Comment on Feynman’s Discussion of Zeno’s Paradox


I got critical feedback on the example I came up with in this post and wrote this blog comment in response:

I think I got mixed up in the following way:

I used an everyday example that varied in some details from the Achilles-tortoise race to check my understanding of Feynman’s explanation of Zeno’s paradox. I imported some background assumptions about how bills work (like lots tend to be regularly repeating on say a monthly basis) but then made the bills deviate from how bills actually work (and also made income deviate from how it actually works) in order to try to fit the ratios of the Achilles-tortoise race. So I basically did a poor rewrite of the original Zeno’s paradox using some superficial elements from the paradox in a new hypothetical where they didn’t make a lot of sense.

original post below

From one of the Feynman lectures on physics:

8–2 Speed
Even though we know roughly what “speed” means, there are still some rather deep subtleties; consider that the learned Greeks were never able to adequately describe problems involving velocity. The subtlety comes when we try to comprehend exactly what is meant by “speed.” The Greeks got very confused about this, and a new branch of mathematics had to be discovered beyond the geometry and algebra of the Greeks, Arabs, and Babylonians. As an illustration of the difficulty, try to solve this problem by sheer algebra: A balloon is being inflated so that the volume of the balloon is increasing at the rate of 100  cm³ per second; at what speed is the radius increasing when the volume is 1000 cm³? The Greeks were somewhat confused by such problems, being helped, of course, by some very confusing Greeks. To show that there were difficulties in reasoning about speed at the time, Zeno produced a large number of paradoxes, of which we shall mention one to illustrate his point that there are obvious difficulties in thinking about motion. “Listen,” he says, “to the following argument: Achilles runs 10 times as fast as a tortoise, nevertheless he can never catch the tortoise. For, suppose that they start in a race where the tortoise is 100 meters ahead of Achilles; then when Achilles has run the 100 meters to the place where the tortoise was, the tortoise has proceeded 10 meters, having run one-tenth as fast. Now, Achilles has to run another 10 meters to catch up with the tortoise, but on arriving at the end of that run, he finds that the tortoise is still 1 meter ahead of him; running another meter, he finds the tortoise 10 centimeters ahead, and so on, ad infinitum. Therefore, at any moment the tortoise is always ahead of Achilles and Achilles can never catch up with the tortoise.” What is wrong with that? It is that a finite amount of time can be divided into an infinite number of pieces, just as a length of line can be divided into an infinite number of pieces by dividing repeatedly by two. And so, although there are an infinite number of steps (in the argument) to the point at which Achilles reaches the tortoise, it doesn’t mean that there is an infinite amount of time.

So the error is reasoning from the infinite divisibility of a particular amount of time in the abstract to the conclusion that there is infinite time in reality. Is that right?

At this point I think I’m not really “solid” on this idea so I want to CHEW it some.

Using your mind you can conceptually cut up a length of time as many times as you want, but it’s still the same total length.

It’s kinda like if you had $100, and you reasoned as follows:
When I get my next bill for $100, I’ll also get another $10 of income.
When I get a bill after that for $10, I’ll also get $1 of income.
When I get a bill after that for $1, I’ll also get 10 cents of income.
And on and on.
I will never run out of money!
In reality, you won’t have infinite time to keep getting the smaller amount of income that lets you handle the smaller bills. At some point it’ll be next month, and the big bill will be do again, and you’ll be in trouble.

Likewise with the Zeno stuff. The tortoise doesn’t have infinite time to keep getting another increment ahead. Eventually he runs out, and Achilles laps him, and that’s that.

Analysis of Wage Rates Scenario in Elliot Temple’s Educational Materials on Reisman’s Marxism Book

Previously, I posted my discussion of Elliot Temple’s educational materials on George Reisman’s most recent book on Marxism. That compilation of discussion does not contain my analysis of the extended wage rates scenario Elliot offers in his materials. I have posted that analysis below. This is only lightly edited (to correct one misstatement I caught through a quick pass through) and I have not yet reviewed the answers he provides.

