Solow Residual - As A Residual Term in The Solow Model

As A Residual Term in The Solow Model

Solow assumed a very basic model of annual aggregate output over a year (t). He said that the output quantity would be governed by the amount of capital (the infrastructure), the amount of labour (the number of people in the workforce), and the productivity of that labour. He thought that the productivity of labour was the factor driving long-run GDP increases. An example economic model of this form is given below1:

where:

• Y(t) represents the total production in an economy (the GDP) in some year, t.
• K(t) is capital in the productive economy - which might be measured through the combined value of all companies in a capitalist economy.
• L(t) is labour; this is simply the number of people in work, and since growth models are long run models they tend to ignore cyclical unemployment effects, assuming instead that the labour force is a constant fraction of an expanding population.
• A(t) represents multifactor productivity (often generalized as "technology"). The change in this figure from A(1960) to A(1980) is the key to estimating the growth in labour 'efficiency' and the Solow residual between 1960 and 1980, for instance.

To measure or predict the change in output within this model, the equation above is differentiated in time (t), giving a formula in partial derivatives of the relationships: labour-to-output, capital-to-output, and productivity-to-output, as shown:

Observe:

Similarly:

Therefore:

The growth factor in the economy is a proportion of the output last year, which is given (assuming small changes year-on-year) by dividing both sides of this equation by the output, Y:

The first two terms on the right hand side of this equation are the proportional changes in labour and capital year-on-year, and the left hand side is the proportional output change. The remaining term on the right, giving the effect of productivity improvements on GDP is defined as the Solow residual:

The residual, SR(t) is that part of growth not explicable by measurable changes in the amount of capital, K, and the number of workers, L. If output, capital, and labour all double every twenty years the residual will be zero, but in general it is higher than this: output goes up faster than growth in the input factors. The residual varies between periods and countries, but is almost always positive in peace-time capitalist countries. Some estimates of the post-war U.S. residual credited the country with a 3% productivity increase per-annum until the early 1970s when productivity growth appeared to stagnate.