Part 1: Solow in Solow’s words

Read Technical Change and the Aggregate Production Function (Solow, 1957). This is a pretty tough paper for undergrads, but I have faith that you will be able to manage it. You’ll need to read it several times. On a first reading, don’t let yourself get too stuck with the mathematics. On second/third readings, you should spend time to make sure you understand the model. It is an extremely important paper-it more or less kicked off all of contemporary macroeconomics, and Solow won the 1987 Nobel for his efforts.

Questions:

What is Solow’s point?
Prior models of the Aggregate Production Function were flawed because they didn’t differentiate between shifts and movements along the function. That is, previous models did not identify whether the rate of change in aggregate production was attributable to increase in available capital or labour, a movement along the model, or due to a technical change in the market, a shift in the model. He suggests that by identifying quantitative time series values for the market output with the respect to labour, the unit relation between capital and labour and the share of capital in the market and plugging these values into his derived equations (2A) or (2B) as appropriate, one can approximate the percentage change in aggregate production attributable to technical change in the economy.

What does Solow mean by neutrality? What is Hicks Neutrality? What is Harrod Neutrality?
Neutrality in Solow’s article refers to any technical change in the aggregate production function which occurs without a corresponding change in the marginal rates of technical substitution for capital and labour.

Hicks neutrality refers to a technical change which is not dependent on the marginal rates of technical substitution between inputs, in this case, labour and capital. Blackorby, Lovell and Thursby (1976) argue that Hick’s neutrality comprises two differentiable concepts; implicit Hick’s neutrality (IHN) and extended Hick’s neutrality (EHN). They suggest that IHN is characterised by an independence of the marginal rates of technical substitution and technical change at constant factor proportions (Define). EHN is alternately characterised by capacity for the production function to be explicitly differentiated into two independent functions; a technical change function and a function of capital and labour inputs. Blackorby et al. (1976) further argue that EHN is not always proved by proving IHN. This problematizes Solow’s equation (1A) which relies on EHN. Ultimately, however, Hick’s neutrality refers to technical change in the economy which increases the efficiency across all factors equally.

Alternately, Harrod neutrality refers to a change in the labour force which increases the efficiency of this input disproportionately to the efficiency of capital. This kind of technical change is therefore effective in reducing necessary labour inputs.

Is his model different to the one we talked about in class?
Yes. The Solow model from the paper aims to differentiate and quantify the percentage change in the aggregate production function as a result of technical change. It does this using real data representing the output generated by each unit of labour, the unit relation of capital and labour and the amount of capital understood to exist in the economy.

The model discussed in the lecture has a different aim and uses different inputs of real data. Specifically, the model aims to identify the percentage change in capital per unit labour. It does this using real economic data describing the capital’s share in output, the savings rate, the population growth rate, the depreciation rate and the initial level of capital for each worker. Therefore, the lecture model has a different aim and uses different data.

What data does he use to test his hypothesis?
Solow uses real economic data from the US in the period 1909-1949. Due to the availability of this data, he uses a gross national product (GNP) time series to represent output per unit of labour. Though Solow notes that a net national product (NNP) would be a more appropriate data input, as it would address the issue of depreciation inherent in GNP data.

To represent the amount of available capital per unit of labour each time point, Solow uses estimates by Goldsmith which aim to quantify how much capital exists at the time. Solow adapts these estimates to exclude public and agricultural data because the efficiency ratios of capital and labour in these areas are expected to differ significantly from the same ratio of the private non-agricultural sector. Solow believes this heterogeneity will impact the integrity of the model. Furthermore, he notes that Goldsmith’s estimates are still a flawed data source as they measure the amount of capital that exists, including capital that may not be in use. He attempts to address this by reducing the data for each time point by the percentage of unemployment in the respective year, arguing that impacts on labour may be an indicator of impacts on capital. Solow notes that this is an imperfect measure.

Finally, to represent the share of capital in the economy, again using private, non-agricultural data, Solow draws information from various published estimates of the share of property in income.

Does Solow find a constant rate of technological progress in the United States?
Solow finds that technical change is relatively consistent over time, with an average 1.5% increase annually across the 40 year period. However, in a post-hoc analysis he identifies a divergence in the data. He notes that in the first 21 years, the average rate of technical change was approximately 0.9% annually, and that after 1929, this increases to approximately 2.25% per year. Solow is however, reticent to differentiate these periods based solely on his post-hoc analysis and instead concludes that the rate is constant over time with minor fluctuations around 1.5% mean.

Does Solow find technological progress to be neutral?
In his applied data, Solow notes that there appears to be no correlation between the capital and labour relation and technical change in the economy during the 1909-1949 period. He therefore argues that, on aggregate, this period can be considered to be a period of neutral technical change.

