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.

    

When NGDP targets are (not) optimal

This may be too technical for non-economists, yet familiar to macroeconomists, but students should find it useful. Its part of a continuing series on NGDP targets

Suppose monetary policy attempts to minimise this combination of excess inflation and the output gap each period, where we ignore discounting for simplicity.

‘c’ is a parameter. The cost function is quadratic to capture the idea that a 1% increase in inflation or the output gap is more costly if inflation or the output gap is already high than if it is low. Let’s abstract from uncertainty, and assume policy can perfectly control the output gap (there is no zero lower bound problem), and the only constraint is a forward looking Phillips curve

where ‘u’ is what is often called a cost-push shock, so it could be an increase in sales taxes for example. (Really it stands for all the things that might influence inflation besides expected inflation and the output gap, which is probably quite a lot.) The parameter ‘a’ measures how sensitive inflation is to the output gap.

Cost push shocks are in some ways the most difficult things for monetary policy to deal with, if we are ignoring uncertainty. Demand shocks that create negative or positive output gaps can in principle be eliminated entirely by an appropriate choice of interest rates. Supply shocks that change the natural rate, if known, can also be dealt with at zero cost by just moving demand to follow supply. A non-zero value of u will however result in some cost whatever policy does.

Let’s look at the case where there is a large positive cost push shock, just in the current period (t=0), and let’s make it equal to 10. This means that if the monetary authorities chose a zero output gap at t=0, then current inflation would be 10% higher. A crucial assumption here is that future inflation, assuming no known future shocks and a zero future output gap, is anchored at target, so excess inflation at t>0 will be zero. We can see in this case that by choosing a zero output gap at t=0 involves a loss of 100 at t=0, but no subsequent loss.

Now suppose all that monetary policy can do is change the output gap in the current period. Perhaps promises to change the output gap in future periods will not be believed. In that case the optimal combination of output and inflation to go for is given by 

where the parameter d=c/a. Now if it so happens that d=1, this policy involves keeping nominal GDP constant. Any excess inflation is exactly matched by the same percentage negative output gap. Now if d=1 because both c=1 and a=1, then using the Phillips curve we can show that the optimal policy will be to have a 5% negative output gap, 5% inflation, with a total cost of 50. That is a lot better than the cost of 10% inflation.

Now allow monetary policy to make promises about the output gap in period t=1 as well. The equation above still describes the optimal policy combination in period t=0, but in addition we have

This is a bit different from the expression involving excess inflation at t=0, because inflation at t=1 will have implications for inflation at t=0, and therefore the choice of output gap at t=0. The neat point is that if d=1 again, this also implies we keep nominal GDP (NGDP) at t=1 constant, because this equation says the growth in NGDP should be zero, and the previous expression made the level of NGDP constant. If a=c=d=1, then we can show that the optimal policy involves having a negative output gap of 4% at t=0 and 2% at t=1. This will produce 4% inflation at t=0 and -2% inflation at t=1, but despite this rather odd combination involving inflation that is too low at t=1, the overall cost is less: 16+16+4+4=40. [1]

A potential problem with this policy is that it is time inconsistent. When period t=1 arrives, it seems better to change the policy and keep the output gap at zero, which implies zero inflation. The total cost would then be only 32. But if that is what monetary policy is going to do, people would not expect future inflation of -2% when t=0, so we would not get the benefits of this expectation. Instead inflation at t=0 would be 6% and the output gap -4%, giving an overall cost of 52, which is worse than if policymakers had only tried to change the output gap at t=0. However if d=1, and policymakers had committed to keeping the level of NGDP constant, then NGDP at t=1 would be too high, so keeping the output gap at zero in t=1 would not be an option. Policy would be committed to following the optimal path, and policy commitments would be credible if NGDP targets were credible. This is the logic behind this post. [2] 

This is why, I would suggest, Michael Woodford is comfortable with targets for the level of NGDP. (A good technical reference on optimal monetary policy is here, where the equation above is an example of his equation 1.21, while his views on NGDP targets can be found here, pages 44-46.) Note that I have not had to invoke anything about the zero lower bound (ZLB) – indeed I have ignored it – which is why I have pointed out before that the case for NGDP targets does not rely on ZLB considerations. 

