*(I have been involved in discussions on this topic on Twitter, starting from an initial contact by Alex Douglas. I have just run across the work of C Trombley, who has written an article on similar lines here -- I Am The Very Model Of A Modern Macro Textbook. It's interesting, but I had been stuck writing out my own chain of logic, and could not respond to his points.)*

## Can't Keep Your Equations Straight Without A Program

This section lists the various equations discussed here, and how to interpret them. They all refer to simplified mathematical frameworks that are seen in a variety of textbook DSGE models.The notation is defined in the previous article. In all cases, the variable $b(t)$ refers to the stock of government debt in real terms, $r$ is the real interest rate, and $s(t)$ is the primary fiscal surplus.

What I refer to as the governmental budget constraint is the following equation.

$$\begin{equation}

b(0) = \sum_{i=1}^{\infty} \frac{s(i)}{(1+r)^i} \label{eq:summation}

\end{equation}$$

This equation says that the summation of the discounted real primary surpluses is finite, and equals the initial stock of debt. The validity of this expression is what we are interested in.

The next equation is the

*1-period accounting identity*:$$\begin{equation}

b(t+1) = (1+r) b(t) - s(t+1). \label{eq:accountident}

\end{equation}$$

This just says that the debt at time $t+1$ is equal to $(1+r)$ times the debt level at time $t$, minus the surplus in the current period ($t+1$). That is, debt will compound by the discount rate, less the primary surplus.

*As an accounting identity, this has to hold.**One annoying habit of some mainstream economists is to also refer to this as a "governmental budget constraint," conflating the dubious ($\ref{eq:summation}$) with the true-by-definition ($\ref{eq:accountident}$).

The next equation is the

*transversality condition:*

$$\begin{equation}

\lim_{t-> \infty} \frac{b(t)}{(1+r)^t} = 0. \label{eq:limit}

\end{equation}$$

This equation says that the stock of debt outstanding cannot grow faster than the discount rate $r$. Transversality is a term that comes from optimisation theory. Since we do not actually apply it, the exact definition does not matter.

Finally, there are couple of workhorse accounting identities that relate the stock of debt at time $0$ and time $t$.

The reasonable-looking forward relation, which tells us the level of future debt based on the initial debt level, and intervening primary surpluses.

$$\begin{equation}

b(t) = (1+r)^t b(0) - \sum_{i=1}^t (1+r)^{t-i} s(i). \label{eq:fwdsum}

\end{equation}$$

There is also the unusual backward relation, which tells us the current level of debt based on a future level. This equation is just an algebraic restatement of the previous. (My first article had a deliberately obtuse attempt to decipher this equation.)

$$\begin{equation}

b(0) = \sum_{i=1}^t \frac{s(i)}{(1+r)^i} + \frac{b(t)}{(1+r)^t}. \label{eq:bkwdsum}

\end{equation}$$

## Attempting to Follow Mainstream Logic

For a variety of reasons, mainstream economists want to use the governmental budget constraint ($\ref{eq:summation}$). Within models, households face a budget constraint, and it would be unfair if governments did not have one (beyond the accounting identity ($\ref{eq:accountident}$)). As I discussed in "On Being Pelted With Peanuts: Part I," they could just assume it to be true.Of course, just assuming something to be true is not too useful. Mexico will pay for that wall, if we assume that they will do so. As such, mainstream economists searched for a reason for it to be true, The usual logic appears to work as follows. (I have never seen a coherent description of the logic behind this, so I had to use guesswork.)

- Starting with the backward relation ($\ref{eq:bkwdsum}$), we can manipulate equations to show that the transversality condition ($\ref{eq:limit}$) implies the accounting identity ($\ref{eq:summation}$). (I did the proof in the previous article.)
- When households search for the optimal solution to their utility maximisation problem in the DSGE model, the optimal solution (allegedly) displays the transversality condition ($\ref{eq:limit}$).
- Therefore, household optimisation preferences will imply that ($\ref{eq:summation}$) holds.

The mainstream economists then go on to wave their hands about future surpluses cancelling out the effect of "debt-financed" fiscal stimulus, and so fiscal policy is ineffective, etc.

Not so fast.

