Government Budget Constraints and Game Theory

I had an interesting chat on Twitter with David Andolfatto (who was very generous), concerning the government budget constraint in macroeconomic theory.

This is a sticking point in the debate between so-called ‘mainstream’ economists and so-called ‘heterodox’ economists, especially those in the Modern Monetary Theory school. There is a lot of confusion about whether MMT rejects the government budget constraint (GBC) and, if so, why. There’s also a lot of unclarity about what the GBC is — whether it’s just an innocent accounting identity or something that restricts policy.

I don’t want to get into these weeds. I’m taking the GBC here to stand for a single assumption — that the government doesn’t default on its debt. I don’t know if MMTists will like me using terms like ‘debt’ and ‘default’ in this way. They might take them to indicate a departure from realism. But I’m really interested in the purely logical aspects of standard macroeconomic models, which depart from realism by their very nature.

I raised with David (I hope he’s ok with first-name basis) the claim that the GBC is derived from household optimisation. He didn’t agree, but I’m not sure I expressed myself well. Let me put it this way. As I see it, it’s in the government’s interest to default on its debt at the time-horizon (specifically, at the second-to-last period). It would mean that it got the real resources that it purchased with its bonds for free, leaving those who bought the bonds with worthless accounting entries (and they’ll be dead after the next period). It shouldn’t default before we reach the time-horizon, of course, because then people won’t buy its bonds without a heavy discount from that period onwards. Right now I’m thinking of a finite time-horizon, but I’ll say something about the infinite time-horizon later on.

Let’s simplify the issue and assume that the government doesn’t issue new money to pay out its bonds. In other words, to repay its bonds, it has to get tax revenue to pay out the bonds. To collect the tax it has to use real resources. And, let’s stipulate, it has to provide real goods and services to the taxpayers for them to tolerate the taxes. So taxation is costly for it, and a final default would be cheaper (a mainstream reason for the same conclusion would be that the government seeks to minimise the tax rate).

Why do I say that household optimisation underlies the GBC? There’s a fairly explicit derivation in a paper by Cochrane, which I linked to in a previous post. But, getting more basic, if it’s in the government’s interest to default, then the reason it doesn’t default must be that private agents won’t let it.

When I raised this with David, he told me that this is just standard game theory: each agent optimises as subject to the behaviour of other agents. But then everything hangs on how we’re setting up the game. In game theory, of course, it matters a lot who (if anyone) has the first move. I think that mainstream macroeconomic models, though not explicitly, set up the game by giving first move to governments (whereas I think the first move should belong to private agents). This is how they justify the GBC.

David pointed me towards this paper, which gives the game-theoretic underpinning to some macro models. If you can access it, I’ll draw your attention towards the subgame perfect equilibrium defined at p.796. The government has a set of possible policies it can follow. Private agents devise their optimal response to each policy, and then the government chooses its policy to optimise its own position, taking into account the response this will evince from the private agents. I say that’s giving the government first move: it must decide what it’s going to do and then the private agents have a chance to respond. The government chooses by backwards induction from their foreseen responses, drawing on common knowledge of rationality, etc. etc.

But let me set up the game of government default very simply like this:

Figure 1

In the first move, the private agents decide whether or not to buy government bonds. In the second (the end of the time-horizon), the government decides whether to default or repay. The government’s payoffs are on the left, the private agents’ on the right. If the private agents choose Buy, the government will optimise by choosing Default. Reasoning by backwards induction, the agents will therefore choose Don’t buy. In this case there would be no fiscal policy at all.

How do we avoid that consequence? We give the government first move. It decides whether to repay its bonds or default at the end of time. The game looks like this:

Figure 2

The government knows that if it chooses Default, the private agents will choose Don’t buy. By backwards induction, the government works out that Repay is the best move.

But time is out of joint in this game. Why should the government’s decision to default or repay come before the decision of private agents to buy or not buy? The only answer I can see is: only to generate the GBC.

Here an economist might pipe up: ‘there is no first move — this is a Walrasian equilibrium — all agents make their decisions simultaneously.’ But in that case, I think we’d have a prisoner’s dilemma:

Figure 3

The government will choose Default, since this dominates Repay, and the private agents, knowing this, will choose Don’t buy. Again, we’ve obliterated fiscal policy altogether.

This is all on a finite time-horizon, and of course the reasoning is very unrealistic. The government doesn’t plan its policy for the End of Time, and private agents don’t decide on the basis of backwards induction from its Apocalyptic policy. And of course agents don’t make a binary choice Buy/Don’t buy; they determine the discount rate and price at which they buy bonds. But I’m trying to cut through to the most basic strategic reasoning here. I’m trying to get at the question of who forces whom (Brian Romanchuk has a post on the same question here).

A nod to realism, surprisingly enough, is the move to an infinite time-horizon. Now we have no final moment in which the government can default. I can’t represent this circumstance in my simple game, since the government’s move was meant to be a choice of whether to default or repay in the end.

But, as I’ve argued several times before, the suspicious move in macroeconomics here is to take the limit of solutions to finite games of increasing length as the solution for an infinite game. I don’t see the logic in that. Once we’ve modelled the game along the lines of Figure 2, we know that any finite extension of the game has its equilibrium at Repay/Buy. That, then, is the limit of solutions to games of any finite length. But it does not follow that it is the solution of an infinite game. In an infinite game, the government doesn’t have a last move to choose at all, and the reasoning by backwards induction that generated the agents’ strategy loses its basis.

An infinite game is a mighty maze, and it’s understandable that game theorists should want to somehow reduce them to finite games. In other instances, they define them out of existence. In this paper on the infinite-horizon version of the Centipede Game (why do they miss a trick and not call it the Infinipede Game?!), it is stipulated that if the game doesn’t end, the payoff to both players is zero (see p4 — the footnote gives the alternative that the payoff from the game not ending is less than the payoff from the earliest possible stopping point). Since the average payoff to both players increases as the game lengthens, it’s not clear why the payoff from an infinite game shouldn’t be infinite. But the point of stipulating a zero payoff for the infinite game is to allow for a solution by backwards induction. It means that both players can stare into eternity and reason that they don’t want to end up there.

Something similar, I think, is going on with macroeconomic models and the GBC. On an infinite-horizon model, it isn’t clear why the government couldn’t just keep borrowing to pay off its debt and borrow more. The private agents keep being repaid, so they have no reason to stop lending. An infinite Ponzi scheme is logically possible; what breaks Ponzi schemes in reality is some external constraint. But here there is no external constraint to be seen except optimisation by the private agents — yet without a final point from which to backwardly induce, they have no optimisation solution. Taking our cue from that paper on Infinipede games, we could stipulate that if the game doesn’t end, the outcome is worse for the private agents than stopping at any point with full repayment. But that would be a stipulation that needed motivating.

A paper by Scott Fullwiler notes that in practice the GBC doesn’t seem to apply. The US government keeps refinancing its debt with more debt, and nobody stops buying the bonds. Nor do bond-prices fall, nor do interest rates rise, nor does hyperinflation rear its head. Applying the mainstream models, this entails that the US Treasury is going to run large and continuous surpluses sometime in the future. If private agents are optimising all the way to the horizon of the future, and they are buying the bonds now, then they must know that sometime such surpluses will be run. But nothing beyond the model bears out that prediction. More likely they’re acting suboptimally, or… they’re disadvantaged by having first move.

The implicit model supporting the idea of the GBC is not, I conclude, empirically validated. Its logic seems ad hoc and unmotivated. This is probably a peripheral point to the debate between mainstream and heterodox economists, but I thought it might be worth making.