Hypothetically suppose there are 20 businesses. This will be a simplified scenario so we can look at how wages are determined. After presenting the scenario, I ask some questions.
Currently the businesses have no employees, but they have the factory floor space and tools to hire up to 100 workers each (and they won’t expand).

So there is factory floor and tool capacity to hire 2000 workers.

There are 800 workers available to take jobs.

So lots of the factory floor and tool capacity is going to go unused (60%).

All work is unskilled – any worker can do any job. All jobs have the same number of hours and equally good working conditions. Workers must make $25/hr or more to have a good standard of living in this society, and $1/hr or more for minimum subsistence. The average rate of profit is 5%/year, so any use of capital with a lower return is inefficient and the capital could be more gainfully used elsewhere (basically you’d be better off investing in stocks and bonds rather than in the business). The business owners take salaries, so all profits of a business are a return on capital.

So the owners have their earnings deducted from sales revenues as expenses of the business like other wage-earners.

As a further simplification, the rate of return on capital from each business is unaffected by the number of workers employed, it only depends on the hourly wage paid – but, nevertheless, each business wants to hire as many workers as possible (up to 100 max) as long as the hourly wage is low enough to get 5%/year profit or more. Additionally, the capital in the businesses is totally liquid – it can instantly be converted into stocks and bonds, or back into a business, for free. Also, everyone lives in the same town and has the same commute to every business, and every worker knows about the job openings at every business, and every business knows about all the available workers. And everyone (both businessmen and workers) are equally good at salary negotiation.
All the workers are equally productive, but the businesses are not equally productive. Some businesses use workers and capital in better ways to create more valuable products. We’ll rank them from 1-20. The best one, #1, can pay workers up to $100/hr and still make a profit of 5%/year on the capital invested in the business. The #2 business can only pay $95/hr, at most, in order to have a profit of 5% or more. The #3 business can only pay $90/hr for workers or else its capital would be better invested in stocks and bonds. And it keeps going down by $5/hr for each business, until the 20th business can only pay up to $5/hr for labor and still make a 5% profit per year.
All the employers are selfish and greedy individually, but they don’t form a cartel or a conspiracy of employers, they just act in their own interests without coordinating with other employers. They try to pay the lowest wages they can and to maximize their profits. And the workers try to get the best wages they can, but they don’t unionize, they just act as individuals. Also, no one starts a new business and no one is self-employed.
What optimally happens in a free market, and why?

The various employers will bid against each other for the available pool of employees. The more productive ones, able to pay more due to their ability to maintain high rates of profit while paying high wages, will outbid the less productive ones.

How many workers are hired by which businesses, and what are their wages?

(Note that I do not think I would have had the same analysis if not for prior discussion on this topic on the FI list.)

The wages the businesses will pay is determined not by the max they are willing to pay, but what is necessary to outbid their next nearest competitor.

Let’s suppose that the “bidding” for employees happens in dollar increments. Business #1 (B1) is able to pay up to $100/hr. But nobody else is offering $100/hr, so it doesn’t actually need to pay that much. (Note that for brevity I’ll refer to the businesses as “B1”, “B2” etc from now on). B2 is willing to pay $95/hr. Assuming, as the scenario specifies, that the workers are all equally productive, B1 doesn’t actually have any incentive to outbid B2 for the workers, since there are sufficient (and equally productive) workers to fully staff both businesses. B1 just has to make sure that it’s bid is high enough to ensure it has a part of the “lot” of 800 workers that is sufficient to staff its business enterprise.

This analysis is the same as we carry on all the way through B8, which pays $65 an hour. B8 does indeed need to outbid B9, which pays $60 an hour. In fact all the business enterprises from B1 to B8 need to outbid B9. If any one of those enterprises only offered $60, then the workers would be indifferent between working for that business enterprise and working for B9. This would be a dumb result, since B1-B8 are more profitable than B9 and thus more able to pay higher wages.