What does chart 4a show?
The scatterplot in Chart 4a shows the relation of output over time divided by technical change over time against the changes in capital over time. For the period 1909-1943, a strong relationship appears to exist between the factors, though between 1943 and 1949, there is a significant shift in this relationship which Solow himself describes as a “mystery.”

The chart appears to suggest there are diminishing returns in the amount of output divided by the amount of technical change in relation to capital. However, the plot does not show a point of capital saturation. Further, a number of different models, including a linear model, fit the data quite well. If a linear model was selected, the capacity for the model to predict capital saturation would be removed.

What is “saturation”?
Saturation refers to a state in the economy at which the gross marginal product of capital falls to the marginal rate of depreciation. At this point further increases in capital will only act to counter the rate of depreciation and will not result in significant increases in output. In this state, the only way to increase aggregate production is through technical change.

References

Blackorby, C., Lovell, C. A. & Thursby, M. C. (1976). Extended Hicks neutral technical change. The Economic Journal, 86(344): 845-852.

Part 2: Simulating shocks in a Solow model

Parameters

Savings rate; s = 0.25
Depreciation rate; \(\delta\) = 0.05
Population growth; n = 0.02
Capital Share of Income; \(\alpha\) = 0.4

Variables

Output = Y
Output per worker = y
Capital = K
Capital per worker = k

At a steady state (derived analytically)

Capital per worker:

\[k = \frac {s}{(\delta + n)} ^{1/(1-\alpha)}\]

\[k = \frac {0.25}{(0.05 + 0.02)} ^{1/(1-0.4)}\] \[k = 8.344\]

Output per worker:

\[ y = \frac {Y}{L} \] \[ y = \frac {K^\alpha*L^{1-\alpha}}{L} \] \[ y = \frac {K^\alpha}{L^\alpha} \] \[ y = K^\alpha\]
\[ y = 8.344^{0.4} \] \[ y = 2.34\]

Model 1: set intial k = 1

# Initialise the exogenous parameters
s <- 0.25
delta <- 0.05
n <- 0.02
alpha <- 0.4

#Initialise two variables
k <- rep(NA, 200)
y <- rep(NA, 200)

#Initialise k[1]=1
k[1] <-1
y[1] <-k[1]^alpha

#Simulate the model
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)

plot.ts(y)

As expected, the model converges with the steady state value of k.

Model 2: Set initial k = steady state

s <- 0.25
delta <- 0.05
n <- 0.02
alpha <- 0.4

k <- rep(NA, 200)
y <- rep(NA, 200)

#Initialise k[1] = steady state value for k
k[1] <-(s/(n+delta))^(1/(1-alpha))
y[1] <-k[1]^alpha

#Simulate model (k[1] = steady state k)
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)

plot.ts(y)

The model is restrained by the limit of the steady state. It is a rising model that has reached its upper limit - the steady state value k, therefore it can not increase further and output becomes a constant.

Model 3: Savings rate of 0.3 from the second period - initial simulation.

delta <- 0.05
n <- 0.02
alpha <- 0.4

#Initialise three news time series; k1, y1 & s1 representing capital per worker, output per worker and savings rating respectively.
k1 <- rep(NA, 200)
y1 <- rep(NA, 200)
s1 <- rep(0.3, 200)

s1[1] <- 0.25
k1[1] <-1
y1[1] <-k1[1]^alpha

#Simulate this model
for (t in 2:200) {
  y1[t] <- k1[t-1]^alpha
  k1[t] <- (1-delta-n)*k1[t-1]+s1[t]*y1[t]
}
plot.ts(k1)

plot.ts(y1)

Model 3: Savings rate of 0.3 from second period. Time Series plot (y, y1)

delta <- 0.05
n <- 0.02
alpha <- 0.4

k <- rep(NA, 200)
y <- rep(NA, 200)

k[1] <-(1)
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*y[t]
}
plot.ts(k)

plot.ts(y)

k1 <- rep(NA, 200)
y1 <- rep(NA, 200)
s1 <- rep(0.3, 200)

s1[1] <- 0.25
k1[1] <-1
y1[1] <-k1[1]^alpha

for (t in 2:200) {
  y1[t] <- k1[t-1]^alpha
  k1[t] <- (1-delta-n)*k1[t-1]+s1[t-1]*y1[t]
  
}

for (t in 2:200) {
   y1[t] <- k1[t-1]^alpha
  k1[t] <- (1-delta-n)*k1[t-1]+s1[t]*y1[t]
  
  df <- data.frame(t = 1:200,y[t],y1[t])


  require(ggplot2)

g<-  ggplot(df, aes(t))                     # basic graphical object
g<- g + geom_line(aes(y=y[t]), colour="red")   # first layer
g<- g + geom_line(aes(y=y1[t]), colour="green")  # second layer
g
}
## Loading required package: ggplot2

g <- g + ylab("Output per worker") + xlab("Time")
g