Simple thoughts on NGDP = RGDP x P

There have been several recent posts in the blogospere arguing about the interpretation of graphs (for several countries, but you can see Canada's here) which show a big fall in NGDP (relative to trend) at the beginning of the 2008 recession. (The latest from Ryan Avent here; Tyler Cowen's collection of links here.)

Here are my simple thoughts.

1. NGDP = RGDP x P. In words: nominal GDP = real GDP times the GDP deflator (which is one way to measure the price level). That's an accounting identity, of course. It tells us how we use words. It doesn't tell us anything about the world. Simple "hat algebra" (differentiating that identity with respect to time) lets us re-write it as NGDP^ = RGDP^ + P^, which in words means: the growth rate of NGPD equals the growth rate of RGDP plus inflation. And if people's expectations are logically consistent, expected NGDP growth will equal expected RGDP growth plus expected inflation.

2. It is easier to measure NGDP (nominal growth) than it is to measure RGDP (real growth) and P (inflation). To say the same thing another way, it is not easy to figure out how to decompose nominal growth into real growth and inflation. Because it's hard to measure things like quality changes. In that sense only, NGDP is more "real" than RGDP.

3. But how people respond to NGDP growth (or expected NGDP growth) will usually depend on how much of that (expected) NGDP growth was (expected) real growth and how much was (expected) inflation. Elasticities with respect to RGDP (or real growth) may be different from elasticities with respect to P (or inflation). If expected NGDP growth falls, the amount by which that causes current NGDP to fall might depend on how much expected real growth falls vs expected inflation falls. Adding expected real growth plus expected inflation together may be a useful simplification, but is only strictly correct if those two elasticities are the same.

4. If we observe NGDP and RGDP both suddenly fall, as they did in many countries in 2008, we cannot say from that fact alone that RGDP would not have fallen as much as it did if the central bank had held NGDP constant (growing at trend). But anyone who denies that assertion would have to believe that P would have increased (relative to trend) by the same amount that RGDP fell (relative to trend) if the central bank had held NGDP constant (growing at trend). Is that belief plausible?

5. A real business cycle theorist, who believes that prices are perfectly flexible, and who believes that monetary policy only affects nominal variables like NGDP and P, and does not affect real variables like RGDP, would believe that to be plausible.

6. Does anyone else believe that is plausible?

7. Update. I used to think that inflation targeting would be reasonably successful in preventing recessions due to deficient aggregate demand. Looking at those graphs for NGDP and P caused me to change my mind. Why? Because I saw that the Bank of Canada had been reasonably successful in keeping inflation close to target. But we still had a recession, in which both NGDP and RGDP fell sharply below trend, and it sure looked like a recession due to deficient aggregate demand. If changes in inflation had roughly matched that sharp fall in RGDP, I would have blamed the recession on the Bank of Canada failing to keep inflation on target. But that is not what happened. So instead I now blame the recession on the Bank of Canada failing to keep NGDP growing at trend, and I want the Bank of Canada to target NGDP.

Comments

[Martin]

I wonder what you have to believe [i]theoretically[/i] if you believe that prices are perfectly flexible. It seems to me that you have to believe that every contract once signed can be renegotiated and if you believe that prices should be perfectly flexible you must believe that contracts should always be renegotiated...

It seems to me that people who believe that or people who think the world should look like that have never heard of the hold-up problem (http://en.wikipedia.org/wiki/Hold-up_problem)... or that they don't think it exists which means they must be working with a model without "K", but "K" is right there in the models usually... I also wonder what it means then that there is a technology shock, why would anyone invest in technology when contracts can be re-negotiated or should be re-negotiated? Oh well.

[Nick Rowe]

Martin: I think that's a very insightful comment, about hold-up costs. Just to make it clear, when Martin talks about a contract being "renegotiated", he doesn't mean breaking the contract. He means doing a side deal because the original price you negotiated is no longer an equilibrium price so the quantity of trades won't be optimal. (At least, that's how I interpret Martin).

[Ritwik]

The central bank could not have held the NGDP on the same path. Let's talk about RGDP later. :)

[David Andolfatto]

Nick,

Glad to see that you've finally boiled the question down to a matter of religion and not evidence.

One small thing to add. To the extent that inflation constitutes a tax on economic activity (many models have this property), the RBC theorist might actually believe that prices would have to rise *by more* than the fall in RGDP to keep NGDP on its path.

