Regarding the first commercial production of his ‘horseless carriage’, Henry Ford supposedly said: “If I had asked people what they wanted, they would have said faster horses.”

(Ford apparently never said that, but even so it’s still a good quote.)

I made this graph below from an article I saw at Wikipedia. It shows the size of largest container ship in the world, in terms of the number of twenty-foot equivalent units (TEUs) it can carry, as a function of the year the ship was launched.

There is one ship, the Emma Mærsk, launched in 2006 with a capacity of just over 15000 TEUs, that stands out from the rest. But otherwise, the data points all seem to fall along a path that generally rises over time. Maybe the path is S-shaped and it seems like the three record-setters launched in 2017, the OOCL Hong Kong, the Madrid Mærsk, and the MOL Triumph, all with capacities greater than 20000 TEUs are nearing the size of the largest container ships that will ever get built.

These ships are monsters – early 400 meters (a quarter of a mile) long and over 16 meters (50 feet) at their deepest. This is twice the size of the largest ships in 2005. I am asking myself: Is this the logical endgame in the shipping industry? Just build a new largest ship, 1000 TEUs larger than last year’s largest, every year forever?

To me, it feels like ‘bigger ships’ is the shipping industry’s equivalent of ‘faster horses’. Maybe we will keep building ever larger ships. Or maybe 3D-printing will allow individuals, small companies or large companies, to make goods in Europe and the US that are now being made in China.

I have been seeing a lot of articles recently about innovations that seem to me to be ‘faster horses’ in agriculture, transportation, electricity and other areas.

Like ships, like largest passenger planes keep getting larger and the longest regular commercial routes keep getting longer. In 2001, Continental Airlines ran a 16-hour flight from Newark to Hong Kong. In 2004, Singapore Airlines flew an 18-hour flight from Newark to Singapore. In 2016, according to the BBC, a flight from Dubai to Auckland, New Zealand, had a scheduled duration of 17 hours 15 minutes. And a planned route from Dubai to Panama City will have a duration of over 17 and a half hours.

Again, ‘longer flights’ feels like ‘faster horses’. Will we still be flying commercial aircraft 500 years from now?

SpaceX is building a rocket that can hold a couple of hundred people and fly from New York to London in less than 30 minutes. They say it should be able to go from any point on Earth to any other in under an hour. Autonomous electric vehicles (AEVs) could replace short-haul flights. That feels more like the horseless carriage compared to the airline industry’s plans for larger planes, small supersonic craft, and longer routes.

Don’t worry about the timeframe – Just ask yourself what is the most logical endpoint, maybe 200 or 500 or 1000 years from now.

For energy, the most logical endpoints I see are solar power or fusion. In 1000 years (or sooner) we will have used up all of the oil, coal and natural gas. In 1000 years, we could also use up all of the uranium for fission. But in 1000 years the Sun will still be shining.

I have seen some articles recently saying that to accomodate more electric vehicles, we need to increase the capacity of electrical supergrids (the large trunk lines that form the backbone of the national-scale energy system). But the more energy we get from solar power, the less we will need giant electricity transmission and distribution infrastructure.

I have seen videos on YouTube of ‘exoskeletons’ – robotic suits that construction workers can wear to assist in lifting heavy weights and moving things. Here’s an article about big mining companies using sensor-equipped hats and helmets to detect fatigue among their vehicle drivers. Won’t these ideas just get wiped out by self-operating construction vehicles and humanoid robots?

I guess all technologies are transitional, and therefore exoskeletons and brain-wave helmets could be useful for a while until the robots arrive. But with technological change getting faster and faster, doesn’t it just make sense to spare the horses and hop on board the new ride?


Back when I was about 13 and the Soviet Union still existed, my social studies textbook contained a photo from a supermarket somewhere in Russia.

I think back about it every now and then. If I recall right, it was a black and white photo down a row of shelves that were almost empty except for a roll of toilet paper. Two women were standing separately in the aisle, each of them looking at that single toilet paper roll.

I realize that a photo is just a moment in time and maybe those women only looked at that roll for a second. But spine of my book had cracked so that basically every time I opened the book, it opened first to that photo. All year long, every time I opened the book, there were the same two women looking at that same roll of toilet paper, never finally deciding whether to buy it or not.

