I was scared of reading Stephanie Schmitt-Grohe and Martin Uribe's paper (pdf). All I knew was that it was a technically demanding paper, and that it had the "Neo-Fisherite" result -- if the central bank increased the rate of interest, inflation would rise.

I have now read it. Sort of. I *think* I now understand the gist of what is going on in their model. But I do not understand all the math. It's an orthodox New Keynesian model, except for the Phillips curve. The Phillips Curve is rigged.

There exists a level of full-employment (potential) output Y*. There exists a natural rate of interest, call it n, that is compatible with output growing at the growth rate of Y*. The central bank sets a nominal interest rate i. There is a ZLB, so i cannot be negative. (There is a Taylor Rule in the model, but I will ignore that, because it doesn't matter.) There is no investment or government expenditure, so output and consumption are the same thing. There is a consumption-Euler equation IS curve in which the planned growth rate of consumption is a *positive* function of the actual real rate of interest r. (Yes, I do mean "positive", and Old Keynesians will find the sign surprising, but that's another story.)

Using hat notation, so that X^ means dX/dt/X (the growth rate of X), we can write that as:

Y^-Y*^ = F(r-n) where F(0)=0 and F' > 0

So far, it's a standard New Keynesian/Neo-Wicksellian model.

Now for the Phillips curve. It's definitely not New Keynesian. For starters, it's kinked at Y*:

1. For Y=Y*, P^ = i - n (The inflation rate does what the theorist wants it to do, to keep planned demand growing at potential. That's the first place where it's rigged.)

2. For Y < Y*, it's complicated. There are two things going on.

2a. The price level P is a positive function of the level of output Y. So if Y jumps down, P jumps down too.

2b. The deflation rate -P^ is a positive function of the gap between potential and actual output. So if Y jumps down, P^ jumps down too.

We can write this segment of the Phillips curve in inverse form as:

(Y*-Y)/Y* = H(-P, -P^) where H(0,0)=0 and H_{1} > 0 and H_{2} > 0 [How the hell do I write that in math? Never mind. It doesn't really matter.]

(That's the second place where it's rigged. ** And it looks like the only reason they put firms in the model, and put a labour market in the model, and distinguished between prices and wages in the model. To rig it so that wage deflation is a continuous function of Y but price deflation a discontinuous function of Y.**)

Suppose the economy is initially in a stationary equilibrium at the ZLB. Output is below potential, but is growing at the same rate as potential output. There is a constant deflation rate, so the real interest rate stays constant, and equal to the natural rate.

Now suppose the central bank makes the nominal interest rate jump up, and pegs it at a new higher level. What happens?

**Output cannot jump.**

Because if output jumps up, the price level would jump up too. And if the price level jumps up, inflation would be plus infinity. And if inflation is plus infinity, the real interest rate would be minus infinity. And if the real interest rate is minus infinity, consumption would jump down. Which is a contradiction.

And because if output jumps down, the price level would jump down too. And if the price level jumps down, inflation would be minus infinity. And if inflation is minus infinity, the real interest rate would be plus infinity. And if the real interest rate is plus infinity, consumption would jump up. Which is a contradiction.

So if output cannot jump, it must either start growing or start falling.

If output starts falling, deflation jumps up, which means the real interest rate jumps up even more than the nominal rate, which means consumption starts rising. Which is a contradiction.

Therefore the only equilibrium is for output to start rising, and for inflation to start rising too.

God I hate [don't like. I've calmed down a bit now] this paper. And I hate having to plough through it to figure out what's really going on. It's like a snowball with a pebble inside. 90% snowjob, and a 10% kernel of wrongness. John Cochrane's papers are much better. He really does try to get to grips with the question of indeterterminacy and multiplicity. This one doesn't.

OK. I never was that great an economist, and I'm past my prime (such as it was), and I never could do math, and my brain is tired. But that is what I think is really going on in that paper. I might be wrong. But if I'm right, there is something seriously wrong with the state of economics today. Where did the economics intuition go? Why am I the one that has to do this?

Exasperated Nick Rowe is fun to read.

Posted by: louis | November 07, 2014 at 08:37 AM

Irving must be having fits in his grave seeing the effort being made to attach the prefix "Neo" to Fisher

Posted by: Marcus nunes | November 07, 2014 at 09:01 AM

Intuition gets harder as the models grow complex, but maybe they need to. We are lucky to have you to help.