[K]

Not so, Martin. An agreed transaction price is not a "price" for the purposes of this discussion. It's a parameter of a contract, the value of which is assumed to vary with the market price (assuming it's a forward contract). To believe that prices are perfectly flexible, you just have to believe that when it comes to negotiating a contract, the agreed price will be the market clearing price. I.e. your current wage rate is totally irrelevant when your contract comes up for renewal.

[Nick Rowe]

Ritwik: "The central bank could not have held the NGDP on the same path."

It is debatable whether the Bank of Canada could have kept NGDP exactly on trend (or have prevented it falling at all). But it is clear that the Bank of Canada could have pushed NGDP back up to trend much more quickly and more strongly than it did. Canada left the ZLB in June 2010. And the expectation that it would return NGDP to trend would have had an automatic stabiliser force, meaning it would have left the ZLB earlier than June 2010 (or might never have hit the ZLB at all).

[Nick Rowe]

David: "To the extent that inflation constitutes a tax on economic activity (many models have this property), the RBC theorist might actually believe that prices would have to rise *by more* than the fall in RGDP to keep NGDP on its path."

I see what you are getting at there, and I agree. RGDP would have fallen further than it actually fell if the central bank had held NGDP constant, if RBC is correct. (But what you actually said doesn't quite come out right.)

[Nick Rowe]

And it's a funny sort of religion, if two graphs that Stephen drew using StatsCan data were a major part of what converted me from one denomination to another.

[Martin]

Nick, thanks, I have to work it out myself as well to get exactly what I mean by that. It just seems to me that for whatever you believe about prices you also have to believe something about contracts, and we simply know that the possibility of side deals is not always desirable.

K, I have difficulty seeing what you mean? As I read you you're saying that all transactions are spot transactions and that all prices are spot prices?

[Saturos]

Nick, NGDP is more fundamental, because (as market monetarists use it) it's a measure of the aggregate flow of money, not a measure of the magnitude of what money is spent on (though others could use it that way). Thus a drop in the aggregate flow of money causes a drop in the magnitude of what can be exchanged for money, if the rate at which other-things-in-general can be exchanged for money is somehow "stuck".

I'm pretty sure Bullard on the other hand would say that since employment fell, obviously RGDP fell, and a central bank told to "hold NGDP constant" would simply have a mandate to arbitrarily raise prices ("the nominal thing that central banks do control") by the same amount that RGDP fell, for some reason. So he's not thinking about the aggregate flow of money and what determines it at all. In fact I suspect even many Keynesians think this way too, where "aggregate demand" is some force that connects directly to real things without having anything much to do with money or its flow, except maybe in the LM-diagram. (Hey, isn't that precisely what you've been trying to disabuse people of on your blog for years now?)

[K]

Martin,

Well all transactions *are* spot transactions, even if they involve future cash flows and services. Think of FX. At any point in time I can do a spot or a forward trade, but in either case I engage into the transaction *now*. They're just two different kinds of transactions. Lets say I buy CAD one year forward at par. As the spot (and forward) price changes, the value of my forward contract changes, because the forward "price" is fixed by the contract. The fact of the existence of forward FX contracts does not constitute "sticky" prices. If on the other hand, the FX spot priced is legally fixed away from the market clearing rate, that does constitute a (perfectly) sticky price.

Same goes for a wage contract. If an employer commits to a long term contract and the clearing wage drops, then present value of the employment contract goes up. I.e. the value of the contract is not sticky. For example, an employer might try to pay off a tenured employee in order to terminate an off-market employment contract. Again, nothing to do with sticky prices. A sticky wage is such that the employer and employee are unable to contract at the level of the market clearing wage for some reason.

RBC guys and their models don't have a problem with contracts. The have a problem with transactions that are conducted at non-market clearing prices. Just ask David Andolfatto if he's still around.

[Nick Rowe]

Saturos: let me edit what you said very slightly, and your "thus" becomes a "logical thus":

'Nick, NGDP is more fundamental, because (as market monetarists use it) it's a measure of the aggregate flow of money, not a measure of the magnitude of what money is spent on (though others could use it that way). Thus a drop in the aggregate flow of money [logically entails] a drop in the magnitude of what can be exchanged for money, if the rate at which other-things-in-general can be exchanged for money is somehow "stuck".'

[Martin]

"If an employer commits to a long term contract and the clearing wage drops, then present value of the employment contract goes up."

