# A formula for maximum wage

(Note: This post has been edited to correct a mathematical error.)

I’ve been thinking about the idea of a maximum wage recently. It seem daft at first, but Google and DuckDuckGo tell me that the idea has been gaining some currency (pun not intended, but welcome) among liberals. It’s also not a new idea – Franklin Roosevelt proposed essentially limiting income to \$25,000 a year for a single person, with any excess going to fund the war effort. This tax-based income limit ultimately did not pass, and really, trying to implement it today would be a disaster, as companies would likely just relocate somewhere that doesn’t limit executive salaries, leaving us with less jobs than before.

Still, in a time of soaring inequality when 1% of people have 50% of the wealth (see page 18), surely something should be done to bring some of that money to the 99% so we can actually buy their products. You know, create the kind of rising tide that libertarians are fond of saying lifts all boats. I, and others on the liberal Internet, think the best way to go about this is to set a maximum wage within each company, proportional to some measure of how the company treats its lowest-paid employees.Before we begin, it must be stressed that performance-related pay would have to be subject to any maximum wage law, otherwise companies will simply use it as a loophole to pay the CEO whatever they want.

## Idea 1: Maximum wage is proportional to the wage of the lowest-paid worker in the company

Seems simple enough. Mathematically expressed:

$M = aw$

where

M is the maximum wage for the company

w is the wage or salary of the lowest-paid employee

a is a constant; a ≥ 1

Unfortunately, this is easily gameable, in the worst way. Let’s consider a small company with 10 employees, and set a to 10.

 Person Salary Alice €42,000.00 Bob €36,000.00 Carol €29,000.00 David €19,000.00 Eve €23,000.00 Frank €32,000.00 Greta €22,000.00 Henry €21,000.00 Irene €30,000.00 John €26,000.00 Highest €190,000.00

OK, so earnings at this company are capped at €190,000. Not too bad, and in fact Alice, the highest-earning person, here gets considerably less than the max. But what if we remove David? Suppose he’s reclassified as an independent contractor, or his job is outsourced.

 Person Salary Alice €42,000.00 Bob €36,000.00 Carol €29,000.00 Eve €23,000.00 Frank €32,000.00 Greta €22,000.00 Henry €21,000.00 Irene €30,000.00 John €26,000.00 Highest €210,000.00

With the minimum wage guy gone, the maximum wage jumps to €210,000. Sure, nobody in this company is earning that much, but it does illustrate that in a company with lots of minimum wage employees where top executives do take home more money than they can spend, there will be an incentive to take the lowest earners off the official payroll in order to allow for higher executive salaries.What else is there?

## Idea 2: Maximum wage is proportional to total number of employees

Once again, this is expressed mathematically as:

$M = ap$

where p is the total number of workers in the company. Thus, the more people the company employes, the more money the top executives are allowed to get paid. This gives the company a vested interest in reducing unemployment, as high employment means more money for them. If a is set to 10,000:

 Person Salary Alice €42,000.00 Bob €36,000.00 Carol €29,000.00 David €19,000.00 Eve €23,000.00 Frank €32,000.00 Greta €22,000.00 Henry €21,000.00 Irene €30,000.00 John €26,000.00 Highest €100,000.00

So yeah, a needs to be set pretty darn high here in order for top executives to take home anything. Handily, if David is removed from official payroll somehow, the maximum wage actually drops:

 Person Salary Alice €42,000.00 Bob €36,000.00 Carol €29,000.00 Eve €23,000.00 Frank €32,000.00 Greta €22,000.00 Henry €21,000.00 Irene €30,000.00 John €26,000.00 Highest €90,000.00

The problem with this scenario is that there is no incentive to pay above minimum wage:

 Person Salary Alice €19,000.00 Bob €19,000.00 Carol €19,000.00 David €19,000.00 Eve €19,000.00 Frank €19,000.00 Greta €19,000.00 Henry €19,000.00 Irene €19,000.00 John €19,000.00 Highest €100,000.00

While it is possible average wages would increase over time as companies big against each other to hire more workers, this is unlikely, as companies usually hire workers when they need more work done, not simply because they have extra money or because executives want to get a higher payslip.What if we combine the two?