I think you meant to write F(0)=0, or just to have Y^ on left hand side.

On a side note, for the Old Keynesians they future is fixed, including future consumption. So a higher growth rate of consumption is the same thing as a lower current level of consumption. The Old and New Keynesian IS curves don't have contradictory slopes, they just live in different spaces. Is that not right?

Posted by: An igyt | November 07, 2014 at 09:44 AM

Without comment on the mathematical aspect, I think the paper's position is well-expressed in the second paragraph of s.6:

> A natural question is whether an interest-rate peg of this type would not make matters initially worse by pushing inflation further down and thereby creating more unemployment. We will show that the answer to this question is no. On the contrary,

raising the interest rate from zero to its intended target lifts agents’ expectations about future inflation.In turn, the expectation of a higher future rate of inflation erodes expected real wages, thereby facilitating employment growth.Clearly, the italicized but forms the core of their argument. I also think that despite falling out of their math, it's not well-supported by natural experiments with the ECB and Sweden.

> There is a consumption-Euler equation IS curve in which the planned growth rate of consumption is a positive function of the actual real rate of interest r. (Yes, I do mean "positive", and Old Keynesians will find the sign surprising, but that's another story.)

I think that falls out of their model, whereby there's no investment good and productivity growth is exogenous. If the interest rate is high, that must mean that people are collectively wanting to consume more in the current period, which in turn increases current consumption.

Beyond the math, I think their "unemployment steady state" falls directly out of their assumptions. They assume that wages are nominally inflexible at full employment, but become more flexible as unemployment increases. Therefore, deflation is consistent with unemployment at the level that permits the reduction of nominal wages.

If wages are sticker than deflation, then I suspect their model would result in a deflationary spiral.

I may also have missed it, but I'm concerned that the model may be over-constraining the amount of bonds. The quantity supplied seems set by the government's budget constraint, but the quantity demanded is set by the household budget constraint and the interest rate (inverse price) is set by the Taylor rule. I'm a bit lost in the notation, so I can't tell if this is reconciled anywhere.

Posted by: Majromax | November 07, 2014 at 10:43 AM

Nick, Good post, my only comment is slightly off topic. In the real world the price level can and does move discontinuously. Let's say that 90% of goods prices are sticky, and 10% are flexible. Call the flexible price goods "commodities." Then if a monetary shock raises commodity prices (such as oil) by 5%, it will immediately raise the GDP deflator by 0.5%. That has no impact on the ex ante real interest rate, because the change occurs before anyone can take advantage of the move. Something similar occurs in the forex markets. For instance, the dollar fell 6 cents against the euro on the announcement of QE1 in March 2009. If interest rates fall on the Fed easing, the the interest parity theorem implies the dollar is expected to regain some (not all) of that depreciation over time. Of course those moves are traditionally explained by the Dornbusch "overshooting" model. Sometimes I wonder whether Dornbusch's insight is working its way into closed-economy macro models.

Posted by: Scott Sumner | November 07, 2014 at 10:47 AM

Main text: "There exists a level of full-employment (potential) output Y*."

Not unique, right?

Main text: "There exists a natural rate of interest, call it n, that is compatible with output growing at the growth rate of Y*. . . . There is no investment or government expenditure, so output and consumption are the same thing."

So we have a growing economy without investment or government expenditure, where we consume all output. So how is this growth accomplished? By some kind of physical or biological growth, right? Increasing population of humans (consumers), increasing population of plants and/or animals (what is consumed)? By increasing human activity (both production and consumption)? How?

Posted by: Min | November 07, 2014 at 11:26 AM

Looking at this briefly. No time to get the bottom of things. This paper is not clearly written and I'm not too familiar with this Keynesian thing. Still, my curiosity was piqued.

Preliminary remark before moving on: On page 8, there must be a typo: the derivative of gamma cannot be negative if the gamma_zero and gamma_one parameters are positive. Right?

So now. Do I understand this correctly?

The real wage, w, is always equal to the marginal productivity of (exogenous!) labor hours, F'(h), while the nominal wage W/P is rigid.

Thus, the model features (exogenous) nominal-wage rigidity but perfect real-wage flexibility... however, the real wage is exogenous (since labor hours and marginal productivity are exogenous), so bottomline everything is exogenous except the adjustment path of the nominal wage, to the extent that it depends on the unemployment rate.

However, since the real wage is exogenous, the unemployment rate is exogenous to the labor market and endogenous only to the output-consumption side of the model (is that right?!), rather than being caused by wage rigidity like I think it's supposed to be in Keynes? The absence of investment in the model is another hint that there is very little of Keynes in the model.

So the recovery is jobless because unemployment is "largely" exogenous (unlike in models like that of Shimer's). Question: Would an Okun correlation graph exhibit anything like the usual coefficients or (if my suspicion is founded) some unrealistically small coefficients?

The model is not general equilibrium on the production side but closed on the monetary side with another built-in source of sluggishness: the bank sets the price index P indirectly via the inflation rate pi, itself via the nominal rate of interest R. So what the monetary feedback rule does is translate movements in output to movements in nominal wages, via the consumption-smoothing of the rational, forward-looking agent, and thereby having a small impact on the nominal wage and on the unemployment rate.

Two sources of sluggishness, presumably bending the curves to make a second equilibrium configuration possible. Other than that, expectations are rational and forward-looking, with agents perfectly aware of the inefficiency of monetary policy and the wage-determination process.

Hard to take anything, good or bad, from this cursory reading of the paper. Plus I've always hated papers that give numerical simulations without algebraic intuition.

Posted by: annoporci | November 07, 2014 at 11:46 AM

Min: "Not unique, right?"

Wrong. It is unique. Whether we get to it is another matter.

"How?" improving technology, that increase labour productivity.

louis: but being an exasperated Nick Rowe, having to struggle through that math, is draining!

Scott: thanks! When I've cleared my head from this stuff, I'm going back to your good post.

Yes, some prices can jump. But as you say, that jump takes place instantly, before false trading can occur, normally. But in this model there's something fishy going on. Because it's sort of perfect foresight. So I don't think they allow P to jump before trade begins. But I am not 100% clear on this.

Majro: "Clearly, the italicized but forms the core of their argument."

Yep, such as it is. But there are also those mathy appendices.

" If the interest rate is high, that must mean that people are collectively wanting to consume more in the current period, which in turn increases current consumption."

No. That's not what's going on. The ratio of current MU(C) to future MU(C) is a function of the relative price of current C to future C (1+r).

"Therefore, deflation is consistent with unemployment at the level that permits the reduction of nominal wages."

Correct.

"If wages are sticker than deflation, then I suspect their model would result in a deflationary spiral."

Wages are sticky down, but prices are perfectly flexible, given wages.

"I may also have missed it, but I'm concerned that the model may be over-constraining the amount of bonds."

My initial hunch was that the stock of bonds was growing at the rate of interest, and that bonds are really money, and that was what was driving the result. But my hunch was wrong. Bonds are money, but we can have an equilibrium in which the stock of money is always zero. Just like in Woodford. People have a chequing account at the central bank, that pays interest i, and can have either a positive or negative balance at the individual level, but aggregates to 0. Red and green money.

Posted by: Nick Rowe | November 07, 2014 at 12:05 PM

Incidentally, this is yet another model where there is a price level but no money. In fact, the word "money" does not appear in the paper at all, and "monetary" only appears twice (once in the abstract as "monetary policy," once in the body as "monetary authority").

I wish that

somediscussion of money was a prerequisite to writing a paper about monetary policy. I suspect that if there was, we'd find that this model would conclude that high interest rates cause expansion of the monetary base.Posted by: Majromax | November 07, 2014 at 12:12 PM

Would interest being paid by the CB not count as government expenditure?

Posted by: Odie | November 07, 2014 at 12:27 PM

@annoporci:

> On page 8, there must be a typo: the derivative of gamma cannot be negative if the gamma_zero and gamma_one parameters are positive. Right?

No, that's okay. Gamma is wage rigidity, and the derivative being strictly negative forces wages to become progressively more flexible with increasing unemployment. From their functional form, d/dx (A*(1-x)^B) = -AB*(1-x)^(B-1) < 0.

However, their functional form is more strict than their conditions require, since gamma(1) = 0 (that is, wages are fully flexible with total unemployment). I suspect their results may not hold if gamma(1) > 0.

> My initial hunch was that the stock of bonds was growing at the rate of interest, and that bonds are really money, and that was what was driving the result. But my hunch was wrong.

It may be worse than this?

Their household budget constraint in eqn 2 involves R(t-1)B(t-1) and T on the "exogenously determined" side, but B(t) on the endogenous side. However, their government budget constraint (unlabeled, on page 9) says that B(t) = R(t-1)B(t-1) - T. That means that B(t) is *not* a free parameter that can be chosen by households to optimize their consumption path over time.

However, R(t) only enters the structure of their model through bonds! The interest rate

has no effectgiven the government budget constraint. I think this is tacitly admitted on page 9, where they suggest that the government could set B(t)=0 for all time.So of *course* they see a strong, short-term Fisher effect: the interest rate is only determined by the Fisher equation. Adding a Taylor Rule here is overdetermining the system.

Posted by: Majromax | November 07, 2014 at 12:34 PM

Addendum (and I apologize for the multiple posts)

Their assumptions actually enforce their conclusion, no appendices necessary.

They assume that wages are perfectly flexible in an inflationary environment. If there is an output gap, the Taylor rule will make the CB set a lower interest rate. By the Fisher "rule", that will give lower inflation.

If that lower inflation is still positive, however, the economy

still sees perfectly flexible real wages. That means that output quickly returns to normal, the output gap is zero, and the Taylor rule gives R=i(target)+r.If that lower inflation is negative, however, the economy sees initially inflexible wages. Unemployment increases to the point where wage flexibility equals the deflation rate. We still see an output gap, but the Taylor rule gives R=i(target)+r-, so i(actual) = i(target)- -- persistently below-target inflation. Tweak parameters and we can get a stable, fixed point with deflation.

Posted by: Majromax | November 07, 2014 at 12:41 PM

> R=i(target)+r-, so i(actual) = i(target)-

This should read "R=i(target)+r-<stuff>, so i(actual) = i(target) - <stuff>"

Posted by: Majromax | November 07, 2014 at 12:42 PM

Marcus and an igyt: I found your comments in the spam filter. Sorry.

Marcus: I tend to agree. But think it's too late now.

an igyt: "I think you meant to write F(0)=0, or just to have Y^ on left hand side."

You are right. I have edited it. well-spotted!

"On a side note, for the Old Keynesians they future is fixed, including future consumption. So a higher growth rate of consumption is the same thing as a lower current level of consumption. The Old and New Keynesian IS curves don't have contradictory slopes, they just live in different spaces. Is that not right?"

I think that's right. (Or it's a good way to interpret the difference). But if the location of the future is what's at issue, like in these sorts of models, we can't use the Old Keynesian assumption.

Odie: "Would interest being paid by the CB not count as government expenditure?"

No. It counts as a transfer payment.

Majro: Agreed. It is (implicitly) a monetary exchange economy. Barter is not allowed (if it were allowed unemployment would be impossible in this model, regardless of r, because the unemployed workers are firms would barter their way back to full employment). "Bonds" are implicitly used as medium of exchange.

" I suspect that if there was, we'd find that this model would conclude that high interest rates cause expansion of the monetary base."

That was my hunch initially. And it is true they would. But you can also construct an equilibrium in which the base is zero initially, and stays at zero the whole time, even though the interest rate is not 0%. So I eventually figured out that is not what is driving the results.

Posted by: Nick Rowe | November 07, 2014 at 12:44 PM

annoporci: MPL=W/P, but MPL is not fixed exogenously. There is diminishing MPL, so you get a downward-sloping labour demand curve, as a function of W/P.

Majro: keep going. You are on a roll, I think.

Posted by: Nick Rowe | November 07, 2014 at 01:16 PM

Majro: " That means that B(t) is *not* a free parameter that can be chosen by households to optimize their consumption path over time."

It is a choice variable for the individual household, but not for all households collectively. It is not possible for all households to save, unless the government dissaves.

("free parameter" means something different. As you probably know.)

Posted by: Nick Rowe | November 07, 2014 at 01:21 PM

> MPL=W/P, but MPL is not fixed exogenously.

I thought labor hours were exogenously fixed and W/P = F'(h), don't have the paper anymore to check.

Posted by: annoporci | November 07, 2014 at 01:41 PM

The labour supply curve is fixed exogenously (perfectly inelastic) but sticky W means we are off the labour supply curve, in the unemployment equilibrium.

Posted by: Nick Rowe | November 07, 2014 at 01:59 PM

> Majro: keep going. You are on a roll, I think.