However is it reasonable to suppose d=1? Quite often macroeconomists assume c=1: a 1% output gap has the same cost as 1% extra inflation. However recent IMF work that I talked about here suggested that the Phillips curve might be very flat at low inflation. Combine the two and d could be much larger than one. This would imply that the optimal policy would be to let most of the shock feed through into inflation. [3] In this case a NGDP target would not be optimal. [4] 

So this is either good or bad news for NGDP advocates, depending on your views of these two key parameters. [5] However there is a lot left out of this simple story, such as what happens if the Phillips curve is more backward looking, or if policymakers and the private sector make mistakes.

[1] Note that the price level at t=2 in this case is just 2% higher than without the shock, compared to 5% higher when we only changed current period output. This gives us a clue to what will happen if we allow output even further ahead to change optimally – see the next footnote.

[2] If we allow policymakers to make promises about the output gap in period t=2 and beyond, we can get the cost of this shock down further still: what we get are more conditions like the last equation, for each subsequent period.  The policymaker promises smaller and smaller negative output gaps, so future excess inflation approaches zero from below. What happens then is slightly magical: by summing all these conditions together, we find that all these future negative inflation rates sum to exactly offset the initial rise in inflation, so (if target inflation was zero) the price level eventually returns to its pre-shock level. More generally the optimal policy involves gradually returning the price level back to its original path. In this sense, long run price level targeting is optimal whatever the value of d.

[3] If c=1 but a=0.5 (and estimated 'a' is normally a lot smaller than this - Charlie Bean in the small model simulated here uses a=0.025), then in the first example the output gap should be -4%, and inflation would be 8%. Although that produces a total cost of 80, that is the best we can do. If the output gap was -5%, inflation would be 7.5%, giving a total cost of over 81. 

[4] Keeping NGDP constant would reduce output by 6 and two thirds per cent. Although, compared to the optimal policy, inflation would be a bit more than 1% lower, output would be nearly 3% lower, and the total cost would be nearly 89.

[5] As I have already begun to read the mind of Michael Woodford, let me continue with my audacity. Michael Woodford first showed us how you can derive the first equation from a standard utility function in a model with Calvo contracts. If you do that, the ‘c’ parameter (which is actually is a function of 'a' along with other parameters) can be a lot less than one, so the chances of ‘d’ being close to one despite small 'a' are greater. The problem I have with this logic is that I do not believe the simple models used to derive these cost functions accurately capture the true costs of output gaps, in part because they typically exclude involuntary unemployment.

Is there a case for inflation targets? The UK versus the US.

One of my projects for the year ahead is to come off the fence (one way or another) on nominal GDP targeting. As an experiment, I’ll try and track my progress on this project with blog posts, although only if my thinking might be interesting to others. It is going to be a long process, because there is a lot involved: levels vs rates of change, GDP deflator vs CPI, nominal GDP versus its price component, and uncertainty about the natural rate to name some of the most obvious issues. However there is also another issue that may be just as important, and that is whether we need targets at all. In this post I just want to think about this last issue in relation to inflation targets, and not any other kind of target.

An obvious way for a macroeconomist to approach this is to imagine a world in which the central bank acts in society’s best interests, and has as good an idea as anyone else what those interests are. (The policy maker is benevolent, and knows the appropriate measure of social welfare to maximise.) Actually, for both the US and UK I do not think that is such a bad place to start. Let’s also suppose, as is standard, that society’s best interests involve getting inflation close to some desired level, and getting the output gap close to zero. Call this a dual mandate if you like. In such a world, why impose a target on the central bank? In other words, why do what is done in the UK rather than do what is done in the US?

By target, I mean something the central bank is required to try and hit. I am not talking about the central bank’s communication strategy. I take it as given that it is a good idea for the central bank without targets to be transparent about what its goals are, including what it thinks the desired inflation rate is. That is why this is effectively a comparison between the UK and US where in both cases transparency is fairly high.

