## Friday, June 9, 2023

### The Fed and the Phillips curve

I just finished a new draft of "Expectations and the neutrality of interest rates," which includes some ruminations on inflation that may be of interest to blog readers.

A central point of the paper is to ask whether and how higher interest rates lower inflation, without a change in fiscal policy. That's intellectually interesting, answering what the Fed can do on its own. It's also a relevant policy question. If the Fed raises rates, that raises interest costs on the debt. What if Congress refuses to tighten to pay those higher interest costs? Well, to avoid a transversality condition violation (debt that grows forever) we get more inflation, to devalue outstanding debt. That's a hard nut to avoid.

But my point today is some intuition questions that come along the way. An implicit point: The math of today's macro is actually pretty easy. Telling the story behind the math, interpreting the math, making it useful for policy, is much harder.

1. The Phillips curve

The Phillips curve is central to how the Fed and most policy analysts think about inflation. In words, inflation is related to expected future inflation and by some measure if economic tightness, factor $$x$$. In equations, $\pi_t = E_t \pi_{t+1} + \kappa x_t.$ Here $$x_t$$ represents the output gap (how much output is above or below potential output), measures of labor market tightness like unemployment (with a negative sign), or labor costs. (Fed Governor Chris Waller has a great speech on the Phillips curve, with a nice short clear explanation. There are lots of academic explanations of course, but this is how a sharp sitting member of the FOMC thinks, which is what we want to understand. BTW, Waller gave an even better speech on climate and the Fed. Go Chris!)

So how does the Fed change inflation? In most analysis, the Fed raises interest rates; higher interest rates cool down the economy lowering factor x; that pushes inflation down. But does the equation really say that?

This intuition thinks of the Phillips curve as a causal relation, from right to left. Lower $$x$$ causes lower inflation. That's not so obvious. In one story, the Phillips curve represents how firms set prices, given their expectation of other's prices and costs. But in another story, aggregate demand raises prices, and that causes firms to hire more (Chris Waller emphasized these stories).

This reading may help to digest an otherwise puzzling question: Why are the Fed and its watchers so obsessed with labor markets? This inflation certainly didn't start in labor markets, so why put so much weight on causing a bit of labor market slack? Well, if you read the Phillips curve from right to left, that looks like the one lever you have. Still, since inflation clearly came from left to right, we still should put more emphasis in curing it that way.

2. Adjustment to equilibrium vs. equilibrium dynamics.

But does the story work? Lower $$x_t$$ lowers inflation $$\pi_t$$ relative to expected future inflation $$E_t \pi_{t+1}$$. Thus, it describes inflation that is rising over time.  This does not seem at all what the intuition wants.

So how do we get to the intuition that lower $$x_t$$ leads to inflation got goes down over time?  (This is on p. 16 of the paper by the way.) An obvious answer is adaptive expectations: $$E_t \pi_{t+1} = \pi_{t-1}$$.  Then lower $$x_t$$ does mean inflation today lower than it was in the past. But the Fed and most commenters really don't want to go there. Expectations may not be "rational," and in most commentary they are either "anchored" by faith in the Fed, or  driven by some third force. But they aren't mechanically last year's inflation. If they were, we would need much higher interest rates to get real interest rates above zero. Perhaps the intuition comes from remembering these adaptive expectations dynamics, and not realizing that the new view that expectations are forward looking, even if not rational, undermines those dynamics.

Another answer  may be confusion between adjustment to equilibrium and movement of equilibrium inflation over time. Lower $$x_t$$ means lower inflation $$\pi_t$$ than would otherwise be the case. But that  reduction is an adjustment to equilibrium. It's not how inflation we observe -- by definition, equilibrium inflation -- evolves over time.

This is, I think, a common confusion. It's not always wrong. In some cases, adjustment to equilibrium does describe how an equilibrium quantity changes, and in a more complex model that adjustment plays out as a movement over time. For example, a preference or technology shock might give a sudden increase in capital; add adjustment costs and capital increases slowly over time. A fiscal shock or money supply shock gives a sudden increase in the price level; add sticky prices and you get a slow increase in the price level over time.

