A theory of the price level is a theory of inflation, but not vice versa

Because of calculus.

Suppose you start with a theory of the price level P(t). Differentiate that theory with respect to time and you get a theory of the inflation rate Pdot(t). If P(t) jumps at some time, which it might in some cases, inflation is infinite at those times, but that's interesting to know.

Suppose you start with a theory of the inflation rate Pdot(t). Integrate that theory with respect to time, and you don't get a theory of the price level. First because you don't have a theory of the constant of integration. Second, because if the price level jumps, your theory will only tell you that it jumps and won't tell you how high it jumps.

(Strictly, inflation is Pdot(t)/P(t), but never mind.)

Microeconomists tell us theories about the price level of apples relative to bananas. If you asked a microeconomist for a theory of the inflation rate of apples relative to bananas, he would probably look at you funny, and then tell you to take the derivative of his theory with respect to time.

Some macroeconomists tell us theories about the price level of apples and bananas relative to money. Other macroeconomists tell us theories about the inflation rate of apples and bananas relative to money. The first group of macroeconomists can always differentiate their theories with respect to time, if they wanted to, and compare their theories to the theories of the second group of macroeconomists. But they would lose some of the information content of their theories if they did so. The second group of macroeconomists cannot integrate their theories with respect to time to compare their theories to the theories of the first group of macroeconomists.

It would be very strange if the second group of macroeconomists said that the first group of macroeconomists were doing it wrong by belonging to the first group. And microeconomists would think it strange too.

The second group of macroeconomists might argue that the price level doesn't matter, but the inflation rate does matter, so any information-content that is lost by a theory that belongs to the second group doesn't matter. I don't think this is right if the price level jumps. It will usually matter how high it jumps, in which case knowing simply that it jumps (inflation is infinite) is not enough. But if it doesn't jump, the second group of macroeconomists do have a point here. Most theories of the first group of macroeconomists have the property of monetary neutrality, in some sense. Which is another way of saying that the price level doesn't matter.

But even here, the fact that the price level doesn't matter is itself an interesting theoretical result, that we need to understand. And I can imagine theories in which the price level does matter. For example, if the good that is used as money has physical properties that do matter, then money would not be neutral in any sense, and the price level would matter. If we used cows as money, for example. We need to understand the differences between using paper money and cow money. And even if we do use paper money rather than cow money, the students need to understand why the price level doesn't matter. "Look students: under these circumstances the price level doesn't matter! If prices were higher that doesn't mean we would be poorer!" Draw the LRAS curve, in {P,Y} space, and explain why it is vertical. That's an important lesson.

The second group of macroeconomists might argue that in some cases the price level is indeterminate, and only the inflation rate can be explained by theory. For example, if the central bank targets inflation by following a Taylor Rule, and if the price level never jumps, and if money is electrons or paper and so neutral in some sense, then the inflation rate can be explained but the price level cannot. You just have to take the constant of integration from history, without explaining it.

But even here, the fact that the price level is indeterminate is an interesting fact, if it is a fact. Draw the AD curve, in {P,Y} space, and show the students that it is vertical. That too is an important lesson. If prices weren't sticky, and if the LRAS curve is vertical too, they would almost always either explode or implode. And if they are sticky, the price level will have a unit root, and long term expectations of the price level will be unanchored. Wicksell really did teach us something important, that the students won't know.

You can always start in {P,Y} space then move to {Pdot,Y} space, if it's useful to do so. You can't do the reverse.

Simon Wren-Lewis' post challenged me and made me think. This is what I thought.

Update: and our students need to be exposed to fallacies too, so we can show them (OK, try to show them) that they are fallacies. Yes, Simon is right that weaker students will think that AD curves slope down for the same reason that micro demand curves slope down, and that AS curves slope up for the same reason that micro supply curves slope up. But they will think this anyway, regardless of what space we use. Switching to {Pdot,Y} space simply evades facing this problem. We need to draw the AD and AS curves in {P,Y} space, and explain that they may not slope the same way as in micro, and that even if they do slope the same way as in micro, they (usually) do so for very different reasons. ("Usually", because the AD curve for a small open economy under fixed exchange rates slopes down for much the same reason that micro demand curves slope down.)

