IdleTheory Joe's Boat

Developed from an original idea of Michael Miller (with whom I disagree). A number of fishermen catch fish. On average, they need 2 fish a day to live. And on average they can catch 3 per day. Joe takes 50 days of free time to build a boat that lasts 100 days to go fishing where the fish are most abundant and catch 30 fish a day.


The Fishermen.

The fishermen all work independently of each other. They make their own hooks and lines, and they eat the fish they catch for themselves. Since they don't depend on each other's efforts, they are a collection of individuals rather than an interdependent society.

The cost of a fish is the amount of time it takes to catch one. For the fishermen the time cost C of of each fish is 0.333 days. The value of a fish is the amount of time it provides. Since the fishermen need 2 fish per day, the value V of each fish is 0.5 days. The Idleness of the fishermen is the amount of free time per day that they have left after they have caught the fish they need for that day. Since they have to work 0.333 days per fish, and each fish is worth 0.5 days, their idleness IF can be calculated as:

IF = 1 - ( C / V )

so IF = 0.333

So the fishermen have 0.333 of a day as free time, during which they can do what they like.


Joe's First Boat.

Joe notices that there seem to be more fish further out to sea than close to the shore. That's where the gulls seem to catch their fish. If he could find some way to get out there, he thinks he might be able to catch fish quicker. After a few experiments with reeds, he decides to build a little reed boat. The boat will be made of bundles of dry reeds, tied together with reeds.

Joe has 0.333 days/day of free time that he can devote to building a boat. Since the boat takes 50 whole days to build, it takes him 150 days to build his first boat. During this period, Joe spends 2/3 of each day catching 2 fish, and the rest working on the boat.

Using the boat, he finds that he can catch 30 fish a day, but he only needs 2 fish, and it only takes him 2/30ths of a day to catch them. So, when the boat has been built, his idleness rises to 1 - 2/30 or 0.933 for 100 days of the lifetime of the boat. So the value of the boat (value-in-use) is the increase in idleness over this period (0.933 - 0.333) multiplied by the boat lifetime 100 days, giving a value of 60 days. Since the cost of the boat was 50 days, and its value(-in-use) is 60 days, he wins 10 days of idle/free time using the boat. So the boat is a "useful tool", because its value is greater than its cost. If he decides to carry on using a boat, he will need to spend 0.5 of each day making the next boat, so that it is ready just in time when the old boat becomes unusable. In addition he works 2/30 or 0.0666 days to catch 2 fish. So his Idleness will be (1 - 0.5 - 0.0666) or 0.433, which is 10% better than his original 0.333 Idleness, and Joe spends 43.3% of his time basking in the sun and watching the clouds go by. And Joe's boat sits at its moorings for 93% of the day.

The other fishermen are not much impressed by Joe's boat, and carrying on fishing the way they used to.


Joe sells fish.

One day it occurs to Joe he could use some of his free time to catch more fish and sell them to the other fisherman. Since it only takes him 1/30 day to catch a fish, and it takes them 1/3 of a day to do the same, then if he priced his fish at 1/4 day, they would do better (i.e. be more idle) to buy fish off him than to catch them themselves. They would pay Joe by working for him for that 1/4 day, doing whatever he asked. He could either get them to do his fishing for him, or have them help with building the next boat, or whatever.

So Joe gives it a try, catching one or two extra fish each day, and offering them for sale. The other fishermen prove eager to buy, because each fish they buy saves them 1/3 - 1/4 or 1/12 day of fishing.

Now Joe's Idleness was 0.433 before he started catching fish. When he starts catching and selling fish, his idleness decreases by 1/30 day for each extra fish he catches, and increases by 1/4 day for each fish he sells. So Joe's Idleness, IJ, if he sells N fish per day

IJ = 0.433 - N/30 + N/4

or IJ = 0.433 + 0.216 x N

So, if he sells just one fish a day, his Idleness increases to 0.649, and if he sells 2 fish per day, it rises to 0.865, and if he sells 3 it rises to 1.081. That is, just selling 3 fish a day he can buy in enough labour to do all his boat building as well as all his fishing, with some left over - and he hardly has to do anything at all He is, in effect, 100% idle.

The price of fish.

As Joe starts selling fish to the other fisherman at a price of 1/4 day per fish, Joe finds that he is besieged by potential buyers when his boat lands, and there aren't enough to satisfy all his customers, and some offer to pay a little more, and together they bid the price upward.

It takes the other fishermen 1/3 of a day to catch fish. But this is the average figure. Some of them can actually catch a fish in 1/4 of a day, and some in 1/2 a day. Those who can catch a fish in 1/4 day are not interested in buying fish priced at 1/4 day, because it doesn't make them any more idle. But those who take 1/2 a day to catch a fish, and who are pretty well continually busy fishing are very interested in fish that cost them only 1/4 day. And they are the ones who bid the price up, because even if the price rises to 1/3 day, they are still better off buying Joe's fish.

What will happen is that the price of Joe's fish will be bid up until there are only emough buyers left to buy the whole of Joe's stock of fish. Thus when Joe first started selling one or two fish each day, the price of these would probably have been bid up to over 1/3, and maybe even almost 1/2. Only as the number of fish landed by Joe rose would the price fall back.

One interesting corollary of this is that Joe's fish would benefit the least idle fishermen first.

However, if all the fishermen take exactly 1/3 of a day to catch a fish, then the price of fish will not change with increasing supply. The highest permissible price of fish will be 1/3 day.

But let's suppose that Joe keeps the price fixed at 1/4 day per fish, despite all attempts to bid up the price.


Joe the boatbuilder.