But we already have sticky prices. This is supposed to be the model, the dynamic model, not a simplified model. Here, inflation lower than it otherwise would be is not the same thing as inflation that goes down slowly over time. It's just a misreading of equations.

Another possibility is that verbal intuition refers to the future, $E_t \pi_{t+1} = E_t \pi_{t+2} + \kappa E_t x_{t+1} .$Now, perhaps, raising interest rates today lowers future factor x, which then lowers future inflation $$E_t\pi_{t+1}$$ relative to today's inflation $$\pi_t$$. That's still a stretch however. First, the standard new-keynesian model does not have such a delay. $x_t = E_t x_{t+1} - \sigma(i_t - E_t \pi_{t+1})$says that higher interest rates also immediately lower output, and lower output relative to future output. Higher interest rates also raise output growth. This one is more amenable to adding frictions -- habits, capital accumulation, and so forth -- but the benchmark model not only does not have long and variable lags, it doesn't have any lags at all.  Second, maybe we lower inflation $$\pi_{t+1}$$ relative to its value $$\pi_t$$, in equilibrium, but we still have inflation growing from $$t+1$$ to $$t+2$$. We do not have inflation gently declining over time, which the intuition wants.

We are left -- and this is some of the point of my paper -- with a quandary. Where is a model in which higher interest rates lead to inflation that goes down over time? (And, reiterating the point of the paper, without implicitly assuming that fiscal policy comes to the rescue.)

3. Fisherian intuition

A famous economist, who thinks largely in the ISLM tradition, once asked me to explain in simple terms just how higher interest rates might raise inflation. Strip away all price stickiness to make it simple, still, the Fed raises interest rates and... now what? Sure point to the equation $$i_t = r + E_t\pi_{t+1}$$ but what's the story? How would you explain this to an undergraduate or MBA class?  I fumbled a bit, and it took me a good week or so to come up with the answer. From p. 15 of the paper,

First,  consider the full consumer first-order condition $x_t = E_t x_{t+1} - \sigma(i_t -E_t \pi_{t+1})$ with no pricing frictions.  Raise the nominal interest rate $$i_t$$.  Before prices change, a higher nominal interest rate is a higher real rate, and induces people  to demand less today $$x_t$$ and more next period $$x_{t+1}$$.  That change in demand pushes down the price level today $$p_t$$ and hence current inflation $$\pi_t = p_t - p_{t-1}$$, and it pushes up  the expected price level next period $$p_{t+1}$$ and thus expected future inflation $$\pi_{t+1}=p_{t+1}-p_t$$.

So, standard intuition is correct, and refers to a force that can lower current inflation. Fisherian intuition is correct too, and refers to a natural force that can raise expected future inflation.

But which is it, lower $$p_t$$ or higher $$p_{t+1}$$? This consumer first-order condition, capturing an  intertemporal substitution effect, cannot tell us. Unexpected inflation and the overall price level is determined by a wealth effect. If we pair the higher interest rate with no change in surpluses, and thus no wealth effect, then the initial price level $$p_t$$ does not change [there is no devaluation of outstanding debt] and the entire effect of higher interest rates is a rise in $$p_{t+1}$$.  A concurrent rise in expected surpluses leads to a lower price level $$p_t$$ and less current inflation $$\pi_t$$. Thus in this context standard intuition also implicitly assumes that fiscal policy acts in concert with monetary policy.

In both these stories, notice how much intuition depends on describing how equilibrium forms. It's not rigorous. Walrasian equilibrium is just that, and does not come with a price adjustment process. It's a fixed point, the prices that clear markets, period. But believing and understanding how a model works needs some sort of equilibrium formation story.

The distinction between rational, or at least forward-looking and adaptive or backward-looking expectations is central to how the economy behaves. That's a central point of the paper.  It would seem easy to test, but I realize it's not.

Writing in May 2022, I thought about adaptive (backward-looking) and rational (forward-looking), and among other points that under adaptive expectations we need nominal interest rates above current inflation  -- i.e. much higher -- to imply real interest rates, while that isn't necessarily true with forward-looking expectations. You might be tempted to test for rational expectations, or look at surveys to pronounce them "rational" vs. "behavioral," a constant temptation. I realize now it's not so easy (p. 44):

Expectations may seem adaptive.  Expectations must always be, in equilibrium, functions of variables that people observe, and likely weighted to past inflation. The point of "rational expectations'' is that those forecasting rules are likely to change as soon as a policy maker changes policy rules, as Lucas  famously pointed out in his "Critique."  Adaptive expectations may even be model-consistent [expectations of the model equal expectations in the model] until you change the model.