Okay: the paper is definitely in error about bonds, since just below the household budget the text explicitly notes that households "choosing C(t) and B(t)". That is directly incompatible with taxes being exogenous.

I think that is the contradiction that supports the rest of the paper. Reduce the model to one with no "taste shocks" and no productivity growth, so we expect full-employment output to be steady-state. Further assume that our initial condition is inflationary, so we indeed have full-employment output.

Now, c(t)=1 (eqn. 4, defined in the paragraph above), required by full-employment. Equation 3 then reduces to 1 = B'R(t) c(t+1)/π(t+1) [notation: B' = Btilde in the paper is the discount rate, R is the interest rate, π is the inflation rate; π* is the target inflation rate].

If we presume that the next period will be inflationary and have full output, then we get π(t+1) = B' R(t). But by the model's Taylor Rule (eq. 10), R(t) = π*/B' + a(π(t) - π*) (the output gap term is zero). That means π(t+1) = π* + B'a (π(t) - π*)

Now, if our current-period inflation is slightly above-target, π(t) = π* + ε, Then next period's inflation is given by:

π(t+1) = π* + B'a(π* + ε - π*) = π* + B'aε.

(Check for consistency: π(t+1) is still above-target, so it is consistent with full employment.)

BUT! Assumption 2 says that B'a > 1. That means that our inflation perturbation will grow with time.What's the saving grace? What I worked on here was the stochastic, expectations-based equilibrium. All I can really say is that if agents see above-target inflation, they will expect above-target inflation in the following period. There is no prescriptive formula for actual next-period inflation. (And there really can't be, since money doesn't exist here and bonds don't have to.)

Note that in the model's non-stochastic equilibrium (sec. 3), full-output R is both π/B' (eq. 11) and π*/B' + a(π*-π) (eq. 18).

I think the model accidentally derives a world where the full-employment equilibrium exists, but it is dynamically unstable to perturbations [or would be if the model had dynamics]. It is telling, I think, that their results only returns to the Taylor Rule after breaking it for a period -- they essentially get to reimplement the Taylor Rule after it does nothing.

Posted by: Majromax | November 07, 2014 at 02:11 PM

Addendum: The paper does use its expectations as a dynamical model for the "results" section. From the top of page 13,

> assume that from period 0 on, inflationary expectations are always fulfilled and that there are no shocks to economic fundamentals.

Posted by: Majromax | November 07, 2014 at 02:22 PM

Majro: " the paper is definitely in error about bonds, since just below the household budget the text explicitly notes that households "choosing C(t) and B(t)". That is directly incompatible with taxes being exogenous."

But what it means is that the **individual** household can choose B(t), but in aggregate B(t) is determined by the government. If T(t)=0, each individual household can choose to save, but in aggregate they cannot save. Each individual can have C > Y or C < Y, but in aggregate Y = C. Standard Old Keynesian stuff, about desired vs actual saving. Y adjusts until desired saving = 0. Y is exogenous to the individual household, but endogenous for households in aggregate.

Posted by: Nick Rowe | November 07, 2014 at 02:40 PM

Second addendum: there's bad math in the appendix on "full employment under negative taste shocks".

To satisfy equation (3) (budget expectation) of the stochastic equilibrium, the appendix sets R0 (the initial interest rate) to be exp(-ξ) π*/B'. However, when they satisfy (10) (the Taylor rule), the appendix (8.7) only requires:

π(0) ≤ π* + (exp(ξ)-1) π*/(a B')

However, the Taylor rule is not an

inequality. To satisfy (10), that ≤ must be precisely equal. That means that the "negative taste shock" must be coupled with precisely the right zero-period inflation rate.Posted by: Majromax | November 07, 2014 at 02:43 PM

Moi: "Not unique, right?"

Nick Rowe: "Wrong. It is unique. Whether we get to it is another matter."

I know that economists like to assume uniqueness, which is why I asked. In this case, that assumption seems quite farfetched to me. Too many variables.

Nick Rowe: "How?" improving technology, that increase labour productivity."

And technology improves without investment or gov't spending -- how?

Posted by: Min | November 07, 2014 at 06:32 PM

Hmm. I'm getting this weird feeling, that maybe Majromax knows some math.

Keep rolling. I don't understand it, but others probably do.

