Which is better a \$1,000 a year raise or a \$300 raise evry six months?

On April 10th, 1992, Cecil wrote a column in which he said that it would be better to take the 300 every 6 months. But where he's demontstrating that, he says that it would be the same as a 600 raise every year (duh), but then he says, two of these a year would come to an annual raise of \$1200, which would rapidly get you ahead of the \$1000 annual raise. This is logical, except how can you get 4 6-month raises in a year? last I checked, there are only 12 months in a year, which means you can only get 2 \$300 per every 6-months raises. Naturally, this would mean a \$600 a year raise, not \$1200, which would put you significantly behind the \$1,000 a year raise. Where did he get the idea of getting \$1,200 a year in raises from \$300 per six-months raises? I guess wishful thinking.

there is no try
-H3 Knuckles-

I think you’re right, but let’s try to figure this out…

Let’s say Company X hires me, offering me 24,000 a year, with the option of a \$300 per year raise every six months, or a \$1,000 per year raise every year.

Let’s say I took the \$300 option.

For the first six months, I am earning \$2,000 per month. Once my first raise kicks in, I am earning an additional \$25 per month (for a total of \$2,025 per month). At the end of the first year, I have earned 24,150. While with the second option, I never got a raise during my first year (since it comes at the end), so I have earned only 24,000 for my first year.

So for the first year only, you are better off with the \$300 raise every six months.

But at the beginning of the second year (with the first option), I begin earning \$2,050 per month. At the end of another six month, I begin earning \$2,075 per month.

The total I would have earned over my second year would be 24,750.

With the second option, I have earned 25,000. So for the second year, you are better off with the \$1,000 per year raise.

And overall, you have earned 48,900 with the \$300 per raise every six months option, but you will have earned 49,000 with the \$1000 per year raise.

That’s pretty darned close. Let’s see what would happen the third year.

For the first six months, I would earn 2,100 per month, and I would earn 2,125 for the last six months, for a total of 25,350. While with the \$1,000 option, I would earn a total of 26,000 that year.

So, yes, it looks like you are better off with the 1,000 per year raise, as long as you’re planning on staying there for two or more years.

And, yes, I am an accountant.

David

The only quibble I have with Cecil’s reasoning about this is the statement of the conditions. The set-up for this problem is the following: Suppose you’re hired for a job at a given salary. Let’s say it’s \$10,000, just to give a number, although it’s not really relevant to the problem. Just to have a definite year for this, let’s say that your first year on the job is the year 2000. You’re given two options: You can have a \$1000 raise every year, or you can have a \$300 raise every six months. It’s clear that a \$1000 raise every year means that you will get \$10,000 in 2000, \$11,000 in 2001, \$12,000 in 2002, etc. What’s not clear is what a \$300 raise every six months means. Does that mean that the amount you make for that six months is raised by \$300 dollars? Or does it mean that the amount you make per year is raised by \$300, so the amount you make for six months is raised by \$150? To make the analysis that Cecil work right, you would have to assume that he means the first of those two, that the amount you make for that six months is raised by \$300.

Here’s the figures: Suppose you ask for \$1000 raise every year. Then the amount of money you make is as follows:

2000 (first six months): \$5,000
2000 (second six months): \$5,000
2000 (yearly total): \$10,000

2001 (first six months): \$5,500
2001 (second six months): \$5,500
2001 (yearly total): \$11,000

2002 (first six months): \$6,000
2002 (second six months): \$6,000
2002 (yearly total): \$12,000

2003 (first six months): \$6,500
2003 (second six months): \$6,500
2003 (yearly total): \$13,000

2004 (first six months): \$7,000
2004 (second six months): \$7,000
2004 (yearly total): \$14,000

2005 (first six months): \$7,500
2000 (second six months): \$7,500
2000 (yearly total): \$15,000

2005 (first six months): \$8,000
2005 (second six months): \$8,000
2005 (yearly total): \$16,000

On the other hand, suppose that you ask for a \$300 raise every six months, and we’ll assume that this means that you will get a \$300 raise in the amount that you get for each six months. Then this works out to:

2000 (first six months): \$5,000
2000 (second six months): \$5,300
2000 (yearly total): \$10,300

2001 (first six months): \$5,600
2001 (second six months): \$5,900
2001 (yearly total): \$11,500

2002 (first six months): \$6,200
2002 (second six months): \$6,500
2002 (yearly total): \$12,700

2003 (first six months): \$6,800
2003 (second six months): \$7,100
2003 (yearly total): \$13,900

2004 (first six months): \$7,400
2004 (second six months): \$7,700
2004 (yearly total): \$15,100

2005 (yearly total): \$8,000
2000 (second six months): \$8,300
2000 (yearly total): \$16,300

2005 (first six months): \$8,600
2005 (second six months): \$8,900
2005 (yearly total): \$17,500

Notice what has happened here. There’s two things going on. In effect, the \$300 per six months option is equivalent to getting a \$1200 per year raise, plus getting a \$300 bonus each year. Clearly that’s better than getting a \$1000 per year raise.