If the backward relation ($\ref{eq:bkwdsum}$) holds, so does the forward relation ($\ref{eq:fwdsum}$). For the sake of argument, assume that the initial real stock of debt is fixed. (We return to this assumption later.) If we look at that equation, we see that the future debt level is pinned down by an accounting identity: the household sector in aggregate cannot alter the future trajectory of the debt by one (real) penny, no matter what optimisation choices it takes. (The models we are discussing feature fiscal policy that is completely unrelated to the state of the economy.) That is, the premises behind logical steps 1 and 2 above are inconsistent. The confusion in "Being Pelted With Peanuts" was the result of the inconsistency in the logic being used.

The next line of defence is to argue that since households have future money ("at infinity") that they will not need, they will spend it now. In other words, they cannot stop the chain of single-period accounting identities which determine the ratio of future debt to the initial level, they can (somehow) change the starting point.

Although that might work for an individual household, that cannot work in aggregate. All purchases made by households in the crippled DSGE modelling frameworks flow right back to the household sector, and there is no way of the household sector reducing its aggregate holdings of government-issued liabilities (other than voluntarily destroying money or bill holdings, which is not optimising behaviour). After all, we were only able to argue that household preferences influenced government debt outstanding because the only sector that held government debt in the model was the household sector. Furthermore, since we are assuming that all households act the same ("representative household"), they would all try to buy at the same time, without any extra supply forthcoming. There is no way of affecting the nominal debt level at time zero.

*The only thing that can adjust is the price level at time zero.*This is not "inflation," as that is the rise in the price level in future periods versus time zero. Instead, the entire price level has to shift instantly, which destroys the real value of financial assets held during the previous period. (Raising interest rates in time zero does nothing to stop this, as this only protects the real value of government bills against the inflation from time period $0$ to $1$.) This is how the assumption that the initial real debt level is fixed is relaxed: by changing the initial price level (which is the only thing that can move).

This adjustment mechanism appears implausible, but it reflects the general under-determination of the price level at $t=0$. Almost all attention is paid on the relative price between current prices and the future, but there is little discussion why the initial price level has to be at any particular level. The only variables with nominal scaling are the inherited financial assets from the previous period. If those debt ratios are "too high," we just scale nominal GDP instantly so that the ratio hits the correct level. (Calvo pricing does not help; firms that are unable to adjust prices to the new starting point get squashed like bugs.)

This effect is the Fiscal Theory of the Price Level (FTPL). The implications of the FTPL are stark: the price level is entirely driven by the state of expectations about fiscal policy. The price level at $t=0$ is entirely determined by fiscal policy expectations at $t=0$. The price level at $t=1$ is entirely determined by fiscal policy expectations at $t=1$.

*This means that monetary policy settings at $t=0$ is utterly irrelevant for the level of inflation at $t=0$.*The FTPL justifies the governmental budget constraint by saying that the private sector will raise (or lower) the price level -- changing the real value of existing debt -- if it ever looks like the budget constraint will not hold. This has nothing to do with "transversality" in optimisations. However, it is once again an assumption about economic behaviour that holds only because we assume that is true. If we assume that other factors influence the initial determination of the price level. the governmental budget constraint disappears.

The FTPL appears to be the only internally coherent class of representative household DSGE models. Unfortunately, the models are fairly degenerate, in that nothing else will really matter for inflation. Furthermore, it seems that their empirical usefulness is highly questionable. (Do we have plausible infinite horizon fiscal forecasts?)

## Further Optimisation Ugliness

Even if we want to ignore the Fiscal Theory of the Price Level, the interaction between the macro constraints and household constraints are worrisome. I have not wasted much of my time looking at microeconomics, but I have severe doubts about how its "laws" have been applied to DSGE macro. We cannot assume that households are "infinitely small"; we need to model $N$ households, and see how they interact with an aggregate macro budget identity. In other words, we cannot use theorem statements that are cherry-picked from micro textbooks without ensuring that all the conditions required by those theorems apply to the model in question.The household sector's financial assets are held in a vise, which is the governmental fiscal surplus. There is no action that can be taken to change the end-of-period holdings, no matter what level of production takes place.