Posted by: Nick Rowe | November 07, 2014 at 06:34 PM

Physicists have given mathematics a good name. It seems that equations can predict everything because they predict so much about the physical world with amazing precision. But, this is a shallow understanding of math and physics. Physicists have carefully investigated the predictions made by their equations and have thrown out all of the "non-physical" results. Math predicts the physical world only because physicists have thrown away all of the math and interpretations which turned out to be wrong, and physics is precisely repeatable.

Macroeconomics reverses this, laughably. Past economic events are amazingly complicated and not repeatable. Mathematics supplies a shiny surface to a rat's nest of assumed quantities and relationships. The assumptions may not and usually cannot represent the part of reality they intend to model. They may not and usually cannot incorporate all of the other important data about reality which they ignore. Then, the math may have many properties which produce unphysical (crazy) predictions. Further, equations do not care about cause and effect. Flip them around a bit and you have probably flipped cause and effect along the way.

One should realize that any error, just one, can result in crazy results when you are dealing with intertwined growth rates and equilibria.

Consider these two assumptions of the model.

(1) "There exists a level of potential output Y at full-employment."

(2) "There is no investment or government expenditure, so output and consumption are the same thing."

These are crazy if the intent is to conclude something about reality. Fiddle with the math as you wish, the equations will not represent reality.

"Potential Output Y" is not well defined. Full employment is not well defined. Y may exist in some philosophic sense, but we have no way of knowing or computing it. If one chooses to manipulate it or conclude from it, you can only do this by assuming that the interest rate affects all of the components of Y proportionally and evenly. Ludicrous.

As for (2), Keynesians don't understand investment; they think that investing kills money flow and makes everyone poorer. This denies even a quick look at reality. Here, the approach is to just ignore investment. I don't have to know anything else about the "model". It cannot represent reality.

Say a cake bakes in 40 minutes at 350F. We look at the physics and see that heat transfer is twice as fast at (say) 450F. We conclude that we can bake the cake in 20 minutes at 450F. I doubt that is a cake you will want to eat.

Similarly, we can conclude all sorts of things from assumed economic variables and the general physics of money flow or interest rates. I doubt you will want to live in the resulting economy after fiddling with those things.

Some advice from the late genius particle physicist Richard P. Feynman. Think of macroeconomics as the science he refers to.

( easyopinions.blogspot.com/#trueScience )

=== ===

“When someone says, ‘Science teaches such and such,’ he is using the word incorrectly. Science doesn’t teach anything; experience teaches it. If he says to you, ‘Science has shown such and such,’ you should ask, ‘How does science show it?’

How did the scientists find out? How? What? Where?’ It should not be ‘science has shown.’ And, you have as much right as anyone else, upon hearing about the experiments and after hearing all the evidence, to judge whether a sensible conclusion has been arrived at.”

=== ===

Posted by: Andrew_M_Garland | November 08, 2014 at 01:10 AM

> The labour supply curve is fixed exogenously (perfectly inelastic) but sticky W means we are off the labour supply curve, in the unemployment equilibrium.

thanks for your feedback,

since the adjustment mechanism is an exogenous function, doesn't that mean that all the action, including changes in unemployment, is on the production-consumption side with nothing meaningful happening in the labor market?

Posted by: annoporci | November 08, 2014 at 05:09 AM

@Majromax

>> On page 8, there must be a typo: the derivative of gamma cannot be negative if the gamma_zero and gamma_one parameters are positive. Right?

>No, that's okay.

Stupid me, thanks.

Posted by: annoporci | November 08, 2014 at 05:33 AM

Andrew: Borges had something interesting to say on that subject.

annoporci: "...doesn't that mean that all the action, including changes in unemployment, is on the production-consumption side with nothing meaningful happening in the labor market?"

Yes. Except, nominal wages are sticky downwards (they can only fall slowly), while output prices are perfectly flexible, and we need the downward-sloping labour demand curve to tell us what P is given W and Y. (Plus the labour supply curve helps us define potential output, of course.)

