DSGE Macro "Proves" There Are No Financial Constraints On Government

In this article, I run through the mathematical logic of standard ("workhorse") dynamic stochastic general equilibrium (DSGE) models, and show how they imply that there is no financial constraint on government in a practical sense. (There is a disclaimer; one can imagine a default constraint, but that is a quite separate issue, as I briefly discuss.) This verbal description aligns with the verbal description of Modern Monetary Theory (MMT) given by Modern Monetary Theorists (MMTists)*. The Fiscal Theory of the Price Level (FTPL) makes an appearance, but as I argue, it is really the Fiscal Assumption of the Price Level.

To be clear, I am describing DSGE models in this article. There is a desire among neoclassicals to "make MMT more rigorous" by attempting to cast them in a DSGE model. If we look at the MMT academic literature, it is a subset of the post-Keynesian literature. It seems safe to say that every single behavioural assumption embedded in DSGE models is viewed as incorrect by at least one post-Keynesian. Making a literature "more rigorous" by ignoring the actual contents of said literature is a very curious position for a scholar to take.

This article is a monologue that ignores pretty much all previous writing about the governmental budget constraint (including my own!). In previous writings, I was uncertain how DSGE researchers were applying their mathematical rules, and so I was hampered by existing verbal descriptions. In my view, previous descriptions were misleading, and so we need to go back to the mathematical basics. I hope to collaborate with Alex Douglas in an article on DSGE macro; if we want to make this text meet academic standards, we would need to relate it to the past literature.

The Cast of (Equation) Characters

My discussion here is based on some generic DSGE models, as well as the Ramsey problem from the textbook by Ljungqvist and Sargent, which view featured in a set of earlier articles (link to first).

In both cases, these are representative agent macro models. There is a wide class of models that will have similar properties, such as the workhorse New Keynesian models with Calvo pricing. I am going to discuss very particular properties of these models, and so any model that shares these properties would be covered. In my view, that is most of the commonly used ones.

(Overlapping generations (OLG) models are more distinct. However, I will reverse an invalid complaint levelled at MMT by neoclassicals who have not read the MMT academic literature: there is no empirical work. Since I am unaware of any empirical work of data sets with time points "one generation" apart, I argue that OLG models are just non-empirical fairy tales.)

Since I need to cover classes of models, I will refer to some of the mathematical components (generally equations) as labels, which are described below. (Some of these labels are used as shorthand in online discussion already.) A mathematically inclined reader will be able to align my labels with the actual mathematics in the models. This list is unfortunately imposing; the reader may wish to skip ahead.

Why not write out the equations? For a number of reasons. Firstly, they will be unintelligible to most readers. Secondly, equations refer to a single model, and so we would need to write multiple equations for each label. Lastly, my argument is that the mathematical notation used in DSGE macro expositions does not meet standard mathematical usage. Since I have never managed to find a reference that gives explicit rules on how to map neoclassical notation to standard mathematical notation, I will not write out the equations as mathematical symbols.

For simplicity, I will assume that money is not in the utility function, and so money holdings are zero. This means that we can refer to government liabilities as "debt", since the money liability is zero. If we want to cover models that have non-zero money holdings, replace "government debt" with "government liabilities."

(HHBUDGET) The household budget constraint. This is an accounting identity describing the household's financial position.

(HHOPT) The household optimisation problem. This is not properly an equation, rather a mathematical entity that consists of a utility function, budget constraint (HHBUDGET), and optimisation statement. That is, find the consumption/work trajectory that gives optimal utility, with wages, prices, and interest rates as exogenous variables.

(HHTRANS) The household transversality condition. It is a first order condition of the (HHOPT), and says that in the limit of time going to infinity, discounted household holdings of government debt is zero. The (HHTRANS) condition is a standard consequence of (HHOPT) based on optimal control theory.

(PROFIT) The business sector profit maximisation problem. Given exogenous wages and prices, and a production function, what number of workers need to be hired to maximise profits? (We would need to add in "rented capital" -- as in the Ljunqvist and Sargent model -- in some cases.)

(GOVTRANS) The government "debt transversality" condition. (This name is a misnomer, since this is not the result of a optimisation problem; the name is just chosen to mirror (HHTRANS).) This just says that if the condition holds (whether it holds is the point of theoretical discussion), the discounted value of government debt tends to zero as time goes to infinity.

