The Magic of Large Numbers

Before you continue reading, perhaps you should make yourself a cup of coffee.

Imagine its warmth in your hands, the aroma rising from the cup and the pleasure of taking that first sip. Perhaps a cup is already steaming beside your computer.

At home, it may have cost you around fifty cents. You probably did not even think about the money. The pleasure of drinking it may be worth much more to you than its price. In a café, you might willingly pay four or even five euros for a similar cup—partly for the coffee, partly for the atmosphere and partly for the pleasure of taking a break.

The cost of something and its value are not necessarily the same.

Coffee in a small company

Imagine a small company with ten employees. It belongs to a single owner, and every employee may drink two cups of coffee free of charge each working day.

Let us assume 20 working days per month and a cost of 50 cents per cup.

Suppose the employees earn an average gross salary of €3,500 per month. The company therefore pays €35,000 in gross salaries. If we add approximately 21% for employer contributions, its total monthly personnel expenditure is about €42,350.

The coffee accounts for less than half of one percent of personnel expenditure.

What does the company receive for its €200?

There is something missing from this calculation. We have counted the cost of the coffee, but not its value.

Every morning and every afternoon, the employees receive a small gift from their employer. They hold a warm cup in their hands, enjoy its aroma, take a short break and perhaps have a friendly conversation with a colleague.

The coffee costs the company only fifty cents per cup, but it can create a moment of pleasure, gratitude and belonging. It says:

The company has thought of us. We are welcome here.

For each employee, the coffee costs the company €20 per month. This is slightly more than 0.5% of a monthly gross salary of €3,500.

But would an additional €20 hidden somewhere in the monthly payslip create the same satisfaction? Perhaps not. After taxes and social-security deductions, the employee might barely notice such a small increase. Free coffee, by contrast, is visible, tangible and enjoyable. Its positive message is repeated forty times every month.

Research supports the broader idea that the form of a benefit can matter as much as its monetary value. In a workplace experiment, employees responded more strongly to a non-monetary gift than to a cash gift of equal value. This does not prove that coffee is better than higher pay, but it shows that a visible gesture can have an effect disproportionate to its financial cost.

Free coffee cannot compensate for an unfair salary, poor working conditions or disrespectful management. Nor should we claim that it is more valuable than a substantial salary increase of two or three percent. But a small salary increase and a small daily gesture are psychologically different. One appears as a number on a payslip. The other becomes part of everyday working life.

The great saving

Now imagine the owner assembling the ten employees and proudly announcing:

I have made an important decision. From tomorrow onward, there will be no more free coffee. This will save our company €200 every month!

Would anyone admire this decision?

Probably not. It would sound petty, stingy and rather ridiculous. The owner would save less than half of one percent of personnel expenditure while taking away a small daily pleasure from every employee. Most people would call the €200 peanuts.

The balance sheet would show €200 less in expenditure. It would not show the disappointment of ten employees, the disappearance of a daily gesture of appreciation or the message conveyed by its withdrawal:

Even your cup of coffee is too much for us.

Now add four zeros

Let us enlarge the company from ten employees to 100,000 employees. Everything else remains exactly the same.

Each employee still receives two cups per working day. Each cup still costs fifty cents. The coffee still costs the company €20 per employee per month.

Two million euros every month—just for coffee!

Suddenly, abolishing free coffee may appear to be a courageous and financially responsible decision. A controller might report:

By eliminating free coffee, I have saved the company two million euros per month.

Two million euros sounds enormous. It no longer sounds like coffee. It sounds like a fortune.

Four houses every month

Suppose the most expensive thing the controller has ever personally bought is a modest house costing €500,000. Beyond amounts of this magnitude, personal experience ends and the numbers become increasingly abstract.

The controller can now make the saving tangible:

My decision saves the company the equivalent of four houses every month.

Now the decision sounds impressive. Perhaps the controller will be congratulated. Perhaps management will praise this example of financial discipline.

