# Why is the Supply of Labour More Elastic in a Single Province/State then it is in the Country as a Whole?

Apparently, the federal government has a greater ability than provincial government to increase tax revenues by raising tax rates because the supply of labour is more elastic in a single province than it is in Canada as a whole.

However, why would supply of labour be more elastic in a single province than in the country as a whole?

Because there is freedom of movement between provinces; and the language barrier is low. When international borders are involved, there is much less freedom of movement, and much greater potential for language barriers.

So it's easier for labour to move between provinces if incentivised to do so by a change in relative taxation rates; than it is to move between countries.

The supply of labour is more elastic in a single province than it is in the country as a whole.

This is simply not possible, if we clearly separate between the static labor supply function and the equilibrium responses of labor supply, that may involve all shorts of changes, supply and demand shifts, income effects, etc.

That it is not possible to hold for the static labor supply is easy to show: denote $L^s_i(w)$ the labor supply at the level of province $i=1,...,n$. Note that the $w$ is not indexed, exactly because we are looking at the static concept, which reflects intended levels of labor supply at different values of theoretical $w$.

Then the supposed inequality can be written as

$$\frac {\partial L^s_i(w)/\partial w}{L^s_i(w)}\cdot w > \frac {\sum_{i=1}^n\partial L^s_i(w)/\partial w}{\sum_{i=1}^n L^s_i(w)} \cdot w,\;\;\; \forall i$$

Simplify and rearrange

$$\frac {\partial L^s_i(w)/\partial w}{\sum_{i=1}^n\partial L^s_i(w)/\partial w} > \frac {L^s_i(w)}{\sum_{i=1}^n L^s_i(w)} \;\;\;\forall i$$

But this cannot hold, because the left side as well as the right side both sum up to unity over $n$.

So for any given wage $w$, some of the provincial labor supply functions have to have lower elasticity than the aggregate supply function.

This answer attempts to clarify that observing higher elasticities at provincial level compared to the national level, relates to equilibrium responses, and has to do with the existence of wage differentials, cross-effects that cause shifts in the supply functions, etc.