If your financial asset holdings is completely outside of your control, how do they matter in an optimisation? It seems that the optimal strategy is to completely ignore financial asset holdings in the utility maximisation problem.**

However, such a step destroys the entire premise of inter-temporal optimisation. Unless the model features real investment, there is nothing (other than the irrelevant financial balances) that link period $t$ and $t+1$. This suggests that the optimal solution to these problems is just to pick the naive point-in-time utility maximisation at every time point. (Insert the production function into the 1-period component of the utility function; find the maximising output.) The fact that optimisation is carried out on an infinite horizon is just a smoke screen; what happens in period 1 has no effect on the solution in period 0.

Finally, since financial asset balances do not matter in these models, the rate of interest does not matter. Once again, the actions of the central bank are entirely irrelevant to the model outcome.

## Concluding Remarks

It is unacceptable that we have to speculate about the solutions to these models. The usefulness of mathematics is that it forces you to think clearly, and crystallise your logic in equations. However, once we lose the discipline of properly solving the equations, we are back to literary speculation.**Footnotes:**

* In some treatments of the topic, this accounting identity has been turned into an inequality, based on logic from financial mathematics. You know a field has completely lost any shred of common sense when the only things that we know hold with equality are turned into inequalities.

** If money does not appear in the utility function, the construction appears straightforward. Assume we have a feasible trajectory $x$, with utility $U(x)$. We then construct $x^*$ with all state variables equal to $x$, other than the primary surplus sequence $s$ and the affected financial asset holdings (normally bills and money). The only restriction on $s^*$ is that it does not somehow bind the household sector's financial constraint by running too large surpluses. This set is non-empty; the time series $s^*(t) = s(t) - 1$ is one such primary surplus series. Importantly, the series $s^*$ is created by only adjusting the taxes imposed; real government consumption is fixed. Since the financial constraints would never bind, $x^*$ is also feasible. Moreover, $U(x^*) = U(x)$. Therefore, we can see that the optimal trajectory is (somewhat) indifferent to the path of financial variables. Of course, there is the problem of attaining a maximum of an optimisation where the set of feasible solutions is not closed and finite, see this article by Alex Douglas. If money appears in the utility function, the set of exogenous primary surplus sequences that we can use is limited to the set that allow the household sector to match the optimal money balance in each period (assuming that we do not allow the household sector to run negative bill holdings, or borrowings from the government).

(c) Brian Romanchuk 2017

This 20 page textbook treatment develops the so-called transversality condition under the heading of Barro-Ricardo Equivalence. On pdf pages 4-5 it states, in essence, the following:

ReplyDeleteThe government debt terms B have disappeared since, at the limit, all government borrowing must be repaid. The condition in equation (14.2), sometimes known as the transverality condition, prohibits the government from always borrowing to repay its debt. At some point in the future all government spending must be backed by government tax revenues.

THIS is the point of departure where the school of functional finance rejects the assumption and assertion that the government must pay down its debt to zero at some time in the future. If the assumption does not fit the historical facts nor an understanding of financial deal flows in actual history then it can be rejected as a false assumption. Then the whole concept of optimization is rejected as an abstraction away from any basis in economic reality.

In theory and practice a bank, firm, or government can roll over and even grow its liabilities for centuries without every having to pay down its accumulated debt. None of these economic units face the finite life horizon of a person facing retirement and planning for a household. Furthermore any unit that can force can force a cash flow in its favor despite being insolvent need not default on its debt service payments. So the question is whether a fiat Sovereign government can "print money" to make its debt service payments or can creditors force it to default on its debt? If inflation is the means of default for a fiat Sovereign government then does that impose a debt ceiling on the government or can it alter its tax and spend policies as it goes along, and authorize a central bank, such that inflation can be contained and there is not necessarily any particular level of debt that can be described as a ceiling? If other sectors of the economy treat Sovereign as the insurance firm and lender of last resort then can it operate in perpetuity with negative net worth?

By the way, when the US budget was "in balance", this only meant that the valuation of government nonfinancial and financial assets was kept roughly equal to amount of government debt such that the net worth would be recorded as approximately zero. Since the federal government in the United States owns vast tracts of land and mineral resources that are not recorded as assets the recorded net worth, kept by the Treasury on behalf of the government, is something of an accounting fiction.

