Wednesday, November 14, 2012
Left to themselves drivers will join lines when they are shorter and avoid them when they are longer. This smoothes out the queue lengths. Odd/Even prevents drivers from doing this. How?
Imagine Bob has an odd plate and is driving on an even day. He sees a shortish line but cannot join it.
Or next day Bob is wondering if he has enough gas. He'd like to wait and see, but the rationing makes waiting risky: he might not be able to wait two days. He'll join the line even if it is long.
Or perhaps there are just more odd-plated drivers where Bob lives, so the lines are longer on odd days.
Odd/Even rationing unbalances queues.
Consider then two queues.
Each driver in the queue takes one time period to get and pay for gas.
This isn't realistic, as some buy more gas, and some piddle around with cash, but it will do as a first approximation.
And let's further assume that new cars arrive at one per time unit at the end of the queue.
This is more subtle.
Drivers are reluctant to queue when the line gets long, and eager to queue when it shortens. So we will generally reach a period of approximate equilibrium where the line reaches a steady length. Not exact but close enough for government work. I'll discuss the other cases below in case this assumption worries you.
We can think of just the people in the queues at a given moment.
So lets put 3 people in queue Even and 5 people in queue Odd
Even: a b c
Odd : P Q R S T
(This is 5-3 split is perfectly fair on my part: I am examining the effects of unbalancing the queues, and a difference of 2 is the smallest relevant difference. S's arrival would lengthen one queue no matter what but T was compelled to join the longer queue when he arrived. Add another Odd driver and it gets even worse. So this example shows the smallest effect of forced unbalancing.)
What's the total amount of waiting and filling for these folks?
In Even it is 1 for a, then 2 for b (he waits for a then spends his time) and 3 for c (who starts when b is done). Total 6.
And in queue Odd it is 1 + 2 + 3 + 4 + 5. Total 15.
Grand total 21.
What if T could move into queue Even? Then we get 1 + 2 + 3 + 4 for each queue.
Grand total 20.
So the inability to smooth out queues has made things worse.
This holds even if the queues keep getting longer until say 9 pm or if some people randomly take longer than others.
Now what happens if new people don't arrive as fast as the queue empties? This will happen late at night for example. Then we get shorter or empty queues and the costs are more obvious. For example after 3 periods queue Even is empty and in queue Odd person T is still waiting when he could have found an empty pump. Segregation has cost T and benefitted no-one.
And what happens when people arrive faster at the tail end? Then the queues keep getting longer. Since our population is finite we have to eventually reach the steady state or the shrinking queue case. We already looked at those.
And what if not everyone uses the same amount of time at the pump? Then it gets a bit trickier and all results become probabilistic. But the main effect remains.
Let people use between 1 and 4 units of time once they pull up to the pump.
Then we get queues that look like this, replacing names with pump times:
Odd: 1 3 3 2 1 4 2 1 4
To calculate the time spent pumping and filling start at tail-end Charley.
He waits for everyone ahead. Here that is 1 + 3 + 3 + 2 + 1 + 4 + 2 + 1.
Then he pumps for 4. So his elapsed time is the sum of everyone's pump time up to and including himself. This is true of driver n, his time is the sum of all times of drivers 1 through n.
If you play around you'll find you want (more or less) queues with even pump times.
So segregation by pump time can help. Think of express lanes at the supermarket: lots of fast shoppers in lane Odd and a few with big baskets in lane Even. Imagine 6 odds at a minute each and three evens at 4 minutes each.
Then we get this:
Odd: 1 1 1 1 1 1
Even : 4 4 4
The totals are 21 and 24, which is pretty closely balanced.
In this example odd/even segregation benefits these drivers. But this example is pretty special. Any arbitrary form segregation can in fact by dumb luck help achieve pump time segregation. Odd/even or bald/hirsute can conceivably give us pump time segregation. Giving blue-eyed drivers priority can also, by blind luck, more the faster drivers to the front of the line. But the chances are remote. In the real world, odd/even is far more likely to make things worse.
It's common sense really, disguised by the time shifting. You ever see the line-up at the women's during intermission?
Monday, November 12, 2012
Thursday, November 8, 2012
SteveI am writing this jointly with Bob Murphy.The discussion of debt-as-burden on FreeAdvice has become so complicated that we cannot even agree on what *your* position is. I am hoping you will be willing to clear this up. Here then is my ( Ken B) understanding:Krugman’s argument is that borrowing rather than taxing imposes no extra burden on unborn Americans in toto, because both are transfers between living persons, and that any burdens are caused not by debt per se but by transfer and incentive effects. (Bob would describe Krugman's position a bit differently, but this is Ken's understanding of it.) Krugman’s argument relies on the constraint that internally held debt represents a flow of assets is within a given pool, and so the pool is not directly diminished by the debt. The debtor and lender are part of the same closed system.Steve endorses but extends this argument. Steve observes that Krugman is missing a point which strengthens and extends his claim: if the foreigner owns a bond because he paid for it we include the payment in the pool's assets the constraint is re-established. (One can also see the initially foreign held cash as a claim against future earnings already, so all we are seeing is an exchange, cash for cash flow, not a new burden.)Steve’s argument relies on the same constraint as Krugman’s, that debt is just exchange amongst living people, not an extraction from the future. This is clear if you take a case where foreign interactions are *impossible*, so that SL's argument reduces to PK's.Bob thinks there's nothing wrong with your own position, but he just thinks it is weird to say "Steve is strengthening Krugman's point" when, in Bob's mind, you are actually advancing a totally different argument. In contrast, I think you are amplifying Krugman's point, by showing that Krugman doesn't even need to focus on whether future Americans "owe it to themselves."We'd appreciate any clarification, and also let us know whether you want your response to remain private.
