Debt and the burden on future generations, Part MMMVIII

I don't want to bore people, but once again this question has come up (see here, here, here, here, here, here, and here for the whole battle royale) , and I thought I'd blog about it, because hey, every econ blog should occasionally do some little "thought experiment" type stuff, even if it doesn't quite as much traffic as does making fun of commenters.

The question, once again, is: "Does government debt impose a burden on future generations?" I took a crack at this question back in January, and my answer is still the same, but I'd like to phrase it more concretely.

Here's how I like to think about this question. In my mind, to "impose a burden on future generations" means  "to decrease the consumption possibilities of future generations". So the question is really whether or not the size of today's stock of government debt reduces the total consumption possibilities of people not currently born. In other words, if government debt is $1,000,000,000 today, does that mean that the consumption of future people must be lower than if government debt were $1 today?

Let's assume a closed economy. In that case, the economy's maximum potential consumption at any point in time is determined by the productive capacity of the economy at that time. Productive capacity is determined by the size of the capital stock, the labor force, the availability of natural resources, and the level of production technology. (For convenience, I'm defining the "capital stock" as including all consumer durables, and defining "consumption" as including the flow of services from those durables.) Now let's assume that the technology level, the labor force, and the amount of natural resources are all completely exogenous, so that the government cannot affect these things (this may not be realistic but we could always drop that assumption later). So the productive capacity of the economy at any point in time is just a monotonic function of the economy's capital stock - more capital at time T means more potential consumption at time T.

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. Since the potential consumption of the economy at any time t is determined entirely by the size of the capital stock at time t, what we are really asking is whether or not the following proposition is true:

∀{D_t},{C_t} ∃T>0 s.t. K_T = f(D_0), where f'(D_0) < 0 

Here K is the capital stock, D is government debt, f is some function, t=0 is today, D_0 is today's stock of government debt, {D_t} is the path of government debt between t=0 and t=T, and {C_t} is the path of consumption between t=0 and t=T. If this proposition is true, then no matter what anybody does in the future, higher debt today necessarily means a smaller capital stock at some point in the future. 

Note that this proposition is not stated as formally as it could be or really should be, for which I apologize.

So now, let's think about what determines the capital stock at a future time T. This is determined by the sequences of consumption and investment from t=0 to t=T-1. In order for K_T to be constrained to be lower than it would otherwise be, it must be the case that K_T-1 is lower than it would otherwise be (this follows easily from the assumption that the production function is monotonic in the level of the capital stock). By backwards induction, the above proposition can only hold if the following proposition holds:

K_1 = f(D_0), where f'(D_0) < 0 

Remember, t=1 means tomorrow. In other words, only if tomorrow's capital stock depends in a negative way on today's stock of government debt can it be true that a higher D_0 forces K_T to be lower at some point in time.

Tomorrow's capital stock depends entirely on today's level of investment (today's level of production is fixed, because today's capital stock is fixed). So our question now reduces to:

Question: If I_0 = g(D_0), where I_0 is today's investment and g is some function, what is the sign of g'(D_0)? 

If g'(D_0) is positive, then a higher government debt stock today means that the economy will invest more today; this means that government debt will impose no burden on future generations.

So is it possible that g'(D_0) > 0? In other words, given two societies that are identical in all respects except that Society 1 has a higher stock of government debt than Society 2, is it possible that Society 1 will invest more today (and consume less today) than Society 2?

Of course it's possible. The investment/consumption choice is entirely behavioral. And when I say "behavioral" I am including the behavior of the government. If Society 1's government chooses to cut welfare and use the money to build a bunch of roads, for example, it could easily invest more and consume less today than Society 2; the high level of D_0 in Society 1 would not prevent it from being able to do this.

So government debt need not be a burden on future generations. It all depends on how economy-wide consumption/savings decisions react to the size of the stock of government debt. And that is heavily dependent on the behavioral model one chooses. Might a higher stock of government debt outstanding induce a society to invest less and consume more (which would constrain future consumption to be lower under certain additional assumptions)? Sure.

