✏️ Formula Example

✏️ $TC = 3q^2 + 5q + 20$
$MC = 6q + 5$
$P = 20$
What is profit?

✔ Click here to view answer

We know that we have to do the same three steps that we covered on Thursday..

Step 1: The first step is setting $P = MC$, as you note. We can do this with our equations…
$P = 20$
$MC = 6q + 5$
$P = MC \rightarrow 20 = 6q + 5$
We can then use algebra to solve for the q which will make $P = MC$.
$15 = 6q$
$q = \frac{15}{6} = 2.5$
Therefore, $q^* = 2.5$

Step 2: Is $P < AVC$. $TC = 3q^2 + 5q + 20$, so $VC = 3q^2 + 5q$, and
$AVC = \frac{VC}{q} = \frac{3q^2 + 5q}{q} = 3q + 5$
When $q^* = 2.5$, then $AVC = 3 \times 2.5 + 5 = 12.5$
$P = 20 > AVC = 12.5$, so we don’t shut down. We produce $q = q^* = 2.5$.

Step 3: $\text{Profit} = q^*(P - AC) = 2.5 \times (20 -$
$TC = 3q^2 + 5q + 20 = 3 \times (2.5)^2 + 5 \times 2.5 + 20 = 51.25$
$ATC = \frac{TC}{q} = \frac{51.25}{2.5} = 20.5$
$\text{Profit} = q^*(P - AC) = 2.5 \times (20 - 20.5) = -1.25$
I won’t shut down, but I’ll be losing money, so I exit in the long run.  ✅

🙋 Can you tell us what happens in the long run and the short run?

✔ In the short run, your fixed costs are fixed. You are are stuck paying them.
$TC = 3q^2 + 5q + 20$
In the short run, you must pay $20 of fixed costs. I like using the example (just for developing intuition) of the fixed cost being a lease. So let’s assume you have a $20 lease.

In the short run, if you produce $q^*$, your profit is -1.25.
In the short run, if you produce 0 (shut down), then:
 $\text{profit} = TR - TC = \$0 - (30 \times 0^2 + 5 \times 0 + 20) = \$0 - (30 \times 0^2 + 5 \times 0 + 20) = -20$
Clearly, in the short run it is optimal to produce $q^* = 2.5$

In the long run, you can also exit. If you exit, $q = 0$, $TR = 0$, $TC = 0$, and $\text{profit} = 0$, because your firm doesn’t exist.
Now you have three options.
if you produce $q^*$, your profit is -1.25.
if you produce 0 (shut down), your profit = -20
If you exit, your profit = $0 - $0 = $0.
Clearly, exiting is the best! In the long run, it wouldn’t be optimal to stay in business and produce $q^*$. Rather, it would be optimal to exit. Exiting is like shutting down because $q = 0$, but it is different because you have no costs at all (your firm doesn’t exist).

Therefore, the firm would just keep on producing $q = 2.5$ units per year until the end of the short run (ie until the lease expires). Then it would exit and produce 0.  ✅