HomeWork: Week 2

Part One: Technical Change and the Aggregate Production Function (Solow, R 1957)

What is Solow’s Point?

Robert Solow is trying to isolate changes in output per capita to identify whether shifts in the production function are due to technical progress or if variations arise from a change in capital per worker available. Solow interprets technical change as any change or improvement that has an impact on the production function. This would account for any speed ups (or slow downs) in the economy, for example, improvements in the education of the labour force or efficiency of a machine’s production. What this paper suggests is a way to identify the optimal amount of capital per worker in an economy, while also introducing the effect on technical change, testing if the increase in output per worker resulting from technical change affected the marginal rates of substitution.

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

Solow defined neutrality as shifts in the aggregate production function that were the result of pure scale changes, leaving marginal rates of substitution unchanged at given capital/labor ratios.

Hick’s neutrality is defined as the ratio of marginal products that remain the same for a given capital labour ratio. The production function can also be defined as \(Y=A_t F(K,L)\) Technical progress here increases the efficiency of all factors to the same proportion.

Harrod-neutral technical progress increases the efficiency of the labour force, allowing producers to make more with the same number of workers available. On a large scale this can spur economic growth, however can also be a contributing factor to unemployment by lessening the demand for labour. The production function can be defined as \(Y=K^a {A_t L^{1-a}}\).

Is His Model different to the one we talked about in class?

Yes, the production function Solow used originally varies slightly from the one we studied in class. The difference is that Solow’s model applies technological change as a multiplicative factor of both capital and labour, whereas the lecture’s model only applied technological change as a labour augmenting factor, leaving capital unaffected by A.

what data does he use to test his hypothesis?

In order to test his hypothesis, Solow required three time series of data; output per unit of labour \((q)\), capital per unit of labour \((k)\), and the share of capital \((r)\). Solow’s data comes from the USA dating from 1909 - 1949, he excludes private non-farm economic activity as well as government output. He uses GNP from the period excluding the above and uses Goldsmith’s data to calculate the labour force assuming labour has unemployment at a fixed percentage throughout the period. This is not a perfect approximation, as it represents the stock of capital goods in existence. Ideally what Solow wanted to measure was the annual flow of capital services, i.e capital in use, not capital in place.

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

No, Chart 2 has the change in technology on the y axis and the years on the x axis. It shows strong changes across the period, a large dip around 1919 could be due to the USA entering WW1, the stock market crash in 1929 resulted in another dip, there are sharps increases these could be due to changes like Henry Ford’s invention of specialization in his Model T ford factory in the late 1920’s, with electricity brought new consumer goods and ways of paying for them in the 1930’s.

Does Solow find Technological progress to be neutral?

The Technical progress during period 1909-49 was found to be neutral on average. Technical progress, the change in A is uncorrelated with the changes in the marginal rate of substitution. Chart 3 shows the time against the change in \(A(f)\). It shows that with 1909 set to equal 1.0, over the next 40 years technical change increased, there were dips after both the world wars, quickly overcame and peaks during the wars as the USA increased output of goods to sell to both sides. Solow observes a leveling off of output in the 1920’s, possibly due to a small recession in 1920 -21 and the drought the USA experience in the mid to late 1920’s.

What does chart 4a show?

There is no chart 4a, the equation 4a is a linear regression of data within Chart 4 which tests the equation of the Cobb-Douglas production function, showing diminishing returns when capital per worker increases, given a level of technical change.

What is Saturation?

Saturation is when the level of marginal productivity of capital reaches zero or when the gross marginal product falls below the marginal rate of depreciation. The wearing out or replacement of capital is greater than the increase in production. For the period Solow analysed 1909-1949, a rate of depreciation of between 3 - 5 per cent and had a capital per unit output estimated between 2 - 3 per cent. Capital saturation will occur whenever the gross marginal product of capital is greater than .03- .05. The gross marginal product of capital rises when the capital is used less efficiently, ie. more capital is required to produce the same amount of output.

Part Two: Simulating Shocks in a Solow Model.

The Solow Model

Initialising k and y.

s <- 0.25
delta <- 0.05
n <- 0.02
alpha <- 0.40
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)

k as a Steady State (8.3445)

s <- 0.25
delta <- 0.05
n <- 0.02
alpha <- 0.40
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)

Estimating a Change in Savings Rates

alpha <- 0.40
delta <- 0.05
n <- 0.02
k1 <- rep(NA, 200)
y1 <- rep(NA, 200)
s1 <- rep(0.3, 200)
s1[1] <- 0.25
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]
}


plot.ts (k1)

plotting y and y1

Redux: Solow Model with Labor-Augmenting Technical Change