Ricardian Equivalence is a theoretical concept that has been used to argue that fiscal policy is not effective. The argument is that increased government spending implies higher future taxes, so households will increase savings to cancel out the increase in government spending. This concept has been heavily disputed; see this blog entry by Bill Mitchell for an example. He argues that various assumptions are too restrictive in the models that show the “Ricardian Equivalence” effect.

In this article, I will introduce the “inter-temporal governmental budget constraint”, which is the equation which provides the justification for Ricardian Equivalence. I also show why this equation does not hold if term premia are non-zero. The chart above gives an example that shows the error in its prediction about the Net Present Value of future primary surpluses can become arbitrarily large. This has obvious implications for models that incorporate this constraint, as well as the usage of the concept in the analysis of fiscal sustainability.

### An Incomplete Version Of The Inter-Temporal Budget Constraint

I will start with an incomplete version of the inter-temporal budget constraint. The reason I start with this version is that this is how I have seen the equation stated in many places on the internet. (In fact, it took me awhile to find the equation stated correctly.)

The incomplete version of the equation is:

(1)Note: The primary surplus is the fiscal balance excluding interest payments (see this post for more details).Amount of government debt outstanding = Net present value of all future primary surpluses.

As an example of a usage of this incomplete version of the equation, it was used in the Wikipedia entry for “Fiscal Sustainability” (at the time of writing). To be fair, the equation is noted in passing, without details on embedded assumptions.

### An Incorrect Interpretation

Equation (1) is often loosely interpreted as meaning that the stock of debt outstanding must be “paid off” by future primary surpluses. This interpretation then leads to a common loose definition of Ricardian Equivalence – if you increase the amount of debt, future taxes will arise to pay back that debt.

This incorrect interpretation would appear to be refuted by the fact that governments roll debt, and so “the debt is never paid back”. However, the equation does not imply that government debt must be reduced to zero; it is equivalent to assuming that the Net Present Value (NPV) of the debt goes to zero as time goes to infinity. In other words, the stock of debt can grow exponentially over time, however the growth rate is less than the discount rate.

### Seigneurage – The Missing Term

The missing term in equation (1) is the effect of seigneurage; which is typically interpreted as the profit the government earns from creating (“printing”) non-interest bearing money or reserves.

The adjusted version of the equation is:

(2)Amount Of Government Debt Outstanding = NPV(future primary surpluses + seigneurage “revenue”).

The addition of seigneurage is obviously necessary. Debt can be cancelled with “printed money” without the need for a primary surplus. However, even if the increase in the amount of money is not permanent, the equation needs to be modified.

As an example, assume a rather small government has $100 in debt outstanding, and the 1-year interest rate is 5%. Let us then assume the government buys back the debt with $100 in the form of money. After one year, it runs a primary surplus of $100, which then cancels out the increase in the monetary base. (Since the debt was bought back, the primary surplus equals the overall fiscal surplus.) The NPV of the sequence of primary surpluses was $100/(1.05) = $95.24. This is less than the initial stock of debt ($100).

Money printing is associated with inflation, as a result of the quantity theory of money. That is beyond the scope of this article. However, the key point to note:

*seignorage has to be accounted for as it creates a liability with an interest rate less than is assumed in the discount curve used to derive the governmental inter-temporal budget constraint*.

### Why The Equation Does Not Hold In The Real World

Equation (2) will not hold in the real world. The reason why is hinted in the previous section: interest costs for the government

*should be expected to be*below the path of forward rates generated by the government curve used to discount the NPV of surpluses.

Take as an example a government with $100 debt outstanding in the form of a 1-year debt issue; it pays annually with a rate of 5%. The following set of operations are expected to occur:

- At year 1, the debt matures and the new outstanding is $105. This debt is rolled into a new 1-year debt issue, and the expected rate is also 5%.
- At year 2, the second debt issue matures, and the outstanding is now $110.25. A primary surplus of $110.25 is used to repay the debt completely.

The NPV of the series of primary surpluses will equal $100 (the initial amount of debt) if and only if the 2-year annual interest rate used to discount the surplus in year 2 is equal to 5%. This implies that the 2-year term premium has to be equal to zero. (See this article for a definition of the term premium.)

More generally, equation (2) will only hold if the term premium is exactly equal to zero for all maturities. However, there is a very large academic literature - affine term structure models – which is predicated precisely on the fact that term premia are non-zero. In fact, the profitability of borrowing short and lending long (which exploits the term premium), is an empirical regularity that has been known about for centuries. (From the point of view of a bond investor, rolling bonds is expected to generate higher returns than rolling T-Bills over long time horizons.) Meanwhile, governments typically have a large amount of short-dated paper that is repeatedly rolled over. This creates a very large amount of debt that will be rolled at rates below the forwards implied by the discount curve.

The errors introduced by ignoring the term premium are not trivial. Since the growth rates of nominal GDP can be near the discount rate, the specification error in the equation can become arbitrarily large. (Making the assumption that debt/GDP ratios remain in a constant range. This assumption is a stock-flow norm, and it appears to be a required assumption for capitalist economies.)

Take as a simple example:

- The country has $100 in 1-year debt (annual pay) outstanding. The interest rate on 1-year debt is 5%, and will always be rolled at 5%.
- Nominal GDP growth is 6%. A primary deficit is run each year, in order to keep the debt/GDP ratio constant.
- After year 2, the discount rate is some value greater than 6%. (Note that the spread of the discount rate over the expected rate of 5% is the term premium.)

- In year 1, the government has to pay $5 in interest, taking the debt outstanding to $105. However, the debt outstanding has to rise to $106 so that the debt outstanding grows at the same pace as nominal GDP (6%). Therefore, a $1 primary deficit is run.
- In year 2, the government has $5.30 in interest expenses (5% of $106). However, it needs to increase its debt by $6.36 (6% growth rate), so that the primary deficit has to be $1.06.
- It can be easily verified that the primary deficit will grow at 6% per year thereafter.

One could argue that I am "cheating" by using a discount rate that is not determined by actual traded instruments (the only bond outstanding has a 1-year maturity). This can be addressed by either:

- assume a swap curve exists, allowing us to impute the discount rates for times beyond the 1-year point; or
- assume some epsilon-small amount of debt is outstanding at each maturity beyond the 1-year point. This creates the discount curve needed. In this case, the forecast errors of the NPV will be slightly smaller than is shown in the chart. Since the error is unbounded as the discount rate tends to 6%, this is non-material.

Finally, one could argue that the errors are truly unknown, since we cannot observe the term premium or long-term growth rates. This may be true, but that just implies that the errors in the inter-temporal government budget constraint are unknown but can be arbitrarily large. This makes it entirely useless as an analytical tool.

### Conclusion

The standard “inter-temporal governmental budget constraint” does not hold under the real world condition of the term premium being strictly positive, and can generate arbitrarily large errors. However, one could attempt to save the concept by:

- making the obviously incorrect assumption that the term premium is identically equal to zero; or
- redefining how to calculate the “net present value”.

Even so, the constraint will still fail as the result of weaknesses in the assumptions regarding the behaviour at infinity.

*Moreover, even if all of these not very sensible assumptions are imposed on the model in order for the constraint to hold, a welfare state will behave in the opposite manner than is commonly supposed by the usual interpretation of Ricardian Equivalence*(hint). I will discuss these points in one or more follow-up posts.

(c) Brian Romanchuk 2013

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