(IGBC) The inter-temporal governmental budget constraint. This equation can get pretty complex if we have things like money, but the summary of the important parts is that the value of the current stock of debt is equal to the summation of the discounted value of expected fiscal primary surpluses from the present to infinity. (The primary surplus is the governmental budget balance excluding interest payments.)

(FTPL) Our final equation is the Fiscal Theory of the Price Level, which sets the price level at the start date of the DSGE problem in terms of the initial debt stock, and the discounted value of future surpluses.

With a bit of algebra, we can show that the (GOVTRANS) and (IGBC) conditions imply each other; that is, there is an "if and only if" relation between those constraints. Whether or not either of them hold is what we would refer to as a "financial constraint on government." We will now investigate the conditions under which (IGBC) holds.

The Fiscal Assumption of the Price Level

The first possibility is that we take the (IGBC) as an a priori modelling assumption. That is, the governmental budget constraint holds because we assume it does.

There is nothing stopping researchers from imposing constraints on mathematical models. However, we need to analyse the effects. In this case, the effects are dramatic, and entirely corrosive for the entire DSGE modelling paradigm.

Meta-Theorem: The (IGBC) holds if and only if (FTPL) holds.

Proof: This is a paraphrase of arguments made by John Cochrane in a number of papers. In this article, I describe a working paper of his that derives the relationship for both a RBC and New Keynesian model.

(UPDATE) I realise that I relied too much on that earlier article. Working from memory, Cochrane's sticky price analysis in that article is done at "time zero," where prices are free to move. Stickiness only shows up later. This is the standard way in which Calvo pricing is described in expositions, based on my sampling of the literature. If we were to repeatedly solve the model -- as I describe below -- we need to incorporate the effect of the fixed price. However, it appears that just gives some intertia to prices, and so the differences are largely cosmetic: we just need a version of the (FTPL) equation that incorporates the effect of the initial conditions.

So long as we assume that the DSGE model is time consistent -- if the (FTPL) holds at time t, it holds at time t+1 -- this result is extremely powerful. The trajectory of the price level is mainly pinned down by the effect of fiscal policy, with the role of monetary policy to determine the compounding rate on government debt levels. If the central bank has an inflation target, it has no degrees in freedom in setting its interest rate rule. Furthermore, what is happening in the private sector does not matter.

There are a few obvious problems with the fiscal theory of the price level.

1. If we just impose the condition by assumption, it is truly the Fiscal Asssumption of the Price Level. Why do we believe this to be true?
2. The model makes a very straightforward claim: the price level should instantaneously shift in response to the change in fiscal stance. Such behaviour does not appear to be confirmed by observed data.
3. There is no plausible way to measure the discounted path of future surpluses. The only way to estimate them would be to back it out from the observed price level. Therefore, as a practical matter, the concept may be called the Price Level Theory of the Price Level.

Can we move beyond just assuming the (IGBC) to be true? Are there plausible microfoundations? The answer appears to be: not really.

A One-Sector Endowment Model

One way to get (IGBC) to hold is to look at a model with a single representative agent, which is a household. I do not have a reference to such a model handy, unfortunately. In such a model, there is a household that is endowed with some asset that produces consumer goods. In the (HHOPT) problem of the stand-alone household, it faces an exogenous price for goods, a tax bill, and it then attempts to optimise utility.

To make this more concrete, imagine a household with an orchard that produces apples (and apples are the only traded good). The government imposes a tax, in money. The household has an initial stock of money, and it can also buy/sell apples for money. (As noted earlier, we assume that all money is immediately placed into government bills, so the money holdings disappear from balance sheets.)

When we close the problem to create the macro problem, we assume that this is the "representative household." All households follow the same path. Since they cannot all buy/sell simultaneously, the only transactions in apples will be the household selling them to the government.

The accounting constraints for this model is very simple: government liabilities outstanding equals household holdings of them. By implication, (HHTRANS) implies (GOVTRANS): the household transversality condition implies the government "transversality" condition. This then implies (IGBC).

However, this implication can only be drawn in the closed macro model. The closed macro model has an important property: the only transactions that occur are tax, and sales of apples to the government. The household can never transfer government liabilities to any other agent.

