Intuition behind value added as TFP measure

Question

In some papers the authors use value-added over capital and labor as TFP measure, what is the intuition behind this? For example, in one paper I just read the authors use

$ \frac{VA}{(p_{K}K + p_{L}L)} $, where VA = value added.

I'm a bit confused, since we normally use a standard Cobb-Douglas production function. Transforming the equation by this following simple step leads to a similar but still different measure of the TFP "A", i.e. here I have gross output in the nominator and in the denominator there is a multiplication and not an addition?

$ Q=AK^{α}L^{1-α} $

$ \frac{Q}{(K^{α}L^{1-α})}=A $

Answer 1

The difference between first and second expressions is that the first expression is measured in dollars using prices.

Value added is nothing else just the prices times quantity of gross output $=p_QQ$. That is why the first expression has $p_K K$ and $p_L L$ in the numerator.

You would use the first expression when you work with values (e.g. when you have data on value of output, value of capital, value of labor). Working with values is often necessary because firm-level data do not record how many units of capital they have in their firm, and also often they do not record the quantities of labor just the total value of the wage bill etc.