Calculating equilibrium and graphing individual firms and their industry counterparts is common in intermediate microeconomics. This post will go over the economics of of a subsidy and its effects on the individual firm's cost structure as well as how the industry as a whole will change with the introduction of a subsidy. Here is the question:

In a competitive industry each firm has total costs C = q^2
+ 16. Demand is D = 80 – 5P. A $4 per-unit subsidy is introduced for this good.
Provide a pair of fully labeled diagrams showing “The Firm” and “The Industry”
to outline the response to this subsidy.

Summary:

1) Solve for the perfectly competitive output level by setting marginal cost equal to average total cost. This gives us the minimum of ATC, which is the long run equilibrium output for a perfectly competitive market (at the firm level).

2) Use this output quantity to solve for ATC and therefore price (price = ATC = zero economic profit). Use this price to find quantity demanded using the market demand function.

3) Figure out what market demand will be, and divide this number by the optimal output per firm to find the number of firms that will produce in this market.

4) Since the subsidy is per unit, you can subtract the subsidy from the price, and resolve for quantity demanded, firms will enter the market to increase the quantity produced.

This can seem like a tricky question, but is actually
straight forward if you know how to approach it. First we know that this is a competitive
industry, so we know that firms can enter or exit the market as they wish. This means that in the long run, firms will
be producing at the minimum of their average total cost (ATC) to ensure that there is zero economic profit. We can find the minimum of the ATC quite
easily by finding equations for both the ATC and the marginal cost (MC) and
setting them equal to each other and solving for quantity (q). This works because MC crosses ATC at its
minimum.

So first we need to find ATC, this is done by taking our cost
function, and dividing by q:

Total Cost = C = q^2 + 16 => ATC = C/q = q^2/q + 16/q

Simplify to get ATC= q + 16/q

To find MC we need to take the first derivative of the cost
function with respect to q. This gives
us:

2q (by use of the power rule, the exponent drops down and
becomes the coefficient and the 16 drops out).

So now we set MC equal to ATC:

2q = q + 16/q

We can subtract q from both sides, and then multiply both
sides by q to get:

q^2 = 16, and then take the square root of both sides to
get:

q = 4.

So the quantity that each perfectly competitive firm will
product at is at a quantity of 4. Note
that this will be the minimum ATC regardless of the subsidy because the subsidy
will simply lower the MC, and ATC by $4 at any point because it is a per unit subsidy. Here is what it looks like on a graph, for
the individual firm:

You can see from the graph, that in both cases, the MC
crosses the ATC curve at a quantity of 4, but the price will change.

At a quantity of 4, the firm will have an ATC of 8, we can
figure this out by plugging in our q = 4 into our ATC equation:

ATC = q + 16/q = 4 + 16/4 = 4 + 4 = 8.

So each firm will charge a price of $8, and at a price of $8
market demand will be:

D = 80 – 5P = 80 – 5*8 = 80 – 40 = 40.

So 40 units will be demanded and if each firm produces 4
units, we will need 10 firms. So without
the subsidy we have a price of $8 and ten firms each producing 4 units.

If we introduce the subsidy of $4 per unit, it drops the MC
and ATC for each firm by $4 for every unit produced. This simply means that the ATC of production
drops from $8 to $4. At a market price
of $4, the firms make an economic profit so in the long run more firms will
enter the market. At a market price of
$4, quantity demanded will be:

D = 80 – 5P = 8 – 5*4 = 80 – 20 = 60.

So if every firm produces at their minimum ATC (a
requirement in long run perfectly competitive markets) then we will need 15
firms (60/4).

This will shift the supply curve to the right (an increase in
supply) because there are more suppliers.
The new equilibrium will occur at a higher quantity and a lower price.