K, the formula for NPV is:

NPV = CF0 + CF1/(1+i) + CF2/(1+i)^2 + etc,

where here CF0 = CF1 = CF2 etc. The only way the NPV of a long term wage contract (CF is the wage) changes here is when "i" is a function of the clearing wage. It suffices to say that this is not necessarily the case for all wage contracts.

[K]

Martin,

The pv is the sum of (Cn-Wn)/(1+i)^n

Where Cn is the contracted wage for year n and Wn is the current market wage for forward employment in year n. The initial value of the wage contract is zero, ie it is entered into at par. There is no sudden windfall just cause you get a job. Then, when the market wage changes away from the contracted wage, the pv of the contract changes. Nothing to do with changes in rates.

[David Andolfatto]

Nick: yes, sorry. What I said definitely did not come out right. And you nailed exactly what I meant!

[Martin]

K, that is not a wage contract, that is what you could call a profit function of the employee. What the employee receives qua employee out of the wage contract is the sum of Cn/(1+i)^n. The sum of Wn/(1+i)^n is what the employee would earn if the labour market was a lot like the financial market and the employee would contract every period.

This brings me to a different point, the desirability and the plausibility of flexible prices seems to be based - as I understand you - on the assumption that all markets should and/or are very much like the financial market. Further the assumption seems also to be that because all markets are or should be like the financial market, that the economy as a whole has, can have, or should have flexible prices. Is this what you're saying?

[K]

Martin,

"What the employee receives qua employee out of the wage contract is the sum of Cn/(1+i)^n."

Wn is the "cost" to the employee of providing her labour in period n, which is what makes the swap initially worthless, ie. par.

"The sum of Wn/(1+i)^n is what the employee would earn if the labour market was a lot like the financial market and the employee would contract every period."

It's the *revenue* the employee would receive.

"This brings me to a different point, the desirability and the
plausibility of flexible prices seems to be based - as I understand you - on the assumption that all markets should and/or are very much like the financial market.  Further the assumption seems also to be that because all markets are or should be like the financial market, that the economy as a whole has, can have, or should have flexible prices. Is this what you're saying?"

Absolutely. There are no *rents* in an RBC model. Employees are indifferent to an hour more or less of labour. Corporations sell their goods at the marginal cost of production. *Nobody* has market power. Therefore, sticky prices are impossible. Meanwhile back in reality, corporations make big, persistent profits, and jobs are valuable and therefore prices and wages can be sticky. Monopolistic competition is a necessary feature of the New Keynesian macro model in order to have sticky prices.

[Nick Rowe]

K: There is no reason you can't have monopolistic competition in an RBC model. And you can (in principle) have sticky prices in perfect competition, but: it's harder to model; and the incentive for an individual firm to cut its price in a recession would be very strong.

[Martin]

K,

whether or not Wn is the cost is beside the point, the formula you gave cannot represent a wage contract. How many wage contracts have you seen where one of the terms of the contract is the cost to the employee and where the cost to the employee is subtracted from the payment to the employee? I get what you're driving at, but what you described is not a wage contract. It's a 'profit function' as I said or 'revenue' minus cost as you said yourself ;).

That said I really don't see your objection to my comment above; it seems to me that they do have to believe something about contracts... All features of the world that can result in sticky prices have to be assumed away; it takes but one feature to get sticky prices though...

[K]

Nick,

You're right. It was anti-fresh water snark. They do, however, have a strong predisposition to pretend equilibrium is efficient.

[K]

Martin,

No! You can have Cournot models with or without sticky prices. Market power doesn't mean sticky prices. It just allows you to introduce sticky prices in a sensible, tractable way. But as Nick says, you can have an RBC model with monopolistic competition. And no sticky prices.

Sticky = market price at which you can transact which for some reason can't change, or can't change quickly enough in response to changing supply and demand. It describes the behaviour of a market price over time. *Not* a historically determined contract parameter.

[Martin]

K,

No to what? I have no idea what you're on about... I never claimed anything about market power... you brought it up, remember? The statement you seem to be responding to with "No!", is the same as:

"the bread costs 1 euro, I have 10 euro in 1 euro coins in my pocket, I just need one of them to pay for the bread, but if we're going to assume that I cannot pay for the bread you have to assume all of those coins away."

Meaning that you can introduce sticky prices in a myriad of ways, but you have to assume all those reasons absent if you're going to assume flexible prices. Forget one of those reasons and you have sticky prices. Better yet, you don't even necessarily have to change the mathematical model to change the reason for sticky prices, just give another economic interpretation to the same construct. The mathematical models, like so many other metaphors exist independently from the economic interpretation given to them.

