How to operationalize agglomeration economies? How most common way to operationalize the concept of agglomeration economies? I am new in research field. I even don't know what conceptual definition means.
Agglomeration economies is a large concept that includes any effect that increases firms' and workers' income when the size of the local economy grows.
The general idea is to estimate the overall impact of density on a local characteristic, such as the impact of local employment density on local productivity. The density estimate captures the agglomeration effect.
As a naive example to operationalize the concept of agglomeration economies, consider the following specification:
$$Y_c = \alpha Density_c + \beta Skills_c + \epsilon_c$$
where $Y_c$ is a measure of the local outcome in a city $c$, say the logarithm of local wage, $Density_c$ is the logarithm of density, $Skills_c$ captures local skills and $\epsilon_c$ is the error term. If there is no correlation between the random component and explanatory variables, then the ordinary least squares estimate of the elasticity of productivity with respect to density ($\beta$) is a consistent measure of total net agglomeration economies. For instance, a value for the elasticity of the local outcome with respect to density of 0.03 means that a city twice as large (knowing that a factor of 10 is often obtained for the interquartile
of local density in many countries) has $2^{0.03} - 1 = 2.1\%$ greater productivity,
because of either pure local externalities or market agglomeration effects that dominate dispersion effects of any kind.
You will find much more, including a discussion of the mechanisms, in Chapter 5 of the Handbook of Urban and Regional Economics, The Empirics of Agglomeration.