## Idea 3: Maximum wage is proportional to both total number of employees and wage of the lowest earner in the company

Mathematically:

$M = apw$

with the same meanings as above. This would combine incentives to increase everybody’s wages with an incentive to increase employment. This time, let’s set a to 1:

 Person Salary Alice €42,000.00 Bob €36,000.00 Carol €29,000.00 David €19,000.00 Eve €23,000.00 Frank €32,000.00 Greta €22,000.00 Henry €21,000.00 Irene €30,000.00 John €26,000.00 Highest €190,000.00

Same as when M is just proportional to w, but that’s due to the value at which a was set.What happens if David’s job is outsourced?

 Person Salary Alice €42,000.00 Bob €36,000.00 Carol €29,000.00 Eve €23,000.00 Frank €32,000.00 Greta €22,000.00 Henry €21,000.00 Irene €30,000.00 John €26,000.00 Highest €189,000.00

In this case, the maximum wage actually goes down as a result of outsourcing and layoffs. On the other hand, if David gets a pay raise:

 Person Salary Alice €42,000.00 Bob €36,000.00 Carol €29,000.00 David €20,000.00 Eve €23,000.00 Frank €32,000.00 Greta €22,000.00 Henry €21,000.00 Irene €30,000.00 John €26,000.00 Highest €200,000.00

Maximum wage goes up!So this seems like a reasonable proposition, but there are still a few loopholes. For example, what happens if the company calculates maximum wage on an hourly rather than annual basis, then halves everybody’s hours and brings in more part-time staff to make up the labour? In that case, the lower-paid workers are taking less home, while the top executives double their income by doubling the number of workers.To counter this, we can refine the formula

$M = {a {hw \over 2,080}}$

Where h is the total number of hours contracted to be worked in a year by employees. ${h \over 2,080}$ thus represents the number of full-time employees required to do all the work in a year, and in any case, ${h \over 2,080} \le p$. Full-time is here defined as 40 hours per week, and salaried employees without set hours are considered to work 40 hours a week for the purpose of this formula. This neatly removes the incentive to hire part-time instead of full-time employees, because 2n half-time employees are worth no more to the top salaries than are n full-timers (and the same is true for all other fractions), and a smaller number of employees requires less training and simpler logistics. Also, since this only counts contracted hours, overtime is excluded from the calculation, thus preserving the disincentive toward overworking employees which overtime provides (and also avoiding a potential loophole in which top execs simply mandate a lot of overtime in order to artificially increase their own salaries). Furthermore, it encourages employee retention – if someone is let go and their duties passed on to other workers, everybody keeps working the same number of hours, and so h goes down by up to 40, leaving M that much lower. As well, it makes zero-hour contracts a liability, since every employee on such a contract does not increase h at all.But what about companies with high turnover, like fast food joints? Maximum wage can’t be calculated on an annual basis, since h is constantly in flux due to people starting and leaving, thus constantly increasing or decreasing the number of hours contracted to be worked. In this case, a further refinement to the formula yields:

$M = {a {tw \over 8}}$

$t = \frac{d_{1}s_{1} + d_{2}s_{2} + ... + d_{p}s_{p}}{5} = \frac{1}{5} \sum\limits_{i=1}^p d_{i}s_{i}$

where

t is the total number of hours which all employees are contracted to work over the time period for which M is being calculated,

p is the total number of employees

si is the mean length of shifts employee i works in a calendar week

di is the total number of solar days on which employee i works each calendar week.

Using this formula, a company can calculate the maximum wage over any period desired, and adjust it on the fly to keep the salary of the highest earner in proportion to that of the lowest.In the case of performance-related and commission-based pay, the value of w is whatever is paid to the lowest-earning person not on performance-related pay. If everybody’s pay is performance-related, w is set as either minimum wagethe base rate of pay (ie how much salespeople are paid even if they don’t bring in a single euro, or in the case of waiters in America, how much they earn before tips), whichever is higher. If there is no base pay, w is minimum wage.Sole traders can earn as much as they want by adjusting their own personal w on the fly, so as to allow themselves to keep whatever they earn less taxes. For a sole trader, $M = w$, as long as $a = 1$) because $t \over {8}$ simplifies to 1 (as workers without set hours are taken as working an 8-hour shift for the purpose of calculating M). Alternatively, we may simply rule that registered sole traders are exempt from maximum wage law, which has the same effect and is probably easier to implement.

