THE TUPY-POOLEY FRAMEWORK
MEASURING THE GROWTH OF KNOWLEDGE WITH TIME

How can we measure whether resources are getting more abundant or less abundant? Historically, many people have done so by estimating the quantity of a resource, such as crude oil, and dividing that estimate by the rate of consumption of that resource. If known reserves of a resource grew at a slower rate than the consumption rate, statisticians could predict the time when a resource were to run out. Or so they thought. This kind of a measurement has to be faulty, for despite millennia of economic activity and hundreds of years of observations and prognostications, humanity has never run out of a single nonrenewable resource. Don’t believe us? Try to think of one.

 

There are many reasons why the measurement of quantities does not yield expected results. First, quantities are often not directly observable. We really have no idea how much oil and gas, for example, we have. That’s because we do not have the know-how to measure the total supply of any given resource on the planet in a cost-effective way. Second, there is a large range of quality, grades, and concentration levels of resources. That makes the costs of discovery, extraction, and refinement of resources highly subjective. Third, because it is so expensive to perform surveys of resource reserves, those surveys usually end up having little or no value. To give an example, after more than a century of the intensive use of fossil fuels, we have more known deposits today of oil and gas than ever before. Fourth, we often don’t know where to look for resources. Although survey technology is improving, the location of new deposits often come as a surprise. Fifth, quantities do not account for substitutes and recycling. History suggests that many substitutes for current resources and ways to recycle those resources are yet to be discovered. Sixth, we can’t predict future discoveries of new deposits of resources and improved extractive technologies that may increase the supply of resources. Seventh, surveyors of resources face asymmetrical costs. If they overestimate resource reserves, companies lose money in a very visible way. If surveyors underestimate resource reserves, exploration does not take place, and the cost to companies is invisible. Finally, the surveyors may also want to underestimate resource reserves to increase their budgets and, consequently, their status.

 

Prices, we believe, are a much better way of ascertaining the relative abundance or scarcity of resources. If prices go up, resources are getting scarcer. If they go down, they are getting more abundant. Where do prices come from? In a free economy, nobody sets prices. They emerge spontaneously through the revealed preferences of individual buyers and sellers. Every time you buy a cup of coffee on the way to work, for example, you incrementally increase the price of coffee beans. Every time you fail to buy your usual morning cup of coffee, you decrease the price of coffee beans by a tiny amount. If everyone stopped buying coffee, the price of coffee beans would collapse. The movement of prices informs both producers and owners of capital about the goods and services that ought to be produced (cars and smartphones) and goods and services that ought to be discontinued (horse-drawn carriages and the telegraph). So the market is a mechanism (or, really, a process) for the gathering and exchange of information. Individual decisions made by billions of buyers and sellers generate useful knowledge that ensures the relatively smooth functioning of an economy (no permanent shortages of desired goods or permanent stockpiles of undesired goods) and the continued generation of wealth.

 

However, note that there are different kinds of prices. The simplest kind of price is the “nominal,” or current price, which you see in the shop when you buy a loaf of bread or fill up your car at the gas station. A more sophisticated, or “real” kind of price, is adjusted for inflation, which accounts for the well-known fact that in the era of fiat currency or government-issued currency, the dollars in your pocket get less valuable every year (i.e., the government tends to print more money than is necessary, thereby lowering the value of the currency relative to the total amount of goods and services in the economy). The real price, therefore, subtracts from the nominal price the rate of inflation, thus making it possible to compare prices or abundance of resources across time. What’s missing from both kinds of prices is the amount of dollars in your wallet. How often have you heard your grandparents complain that a gallon of gas cost 50 cents and a loaf of bread 5 cents “back in the good old days”? “True, Grandma and Grandad,” ought to be your response, “but what happened to your incomes over your working lives?”

 

Typically, though not always, individual incomes increase at a higher rate than inflation. That’s because people tend to grow more productive (i.e., they use new knowledge or inventions to generate more value per input, such as an hour of work, an acre of land, and amount of capital available) over their lifetimes and across time. Just think of the economic output or productivity of a worker with a shovel versus that of a driver of a giant excavator. It is that productivity growth that has resulted in our incomes today being much higher than that of our grandparents (adjusted for inflation, the real median personal income in the United States rose from $25,326 in 1974 to $35,805 in 2020). To correct for the problem of the missing wage increases, we have come up with the concept of “time prices.” Whereas nominal and real prices are measured in dollars and cents, time prices are measured in hours and minutes. To calculate a time price, all you need to do is to divide the nominal price of a good or service by your nominal hourly income. That tells you how long you must work to afford something. So long as your nominal hourly income increases at a faster pace than nominal prices do, goods and services get more abundant.