I guess what has stayed with me all these years about that photo is that, at the time, we were told that the Soviet Union was the world’s only other superpower besides the US. (They had beaten us into space!) And here with a basic staple item that is not particularly difficult to manufacture, transport or store, and that does not spoil, the Soviets forced to ladies to stand there, staring at a single roll of toilet paper forever.

Tony Robbins talks about the effect of taxes and fees on saving capital in his latest book, Unshakeable, and I thought one example he gives in his Unleash The Power Within seminar is so remarkable, simple as it is, that I had to recreate the numbers myself to check how it works.

Say that you start with $1 on the first of January and can put your money (capital) into an investment that will double one year later. At the start of year 2 you will have $2. At the start of year 3 you will have $4. At the start of year 4 you will have $8. How much money will you have after 20 years (the start of year 21)?

You can see in the table below that doubling your money each year for 20 years will leave you with over $1 million dollars. (A 100% return every year for 20 years is completely unrealistic, but it makes the math easier to follow).

What if your money doubles each year, but at the end of the year, someone (a tax collector or a professional money manager) takes 30% of the increase in your capital. So, when your initial $1 doubles to $2, the taxman (or money manager) takes 30 cents (30% of the $1 increase, not 30% of the full $2). How much will you be left with at the end of 20 years?

Again, we can see in the table. The first year, the fee will be $0.30, leaving you with $1.70 instead of $2.

The second year, your $1.70 will double to $3.40, but the fee will be 30% of the $1.70 increase (or $0.51), leaving you with $2.89.

The third year, your $2.89 will double to $5.78, but the fee will be 30% of the $2.89 increase (or $0.87), leaving you with $4.91.

So, by the start of year 4 you will have $4.91 instead of $8 and by the start of year 5 you will have $8.35 instead of $16. This 30% tax (or fee) has essentially left you one year behind by the start of year 5.

How much will you have in this scenario by the end of year 20 (the start of year 21)?

You can see from the table that it will be less than $41,000 ($40,642.31). That 30% fee leaves you in the end with less than 4% of what you would have had had there been no fee at all.

Where did the other 96% go? If we add up the fee collected each year, it only amounts to $29,610.40. In other words, you get about $41,000, the fee collector gets about $30,000 (for a total of about $71,000) and the other 93% of the money – $978,323.29 – simply disappears because it was never in the interest-generating pile of capital.

Now this example is extreme because no investment returns 100% per year every year for 20 years and taxes on capital gains (in the US) are more like 15%, taken only when you sell the asset at the end, not 15% of each year’s gain. However, some hedge funds charge ‘2 and 20’ (2% of your total capital, plus 20% of the increase), so a more realistic example might be 10% growth per year with a 15% annual fee.

Even in that case, you end up with about 76% of what you would have had if there had been no fee, the total fee amounts to about 12% of what you would have had, and then other 12% is simply destroyed.

I guess the same principle applies to non-financial matters as well. If the work you do builds on itself over time, and 15% or 30% of your effort each year goes toward unproductive tasks, then what you are left with at the end of 20 years or so could realistically be only a very small fraction of what it otherwise could have been.

Sales Velocity, the rate at which a sales team brings in money, is a common key performance indicator (KPI) of sales team effectiveness.

But, there are several ways to increase Sales Velocity and it is not necessarily obvious which ways are the best in a given situation., a software company that helps you to “identify the most valuable sales prospects for your business”, has produced a short ebook called ‘Mathematics for B2B Sales‘, which deals with Sales Velocity and ways to increase it.

They give the equation for Sales Velocity (amount of sales per time) =


Leads in Progress x Hit Rate x Average Deal Size / Sales Cycle Time


To show how it works, they give the example of a company that can acquire new customers through outbound marketing.

Outbound sales opportunities are generated by the company’s 10 salespeople, each of whom contacts 5 prospective customers per week. It takes prospects 4 weeks, on average, to either decide to become a customer or to decline to make a purchase. 10% of prospects (the ‘hit rate’) become customers while the other 90% decline. Each sale is worth $9000 on average.

