# Identify absolute ability from a wage equation

Consider a simple model where the wage of a worker $i$ in period $t$ is:

$$w_{it} = \theta_{t} a_{i}$$

where $\theta_{t}$ is the wage rate or efficiency wage per unit of ability, and $a_{i}$ is the workers' ability endowment.

For simplicity let us assume no other factor affects a workers' wage (no experience, education, or any the usual Mincer stuff).

If I run a panel data regression such as (also known as a two-way effects model):

$$\ln w_{it} = \ln \theta_{t} + \ln a_{i} + \epsilon_{it}$$

then, according to my model, the estimated effect for the individuals correspond to (the log of) absolute ability. However, in the literature you always see that predicted unobserved heterogeneity is not ability per se but ability together with it's price. This is, there is always a coefficient like in $c\ln a_{i}$ or in $\ln c a_{i}$, and then they move to calculate relative ability by just comparing the ratio of predicted heterogeneity for two workers.

I just can't see why it is not possible to identify absolute ability. Where is the missing link in my model?