The differential equation y′=by1+α, for positive α and b, has solution
y=1(t0−t)1/α
(after changing the units). The Roodman report argues that our economy follows this hyperbolic growth trend, rather than an exponential one.
While exponential growth has a single parameter — the growth rate or interest rate — hyperbolic growth has two parameters: t0 is the time until singularity, and α is the “hardness” of the takeoff.
A value of α close to zero gives a “soft” takeoff where the derivative gets high well in advance of the singularity. A large value of α gives a “hard” takeoff, where explosive growth comes all at once right at the singularity. (Paul Christiano calls these “slow” and “fast” takeoff.)
Paul defines “slow takeoff” as “There will be a complete 4 year interval in which world output doubles, before the first 1 year interval in which world output doubles.” This corresponds to α≤2. (At α=2, the first four-year doubling starts at 163 and the first one-year doubling starts at 43 years before the singularity.)
So the simple hyperbola y=1/(t0−t) with α=1 counts as “slow takeoff”. (This is the “naive model” mentioned in footnote 31 of Intelligence Explosion Microeconomics.)
Roodman’s estimates of historical α are closer to 0.5 (see Table 3).
Hyperbolic growth
The differential equation y′=by1+α, for positive α and b, has solution
y=1(t0−t)1/α
(after changing the units). The Roodman report argues that our economy follows this hyperbolic growth trend, rather than an exponential one.
While exponential growth has a single parameter — the growth rate or interest rate — hyperbolic growth has two parameters: t0 is the time until singularity, and α is the “hardness” of the takeoff.
A value of α close to zero gives a “soft” takeoff where the derivative gets high well in advance of the singularity. A large value of α gives a “hard” takeoff, where explosive growth comes all at once right at the singularity. (Paul Christiano calls these “slow” and “fast” takeoff.)
Paul defines “slow takeoff” as “There will be a complete 4 year interval in which world output doubles, before the first 1 year interval in which world output doubles.” This corresponds to α≤2. (At α=2, the first four-year doubling starts at 163 and the first one-year doubling starts at 43 years before the singularity.)
So the simple hyperbola y=1/(t0−t) with α=1 counts as “slow takeoff”. (This is the “naive model” mentioned in footnote 31 of Intelligence Explosion Microeconomics.)
Roodman’s estimates of historical α are closer to 0.5 (see Table 3).