Because aggregating across people alive at a particular time is not the same as aggregating across time periods in the lifetimes of a particular cohort of people. (See JP Koning on the Borges Problem  or Scott Sumner).

[Update: or does Noah get my point? because later on in the comments to his post he says: "Nick Rowe's point is that if the government borrows to consume, then it
can effectively be shortchanging the future generation, whether that
shortchanging is accomplished by reduced investment or by durable
government commitments to future net transfers. This is also a good and valuable point." Dunno.]

[Update 2. OK, Noah does get my point. Sorry Noah. But there is some strange stuff going on in his model. See my discussion with Noah in the comments below.]

Noah Smith is a very smart guy, and a very good economist. And Noah totally misses my point about the burden of the debt on future cohorts. But because Noah is a very smart guy, and a very good economist, he is also beautifully clear in exactly why he misses my point. And I bet lots of other very smart people and very good economists are missing my point for exactly the same reason.

Here's Noah: "Now let's define "burden on future generations". That means that at some time T > 0 (t=0 being today), the potential consumption of the economy will be lower."

NO! That is NOT how we should define "burden on future generations". We should define it as "lower lifetime consumption (or lower lifetime utility) of future cohorts of people".

In comments, Andy Harless (who gets my point) tries to explain this to Noah. Noah responds to Andy:

"You're right that I could use
that alternate definition of "generation". But the result would be the
same, since a negative result for future times implies a negative result
for future generations, since future generations live only during
future times…..Also note that to state my point
more completely, in my response above I should have written "a future
cohort's consumption is just the union of the consumption values for various (i,t>0) pairs.""

NO! It's NOT the same. A future cohort's consumption is NOT just the union of the values for aggregate consumption at various future times when that cohort is alive.

When I was explaining my point about the burden of the debt on future generations, I deliberately chose an example in which aggregate consumption at all future times was constant, and unaffected by the debt. In other words, I deliberately assumed there was no "burden" in the sense in which Noah defines "burden". But I showed that there was nevertheless a burden to a future cohort.

Here is another example in which aggregate consumption is constant at all times and is unaffected by the debt. But the current cohort gains at the expense of ALL future cohorts:

Assume all output is consumed.
Assume people live for 2 periods, produce 100 when young and 100 when
old. Assume one person per cohort (so population is always 2). So
aggregate consumption is 200 at all time periods.

In the baseline scenario each person produces and consumes 100 each period, so lifetime consumption is 200, and aggregate consumption per period is 200.

Now suppose:

In period 0, the young in cohort A consume 100. (Same as baseline)

In
period 1, the old in cohort A consume 200, and the young in cohort B
consume 0. Aggregate C=200. Lifetime consumption of cohort A is 300.

In
period 2, the old in cohort B consume 150, and the young in cohort C
consume 50. Aggregate C=200. Lifetime consumption of cohort B is 150.

In
period 3, the old in cohort C consume 125, and the young in cohort D
consume 75. Aggregate C=200. Lifetime consumption of cohort C is 175.

In
period 4, the old in cohort D consume 112.5, and the young in cohort E
consume 87.5. Aggregate C=200. Lifetime consumption of cohort D is
187.5.

And so on.

The government borrowed 100 and gave it to cohort A to consume, then slowly reduced the debt at the rate of 50% per period.

Every single future cohort has lower lifetime consumption, even though aggregate consumption is always 200 at all time periods. There is a burden on ALL future cohorts (though that burden asymptotically approaches zero as we go into the very distant future).

Here is a second example, where all future cohorts have lower lifetime utility, even though their lifetime consumption stays at 200:

Same as the first example, except the government only borrows 50, and then taxes all future generations to pay the interest on the debt, but leaves the debt unchanged at 50 forever. So all future cohorts consume 50 when young and 150 when old. But they have a utility function U=consumption when young x consumption when old. So their lifetime utility will be 50×150, which is less than 100×100.

Update: And if the rate of interest is positive (above the zero growth rate in this economy) then the government must (eventually) raise taxes to pay at least the interest on the debt, or the debt/GDP ratio will grow without limit until the young cannot afford to buy the bonds.

If Noah doesn't get it after this, then I will despair.

Does this mean that debt always creates a burden on future generations? Emphatically no. It is possible to construct different examples where lifetime utility goes up even if lifetime consumption stays the same (because the rate of interest is always negative). Or where lifetime consumption and utility of all future cohorts goes up even though aggregate consumption is unaffected (because the rate of interest is always less than the growth rate). Or the deficit could finance investment that increases future cohorts' consumption. There may be benefits as well as costs for future generations. But until people get my point, we can't talk about the benefits and costs sensibly.