Scarcity occurs when a resource has very limited availability. In other words, scarcity occurs when the supply of a good does not meet the demand of that good.
The most likely effect of ivory scarcity in the Ancient World, thus, was a reduction in the supply of ivory when compared to the demand for the good. Scarcity did not necessarily reduced demand, but it did reduce supply. This very likely made ivory a very expensive good at the time.
