# Transversality and Limits Again

I’ve been chasing equations around textbooks on monetary theory again, to try to understand what it is that confuses me about them. I’ve decided that the problems start with household optimisation. Here we go.

In previous posts (e.g.) I worried about the ‘transversality condition’ on government debt. Standard DSGE models of government debt include something like this accounting identity:

Here, the real value of bonds, b(0) is equal to the sum of future surpluses (the first term on the right-hand side) plus the outstanding debt at the final period (the second term on the rhs). I’ve ignored discount rates for ease of presentation. Stick some betas into the equation if you really care.

Then there is the ‘transversality condition’, which in effect stipulates that the final term — the outstanding debt — converges to zero as time approaches infinity:

This gives us (the notation is ugly but standard):

The initial real value of bonds equals the whole stream of future surpluses ‘to infinity’. This is the transversality condition. It isn’t violated in the ‘Fiscal Theory of the Price Level’; all the FTPL adds is that the government can satisfy the condition by setting the price level so as to deflate or inflate the nominal value of its initial bonds until the condition holds.

Where does this condition come from? The standard implicit answer is that it comes from household optimisation — you can see the derivation explicitly on p.20 here. When a household optimises, its budget constraint holds with equality. Optimising households won’t hold wealth they’ll never spend, so that as time approaches infinity they will converge towards having run down their whole stock of bonds. More accurately, as they converge towards holding no bonds, then the government must, by accounting identity, converge towards having no bonds outstanding.

First, let’s think about this in terms of a finite time-horizon. We have:

So what happens at N? At this point, households hold no outstanding bonds; government surpluses have paid them all out. Is this plausible? Why couldn’t the government just default at N? If the households and the government are in a ‘Walrasian equilibrium’, we can imagine some supernatural mechanism leading the households to behave in a ‘Ricardian’ fashion. Suppose the households have perfect foresight. The Walrasian ‘auctioneer’ takes various offers from the government for paths of surpluses, and the households choose one satisfying the condition above, to maximise their and their descendants’ lifetime utility.

The Walrasian auctioneer is a way of giving households the first move in a game between themselves and the government — call this the ‘Bond Game’. If the government doesn’t offer a path of surpluses equal to the value of the initial bond offering, the households won’t take it.

Operationally speaking, however, it seems like the government has the first move in the Bond Game, provided it controls the issuance of the currency needed to pay taxes. This is the neo-chartalist point. All it has to do is tell the households that the currency they need to pay taxes, which it imposes, can only be got by trading in government bonds; indeed it could nominate the bonds themselves as the currency needed to pay tax. Now it issues bonds to the households, to buy goods and services from them. The households have to take the deal, because they *might* need the bonds to be able to pay their future taxes. They have to buy the bonds at the price the government nominates or risk the penalties of tax-evasion (which the government also sets).

Household optimisation is thus pushed out of the picture. The households can only optimise if the government lets them.

Now take the infinite time-horizon. We no longer have a period N in which the government can default. We have, rather:

What do we make of optimisation in this case? Let’s allow that households optimise over this infinite time-horizon — that is, the above condition is satisfied. Does this mean that the households can’t hold onto bonds to infinity? Well, what does the above formula mean? I’ve argued before that it isn’t really an equation in the ordinary sense; it’s a proposition of analysis of the following form (if you care, it represents a higher-order function taking first-order equations as its arguments):

Notice that this is *consistent* with the following proposition:

Since N ranges over the positive integers, if, at i=N for any value of N, there are outstanding bonds, there will still be an infinity of values N+1, N+2, etc. such that as i reaches them the sum of the stream of surpluses can be run such as to pay out the outstanding bonds.

Even though the (time-discounted, if you like) sum of surpluses must equal the value of the initial bond issuance approaching infinity, this is consistent with there being no time at which the value of the sum of surpluses is anywhere near the value of the initial bond issuance.

This result might look contradictory, but that’s what happens when you try to think about infinity using arguments that only range over finite values.

Here’s a slightly crude way of trying to break out of that constraint. Think of the time-argument, i, taking the finite integer series (1, 2, …) as its model. Now rearrange the series to put all the odd numbers first and all the even numbers after: (1, 3, 5, … , 2, 4, 6, …). Let the household increase its bond holdings steadily during the odd-number portion of the series and then reduce them towards zero during the even-number portion.

Now two things are true. Over an infinite time-period, the household increases its net bond holdings without running them down. Also, over an infinite time-period, the household runs down its net bond holdings towards zero. Worse still, the infinite time periods are the same length: the cardinality of the set of positive integers is equal to the cardinality of the set of only the odd integers. Transversality is consistent with a situation in which the government rolls over its bonds without paying them down over an infinite period.

So this is what we have. Over a finite time-horizon, there are powerful neo-chartalist arguments to suggest that the government can head towards a final default if it wants, and the households can’t stop it. They don’t have first move in the Bond Game. Over an infinite time-horizon, *even if transversality is satisfied*, this is consistent with the government failing to pay down its whole debt over an infinite time-period.

Either transversality doesn’t hold as a necessary condition, or it doesn’t mean what the models want it to mean.

For more on the logic and mathematics of DSGE models and transversality, I recommend Brian Romanchuk’s posts, e.g.:

Does The Governmental Budget Constraint Exist?

You might also be interested in Neil Wilson’s posts discussing the neo-chartalist perspective, e.g.: