Friday 8 January 2016

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.


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)          yt = E[yt+1] . ( β . it+1 / E[πt+1] )

(2)          πt = E[πt+1]β . ytκ

(3)          it+1 = i* + φπ . ( E[πt+1] - 1 ) + φy . ( E[yt+1] - 1 )

(4)          bt = bt-1 . it / πt - τt

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, φπ = 0.5, φy = 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.


  1. Ricardian Equivalence includes the assumption that increased debt will eventually result in increased taxes. It is unclear whether the future increased taxes might be the result of a decision to increase tax rates or the result of increased collection due to increased economic activity at a constant tax rate.

    There is an alternative to the Ricardian Equivalence theorem. Government can increase spending over a very long time period (over many years) by funding the spending with ever increasing amounts of debt. In my opinion, the United States has done this nearly continuously since the early 1970's.

    Inflation has occurred over the same period of years but it is unclear in my mind whether the borrowing, the labor laws, the change in the money supply (which may be measured in many ways) or some other reason might be factors in causing the observed inflation. With a pattern of borrowing and inflation occurring over so many years, it is hard to believe that spending has been purposely delayed in anticipation of future taxation.

    Turning to the governmental budget restraint, I think it is important to observe that when government expenditures are financed by debt, the entire amount of debt increase will appear as a nominal private wealth increase . [This is necessary to fulfill the economic theorem that once created, money will exist until it is destroyed.]

    At this time, much of the debt of the U.S. Government is held by the Central Bank (which is owned by government). Why does this debt ever need to be paid back? What happens if the Central Bank simply forgives this debt?

    Well, if I loan a grandchild money and receive a promissory note in return, the grandchild will pay on that note until the money is repaid. What happens if I give the note (as a gift) back to the grandchild after several years of faithful repayment performance? The grandchild simply stops making the payments and gratefully says "thank you".

    The same thing would happen if the Central Bank forgave the U.S. Government it's debt. But, and it's a big BUT, the Central Bank would have no assets. The Central Bank would be broke. What good is a central bank with no assets?

    One lesson to take away from my comment: The main action resulting from new debt comes from the new spending at the time of creation. Subsequent debt repayment and interest are typically incremental payments that distort and shape future economic activity.

    1. Hi Roger,

      "It is unclear whether the future increased taxes might be the result of a decision to increase tax rates or the result of increased collection due to increased economic activity at a constant tax rate."

      In this case, it doesn't make any difference for the same reason that it doesn't make any difference when the tax arises - only when the announcement is made, i.e. only when people have to revise their expectations.

      "Government can increase spending over a very long time period (over many years) by funding the spending with ever increasing amounts of debt."

      There are certainly models, where this can happen, but in the model here (basically a NK model) that would conflict with the assumption that households were maximising utility.

      As regards the central bank holding the debt, that is a detail not included here. One way to think about is to note that the private sector then holds a roughly equivalent amount of interest bearing reserves instead, which can be considered the same thing here.

  2. Thanks for your comment.

    I thought that some additional perspective was needed to give some background for my previous comment. A figure was needed which introduced the need for a blog post. The results can be found at

    I refer to your post (and quote from it) to lay the framework for the need for a closer examination of how debt influences the cash flow economic stream. My explanation is certainly sketchy, and certainly incomplete, but I think it sets a basic pattern that can be replicated over many years to obtain your results.

  3. I am continuing to think about your post and my subsequent illustration of the anticipated shape of the cash flow pattern resulting from new debt.

    First, I question my assumption that the cash flow curve is concave. When we remember that the principal portion of loans is typically displaced toward the end of the loan period, the effect of this delay would be to make the cash flow curve CONVEX.

    Secondly, you make the 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."

    Could you explain what this means? Are you trying to introduce the effects of inflation with this assumption?

    To sum the thrust of this comment, the concave nature of the curve you present may be seriously incorrect (may be CONVEX) depending upon the cumulative effects of the underlying repayment schedules and how those repayment schedules are financed.

    1. That the budget constraint is binding is not an assumption - it's a consequence of the assumptions. There's some explanantion of the budget constraint here: Inflation comes in here, because we need to use the real interest rate, which is the nominal interest rate adjusted for inflation.

  4. Thanks for your additional comment(s). They all helped.

    I took a look at the Fiscal Sustainability reference. A very interesting presentation.

    After a lot of further thought, I think that you are melding three theories: Fiscal Sustainability, Ricardian Equivalence, and "real" interest rates (as compared to "nominal" interest rates). What all three theories have in common is a method of discounting (or anticipating) inflation, which is a form of unstable prices.

    We can look at your figure and see concave curves. Concave curves demonstrate that the effect of a one time (but sustained) decrease in tax rate results in a stimulated economy with the effect of stimulation declining over time (becoming the status quo). I would agree that any one-time sustained change in taxation will result in an economy that must re-stabilize to a new economic level.

    If I were to argue with your figure, it would be on increased interest rates. Any influence on interest rates should depend upon the method of financing the increase in government debt. If the debt is internally financed (Central Bank generates the new money), interest rates should drop due to more money becoming available for loan. If the debt is borrowed from the private sector, the curve should be as you show.

    Thanks for a post that (certainly) generated further study and thoughtful effort by this reader.

  5. Replies
    1. Thanks. I'd need a bit of time to go through it properly, which I don't have right now, but my initial impressions are: 1) the difference between tax and debt finance here is simply due to the fact that the pattern of distortionary tax is different; 2) this is not necessarily a case for stimulus, since the initial boost is followed by a period of reduced output and I would expect that the overall effect is one of reduced welfare. That said, I have a lot of sympathy with the idea that looking at models with lump sum taxation can be very misleading.

  6. Replies
    1. I haven't got the time to go through this properly, including the comments that prompted it, (which is why I'm replying here not on David's blog) but...
      I think David is right that it matters that people make decisions under conditions of uncertainty and do not make all future purchases under some kind of forward contract. That said, I'm not sure this is an issue of budget constraints, which are really just accounting identities. An intertemporal budget constraint is actually just a budget constraint. An IBC over ten succesive one year periods is just a BC for a single ten year period. Same point for a lifetime, or for eternity.

  7. Nick, are there SFC models with a non-zero money multiplier? Do you have a simple example?

    1. Tom,

      The basic meaning of money multiplier is just the ratio of commercial bank money to central bank money. It simply reflects the general position that the amount of central bank money required is much smaller than the amount of central bank money. I would expect any SFC model which included both central bank money and commercial bank money to show the latter as much greater, hence a large multiplier. A multiplier of zero would suggest zero bank money.

      When people object to notions of the money multiplier, what they really mena is that they objest to the idea that the multiplier is independent of the quantity of central bank money - i.e. that the central bank can manipulate the stock of base money and expect the multiplier to remain unchanged.

    2. Nick, thanks! You write:

      "It simply reflects the general position that the amount of central bank money required is much smaller than the amount of central bank money."

      Was that 2nd "central bank money" supposed to be "total money, i.e. central bank money + commercial bank money" or perhaps just "commercial bank money?"

      Also, you *might* be interested in the last bit of this thread, starting here. (On my machine, I have to give my browser a couple of seconds to advance to the comment of interest once the page loads).

    3. Note: that comment I linked you to above is in regards to G&L's SIM model. Looking at the response(s) to my comment(s), it seems to me that there should be more advanced SFC models (like you allude to) which can accommodate his assumptions about SIM explicitly.

    4. ... but I don't know. I don't know much about SFC models. Pretty much nothing actually... but I have spent some time with SIM! =)

    5. Yes, you're right - that is what I meant.

    6. I did have a quick look at the post (and the comment) in your link. I also (somewhat against my better judgement) left a comment on that post as my own model was mentioned at one point.

    7. Nick, thanks for your comment.

      BTW, you might like this. Follow this link to a online circuit model that I've tuned to represent SIM. Just click on the switch in the upper left hand corner to close the switch, and current starts flowing. Across the bottom there are four plots (there's a key here: technical problems with adding labels to the model itself), left to right:

      1. The voltage across the circuit representing G
      2. The current through the battery representing Y
      3. The current through the upper resistor on the bottom: T
      4. The current through the inductor representing H

      There's more explanation here.

    8. You can see each of the curves reach their steady state value:

      1. G steady state = 20 (immediate, i.e. the time constant Tc = 0)
      2. Y steady state = 100
      3. T steady state = 20
      4. H steady state = 80

      All but G have Tc ≈ 6 (where I have 1 period = 1 second). This is using the default SIM parameters.

    9. I'm afraid I haven't looked at this in any great detail, but it immediately strikes me as problematic to have H (a stock) as having the same dimension as G, T and Y which are flows.

      Stocks like H only have an existence at a point in time. Flows have no existence at any point in time. They only exist between points in time. So we can ask what is H at time t1 and we can ask what is Y between t1 and t2, but not the other way around.

      If we use a continous time model, then this will imply values for flows at particular points in time. In my opinion, it is OK to use continous time models if it is useful to do so, but we need to bear in mind that we are introducing a fiction in doing so. I think Jason has got horribly confused about this.

      I'm not great on circuits, but my initial impression is that it would be hard to map the dynamics of an economy to that of an electronic circuit.

    10. Nick, you're absolutely correct. In fact I am planning to do a post on that very concern. There are many different continuous time models that will map to the discrete sample points of SIM, however, there is one in particular that suggests itself as the simplest one.

