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.