what is the marginal cost divided by the average cost Marginal cost and average cost are both a u-curve. They cross in the competitive equilibrium. If we divide one by the other, is the ratio the degree of competition?   Ratio is 1 in perfect competition. What is the name of this quantity, MC/AC?
$MC/AC$ is not a quantity, it is a function. More precisely it would be written as $MC(y)/AC(y)$.

It is true that in a competitive setting in the long-run equilibrium $MC(y^*) = AC(y^*)$.

However, this can also be true in a monopoly setting. For example in the special case when $C(y) = c y$ where $c \in \mathbb{R}_+$, you have
$$
\forall y: \ MC(y) = c = AC(y).
$$
Also the monopolist's profit-maximizing $y^*$ is independent of any fixed costs $F$, but $AC(y^*)$ is not. For example if the inverse demand function is $p(y) = 1 - y$ and the cost function is $C(y) = y^2 + F$ then
\begin{align*}
y^* &amp; = \arg\max_y (1 - y)y - y^2 - F = 1/4 \\
MC(y^*) &amp; = 2y^* = 1/2 \\
AC(y^*) &amp; = y^* + F/y^* = 1/4 + 4F. 
\end{align*}
In this case
$$
\begin{array}{lcl}
MC(y^*)/AC(y^*) &lt; 1 &amp; \mbox{ if } &amp; F &gt; 1/8 \\
MC(y^*)/AC(y^*) = 1 &amp; \mbox{ if } &amp; F = 1/8 \\
MC(y^*)/AC(y^*) &gt; 1 &amp; \mbox{ if } &amp; F &lt; 1/8.
\end{array}
$$
Yet there is exactly one firm in the market, so there is no competition in the classical sense. Nor is it the case that the firm is a natural monopoly at $F = 1/8$. This could mean that this is the 'efficient level of competition', but it is not the case, because when  $F = 1/8$ we have
\begin{align*}
y^* &amp; = 1/4 \\
p(y^*) &amp; = 3/4 \\
AC(y^*) &amp; = 1/2 \\
\text{Profits} &amp; = 1/16 &gt; 0. 
\end{align*}