As a result, the "money" in the model is in fact mathematically equivalent to non-negotiable tax credit, and "government debt" is a discounted non-negotiable tax credit. It is entirely reasonable that the (IGBC) is equivalent to the (FTPL). John Cochrane argues that (FTPL) equation is a valuation equation, but I would qualify the description as it is the valuation equation for non-negotiable tax credits. (The fair value of a tax credit is the discounted value of the taxes it will offset. Since it cannot be traded in a market, there are no other factors to influence pricing.)

One often sees MMTists refer to government money as "tax credits." However, this model setup is different: the point to money is that it is negotiable; in this case, it is not. We should expect an economy with money to function differently than an economy with no money, only non-negotiable tax credits.

In other words, we need more distinct optimising agents to generate a demand for money.

Standard Two Sector Models

The next type of model adds a business sector, that optimises profits (PROFIT). One of the simplest versions is found in Chapter 2 of the Gali textbook on monetary policy (although fiscal concerns are eliminated from that model).

The key to note is that under standard assumptions for the production function, profits in each time period are either strictly positive, or zero, if no workers are hired (and hence, no production). Under the usual assumptions for the production function, the profits will be strictly positive for equilibrium states, barring the choice of pathological parameters elsewhere in the model.

However, if we examine (HHBUDGET) in Gali (and in similar models), the profits do not return the household. (There may be "dividends" or "capital rental", but these are not based on the actual business profits.)

We can then understand why post-Keynesians complain about neoclassical models not respecting accounting constraint: since the text does not include the business budget constraint, the "money" that shows up as profits literally disappears from the published model.

Since the business sector budget constraint is not interesting for the (PROFIT) problem, we can just assume that it was omitted to save space. Whatever.

In any event, we can now look at the transversality condition.

One of the national accounting identities is:

(Government debt outstanding) = (Household Holdings) + (Business Holdings),

which is a statement that is true for any time t.

If we denote the discounted limit of a series as LIMIT(), we see that:

LIMIT(Government debt outstanding) = LIMIT(Household Holdings) + LIMIT(Business Holdings).

We then apply (HHTRANS) to see that LIMIT(Household Holdings) = 0. Hence,

LIMIT(Government debt outstanding) = LIMIT(Business Holdings). 

As a result, (GTRANS) is only true if and only if the LIMIT(Business Holdings) = 0. Since the business sector does not have a consumption equation, there is no transversality condition to appeal to. And as noted earlier, the business sector acts as a black hole for government liability holdings, so it will accommodate any fiscal trajectory.

In other words, in workhorse models, the MMTists are correct: there is no financial constraint on government.

Ljungqvist/Sargent No Profit Model

In order to micro-found (IGBC), we need a model in which the discounted limit of debt holdings for all private actors is zero, and there has to be at least two distinct actors (as otherwise we have a non-negotiable tax credit economy). This could be done by having multiple distinct household actors, or business sectors that respect the "transversality" condition. Since having a consumption function for a business sector appears implausible**, we need to force profits to zero.

This was done in the model in Section 16.2 of Recursive Macroeconomic Theory by Lars Ljungqvist and Thomas J. Sargent (and discussed in earlier linked article). The zero profits condition was achieved by having a linear production function, so that the equilibrium profits are zero.

Although this resurrects the (IGBC), there are a number of issues.

• If production functions are linear, then I'm not sure how much would be left of neoclassical theory of the firm.
• The model predicts that there must never be a net financial transaction between the household sector and the business sector, which still keeps "money" looking like a non-negotiable tax credit.
• The (FTPL) returns, with the implausibilities it implies. 

Two Household Models

Since there are no plausible micro-foundations for businesses that pop into existence with the objective of not making a profit, we are stuck with a model that describes the travails of economies of two (or more) distinct households. The only private sector agents in the model have (HHBUDGET) constraints, and so all face (HHTRANS).

My impression is that the only way to get DSGE macro models to say anything about government finance, one needs to appeal to such models.

The objection to such a model is straightforward: it is obviously a contrivance to get a desired result. The set of models is infinite; we should be able to find a model that fits any internally consistent view. Do we really believe that industrial capitalist economies are best characterised by a model that corresponds to two individuals trading coconuts and fish on a tropical island? Where is the empirical evidence for such models?

Finally, we just end up with the (FTPL), with the empirical issues it raises.

Concluding Remarks

It seems very clear that plausible DSGE macro models do not imply a financial constraint on government, other than the implausible (and non-falsifiable) Fiscal Assumption of the Price Level. If there is a constraint, it is the risk of default.