Also I don't get your whole "contract parameter"-thingy; the price and quantity at time t=1 is a term of the contract, and so can be the price and quantity at t=2 etc.

[K]

Martin,

"How many wage contracts have you seen where one of the terms of the contract is the cost to the employee and where the cost to the employee is subtracted from the payment to the employee?"

Wn is the negative value to the employee of having to do the work. It is the price at which he is indifferent to doing it or not doing it. Initially the positive value of future wages exactly offsets the negative value of future labour. Then the market price of labour changes, and the wage contract becomes off par. It's not a "profit function". It's the npv of the labour contract.

" it takes but one feature to get sticky prices though..."

That's the part I said "no" to. It takes sticky prices to get sticky prices. Market power doesn't do it. Nor does contracting. It takes some weird mechanism that prevents people from freely entering into new transactions at whatever price they choose. For some strange reason they all have to trade at some other price. In the NK model you have to wait for a visit from the Calvo fairy.

[Martin]

K,

It's not a labour contract/wage contract what that formula described, repeatedly stating that it is won't make it so. The definition you're using is a different one from what you'll find in the labour economics literature and different one that you'll find in the law literature, and it is different from anyone's understanding of what a labour contract is.

Look for example in Gibbons & Waldmann (1999), you'll find a wage contract described as w = s + b*y, where y = e + epsilon, where s is the fixed part, b, the part contingent on income generated, y, income, e, effort, epsilon a random variable. The employee faces a cost of effort of c(e) = e^2 and maximizes Eu(e) with a participation constraint. What you describe as the wage contract is closest in description to the utility function of the employee.

Similarly, in the law literature you will find that a wage contract is contract for a particular service, where the employee is subordinate to the employer, and receives a wage because of that. The outside option, by you described as Wn, does not even come into it.

You can make up all the definitions that you want, but nobody is going to be able to understand what you're talking about. If you're going to consistently assert that green is really "red", and that red is really "green".

As for the sticky prices: there is not only way to get sticky prices, some are time dependent, others are state dependent... and if you don't get what I said earlier you'll be really surprised by menu costs (http://en.wikipedia.org/wiki/Menu_cost).

[Peter N]

The problem of sticky wages is only a problem if you make simplifying assumptions about the bargain between employer and employee and you deny any group effects.

An employer isn't trying to minimize the individual costs of each member of a set of of generic employees. The goal is to maximize the company's return on certain metrics considering all the costs of employment and all the contributions of the employees.

1) The hiring and training is expensive. This represents a premium against the salary of a new hire.

2) Companies gain great economies from flattening the management hierarchy giving lower level employees more responsibilities and expecting them to show initiative and internalize the goals of the company.

3) This means that morale is extremely important, because only a relationship of trust and a belief in shared interest can motivate this kind of behavior at an acceptable cost.

4) There are fields like financial services which use primarily financial incentives to motivate, but salaries and bonuses are much too high for this to be practical for most businesses.

5) Furthermore the pure financial approach can have long term negative consequences. Large bonuses are more common in Europe, and have become a problem.

6) Employees live in a nominal world. Their costs are in nominal dollars. They see wage cuts as real, and they are. They don't live in the long term.

7) Businesses prefer to concentrate the pain and lay off employees. You have fewer unhappy people, and those are no longer around to cause trouble. This is exactly the opposite of the result of wage cuts.

8) Where the business model doesn't involve employee autonomy and identification with company goals, you do, in fact, see flexible wages. A good example would be a telemarketing call center. This may embody the purest of theoretical economic principles, but nobody, given any choice, wants such a job.

10) It seems only reasonable that as automation improves productivity, the remaining jobs in the organization will become stickier. Each employee is a larger percentage of the company workforce and thus of its total responsibility and initiative, while at the same time labor costs are a smaller share of company total costs.

[K]

Martin,

We are talking about an economic model, not law. So when looking at the value of a company should I subtract the pv of all contracted future wages and ignore the future labour services provided under those same contracts? In your way of seeing things, companies will never sign labour contracts since it represents a big loss. In the model, you have to assume they are buying something of value. Anyways, OT. The issue is whether labour contracts pose a stickiness problem for rbc models, which they don't. It all started with you saying that contracts are inconsistent with flexible price assumptions, which, for the purposes of the models we are discussing, is not the case. Anyways, enough tying up Nick's post.