Comments

[Doug M]

Price changes are always discontinuous. One day, Mr. Shopkeeper decides to raise his prices and the price of a wigit goes from $1 to $1.25. It was never $1.10 or 1.15.

strictly speaking dP/dt is always either 0 or undefinded. delta P/ delta T is defined for greater time differentials. We can smooth the price function to make it apear more or less continuous, and then derive dP/dt. And we can build a model of inflation and integrate it to a smoothed price fuction (assuming we know our intitial conditions).

Philisophically, do we care more about the state of affairs or the dynamics?

Most (theroritical) models attempt to solve for the equilibria, and then assume that there exists a path from one equilbrium to the next. They care about the level of prices. Most of the "Wall Street" economists, have no clue what the equilbrium is, but they do have a sense -- or sell their skills as having a sense -- that they can model the direction that prices, and output will move. They care about dynamics. They care about the rate of change. They care about inflation.

[Sam H]

Question on the AD/AS picture for Canada... Scott Sumner wrote a post in March called "Did a supply shock cause the Canadian recession? And did tight money make it worse?" He argued that plunging exports to the US can only be interpreted as a negative supply shock. AS shifted to the left should've meant a higher rate of price level growth, but the BoC's 2% inflation target forced a negative AD shift as well (tighter money via higher money demand), leading to lower real output than the pure supply shock.

In reference to your earlier posts puzzling over the disinflation Canada's just now getting, is that slower price level growth not consistent with Sumner's theory, bought on by reviving exports and the AS shifting towards its initial position, moving Canada's economy down along AD? Thanks for your insight.

[Sam H]

Canada-wide, Wood Products were some of the exports that were hit the hardest at the onset of the recession. Thus the AS shock was greater in some provinces than others - namely the ones with the highest share of wood-product exports. Given what I conjectured above, now that housing starts are up and the US construction market is recovering (i.e. AS is moving right) you would expect the lower price level growth particularly in those provinces. Not surprisingly then, BC and NB, both heavy wood exporters, have experienced the lowest price growth over the last 12 months. Model confirmed?

[Scott Sumner]

This is probably a stupid question, but I'm a bit confused as to why so many people want to draw AD curves conditional on some sort of policy regime, like inflation targeting. I use AS/AD to show policy failures, events such as the Depression. But if AD reflects a policy regime, then you can't use AS/AD to show demand-side policy failures, can you?

[nottrampis]

So many great articles from each of you in so many different categories. This blog is a goldmine.
Maybe I am predjudiced because the man actor and actress from Castle are both from Canada!

I hope you are gaining more readers from downunder!

Keep it up please

[Tom Brown]

Mathematicians solved the problem of differentiating across a step function long ago, and engineers (at least) have learned how to use that:

differential of x wrt t with {x(t) = 0, t < 0; x(t) = a, t >= 0} is {x'(t) = a*dirac_delta(t)}

The good old dirac delta function! Just scale it for the size of the step (and shift it for the position)! As for the rest... why won't initial conditions do? That's the usual method for solving x'(t) = A*x(t) + B*u(t). As long as you have x(0) and u(t) you're good!

As an aside, you can also try Lebesgue integration instead of Reimann if normal integration doesn't work. Lebesgue lets you integrate highly discontinuous functions such as the "grass" function: defined as 1 at every rational number and 0 otherwise (there's an infinite number of "spikes" on any open interval, so it's pretty discontinuous!).

[Edward Lambert]

The demand curve slopes down because a higher price level implies less aggregate output demanded. What is the mechanism behind this? The strength of demand is based on labor income to buy finished goods. So let's look at labor share.

labor share = unit labor costs/price level

As the price level rises, the balance of labor income declines. This is not difficult, but you are missing it, because you don't have the model.
The supply curve slopes up. According to my model, FWIW, capacity utilization, which increases with output, is measured in relation to labor share.

real output = potential RGDP + a * (capacity utilization - labor share)/labor share

As price level rises holding other variables steady, labor share falls, which increases real output. Do the other variables hold steady? No... That is why the AS curve shifts right when there is spare capacity and when there is sudden jump in demand.

