Tom Kemp 1214081001|%e %b %Y, %H:%M %Z|agohover
I was with a client who said, "So, I can spend a dollar to buy something; I can spend a day doing something; how can I spend a dollar-day?".
This started me thinking about what exactly is a dollar-day. I think it's simply a day in the life of a dollar bill. Imagine this drawn as a little green rectangle. Then a week in the life of that dollar bill is 7 such rectangles, which we can arrange in a line. Now, on each day the bill belongs to someone. We can mark each rectangle with that person's name. If I own the bill on Monday and Tuesday then those two rectangles show 'Tom'. If I then buy a cup of tea for a dollar from Alice then Wednesday's rectangle shows 'Alice'. In fact, Alice saves the dollar and so the remaining rectangles (as far in the future as we care to look) all contain her name.
The more rectangles which have my name on (ie the longer I possess the dollar bill), the more choice I have. My choice is the same every day: spend the dollar or save the dollar. We are calling each of these little rectangles a "dollar-day". So the more dollar-days I possess the more choice I have in my life. I could say to Alice, "You can have the dollar bill today or at the end of the week.". What will she do? She is guaranteed to take the money earlier rather than later. Why is it so important to have the money as soon as possible? It's often instructive to think of extreme cases. In one case we give a baby a dollar bill and in the other we give a dollar to a person on their death bed. In the first case we are being much more generous to the person: the baby will grow up and could choose any day during his or life to spend the dollar.
What if Alice takes a couple of days to make my cup of tea (yes, I'll get very cold tea). I wanted to pay her on Wednesday but she now has to wait until Friday to get her money. She has missed possession of two dollar-days. The dollar still exists but the dollar-days belonged to me during a time when she should have had them. From Alice's point of view these are lost throughput dollar-days. I'm annoyed because I don't have the tea when I was promised it. You might think I would be happy to have possession of two more dollar-days than I was expecting. However, my choice has been denied: I wanted to spend my dollar on Wednesday; in fact, I didn't want those two dollar-days.
The reason for lining up dollar-days as a sequence of rectangles is that we can reason about sets of dollar-days geometrically. For example, the month-long history of ten dollars would be ten sequences of small rectangles stacked on top of each other. This is a large rectangle with dimensions 10 (dollars) by 30 (days). I find it useful to teach the arithmetic of dollar-days by manipulating such large rectangles.
So, I told the client, you can't spend a dollar-day; you can either own it or not own it.