The graphic below shows the mechanics of how this process works. Sales people generate Prospects who make a purchase decision, on average, in 4 weeks. 10% of these prospects buy a unit of the product, with an average deal size of $9000.

Since salespeople each generate 5 prospective customers per week and the decision-making time is 4 weeks, there are 20 active prospects (from outbound marketing) per salesperson at any given time.

Vainu’s ebook lists several ways to increase the company’s Sales Velocity. Some of these suggestions include:

  • hire more salespeople
  • improve the hit rate
  • shorten the sales cycle time

However, not all of these methods are equally effective. We have actually already seen this basic model before, in my earlier post ‘Deeper Into The Equation that Governs Your Sales Team’s Effectiveness‘.

The graph below shows what happens to Sales Velocity when in week 10 we decrease the sales cycle time from 4 weeks down to just 2 weeks.

The Sales Velocity (dollars in sales per week) spikes as the large number of Prospects in Progress make their purchase decision. But, as these people make their decision, the number of Prospects in Progress drops, and with it, the Sales Velocity returns to its base value.

That is, the ‘sales cycle time’ and the number of active prospects are mechanically related to each other. Changing either one of these numbers changes the other.

The term ‘Sales Velocity’ sounds impressive and the equation above makes it look like there are four key levers you can pull to increase it. But ultimately, the term ‘Sales Velocity’ is just another word for the dollar rate of sales ($ in sales per week) and the sales rate must be limited by the rate of generating prospects (the inflow pipe to the pool of Prospects in Progress), not the amount of time it takes for a new prospect to make a decision.

There might be some good reasons to reduce the sales cycle time, but hoping that doing so will increase sales or profit are not among them.

App in the Air is a mobile phone app for frequent fliers that includes flight information updates, airport maps from around the world, local weather conditions and currency exchange rates. It works on a freemium monthly subscription model: a basic version of the product is available for free, while access to a version with more features can be purchased for a monthly fee.

Businesses based on apps like these often share many of the same problems: With thousands of popular apps available for free (or low cost), it can be difficult to get potential users to hear about and download your product. Many people download an app and try it a few times, but then quickly lose interest and never use it again. Other people become regular users of the free version of the product, but never upgrade to the paid version. And many people who are paying customers let their subscriptions lapse after just a few months or a year.


Description of the Model

App in the Air’s CEO, Bayram Annakov, built a simulation model to capture much of this process. The diagram below is a simplified version of his work. There are two main portions: the main chain that tracks Potential Users becoming New Users, Retained Users and finally Paying Users. The second portion tracks Cash, income and expenses.

The stock-and-flow structure of the ‘App in the Air’ business, including some important parameter values.


‘Potential Users’ (near the upper-left corner of the diagram) download the app, making them New Users. Only about 20% of these new users retain using the app after the first month, becoming ‘Retained Users’. The other 80% leave the system, never to download or use the app again. Each month, some portion of the retained users who have the free version choose to upgrade to the monthly payment plan, making them ‘Paying Users’ and some portion of the Paying Users each month cancel their subscription.

It is Paying Users who generate the company’s income, by paying a monthly subscription price. One of the business’s main expenses is advertising, which raises awareness of the product and therefore helps to drive downloads. In advertising parlance, the amount of money spent to acquire a new customer is the ‘cost per acquisition’, or ‘CPA’. If it currently costs $50 on average to acquire a new customer, then we can find the number of downloads driven by advertising by dividing the ‘monthly advertising spending’ ($20,000) by the CPA. As the pool of potential users dries up, the cost per acquisition (CPA) of a new user should tend to go up. In this model, we simply divide the initial number of potential users by the current number. So, when the number of potential users falls to half of its original value, the CPA rises to $100. When the number of potential users remaining falls to one-quarter of its original value, the CPA doubles again.

The second means for acquiring new users is from referrals by existing users (both the ‘Retained Users’ who have the free version of the product and the ‘Paying Users’). We represent the rate of ‘downloads from referrals’ is by analogy to the rate of infection of a disease. The combined number of retained and paying users essentially ‘infect’ the remaining potential users. The rate of this is: 0.4 (Retained User + Paying Users) (Potential Users / INIT(Potential Users))


Identifying Leverage Points

What can we do to have the best impact on the company’s business? With our model, we can simply test these changes and compare the results.