[Frank Restly]

But suppose you start with a theory of both the inflation rate Pdot(t) / P(t) and the price change rate Pdot(t)

Pdot(t) = f(t)
P(t) - P(0) = Integral ( f(t) ) - F(0)

Pdot(t) / P(t) = g(t)
ln ( P(t) ) - ln ( P(0) ) = Integral ( g(t) ) - G(0)

P(0) = P(t) - Integral ( f(t) ) + F(0) = P(t) / exp [ G(0) - Integral ( g(t) ) ]

P(t) = [ Integral ( f(t) ) - F(0) ] / [ 1 - exp [ Integral ( g(t) ) - G(0) ]

If we know what f(t) is and we know what g(t) is, we can solve for P(t) directly.

[Dan Kervick]

But if it doesn't jump, the second group of macroeconomists do have a point here. Most theories of the first group of macroeconomists have the property of monetary neutrality, in some sense. Which is another way of saying that the price level doesn't matter.

Right. Isn't this basically what the physicists gauge symmetry"?

By the way, to go from a theory of inflation to a theory of the price level, do you really need a theory of the constant of integration? You just need to know the value of that constant at any one particular time in the domain of times over which you are integrating and the rest follows.

Also, if we used a primary medium of exchange that was itself a commodity, and whose inherent value as a commodity was a factor in its exchange value, then I would suggest that what economists should seek is a theory of the price level and inflation that uses a fixed measure of price against which changes in the prices of all commodities including the primary medium of exchange. If we used chocolate bars as money, we would want to be able to study the difference between changes in the price level as measured in chocolate bars that were due to an increase in the production and circulating number of chocolate bars, and changes in price level that were due to changes in the inherent desirability of chocolate.

[Patrick]

"Isn't this basically what the physicists gauge symmetry"

No. Macro models don't look much like a field theory, and gauge symmetry is a property of field theories. Besides, the price level is observable, unlike e.g. a potential where only differences are measurable but the potential is a real thing and matters - it just can't be measured.

[Nick Rowe]

Doug M: "Price changes are always discontinuous."

That might be true for some individual prices (but even there stock prices are as close to continuous as we can see). But if there are lots of individual prices the aggregate price level might be very close to continuous. Depends if we model in continuous or discrete time of course. But what matters here is that models may or may not allow for jumps.

"Philisophically, do we care more about the state of affairs or the dynamics?"

Let's translate that into "Economically, do we care about the price level or the inflation rate?"

Sam H: I think that there's a lot of sense to Scott's interpretation. Whether it fully fits, and fully explains, the facts when we look more carefully, is I think still an open question. Do *all* the facts fit Scott's interpretation? Are the effects big enough to explain everything? Dunno.

[Nick Rowe]

Scott: I've been thinking over your question (which means it's not *obviously* stupid!).

When we draw an AD curve (or write down an AD function) we must be holding *something* constant. If we don't hold monetary policy constant, what do we hold constant instead? You might answer "NGDP" but then your AD curve will shift if the central bank doesn't target NGDP. But some other economist might instead hold M constant, or i constant, or whatever. And there wouldn't seem to be any way to say which assumption is best.

" But if AD reflects a policy regime, then you can't use AS/AD to show demand-side policy failures, can you?"

I think you can.

Let's compare NGDP to Price level targeting: the first has a rectangular hyperbola AD and the second a flat AD curve. If AS shifts, they give different results.

Or, let's compare NGDP to M targeting. The first AD curve only shifts if the CB makes a mistake and misses its NGDP target. The second shifts if the CB makes a (different) mistake and misses its M target, or if any one of a load of other things changes, even if the CB predicts those changes and keeps M on target. Same if we compare NGDP targeting to fixed exchange rates. All sorts of things will shift the second AD curve, but the first shifts only if the CB makes a mistake and misses its target.

[Nick Rowe]

Tom Brown: I'm afraid your comment went way over my (math-challenged) head! But probably other people reading this will find it a useful comment.

Edward: "labor share = unit labor costs/price level

As the price level rises, the balance of labor income declines. This is not difficult, but you are missing it, because you don't have the model."

Of course I don't have a model *in this post*, because this post is not about my model.

And what you have said there is not a model. It's an assertion based on some accounting plus some hidden assumptions. (like assuming W is constant, presumably! Plus some assumption about the relative propensities to spend out of wage vs non-wage income).

(Please don't reply with yet more links to your model. I have already seen those links everywhere in the blogosphere.)

Frank: you lost me, as usual.

Dan: "Right. Isn't this basically what the physicists gauge symmetry"?"