If society does not care that the government liabilities exceed its assets for long periods of time then there is no transversality condition and if the government can perpetually roll over its debt while avoiding civil unrest then there is no transversality condition.

Historical evidence does not validate the theory of transversality although the collapse of a government does mean it defaults on its debt. The question is then how does a government remain a "going concern" that can rollover its debt and is it debt that kills the government or some other process such as political strife or conflict with outside forces?

Link that was left out to a reference:

Deletehttp://faculty.wcas.northwestern.edu/~mdo738/textbook/dls_ch14.pdf

"All debts must be repaid." At what finite N is this happening, asks the hardline mathematician who does not want to mess around with this aleph-null stuff?

DeleteN goes to infinity in the continuous time model as you point out. Since infinity is "too big to count" this means the debt is not paid down to zero in any model in which we count actual periods.

DeleteThe idea of Barro-Ricardo Equivalence is derived from this paper published by Barro in 1974 Are Government Bonds Net Wealth?:

https://dash.harvard.edu/bitstream/handle/1/3451399/Barro_AreGovernment.pdf?sequence=4

This paper defines net wealth of government securities as the capitalized value of future tax streams. I am in the process of reading so have no comments here. I think you might find it of interest regarding ideas about deficit spending, employment, and the ability of government to stimulate an economy, and perhaps other points of interest in comparison to MMT analysis.

Is it of any use to extend the argument about the possibility of a government budget constraint out to a time infinity? Is there even one country that has not changed the terms of its monetary system in just the last 100 years? Maybe there are a few that were never on a gold standard, or didn't peg their currency to another country that hasn't itself changed, or has never experienced a complete change in government?

ReplyDeleteIt looks crazy, but they need an endpoint (allegedly) for the optimisation. (As I argue, the financial assets have no effect on the optimusation, but that's another issue.)

DeleteIf we say that the world ends in N years, we get funny behaviour at period N in an optimisation. Everything gets consumed, investment stops, etc. Pushing the boundary to infinity creates more realistic behaviour. Since things like this are uncertain, we just assume that steady state behaviour just keeps going. The only alternative is to assume that "steady state" is reached by some finite time N, but that will still create funny discontinuities between N-1 and N.

In other words, I have no complaints about infinite time.

In some yield curve models I have seen, the maturity axis was the entire real line, and the boundary condition at infinity determines behaviour. Looks silly, but so does terminating the curve at the last bond maturity. By putting it at infinity, we have smooth forwards, etc.

It makes the mathematics difficult to work with - if you do it properly. (My previous article shows how you have to work with infinity.) However, mainstream economists like waving their hands around, and not actually dealing with the mathematics they are supposedly solving. I wrote three articles freaking about the mathematics thay the mainstream jumps over with a couple of lines at the beginning. Their proofs go downhill from there.

I was going to respond to your reply to my comment on your previous post, but you seem to have expanded on the point here.

ReplyDeleteThe restriction on lim b(t) > 0 is not the same sort of thing as the restriction on lim b(t) < 0. The latter is a result of an assumed restriction on household borrowing that prevents discounted household spending being more than the discounted income. The former is a reflection that if households spend less than they earn (suitably discounted), then they are not optimising - it is not based on any ideas about government finance.

I agree that if we set lim b(t)=0, then something else has to give. In certain FTPL models, then the only thing free to move is the current price level as you describe. But actually I think the usual mainstream story is something more like this. If lim b(t) would otherwise be greater than zero, households will spend more. This pushes up prices. Since the government is inflation targeting, it will have to respond by raising interest rates. However, raising interest rates does not actually fix the problem of lim b(t) > 0, so "eventually" the government will have to increase surpluses. In other words, I think the normal thing is to treat surpluses as endogenous, given the assumption of an inflation targeting regime.

Needless to say, although I think there is something to be said for this story, there are a number of issues with it.

That sounds like what they usually write, agreed. That is, you are just the messenger.

DeleteI'm a hard line Bourbaki-an: everything in nathematics is a set, an element of a set, statements about sets, or operations on sets. Try taking that mainstream Calvinball, and convert it to statements about sets.

1. The primary surplus series s is exogenous, that is a fixed (infinite) vector.

2. But whoops, no it's not, because we say so.

Once again, the fiscal constraint only holds by assumption.

