## DSGE macro models criticism, a round up. Part 5, the intertemporal government budget constraint

Guest post by **Brian Romanchuk** (see also his blog Bond Economics)

Earlier posts in this series, which consists of concise posts looking at DSGE models using the lens of statistical concepts, were about money, market fundamentalism, unemployment and capital.

Problems With Fiscal Policy In DSGE Models

Dynamic Stochastic General Equilibrium (DSGE) models suffer from a great many defects, but the specification of fiscal policy in standard models stands out. Given the ongoing wave of publications of these models, it is hard to generalise about them. My statements here are based on how fiscal policy is represented within the models presented in the Chapter 12 of the text *Advanced Macroeconomics* by Paul Romer (fourth edition). The models in Romer are fairly indicative of much of the literature, but my criticisms here will not apply to all of them.

Behavioural Assumptions

One of the key results used within DSGE models is that the timing of taxes has no effect on household behaviour. This assumption is known as “Ricardian Equivalence”. There is an academic literature which questions this behavioural assumption. Although this is an interesting debate, my view is that the specification of fiscal policy within DSGE is internally inconsistent, and so the quality of the behavioural assumptions around Ricardian Equivalence appears to be a secondary issue.

Poor Specification Of Fiscal Policy

The standard specification of fiscal policy (if it is present at all) is as a series of exogenous primary fiscal balances. (The primary fiscal balance is defined by: (fiscal balance) = (primary balance) + (interest expense).) By stripping out interest payments, it appears that fiscal policy is completely decoupled from interest rate policy.

This makes little sense, for three reasons. Firstly, taxes are paid on interest receipts from government, and so changes in the interest bill will affect the primary balance, even if we hold all else equal. Secondly, unless we assume that interest income has no effect on household spending, interest costs have a multiplier effect which will affect non-interest spending, such as unemployment insurance. Finally, it makes no sense to make primary balances entirely exogenous to the state of the economy (although it should be noted that some DSGE models relax that assumption).

The Accounting Identity

Within a mathematical system, constraints determine the possible solutions to the system. Within DSGE models, there are two types of constraints: firstly, an accounting identity, and secondly, a constraint that allegedly sets the limits on the feasible set of future primary surpluses. The second constraint is the subject of the article, but some DSGE treatments will lump these two types of constraints together.

The accounting identity used within DSGE models is obviously correct. (Although it is sometimes expressed as an expectation, which is rather bizarre, as the probabilistic framing implies that there is some probability that the accounting identity will be violated.)

The accounting identity is abused sometimes in non-mathematical discussions of fiscal policy, as it is restated as:

Government spending = (Increase in debt) + (Increase in money) + (Taxes).

This is converted to the verbal formulation that “Government spending is financed either by debt, money-printing, or taxes.” This then leads to discussion how the different modes of finance have different effects. Such discussions are not supported by the theory, and are just another example of how verbal discussions of accounting constraints break down (the S=I identity is the classic example of how projecting beliefs onto accounting identities leads to confusion).

The Infinite Horizon Constraint

If we assume that there are no money holdings, the infinite horizon constraint is:

Present Value of Government Debt = Expected Discounted Value of Projected Primary Surpluses.

This condition is equivalent to stating that the present value of the stock of government debt will go to zero as time goes to infinity. If we add in money holdings, one needs to add corrective terms for the effect of money growth (although this is often ignored).

The problem with this constraint is that it is either trivial or wrong.

- If the rate of interest rate on debt is higher than the growth rate of the economy, all the constraint says that the debt-to-GDP ratio will not allowed to become arbitrarily large. Primary surpluses are needed to act as a brake on debt growth. However, having the debt-to-GDP ratio becoming arbitrarily large is an outcome that would never happen within a sensible model framework. This has been analysed in the Stock-Flow Consistent modelling literature; Marc Lavoie’s tekst
*Post-Keynesian Economics: New Foundations*gives an overview of this in Section 5.6.3. Therefore, this is not really a “constraint”. - If the rate of interest is less than the growth rate of the economy, the constraint implies that the debt-to-GDP ratio will tend to 0 as time goes to infinity. Since government debt is needed for liquidity management within the financial system, such an outcome seems implausible.

It should be noted that not all models impose this constraint; for example, overlapping generations models with infinite generations do not. See [Fullwiler 2006] for a longer discussion.

Moreover, it is unclear why the constraint is supposed to hold.

The justification is based on a “no-Ponzi” microeconomic logic: households will lend to other entities, but they will not finance a “Ponzi scheme”. For example, in Romer (page 589):

*And if there are a finite number of agents [households], at least one agent must be holding a strictly positive fraction of this debt. This means that the limit of the present value of the agent’s wealth is strictly positive; that is, the present value of the agent’s spending is strictly less than the present value of his or her after-tax income.*

However, this appears to be a fallacy of composition. Since the household sector can juggle between money and debt holdings, and the price of debt varies based on a varying interest rate, it is difficult to describe the full solution. In order to simplify matters, we can consolidate government liabilities into a single variable *L(t), *which is equal to money holdings at time *t* plus the present value of government debt at time *t.* Although an individual household can reduce its financial assets by buying goods, this implies that its counterparty will increase its financial assets accordingly. There is nothing that the household sector can do at time *t* to affect the level of *L(t).*

We can then apply the governmental accounting identity to realise that *L(t+1)* is greater than or equal to *L(t)* plus the primary deficit. (It is equal if the interest rate is zero, and greater if interest expense is greater than zero. I assume that interest rates are greater than or equal to zero.) *There is nothing that the household sector can do about this, since the fiscal balance is assumed to be exogenous to the state of the economy (or nearly so) in the specification of fiscal policy within the DSGE model.*

Since the government can set an arbitrary lower bound to the amount of debt outstanding within a DSGE model on any finite horizon, it contradicts the assumption that the present value must tend to zero on an infinite horizon. This contradicts the infinite horizon budget constraint.

