Sticky Prices, Unexpected Inflation and Ricardian Equivalence

An interesting point came up in my recent discussion with Stephen Williamson (referred to in my last post, but there's no need to read that to understand what I'm going to say here.)  It is to do with the interpretation of Ricardian Equivalence, when there are unexpected policy changes.  I actually thought the point was obvious, but from Williamson's response I am left thinking that it's either not obvious or I'm wrong.  If anyone wishes to enlighten me either way, I'd be grateful.

To look at this, I need to make some of the standard assumptions for Ricardian Equivalence to apply, so I'm going to assume homogenous households with infinite horizons and no issues like liquidity constraints.  Under this assumption, the long-run government budget constraint is binding.  This says that the present value of taxes cannot be less than the value of current debt plus the present value of government spending.

The usual way to interpret this is to suppose that any change in tax now must be offset by a change in tax at some future time.  So, for example, if there is a one-off tax reduction today, then this will need to be financed by issuing debt.  That additional debt, plus the interest, must be repaid at some point and this requires future taxes.

This analysis is usually set out in real (non-monetary) terms, with bonds paying a real rate of interest.  In this case, there is no way other than a future tax increase for the government budget constraint to be met.

However, in reality, public debt is generally issued in nominal terms, paying a nominal rate of interest.  This means that the actual rate of return will vary from the expected rate of return if there is unexpected inflation.  What matters for the government's budget constraint is the actual rate of return.

So, consider a case where the government makes an unexpected tax cut.  One possibility we have to consider is that this will lead to a higher inflation than was expected before the tax cut was known about.  The result would be that the actual rate of return on outstanding government debt is lower than previously expected.  This represents a kind of unexpected inflation tax on bondholders, which must be factored into the government budget constraint.  The need for future taxes is correspondingly lower.

In a model with perfectly flexible prices, we might have a position where an immediate jump in the price level produces an inflation tax that completely offsets the impact of the tax reduction.  In this case, the tax reduction would have no impact, simply because it would make no net difference to households' real position.  The Ricardian Equivalence result holds, albeit somewhat trivially.

With sticky prices, it's a different matter.  Any inflation shock that occurs in response to a tax reduction must be correlated with inflation in subsequent periods, whether under a forward-looking or backward-looking relationship.  So, if a current tax reduction is to be recouped through an inflation tax, that must take place over time.  And, throughout the period of higher inflation, expected real interest rates are higher, which will impact on the pattern of household spending.  So, in this case, the choice between taxation or bond issuance does make a difference.

It needs to be stressed that this is fully consistent with rational expectations, provided we are considering an unexpected change in taxation (or equivalently a change in actual taxation, following a known rule, but in response to some other unexpected event).  The standard New Keynesian Phillips curve has current inflation as a function of expected inflation.  We can equate expected inflation to the actual outcome for every point in time where there are no unexpected tax policy changes.  But, if a new policy is announced we have to allow for expectations to be revised accordingly.

In a standard New Keynesian model, therefore, we seem to have two options as to how an unexpected tax reduction might play out:

We can assume that people do not revise their inflation expectations and that they believe that the government will increase taxes again in the future to pay off (with interest) the debt incurred to finance the tax cut.  In this case, the model-consistent expectation solution is that there is no inflation shock, and indeed the only way the government budget constraint can be met is by increased future taxes.

Alternatively, we can assume that people do not believe that the government will raise future taxes and that they revise their inflation expectations to take this into account.  In this case, the model-consistent expectation solution is an (unexpected) jump in inflation, followed by a steady fall in inflation back to its baseline level.  This inflation erodes the value of outstanding debt and the government budget constraint is indeed met without any future taxes.

If we were using a non-monetary model, we would have no choice but to go the first route, because there is no unexpected inflation.  But once we think in terms of monetary debts, there is no good reason for saying that the former should hold rather than the latter.  The first gives us the classic Ricardian Equivalence result; the second does not.

