The ULF is based on Equation (7) expanded with [[beta].sub.3] (1/ln participants) + [[beta].sub.4] (l/(In participants) (2)), while the HACD cost model is equal to Equation (7), where the second output term is replaced by [[beta].sub.2] (1/participants).
The third cost function considered is the hyperbolic HACD model (see column 3 in Table 6).
The simplified ULF (SULF) and the TCF have AIC values at almost equally low levels (38,366 and 38,365, respectively), while the HACD's AIC, at 38,465, is "significantly" higher, so that the HACD model appears to be less well-suited.
The estimated cost elasticities for the mean pension fund size show a similar pattern for all considered cost functions: fluctuation somewhat around 0.71 (TCF and ULF) or around 0.74 (HACD), rising somewhat to a higher level during 1997-2005, and sliding back in recent years.
For the HACD model Table 7 shows scale economies only, because this model does not indicate an optimal scale.
For the reciprocal term in the HACD model, arguments (to replace o) exist for both the geometric and arithmetic mean.