# Growth accounting

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.[1]

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 labor—and that which cannot be accounted for by observable changes in factor utilization. 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 labor force growth rates. Hence, technological progress plays a key role in the economic growth of nations, or the lack of it.

## Example

Decomposing increase in output into that due to technology and that due to increase in capital (click to enlarge)

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 labor 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 labor is given by the growth rate of labor weighted by labor's share in income. If capital's share in output is , then labor's share is (assuming these are the only two factors of production). This means that the portion of growth in output which is due to changes in factors is .06×()+.01×()=.027 or 2.7%. 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.

## Technical derivation

The total output of an economy is modeled as being produced by various factors of production, with capital and Labor 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:

$Y=F(A,K,L)$

where Y is total output, K is the stock of capital in the economy, L is the labor 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 labor are used in production.

Standard assumptions on the form of the function F(.) is that it is increasing in K, L, A (if you increase productivity or you increase the amount of factors used you 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 you double both K and L you 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:

${dY}/{dK}=MPK=r$

${dY}/{dL}=MPL=w$

where MPK denotes the extra units of output produced with an additional unit of capital and similarly, for MPL. Wages paid to labor 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;

$dY=F_A dA+F_K dK+F_L dL$

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

$dY=F_A dA+MPK dK+MPL dL=F_A dA+r dK+w dL$

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

${dY}/{Y}=({F_A}A/{Y})({dA}/{A})+(r{K}/{Y})*({dK}/{K})+(w{L}/{Y})*({dL}/{L})$

or denoting a growth rate (percentage change over time) of a factor as $g_i={di}/{i}$ we get:

$g_Y=({F_A}A/{Y})*g_A+({rK}/{Y})*g_K+({wL}/{Y})*g_L$

Then ${rK}/{Y}$ is the share of total income that goes to capital, which can be denoted as $\alpha$ and ${wL}/{Y}$ is the share of total income that goes to labor, denoted by $1-\alpha$. This allows us to express the above equation as:

$g_Y={F_A}A/{Y}*g_A+\alpha*g_K+(1-\alpha)*g_L$

In principle the terms $\alpha$, $g_Y$, $g_K$ 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 $\frac {F_A A} {Y}*g_A$ however is not directly observable as it captures technological growth and improvement in productivity that are 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:

$Solow Residual = g_Y-\alpha*g_K-(1-\alpha)*g_L$

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

$Solow Residual=g_{(Y/L)}-\alpha*g_{(K/L)}$

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.

## Notes and references

1. ^ Solow, Robert (1957). "Technical change and the aggregate production function". Review of Economics and Statistics 39 (3): 312–320. JSTOR 1926047.