 

There are at least four reasons why time prices are better at measuring abundance than money prices. First, as mentioned, time prices contain more information than money prices do. Since new knowledge, which is to say innovation or productivity, both lowers prices and increases wages, time prices more fully capture the benefits of that new knowledge. To just look at prices, without also looking at wages, only tells half of the story. Time prices make it easier to see the whole picture. Second, time prices transcend the complications associated with converting nominal prices to real prices. They avoid subjective and disputed adjustments such as the consumer price index (CPI), the GDP deflator, and the purchasing power parity (PPP). Time prices use the nominal price and the nominal hourly income at each point in time, so inflation adjustments are not necessary. Third, time prices can be calculated for any product in any currency at any time and in any place. That means you can compare the time price of bread in France in 1850 to the time price of bread in New York in 2022. Fourth, time is an objective and universal constant. The International System of Units has established seven key metrics, six of which are bounded in one way or another by the passage of time. As the only irreducible element in the universe, time cannot be inflated or counterfeited. With 24 hours in a day, everyone has perfect equality of time. These four reasons make using time prices superior to using money prices for measuring resource abundance. Time prices are elegant, intuitive, and simple. They are the true prices we pay for the things that we buy in life.

 

Our analytical framework was inspired by the works of the late University of Maryland economist Julian Simon, Stanford University’s Hoover Institution economist Thomas Sowell, and the American bestselling author and economist George Gilder. Simon was a rigorous empiricist who found that, in the long run, resource prices declined; he postulated that the ultimate resource is the minds of free-thinking people. Sowell argued that economic growth is not about the quantity of atoms, which are finite, but about the increase in human knowledge, which is infinite. As he put it, “The cavemen had the same natural resources at their disposal as we have today, and the difference between their standard of living and ours is a difference between the knowledge they could bring to bear on those resources and the knowledge used today.” Finally, Gilder offers three propositions: wealth is knowledge, growth is learning, and money is time. From these propositions we derived a theorem, which states that the growth in knowledge can and should be measured with time. This measurement can be performed at individual and global levels. Let us show you how.

The Framework

The Tupy-Pooley Framework is a way of measuring the growth of knowledge and, consequently, innovation, productivity, and standards of living, with time. We analyze abundance at the personal or individual level and at the global or population level. The framework includes 13 basic equations: 6 concerning the Personal Resource Abundance level, 4 concerning the Population Resource Abundance level, and 3 concerning elasticities.

Personal Level

1. Time Price = Money Price ÷ Hourly Income 

Time price denotes the amount of time that a buyer needs to work to earn enough money to be able to buy something.

Consider bananas. If bananas cost 50 cents a pound and you earn $10 an hour, a pound of bananas will cost you 1/20th of an hour or three minutes of work:

 

Time Price = $0.50 ÷ $10.00 = 0.05 hours = 3 minutes

2. Percentage Change in the Time Price = (Time Price[end] ÷ Time Price[start]) - 1

The percentage change in time price over time provides more valuable information than individual time prices can provide. The time price is like a picture. It provides a snapshot of people’s standard of living at any given point in time. The percentage change in time price is like a movie. It allows us to observe long-term patterns in the abundance of goods and services.

 

To see if you are better off today than in the past, let’s compare time prices over time. If the money price of bananas increases to 60 cents a pound, but your income increases to $18 an hour, a pound of bananas will now cost you only two minutes of work. The time price of bananas has decreased by 33 percent:

 

Percentage change in the time price = (2 ÷ 3) - 1 = 0.66 - 1 = -0.33 = -33%

 

The most important thing to remember is that so long as the hourly income increases at a faster pace than the money price, the time price will decrease.

3. Personal Resource Abundance Multiplier = Time Price[start] ÷ Time Price[end]

The personal resource abundance multiplier tells you how much more of a resource you can get for the same amount of labor between two points in time. You can also think of the multiplier as showing how much more or less abundant a resource has become from the perspective of an individual over time.

 

The personal resource abundance multiplier is the ratio of the start-year time price over the end-year time price. With respect to our banana example:

 

Personal resource abundance multiplier = 3 ÷ 2 = 1.5

 

In other words, the working time required to earn enough money to buy a banana in the start year of the analysis will purchase one and a half bananas in the end year of the analysis. So long as the multiplier is greater than one, your personal resource abundance is increasing.

4. Percentage Change in Personal Resource Abundance = Personal Resource Abundance Multiplier - 1

The percentage change in personal resource abundance tells you how much better off you have become between two points in time.

 

Using our banana example, we divide three by two and then subtract one:

 

Percentage change in the personal resource abundance = 1.5 - 1 = .5 = 50%

 

The abundance of bananas has increased by 50 percent between the start-year of the analysis and the end-year of the analysis. The same amount of time it took in the start-year to earn one pound of bananas will get you one and a half pounds of bananas in the end-year.

