# Growth Accounting

Growth accounting is the process used to attribute economic growth to growth in labor, capital accumulation and technological progress.

Let’s consider High Garden whose sole product is tulips. In Period 1, a total of 1,000 people worked with 100 harvesters to produce 40 million tulips. In Period 2, total production rose to 45 million tulips when 1,050 people worked with 105 harvesters. Let’s assume High Garden’s marginal product of capital (MPK) and marginal product of labor (MPL) are 200,000 and 30,000 units of tulips respectively.

Since the marginal product of capital i.e. additional production that results from each additional harvester is 200,000 tulips, the increase of 5 harvesters explains additional production of 1 million tulips (5 × 200,000). Similarly, increase of 50 workers causes increase in production of 1.5 million tulips (50 × 300,000). The residual increase of 2.5 million tulips (5 million – 1 million – 1.5 million) is not explained by increase in capital or labor. This factor is called Solow residual and it represents general improvement in productivity i.e. increase in total factor productivity.

## Growth Accounting Equation

For the High Garden, the following equation explains the increase in production (∆Y) from Period 1 to Period 2 as the sum of (a) product of change in capital (∆K) and marginal product of capital, (b) product of change in labor (∆L) and marginal product of labor and (c) change in total factor productivity (∆A).

$$\Delta \text{Y}=\Delta \text{K} \times \text{MPK}+\Delta \text{L} \times \text{MPL}+\text{F}(\text{K,L}) \times \Delta \text{A}$$

$$\text{5,000,000}\\=(\text{105}-\text{100})\times\text{200,000}\\+(\text{1,050}-\text{1,000})\times\text{30,000}\\+\ \text{2,500,000}$$

If we divide the above mathematical equation by Y = A × F(K, L) and do a bit of mathematical manipulation, we get a relationship between growth rates:

$$\frac{\Delta \text{Y}}{\text{Y}}=\frac{\text{MPK} \timesK}{\text{Y}}\times\frac{\Delta \text{K}}{\text{K}}+\frac{\text{MPL} \timesL}{\text{Y}}\times\frac{\Delta \text{L}}{\text{L}}+\frac{\Delta \text{A}}{\text{A}}$$

The first factor i.e. ∆Y/Y is the GDP growth rate, the ratio of (MPK × K) to Y equals capital’s proportion in total production, ∆K/K is the percentage change in capital, the ratio of (MPL × L) to Y equals labor’s proportion in total’s production, ∆L/L is the percentage change in labor and ∆A/A is the Solow residual.

If an economy has constant returns to scale and the proportion of capital in total production is α, the proportion of labor is 1 – α.

$$\frac{\Delta \text{Y}}{\text{Y}}=α\times\frac{\Delta \text{K}}{\text{K}}+(\text{1}-α)\times\frac{\Delta \text{L}}{\text{L}}+\frac{\Delta \text{A}}{\text{A}}$$

The formula above is the growth accounting equation, a mathematical representation of the relationship between economic growth, capital accumulation, labor growth rate and growth in total factor productivity. In plain English, it can be written as follows:

$$\text{Growth Rate}\\=\text{Capital Share}\times\text{%\ Capital Growth}\\+\text{Labor Share}\times\text{%\ Labor Growth}\\+\text{Technological Progress}$$

## Growth of GDP per Capita

Economic growth is generally defined as the percentage increase in real gross domestic product of an economy. Growth rate of GDP per capita differs from growth rate (of GDP) because GDP per capita also depends on the growth rate of population.

If we define y as output per worker i.e. Y/L and k as capital per worker i.e. K/L, the growth accounting equation discussed above can be derived for per-capita GDP

$$\frac{\Delta \text{y}}{\text{y}}=α\times\frac{\Delta \text{k}}{\text{k}}+\frac{\Delta \text{A}}{\text{A}}$$

It shows that percentage increase in GDP per worker (∆y/y) is the sum of (a) product of capital’s share in production (α) and percentage increase in capital per worker (∆k/k) and (b) technalogical progress (∆A/A).

## Capital-Labor Ratio

The expression ∆k/k in the growth accounting equation for GDP per capita represent the percentage increase in capital to labor ratio. The capital to labor ratio is the ratio of capital to workers i.e. K/L. It shows the extent of capital-intensiveness of an economy

The implication of the growth accounting equation in per-capita form is that the improvement in standard of life of people in an economy depends on availability of capital and technological progress and not on the percentage increase in employment. It is because even though employment increases total production, since its share in production is lower than 1, the resultant increase in per-capital GDP is lower.