# Is it true that if marginal cost is constant, then average variable cost is also constant and equals marginal cost?

I'm inclined to think yes because marginal cost only depends on variable cost (fixed costs don't matter), but I'm not 100% certain.

Basically, my thought process is that marginal cost of producing one additional unit is the change in total variable cost to produce that unit. So, my current idea is that if marginal cost is constant, that must mean that average variable cost (total variable cost/output) is also constant. But I'm still not entirely convinced.

Your intuition is correct.

First, you're right that "marginal cost only depends on variable cost", since $$$$MC(q)=\frac{\mathrm dTC(q)}{\mathrm dq}=\frac{\mathrm d(FC+VC(q))}{\mathrm dq}=\frac{\mathrm dVC(q)}{\mathrm dq}.$$$$ Next, if marginal cost is some constant $$k$$, then variable cost must be $$VC(q)=kq$$, because we can integrate $$MC$$ to obtain $$TC$$, where the term that varies with $$q$$ is $$kq$$: $$$$MC(q)=k \quad\Rightarrow\quad \int k\;\mathrm dq =\underbrace{kq}_{VC(q)} +\underbrace{K\vphantom{q}}_{FC}$$$$ where $$K$$ is a constant resulting from the integration. Lastly, $$$$AVC(q)=\frac{VC(q)}{q}=\frac{kq}{q}=k,$$$$ which is a constant, and same as $$MC$$.

• This is an excellent answer - thank you. One clarifying question: Sometimes in class our teacher will describe marginal cost as being "the cost of producing one additional unit." How does that relate to the calculus-based definition of marginal cost?
– Will
Oct 7 '20 at 7:10
• If you look at the first derivative in the answer from @herr-k, notice that it's dTC/dq, which is the change in total cost w.r.t a one-unit change in quantity. That's the definition your teacher gave.
– Jeff
Oct 7 '20 at 15:06
• @Will the cost of an extra unit is the discrete version of marginal costs and easier to understand as concept. For discrete production we can calculate the marginal cost only by determining what it costs to produce one additional unit. When quantity is continuous we can calculate the marginal cost at a point by calculating the cost of producing an infinitesimal extra unit. Oct 7 '20 at 15:16
• @Will: Thanks. In addition to the above two comments, you may also refer to my answer to a related question on this topic. Oct 7 '20 at 15:39