,200)
k[1] <- (s/(n + delta))^(1/(1-alpha))
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, main='Captial Per Worker Over Time if k[1] = k*', xlab='time (t)', ylab='capital per worker (k)', ylim=c(0,10))

Clearly, there is a difference between the Graphs with different initial k values.

# let s1 <- rep(30/100, 200) & s1[1] <- 25/100
alpha <- 0.4
delta <- 0.05
n <- 0.02
s1 <- rep(0.3, 200)
s1[1] <- 0.25
k1 <- rep(NA,200)
y1 <- rep(NA,200)
k1[1] <- (s/(n + delta))^(1/(1-alpha))
for(t in 2:200){y1[t] <- k1[t-1]^alpha
  k1[t] <- (1-delta-n)*k1[t-1]+s1[t]*y1[t]}
t <- 1:200
df <- data.frame(y, y1)
# the blue line refers to y1 & the red line refers to y
plot.ts(t, y1, type='l', main='The Relationship between Saving & Output Per Worker', xlab='time (t)', ylab='capital per worker (k)', ylim=c(0,10), col='blue')
lines(t, y, col='red')

Responce to Bonus Question

# returning to the initial asumption, i.e., k[1] <- 1, lets now assume the change in A is 1.5% per year N.B. Jones & Vollrath (2013, p.37) denotes rate of change in A as 'g'.  
alpha <- 0.4
delta <- 0.05
g <- 0.015
n <- 0.02
s <- 0.25
k <- rep(NA,200)
y <- rep(NA,200)
A <- rep(NA,200)
A[1] <- 1
k[1] <- 1
for (t in 2:200) {y[t] <- k[t-1]^alpha*A[t-1]^(1-alpha)
  A[t] <- A[t-1]*(1+g)
  k[t] <- (1-delta-n)*k[t-1]+s*y[t]}
plot.ts(k, main="Captial Per Worker Over Time if k[1] = 1 & g = 0.015", xlab='time (t)', ylab='capital per worker (k)', ylim=c(0,120))

plot.ts(y, main="Output Per Worker Over Time if k[1] = 1 & g = 0.015", xlab='time (t)', ylab='output per worker (y)', ylim=c(0,120), col='red')

Becuase technological progress, i.e., g, constantly positive over the long term this new model does not exhibit capital nor output convergence.

\[k^{*} = \left( \frac {s}{\delta+n+g} \right)^\frac {1}{1-\alpha}\] \[k^{*} = \left(\frac {0.25}{0.05 + 0.02 + 0.015}\right)^{\frac{1}{1-0.4}}\] \[k^{*} = 6.037657\]

\[y^{*} = {k^{*}}^\alpha A^{1-\alpha}\] \[y^{*} = 6.037657^{0.4} 1^{1-0.4}\] \[y^{*} = 2.052803\]

# let k[1] <- (s/(n + delta+ g))^(1/(1-alpha))
alpha <- 0.4
delta <- 0.05
g <- 0.015
n <- 0.02
s <- 0.25
k <- rep(NA,200)
y <- rep(NA,200)
k[1] <- (s/(n + delta + g))^(1/(1-alpha))
for (t in 2:200) {y[t] <- k[t-1]^alpha*A[t-1]^(1-alpha)
  A[t] <- A[t-1]*(1+g)
  k[t] <- (1-delta-n)*k[t-1]+s*y[t]}
plot.ts(k, main='Captial Per Worker Over Time if k[1] = k* & g = 0.015', xlab='time (t)', ylab='capital per worker (k)', ylim=c(0,120))

Again, there is a difference between the Graphs with different initial k values.

# let sII <- rep(0.3, 200), sII[1] <- 0.25 & g <- 0.015
alpha <- 0.4
delta <- 0.05
g <- 0.015
n <- 0.02
s <- 0.25
y <- rep(NA,200)
k[1] <- (s/(n + delta + g))^(1/(1-alpha))
for (t in 2:200) {y[t] <- k[t-1]^alpha*A[t-1]^(1-alpha)
  A[t] <- A[t-1]*(1+g)
  k[t] <- (1-delta-n)*k[t-1]+s*y[t]} 
sII <- rep(0.3, 200)
sII[1] <- 0.25
yII <- rep(NA,200)
kII <- rep(NA,200)
kII[1] <- (s/(n + delta + g))^(1/(1-alpha))
for (t in 2:200) {yII[t] <- kII[t-1]^alpha*A[t-1]^(1-alpha)
  A[t] <- A[t-1]*(1+g)
  kII[t] <- (1-delta-n)*kII[t-1]+sII[t]*yII[t]}
t <- 1:200
A[1] = 1
dfII <- data.frame(y, yII)
# the blue line refers to yII & the red line refers to y
plot.ts(t, yII, type='l', main='The Relationship between Saving & Output Per Worker if g = 0.015', xlab='time (t)', ylab='capital per worker (k)', ylim=c(0,120), col='blue')
lines(t, y, col='red')

Bibliography

Jones, CI & Vollrath, D 2013, Introduction to Economic Growth, 3rd edn, W. W. Norton & Company, New York.

Solow, RM 1957, ‘Technical Change and the Aggregate Production Function’, The Review of Economics and Statistics, vol. 39, no. 3, pp. 312-320.