[K]

In fact, come to think of it, complete markets are a common assumption of DSGE models. Why? Because it allows the modeler to assume that all agents contract for *all* future contingencies which lets us ignore heterogeneous preferences and use a representative agent. Far fetched? Yes. But contracts, rather than posing an obstacle to flexible price models, are in fact the *only* way future transactions are performed in many of those models.

[Martin]

K,

I clearly stated that what you wrote down is not a wage contract and is not what would be recognizable as such to:
I. a labour economist, nor
II. to a lawyer, nor
III. anyone else

This is not about law versus economics, it seems more that it is you versus anyone who has anything to do with wage contracts. That should have told you that your definition of a wage contract and your representation of it is quite off...

You state that what you wrote down is the PV of the wage contract from the perspective of the employee, which is simply wrong, for what you wrote down is not what is recognizable as a wage contract, rather it is, as I said, the 'profit' or 'utility' if I try to interpret Wn favourably to you.

Now you state that what you wrote down is the wage contract from the perspective of the firm, which is (mostly) wrong too, but I am not interested in getting into that as they key problem is not why it is right or wrong. The key problem is the approach you seem to be taking, for what you appear to be doing is giving a tortured interpretation of finance models to fit them to other areas of economics: in other words you're trying to fit a square peg into a round hole.

Economics as a science has a unified approach to solving particular problems and explain particular phenomena, but this does not mean that you can use models to solve particular problems or explain particular phenomena, to solve different problems or explain different phenomena. A model of one market is only like every other market in that we can use the same approach; similarly an employee is like a firm only in that these are two economic actors that we can represent as maximizing something.

Just because contracts are used in a particular way in one market does not mean that all contracts in all markets are like that....

[K]

Martin,

I have no idea how you balance global budget constraints if contracts have radically different values to the two parties to the contract. Law of One Price, non-arbitrage, and all that. But anyways... It seems I'm just not capable of understanding what you're saying about the relationship of contracts to flexible price models. Nick seemed to think it was something important, so maybe he'll explain it to me. I don't get it and we don't seem to speak the same language.

[K]

Actually, more correctly, if a contract has different value to two agents, why don't they trade? You are not describing an equilibrium model. Nick?

[Saturos]

"a drop in the aggregate flow of money [logically entails] a drop in the magnitude of what can be exchanged for money"

Almost. I see what you're saying: if money "flows" then it is being exchanged for other stuff, so it's logical equivalence and not physical cause. But for me to make a purchase of something with money, that money has to flow through to me. So if for some reason half the banknotes disappear off the face of the earth, less purchases will be made, not only (tautologically) because less money will be spent on stuff, but also because less money will be received from selling stuff, and hence available to spend on stuff. So scarcity of money begets scarcity of money - or as you say, demand creates demand (http://worthwhile.typepad.com/worthwhile_canadian_initi/2011/02/but-where-will-the-demand-come-from-in-praise-of-older-keynesians.html).

[Greg Ransom]

NGDP is a social construct -- a constructed statistical measure than can be constructed in a hundred alternative ways

RGDP is a theoretical fiction -- it isn't even a social construct and efforts to come up with a social construct stand in could be constructed in a hundred different ways.

P is likewise a theoretical fiction -- it isn't even a social construct and efforts to come up with a social construct stand in could be constructed in a hundred different ways..

We need to keep in mind and be serious about the fact that NGDP is an arbitrary social construct with all the messy hodge-podge of contradictory measurements well noted in the literature by Stiglitz and generations of other economists.

[rsj]

In the absence of an auctioneer, there *is* a tension between believing that everyone is a price taker and believing that prices are perfectly flexible. We do not have access to the Walrassian magic that reduces that tension.

You cannot argue that prices contain useful signaling information, e.g. for planning, and at the same time believe those prices are always unknown ex ante. As soon as they are known ex-ante, they are stuck for the period of time in which they are known. So workers get hired with an annual pay and each year there is a performance review and the pay is changed. Auto makers announce their models for the year, etc.

Instead of thinking about how prices are sticky, we should be thinking about how they can be flexible. The auto maker might have some wiggle room for dealer incentives. The employer might set aside some resources for bonuses or engage in some profit sharing, etc.

But the base case is always that the price is known ex-ante for an announced period of time, at a minimum the time corresponding to the time period necessary for planning, production, and sale.

