dr of anit die ffi am Aen, (A9.2) dg dg dtdg dtdg g g Ag Ag
drK, or Ark, give free growth as a flow, while —dfK, or —AdfK,, give the flow of
thrift.
Define the “productivity index” or “free growth index” @ (phi)as r/g or Ar/Ag,
and the “thrift index” @ (theta) as -f /g or —Af /Ag.(A9.2) can then be put as
g+@=1, (A9.2a)
in either the continuous time or discrete period sense.
Free growth theory is the prediction that @ at the collective scale will average unity (the number one), implying that 0 averages zero, when @ or @ is measured for each year or for shorter periods if practical. Thrift theory makes the opposite prediction @—1 and g—>0. The point is to compare simultaneous changes in acceleration and thrift, and then find the long-term average of these simultaneous observations, rather than compare long-term changes in the first place. If free growth is right, they will prove uncorrelated. That is exactly what the charts and tables show whenever data are available. Acceleration is as likely to coincide with
unthrift, meaning increase in consumption rate C/K, as with thrift.
Division of (A9.1) by acceleration was not essential to the logic. It added the
convenience of index numbers totaling unity.
The test should be as fine-grained as practical. If the Piketty-Zucman website showed quarterly or monthly data revealing any two of r , f and g,I would have
averaged the largest number of shortest periods. What I try to compare is ex ante
APPENDIX A: The Argument in Notation 3/7/16 19
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