First, let us consider the ‘base case’:

(A) 80% ‘churn rate’ for New Users, 2% of Retained Users become Paying Users each month, and 5% of Paying Users cancel their subscription.

Maybe with some effort we can do one of the following:

(B) reduce the ‘churn rate’ (the fraction of new users who stop using the app within the first month) from 80% to 76%.

(C) decrease the ‘subscription cancel ratio’ (the fraction of Paying Users who give up their subscription each month) from 5% to 4%.

(D) increase the fraction of Retained Users who become Paid Users each month from 0.02 to 0.04.

Which of these four things would have the best impact on the business?

The figure below shows the amount of Cash the business would have over time for each of these four cases. (For each graph, we have stopped drawing at the point of maximum cash on hand. After that point, expenses are always greater than income and the business begins permanently losing money.) In the base case, in the first months the company would lose money, since initially there are no Paying Users. The company’s expenses would exceed its income and it would see its initial pile of $500,000 in cash dwindle to just under $67,000 before the company became profitable. The amount of Cash on hand would peak at about $5.7 million at month 189.

The figure also shows the effect of (B) reducing the churn rate from 80% to 76%, (C) decreasing the subscription cancel ratio from 5% per month to 4% per month, and (D) increasing the fraction of Retained Users who become Paid Users from 2% per month to 4% per month.

Which of these is the best option for the company? Surprisingly, they all are.

For case C, decreasing the subscription cancellation ratio (the fraction of Paying Users who stop their paid plan) from 5% to 4% has obvious economic benefits. It is the equivalent of increasing the average lifetime of a customer from 20 months to 25 months. At a subscription price of $25 per month, this increases the lifetime value of a customer (LTV) by $125. The result is that the maximum amount of cash the company holds would now reach a peak of over $7.8 million around 200 months after the product launch. This is an increase of more than $2 million over the life of the company compared to the base case.

If our only goal is to maximize the total profit the company makes, then it looks like reducing the subscription cancel ratio is the best place to focus our efforts. However, both the base case and case C result in the company early in its life reaching a cash position of less than $80,000. If any of our estimates from the model – the initial pool of Potential Users, the churn ratio, the  rate of downloading the app – are wrong, or if there are any large unexpected expenses in the first few years of the company’s operation, then the company’s cash position could reasonably drop to zero and it would go bankrupt.

To minimize the chance of the company going bust, we should want the company’s minimum cash position to be as large as possible. For the cases we are considering here case D, increasing the fraction of Retained Users who become Paid Users each month from 2% to 4%, results in the company reaching profitability without the cash position ever going below $165,000. The maximum cash position for this case would be $6.55 million in month 158. So, increasing the ‘conversion to paid’ fraction results in less total profit overall, but with a lower chance of going bankrupt. If our goal is to minimize the chance of going bankrupt, case D, focusing on increasing the rate at which Retained Users become Paying Users, is our best option.

But if our goal is to maximize the rate at which we earn profit, then we are better off with case B, reducing the churn rate from 80% to 76%. At month 125, the cash position will be $6.35 million, for a net profit of $5.85 million in only 125 months, or $46,800 per month. We could quit (or sell) the business at that point and move on to the next venture, leaving someone else to squeeze the last profit out of a declining business over its last five or six profitable years.

So, depending on whether our goal is to maximize the company’s total profit, minimize the chance of going bankrupt, or maximize the rate per month at which we earn profit, we should focus our efforts on improving three separate aspects of the company’s business.

I found this article from 1998 online, ‘The Simple Economics of Easter Island‘, by James Brander and M. Scott Taylor.

The authors consider some small South Pacific islands, especially Easter Island, and have built a simple mathematical model that they claim helps explains the island’s growth and decline in human population from its settlement around the year 400 AD until the arrival of Europeans in the 18th Century.

The model is a version of a predator-prey system with people as the hunter and forest as the prey. The forest would naturally grow exponentially at a rate of about 4% per decade, but its actual growth rate is constrained by a ‘carrying capacity’, the maximum area of trees the island can sustain when it is completely covered in forest. As the forest cover increases toward the island’s carrying capacity, the growth rate of additional forest slows down.