That observation is important in the current policy debate. The proposition that interest rates must be higher than current inflation in order to lower inflation assumes that expected inflation equals current inflation -- the simple one-period lagged adaptive expectations that I have specified here. Through 2021-2022, market and survey expectations were much lower than current (year on year) inflation. Perhaps that means that markets and surveys have rational expectations: Output is temporarily higher than the somewhat reduced post-pandemic potential, so inflation is higher than expected future inflation ($$\pi_t = E_t \pi_{t+1} + \kappa x_t$$). But that observation could also mean that inflation expectations are a long slow-moving average of lagged inflation, just as Friedman speculated in 1968 ($$\pi^e_t = \sum_{j=1}^\infty \alpha_j \pi_{t-j}$$). In either case, expected inflation is much lower than current inflation, and interest rates only need to be higher than that low expectation to reduce inflation. Tests are hard, and you can't just look at in-sample expectations to proclaim them rational or not.

Rational expectations change when policy deviates from a rule, or when the policy rule changes. That's their key feature. We should talk perhaps about rational vs. exogenous expectations.

5. A few final Phillips curve potshots

It is still a bit weird that so much commentary is so focused on the labor market to judge pressure on inflation. This inflation did not come from the labor market!

Some of this labor market focus makes sense in the new-Keynesian interpretation of the Phillips curve: Firms set prices based on expected future prices of their competitors and marginal costs, which are largely labor costs. That echoes the 1960s "cost push" view of inflation (as opposed to its nemesis "demand pull" inflation). But it begs the question, well, why are labor costs going up? The link from interest rates to wages is about as direct as the link from interest rates to pries. This inflation did not come from labor costs, maybe we should fix the actual problem? Put another way, the Phillips curve is not a model. It is part of a model, and lots of equations have inflation in them. Maybe our focus should be elsewhere.

Back to Chris Waller, whose speech seems to me to capture well sophisticated thinking at the Fed.  Waller points out how unreliable the Phillips curve is

What do economic data tell us about this relationship? We all know that if you simply plot inflation against the unemployment rate over the past 50 years, you get a blob. There does not appear to be any statistically significant correlation between the two series.

In more recent years, since unemployment went up and down but inflation didn't go far, the Phillips curve seemed "flat,"

the Phillips curve was very flat for the 20-plus years before the pandemic,

You can see this in the decline of unemployment through 2020, as marked, with no change in inflation. Then, unemployment surged in 2021, again with no deflation.  2009 was the last time there was any slope at all to the Phillips curve.

But is it "flat" -- a stable, exploitable, flat relationship -- or is it just a stretched out "blob", two series with no stable relationship, one of which just got stable?

In any case, as unemployment went back down to 3.5 percent in 2022, inflation surged. You can forgive the Fed a bit: We had 3.5% unemployment with no inflation in 2020, why should we worry about 3.5% unemployment in 2022? I think the answer is, because inflation is driven by a whole lot more than unemployment -- stop focusing on labor markets!

A flat curve, if it is a curve, is depressing news:

Based on the flatness of the Phillips curve in recent decades, some commentators argued that unemployment would have to rise dramatically to bring inflation back down to 2 percent.

At best, we retrace the curve back to 2021 unemployment. But (I'll keep harping on this), note the focus on the error-free Phillips curve as if it is the entire economic model.

Waller views the new Phillips curve as a "curve," that has become steeper, and cites confirming evidence that prices are changing more often and thus becoming more flexible.

... considering the data for 2021... the Phillips curve suddenly looked relatively steep.. since January 2022, the Phillips curve is essentially vertical: The unemployment rate has hovered around 3.6 percent, and inflation has varied from 7 percent (in June) to 5.3 percent (in December).

Waller concludes

A steep Phillips curve means inflation can be brought down quickly with relatively little pain in terms of higher unemployment. Recent data are consistent with this story.

