Draft:Pooley-Tupy theorem

The Pooley-Tupy theorem is an economics theorem which measures the growth in knowledge resources over time at individual and population levels.

$$Percentage \ Change \ in \ Knowledge \ Resources = \left ( \frac{Time\ Price_t}{Population_t} \right ) \div \left ( \frac{Time\ Price_{t+n}}{Population_{t+n}} \right )-1$$

The theorem was formulated by Gale Pooley and Marian Tupy who developed the approach in 2018 in their paper: The Simon Abundance Index: A New Way to Measure Availability of Resources

The theorem is informed by the work of Julian Simon, George Gilder, Thomas Sowell, F. A. Hayek, Paul Romer, and others.

Gilder offers three axioms; wealth is knowledge, growth is learning, and money is time. From these propositions a theorem can be derived: The growth in knowledge can be measured with time.

While money prices are expressed in dollar and cents, time prices are expressed in hours and minutes. A time price is equal to the money price divided by an hourly income rate.

$$Time \ Price = \frac{Money \ Price}{Hourly \ Income}

$$

The Pooley-Tupy theorem adds changes in population as an additional variable in their formulation. In the case of an individual, population is equal to 1 at $$t$$ and $$t+n$$.

Examples
If knowledge resources were being evaluated at the individual level and the time price was 60 minutes at $$t$$ and 45 minutes at $$t+n$$, the percentage change in knowledge resources would be:

$$= (60 \div45) - 1 $$

$$= 1.33 -1 $$

$$=0.33 =33%$$

If population at $$t$$ was 100 and 200 at $$t+n$$, the percentage change in knowledge resources would be:

$$= (60 \div100) \div (45 \div200)-1

$$

$$= (.6) \div (.225)-1$$

$$= 2.666-1$$

$$= 1.666 =166.6%$$

Other equations
The Pooley-Tupy Theorem is part of an analytical framework that uses several other equations for analysis. This framework is described in their book, Superabundance: The story of population growth, innovation, and human flourishing on an infinitely bountiful planet.

The percentage change in a time price over time can be expresses as:

$$Percentage \ Change \ in \ Time \ Price = \frac{Time\ Price_{t+n}}{Time\ Price_{t}}-1 $$

The resource multiplier indicates how much more or less of a resource the same amount of time can buy at two points in time.

$$Resource \ Multiplier = \frac{Time\ Price_{t}}{Time\ Price_{t+n}}$$

The percentage change in the resource multiplier is just the resource multiplier minus one.

$$Percentage \ Change \ in \ Resource \ Multiplier = Resource\ Multiplier -1$$

The compound annual growth rate or CAGR can be calculated as:

$$Compound\ Annual\ Growth\ Rate = Resource\ Multiplier^{1/n} -1$$