People harvest the forest as one of their resources. In the absence of any harvesting, the human population would decline 10% per decade. Population is gained in proportion to the amount of resource harvesting.

The equations in the model are shown below

‘Resource’ is the number of (acres of) trees on the island and ‘Population’ is the number of people on the island. ‘b’ is the fraction of people involved in harvesting trees (so, ‘b x Population’ is the number of people involved in harvesting trees) and ‘a’ is their ‘harvesting productivity’, according to the authors.

Why does the harvesting rate depend on the number of trees? One reason is that the higher the number of trees, the easier it is to harvest them (and vice-versa). As the number of trees drops, workers must go further from their homes to access the remaining trees and use more time to bring the wood back to the village.

The authors suggest that 40% of the population is involved in harvesting resources. “Various pieces of evidence suggest that the resource sector probably absorbed somewhat less than half the available labor supply. A value of 0.4 for B is probably in the reasonable range.”

These equations and parameter values reproduce what the authors claim was the basic population dynamic over 1500 years of Easter Island history. A small population grew to about 10,000 people by the year 1200 and then slowly declined to around 3000 people by the time of first contact with Europeans.

I am interested in the values of Population, Resources and ‘harvesting Resources’ if we run the simulation for a long time. For the case of b=40%, the Population will eventually settle down to around 4790 people and the resource reaches about 6250 acres of trees (just over half the carrying capacity) if we run the calculations long enough. Thus, there are about 1.3 acres of trees per person. The long-term harvesting rate is about 0.025 acres (about 10 meters by 10 meters) per person per year.

What if the value of ‘b’ (the portion of the population involved in resource harvesting) was some other value, like 20% or 80% or 100%? What would happen to the Population, and Resources per person?

The graph below shows the long-term (that is, ‘equilibrium’) Population for various values of ‘b’.

If ‘b’, the fraction of people who are involved in harvesting resources, is less than about 20%, the population eventually falls to 0, as the rate of losing population always exceeds the rate of gaining population. For this model, the largest sustainable population is about 4800 people when ‘b’ is 42% and it falls to about 3166 as ‘b’ rises to 100%.

One measure of the prosperity of the Easter Island society is the number of acres of forest per person (that is, the quantity ‘Resources’ divided by ‘Population’). The forest serves not only as a source of wood, but also as a habitat for birds and nuts the islanders can eat, so more trees per person is, in itself, a measure of how prosperous the society is.

The graph below shows the long-term (‘equilibrium’) ratio of Resources / Population for various values of the ‘b’ parameter. When 40% of the population is involved in harvesting resources, the Population eventually reaches its largest possible long-term value. That point is marked with a dark blue dot on the graph.

Raising the portion of the population harvesting resources above 42% causes the equilibrium population to decrease and also causes the Resources/Population to decrease. When 100% of the population is involved in harvesting resources, the population settles down to about 3166 and the Resources/Population to about 0.8. (That point is marked with an open circle.)

There is another value of ‘b’ that also results in a long-term population of 3166 and that is when the portion of people harvesting resources is around 26.3%. In that case, the Resources/Person is about 3.

Another measure of the prosperity of the society is the per-person rate of harvesting resources (‘harvesting Resources’ / Population). The forest is a source of wood, which is useful for making houses, canoes, tools and fire. So, the amount of forest cut down per time tells how much wood is available per person. Interestingly, for all of the different ‘b’ values above 20%, the long-term ‘Harvesting per Person’ is exactly the same: 0.025 acres per person per year.

So, depending on the value of ‘b’, two different Easter Islands are possible. In one, the portion of the population harvesting resources is at or below 40%, the resources per person is 1.3 or more, and the forest covers at least half of the island.

In the other version, the portion of the population harvesting resources is above 40%, the resources per person is below 1.25, and the forest covers less than half of the island.

Which version of the island would you rather live on?

The good news is that, in this model, is it possible to reversibly switch from one version of Easter Island to the other, although it can take several hundred years for the system to settle down and the population can change (read: drop) dramatically during the transition.

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