Isn't that nice -- from horizontal to vertical all on its own, and in the latest data points inflation going straight down.

Still, perhaps the right answer is that this is still a cloud of coincidence and not the central, causal, structural relationship with which to think about how interest rates affect inflation.

If only I had a better model of inflation dynamics...

1. re: the Global Slack Hypothesis

"The problem is that the standard, domestic, accelerationist Phillips curve no longer captures the inflation process. Figure 1 shows the US output gap vs. change in core PCE inflation over four periods: 1960-1973, 1973-1990, 1990s, 2000-2016. We have used quarterly data from the CBO and FRED. We see that the model worked from 1960 through 1990, weakened in the 1990s, and then disappeared after 2000.

The competing explanation is the Global Slack Hypothesis which says that due to the integration of global markets, what now drives inflation is not domestic slack but rather global slack. Due to competition from global rivals, domestic producers in the tradable sector cannot raise prices when the domestic labor market tightens and wage pressures build. Instead, they either rebalance their global supply chains and off-shore production; or they lose business to their foreign rivals. In either case, domestic inflation is determined as much by global slack as by domestic slack.

The evidence is mounting that the second explanation is the right one. What is especially compelling is the evidence that global slack is statistically significant in ALL countries for which data is available while domestic slack is significant is NONE since 2000. See Figure 3. The last column corresponds to global slack (“foreign gap”); no stars means the variable is not significant; three means it is significant at the 1 percent level."

source: https://policytensor.com/2016/12/17/global-slack-us-inflation-and-the-feds-policy-error/

2. Thanks John, enjoyed that you highlighted the role of expectations. Since I am unpersuaded about the conclusions of your FTPL, but understand that it is coming from the math, I always thought that the problem must be the structure of expectations. In some sense I read your pieces as very strong argument against long-run iterative forward looking expectations.

3. The NK-DSGE IS curve is a optimality condition, not a state equation. What the equation is telling us is that a necessary, but not sufficient, condition for realizing an optimal policy of maximizing the present value of the utility of consumption over time (finite or infinite) requires that the present rate of consumption equals the present value of expected future consumption one period ahead. The one period discount factor is a function of the social rate of time preference ( - ln(β)), the nominal rate of return on private investment expenditures, ln(Rₜ), the expected rate of inflation in the period ahead, πₜ₊₁, and reciprocal of the coefficient of risk avoidance, σ.

Confusion arises in two areas: (1) the assumption of small perturbation linearization (log-linear approximation), and (2) the assumption that consumption, C, equals national income, Y, (represented by nominal GDP). The second assumption gives rise to relation X = ln(Y) - ln(Y⁺), where Y⁺ is the theoretical nominal output capacity of the economy in the absence of frictions (of which 'sticky prices' is but one). X is styled the 'output gap'. The first assumption allows the modeller to study small perturbations from steady-state trend using a linear model of the system. Linear mathematics is defined by the applicability of the superposition principle. Real life is non-linear (i.e., superposition principle is inapplicable). When the modeller adopts a log-linear model of a non-linear system, she is making a statement about the region of convergence (application) of the model. When the linear model of the system becomes unstable (e.g., a pos. eigenvalue in a linear 2nd order dynamic system—in control engineering terminology, a pole in the left-hand half of the s-plane), linearity is violated (the perturbation is no longer ‘small’) and transformation to a new steady-state is indicated unless the non-linear system on which the log-linear approximation is based is itself unstable (in which case a steady-state may be unattainable).

Linearization is not a necessary condition for solving economic models. See, for example, Karl Shell (1968) Application of Pontryagin's Maximum Principle to Economics--Notes based on lectures given at the International Summer School for Mathematical Systems Theory and Economics, Villa Monastero, Varenna, Italy, June 1-12, 1967. Cambridge: Mass. Inst. Tech.

But, the NK-DSGE model as a linear model thus must rely on small perturbation theory to approximate the underlying socio-econ. system. The derivation of the log-linear IS-curve (equation) is straight-forward. With the exception of the assumption that C = Y (violated in the case of a large open economy such as the U.S. economy), the IS-curve does not require the assumption of small perturbation linearization for its applicability. It does require the assumption of a “representative household” economy (not applicable to the U.S. economy).