 

Note that since the personal resource abundance multiplier always equals one in the start-year of the analysis, you can simply subtract one from the personal resource abundance multiplier and get the same answer.

5. Compound Annual Growth Rate in Personal Resource Abundance =

[Personal Resource Abundance Multiplier ^ (1÷ Years)] -1

Once we have the personal resource abundance multiplier, we can then calculate the compound annual growth rate of personal resource abundance (i.e., the speed at which personal resource abundance is growing).

 

The compound annual growth rate in personal resource abundance is calculated by the end-year personal resource abundance multiplier, raised to the power of one divided by the number of years, minus one.

 

In our banana example, the personal resource abundance multiplier of bananas increased from one to one and a half over a 10-year period. So, to calculate compound annual growth rate in personal resource abundance:

 

Compound annual growth rate in personal resource abundance = [1.5 ^ (1÷ 10)] -1 = 1.04138 - 1 = 0.04138 = 4.138%

 

You can also calculate the compound annual growth rate in personal resource abundance value using the RATE function in a spreadsheet like Excel or Google Sheets or by using a financial calculator.

6. Years to Double Personal Resource Abundance = 70 ÷ Compound Annual Growth Rate

The doubling period refers to the length of time required for a resource to become twice as abundant. To calculate the doubling period, we can use the rule of 70s. To calculate the number of years needed for personal resource abundance to double, we need to divide 70 by the compound annual growth rate in personal resource abundance.

 

In our example, abundance of bananas increased from one to one and a half in 10 years, indicating a compound annual growth rate in personal resource abundance of 4.138 percent. Therefore:

 

Years to double personal resource abundance = 70 ÷ 4.138 = 16.9 years

 

You can also calculate this value using the NPER function in a spreadsheet like Excel or Google Sheets, or by using a financial calculator.

Population Level

We extend our analysis of resource abundance from the personal level to the global level by adding population as a factor. You can think of the difference between the personal and population levels analysis by using a pizza analogy. Personal resource abundance measures the size of a slice of pizza per person. Population resource abundance measures the size of the entire pizza pie. The size of the pie is calculated by the size of the individual slice or personal resource abundance multiplied by the total number of slices or population.

7. Population Resource Abundance Multiplier = Personal Resource Abundance Multiplier x Population [end] ÷ ((Personal Resource Abundance Multiplier x Population) [start])

The population resource abundance multiplier tells us how much more of a resource a population can get for the same amount of labor between two points in time.

 

Let’s think, once again, about our bananas. If there are four people and each of them has one banana, then you have a total of four bananas. If a decade later the time price of bananas falls by 50 percent, you now get two bananas for the time price of one. Personal resource abundance of bananas has increased by 100 percent.

 

If during the same period the population increases from four to six people, then population has increased by 50 percent. The population resource abundance of bananas will go from one times four to two times six. Therefore:

 

Population resource abundance multiplier = (2 x 6) ÷ (1 x 4) = 12 ÷ 4 = 3

8. Percentage Change in the Population Resource Abundance =

Population Resource Abundance Multiplier - 1

Once we have the population resource abundance multiplier, we can then calculate the percentage change in population resource abundance over time. The percentage change in population resource abundance tells us how much better off the whole population has become between two points in time.

 

To calculate the percentage change in population resource abundance of bananas between two points in time, subtract one from the population resource abundance multiplier. Convert this value to a percentage by multiplying it by a hundred.

 

So, in our banana example, the percentage change in population resource abundance of bananas will increase by 200 percent:

 

Percentage change in the population resource abundance = 3 - 1 = 2, or 200%

9. Compound Annual Growth Rate in Population Resource Abundance =

[Population Resource Abundance Multiplier ^ (1÷ Years)] -1

The compound annual growth rate in population resource abundance tells us the rate of improvement in population resource abundance between two points in time.

 

For example, if the population resource abundance multiplier of bananas increased from one to three over a 10-year period, then the compound annual growth rate in population resource abundance equation would amount to three raised to the power of 1 divided by 10, or 0.1. That amounts to 1.1161 minus 1, or 0.1161, which equates to 11.61 percent.

 

Compound Annual Growth Rate in Population Resource Abundance =

[3 ^ (1÷ 10)] -1 = 1.1161 - 1 = 0.1161 = 11.61%

 

You can also calculate this value using the RATE function in a spreadsheet like Excel or Google Sheets or by using a financial calculator.

10. Years to Double Population Resource Abundance =

70 ÷ Compound Annual Growth Rate in Population Resource Abundance

The doubling period refers to the length of time required for a resource to become twice as abundant at a population level. To calculate the doubling period, we can use the rule of 70s. To calculate the number of years needed for population resource abundance to double, we need to divide 70 by the compound annual growth rate in population resource abundance.

