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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 $$Q_3(p_1)=\frac{(\beta-\alpha)(Q_1(p_1)+Q_2(p_1))}{\alpha}$$

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