The Phillips Curve is not a dynamic equation of state. If it has any applicability at all, it is as a first order necessary, but not sufficient, condition of optimality. Then it states that on the optimal path, the current period rate of price inflation equals the discounted rate of price inflation one period ahead plus the current period scaled output gap (X). It is not an equilibrium condition. Off the optimal path (policy), the NK-DSGE PC equation does not hold.

What you’re struggling with is the determination of the optimal policy for the model economy as a function of time. The NK-DSGE IS-curve and PC-curve provide the necessary first-order conditions for optimality. Missing are the sufficient conditions for optimality, and the dynamic state equations which govern how the state transforms through time. The interest rate is exogenous to the model. The forward period expectations are endogenous, but unmodelled. C = Y should be understood as δC = δY, i.e., small perturbations consistent with the (log-)linear model assumption.

4. you can't think of interest rates alone. If you have a Federal Reserve printing $9 trillion Dollars for Bankers, Federal Government printing$2 Trillion a year....The Federal Reserve PRINTED $400 Billion in March alone. They can raise interest rates to 20%....but if they PRINT money to accelerate the economy...it will make little NO DIFFERENCE! You have to factor in profligate spending Didn't the Federal Reserve and Treasury....just BACK EVERY BAD Bank and Loan? Banks pay for the FDIC guarantee account holders for$250,000....but if you are a bank run by Democrats....you have an UNLIMITED guarantee. Try factoring those into inflation...where profits are private and losses are covered socially unlimited? Infinite is a BIG Number! Capitalism should have a RISK...that RISK is gone for politically connected!

5. Great discussion. Perhaps the real problem is the lack of a dynamic/equilibrium model of how the Federal Reserve Board will behave. After all, the past data include Fed actions—some quite unusual, such as under Paul Volcker—and arguably these change the macro dynamics, so rational expectations require an explicit representation.

6. It can be shown that the change in the rate of inflation, period to period, is given by Δπₜ = ( β + σ·κ )·𝔼ₜ{Δπₜ₊₁} + κ·𝔼ₜ{Δxₜ₊₁} – σ·κ·Δiₜ , where, e.g., Δπₜ = πₜ – πₜ₋₁ , Δπₜ₊₁ = πₜ₊₁ – πₜ , Δxₜ₊₁ = xₜ₊₁ – xₜ , and Δiₜ = iₜ – iₜ₋₁ . This formulation provides a better 'sense' or 'feel' for the dynamics of the model than the usual static Euler equation formulae. The output gap (IS-eqn.), to continue the reformulation, becomes Δxₜ = σ·𝔼ₜ{Δπₜ₊₁} + 𝔼ₜ{Δxₜ₊₁} – σ·Δiₜ . From this it is seen that it is not necessary for iₜ > 𝔼ₜ{πₜ₊₁} in order for the FOMC's policy to affect πₜ and xₜ . The change in policy is sufficient. And, as long as Δiₜ > 0, for t = 1, 2, ..., n, ... , πₜ and xₜ and 𝔼ₜ{πₜ₊₁} and 𝔼ₜ{xₜ₊₁} will be adjusting, ceteris paribus.

The Latin phrase, ceteris paribus, is key to this observation because when the fiscal authority (Congress, and/or the administration) is pursuing an expansionary policy (e.g., continual or increasing deficit spending) then 𝔼ₜ{πₜ₊₁} and 𝔼ₜ{xₜ₊₁} will be influenced by the trajectory of fiscal policy. It is through the expectations, 𝔼ₜ{πₜ₊₁} and 𝔼ₜ{xₜ₊₁}, t = 1, 2, ..., n, ..., that the feed-back loop from fiscal policy is effected in the NK-DSGE model. The basic premise of the FTPL is that primary deficits affect the price level through gov't borrowing; gov't borrowing is tantamount to increasing the money supply unless the gov't commits to producing primary surpluses in the future to repay the monies borrowed in an earlier period, and in the current and future periods. For FOMC policy to be effective, fiscal policy must be neutral. If fiscal policy is expansionary during a period when FOMC policy is contractionary, then Δiₜ > 0, must be positive for longer in order for the FOMC to realize its goal of price stability. The interplay is the key to the future evolution of πₜ and xₜ .

