# Supply elasticity on a piece-wise linear supply function

The supply elasticity at a point is the willingness to produce more or less after the price has changed (due to whatever reason?).

$$\mu_{Q,P}=\frac{\partial Q_S/Q_S}{\partial P/P}=\frac{\partial Q_S}{\partial P}\cdot\frac{P}{Q_S}$$

Where $$Q_S$$ is the quantity supply function, $$P$$ is the price. $$\frac{P}{Q_S}$$ is the current ratio (the one at the point of measurement or at equilibrium) of price to quantity.

Imagine there are three different suppliers with different supply functions $$Q_1,Q_2,Q_3$$ For low prices some suppliers won't be able to afford to produce so there will only be one supplier for example. for a certain range $$(p_0,p_1]$$ we can have that two firms are producing, and for $$P\in(p_1,\infty)$$ all three firms are producing.

In this example it just so happens that the aggregate supply function is continuous.

Say the equilibrium price is $$p_1$$ and the aggregate supply is $$Q_S(p_1)=Q_1(p_1)+Q_2(p_2)$$ (only two out of three firms produce for this price). If the slope of $$Q_S$$ is some number $$\alpha$$ at that point, (on the left, because on the right supplier 3 also produces and so the slope is different, say $$\beta\ne\alpha$$), then the supply elasticity is $$\alpha\cdot \frac{p_1}{Q_1(p_1)+Q_2(p_1)}$$.

Since the slope on the right of $$p_1$$ is beta, and the aggregate supply function is continuous (that implies $$Q_1(p_1)+Q_2(p_1)=Q_S(p_1)=\lim\limits_{p\rightarrow p_1\\p> p_1}Q_1(p)+Q_2(p)+Q_3(p)$$) the supply elasticity is $$\beta\cdot \frac{p_1}{Q_1(p_1)+Q_2(p_1)+Q_3(p_1)}$$

Can the "right and left" elasticities be different in absolut value?

How should we understand a market where supply elasticity is not continuous at the equilibrium?

NB: if we had to impose that the elasticity has to be continuous that would be imposing $$Q_3(p_1)=\frac{(\beta-\alpha)(Q_1(p_1)+Q_2(p_1))}{\alpha}$$