[K]

rsj,

But a price is a level that you can trade at. Once you transact you have a term contract. But the parameters of that agreement are not a price. They are just the terms of a signed contract. It's interesting that we have contracts. But the economic consequences are very different from stickiness, ie the inability to enter into a new trade at whatever price you the parties might want to agree to, but instead being required to enter into new trades at yesterday's price.

[rsj]

K,

Yes, you can enter into new trades at new prices. Just try it. Open up a coffee shop, and sell off the coffee by auction. See how many customers you get. See how much your creditors will lend to you to open up a coffee shop if you have no idea how much you will be able to sell the coffee for. See how many workers you can hire if you have no idea how much you can pay them, etc.

[K]

rsj,

Yup. Prices are sticky. And people like contracts.

[Too Much Fed]

RGDP 3%, P 2%, NGDP about 5%.

“Something happens”. RGDP 1%, P .5%, NGDP about 1.5%.

Next, RGDP 1%, P 2%, NGDP about 3% and stays there.

What happens if P is raised to 4% and RGDP starts falling? Amy Pond says explain.

[Philonous]

“It is debatable whether the Bank of Canada could have kept NGDP exactly on trend . . . .” But if it had been targeting the forecast, it could have kept *expected* NGDP (one or two years into the future) almost exactly on trend; and this is what it should have been doing. (By ‘expected’ I mean *expected by the market*; this can be estimated pretty accurately even in the absence of a Sumnerian futures market.)

[K]

Philonous,

How do you back out implied forward GDP from financial markets?

[Sergei]

Question.

We have NGDP(t) = RGDP(t) x P(t) from which we get NGDP'(t) = RGDP'(t) + P'(t).

However the limit of both NGDP(t) and RGDP(t) as times goes to 0, i.e. the definition of derivative, is ... zero. Because GDP is a flow and therefore needs time to accumulate. If you do not give it time to accumulate, it will be zero. So obviously this is NOT what you mean, is it correct?

What do you mean then?

You probably mean another functional dependence of NGDP and RGDP on time. The one which is described by the structure of GDP (so you can forget the flows) and which can change over time in a very possible fancy way. And if the functional form of structure depends on P(t), which it very surely does, then there is no guarantee you can have a solution. But nevermind.

What I am trying to understand is where you actually talk about the structure of GDP which you want to differentiate in the first place (the equation above). Because what matters for people is not the size of GDP over a some time but their small individual slice of the structure of GDP. For instance, to stop freaking out about the pension system and related liabilities you can politically legislate that every person of retirement age is euthanized either by will or political necessity (I remember there was a referendum 10 years ago and pensioneers lost on this case). Such act will not change the NGDP or RGDP because pensioneers do not produce anything. But will change the structure making everybody else economically happier.

The NGDP formula is happy with both RGDP of 10 and price level of 10 as well as RGDP of 1 and price level of 100. If the central bank owns the whole economy and can set all prices then it has full control of *N*GDP. Because it controls N. But how does to steer the *R* part of the formula? And if does not, how can we be sure that we do NOT get the latter "unstable equilibria"?

I hope my point and question are clear.

[Saturos]

Sergei, your point and your question are not clear. I don't know why you are bringing up derivatives, but d/dx (RGDP(t)P(t)) = RGDP'(t)P(t) + P'(t)RGDP(t) [the product rule], not RGDP'(t) + P(t). And the definition of the derivative is lim t->0 (deltaNGDP/deltat), not lim t->0 (NGDP(t)). Hopefully that helps with the rest of your question, whatever it is.

[Saturos]

lim deltat->0 (deltaNGDP/deltat), I meant.

[Sergei]

Saturos, I was really sloppy in my comment. Let me try again.

GDP is a flow. GDP is output produced over time. Lets define in my sloppy terms (I hope it is more clear now) as GDP = productivity * time. Then GDP' = productivity. But productivity as t->0 is meaningless because any production requires time and productivity as such does not have a time dimension. So why should it grow or decrease over time? There must be other reasons which are not time. I called them structure. However poorly constructed, what I was trying to show is this connection, i.e. that approaching the problem of economics with differentials/derivatives is not always a good one. Yes, GDP has a time dimension because it is a flow of product over time but it does not mean that taking time derivative gives us lots of insight into nature. What is important for GDP is not time.

So why does productivity change and how? We can have both RGDP of 10 and price level of 10 as well as RGDP of 1 and price level of 100. But how do we ensure that above all we get to RGDP of 100 and P of 1? What processes do we have in place for that? Invisible hand?