So B1-B8 will pay the wage necessary to keep workers from going to B9 or below. Assuming dollar wage increments, they will bid up the wage for all their workers to $61/hr. Note this is way higher than subsistence 😉

What is the unemployment rate?

The involuntary unemployment rate is 0%. Because of the number of profitable business enterprises relative to the number of workers, there is full employment.

In what ways does worker need matter (like the minimum wages needed for a good standard of living, or for minimum subsistence)?

Minimum subsistence plays no role in the determination of wages. The market wage is in fact way way above minimum subsistence, but could have been below or at subsistence given different market conditions.

Likewise, the minimum needed for a good standard of living plays no role in the determination of wages.

In what ways does employer greed matter?

Employer greed matters in that the employers act in order to maximize their own profit. One way to look at labor is an input to the production process. With other inputs (like steel or electricity), the business owners want to make sure they pay enough to outbid their next nearest competitor for a given resource. Doing so allows the business owners to maximize their own profit. So rather than driving down wages, employer greed causes wages to be bid up to the point where the next nearest competitor can’t afford the labor.

A new business is created. It’s able to offer wages of up to $25/hr for up to 100 workers (while still making a 5% return on his capital or higher). What affect does this have on wage rates and how many workers work at each business?

I believe this would have no effect on wage rates. There’s already a business (B16 I think?) that offers that wage rate and is currently being outbid by businesses 1-8. The addition of another business offering below-market wages when there is full employment at the market wage.

Another new business is created. This businessman invented a really great new product which is easy to make. He’s able to offer wages of up to $500/hr for up to 100 workers. What effect does this have on wage rates and how many workers work at each business?

This new business (BN) would be paying at the top of the market, and would hire the “first” 100 of the fungible employees specified in the hypo. The BN + B1-B7 would collectively hire the entire labor pool.

B8 wants workers and is willing to pay $65/hour. BN + B1-B7 need to outbid B8 but not each other. Assuming dollar bidding increments, the entry of BN results in the market wage being bid up to $66/hr.

What happens to wages if another 200 workers enter or exit the situation?

According to general economics principles, an increase in the supply of something, all things being equal, should lower the price. Let’s look at the details.

The entry of 200 workers into the original hypothetical situation means that enterprises B9 and B10 will be able to staff their enterprises. This means that B1-10 need to outbid B11 and below (but not each other). B11 pays $50/hr. So assuming dollar bidding increments, the entry of 200 workers lowers the market wage to $51/hr.

What if the #1 business expanded in size and could now hire up to 400 workers instead of 100?

Collectively, B1-B5 would be able to hire the entire labor pool. This means they would need to outbid B6, which pays $75. Assuming dollar bidding increments, the expansion of B1 would result in a market wage of $76/hr.

What if the #20 business expanded to be able to hire up to 400 workers?

No effect on the market wage, since they’re being outbid.

In the original scenario, what happens if the government passes a minimum wage law of $25/hr?

No effect, since the market wage is higher.

What about a minimum wage law of $80/hr?

I think one way to look at this is that employers who are below the minimum wage cannot participate in the auction for labor. If they did participate, then the bidding process to beat the next-nearest competitor for the available pool of labor would be driven below the minimum wage.

The cut off point here is B5. B5 is the business that pays exactly the minimum wage. In the original scenario, B6-B8 also participated in the auction for labor. Now, they cannot.

Since the employees are fungible, each concern has no incentive to pay higher than the minimum wage. So the government-controlled wage winds up being $80 for those still employed. However, only 500 of the original labor pool of 800 are employed in the new situation. There are now 300 people unemployed. The unemployment rate has gone from 0 to 37.5%. The government has effectively outlawed the employment of these people.

