The Derivative of a Constant is 0

William Stanley Jevons

We are going to start today’s lesson with some mathematics, at the end we are going to fully understand the roots of why a basic income will not impact the amount people work.

What is change?

In order to understand the impact of basic income on unemployment we need to start with some fairly basic mathematics.

Change in math is answering the question of how much will one thing change if another changes? If I turn this knob on my bathtub, how much water will come out of the spigot? If I tilt my milk carton, what will be the velocity of the milk? If I shoot a gun in the middle of a field, what is the velocity (change of location over time) of an unladen swallow? It is important for people to understand these very real life scenarios so you don’t flood your bathroom, spill your milk, or be catapulted into a bottomless valley. This is the idea of change. In mathematics we call it the slope of a line.

How do we calculate slope? Well, you do it all the time. If you drive onto the freeway it will say that the speed is something like 100 km/hr. This sign is telling you that if you were to drive on this freeway at the speed limit for one hour you will travel 100 km. This is boring, but it is very important. Slope can then be written as a change in distance over a change in time.

Infitesimal and derivative

The roots of marginal analysis

The next idea of slope is the idea of an infitesimal. We can find the average rate of change of any two slopes with that fairly simple equation in the previous section, but in order to proceed we need to be able to find the slope of any line at any point.


This is the basic idea of a derivative. This is useful in almost any scientific field you can imagine. You can plug a general equation into that definition with the limit at the bottom and find the correct equation for any equation in the universe if it has a derivative.

For example:


This is the essence of calculus.

Why do people do what we do? The essence of economics which is Marginal Analysis

So let’s say you went to the store this week like many people do. Why did you buy what you did? You reached the dairy aisle and you were confronted with several choices, you have cows milk, almond milk, soy milk, and other options. Within each of those categories you have whole, half and half, 2%, and fat free cows milk, and many flavors for all of your vegan milks and milk from other animals. Why will you buy your whole cows milk over vanilla soy milk? The most likely answer is because you prefer it, or more importantly, you think you will enjoy drinking whole cows milk more than vanilla soy milk based on the information you have.

These are three very important ideas we use in economics to describe and understand behavior.

  • Perceptions
  • Information
  • Preferences

First of all, a perception is based on the information someone has on an item and how it will impact  them and the world around them. The amount of information people have is constantly changing, and it will impact how people choose what they buy and do not buy.

Preferences is very simply someone saying they want one item over another and they will choose it over another.

What are they weighing against? Well, they are trying to improve their total happiness. We call this utility in economics, because the idea was founded by the Utilitarians of the early 19th century.

In short, people are always trying to maximize their utility in everything they do day after day.

The second important concept is not just how much utility doing this will give me but also the cost of doing one activity over another. There are two types of costs people will consider. They will consider their absolute costs, such as I would love to travel to the Great Barrier Reef, but it will cost me about $2000 round trip from where I live for just the airfare. That $2000 could have been spent in other ways, and if I default on my rent I will be homeless, meaning my overall pleasure is reduced. The cost relative to what else we could have done with that money definitely impacts the decisions people make from day to day.

The next question is while we have the understanding that people want to maximize their pleasure, minimize pain, and reduce costs (mainly because of the trade offs), our final question is how those relate to each other. This forms our demand curve, which represents the relationship between the cost of one good and the amount we will buy. Also, if the price of a ticket goes down, people will be more likely to buy more of it, and vice versa. If you see an item you like is on sale, you will likely buy more because that maximized your utility. Essentially, as the price goes up, the amount people are able and willing to buy declines without sacrificing other goods which will reduce their overall wellbeing, and vice versa.


Supply works the same way, so in the labor market the amount of hours people are willing to work for a certain rate is proportional. It is not going to be directly proportional however, because as people get more money they will need to be paid to put in another hour of work is going to be larger because the marginal benefit of every additional dollar is relatively smaller. If I already have a million dollars saved up, a nice house, a fancy electric car, and I can travel whenever I want, why would I go do a tedious job for a measly $20 per hour? I would absolutely be willing to spend an entire week in Yosemite hiking with some wonderful friends and sacrifice that $800 since my million dollars in the stock market is going to likely grow more in that time than the offered wage. For that reason, the pleasure gained form the added income for doing a job will always have to be greater than if that individual had been doing recreation. Ever wondered why CEOs and stock traders get paid such lavish salaries? This is why. As long as the value of that works is worth that much money, the stock holders will be more than willing to pay it. This means the supply of labor offered for each wage will look like so:

This graph is with the total number of hours worked on the bottom, and the wage is going up to something like a million dollars per hour. After we add up all of these curves for every individual in the market, and for the analysis of how it will effect 99% of people it essentially looks something like this:

Basic Income

This is where the derivative, basic income, and demand come together into one very powerful lesson.

If a socialist state were to say they were going to give their people a set amount of money every month, will this impact the amount people are willing to work? Well, this is where calculus comes in to solve the problem.

Let’s say a company were to give every employee a $500 bonus. Will this inevitably lead to people taking more time off? Probably not a significant amount. People think on the margin, and the overall benefit  of a reasonable amount of money gained through a basic income program is almost certainly not going to be at the level which will cause people to start taking time off of work in droves. This is essentially a basic income for your employees of that company. As long as the marginal benefit of seeking extra work with the company is greater than the marginal benefit of going on that additional vacation, the person will continue to work.

The same will happen at a state or national level as well. If people see an extra $1000 in cash, this does not impact the calculus of the slope of determining whether people will offer significantly more or less work compared to now. The equation for how many hours people will supply their labor is for all practical purposes a linear equation, where the amount of hours they spend is reflected by the wage paid. Basically, the equation looks like so:

Total wage = hourly rate * hours worked + bonus

If you take the derivative of the total wage with respect to hours worked ( in short, what will determine the amount of hours worked by an employee), you find that the bonus is a constant, so the derivative becomes:

δ wage/δ hours worked = hourly rate

The bonus does not impact the number of hours worked in the short run.

Basic income has no impact on either the wage paid per work hour so it does not impact the overall amount of hours which will be spent working in society. As long as it is not large enough for the relative pleasure of an additional vacation to be worth more than working an extra week, it will not effect employment. No country has ever gotten to that level.

Practically that means if a bonus was theoretically an amount large enough to provide enough income from investing the bonus in a short amount of time you would see it effect the amount of hours worked. No serious proposal for basic income, or actual implementation of one has ever been that large.

If basic income did make an impact on how much people worked, than companies would not give out bonuses to workers. Most of corporate America has some sort of bonus structure, and many co-ops will give dividends out to all of their members regardless of the amount of work they put in based on the profit of the firm on a regular basis. If that were to impact the amount of work employees put in they wouldn’t have such policies.

This is the full reason why basic income has no impact on employment.

This is why the $600 unemployment bonusus did not lead to people working less.

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