### Motivation On Why I Am Looking At This Theory

By way of background, I read this article by Professor Simon Wren-Lewis of Oxford on public investment. He cited an article he co-authored with J. Portes (available at the NIESR), Issues in the Design of Fiscal Policy Rules. I ignored the literary conclusions and jumped to the first mathematical model, and I could not see how it worked. I ended up having to do a small survey of the literature of optimal fiscal rules (which I had not paid much attention to previously). I found an earlier paper, Debt Stabilization in a Non-Ricardian Economy (by Campbell Leith, Ioana Moldovan and Simon Wren-Lewis), which was more explicit in the derivation.

The benchmark model for "optimal debt" is described in these articles as the following mathematical system. I will express the mathematics in English. We need to find the sequence of tax rates (as a percentage of GDP), such that:

- we minimise the discounted infinite sum of the square of the tax rate;
- subject to the constraint that government debt follows the dynamics:

*(Change in government debt) = (Government Consumption) - (Taxes) + (Interest Cost).*

The dynamic constraint only implies that the government does not default.

The papers discuss the use of the Lagrange multipliers to find a solution, but that misses the fact that the solution to the problem as stated is to set the tax rate to zero. Of course, the debt level would spiral out of control under normal circumstances. This was obviously not intended, and so I ended up having to trawl further through the literature to see what was missing in the statement of the mathematical system.

As I discuss below, I have some misgivings about this literature, so it is unclear how much deeper I intend to delve.

As I discuss below, I have some misgivings about this literature, so it is unclear how much deeper I intend to delve.

### "Optimal" Fiscal Policy Cannot Be Separated From Monetary Policy

Anyone who has read Abba Lerner could guess what was missing - the mathematical problem completely misses the real constraint on fiscal policy - inflation. In other, more complex papers that I found, the problem is recast in the following form:

- Policymakers want to minimise an objective function which is based on the square of the tax rate, as well as a penalty on inflation;
- In addition to the period budget constraint, the Inter-temporal Governmental Budget Constraint (IGBC) is assumed to hold.

I have previously noted my complaints about the IGBC; I will set those for now. But if we ignore that, we still have the issue that the problem is too complex to truly solve, and we end up having to take linearisations. Since we are presumably interested in trajectories where economic variables make large deviations, this step is questionable. But the advances of this literature (relative to other DSGE models) appear to be:

- Tax Smoothing. Since there is a penalty on the square of the tax rate, the optimal solutions feature smooth tax rates over time. This forces fiscal policy to be more realistic than is the case for DSGE models that only look at monetary policy. In those models, fiscal policy has to passively accommodate monetary policy (as a result of the Fiscal Theory of the Price Level). In those models, tax rates will be forced to jump all over the place if central banks react aggressively to a shock. But given the reality of democratic politics, it seems unlikely that tax rates can be anything other than smooth.
- Fiscal policy is somewhat more realistically described. Taxes are specified independently of government consumption, making the primary surplus endogenous.
- The trade-off between tax policy and inflation is brought into focus. Models that ignore fiscal policy offer an incomplete view of overall government policy.

### What Discount Rate?

One problem the literature raises is the choice of the discount rate for policymakers. In the standard DSGE model framework, the discount rate is used by the households is used to discount future consumption, and operationally appears as the real rate of interest. This discount rate presumably can be measured using econometric analysis.

For policymakers, the choice is essentially arbitrary, and it affects the results.

As a basic example, assume we have a no-growth economy where the tax rate is 20% of GDP, which matches government consumption. We then look at a simple shock to tax rates, where:

- The tax rate at period
*t*is set to*0.2 + d*. - The policy is reversed at time
*t+1*, and the tax rate is set to*0.2 - (1+r)d*. - The tax rates revert to
*0.2*at time*t+2*, and all times thereafter.

(The reversal is done to keep the experiment consistent with the IGBC, to avoid controversy.)

What we see is:

- If the discount rate in the policymaker utility function is greater than the rate of interest
*r,*cutting taxes (*d*negative) leads to an improved (but presumably suboptimal) solution. - If the discount rate is less than the rate of interest, the preferred policy to raise taxes now, so that they can be cut in the future.

(I am skipping the algebra, but note that the shock to the tax rate would be small in either case.)

In either words, either cutting or raising taxes is "optimal". This highlights the general uselessness of optimisation in decision-making - any decision can be found to be optimal, by a judicious choice of an objective function. (I discussed the engineering take on "optimal control laws" earlier.)

### Strange Optimal State

The DSGE representative household framework breaks down if government debt is negative. What could that possibly represent? How would the economy function without government liabilities for liquidity management? Arguably, such a situation should be excluded as being outside of the sensible operating bound of an estimated model.

Unfortunately for the "optimal fiscal policy" rule literature, that is exactly where the government allegedly wants to be. If the government has a very negative amount of debt, it could use the interest it receives to pay for consumption, allowing the tax rate to be zero.