One of the most difficult tasks in predicting the future course of the modelled economy (surrogate for the real economy) is determining 𝔼ₜ{Δπₜ₊₁} and 𝔼ₜ{Δxₜ₊₁}. The FOMC's models include sub-routines to estimate these two factors endogenously. The NK-DSGE 2-equation model used in the article omits these two endogenous variables, relegating these variables to the status of exogenous inputs. Since the two equations are merely first-order necessary conditions, the theory does not need to consider the expectations provided that is understood that the expectations are to be supplied by the modeller later (which never comes, almost surely).

1. no expectations are not exogenous, they're rational which means endogenous. Look at equation 7 of the expectations and neutrality paper, if you can somehow prevent the expected future inflation term on the left from changing then you get that i and pi move in opposite directions. It's fiscal policy that does, or does not, fix that value. The left side of equation 7 is not exogenously fixed or it would always be the case that i and pi moved in opposite directions.

2. My remarks did state that, but without reference to the "expectations and neutrality paper".

3. The NK-DSGE 2-equation model used in the article omits these two endogenous variables, relegating these variables to the status of exogenous inputs.

Actually your remarks stated the exact opposite.

4. I suggest you re-read the sentence you quote above. It doesn't support your assertion. Here is the sentence with emphasis added
"The NK-DSGE 2-equation model used in the article omits THESE TWO ENDOGENOUS VARIABLES, relegating these variables to the STATUS of EXOGENOUS INPUTS." (UPPER CASE used for emphasis).

5. Yes that's what you said, but the article does no such thing. The article is assuming rational expectations.

6. Rational expectations are the ne plus ultra of new Keynesian economics. But, the devil is in the details. How are rational expectations formed? Whose rational expectations are at the heart of the nK-DSGE 2-equation model? It would be the expectations of the representative household, would it not? Of course it would, because that household along with the myriad imperfect competitive firms' owners who set prices according to the theoretical model underlying the Phillips curve, determine their expectations rationally (Lucas critique)--but not within the model. For a variable to be endogenous, the variable must be determined within the model. If a variable is not determined within the model, that variable is exogenous. Variables 𝔼ₜ{πₜ₊₁} and 𝔼ₜ{xₜ₊₁} are not determined within the model. Consequently, we must consider them to be exogenous. John makes the assumption that 𝔼ₜ{πₜ₊₁} and 𝔼ₜ{xₜ₊₁} can be replaced by πₜ₊₁ and xₜ₊₁ plus terms representing the forecasting errors, 𝔼ₜ{πₜ₊₁} – πₜ₊₁ and 𝔼ₜ{xₜ₊₁} – xₜ₊₁. But he goes further -- he assumes that the forecasting errors are distributed in probability ~N(0,s_π) and ~N(0,s_x), resp. Torsten Zlok has demolished this assumption by observing the forecasting errors of professional economic forecasters in the periodic surveys published in the U.S. are grossly off in a systematic way. There are other issues with the assumptions as well, but those are technical and not germane to this discussion.

7. "Variables 𝔼ₜ{πₜ₊₁} and 𝔼ₜ{xₜ₊₁} are not determined within the model."

yeah, they are.

Furthermore, while I always enjoyed seeing Torsten when he was at DB I don't think he's demolished any assumptions, one of the funny and often ignored part of rational expectations are that they're taken with the respect to the risk neutral measure. Showing a bias to the physical measure doesn't actually test anything at all.

8. "Variables 𝔼ₜ{πₜ₊₁} and 𝔼ₜ{xₜ₊₁} are not determined within the model."

"yeah, they are."

Then point to the equation in the model that determines the variables 𝔼ₜ{πₜ₊₁} and 𝔼ₜ{xₜ₊₁}. If those variables are not determined within the model then those variables are exogenous to the model and not influenced by the model's dynamics. That's a standard definition, by the way.

9. Equation (7) of the paper "Expectations and the Neutrality of Interest Rates", June 7, 2023, is predicated on several assumptions of which the assumption that 𝔼ₜ{xₜ₊₁} = 0, on page 3, is a strong assumption, i.e., it is a very restrictive assumption. That assumption states that the output gap in period t + 1 is zero for all t > 0. This assumption lets John proceed to Eqn. (7) which you assert demonstrates that 𝔼ₜ{πₜ₊₁} is determined endogenously.