Posted by: Nick Rowe | November 08, 2014 at 06:59 AM

Good post Nick. I'm having a different problem with the paper, based on (no doubt) having struggled with it much less than you. In Section 6, they introduce a permanent interest rate peg. They say they do this "for simplicity", but I don't quite understand why, after the interest rate is pegged permanently at R^*, the model doesn't become indeterminate, as per Howitt (1992) "Interest Rate Control and Nonconvergence to Rational Expectations", JPE. I think it has to do with their assumption that inflation expectations are "well anchored," and everyone is assuming that inflation will converge to the high-employment inflation level pi^*. This is tantamount to exogenous rather than rational expectations in the face of an interest rate peg. This is obviously related to your remark towards the end concerning coming to grips with indeterminacy.

I did see Frederico Ravenna (at the Bank of Canada, yesterday) give an intuitive graphical explanation of their result, but it doesn't involve a permanent interest rate peg, just a temporary one. Take the Benhabib, Schmitt-Grohe and Uribe graph from (I think) "Perils of Taylor Rules" with the kinked Taylor rule (kink at zero) that intersects with the i = r + pi Fisher relation at two points, one being the good steady state and the other the bad steady state with i = 0 and pi = -r. Now get rid of the part of the Taylor rule curve at some point before it hits the zero lower bound and assume that at that point i jumps to a point above the Fisher line and stays at that level as you travel to the left. This gets rid of the bad deterministic steady state with pi = -r. Then the real question becomes the following. Does this give you determinate dynamics so you will get back to the good steady state no matter where you start? I'd like to see this worked out without just imposing "anchored" inflation expectations.

Posted by: Steve Ambler | November 08, 2014 at 10:17 AM

Steve: Thanks! I agree totally. But that's not a different problem to the one I am having. They simply fail to address those questions. They ignore them.

They might just as well have said: "Assume perfectly flexible prices, and rational expectations. Therefore the real rate is pinned down by savings and investment at full employment, and the Fisher equation then tells us that if the central bank raises the nominal rate that must cause the inflation rate to rise. But if the central bank is so stupid as to set a zero nominal interest rate, there would need to be deflation to get the real rate **high** enough for full employment equilibrium, and if prices are sticky down we will only get deflation if there is unemployment, so that must cause unemployment."

Posted by: Nick Rowe | November 08, 2014 at 11:08 AM

Mr. Rowe,

Thank you for your link to the thoughtful defense of the methodology in the Schmitt-Grohe and Uribe paper.

I learned that the "potential output Y at full-employment" is measurable and well understood in detail. Also, that investment doesn't matter in describing the economy. I see, that paper is only "a map", a static and necesssarily imprecise representation of reality. We may bump into a few trees by following it, but certainly will not fall into any gulches. We know this because that is the way maps are. If there are any big problems, we will draw them onto the map along the way after pulling our wagon out of the ditch. Thus, macroeconomics ever progresses to greatness.

The above is sarcastic, but certainly no more sarcastic and insulting than a link to the well cited Spanish economics paper "On Rigor in Science", a one-paragraph short story by Jorge Luis Borges about the map/territory relation, written in the form of a literary forgery. It seems that this paper represents well the "conclusion by analogy" technique of modern macroeconomics.

I am puzzled, because you also don't like the assumptions in the Schmitt-Grohe and Uribe paper.

Posted by: Andrew_M_Garland | November 08, 2014 at 02:08 PM

Majromax: "Their household budget constraint in eqn 2 involves R(t-1)B(t-1) and T on the "exogenously determined" side, but B(t) on the endogenous side. However, their government budget constraint (unlabeled, on page 9) says that B(t) = R(t-1)B(t-1) - T. That means that B(t) is *not* a free parameter that can be chosen by households to optimize their consumption path over time."

Nick Rowe: "But what it means is that the **individual** household can choose B(t), but in aggregate B(t) is determined by the government. If T(t)=0, each individual household can choose to save, but in aggregate they cannot save."

Majromax is right. Aristotelian logic still applies. Each individual household can

attemptto save, but if they all attempt to save, some of them will fail. What we may have here is a kind of typo, where the same variable is used for different things. Sometimes that is innocuous, but such a typo can become a thinko.Posted by: Min | November 08, 2014 at 04:48 PM

Majormax@02:11PM "That means that our inflation perturbation will grow with time"

I think they themselves have shown that in Benhabib, Schmitt-Grohe and Uribe (e.g. Figure 2).

