it is elastic when the price drops?
It could be so.
Elasticity is a local, pointwise notion, or relative to a specific interval, that is the elasticity of a curve in general varies from point to point of the curve and from portion to portion of the curve.
We must remember the formula that defines elasticity, in our case, elasticity of the demand of a good with respect to its price.
Elasticity of demand $Q$ with respect to price $p$ is defined, if $\Delta p$ and $\Delta Q$ are finite differences, as:
$$\epsilon= |\frac {\Delta Q} {\Delta p} \frac {p} {Q}| \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (1)$$
or
$$\epsilon = |\frac {dQ} {dp} \frac {p} {Q}| \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (2)$$
if $Q$ and $p$ are continuous variables and the change is ‘infinitesimal’, where $\frac {dQ} {dp}$ is the derivative of the demand function with respect to price$^1$.
$\frac {dQ} {dp}$ and $\frac {\Delta Q} {\Delta p}$ can vary from point to point of the function, and also $\frac {p} {Q}$ can vary from point to point.
Therefore, when speaking of elasticity, we must specify the point in which we are considering the elasticity or, in case of finite changes of the variables as in $(1)$, the interval we are referring to.
Also in the case of the simplest form of a demand function, a linear demand function, the elasticity of demand is different at each point, and we have a section of the demand curve that is elastic and a section of the curve which is anelastic.
This can be easily shown using a graph for a linear demand function
$$Q= a-bp \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;a>0, b>0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (3)$$.
The slope of the demand curve in the above picture is, in absolute value, 1/b$, ^2$ and it is a constant for each point of the demand curve. The ratio $p/Q$ is represented by the red lines. As you can see, their slope decreases as we go toward the right part of the demand curve. Therefore, there is a point where $\epsilon =1$ (when the line $p/Q$ is perpendicular to the demand curve), on the right of it we have $\epsilon <1$, (the demand is anelastic) and on the left $\epsilon >1$ (the demand is elastic).$^3$.
A similar, but opposite, situation can occur for other forms, not linear, of demand curves, as in the picture below.
Therefore, it is possible that the demand of a good is elastic for some range of the price, and not elastic for another range: in the picture, is anelastic for a price $P>P_0$ and elastic for $P<P_0$.
A similar situation can hold for the demand of diamonds.
Until now, I spoke of elasticity at one point.
If, as you argue, we have a huge decrease of the diamond price, we have to speak of elasticity in an interval, that is a finite change of the price $\Delta P$.
Sometimes, when we are confronted with a big change of the variable $p$, economists use the concept of arc elasticity, a convention that can be used to avoid some ambiguities about the assessment of the elasticity on an interval. Arc elasticity is based on an average of initial and final values of quantity and price, $Q_0, Q_1, P_0, P_1$, and the formula becomes:
$$\epsilon= |\frac {\Delta Q} {\Delta p} \frac {p_0+p_1} {Q_0+Q_1}| \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (4)$$
$$***$$
Of course, what in practice happens in the market of diamonds, if demand is elastic or not, and for which intervals of the price, is an empirical matter, that cannot be resolved by the above formulas.
$^1$ Notice that in $(1)$ and $(2)$ we are taking the absolute value, this is the usual convention, because it is more comfortable to avoid to keep the minus sign before any formula.
$^2$ We must remember that in the graph we have the inverse demand function, that is $p= a/b-1/b Q$, as we have $P$ on the ordinate axis and $Q$ on the abscissa axis.
$^3$ By definition, if $\epsilon >1$, the demand is said elastic, if $\epsilon <1$ the demand is said anelastic.