So there’s that, there’s still the issue of bonuses. Banning bonuses altogether is a bad idea – while we only ever hear about corrupt bankers receiving a bonus the size of a small country’s GNP, most bonuses are relatively modest extras given to all employees at Christmas, and others are genuine, reasonable rewards for improving the company’s efficiency or profitability. So what needs to be done here is to make bonuses sensible without obliterating them.In the cases of bonuses for improving the business, one solution would simply be to require that the bonus be some fraction of the money the company has which it would not have had if the improvement hadn’t happened. The problem here is that technically any successful deal a banker pulls off can be considered an improvement, and so they pursue increasingly risky and complicated financial instruments in the hope of more profits and thus higher bonuses, which is precisely what led to the current financial crisis. Thus, the sensible solution to the problem of bonuses is to make a bonus proportional to the salary of the top earner in the company. Mathematically expressed,

$B = {T \over c}$

where

B is the maximum allowable bonus

T is the highest wage or salary the company pays, which may be defined as a weekly or annual payment as appropriate; TM. In the case of a company where everybody is on performance-related pay, T would be the highest of everybody’s average pay over a set time period.

c is a constant.

Thus are bonuses reined in, but there’s still the issue of paying people with stock. Since stock isn’t actually money, it could be used in place of money to get around maximum wage legislation.In the case of a publicly-traded company, this is easy to fix – consider a share to be an amount of money equal to its current market price, and include stock payments when calculating an employee’s real salary or bonus.

If a company is not publicly traded but has sold some stock to investors, a share is considered equivalent to an amount of money equal to the average price for which a single share was sold in the most recent round of selling.

For a non-traded company, things are trickier. Market valuation is complex, subjective, and open to abuse. Shares can’t be defined in terms of the value of the company’s assets, because companies can employ byzantine accounting practices to make liabilities look like assets (again, see the credit crunch). Requiring that they be valued by an impartial external appraiser is also dicey due to the possibility of bribery. Thus, if no stocks have yet been sold to outside investors, a single stock should be taken to be worth an amount of money equal to $k \over z$, where k is the total amount of money the company holds at the time of issue and z is the total number of shares in existence.

Then there’s franchises. How would a maximum wage affect, for example, McDonald’s? Legally, each McDonald’s outlet is an independent business that pays a regular fee and agrees to follow certain rules in order to keep using the name and certain trademarks. If one franchise operator decides they don’t want to keep paying, they are entitled to keep on doing the same business as long as they drop the trademarked aspects, as happened when Brian Dunne changed the names of the Eddie Rocket’s restaurants he runs to Rockin’ Joe’s.In the case of a franchise, each instance would be counted as a separate company, with maximum wage calculated for each individually. Thus, if Katie and Liam both own a McDonald’s each, Katie can earn a higher maximum wage than Liam as long as she employs more people or pays them better.

Finally, there is the problem of multiple incomes. Should income from stock dividends count towards what someone earns at their day job? What about if somebody works as a janitor and keeps bees as a hobby makes a bit on the side selling their own honey? Someone who works two part-time jobs? How about someone who owns four McDonald’s?First up, income from stocks and dividends, and capital gains, would not count towards earnings when calculating maximum wage. This means the incentive for venture capitalists to invest in startups is still there, and those who live entirely on stock payouts get to keep their incomes.If somebody, let’s call them Mary, owns multiple instances of a franchise (for example, four McDonald’s), that’s all counted as a single metacompany. As such everybody who works at any of Mary’s McDonald’s is considered an employee of Mary, and her maximum wage is based on every single person she employs and how much each of them earns.Otherwise, each source of income is evaluated separately. The janitor who sells his own honey can make as much from the honey as he wants without it affecting the maximum wage at his day job. The person with two part-time jobs has two separate maximum wages, though is unlikely to hit either.

# The End

Wow, that as quite a post, eh? There are probably all sorts of problems and ideas I didn’t even consider. If you think you can improve this, or just want to point out a flaw, please do comment.