Dunno. My physics stopped 40 years ago, in high school. But I sort of doubt it's the same thing.

" You just need to know the value of that constant at any one particular time in the domain of times over which you are integrating and the rest follows."

Yes. But if we want to *explain* P(t), from Pdot(t), we would also want to *explain* (not just observe) P(t-1). So we might get into an infinite regress, going further and further back into history. Now, in some cases, you might say that that infinite regress is inevitable, because history might matter permanently, so that where we are today really does depend on historical accident in the very distant past. But even then, if that is true, that is an interesting fact to know, and to explain why that is true. E.g. is it really true that it is purely arbitrary which particular words we use to mean various things, so it's pure historical accident, and we can only explain why words change meaning over time? That seems to me like an interesting question that we might want to try to answer, or if we know the answer, explain it to our students.

"...what economists should seek is a theory of the price level and inflation that uses a fixed measure of price against which changes in the prices of all commodities including the primary medium of exchange."

A lovely philosophical question (even if off-topic here, I think).

Most economists will say that's a purely metaphysical quest. All we ever observe are exchange ratios between different goods (including money). A second group would say that utility is the ultimate measure of value, and stat a long argument on whether utility is cardinal (in various senses), whether interpersonal comparisons of utility are meaningful, and whether utility is directly measurable (surveys etc.). A third group would assert that labour is the ultimate measure of value, and start yet another argument about why labour is privileged over other scarce resources, and about interpersonal comparisons of labour, and whether it matters if some types of labour give disutility while others are fun, etc.

But I'm staying out of that one here, other than to say that there is nothing about money that would make it a good candidate for being the ultimate measure of value, even if that concept is meaningful. And if money is something like paper, or electrons, you could even argue that money prices would be the worst possible candidate, precisely because they are so irrelevant. The quantity Theory of Money (and its sister theory the neutrality of money) are based on the idea that monetary magnitudes are irrelevant, expect as ratios to each other.

[Joseph]

As Nick noted in the first sentence, the general problem with integrating from a pure theory of inflation to theory of the price level is that your theory of inflation does not supply enough constraints on your theory of the price level. A theory of inflation wouldn't provide any suitable boundary conditions for the price level other than possible either 0 or infinity at time plus/minus infinity, and only then if you can show that the average inflation rate is nonzero over sufficiently long periods, which is not very useful. If one were to utilise a delta function, as per Tom's suggestion, then you need either a suitable second boundary condition or some other condition to impose. To go from your theory of inflation to a theory of the price level you need to make some set of assumptions on the properties of the price level or use historical data to calibrate your model, but that has the result of making it specific to the data you used instead of a general theory.

[Frank Restly]

Nick,

Suppose we have a data set of inflation rates for a good measured over time:
g(t) = { 20%, 8.333%, -23.0777%, 40%, etc. } - All on annualized basis

And suppose we have a data set of price level changes for the same good measured over time:
f(t) = { $100/year, $50/year, $-150/year, $200/year }

Both are dynamic measurements of price change. Using both data sets, we can calculate the price level:

P(1) - P(0) = $100, P(1) = P(0) + $100

( P(1) - P(0) ) / P(0) = 0.20
( P(0) + $100 - P(0) ) / P(0) = 0.2
$100 / P(0) = 0.2
P(0) = $100 / 0.2 = $500.00

[rsj]

The price level is not a value of interest, nor is it an economic quantity that should be explained by a theory.

Countries can and do renumber, lowering the overal price level by a factor. One nation, which starts out later than another, might have a lower price level even if both have historically had the same inflation rate. Economically the two nations are the same. There is no value in trying to predict the price level above and beyond being able to predict inflation.

Physics focuses on velocity and acceleration, because the laws of physics don't involve position. Position is whatever it happens to be as a result of the actual laws the system running for some period of time. This is a virtue, not a curse.

The price level doesn't appear in any economic laws (ignoring crude monetarism) for the same reason, nor should it appear, because in a dynamic economy people react and decide whether to raise or lower prices, whether to charge more or less than a competitor. The laws have to be about how changes in price create changes in quantity sold. If the driver of economic decision making is competition and trade-offs, then whatever laws come out of that are going to be formulated in terms of derivates of gross quantities being supplied and derivatives of prices. Then, people who are concerned about the price level can go back to when that currency regime started, making adjustments for all splits along the way, etc.