But is it really any wiser than the small company owner’s decision?

What about the salaries?

The large company pays €350 million in gross salaries every month. Including approximately 21% in employer contributions, its total monthly personnel expenditure is around €423.5 million.

The company’s monthly personnel expenditure corresponds to the price of approximately 847 houses. Over an entire year, that becomes 10,164 houses—enough homes for roughly 20,000 people at Germany’s average of about two people per household. That is almost the population of the city of Donaueschingen.

The annual coffee bill, by comparison, amounts to €24 million: the equivalent of 48 such houses.

The percentage has not changed

Nothing fundamental has changed.

The coffee costs exactly the same amount per employee. It represents exactly the same percentage of personnel expenditure. The employees receive exactly the same small daily benefit.

The only difference is the number of employees—and therefore the number of zeros in the total.

If cancelling the coffee looks petty in the small company, it does not automatically become wise in the large company. Multiplication changes the total, but it does not change the relationship.

Why does the large saving impress us?

Human intuition is well adapted to small numbers. We understand fifty cents because we use coins. We understand €200 because it might be the price of a dinner, a household purchase or a repair. We can perhaps imagine €500,000 because it resembles the price of a house.

But at some point, our personal experience ends. Very few people have an intuitive understanding of €2 million, €423 million or several billion euros. We can read these numbers and calculate with them, but we can no longer truly feel their proportions.

The controller understands the price of a house. Four houses therefore feel like an enormous saving. But the controller has probably never had to pay 100,000 salaries.

The comparison with private experience is vivid—but misleading. A corporation with 100,000 employees cannot sensibly be judged using the financial scale of a private household.

Another way to describe the same amount

The statement “The company spends two million euros on coffee every month” creates one impression.

But exactly the same reality can be described differently:

The company spends one euro per employee per working day on two cups of coffee.

Now it no longer sounds extravagant.

For one euro per day, the company offers each employee two small moments of pleasure. It provides an occasion for informal meetings and conversations. It creates a modest but visible sign of hospitality and appreciation.

When the company abolishes free coffee, it does not merely save €2 million. It also disappoints 100,000 people—every working day.

That disappointment will not appear in the controller’s calculation. Even if the decision produces only a tiny decline in satisfaction, goodwill or motivation, that effect is also multiplied by 100,000. Large numbers work in both directions.

Two correct statements

The controller can truthfully say:

I have saved the company two million euros per month—the price of four houses.

An employee can reply, equally truthfully:

You have reduced personnel-related expenditure by less than half of one percent and taken away two cups of coffee per day from every employee.

Both statements are mathematically correct. But they create entirely different impressions.

This is why totals should rarely be considered in isolation. Whenever we encounter an impressively large number, we should ask:

• For how many people?
• Over what period?
• How much is it per person?
• What percentage of the relevant total does it represent?
• What value is received in return?
• Which consequences are missing from the calculation?

Peanuts remain peanuts

Two million euros is undeniably a large amount of money. But size alone tells us almost nothing about whether an expense is reasonable.

For a private individual, €2 million is a fortune. For a company whose monthly personnel expenditure exceeds €423 million, it is less than half of one percent.

The small company owner who boasts about saving €200 by cancelling free coffee looks stingy and ridiculous. The controller who boasts about saving €2 million in the large company may be celebrated as a prudent manager.

Statistically, however, they have made exactly the same decision.

Both have saved €20 per employee per month. Both have reduced personnel-related expenditure by approximately 0.47%. Both have taken away a visible daily benefit whose value to the employees may be considerably greater than its cost to the company.

Sources and assumptions

• Sebastian Kube, Michel André Maréchal and Clemens Puppe, “The Currency of Reciprocity: Gift Exchange in the Workplace”, American Economic Review, 2012.
• City of Donaueschingen: Facts and figures.
• German Environment Agency: Population and private-household structure.

The coffee price, salary, employer contributions, working days and house price are deliberately simplified assumptions for this thought experiment. Actual costs vary.