Steve kindly replied and his answer is below, with permission:
Monday, November 5, 2012
Krugman argues that internally held debt cannot in any direct way impoverish America as a whole. Internal borrowing rather than taxing imposes no extra burden on any future America in toto, because both are transfers between then living Americans. Internally held debt represents a flow of assets is within a given pool, and so the pool is not directly diminished by the debt. Krugman gives a good analogy: Imagine a "Santorum tax" that just transfers funds between Americans. Leaving an internally held debt is just like a Santorum tax.
I think Krugman's argument is sound, convincing, and correct. But Bob Murphy and Nick Rowe insist, reasonably enough, that the claim be tested against some models to see if they can disprove it. Bob presents his model, and claimed counter-example, here. The rest of the discussion takes place within the context of this model.
At first blush it looks like Bob has made a point. We do indeed see people who eat fewer apples. But in fact this is not a counter example at all. I'll come back to why but we must first translate Krugman's claim into this model. Krugman talks about future states of America.
The natural and correct interpretation of Krugman here is that a state of the whole country at any moment corresponds to a row in the table, and a period of time corresponds to a contiguous block of rows. (We know this is what he means because he tells us so with the Santorum tax). Krugman's assertion is that at no point in time and for no period of time is a future America impoverished. This is plainly true in the model: each row (point in time) has 200 apples. The period claim follows from the point in time claim. If that's what a future state really is, then Krugman's claim holds in this model.
So the question reduces to: is this the appropriate definition of future state? Bob's example assumes not. Rather than level sections, Bob presents diagonal sections. He includes for example Young Frank, Old Frank, Young George, Old George but not Young Hank or Old Eddy, slicing diagonally through the table. In any of his slices there is always one person from the first level and one from the last level omitted. And this is the problem. At no point in time and for no period of time are the people and assets of a diagonal section co-extensive with the population and assets of the Island. In the real world a slice like this might include all the people whose names end in vowels who were born before 2012 and all those who names do not end in vowels who were born after 1970. That diagonal section never, not for a moment, not for a period, is all and only the population of the country. A diagonal section is not a state of the country. Bob's diagonal sections may be interesting -- I will return to this point -- but they do not correspond to a future state of the country. Since they do not refer to what Krugman is talking about -- future states -- they cannot be counterexamples to Krugman's claim.
So Bob's construction is not a counterexample after all. But don't Bob diagonal sections show something? After all he seems to show a loan impoverishing 65 generations. Doesn't that prove something? No, not about the aggregate consequences of debt anyway.
First Bob's example does not show one loan affecting 65 generations. He shows 65 consecutive loans affecting 65 consecutive generations. In each case the whole daisy chain would be ended unless each generation takes from the next while both are alive. Gene Callahan put this well: I can eat my son's desert, and when I am dead he can eat his own son's desert, but that doesn't mean I ate my grandson's desert. I explicate this point in detail in comments on that thread. Borrowing from Young Hank does not burden the future, it burdens Hank.
Do Bob's diagonal sections correspond to anything meaningful? Perhaps they do, but not for the purpose of looking at the aggregate effect of debt or transfers. I illustrate this with an example where imposing a debt on a diagonal section seems to enrich it. This is pretty odd if you want to argue the section corresponds to something meaningful, which is burdened by the infliction of a debt. But a diagonal section is really just an incomplete set of accounts. There is no mystery why you can make an incomplete set of accounts seem unbalanced in odd ways.
So Krugman is right, in detail, and Rowe and Murphy have merely exhibited a combinatorial artifact, which they misinterpret.
Here it is. I hope the formatting is decent:
Can a debt *enrich* the future? By Bob's criteria it can.
Let's look at the example of Apple Atol, just a few miles from Apple Island where their policies differ.
In period 1 govt borrows 1 at 100% from young A for old Z.
In period 2 govt borrows 2 at 100% from young B and pays off the loan to old A.
In period 3 govt borrows 4 at 100% from young C and pays off the loan to old B.
In period 4 govt borrows 8 at 100% from young D and pays off the loan to old C.
In period 5 govt borrows 16 at 100% from young E and pays off the loan to old D.
At this point the Atol switches policies to get a steady state. There are many ways to do this.
Govt can tax the youngster 8 and borrow from the youngster 8 at 100%, or borrow from the youngster 16 at 0%.
Let's use the latter.
In period 6 govt borrows 16 at 0% from young F and pays off the loan to old E.
In period 7 govt borrows 16 at 0% from young G and pays off the loan to old F.
In period 8 govt borrows 16 at 0% from young H and pays off the loan to old G.
I lack Bob's mad table skills but let me show the resulting data
oq 100 yz 100
oz 101 ya 99
oa 102 yb 98
ob 104 yc 96
oc 108 yd 92
od 116 ye 84
oe 116 yf 84
of 116 yg 84
oh 116 yi 84
Now let's do Bob's diagonal sums.
Everyone ends up with at least 200, and some end up with more than 200. C gets 204.
The future, by Bob's critria, has been enriched.
A few other things to note. This example runs on pure debt, not defaults and not taxes, but there are several ways to produce such examples.
Old Z is better off too, no-one is harmed.
By the Krugmanite criteria, the Atol is not enriched.
For style points, the govt can just destroy the apple it borrows from ya.
Then the chart changes to "oz 100 ya 99"
That leaves Z out of the windfall, but the Bob-defined enrichment still works with no-one losing out *even with an apple destroyed*.
By the Krugmanite criteria the style points apple destruction impoverishes the present, without aiding the future. Broken windows anyone?
I suggest this example shows Bob's crtiteria are wonky, and that the limits of integration are inappropriate.