The path of output, inflation and (nominal) interest rate in this second case would be something like that shown below (model details at end of post).  The shock occurs when the announcement is made.  It makes no difference whether the tax reduction is actually applied now or in the future.  In that sense, the result is still Ricardian.

A key point here is we are relying on the fact that, under the government budget constraint, a current tax reduction can be made whole either by an increase in future taxes or by a change in the rate at which future taxes are discounted.  The usual approach is to assume that there needs to be a rise in futures taxes.  This leads to a self-fulfilling loop:

a) Because there is an offsetting rise in future taxes, there is no change in household spending and real interest rates remain unchanged.

b) Because real interest rates remain unchanged, there needs to be a increase in future taxes to meet the government budget constraint.

The trick is to recognise that when we have expectation shocks and sticky prices, there is an alternative:

a) Because there is no offsetting rise in future taxes, households spend more currently, causing a temporary, but drawn out, reduction in real interest rates.

b) Because of the reduction in future real interest rates, the present value of future taxes is increased and so there is no need for additional taxes to meet the government budget constraint.

In my view, this is actually quite an important point of departure between mainstream and more heterodox approaches.  If we come at this from a mindset that sees the government budget constraint as something that dictates policy, rather than simply an accounting identity, then we might naturally assume the former.  But, if we take the view that the government will just do what it thinks appropriate from time to time and we just need to work out how the economy responds, then we might be more inclined to the latter.  And, in my opinion, the latter is more plausible.

Model

The model I've used here is a standard three equation NK model with an additional equation showing the evolution of the government debt and the further condition that the level of debt is bounded (this is equivalent to the government budget constraint). 

(1)          y_t = E[y_t+1] . ( β . i_t+1 / E[π_t+1] )^-σ

(2)          π_t = E[π_t+1]^β . y_t^κ

(3)          i_t+1 = i* + φ_π . ( E[π_t+1] - 1 ) + φ_y . ( E[y_t+1] - 1 )

(4)          b_t = b_{t-1 .} i_t / π_t - τ_t

where y is output, π is inflation (both normalised to 1), i is the nominal interest rate, τ is tax and b is the real value of government debt.  Parameter values used are β = 0.97, σ = 1, κ = 0.2, φ_π = 0.5, φ_y = 0.4 and i* is set at 1/β.  The baseline level of tax is 0.2 and the shock is a one-off reduction by 0.05.

The usual approach would be to take equation (4) and the boundary condition simply to place a constraint on the level of taxes, in which case it has no bearing on the other variables.  What we are doing here is allowing taxes to be freely set, letting expectations adjust when a change in tax policy is announced and using equation (4), plus the boundary condition, to pin those expectations down.

Fiscal Policy in Neo-Fisherite Versions of New Keynesian Models

This post is intended to enable me to explain in a bit more detail something I have been describing in an exchange with Stephen Williamson on his blog.

The context is how fiscal policy might impact on the results of models of interest rate changes, particularly in the impulse response on output.  I have in mind here a model based on John Cochrane's, which I've looked at before.  The two main equations are a New Keynesian IS curve and Phillips curve.  There are also assumed to be lump sum taxes and single period government bonds, so there needs to be a government budget constraint which determines the evolution of the stock of bonds.  The three equations in y (output), π (inflation) and b (the real end period value of the stock of bonds) are:

(1)          y_t = E[y_t+1] . ( β . i_t+1 / E[π_t+1] )^-σ

(2)          π_t = E[π_t+1]^β . y_t^κ

(3)          b_t = b_{t-1 .} i_t / π_t - τ_t

where i is the interest rate and τ is the level of real lump sum taxes.

Here, monetary policy is an interest rate peg with i as the policy tool - we have no central bank reaction function.  We want to consider the impact of a 1% cut in this rate, announced five period in advance.  In the first instance, we will assumes that fiscal policy involves keeping τ constant.

The results of this are shown below (showing deviations from starting values).

Inflation starts to decline immediately and eventually reflects the drop in nominal interest rates.  This means that, prior to the actual rate cut, real interest rates are higher, which implies that output rises.  Once actual nominal rates are cut, real rates are lower and output declines again.

