# Dimensional analysis of GDP

The GDP is sometimes given by $$GDP=P\cdot h\cdot e \cdot F$$

Where $$P$$ is the Productivity, $$h$$ is the number of hours worked, $$e$$ is the employment rate and $$F$$ is the size of the labor force.

If we do a dimensional analysis, we see that the left hand side of that equation i $$\left[ GDP \right]= \text{value / time}$$

On the right hand side we have:

$$[P]=\text{value/time}\\ [h]=\text{time}\\ [e]=\text{dimensionless}\\ [F]=\text{dimensionless}.$$

This analysis yields that the dimension of the RHS is value.

Hence, the dimension of the left hand side and the right hand side is not equal, which would constitute a contradiction.

Does anyone know why that is?

If GDP is measured (as it usually is) for a period such as a year, then number of hours worked ($$h$$) should be measured for the same period. Thus $$h$$ is dimensionless and the dimension of the right hand side is also value / time.