[Nick Rowe]

rsj: "The price level is not a value of interest, nor is it an economic quantity that should be explained by a theory."

Compare David Hume:

"It was a shrewd observation of ANACHARSIS the SCYTHIAN, who had
never seen money in his own country, that gold and silver seemed to
him of no use to the GREEKS, but to assist them in numeration and
arithmetic. It is indeed evident, that money is nothing but the
representation of labour and commodities, and serves only as a method
of rating or estimating them. Where coin is in greater plenty; as a
greater quantity of it is required to represent the same quantity of
goods; it can have no effect, either good or bad, taking a nation
within itself; any more than it would make an alteration on a
merchant's books, if, instead of the ARABIAN method of notation, which
requires few characters, he should make use of the ROMAN, which
requires a great many."

rsj again: "The price level doesn't appear in any economic laws (ignoring crude monetarism)..."

That is almost the exact inverse of the truth. Crude monetarism says precisely that we should formulate all economic laws without any reference to nominal variables, like the price level or quantity of money.

What you are striving towards is exactly the idea of the neutrality of money/quantity theory of money/economic irrelevance of the price level, that is the cornerstone of monetarism.

rsj again: "... making adjustments for all splits along the way, etc."

The quantity theory/neutrality of money is based on precisely that same idea that says that stock splits don't/shouldn't matter.

Welcome to The Club!

But first we have to explain to the students precisely why (and under what circumstances) the price level doesn't matter. (It wouldn't be true if we used cows as money, for example).

[Frank Restly]

RSJ,

"The laws have to be about how changes in price create changes in quantity sold."

Um, no. New goods are invented / created every day (see entertainment / software industry). The laws must allow for the first new good to be sold.

"Physics focuses on velocity and acceleration, because the laws of physics don't involve position."

In physics, the laws of motion rely on conservation of momentum / energy and those are unchanging. With monetary economics the only constraints are legal constraints agreed to by a voting public and those constraints can be changed at the ballot box.

[rsj]

Frank,

Then that is just a small change from zero. Seriously, this is not a hard concept and I didn't realize I would need to be arguing with Xeno here about the sufficiency of derivatives to describe changes in position.

[Frank Restly]

RSJ,

"Then that is just a small change from zero."

It may not be. In the electrical and electronics industry, many times a new product will have a high introductory price ( televisions, stereo systems, media players, phones ,etc.). Then as economies of scale take over, the price tends to fall. So that "small change" from zero actually will be the largest positive change in price that the good has.

I don't disagree with you that from a monetary policy standpoint, you should focus on price changes rather than price level. However, there is more to money than monetary policy.

[Dan Kervick]

I still thing the idea of gauge invariance is a very good analogy. Take any plausible model of a monetary economy evolving over time according to some combination of laws. Now multiply all nominal quantities in the model by some arbitrary constant. Won't the resulting model still satisfy all of the laws? Won't it in fact be a description of the exact same economy?

[Nick Rowe]

Dan: "gauge invariance" is a new concept for me, but I think I get it.

"Take any plausible model of a monetary economy evolving over time according to some combination of laws. Now multiply all nominal quantities in the model by some arbitrary constant. Won't the resulting model still satisfy all of the laws? Won't it in fact be a description of the exact same economy?"

EMPHATICALLY YES!

And that is the important insight that is what Hume was driving at, and that lies behind the Quantity Theory/Neutrality of Money.

But the QT/NM asserts a little more than that, and that's where it gets tricky. Because if we start in one of those possible world/economies, and *in real time* the central bank multiplies one of those nominal quantities by some arbitrary constant, will the economy traverse in finite time to that other possible world/economy, that had been there all along? If some of those nominal quantities are "sticky" it certainly won't do that traverse instantly. Or if it takes time for expectations of those nominal quantities to adjust it won't do the traverse instantly either. And if the initial conditions matter permanently, it won't ever do that traverse. Or if those combinations of laws have a multiplicity of solutions/equillibria, it might not traverse to the same one. And if you keep on repeating that experiment, by multiplying that same nominal quantity by the same constant every year, so it doubles and then redoubles again, and again, is it still neutral ("superneutrality")?

[Nick Rowe]

And BTW, your four sentences there are a beautifully clear way of putting it.

