# Total Factor Productivity

Total factor productivity (TFP) is a measure of productivity calculated by dividing economy-wide total production by the weighted average of inputs i.e. labor and capital. It represents growth in real output which is in excess of the growth in inputs such as labor and capital.

Productivity is a measure of the relationship between outputs (total product) and inputs i.e. factors of production (primarily labor and capital). It equals output divided by input. There are two measures of productivity: (a) labor productivity, which equals total output divided by units of labor and (b) total factor productivity, which equals total output divided by weighted average of the inputs.

$$\text{TFP}=\frac{\text{Total Product}}{\text{Weighted Average of Inputs}}$$

The most widely used production function is the Cobb-Douglas function which is as follows:

$$\text{Q}=\text{A}\times \text{K}^\alpha\times \text{L}^\beta$$

Where Q is total product, K is capital, α is output elasticity of capital, L is labor and β is the output elasticity of labor.

Q is the total product and the product of Kα and Lβ is the weighted average of inputs. If we rearrange the Cobb-Douglas function, we get the following formula for total factor productivity:

$$\text{TFP}=\text{A}\ =\frac{\text{Total Product}}{\text{Weighted Average of Inputs}}=\frac{\text{Q}}{\text{K}^\alpha\times \text{L}^\beta}$$

TFP represents the increase in total production which is in excess of the increase that results from increase in inputs. It results from intangible factors such as technological change, education, research and development, synergies, etc.

It is more useful to look at productivity increase over a period instead of the absolute value of total factor productivity. The following growth accounting equation gives us the relationship between growth in total product, growth in labor and capital and growth in TFP:

$$\frac{\Delta \text{Q}}{\text{Q}} = α\times \frac{\Delta \text{K}}{\text{K}}+β\times \frac{\Delta \text{L}}{\text{L}}+\frac{\Delta \text{A}}{\text{A}}$$

## Example

Consider the following production function for mining industry in Andalusia:

$$\text{Q}=\text{A}\times \text{K}^{\text{0.70}}\times \text{L}^{\text{0.45}}$$

If the growth in total output is 3% in a period in which capital and labor grew by 1.5% and 2%, determine the growth that is attributable to total factor productivity.

We need to isolate the increase in total product that is not explained by the increase in inputs i.e. capital and labor. Let’s just punch the available data in the growth accounting equation above:

$$\text{5%}=\text{0.70}\times\text{1.5%}+\text{0.45}\times\text{2%}+\frac{\Delta \text{A}}{\text{A}}$$

$$\frac{\Delta \text{A}}{\text{A}}=\text{5%}- \text{0.70}\times \text{1.5%}-\text{0.45}\times \text{2%}=\text{3%} - \text{1.95%} = \text{1.05%}$$