On the other hand, suppose that the condition that you get a \$300 raise each six months means that you get a raise in your yearly salary of \$300, so each six months you get a raise of \$150. (This strikes me as just as reasonable an interpretation of the condition as the other one.) This works out as follows:

2000 (first six months): \$5,000
2000 (second six months): \$5,150
2000 (yearly total): \$10,150

2001 (first six months): \$5,300
2001 (second six months): \$5,450
2001 (yearly total): \$10,750

2002 (first six months): \$5,600
2002 (second six months): \$5,750
2002 (yearly total): \$11,350

2003 (first six months): \$5,900
2003 (second six months): \$6,050
2003 (yearly total): \$11,950

2004 (first six months): \$6,200
2004 (second six months): \$6,350
2004 (yearly total): \$12,550

2005 (first six months): \$6,500
2000 (second six months): \$6,650
2000 (yearly total): \$13,150

2005 (first six months): \$6,800
2005 (second six months): \$6,950
2005 (yearly total): \$13,750

So this would be the equivalent of getting a \$600 per year raise, plus a \$150 bonus per year. After the first year, this is not as good as getting a \$1000 raise per year.

Despite the hour’s difference, KingDavid8’s post and mine were simul-posts. He’s assumed that a \$300 raise per six months means that one receives \$300 more per year and hence actually only \$150 more per six months. He’s calculated, as did I, that this would be be better than a \$1000 per year raise only in the first year. As I noted though, if you assume that a \$300 raise per six months means that one receives \$300 more for each six months, it is better than a \$1000 raise.

http://www.straightdope.com/classics/a3_198.html

(emphasis is mine)

First time I read this, it made my head spin so much that I gave up. I read it again, and it still makes my head spin. I guess the idea is, the \$300 raise applies to only half the year, for an overall yearly effect of \$600 a year. Of course, you get two of those a year. Doesn’t seem right to me either. Maybe if you had a raise every four months, you would get 27 (3 cubed cubed) of them a year? Like I said, my head is still spinning.

What part of “I don’t know” don’t you understand?

The problem isn’t Cecil’s, the problem is Marilyn’s. She’s the one who came up the bizarre definition that a \$300 raise every six months means “a \$300-per-six-month raise every six months.”

It is too clear, and so it is hard to see.

Where the “2 such raises per year” comes from…

The first \$300 raise per six months means that you will then get \$300 for the next six months, and \$300 for the six months after that, ad infinitum, from that one raise. That is the \$600 raise per year off a \$300 raise per 6 mo. (The second raise coming at the end of the year, so technically the money doesn’t come to you until the following 6 mo.)

Then you get another \$300 per 6 mo raise at the end of the year on top of the \$300 per 6 mo raise you got last time. So that’s an additional \$300 every 6 mo.

Go look at Wendell’s numbers with that in mind.

(Wendell, your year designations got all messed up, but you were consistent, so it doesn’t effect the outcome of the answer.)

So I want an increase in paycheck of \$300 per six months over \$1000 per year.

The column being referenced would be What’s better, a \$1,000 raise each year, or a \$300 raise every six months? (10-Apr-1992).

To algebraically state Wendell Wagner’s argument:

First six-month period, salary = x
Second six-month period, salary = x + 300
Third six-month period, salary = x + 600
Fourth six-month period, salary = x + 900
nth six-month period, salary = x + (n - 1) * 300
after nth 12-month period, total salary = nx + ((2n) (2n - 1) / 2) * 300

First 12-month period x
Second 12-month period x + 1000
Third 12-month period x + 2000
after nth 12-month period, total salary = nx + (n (n - 1) / 2) * 1000

So after n 12-month periods, the first total salary (the one where you add \$300 every six months) will exceed the second total salary by
(2n (2n - 1) / 2) * 300 - ( (n (n - 1) / 2) * 1000) = 100 n^2 + 200 n.

e.g. after 3 years (3 12-month periods) the total salary with the \$300 raise every six months will exceed the \$1000 yearly raise salary by
100 (3^2) + 200 (3) or 1500.

Seems to me this one’s less about numbers than nomenclature.

Generally, wages/salaries (and raises thereto) are defined in terms of some standard time-period, be it hourly, yearly, or whatever. And it’s generally clear from context just what time period we’re talking about.

This is why, IMO, Marilyn was wrong to interpret the \$300 raise as earning \$300 more for the 6-month period after the raise. Nobody thinks of their salary in terms of how much they’re making in 6-month periods. If someone tells you they’re making \$35,000, you assume that means annually, unless you’re talking to Bill Gates, in which case it might mean ‘per second.’

Ok, flat out, you will NOT get a \$1200 increase in your salary if you get a \$300 raise every 3 months. Here is why:

At the end of each 6 months, your salary will be increased by \$300. Therefore, your salary will always be \$300 more than it was in the previous 6 months. It is absurd to think that because your annual salary increase is \$600 and because there are 2 increases per year, the net increase is \$1200. The \$600 annual salary increase already takes into account the fact that there has been 2 increases in the year.

I have made an excel spreadsheet that shows the difference between getting a salary increase of \$1000 once per year and getting 2 \$300 increases per year. The spreadsheet shows the variation over 10 years, and it can be viewed in HTML format at:
http://www.msu.edu/user/hoppetod/salary.htm

I stand corrected. A simple clerical error I made in Excel led me to the wrong answer. I have adjusted the spreadsheet to show the correct numbers.

It appears that this works in a similar way to compound interest. Just check the spreadsheet to see it working. That is the easiest way to explain it.
http://www.msu.edu/user/hoppetod/salary.htm