This is an inference that is materially wrong. Setting 𝔼ₜ{xₜ₊₁} to zero, does not make 𝔼ₜ{πₜ₊₁} an endogenous variable in the NK-DSGE model. 𝔼ₜ{πₜ₊₁} is not determined by Eqn. (7) in the general case. It is doubtful that it is determined by Eqn. (7) in the special case when 𝔼ₜ{xₜ₊₁} = 0.

The risk-neutral measure you refer to is limited to applications in financial mathematics. It is not found in macroeconomics. This can be seen by replacing the iso-elastic utility function that is found in the integrand of the cost function of the representative household used to derive the NK-IS equation (Eqn. (1) of John's paper, on p. 2). Eqn. (1) states, xₜ = 𝔼ₜ{xₜ₊₁} – σ·( iₜ – 𝔼ₜ{πₜ₊₁} – ρ). Note: I have added back the time preference rate of discount, ρ, to Eqn. (1) that John has dropped informally. The elasticity of intertemporal substitution is represented by the parameter σ. It is assumed that σ is positive and bounded. This tells us that the representative householder is not a risk-neutral investor. If the representative householder is a risk-neutral investor, then his utility function is linear, i.e., U(x) = a x - b, and the elasticity of intertemporal substitution is infinite, i.e., σ = ∞. But, we know from experience that the output gap is always bounded, i.e., |xₜ| < ∞, and so too is |𝔼ₜ{xₜ₊₁}| < ∞. Therefore, it follows that ( iₜ – 𝔼ₜ{πₜ₊₁} – ρ) = 0 when σ = ∞.

This gives us a Martingale, i.e., 𝔼ₜ{xₜ₊₁} = xₜ , when the representative householder is risk-neutral. We can then extend this result to the general price level and state that when the representative household is risk-neutral, 𝔼ₜ{Pₜ₊₁} = Pₜ , a Martingale that you will recognize from financial mathematics. If 𝔼ₜ{Pₜ₊₁} = Pₜ , then 𝔼ₜ{πₜ₊₁} = πₜ which is also a Martingale. This then is your connexion to the risk-neutral measure.

But what is the real rate of interest, r , in this case? Use the Fisher equation to find that ( iₜ – 𝔼ₜ{πₜ₊₁} ) = r = ρ. In other words, the risk-free real rate of interest equals the time preference rate of discount. Since the central bank cannot determine the value of time preference rate of discount, ρ , through open-market operations, it cannot determine the real rate of interest, r , when the representative householder is a risk-neutral investor. But, this is a restrictive case, and does not hold in general. Macroeconomics does not assume that the representative householder is risk-neutral, but assumes that he is risk-averse, i.e., σ < ∞ and strictly bounded. In the general case, 𝔼ₜ{xₜ₊₁} and 𝔼ₜ{πₜ₊₁} are exogenous variables, notwithstanding John's lead up to Eqn. (7) in his June 2023 revision of his paper.

7. That change in demand pushes down the price level today, and hence current inflation, and it pushes up the expected price level next period and thus expected future inflation.

A higher real rate reduces demand today yes, but does it automatically push up demand next period? Surely demand next period will depend on next period's real interest rate, would it not?

8. I did research on the Phillips Curve with Dick Lipsey and Chris Archibald in 1966, and was to be credited as a co-author. Alas, that didn't happen. But I still think that Lipsey is a great guy, going strong at 96. Michael Cunningham

9. Gov. Chris Waller has fallen into the same 'trap' (if you will) of imputing to the central bank the ability to trade-off inflation against employment, and vice-versa. The power law relationship between changes in the wage rate and unemployment in England, described by A. W. Phillips in his 1958 paper, and briefly discussed by P. A. Samuelson and R. M. Solow in their paper “Analytical Aspects of Anti-Inflation Policy” [American Economic Review Papers and Proceedings (1960), 50(2), pp. 177-94] simply doesn't stand up to close scrutiny.