 

In our example, the population resource abundance of bananas increased from one to three in 10 years, indicating a compound annual growth rate in the population resource abundance of bananas of 11.61 percent. Therefore,

 

Years to double population resource abundance = 70 ÷ 11.61 = 6.02 years

 

You can also calculate this value using the NPER function in a spreadsheet like Excel or Google Sheets or by using a financial calculator.

Elasticities

In economics, elasticity measures one variable’s sensitivity to a change in another variable. If variable X changes by 10 percent and variable Y changes by 5 percent, then the elasticity coefficient of X relative to Y will equal to 10 divided by 5, or 2. That means that a 2 percent change in X corresponds to a 1 percent change in Y. We have three elasticity equations in our framework.

11. Time Price Elasticity of Population = Percentage Change in the Time Price ÷ Percentage Change in the Population

We can use the concept of elasticity to estimate the sensitivity of changes in time prices to the changes in population. If time price decreases by 50 percent and population increases by 75 percent, then the time price elasticity of population will equal to -50 divided by 75, or -0.66. Therefore, we can say that every 1 percent increase in population corresponds to a -0.66 percent decrease in time price:

 

Time price elasticity of population = -50% ÷ 75% = -0.66

12. Personal Resource Abundance Elasticity of Population =

Percentage Change in Personal Resource Abundance ÷ Percentage Change in the Population

The personal resource abundance elasticity of population measures the sensitivity of personal abundance of resources to population growth. Thus, if personal resource abundance increases by 100 percent and the population increases by 75 percent, then the personal resource abundance elasticity of population will equal to 100 divided by 75, or 1.33. Therefore, we can say that every 1 percent increase in population corresponds to a 1.33 percent increase in personal resource abundance:

 

Personal Resource Elasticity of Population = 100% ÷ 75% = 1.33

13. Population Resource Abundance Elasticity of Population =

Percentage Change in Population Resource Abundance ÷ Percentage Change in the Population

The population resource abundance elasticity of population measures the sensitivity of population abundance of resources to population growth. Thus, if population resource abundance increases by 200 percent and population increases by 75 percent, then the population resource abundance elasticity of population will equal 200 divided by 75, or 2.66. Therefore, we can say that every 1 percent increase in population corresponds to a 2.66 percent increase in population resource abundance:

 

Population Resource Elasticity of Population = 200% ÷ 75= 2.66

Visualizing Population Resource Abundance

We can now use our banana example to illustrate the growth in population resource abundance over time. The goal of this visualization is to help you see the magnitude of the difference in population resource abundance of bananas between the start-year of the analysis and the end-year of the analysis. We will use a red box to indicate population resource abundance of bananas in the start-year of the analysis and a green box to indicate population resource abundance of bananas in the end-year of the analysis. In the visualization, the horizontal axis denotes population and the vertical axis denotes personal resource abundance of bananas. The start-years and end-years of the respective analyses are depicted in the boxes themselves.

 

To begin, we have indexed personal resource abundance and population size in the start-year of the analysis to one. Consequently, the red box depicting the start-year of the analysis is a square measuring one by one. That way, we can use the same scale to compare relative changes in population and personal resource abundance of bananas.

 

Next comes the green box, which depicts population resource abundance of bananas in the end-year of the analysis. The green box has increase along the horizontal dimension based on the percentage increase in population and along the vertical dimension based on the increase in personal resource abundance of bananas. The area colored in green thus represents all the “new” population resource abundance that was created between the start-year and the end-year of the analysis.

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Superabundance

In our book, we define “superabundance” as a situation where personal resource abundance increases at a faster pace than population growth. To ascertain how widespread that phenomenon is, we have looked at 18 different datasets, containing hundreds of commodities, goods, and services spanning two centuries. To our surprise and delight, all 18 datasets fell into the superabundance zone, where personal resource abundance increases at a faster pace than population growth. That tells us that, on average, every additional human being creates more value than he or she consumes. This relationship between population growth and abundance is deeply counterintuitive – if you’ve been thinking like an accountant instead of an economist – yet it is true. Economists, such as Julian Simon, Thomas Sowell, and George Gilder, understood that more people produce more ideas, which lead to more inventions. People then test those inventions in the marketplace to separate the useful from the useless. At the end of that process of discovery, people are left with innovations that overcome shortages, spur economic growth, and raise standards of living.

 

But large populations are not enough to sustain superabundance―just think of the poverty in China and India before their respective economic reforms. To innovate, people must be allowed to think, speak, publish, associate, and disagree. They must be allowed to save, invest, trade, and profit. In a word, they must be free.

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