At moment z, self-invested output ends and all change in value is explained by depreciation alone. It equals the entire accumulated value of dPV at final moment z.
That is,
D(z)dz=—dPV’(z)dz= dPV(z)=dPV(x)e™"™, (A8.3)
The following table shows some illustrations:
Depreciation Factor e’~™ if z—x is 50 Years
Interim z—x (years): 0 10 20 30 40 50 Factor if r(x) = .035: 174 247 350 A497 705 1 Factor if r(x) = .065: 039 074 142 Zits 522 1
This exactly reverses the analysis applied in national accounts, which models the
factor as decreasing rather than rising exponentially.
It should be stressed that these equations and this table describe each successive differential increment of outside investment (transfer in), not assets overall or groups of them. If transfer in were constant and continuous in an asset or group,
other things equal, overall depreciation would show as linear.
Free Growth Theory
By the total return truism (A1.6a), showing r = g + f, we derive
g=r-f, dg=dr-—df, and Ag=Ar-Af. (A9.1)
dg or Ag is “acceleration”, dr or Ar is “productivity gain” or “free growth rate”
and —df or —Af is “thrift gain”. Divide by acceleration to reach
APPENDIX A: The Argument in Notation 3/7/16 18
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