Growth Accounting - Technical Derivation

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:

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:

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;

where 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:

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

or denoting a growth rate (percentage change over time) of a factor as we get:

Then is the share of total income that goes to capital, which can be denoted as and is the share of total income that goes to labor, denoted by . This allows us to express the above equation as:

In principle the terms, and 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 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:

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:

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.