Inflation targeting central banks only get you so far. They are charged with future inflation rates; the initial price level is out of their mandate.

They should only be taken seriously if they can demonstrate that they have a solution, and can show its properties. Considering that they are allegedly optimising over multiple state variables for an infinite number of future states, I am not holding my breath.

Also: what can the central bank do? Once they no longer have bills on their balance sheet, they can no longer sell the things. The belief that interest rates can be infinite is another piece of non-mathematical hand-waving.

DeleteIn my somewhat simplified agent-based models when the aggregate bank and financial sector attempts to expand balance sheets, as should correspond to a period of significant inflation, then the central bank can more or less set the overnight interest and force financial intermediaries to pay more when rolling over short term liabilities. I have not considered whether a central bank could run out of bonds to sell before putting this rate as high as necessary to induce a reduction in the pace of credit formation? I simply assume it can jack up the short term interest rates and thereby bankrupt some banks, financial intermediaries, and non-banks and thereby end softly or more abruptly a credit-fueled inflation cycle.

DeleteMark Thoma discusses the combined central bank and treasury budget constraint, and the so-called inter-temporal budget constraint, in this excellent 30 minute video on the subject:

Deletehttps://www.youtube.com/watch?v=en5Biad0VZo

In the video he writes the same transversality condition published above as Equation (3) and near the end of the video he says there is no mechanism to enforce this condition on a consolidated government.

The 23 page paper under the following link becomes interesting on nominal page 4:

http://econ.ucsb.edu/~bohn/papers/BohnDebtConsequences.pdf

where near the end of nominal page 5 the author says the US deficit (negative surplus) averages 0.3% of GDP from 1792-2010 and 1.2% for 1915-2010. This author makes an effort to study the deficit and surplus using historical data and fit models for fiscal policy to the data. On nominal page 9 this sentence appears: "In summary, the foundation of U.S. debt policy is the promise of safety for bondholders backed by primary surpluses only in response to a high debt-GDP ratio." This is close to the position articulated by Hyman Minsky in Chapter 13 of Stabilizing an Unstable Economy.

With respect to your earlier question, I am worried about *infinite* rates (bill prices of zero). I discuss this issue in the next post "Fun with Central Bank Calvinball." Otherwise, yeah, the central bank should be able to set rates. But even then, if we are serious about how markets clear, it's something to look at. From the perspective of standard models, the upper limit is determined by the money demand function. The central bank has complete control only for deposit rates, but that destroys the assumption that there's a market for Tbills; the central bank is the price setter, and the only choice for the private sector is to take whatever the central bank is offering, or hold money.

Delete"Inflation targeting central banks only get you so far. They are charged with future inflation rates; the initial price level is out of their mandate."

ReplyDeleteAgreed and that is one of my issues with this story. In fact, I would go further and say that although inflation targeting is currently the way monetary policy is conducted, this is not set in stone. We can't asssume monetary policy will not be altered if circumstances change.

"This just says that the debt at time t+1t+1 is equal to (1+r)(1+r) times the debt level at time tt, minus the surplus in the current period (t+1t+1). That is, debt will compound by the discount rate, less the primary surplus. As an accounting identity, this has to hold.*"

ReplyDeleteIt seems to me that we have a curious mixing of two temporal states when we mix existing debt with calculated future debt. Such mixing occurs when we use an interest rate to calculate a future debt value.

Please let me try to explain: Economist often claim that new bank debt creates money. Let's think about this. When government writes debt to borrow money in existence at a bank, the money borrowed must either be "in vault" or newly created.

If "in vault", someone's deposit will be decreased an amount identical to the amount borrowed by Government. It is then logical to say that no new money has been created. Hmmm, that would deny the economist claim that money is created when banks lend money.

On the other hand, if the bank, when lending, merely creates a new deposit and allows government to spend money, then the economist claims (that banks create money when new loans are made) are supported.

OK, what about interest as used in the present value equation to justify the "governmental budget constraint"? Well, does government borrowing from banks create money or not?

If money is created by bank borrowing, a single borrowing by government only creates enough money to pay back the original amount borrowed. Any increase (in the amount promised for repayment) caused by promised interest would be impossible to fulfill because the money to complete the promise simply would not exist. Completion of the promise to pay interest would require additional creation of money by future borrowing.