The standard academic story involves time inconsistency and inflation bias. (Those familiar with this can skip the rest of this paragraph.) Essential to the inflation bias story is that a positive output gap (output above trend) is better for society than a zero output gap, for given levels of inflation. If you think this is obvious (more output is always better), remember more output means people working longer hours. If you think a zero output gap must be best, think about monopoly distortions or the impact of distortionary taxes. If a positive output gap is best, then a central bank may be tempted, once inflation expectations are formed, to try and temporarily raise output above the natural rate, knowing that the impact on inflation will be modest because inflation expectations are given. However rational agents will anticipate this, and the implication that their expectations about inflation will therefore be wrong. So they raise their inflation expectations above the central bank’s desired level, to a point at which the central bank no longer wants to raise inflation still further to get a positive output gap. The difference between this level of inflation and its desired level is inflation bias.

Although there is a huge literature on this, I have never been that persuaded of it’s continuing relevance in a world of long standing independent central banks. Central bankers, or academics on the UK’s Monetary Policy Committee, know that it is foolish to try and go for a positive output gap in this way, so they will avoid doing so. If the public nevertheless thought otherwise and therefore set inflation expectations above desired levels, the central bank would not settle for the inflation bias equilibrium (the time consistent or discretionary equilibrium), but would deflate the economy to get inflation down. This would soon convince the public that it was not trying to achieve a positive output gap.

Even if I’m right on this, we can still use the inflation bias argument in reverse. By this I mean that the public in ignorance will want the central bank to raise output above the natural rate, and the inflation target protects the central bank from this pressure. I mention this not because I think it is that convincing, but because this ‘using targets to protect the central bank from public pressure’ argument may have much more validity when we come to level targets. Of course the time inconsistency problem is more general than just inflation bias, but effects how the monetary authority responds to shocks (often called stabilisation bias), but here again levels targets may be more useful than inflation targets. 

I suspect the actual reason for inflation targets where they exist is more political. They increase the accountability of the central bank, and in some cases (like the UK) they allow politicians to set the target. These may be important advantages, particularly at the beginning of a new policy regime.

I also suspect that many macroeconomists have traditionally assumed (as I did) that the costs of inflation targeting were small, because if that target was achieved flexibly, it was quite compatible with optimising some combination of inflation and the output gap. The reason is of course the Phillips curve, which says inflation cannot be stable in the medium to long term if the output gap is non-zero. So a regime that targeted some fixed inflation target over the medium term would automatically achieve a zero output gap over the same time horizon.  Flexibility means leaving the choice of any particular short term combination of excess inflation and non-zero output gap up to the central bank.

This rather sanguine attitude has been tested by recent events. Some countries have experienced a whole series of positive inflationary ‘shocks’, some of which just reflect fiscal policy decisions. In the UK this has exhausted any flexibility that the MPC may have felt they had in not meeting the inflation target, so that their plans now involve meeting that target (or, indeed, expecting to slightly undershooting it), even though they forecast a large negative output gap to persist. Aiming to achieve the inflation target conflicts with what a benevolent policymaker would do. In contrast, the Fed in the US has (albeit only recently) explicitly countenanced exceeding their desired inflation level in an effort to get the output gap down. In other words, the inflation target in the UK is stopping the MPC doing what the Fed signal they are prepared to do.  

As a result, monetary policy in the US is better than in the UK, as a direct result of the impact of the inflation target in the UK. A related problem is the measure of inflation used. As I have pointed out before, the CPI is particularly susceptible to inflationary shocks like tax changes or higher commodity prices. As it is not obvious what the correct measure of inflation is from a welfare point of view, focusing on a measure that over a period is persistently higher than others may be distorting policy. The more this bias is hard wired in through mandated targets, the more sub-optimal policy may become.

So, my own view at the moment is that I prefer the flexible dual mandate approach in the US to the explicit inflation targeting regime in the UK.[1] Now of course this view is predicated on US monetary policymakers being fairly close to the benevolent ideal. If instead policymakers without a mandated target acted as if they all they cared about was CPI inflation (as in the ECB, for example), the disadvantages of an inflation target fall away. Nevertheless, what my view implies is that – all other things equal – the case for a nominal GDP target relative to the current regime is rather stronger in the UK than it is in the US right now. 

[1] A possible half way house has recently been suggested by Kate Barker, who was a member of the MPC, This would be to target a range (say 1%-3%), where the point chosen within that range by the MPC would depend on other factors, like the output gap. 