So the answer to the question is: It depends. What does it depend on? It depends on how consumption/savings decisions react to the size of the stock of government debt, which depends on the behavior of the government, firms, and households. Modeling that behavior is a major challenge.

Also, note that this does not answer the question of "Does government borrowing impose a debt on future generations?" This is because the economy's consumption-savings choices may respond differently to changes in debt than to levels of debt. But in general, the answer will have the same form.

So to sum up:

• Must higher government debt today lead to lower potential consumption sometime in the future? No.

• Does higher government debt today lead to lower potential consumption sometime in the future? Maybe; I don't know.

• Does higher government debt today lead to lower actual consumption sometime in the future? Maybe; I don't know.

• Must higher government borrowing today lead to lower potential consumption sometime in the future? No.

• Does higher government borrowing today lead to lower potential consumption sometime in the future? Maybe; I don't know.

• Does higher government borrowing today lead to lower actual consumption sometime in the future? Maybe; I don't know.

(Just in case you were wondering: The example Nick Rowe creates here is a case of higher government borrowing today leading to lower actual consumption in the future. He uses a "fruit-tree economy" with no capital (or if you prefer, with K fixed), so potential consumption in each period is fixed. In that sort of economy, it is impossible for anything to "impose a burden" on any cohort, using my definition of "imposing a burden".) 

Update: More interesting conversation between me and Nick over at his blog, as well as in the comment section of this post. We look deeper into the issue and get some more interesting results.

Update 2: Nick and I have been discussing the issue. I think we agree on everything now, and a number of interesting conclusions have emerged. Let me see if I can translate them into plain English...

The "Burden" Result: It is possible that the existence of past government transfers can ensure that either currently living people or as-yet-unborn (or both) must get screwed, relative to the baseline in which no transfers occurred. These past government transfers can be accomplished by government borrowing and spending; in that case, the past government transfers will affect the value of today's government debt. This is the upshot of Nick's model.

The "No Future Burden" Result: However, no matter what transfers happened in the past or how much government debt we have today, then given some simple assumptions, it is always possible to get away with only screwing people who are currently alive (and yes, you can quote me on that!). This is the upshot of my proof.

Note that these two results are not incompatible at all. So Nick and I don't disagree.

The "Dues Paid" Result: Given some more simple assumptions, it is always possible to limit the total amount of screwage (in consumption terms, not utility terms) to the amount of consumption that was, in the past, transferred away from people who are currently alive. In other words, the total amount of screwage never has to be bigger than the total "dues" already paid by currently living people. This is something I realized while talking to Nick over at his blog. I think it's kind of interesting.

The "Debt Does Not Equal Burden" Result: This means that the govt. debt number may not equal the burden number (and in general does not). The size of the current stock of government debt may be much larger than the total amount of the aforementioned screwage. In other words, govt. debt may be $10,000,000,000 today, but the total amount of necessary screwage might be much smaller, or might even be zero. This can happen, for example, if the government spends money on the same people it taxes, or if people leave government bonds to their children in a certain way. So debt is not a book-keeping device that faithfully records the amount of necessary future screwage.

(Note that this means that government debt's effect on society is very different from the effect of one household's debt on that household. If I borrow $10,000 and spend it today, I'm going to need to take a $10,000 hit in the future in order to pay it back. But if the government borrows $10,000 today, it's quite possible that nobody ever has to take a hit at all. I am not sure, but I think that this might be Paul Krugman's main point.)

(Update: Antonio Fatas thinks that this last result should be the main takeaway from the debate.)

In conclusion: When you ask "Does debt impose a burden on future generations?", you have to be very careful about exactly what you mean when you ask that question. But if you are careful - if you use math in your explanation, state all definitions and assumptions clearly, and above all think clearly and don't get mad - then the truth will out.