I think this logic foils the government constraint theorem before it gets off the ground. The only validity that the theorem has is to observe that government debt that includes an interest promise obligates higher future government debt levels. The theorem has nothing to say about government ability to repay debt with money in existence due to creation of the initial debt.

Looks like I'm making an argument that the government budget constraint does not exist. If government can not find the money it needs, government just borrows. It even borrows from itself (from the central bank) if the need to spend is great enough.

Roger,

DeleteWe can think of the economy as having three sectors: banks, nonbanks, and Sovereign. The Sovereign combines a federal Treasury and central bank balance sheet. The cash flow analysis of the Sovereign over a period is given generally by an identity:

(1) G + i = T + dB + dH

Sovereign spending G and interest payments i over an accounting period equal taxes and other revenue T plus the change in the float of securities held by banks and nonbanks dB plus the change in the float of high powered money held by banks dH.

If the Sovereign keeps dH = 0 for a period then:

(2) dB = G + i - T

so the Sovereign reduces the float of federal securities if T > (G + i) and it increases the float of federal securities if (G + i) > T. If all securities transactions are made with nonbanks then there is no change in high powered money dH = 0 and no change in bank liabilities because the government is spending to some nonbanks in the same amounts that it borrows and taxes other nonbanks.

If the Sovereign keeps dB = 0 for a period then:

(2) dH = G + i - T

and this is not different then Treasury selling new securities which are bought by the central bank in markets, or no different then Treasury borrowing directly from central bank. This would increase the assets of the aggregate bank by dH and increase the liabilties of the aggregate bank (assuming the vast majority of government spending is to nonbanks) also by the magnitude of dH.

In the United States, if Congress would abolish the debt ceiling, then Treasury would always cover the deficit by issuing more Treasuries and allow the Federal Reserve to buy or sell Treasuries in open markets to keep the volume of high powered money H at whatever level necessary to implement monetary policy. Banks would have the option to buy or sell Treasuries in markets or at auction and if this changes dH in a way that adversely impacts monetary policy then Fed would operate in markets doing deal with nonbanks to force banks to hold the imposed level H.

So the Sovereign in an ideal system would not add or drain funds from the aggregate bank balance sheet unless undertaken by the monetary authority or central bank to implement monetary policy. Instead the Sovereign would increase the float of securities to cover a deficit and decrease the float of securities to dispose of a cash flow surplus. This leaves the aggregate bank balance sheet unchanged to a first approximation.

If government securities are considered economic substitutes for bank liabilities, however, then a deficit does increase the sum of bank liabilities and government securities, and a surplus does reduce the sum of bank liabilities and government securities in the float of financial assets held by nonbanks. One can actually think of banks that offer federally insured deposits as off-balance sheet branches of government or think of the government as the lender and securities dealer (bank) that insures all other banks by offering the combination of H + B (high powered money and Treasury securities).

I agree with you that the government pays interest by selling more securities when operating a deficit, however, it pays interest by collecting taxes and other revenue when operating at a surplus. Historically deficits are greater than surpluses so the government pays interest by increasing its debt in general. When government borrows from nonbanks and matches its spending to nonbanks no money is added or drained from the aggregate bank.

Shouldn't a budget constraints be measured against its effect on real resources? If the government buys all food to stockpile for a future "emergency" then people won't eat, and you just found a budget constraint. I feel like this is a silly question given all the math presented.

ReplyDeleteFrom the perspective of economics jargon, that would not be a "budget constraint." It might be called a "real (resources) constraint", but that term is non-standard (I believe). People often interpret "real constraint" in a qualitative way (such as its usage in Functional Finance), which would confuse things.

DeleteA budget constraint in economics is an accounting identity on monetary flows.

You could have similar "accounting" identities for quantities of real goods, but the terminology is less standard.

(For example:

New inventories = (old inventories) + (production) - (sales),

for quantities of a uniform commodity (oil, grain,...).

In simple macro models without inventories and investment, we have Y = C + G, which means:

Production = (household consumption) + (government consumption).

This equation will appear in the model, but is not usually called a constraint (it could be, from the perspective of mathematical jargon).