It should be abundantly clear that DSGE macro models aimed at central bank consumption cannot be used to model default risk. The central bank sets the discount rate on government debt, and so deficits will always be "financed" at the rate set by the central bank. Moreover, if we apply the (FTPL), inflation-targeting looks silly.

To model government default, we need to dig into the legal structures surrounding government debt operations. The MMTists have covered that ground thoroughly years ago, and it will be difficult for the neoclassicals to catch up if they ignore that previous literature. (Why scholars would ignore papers published by academics in journals is an interesting question for those who study the history of academic thought.)

This also explains why MMT has generated a following among rates market participants. Bond market participants have learned -- possibly via the example of the Widowmaker Trade -- that financial flows are circular. Central government default is the result of some factor breaking those flows, which in practice is usually the result of some form of currency peg. MMTists are the only researchers that offer any theory that explains this reality. The neclassical obsession with DSGE macro has created a theoretical background of what happens if we assume default out of existence -- which sort-of matches empirical reality for floating currency sovereigns -- but default risk cannot be easily shoe-horned into the framework. Researchers may attempt to create a DSGE "teaching model" where default is possible, but these are toy models that cut off from whatever empirical results DSGE macro appeals to. Moreover, I have severe doubts about the mathematical validity of any such model.***

Technical Appendix: Equilibrium and the FTPL

The FTPL does allow us to clear up some of the mysticism around DSGE models. In the verbal descriptions of the models, a great deal of time is spent discussing "equilibrium selection" or "out-of-equilibrium" states. If we believe that mathematics is the description of operations on sets, these discussions appear meaningless (unless they can be formally incorporated into the model). In practice, "equilibrium" is just what would normally be called the "model solution" in standard applied mathematics.

We can then turn to one possible justification for the (IGBC): if it is not obeyed, there will be no "equilibrium." In standard mathematics, this is just saying there is no model solution. Why should anyone in the real world care if some neoclassical writes down a model for which they cannot find a solution? If there is no solution, the model tells us literally nothing.

If we accept the equivalence of the (FTPL) and (IGBC), we can come up with a more rigorous discussion.

We first start with a policy that conforms to the (IGBC). We know that the summation of the discounted primary surpluses is some number M. We can then back out the price level, using (FTPL). We then look at what happens when we have a tax cut at time zero of size d. The summation of discounted primary surpluses is now (M-d), and since we divide by the discounted primary surpluses to get the price level in (FTPL), the price level rises relative to the previous by (M)/(M-d). That is, a loosening of fiscal policy relative to some baseline causes an immediate rise in the price level.

What happens if d converges towards M? The initial price level tends to infinity. Therefore, we have an insight derived from the mathematical model as to what the (IGBC) really represents: if we loosen fiscal policy to a level that threatens the (IGBC) condition, the initial price level will immediately launch to extremely high levels.

This again dovetails with the verbal description of MMT views, as one could pretend that this is equivalent to the Functional Finance component of MMT. (However, the actual MMT position on inflation has moved beyond just crude Functional Finance, so this is "strawman MMT".) The issue with that interpretation is that MMTists reject DSGE model assumptions (and thus (FTPL)), and once again, this is a rather implausible description of how the real world reacts to fiscal policy loosening. It works for determining the fair value of non-negotiable tax credits, but that is about it.

Footnotes:

* I used to write "MMTers" as shorthand for "fans of Modern Monetary Theory," but "MMTists" probably makes more sense as a contraction.

** Why not sweep business holdings of government liabilities back to the household sector? In fact, that is what a stock-flow consistent model would do. This is not possible in the DSGE macro framework, as that breaks the encapsulation of the optimisation problems. The dividends would show up in (HHBUDGET), and that would imply that (HHOPT) becomes dependent upon the entire macro outcome, in which case this creates constraints on variables that result in the standard first-order conditions being invalid.

*** I will go over my doubts in a later article; very simply, I have severe doubts about the interpretation of random variables in DSGE models. However, it is straightforward to see that a default model is mathematically intractable. Let us assume that default is in some sense random. A default will have a massive effect on (HHBUDGET). The effects are highly nonlinear, and so the resulting probability distributions of variables are effectively unknown. This raises the issue of what market clearing conditions, and casting (HHBUDGET) in terms of just the expected value misses the critical issue of the probability distribution for the household wealth sequence.

(c) Brian Romanchuk 2019