[Nick Rowe]

And when we draw the vertical LRAS curve, what we are trying to say is exactly what Dan said in those 4 sentences above. And when we draw the SRAS curve as non-vertical, what we are trying to say is that it won't work exactly and immediately if the central bank changes one of those nominal variables in real time.

All those posts where I tried to explain the Quantity Theory failed. In this one post where I was not trying to explain the Quantity Theory, rsj and Dan figure out the basic idea behind it by themselves. And then Dan states that idea more clearly than I ever did. I give up.

[Scott Sumner]

Nick, I see your point, but let me come at this from another perspective. Instead of focusing on policy, we focus on the "classical dichotomy" as taught in intro texts. There are nominal shocks and real shocks, and they have very different effects on the economy. You can consider an AD curve that is based on stable NGDP to be the curve for showing "nominal shocks" of whatever cause. Thus it might be an explicit policy failure, such as a big exogenous change in the money supply, or an implicit policy failure, such as not adjusting the money supply when V changes. Either way, these nominal shocks have effects on the economy that are clearly explained using a simple short run/long run framework. And then we also have a set of real shocks, which show up as shifts in the AS curve.

In contrast if you draw the AD curve as a horizontal then changes in the AD curve might reflect either real or nominal shocks, with vastly different implications. Thus it might be a policy failure such as 2009, when the Fed let prices fall unintentionally, or it might be a supply shock combined with a central bank policy of "flexible inflation targeting." In other words, the shift in the AD curve becomes very ambiguous, it might be a nominal shock, or it might be a real shock.

I admit that even NGDP is not perfect, but it's more likely to be the case that the move in NGDP represents a nominal shock that disequilibrates the labor market, than is the case with a movement in P. One can more easily imagine P moving when there is no nominal shock at all, no involuntary unemployment.

And the same for M. A change in M that merely offsets a change in V is not really a nominal shock. So I guess I'm saying that the best way to describe the AD curve is not to link it to central bank policy, but rather to draw it in such a way that if it doesn't shift the economy would not be hit by a nominal shock of the sort that disequilibrates the labor market.

[W. Peden]

Nick Rowe,

"All those posts where I tried to explain the Quantity Theory failed. In this one post where I was not trying to explain the Quantity Theory, rsj and Dan figure out the basic idea behind it by themselves. And then Dan states that idea more clearly than I ever did. I give up."

Perhaps it's something that can be seen, but not explained.

[Patrick]

Dan, not really. Gauge theory has a specific definition. There might be (tenuous) analogies but that's about it.

[Patrick]

Dan, you might enjoy the Wikipedia article about Noether's theorem.

[Edward Lambert]

Nick, Sorry about the links... sometimes there is so much to say, it is better to link than to write a big comment... So I will keep this short.

The equation I use for the Effective demand curve is this... (effective demand replaces aggregate demand.)

Price level of ED = (real GDP*effective unit labor costs)/(capacity utilization*(1-unemployment)*effective demand) - 1

Since effective demand is in the denominator... as effective demand decreases, price level rises. However, the price level is determined at effective demand = real GDP, thus we can reduce the equation.

Price level of ED = (effective unit labor costs)/(capacity utilization*(1-unemployment)) - 1

This curve crosses the real aggregate supply curve at the NAIRU point of factor utilization.
The difference between the price level of ED and inflation will determine the NGDP level of output.
I'll leave it at that...

[Nick Rowe]

W. Peden: "Perhaps it's something that can be seen, but not explained."

Or perhaps you are trying to be nice to me, when I am feeling disheartened about my teaching/explaining abilities!

Sometimes I think that we can never really teach anyone anything. They always have to figure it out by themselves. What is the "Socratic Method", anyway?

Or maybe you are right. There is something about the classical dichotomy between real vs nominal variables, that we just intuit, but can't really explain *why* it is true. God only knows. Here was one of my early attempts.

Scott: I'm still thinking through your most recent comment.

One point: if we draw the AD curve as horizontal, the most natural interpretation would be that the central bank is doing *strict* price level targeting. And then an AS shock would cause no change in P but a big change in Y. A monetary policy of "flexible inflation targeting" (or flexible price level targeting) isn't really a horizontal AD curve. It's more downward-sloping, at least in the short run, since the CB trades off P against Y.