As Joe starts selling fish, he finds that he needs to work less and less. In fact, just selling 3 fish a day buys him enough labour to do both his fishing and boat building. But since his boat, manned by other fishermen, can actually catch 30 fish a day, he can in fact sell 28 fish a day, keeping the remaining 2 for himself. And 28 fish, each selling at a price of 1/4 day, will buy him 7 days of labour every day.

So Joe's boat is set catching 30 fish a day. Joe goes out fishing all day, returning with 30 fish. He eats 2 of them, and sells the other 28. And he employs the 7 days of labour he buys with those 28 fish to work on building his next boat.

However, he now has in effect 7 men working every day on the new boat, and it takes 50 man-days to build a boat, the new boat is completed in just over 7 days, rather than in the 100 days it took him to build one, working half a day on it each day.

With the second boat completed long before the first boat's lifetime is up, Joe now has two boats. Rather than leave the boat idle, he uses his available labour to send the new boat out fishing all day with a hired crewman. Since he sells fish for 1/4 day of labour each, he can buy a day of labour with 4 fish.

So each day the second boat goes out, and comes back with 30 fish. The hired crewman is paid 4 fish, and the remaining 26 fish are sold to buy 6.5 days of labour.

If both boats are then operated on the same basis, with a hired crew, while Joe stays ashore supervising boat-building and selling fish, he now has 13 man-days of labour available each day to build boats. The next boat is completed in under 4 days.

He now has 3 boats each landing 30 fish, and with 26 of the fish being sold for 1/4 day of labour, he now has 19.5 days of labour available each day to build boats. The fourth boat rolls out of the yards after barely 2.5 days.

Now if the population of fisherman is 100, each requiring 2 fish per day, the total daily demand is 200 fish per day. Since his boats bring in 30 fish per day each, he can completely meet the demand with 7 boats. The 8th boat he builds will see its catch go unsold.

Now these 7 boats have appeared very quickly. And they did so because Joe used the labour he could buy with his increasing haul of fish to build new boats. He had needed to start building a new one anyway to replace the old one, but within a matter of 2 or 3 weeks he finds that he has 7 boats. This hadn't been Joe's plan. It just happened pretty much of its own accord/

Over those few, Joe had transformed the community of independent fishermen into employees who worked to man his boats and to build new ones. Their idleness has risen from 0.33 to 0.5. Now half their day was idle, and half was spent working for one man - Joe.


Joe's palace and gardens.

Given that 7 ships is the maximum size of his fleet, Joe now maintains the fleet at this size. He has an income of nearly 50 man-days of labour every day. Of this, he spends 7 man-days each day for his boats' crew. And since he has to build another seven ships every 100 days, at a cost of 50 days labour each. So he spends a further 3.5 man-days each day on boat building. So he has nearly 40 man-days of labour available for other work.

Since his 7 boats bring in 210 fish per day, and he only sells 200, Joe has the extra 10 fish dried and stored.

Now Joe wonders what to do with his income of 40 man-days of labour. There's no point in maintaining any thing larger than a fleet of 7 boats. So he decides to build a palace for himself. The 40 man-days are use to build a large palace surrounded by a substantial garden. When these are completed, the 40 man-days are used for gardeners, butlers, cooks, masseurs. Joe enjoys a luxurious, indolent life.

Joe's competitors.

At first, when the idleness of the community was rising, during Joe's boat-building programme, everybody was delighted, and Joe was very popular. With half a day of free time, life was better than it had been in the old days, when only 1/3 of the day had been free time. But after Joe had built his palace, and settled into a life of luxury, with everyone working a half day for him as boat-builders, boat crews, masons, gardeners, and artists, the euphoria gradually wore off.

So while Joe lazed in his palace, a few people began to plan a new departure. They proposed to emulate Joe, and build their own boats, catch fish, and sell them cheaper than Joe.

And since there were 10 of them, each with half a day of free time available each day, they could together build a boat in 5 days. They would then take it in turn to crew the boat every 10 days. Of the 30 fish caught each day, the company owners would take 2 each, leaving 10 to sell. But instead of pricing their fish at 1/4 day per fish, they'd set the price at 1/5 day, undercutting Joe. And 10 fish priced 1/5 day would bring in 2 days of labour each day, which would be employed building a second boat

And so the boat was built. Since the company owners now worked only one day in 10, their Idleness now rose from 0.5 to 0.9. And those who managed to buy a couple of fish from them saw their idleness rise from 0.5 to 0.7. And 25 days later, a second boat appeared.

The income of 2 days labour per day from the first boat was now used to provide a crew for both boats. The company founders now did no work, and their 2 boats brought in 40 fish each day, and an income of 8 days of labour. After the 2 boat crews had been subtracted, this left 6 days of disposable labour, which was used to start a third boat.

The third boat appeared 8 days later.

The price war.

Joe woke up from his slumbers one day to be told that half the fish his boats were landing weren't selling. He was also told that his gardener and his chef were no longer working for him. And work had stopped on the swimming pool extension. Joe wondered what to do.

If he did nothing, he would soon find that the competition would take all of his customers, and his income of labour would fall to zero. His palace and gardens would fall into ruin, unrepaired and untended.

If he dropped his prices, however, the competition would drop theirs, and prices would rapidly spiral downwards to the point where fish were selling at cost. Something like 1/30 of a day of labour. That way, his palace and gardens would also fall into disrepair.

But Joe saw another possibility. Over the past months, he had always had 10 fish unsold each day, so that now he had 3000 dried fish stocked up. These fish could now come in very handy.

(To be continued)

Idle Theory

Author: Chris Davis
First created: 16 Jan 2001