When we teach students economics, we sometimes use numerical examples to help them understand general principles. Sometimes we make them work through those numerical examples by themselves, as an assignment. It's the only way they will really get it, and see what's really going on. But once they get it, they don't need the numbers any more, they can see the general principle that doesn't depend on any particular numbers.
As an undergraduate at Stirling I took a course on Merleau-Ponty's Phenomenology of Perception. One book for one course. Looking back on it, I have no idea what it was about. I think I learned something, but I couldn't really tell you what it was.
But I remember (I think I remember) one of Merleau-Ponty's examples. There is a row of light bulbs. The first bulb lights up, then goes out. Then the second bulb lights up, then goes out. Then the third, and so on. And we see a light moving from left to right along the row, even though none of the bulbs is moving.
I think it was Roger Farmer who first explained Samuelson 1958 to me, back in grad skool at Western. He used a numerical example. I think it went something like this:
1. Imagine an infinite line of people, each holding one beer. One equilibrium is where each person drinks one beer. But there is a second equilibrium, where each person gives his beer to the person in front. The person at the front of the line drinks two beers, and everyone else drinks one. The second equilibrium is Pareto Superior to the first, because the person at the front of the line drinks more beer, and everyone else is the same. You can imagine the first person in line giving the person second in line a bit of paper, in exchange for the beer. That bit of paper (money) travels down the line in exchange for the beers traveling up the line.
2. Now change Roger's example, so there is one person first in line, two people second in line, four people third in line, and so on. The population doubles every generation. If every individual gives his beer to the person in front, every individual can drink two beers. The paper money becomes twice as valuable every generation, or pays 100% interest per generation (same thing). But if the population ever starts to decline, somebody is going to be stuck drinking less than one beer.
3. Now change the example again. Let the population be constant, and assume each person lives two periods. He produces 100 beers when young, and none when old. He would like to save half his beer to drink when old, but beer does not keep. If each young person gives 50 beers to the old person in front, everyone is better off. The old people at the front of the line get to drink 150 beers over their lifetime (100 when young, and 50 when old). Everyone else gets to drink the same 100 beers over their lifetime, but now drinks 50 when young and 50 when old, which they prefer.
The total number of beers consumed over one's lifetime tells us something, but it doesn't tell us everything. If beer has diminishing Marginal Utility, and if people do not discount future utility, they will always prefer drinking 50 when young and 50 when old to any other combination of drinking 100 beers over their lifetime. For example, let:
lifetime utility = log(beers consumed when young) + log (beers consumed when old)
4. Now change the example again. Same utility function, constant population, but now assume people produce 50 beers when young and also produce 50 beers when old. If each individual consumes what he produces, and consumes it when he produces it, it looks like this:
A 50 50
B 50 50
C 50 50
D 50 50
E 50 50
Each row represents a generation (or cohort). Generation A is followed by generation B is followed by generation C and so on, forever.
Each column represents a period in time. By assumption, each column must add up to 100 beers, because it is impossible to make beers travel in time. You are not allowed to move beers horizontally.
But you can move beers vertically. You can take 10 beers away from the young in generation B and give them to the old in generation A. And you can imagine that you give the young a bit of paper in exchange for their 10 beers. And you can repeat this, so that the young in generation C give 10 beers to the old in generation B. And then stop, so that the young in generation D give no beers to the old in generation C (the debt is redeemed).
You can make it look like those 10 beers are traveling back in time from generation C to generation A. Just like Merleau Ponty's row of light bulbs.
There are two ways to aggregate: aggregate by columns (years); aggregate by rows (generations). You get different perceptions of reality depending on which way you aggregate. You can't see beers travel back in time horizontally, but you can see beers travel back up the generations vertically.
[Update: if we use natural logs, then lifetime utility is 7.824 for every row; and the aggregate utility in any period (if you are comfortable with adding two people's utilities) is also 7.824 for every column.]
Play with this numerical example by yourself. Because I can't teach you anything. You can only teach yourself this stuff. And the best way to teach yourself is by playing with numerical examples. That's how I learned it. That's how Bob Murphy learned it. See what happens to the lifetime consumption, and lifetime utility, of future generations, if you use debt to finance transfers to the old in generation A. There is no way you can do it without making some future generation have lower lifetime utility.
[The equilibrium rate of interest in this model will be determined by:
1+r = (consumption when old/consumption when young)
Because Marginal Utility of consumption = dlog(C)/dC= 1/C
At the initial equilibrium drawn above, r=0. But if the debt is positive, r will be strictly positive.]
I would be indebted to anybody who knows how to set this thing up on a spreadsheet, with the lifetime utility calculator built right in, and can put it up on the web somehow. Young people can do that sort of thing, right? [Update: thanks to rpl for creating this spreadsheet; which you should be able to copy and edit! I think rpl used base 10 logs, which is why his utility numbers are different from mine.]