I will never understand why you still discuss with “statistics” and so on in a model which is NON SENSE. Before doing that, you have to justify the main hypothesis of the model – there is a function that is maximized (the “Maupertuis principle”).

I don’t see it in the “real world”.

I do not understand why you want to play in the same field that the “orthodox” have chosen. You are sending the signal that DSGE are relevant (you don’t confront to reality an irrelevant model) and that we (teachers, students, etc.) HAVE TO LEARN a lot of mathematics and micro. to be able to understand (and criticize) them.

Exactly the opposite from the “postautistic” movement, at its beginning.

Dear Guerrien,

what I want to show with this series is exactly that DSGE economists and economic statisticians are not playing the same ballgame. There are fundamental differences between the concepts and kind of definitions of DSGE models and the concepts and definitions of statisticians. However – the difference between maximization and accounting identities is indeed a nice topic to show this!

Hello Guerrien

As someone with training in applied mathematics, I am more comfortable dealing with a theory that is built around mathematics. My problem with the DSGE framework is that the mathematics appears contradictory. Although I find this topic somewhat amusing, I accept that not everyone would see it that way.

As for a strategy for a development for economics, I would argue that heterodox economists need to forge ahead and develop their theories in parallel. But given the dominance of mainstream macro in institutions, I doubt that it can be completely ignored in discussions. I write for a general audience about macroeconomics and finance, and it would appear bizarre to completely ignore the writings of people like Bernanke, Krugman, and Summers.

Brian,

“This condition is equivalent to stating that the present value of the stock of government debt will go to zero as time goes to infinity.”

I’m not sure I understand this. In the simplest case, if the primary surplus is equal to X per period from now to infinity and the discount rate is r, then the equilibrium present value of the debt is X/r and will remain so till infinity.

The problem here is that the GBC is just the accounting identity that applies in non-explosive situations, but it is interpreted as being a restriction on behaviour. Romer says that “… the government must run primary surpluses large enough in present value terms to offset its initial debt.” This is true, but it is very misleading.

If you take an SFC model of the G&L type, Romer’s statement will remain true. The present value of primary surpluses out to infinity must equal the initial debt. The point is though that this has nothing to do with fiscal policy – it is inevitable. Whatever the rates of spending and tax are, NGDP levels will adjust so the surpluses are at the level required to fulfil the condition. So there’s no kind of constraint there.

It only creates a problem in models which seem to allow the government scope to break the accounting condition. This happens if you try to make the surplus itself an instrument of policy (together with certain other conditions), rather than letting it be endogenous.

In fact, the GBC is not really a constraint, as indeed the household budget constraint is not really a constraint. Households are not restricted by their budget constraint – they are restricted by their borrowing capacity.

Hi,

Not sure I follow what you mean by explosive situations. In all reasonable cases, the debt level is going to infinity. It’s just a question whether it grows faster than GDP or not.

I noted that the condition is trivial if the compounding interest rate is greater than the growth rate of GDP; the condition must hold in order to prevent the debt-to-GDP ratio from going to infinity. As you say, the debt-to-GDP ratio will not go to infinity in a SFC model.

The problem is what would happen if the growth rate is greater than the interest rate. The implication is that the debt to GDP ratio goes to zero (the level of debt may still be growing). This would cause liquidity issues within the financial system. This raises the question: why must the condition hold? The DSGE modellers argue that there is a microeconomic rationale which forces the condition to hold, which I see as contradictory.

I guess I don’t see it as a condition, i.e. something that restricts fiscal policy options – it’s just something that tells you what surpluses and real discount rates will turn out to be. It’s only when you try to treat surpluses as something that can set arbitrarily (as many in the mainstream are inclined to do) that you need some kind of “condition” to explain why you can’t do this.

Yes, that is what I am referring to when I said that it makes little sense to treat the primary balance as exogenous. I did not expand that too much, as I wanted to keep the overall length of the article reasonable. I think I wrote a 1000 word article just on that issue alone.

Norway has the best setup. Keep socking away a decent chunk of the income from the depletion of your natural resources , invest those funds diversely and globally , and eventually you’ll have enough to fund the government and social benefits with low to zero taxes. If gov’t bonds are needed by the financial system , they can create them at will and dump the proceeds into the SWF.

If the SWF becomes big enough , and global growth is continuous and at reasonable rates , it seems to me that both the private and public sectors in Norway could be running positive financial balances at some point in the future , entirely at the expense of the ROW , via SWF returns.

The sovereign wealth fund is a great set up, but it relies upon taxing foreigners. The beauty of having oil wealth is that the profit margins on it are so high that you can stick a tax on it and not affect your competitiveness much. Unfortunately, not many other businesses have that feature.