Analysis of Quote from “Liberty and Property” by Ludwig von Mises

The speech by Mises called “Liberty and Property”, was mentioned in Elliot Temple’s latest newsletter. I quote from it below and analyze some:

The socialists must admit there cannot be any freedom under a socialist system. But they try to obliterate the difference between the servile state and economic freedom by denying that there is any freedom in the mutual exchange of commodities and services on the market. Every market exchange is, in the words of a school of pro-socialist lawyers, “a coercion over other people’s liberty.” There is, in their eyes, no difference worth mentioning between a man’s paying a tax or a fine imposed by a magistrate, or his buying a newspaper or admission to a movie. In each of these cases the man is subject to governing power. He’s not free, for, as professor Hale says, a man’s freedom means “the absence of any obstacle to his use of material goods.”6

Professor Hale ignores the requirements of producing material goods on a large scale. Producing material goods on a large scale requires things like property rights so that people can have predictable access to means of production, and so that consumers can know they’ll have predictable access to consumer goods that they buy. In a world where there are no predictable access to means of production, long range economic planning becomes impossible, and that restricts what can be produced. And in a world where there are no property rights, consumers can’t rely on having predictable access to the things they buy (cuz the government might take it, or gangs or whatever). So then there is less incentive to work and be productive, cuz anything you can’t defend with force might get stolen anyways. And this is just a tiny subset of the issues with getting rid of property rights.

There’s also a whim-worship element here, to look at it from an Objectivist lens. It’s basically saying, “freedom means my whims are satisfied regardless of context, even if that contradicts the nature of reality and leads to conflicts with other people’s rights.”

This means: I am not free, because a woman who has knitted a sweater, perhaps as a birthday present for her husband, puts an obstacle to my using it. I myself am restricting all other people’s freedom because I object to their using my toothbrush. In doing this I am, according to this doctrine, exercising private governing power, which is analogous to public government power, the powers that the government exercises in imprisoning a man in Sing Sing.

With private ownership you can offer people money for sweaters and toothbrushes. The government, OTOH, points guns at people.

Those expounding this amazing doctrine consistently conclude that liberty is nowhere to be found. They assert that what they call economic pressures do not essentially differ from the pressures the masters practice with regard to their slaves. They reject what they call private governmental power, but they don’t object to the restriction of liberty by public government power. They want to concentrate all what they call restrictions of liberty in the hands of the government.

Even if they were right that there were lots of private restrictions of liberty (and they’re not), concentrating all that power in the hands of the government sounds dangerous! Better a thousand petty tyrants than one uber-giga tyrant.

They attack the institution of private property and the laws that, as they say, stand “ready to enforce property rights—that is, to deny liberty to anyone to act in a way which violates them.”7

A generation ago all housewives prepared soup by proceeding in accordance with the recipes that they had got from their mothers or from a cookbook. Today many housewives prefer to buy a canned soup, to warm it and to serve it to their family. But, say our learned doctors, the canning corporation is in a position to restrict the housewife’s freedom because, in asking a price for the tin can, it puts an obstacle to her use of it. People who did not enjoy the privilege of being tutored by these eminent teachers, would say that the canned product was turned out by the cannery, and that the corporation in producing it removed the greatest obstacle to a consumer’s getting and using a can, viz., its nonexistence.

I really liked this point. It was the reason I wrote the post. It was a very Objectivist-y type focus on CONTEXT. The can literally wouldn’t exist if not for the corporation. Is the private housewife gonna figure out how to deal with acquiring and processing tin and dealing with canning and all that crap? No. She’ll just make soup in a pot like her grandmother did. The corporation is offering another option that is HELPFUL AND LIBERATING, not oppressive.

The mere essence of a product cannot gratify anybody without its existence. But they are wrong, say the doctors. The corporation dominates the housewife, it destroys by its excessive concentrated power over her individual freedom, and it is the duty of the government to prevent such a gross offense. Corporations, say, under the auspices of the Ford Foundation, another of this group, Professor Berle, must be subjected to the control of the government.8

I liked Mises’ casual trashing of the Ford Foundation. People recognize some stuff as activist groups, but these left-wing “do-gooder” non-profits with vague mission statements about advancing human welfare have been a big cultural/social problem for a long time. And lots of people are fooled by them, in a similar way to how they were fooled by stuff like the communist-founded ACLU being “non-partisan” up until the past year or two.