This change results in a temporary increase in the real stock of bonds.  This happens because real rates have risen and real rates are a key factor in determining how the stock of bonds changes over time.

So I want to consider an alternative fiscal policy rule; one where the level of taxes, τ_t, is adjusted to prevent b_t ever rising above b₀.  We therefore need a fourth equation which is:

(4)          τ_t = GREATER OF { τ₀ } OR { b_{t-1 .} i_t / π_t - b₀ }

Adding in this equation gives us the results shown below (b_t = b₀ for all t, so I have not included it in the chart).

Under this fiscal policy rule, both output and inflation drop sharply on the announcement, and then both rise until the actual rate change takes place.  These paths clearly satisfy equations (1) and (2) for every period where there is no surprise.  Output is rising when inflation is lower than nominal interest rates; inflation is rising when output is below its equilibrium level.  Equations (3) and (4) selects this equilibrium path from multiple others that would also satisfy (1) and (2) (although I cannot be sure that there are no other alternative paths that would satisfy all four equations).

Note on parameters.  I have used Cochrane's values of β = 0.97 and κ = 0.2, (as well as assuming a τ₀ equal to 20% of y₀).  These parameters suggest time periods in the order of a year, so we are talking about changes announced five years in advance.  If we adjust the parameters to reflect shorter periods (like a month), then announcing five periods in advance has a more limited impact (although still the same pattern).

Money as a Good and Money as a Record

In a recent post, Nick Rowe has been playing around with how to think about money when measured money can be both posive and negative.

It is worth distinguishing two models of money, which on the face of it are very different.

In the first model, money is a type of good.  Monetary exchange means that every purchase involves the buyer delivering the agreed amount of money to the seller in return for goods and services.  It's useful to imagine this as a physical delivery - the buyer hands over some gold coins or paper notes to the seller.

In the second model, monetary exchange takes place through a series of accounts kept by a central registry (the Bank).  Each agent holds an account, which is nothing more than a record of a balance at the close of each day.  When a purchase takes place, the buyer requests the Bank to debit the purchase price from his own account and credit it to that of the seller.

In each case there is a limit on how much people can spend.  In the first model, this is a physical limit.  The buyer cannot hand over more paper notes than he actually has - you can't have negative notes.

In the second model, there is no physical limit.  If the buyer spends more than he holds, his balance simply becomes negative.  However, the Bank will want to impose some limits.  The more an account goes negative, the greater the risk to the Bank if the accountholder fails to get it positive again.  So the Bank will decline payment requests which will make accounts more negative than it wants.

So in the first model, the limit is necessarily zero.  In the second model, it may be any non-positive number and will depend on the Bank's view on credit risk.

This question of limits is the essential difference between the two models, rather than any question of whether money is in bearer or registered form.  Of course, the form money takes may dictate the limit - you can't have negative bearer notes.  But it's the limits that matter.

This allows us to see that the first model is simply a special case of the second model.  In the second model, the limit need only be non-positive.  In the first model, it must be zero.

Negative account balances in the second model are debts of the accountholder.  We can have debts in the first model as well, where the Bank (or any other agent) lends notes to someone.  However, there is a clear distinction here between debts and money.  Although the paper notes may constitute debts of the Bank, as far as the non-banking sector is concerned debts and money are quite different things.

This makes it easy to isolate a particular thing - in this case the collection of paper notes - and identify it as money or the medium of exchange.  Helpfully, such a thing can be captured in a single aggregate measure.  So, it's often easier to think through monetary mechanics using the first model, and often this is OK.

However, in the general case the distinction between medium of exchange and debt becomes more blurred.  We can no longer think of money as a simple aggregate; we need to start to think about the general liquidity position.  This manifests itself in the way counterparties to monetary contracts - both banks and non-banks - agree the terms under which those contracts may be settled on demand.

The second model is a closer reflection of the real world.  We can learn useful things from thinking about models where money is a good with a strictly positive value.  But we should not be misled into believing that everything we see in those models will translate neatly to the real world.