If we know that the long-run number of firms in a competitive market is $50$, the total short-run cost (associated with the long-run equilibrium output) is $C_s(q) = 0.5q^2 - 10q + 200$, can we say the following:
Industry SR supply is $P = \frac{Q}{50} -10$ since $P = SRMC = q-10$? Is the direct substitution of $Q = 50q$ allowed?
Short-run total cost function of the industry is $C(Q) = 0.5(Q/50)^2 - 10(Q/50) + 200$? Again, is the substitution allowed?
Answer to 1
In perfect competition all firms are identical by assumption so you can say that total supply is sum of all individual supplies or:
$$Q_S=\sum{q_i}=nq$$
The last equality holds because all firms are the same. So if individual firm supply is:
$$q=p+10$$
where we use the fact that $p=MC$ then the industry supply will be:
$$50q=Q_s=50p+500$$
and inverse supply curve will be given by:
$$p=\frac{1}{50}Q_s -500$$
Answer to 2
Total industry costs are just sum of all individual costs so they are:
$$TIC = \sum TC_i = n(TC)$$
where the second equality again takes advantage of the fact that all firms are same.
So then the total industry costs are:
$$50TC= 50(0.5q^2−10q+200)$$
Every firm produces the same $q$ so all you need to do is to find TC for single firm and multiply by 50. Be aware that this only holds when all firms are identical and produce identical quantity. You can't do this outside perfect competition or other models that have identical firms producing identical quantities.