With NGDP discussions without answering the question "how" I rather believe that one thing invisible hand will ensure is that the NGDP authority will be arbitraged for its negligence of RGDP and P processes. There will be smart people who will figure out the "how" part for themselves and make a (redistribution) profit because NGDP will be always guaranteed. And monetary profit is required to get a real benefit. Just like today where we have a huge financial sector which has very successfully learned how to arbitrage the inflation targeting monetary policy authority which believes in invisible hands and is ignorant of all important processes in the economy.

[Nick Rowe]

X^ means (dX/dt)/X. It's the *percentage* rate of change of X, per unit of time.

GDP per year is 12 times GDP per month, which is 30 times GDP per day, which is 24 times GDP per hour, etc.

Strictly speaking (I think I've got the math right) NGDP^ = RGDP^ + P^ + RGPD^*P^. Just ignore the cross-product term RGDP^*P^. It's too small to bother with, unless you measure time in decades, or you are in hyperinflation.

(BTW, "productivity" means the flow of output divided by the flow of input. It does not mean the instantaneous flow of output per limitingly small unit of time.)

I think I'm agreeing with Saturos, but I never truest my math.

[Nick Rowe]

Philonous: "But if it had been targeting the forecast, it could have kept *expected* NGDP (one or two years into the future) almost exactly on trend; and this is what it should have been doing."

Agreed. But the graphs show actual NGDP, not expected.

[Sergei]

Nick, right, it is a derivative of logarithm and that is why you get a sum. But it is not important. And yes, productivity is also a bad term. Sorry, do not know how to call it better so please ignore any direct meaning and try to look beyond it. It was not my question but rather a sloppy way to express my thinking.

Time derivative of GDP, whether percentage or absolute, is not an interesting question or challenge. "Why" - this is THE challenge. However if you ignore "why" then the derivative and its result will also become a challenge. But a wrong one. So my question is about how we ensure that we get to equilibrium with R=100 and P=1 instead of equilibrium with R=1 and P=100? I do not want to get into the latter one. How do you make it NOT happen?

Why should I vote for you if that is a right analogy :)

[Nick Rowe]

Sergei: if RGDP grows 3% in one year and the price level grows 2% in one year, then NGDP grows 5.06% in one year. Forget the 0.06. Don't sweat the small stuff.

[Peter N]

" But productivity as t->0 is meaningless because any production requires time and productivity as such does not have a time dimension. So why should it grow or decrease over time? There must be other reasons which are not time. I called them structure."

I'm sorry, this is ridiculous.

It's nice to have something that is a flow sometimes and the integral of a flow at others, and I hope you're paying your terminology extra for all its hard work.

However:

1) to repeat Greg Ransom verbatim for those who weren't paying attention:

"NGDP is a social construct -- a constructed statistical measure than can be constructed in a hundred alternative ways

RGDP is a theoretical fiction -- it isn't even a social construct and efforts to come up with a social construct stand in could be constructed in a hundred different ways.

P is likewise a theoretical fiction -- it isn't even a social construct and efforts to come up with a social construct stand in could be constructed in a hundred different ways.."

2) The components of NGDP are discrete, It is only notionally a flow, since it isn't observable at that level, or even well defined, since:

a) it is the result of mixing figures from different systems of accounting without any hope of actual reconciliation.

b) it contains a large percentage of imputations and arbitrary choices whose period of validity (to the extent they are "valid") doesn't and can't ever match the periods of observation. You might take a look at accounting for imports, for example, or implied rent.

3) So if you want to make the simplifying assumption that it's a flow, fine. But then its time derivative is exactly as meaningful as your assumption. They are joined at the hip.

4) Since GDP doesn't include nonproductive expenditure, and there is no isolation between the realms of productive and nonproductive expenditure (work not for pay, exchanges of existing assets and the underground economy), constructing an isolated deflator is impossible, since you're ignoring a large part of the economy

5) If N = R*P then R = N*(1/p) -> N' = R'P + P'R and R' = N'/P - N/P**2*P' these are equivalent since we can transform the second.
R'P = N' -(N/P)*P' substitute R*P for N -> R'P = N' - RP' add RP' to both sides giving-> R'P + P'R = N'.

6) Thus we can treat R as derived from N, which is the actual observable just as easily and with a great deal more sense.

7) After all, the choice and calculation of the deflator requires an additional tower of assumptions and approximations.