Phillips faired a power law curve having the form Y(X) + a = b X^c through the data he studied. X is the unemployment rate; Y(X) is the change in wage rates as a function of unemployment. It was a hand-fitted curve that has parameter values for a, b, and c of a = 0.90, b = 9.638, and c = -1.394. The negative value of the exponent c is what gives the Phillips Curve its attractiveness for central bankers. Samuelson and Solow reviewed Phillips's paper in their 1959 address to the AEA (publ. 1960) and made their own power law curve by fitting three points picked out of U.S. data measured from 1948-1957 finding (or, reading into the data) a negative value for the exponent, without doing a proper OLS regression on the data. Solow in later years contended that they always intended on revisiting the conclusions when they had an opportunity, but that the opportunity never arose.

If Phillips's power law model Y(X) + a = b X^c is fitted to U.S. monthly seasonally-adjusted data of Trimmed Mean PCE Inflation Rate (Percent Change from Year Ago, Monthly, Seasonally Adjusted), Y, versus Unemployment Rate (Percent, Monthly, Seasonally Adjusted), X, for the period 1978-Jan through 2023-April, one finds the following parameter values: a = free choice, e.g., 0; b = 2.014; and c = +0.1544. Note the positive sign for the exponent c. The explanatory power, r^2, is low (0.007), F-stat = 4.306, deg. of freedom = 542; T-statistics for b and c are 5.24, and 2.075, respectively. Data source: https://fred.stlouisfed.org/graph/?g=162Vf.

Gov. Waller's observations about the 2020-2023 data of inflation vs. unemployment is readily explained: (a) the inflation rate rise is the effect of pandemic fiscal stimulus (deficit funding of transfers to households and businesses during 2020) coupled with the Biden administration's legislative stimulus deficit spending during 2021-2, coupled with (b) the FOMC expansionary monetization of the fiscal deficit funding during 2020 and the accommodative FOMC policy change adopted by the Fed on 8-27-2020; (c) the unemployment rate jumped to 10% during Q1-2020 and then as the economy adapted to the pandemic conditions and reopened in Q4-2020 and Q1-2021, the unemployment rate declined gradually declined as businesses resumed hiring and discouraged unemployment workers left the work-force. A similar pattern occurred following the 2008-Q4: 2009-H2 financial crisis, and this is seen in the plots of inflation rate vs. unemployment in subsequent periods. But outside those periods, the power law relationship first described by Phillips is absent in the data. Whatever the reason behind absence (e.g., “re: the Global Slack Hypothesis”, Fares, A., 6/9/2023 this blog article), the fact remains that there is no leverage available to the central banker to trade-off inflation for employment. To think so is a major misapprehension, and a serious distraction to the work of the Fed/FOMC going forward.

10. Nice work Dr. Cochran.

Intuitively there is also probably some linkage in corporations on the microeconomic side. That is, as a 3% loan received 2 years ago rolls over at say 7%, a company with fairly inelastic goods pricing can raise prices to pass along the expense delta. Adds to inflation at the margin. Who is researching this factor?

- Ross M.

11. Two papers from 1999 shed important light on the New Keynesian 'Phillips Curve' model:
(1) Gali, J., and M. Gertler, "Inflation dynamics: A structural econometric analysis" (1999), J. Monetary Econ. (28 pp.)
https://www.sciencedirect.com/science/article/pii/S0304393299000239
and,
(2) Clarida, R., J. Gali, and M. Gertler, "The Science of Monetary Policy: A New Keynesian Perspective" (1999). Cambridge, MA: National Bureau of Economic Research. Working paper 7147. (105 pp.)
https://www.nber.org/system/files/working_papers/w7147/w7147.pdf

Each paper starts with the derivation of the 'Phillips Curve' and then discusses why that particular derivation can be improved by considering lagging indicators and current indications, and applying econometric methods to estimate the unknown parameters. In the second paper, the authors examine whether the central bank can manage the rate of inflation by manipulating the bank rate of interest. The authors conclude that the central bank can do so, if and only if, inflation is 'cost-push' inflation, but that it cannot change expectations in future periods and, as a consequence, the central bank cannot affect demand-pull inflation through manipulation of the bank rate.

Of the two papers, the second is arguably the more relevant paper to this blog's postulates.

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