Austerity and not wasting a crisis

                In the austerity versus stimulus debate, I argue from a macroeconomic point of view that this is a false choice if we are talking about economies where markets are eager to buy government debt i.e. pretty well everywhere apart from the Eurozone. In the short term we need stimulus, whereas fiscal discipline is a long term problem. We need to raise taxes and cut spending when times are good, not when times are bad. The Eurozone 2000-2007 is a clear example of when this should have happened.

                The response is often to concede the macroeconomic logic, but say that promises of austerity tomorrow cannot be trusted. It is easy to promise, but when tomorrow comes governments will break that promise. This resonates, but it is less clear what the underlying logic is. We can all agree that governments are subject to deficit bias. There are a number of reasons for this: the common pool problem, over optimistic forecasts, political impatience stemming from elections etc. (See Calmfors and Wren-Lewis, 2011, for a more complete account.) However bias stemming from these causes is fairly constant: it does not disappear when deficits are high and only materialize in good times.

                I sometimes put it this way. Suppose the government did go for fiscal stimulus today, and this helped lead to a macroeconomic recovery. Once the recovery was complete, the underlying fiscal position (i.e. the structural or cyclically adjusted deficit) would be a little worse than today, because the stimulus would have added to debt. So if a politician is prepared to undertake austerity today, they will be even more willing to undertake it tomorrow. Of course following a recovery the actual (rather than structural) deficit may be smaller, but is our macroeconomic discourse that naive?

                In economics we are of course used to the optimal policy changing solely as the result of the passage of time (time inconsistency). A central bank may promise to keep interest rates high for some time to reduce inflation today, but come tomorrow when the policy has done its job it becomes optimal to reduce interest rates. However when it comes to fiscal policy and deficit bias, the role of time inconsistency seems less central. The problem is not that we have a benevolent policy maker that is subject to a time inconsistency temptation; it is that we do not have a benevolent policy maker.

                Having said this, I think many feel that, when governments are prepared to act ‘out of character’ and impose restraints on themselves, this chance should be grabbed before it disappears. It is often possible, when looking at countries where government debt is not a major problem, to trace this back to changes made following a fiscal crisis. Lars Calmfors has a nice account of the Swedish case here. So can we rationalise this idea? Perhaps a crisis, and therefore the chance to act, is not governed by the level of debt (funding problems aside), but by its rate of change. So governments became prepared to impose austerity when debt started rising rapidly following the recession, but if nothing was done and debt remained high, the political imperative would gradually disappear. Higher debt would become the new normal.

                However I still have problems with this line of reasoning. As Lars emphasises, Sweden was successful because it instituted a comprehensive set of reforms following the crisis, including a fiscal target to be achieved over the course of the cycle. It was also successful because a large nominal depreciation boosted output growth, more than offsetting the deflationary impact of fiscal consolidation. The world as a whole cannot use this trick, so an important condition for successful global fiscal consolidation is missing. However countries could establish rules today that were designed to only begin operating when the economy has recovered. Use the current crisis to get the rules established, but recognise that their implementation needs to be delayed because of the recession.

                It is often said that instituting rules that only operate once the recovery is complete will ‘not be credible’. Here is the time inconsistency analogy again, and (funding crisis aside) it seems equally inappropriate. We just need to ask: credible to whom? Is a government that commits to future austerity that begins tomorrow, any more likely to renege in the future than that same government that commits to long term austerity and starts it today? We could argue the opposite: an austerity plan in conjunction with a recession is more likely to come unstuck.

                I think a mistake we can make here is to imagine we are trying to discover the preferences of two different governments. In that case, a government that just promises future austerity tells us very little, because it could just be cheap talk. If a government undertakes austerity now, we know more through its actions. However that is not what the current debate is about. In practice we have governments that are prepared to undertake austerity now. They have demonstrated that they take the debt problem seriously. Are these preferences going to change if we delay austerity until after the recovery?

                So let us, by all means, not waste a fiscal crisis. Use it to set out a combination of rules and institutional changes that avoid deficit bias in the future. However I do not see why it is politically impossible to delay the implementation of those rules until after the recession is over. But maybe I’m just displaying my ignorance of political science.