Delong has once quibbled their conceptualization (or Bullard's conceptualization in 'Seven Faces of "The Peril"' paper, which was inspired by their paper) in his blogpost, and I commented as follows:

"They used the natural interest rate as discounting rate of consumers. So, the consumer's choice between current and future consumption is not affected by the monetary policy. The monetary policy only affects return from financial assets. So, if the central bank hikes interest rate, the consumer's interest income goes up, but the consumer's discount factor remains unchanged. That leaves inflation rate as the only adjusting factor. Now, how realistic does it sound?"

The equation Majormax derived, π(t+1) = B' R(t), exactly expressed this relationship, I think.

Posted by: himaginary | November 08, 2014 at 09:30 PM

Andrew: a good map for canoeists would include portage trails, but a good map for truckers would exclude them. Same with models. And if you consider my recommending the brilliant Borges an insult, which it wasn't, then you probably deserve it. Go away.

Min: "Each individual household can attempt to save, but if they all attempt to save, some of them will fail. What we may have here is a kind of typo, where the same variable is used for different things. Sometimes that is innocuous, but such a typo can become a thinko."

That isn't a thinko in this paper. Because in equilibrium no household attempts to save.

himaginary: we are on the same page. The assumption that the inflation rate is what adjusts endogenously, to reconcile the exogenous nominal rate with the exogenous natural rate, is not just unrealistic, it is precisely what's at issue. It assumes the conclusion, with zero explanation.

Posted by: Nick Rowe | November 09, 2014 at 08:28 AM

Nick Rowe: "That isn't a thinko in this paper. Because in equilibrium no household attempts to save."

Sounds like heaven. There is pie in the sky in the great bye and bye. And it's a growing pie at that, thanks to immaculate technology that falls like manna without investment or expense.

Posted by: Min | November 09, 2014 at 12:27 PM

@Nick:

> They might just as well have said: "Assume perfectly flexible prices, and rational expectations.

I think that is what they said, just with a few appendices of math as wrapper.

@Min:

> Each individual household can attempt to save, but if they all attempt to save, some of them will fail.

The physicist in me is disconcerted by the notation for endogenous household bonds, since this is a representative agent rather than an aggregate. The econ-blog-reader in me is comfortable with the idea that individuals can save if the aggregate doesn't.

As a practical matter, the question is moot. Bonds don't show up in the remainder of the model, so their stock or flows are irrelevant. (Aesthetically, that's one thing that bugs me about the paper: irrelevant parameters and cumbersome notation.)

@himaginary:

> I think they themselves have shown that in Benhabib, Schmitt-Grohe and Uribe (e.g. Figure 2).

Ooh, thanks for the link. That definitely confuses me, since a graph like that which suggests that the targeted equilibrium is unstable immediately raises the question of where our

~~Earth-shattering kaboom~~hyperinflation has been.I also think (but haven't mathed) that this paper is assuming both a fiscal theory of the price level (bottom of p. 49, talking about the government budget constraint wrt the price level) and an interest-rate control mechanism. Both can't be independently true.

@Nick:

> a good map for canoeists would include portage trails, but a good map for truckers would exclude them.

I wonder just what a good map for monetarists would include?

In rough order of certainty:

*) I'm reasonably certain that we'll never have a good model for inflation from a model that doesn't explicitly include money

*) I suspect that such a model would also have to include a dynamical story, beyond simple equilibrium solutions.

*) I wonder if the model wouldn't also have to include less-liquid, long-term debt obligations. Perhaps also endogenous investment and productivity growth.

(As a rough sketch: Households supply labour to the market, and they have consumption and savings preferences based on utility maximization. Firms employ labour and make investments by issuing long-term debt, based on the long-term interest rate. Exogenous government issues long-term debt to resolve its budget constraint. The central bank acts via setting the interest rate on short-term money (interest-on-money) and by conducting OMOs to swap money for long-term debt and vice versa.)

Posted by: Majromax | November 10, 2014 at 09:58 AM

I looked at the paper a long time ago, but if memory serves, it is rigged in only one spot: the taylor rule.

In a normal NK, causality runs from inflation to interest rate target: i_t=f(p_{t-1})

but in this paper, causality runs from interest rate target to inflation, Cochrane-style: i_{t+1}=f(p_t)

Posted by: Matthew | November 10, 2014 at 11:42 AM

oops, got the subscript wrong there. Second one should be i_{t-1}=f(p_t) That is, the interest rate is set based on expected inflation, not observed inflation

Posted by: Matthew | November 10, 2014 at 11:46 AM