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

Question

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.

Herr K. · Accepted Answer

Your intuition is correct.
First, you're right that "marginal cost only depends on variable cost", since
begin{equation}
MC(q)=frac{mathrm dTC(q)}{mathrm dq}=frac{mathrm d(FC+VC(q))}{mathrm dq}=frac{mathrm dVC(q)}{mathrm dq}. 
end{equation}
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 $kq$:
begin{equation}
MC(q)=k 
quadRightarrowquad 
int k;mathrm dq =underbrace{kq}_{VC(q)} +underbrace{Kvphantom{q}}_{FC} 
end{equation}
where $K$ is a constant resulting from the integration.
Lastly,
begin{equation}
AVC(q)=frac{VC(q)}{q}=frac{kq}{q}=k,
end{equation}
which is a constant, and same as $MC$.

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

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