  Product diversity, which is highly important in economic systems, has been
highlighted by recent studies on international trade. We found an empirical
pattern, designated as the "S-shaped curve", that models the relationship
between economic size (logarithmic GDP) and export diversity (the number of
varieties of export products) on the detailed international trade data. As the
economic size of a country begins to increase, its export diversity initially
increases in an exponential manner, but overtime, this diversity growth slows
and eventually reaches an upper limit. The interdependence between size and
diversity takes the shape of an S-shaped curve that an be fitted by a logistic
equation. To explain this phenomenon, we introduce a new parameter called
"substitutability" into the list of capabilities or factors of products in the
tri-partite network model (i.e., the country-capability-product model) of
Hidalgo et al. As we observe, when the substitutability is zero, the model
returns to Hidalgo's original model but failed to reproduce the S-shaped curve.
However, in a plot of data, the data increasingly resembles an the S-shaped
curve as the substitutability expands. Therefore, the diversity ceiling effect
can be explained by the substitutability of different capabilities.
