What is the relation between Solow Residual and total factor productivity?

Relation between Solow Residual and total factor productivity

Growth accounting is a procedure used in economics to measure the contribution of different factors to economic growth and to indirectly compute the rate of technological progress, measured as a residual, in an economy. This methodology was introduced by Robert Solow in 1957.

Growth accounting decomposes the growth rate of economy’s total output into that which is due to increases in the amount of factors used – usually the increase in the amount of capital and labour -and that which cannot be accounted for by observable changes in factor utilisation.

The unexplained part of growth in GDP is then taken to represent increases in productivity (getting more output with the same amounts of inputs) or a measure of broadly defined technological progress.

The technique has been applied to virtually every economy in the world and a common finding is that observed levels of economic growth cannot be explained simply by changes in the stock of capital in the economy or population and labour force growth rates. Hence, technological progress plays a key role in the economic growth of nations, or the lack of it.

As an abstract example consider an economy whose total output (GDP) grows at 3% per year. Over the same period its capital stock grows at 6% per year and its labour force by 1%.

The contribution of the growth rate of capital to output is equal to that growth rate weighted by the share of capital in total output and the contribution of labour is given by the growth rate of labour weighted by labour’s share in income.

This means that the portion of growth in output which is due to changes in factors. This means that there is still 0.3% of the growth in output that cannot be accounted for. This remainder is the increase in the productivity of factors that happened over the period or the measure of technological progress during this time.

The total output of an economy is modeled as being produced by various factors of production, with capital and labour force being the primary ones in modern economies (although land and natural resources can also be included). This is usually captured by an aggregate production function:

Where Y is total output, K is the stock of capital in the economy, L is the labour force (or population) and A is a “catch all” factor for technology, role of institutions and other relevant forces which measures how productively capital and labour are used in production.

Standard assumptions on the form of the function F (.) is that it is increasing in K, L, A (if we increase productivity or you increase the amount of factors used we get more output) and that it is homogeneous of degree one, or in other words that there are constant returns to scale (which means that if we double both K and L we get double the output). The assumption of constant returns to scale facilitates the assumption of perfect competition which in turn implies that factors get their marginal products:

In principle the terms a, g_Y, g_K and g_L are all observable and can be measured using standard national income accounting methods (with capital stock being measured using investment rates via the perpetual inventory method).

The term where MPK denotes the extra units of output produced with an additional unit of capital and similarly, for MPL. Wages paid to labour are denoted by w and the rate of profit or the real interest rate is denoted by r.

Note that the assumption of perfect competition enables us to take prices as given. For simplicity we assume unit price (i.e. P = 1), and thus quantities also represent values in all equations.

If we totally differentiate the above production function we get;

Where denotes the partial derivative with respect to factor i, or for the case of capital and labour, the marginal products. With perfect competition this equation becomes:

If we divide through by Y and convert each change into growth rates we get:

However is not directly observable as it captures technological growth and improvement in productivity that is unrelated to changes in use of factors. This term is usually referred to as Solow residual or total factor productivity growth.

Slightly rearranging the previous equation we can measure this as that portion of increase in total output which is not due to the (weighted) growth of factor inputs:

Another way to express the same idea is in per capita (or per worker) terms in which we subtract off the growth rate of labour force from both sides:

Which states that the rate of technological growth is that part of the growth rate of per capita income which is not due to the (weighted) growth rate of capital per person?

The Solow residual is a number describing empirical productivity growth in an economy from year to year and decade to decade. Robert Solow defined rising productivity as rising output with constant capital and labor input.

It is a “residual” because it is the part of growth that cannot be explained through capital accumulation or the accumulation of other traditional factors, such as land or labour. The Solow Residual is procyclical and is sometimes called the rate of growth of total factor productivity.

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 below:

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 generalised 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 (i), giving a formula in partial derivatives of the relationships: labour-to-output, capital-to-output, and productivity-to-output, as shown:

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,

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.

In the equation given above a higher value of A means that the same inputs lead to more output and vice versa. It shows how efficiently that input is being used to further the interests of the economy and it is the productivity of the capital and labour investment.

Total Factor Productivity is considered to be the actual determining factor in the growth of an economy as both capital and labour cannot continue to be invested indefinitely.

Moreover, the growth of economy, if depended solely on capital and labour would decline as soon as these investments in these inputs are reduced and vice-versa. Thus it is not a stable growth. Hence increased Total Factor Productivity is the only way that an economy can maintain a stable growth.

Also, the Law of Diminishing Marginal Returns, tells us that a sustained influx of Labour and Capital will not achieve long term growth as the value of the inputs get maximised; they onset to deliver lower returns over a period of time.

Thus the only way growth can be ensured and sustained is to maximise the efficiency of these inputs and to work on improving the quality and quantity of returns for the same amount of inputs i.e., to increase Total Factor Productivity.