Editing Growth accounting (section)

==Technical derivation==
{{Refimprove section|date=October 2007}}
The total output of an economy is modeled as being produced by various factors of production, with capital and labor 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]]:<ref>{{cite journal | last1 = Zelenyuk | year = 2014 | title = Testing Significance of Contributions in Growth Accounting, with Application to Testing ICT Impact on Labor Productivity of Developed Countries | journal = International Journal of Business and Economics | volume = 13 | issue = 2| pages = 115–126 }}</ref>

<math>Y=F(A,K,L)</math>

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 number of factors used you get more output) and that it is [[Homogeneous function|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:

<math>{dY}/{dK}=MPK=r</math>

<math>{dY}/{dL}=MPL=w</math>

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;

<math>dY=F_A dA+F_K dK+F_L dL</math>

where <math>F_i</math> 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:

<math>dY=F_A dA+MPK dK+MPL dL=F_A dA+r dK+w dL</math>

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

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

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

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

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

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

In principle the terms <math>\alpha</math>, <math>g_Y</math>, <math>g_K</math> and <math>g_L</math> are all observable and can be measured using standard [[National accounts|national income accounting]] methods (with capital stock being measured using investment rates via the [[Capital formation#Perpetual Inventory Method|perpetual inventory method]]). The term <math>\frac {F_A A} {Y}*g_A</math> 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:

<math>Solow Residual = g_Y-\alpha*g_K-(1-\alpha)*g_L</math>

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:

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

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.