Winner of the New Statesman SPERI Prize in Political Economy 2016

Friday 28 March 2014

Time inconsistency and debt

For macroeconomists

In recent posts I’ve talked about empirical work I did a decade ago on exchange rates, a non-technical piece on policy I wrote a few years ago, and some recent microfounded analysis undertaken by others. So for completeness, here is something on a pretty technical, thoroughly microfounded article that I wrote with Campbell Leith that recently came out in the JMCB.

The place to start is a result that is relatively familiar. When governments can commit and follow an optimal policy, steady state debt follows a random walk. If we start out from a position where the debt stock is at its optimal level, and then there is a shock that causes debt to rise, the optimal response is to let debt stay permanently higher. This is essentially a variant of the tax smoothing idea. Taxes could rise to bring debt back down to its pre-shock level, but that incurs current costs (higher distortionary taxes) for future benefits (less debt, therefore less debt interest, therefore lower taxes). If the real interest rate equals the rate of time preference, then tax smoothing implies it is better to smooth taxes, which in turn implies it is better to let debt stay higher.

I discussed this result in this post based an earlier EJ article written with Tatiana Kirsanova. That paper focused on simple fiscal rules combined with optimal monetary policy. This JMCB paper just looks at optimal policy, where the government jointly controls monetary and fiscal instruments, in a very conventional New Keynesian model.

In this kind of model the description above of the optimal response to a debt shock is not quite complete. In the initial period, governments will act to reduce debt by a small amount. It is optimal to engineer a small burst of surprise inflation to reduce debt. This only occurs in the initial period, and it has a hardly noticeable impact on debt. However once that period has passed, the same incentive exists to generate a bit of surprise inflation in the new current period to further reduce debt. So the policy is time inconsistent for this reason.

The time inconsistency problem is similar to the familiar inflation bias case for monetary policy. There, the optimal policy would be to achieve the inflation target, but if the natural rate of output is inefficiently low there is an incentive to generate an initial burst of surprise inflation. The only way of removing this temptation in future periods is to run inflation well above target, which is inflation bias.

So how do we remove the incentive to keep cutting debt a little bit? The answer is obvious once you state it, but it is unfortunately non-trivial to prove, which is what the paper does. The incentive to initiate a small amount of surprise inflation to reduce excess debt exists as long as there is excess debt. To remove the incentive to cut debt by a little, you have to cut debt by a lot. The discretionary, time consistent response to a positive shock to debt is to bring debt very quickly back to the pre-shock level.
So the first best policy, if the government can commit, is to let debt stay higher. The inferior policy that results from a lack of commitment is that debt is brought back down very quickly. If this result seems strange, it may be because we have in the back of our minds the real world problem of deficit bias and potential default. However neither is present in this model: the government is benevolent, and there is nothing in the model to make high levels of debt problematic. 

The paper calculates welfare in both the commitment and discretionary cases. The welfare costs of any shock to the public finances are much greater under discretion, as you might guess for a policy that immediately brings debt back down to its original level. Finally the paper looks at ‘quasi-commitment’, which puts some probability on plans being revised.

The paper takes an idealised set-up (benevolent governments) in a simple, idealised model (e.g. agents live forever), so it is a long way from practical policy concerns. (If you want something along those lines, see this paper I wrote with Lars Calmfors.) However what this paper does show is that there is no necessary linkage between the problem of time inconsistency and the lack of debt control. In a simple New Keynesian model lack of credibility can lead to excessive control of debt.



  1. 2nd para: “we start out from a position where the debt stock is at its optimal level, and then there is a shock that causes debt to rise, the optimal response is to let debt stay permanently higher.”

    I’m puzzled. If the debt is higher than it need be, a relatively high rate of interest will have to be paid to induce debt holders to keep holding that debt. I’m baffled as to why that should be “optimum”.

    A couple of sentences later: “Taxes could rise to bring debt back down to its pre-shock level, but that incurs current costs (higher distortionary taxes)…”

    Taxes CAN BE “distortionary” but they don’t need to be. E.g. raising tax on everyone’s income by the same percentage is pretty distortion free, and the same sales tax placed on all goods and services is distortion free.

    Re the idea that those taxes constitute “current costs”, that would not be true if the debt were too high. That is, as Keynes pointed out, if the private sector’s stock of base money and debt (they’re much the same thing) is inadequate, the private sector will save and we get paradox of thrift unemployment. Conversely, if that stock is more than the private sector wants, it will try to spend it away (the hot potato effect) and we get excess demand. Extra tax in that circumstance is not a “cost”: it’s simply a deflationary measure designed to prevent excess inflation. It does not reduce real GDP, so it cannot be said there is any real “cost” there, seems to me. Indeed, in that that tax prevents excess inflation, the tax could be argued to INCREASE real GDP.

    “If the real interest rate equals the rate of time preference…”. I smell a circular argument: the rate of interest it’s necessary to pay someone to abstain from current consumption (rate of time preference) varies with how much stored up consumption (to coin a phrase) they already have. E.g. if they already have a large stock of government debt, it will be necessary to pay them a relatively high rate to increase that stock.

    “It is optimal to engineer a small burst of surprise inflation to reduce debt.” I’m baffled as to what is “optimum” about excess inflation. If the private sector’s stock of debt and base is such as to induce it to spend too much, strikes me the “optimum” response, as I suggested above, is to raise taxes (ideally on the better off, i.e. those who tend to hold debt). A general rise in inflation inconveniences the ENTIRE population: not very “optimum” I suggest.

    Of course, to enable the better off to pay the above taxes, it might be necessary to turn some debt in to base money, but that’s easily done via QE.

  2. Is that higher in nominal terms? That seems reasonable. Percentage terms would seem too much. Counter cyclical would reasonable if the real interest rate were allowed lower for a longer while.

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