      There's actually enough circuit elements on that site to do it: what it will show (when I'm done) is the SIM sample points displayed next to their continuous time counterparts, and the wires in the circuit themselves will be enforcing the accounting identities. Instead of amps = dollars, I'll have couloumbs = dollars, and the continuous time version I have in mind will obey the accounting over any interval of time (not just at the sample times).

      What I've produced there (in the link above) is a version that happens to match SIM's outputs at the sample times. Y and T are continuous functions which represent values accumulated during a sample interval. In the new scheme I'm thinking of they will represent instantaneous rates, and I'll have a sliding window integrator circuit to produce SIM outputs.

      Why? For my own amusement.

      And you're correct also that SIM makes it easy to replicate it with a circuit, but it's not generally the case that this will be true (SIM is a linear system). However, there are non-linear circuit elements in the library, and it's possible that other SFC models (which aren't necessarily linear) can be represented as well.

      Again, why? ... this all follows from a discussion Roger Sparks and I have been having. It's purely for amusement. I'm not sure it serves any higher purpose.

    11. ... I suggested a mechanical analogy to Roger for SIM involving springs and dashpots, but the circuit analogy is easier to diagram and demonstrate in action (because of that website). He's a HAM radio guy, so a circuit is as good as springs and dashpots I think.

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    14. I constructed a new circuit model of SIM with 1 dollar = 1 coulomb of charge and Ts := 1 default (AKA "original") SIM period = 1 second which I think addresses some of your concerns regarding stocks vs flows. Click on the switch at the top center to activate the circuit. Here's a screen capture with annotations to tell you what's going on (I can't put labels the simulation). It may take a second or two to get going, as it's synchronized to the leading edge of a clock marking off the sample periods.

      I realize this continuous time model is not unique and thus represents a fiction, since I'm choosing to fill in the details between sample times in a particular manner. Also I realize that analog circuit elements may not be able to represent SFC models in general because there may be no non-linear elements to match non-linearities present in SFC models. SIM, being linear, is thus a special case.

      This particular continuous time version of SIM uses what I call "instantaneous flows" (or rates) that should satisfy the accounting identities over any arbitrary interval of time. For these "instantaneous flows" (expressed in dollars per default SIM sample period) I use the symbols g, y, τ, yD and c corresponding to SIM flow variables G, Y, T, YD and C respectively. I also integrate these instantaneous rate variables over 1-period (1 second) windows to demonstrate they match SIM's default outputs (really I just do this for Y, but the others would use the same technique).

      The scopes across the bottom of the screen represent the following from left to right: Y[p]/Y(t) over y(t), then τ(t), yD(t), c(t), H[p]/H(t), g(t).

      H (a stock) is measured in coulombs of charge (= dollars) which is accumulated on the 1-farad (1-F) capacitor at the top center of the circuit while g, y, τ, yD and c are measured in coulombs / sec (AKA amperes). These instantaneous rates must be integrated in 1-period length windows of time to recover SIM's discrete time "flow" outputs (as I demonstrate with Y in the circuit). They are integrated using integrator circuits also utilizing 1-F capacitors (in the upper left part of the circuit), so the voltage (with that choice of capacitance) across all capacitors in the circuit are identical to total charge (measured in coulombs) stored in these same capacitors. I need two such integrator circuits (one working on even sample periods and the other on odd sample periods) due to a limitation of the circuit simulation software to discharge the capacitors fast enough at the end of each sample period, and in reality, such a "ping-pong" integrator arrangement would probably make the most sense in actual hardware. These integrator circuits produce a saw-tooth looking output (the curve on the left labeled Y(t)), the peaks of which (Y[p]) correspond to G&L tabulated SIM output values for Y (in particular, the "p" index on the discrete time Y stands for "period" in G&L's table of numerical results, with the relation being Y[p] = Y(t = (p-1)Ts) where Ts = the sample period length. The upper scope on the left reports peak values (since the last simulation "Reset") in its upper left corner, and because the peaks of each saw-tooth increase with the default SIM g(t) function (g(t) = (20 amperes)(u(t)) where u(t) is the unit step function) , you can pause the simulation with the "Stopped" checkbox in the upper part of the right side-bar and verify that indeed each of these peaks correspond to G&L's SIM outputs for each period (which I have more of tabulated in the embedded spreadsheet here as well).

      Here's a corresponding post with more information.

    15. I have had a look at this, but I'm afraid my knowledge of circuits is not up to it. I can't really work out what is going on, so I can't say how it might relate to economics. Sorry.

    16. Thanks for taking a look Nick. The circuit business is a bit of a tangent... mostly the point is that I'm implementing an analog computer (or rather a digital simulation of one) to compute a continuous time version of SIM using "flows" which are defined at an instant of time. I call those "instantaneous flows" (and they are represented by instantaneous rates of the flow of charge (amperes)). These instantaneous flows must be integrated over each sample period (accounting periods) to produce an SFC-like flow (which the circuit also does).