It is clear that the government cannot have negative debt, rather it can accumulate private sector assets. I am not a libertarian, but the idea that the government should accumulate large asset positions to pay for its activities seems questionable. The only cases where we have seen that achieved is in oil-rich countries, where it is possible to effectively tax foreign consumers of petroleum products, and the proceeds can be invested in much larger foreign asset markets. This situation is not easily replicable in a closed-economy model, or by real-world developed states.

A mathematical framework that pushes towards to an obviously degenerate solution probably needs a re-think.

### A Better, But Still Incomplete Welfare State Model

By breaking the primary fiscal balance into taxes and government consumption is a baby step in the right direction, but it still misses half of the automatic stabilisers of the welfare state. "Welfare" spending - unemployment insurance, welfare, and even state pensions - will tend to rise as people become unemployed. Since these flows are presumably to the households with the highest propensity to spend out of income, this spending will have a high multiplier. However, they are transfers and impossible to model properly within a representative household model.

And if households are rational, the existence of these stabilisers would probably be more effective in holding business cycle expectations steady than relatively weak monetary policy.

### Interest Is Taxable

The description of tax policy within these models is incomplete - the tax rate is not applied to interest paid on government debt.

This is not trivial in the theoretical context of DSGE models. Since the growth rate of compounding government debt is less than the discount rate, tax rates do not need to be adjusted in order for the "transversality condition" to hold. (This is the assumption that the long-term growth rate of government liabilities is less than the discount rate.) Primary surpluses may occur, but they are the result of taxes on interest.

### A Simple Look At Fiscal Policy Trade-Offs

Since the full solution to these problems are too complex to actually solve properly, I will look at a very simple example. Imagine an economy with the following characteristics:

- nominal GDP grows at 4% per year;
- government debt-to-GDP is steady at 50%;
- a fiscal deficit is constant at 2% per year.

Now imagine that the government cuts taxes by 1% of GDP, so that the deficit expands to 3% of GDP and stays there. We could imagine two possible outcomes:

- The tax cut is completely saved, and so nominal GDP growth remains at 4% of GDP. The government debt-to-GDP would asymptotically rise to 75% of GDP.
- The tax cut raises nominal activity, but stock-flow norm behaviour keeps the debt-to-GDP ratio constant. By implication, nominal GDP growth would have to rise to 6% per year. If we assumed the economy was at full capacity (
*which it never is in practice*), this would correspond to a 2% rise in inflation.

Obviously, an outcome in an actual economy could be something between those outcomes. And if the economy was not at full capacity, the move could lower the unemployment rate, and so it may raise real growth. At the extreme, it would be a free lunch (inflation is steady, but real GDP growth rises).

We would need a reliable short-term economic model to assess the outcome. And in this case, given the political nature of the analysis, it is even harder to find neutral analysis. But this is exactly the sort of analysis that needs to be reliable if we want to attempt to analyse the trade-off between tax rates and inflation.

### My Discontent With the IGBC

This article explains some of my concerns with the inter-temporal governmental budget constraint. If that constraint is called into question, the formal mathematical models used in the literature will break down.

In practice, I view "fiscal sustainability" to be largely a non-issue. A model that properly models the welfare state and stock-flow relations will find that any reasonable set of policies will result in government debt stabilising at some finite debt-to-GDP ratio. For a sovereign that controls the currency it borrows in, the debt-to-GDP ratio is essentially a piece of trivia - see Japan as an example. Very simple, debt holders will eventually spend out of their government bond holdings, increasing the size of nominal GDP versus the stock of debt.

### Not A Unified Fiscal Theory

Finally, the theory treats government consumption as fixed outside the model (exogenous). The only question is how to set tax rates so that debt and inflation dynamics meet some desired behaviour. And if there was a strong relationship between deficits and inflation in the long term, setting an inflation target largely eliminates whatever flexibility there is in tax rates (if spending is fixed).

Therefore, this literature, despite its mathematical complexity, has nothing to say about the impact of increasing or decreasing a government programme, or a change of mix in spending. In other words, most of the topics within the area of fiscal policy that may actually be of interest.

In another article, Simon Wren-Lewis notes the ideological overtones of the usage of "distortionary" in describing taxes. Calling taxes "distortionary" is just as redundant as calling them "taxing" - distorting market outcomes is exactly why the government has to impose taxes. Taxes need to change behaviour in order to allow the government to divert resources towards government consumption (or transfer payments). If private sector behaviour was not changed, the economy would be driven beyond full capacity, as the private sector would make no allowance for the government's demands for resources.

### A Final Comment On Terminology

In another article, Simon Wren-Lewis notes the ideological overtones of the usage of "distortionary" in describing taxes. Calling taxes "distortionary" is just as redundant as calling them "taxing" - distorting market outcomes is exactly why the government has to impose taxes. Taxes need to change behaviour in order to allow the government to divert resources towards government consumption (or transfer payments). If private sector behaviour was not changed, the economy would be driven beyond full capacity, as the private sector would make no allowance for the government's demands for resources.

**See also:**

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