# Why does short run market supply become more elastic with more firms

I have heard a justification along the lines of this: The supply curve becomes flatter (more elastic) with more firms in the market, because a given increase in price calls forth more production when there are many firms rather than one. This however does not make sense to me as elasticity only cares about proportional change. Mathematically is there are $$N$$ firms with supply functions $$q(p)$$ and a market supply of $$Q(p) = Nq(p)$$ then $$\frac{p}{Q}\frac{\mathrm dQ}{\mathrm dp}=\frac{p}{Nq}\frac{\mathrm dNq}{\mathrm dp}=\frac{p}{q}\frac{\mathrm dq}{\mathrm dp}$$

• this is not full answer, but depending on what the market structure is $p$ and even $q$ will be functions of $n$ for example in perfect competition. In addition, in equilibrium $Q_d(p) = nQ_S(p)$ but by the same token also $p(Q_d)=p(nQ_s)$ so even though I dont have proof for that statement you are asking for the elasticity calculation where $n$ cancels is not generally valid – 1muflon1 Dec 5 '20 at 11:17
• I agree with your calculation, the elasticity can stay constant (increase or decrease) with $N$. When the number of firms increases, however, the supply curve becomes flatter, in the sense that the derivative $Q'(p,N+1)=Q'(p,N)+q_{N+1}'(p)>Q'(p,N)$. A given increase in price triggers a bigger increase in produced quantity $Q$ when $N$ is big, than when $N$ small. Regarding the elasticity, anything can happen. – Bertrand Dec 12 '20 at 17:17