8) Since R and P are basically notional, the question of whether they are continuous and differentiable is more useless than angel's dancing on the point of a pin (which had meaning in the context of the period). They are if you want them to be and use them consistently and within their domain of validity, which is true about most macro variable. Outside their domain of validity they are useless, like any other mathematical entity similarly abused.

9) There's nothing more wrong with instantaneous productivity or instantaneous inflation than with instantaneous RGDP and no less either. You pays your money and you takes your choice.

10) And guys, can we lose the damned hats "^" and use "'" for derivatives or dx/dt. If you want to use them, use them for exponentiation where I've used the Fortran **.

And Nick, I think part of the answer to your questions may be that both P' and N' are meaningful in the context in which you use them. After all the sums of the products don't equal the products of the sums. Certainly they don't move in lockstep before they are aggregated, and multiplying aggregates will usually get you in trouble (but we've been over that ground a few times). There's no theoretical justification for such aggregate abuse, you have to argue statistically based on regressions (emergent behavior). Has anyone done this, and if so, what did they think they found.

[Sergei]

I appreciate all the efforts to explain the lousiness of my terminology. Really. But it was not worth it. Why such a deliberate intention to focus on anything but my question? Yes my awkward preamble was super lousy but lets get past it. The question is not a numeric one of derivatives, aggregation, or multiplication and related issues. It is a very *qualitative* question.

So my question is as simple as: NGDP = RGDP * P. If you do not like RGDP and P because they are theoretical, not aggregateable, not instantaneous or whatever then lets talk NGDP = X * Y. Whatever stands behind NGDP it is something we can measure. X indicates some (very likely artificial, i.e. theoretical but still) measure that supposedly should be "value" positive while Y indicates something (very likely artificial, i.e. theoretical but still) which is "value" negative.

Now we have a policy which ensures a certain outcome of NGDP. How under this policy do we steer X and Y so that we get more of X and less of Y? Is there anything which says "we do it like this and that"? Why should I vote for Nick and his NGDP? Where is the "guarantee" that we do not get smaller X and larger Y and additionally a much more unequal distribution of the ever smaller X? This is not an UNfounded fear. This is a fear which is well founded in the existing reality.

[Nick Rowe]

Sergei: "Now we have a policy which ensures a certain outcome of NGDP. How under this policy do we steer X and Y so that we get more of X and less of Y?"

Why not just say "Why do you think NGDP targeting would be better than other monetary policies?"?

I have done other posts addressing that question. One here. Another here. There are more. I also address it here in my point 7 above.

[Sergei]

No, Nick, this is a wrong answer to my question. "Better policy" is a very dangerous path. I want the one which works! And by that I mean the one which delivers better X and not better NGDP.

At the moment we have a policy which, in a broad sense, tries to deliver better X, whether it works or not. If the current policy of delivering better X fails then there is a clear pressure to "fix" something. And the theoretical basis worked until now. However bad the theoretical justification was/is there is a clear link between the facts of life (development of X) and actions triggered by these facts of life. In your policy proposal you simply wash your hands. You stop caring about life and start caring only about your theory.

I will not vote for you unless you tell me how your policy ensures progressively better X. If you are not able to do it then you just stick yourself into a dark room with Chuck in it. You will by definition face a political party which will shut down your nice experiment overnight regardless of how supercool and deeply justified your theory is.

[Peter N]

Sergei,

Once it occurred to me the hats were supposed to display over the variable, I looked at Wolfram. I believe the idea is that if Z = X * Y -> Ln(Z) = Ln(X) + Ln(Y). Since d(Ln(X))/dt = (1/X)*dX/dt. If you symbolize this as X^ [let's pretend the ^ is over the X] -> X^ = Y^ + Z^.

This works fine, except suppose X(y,Z,t) = X(Y,t)*Y(X,t). I believe this is your question. You are quite correct.

The underlying structure is a partial differential equation. NGDP(t) = RGDP(t) * P(t) is assumed to be a faithful (approximate) representation of this equation (which is almost certainly both unknown and insoluble in closed form). You can see, this representation eliminates what you are worrying about, but is this a faithful representation, and if so, why? My guess is that is within limits, and we don't know what the limits are, but this is just a guess.

It is also may very much a question of mathematics abuse, for instance, if operating on aggregates isn't equivalent to aggregating the results of individual operations (which it isn't without counterfactual assumptions). While you're looking at your qualitative question, the economagician is using his mathematical other hand to put the rabbit into the hat.

This is a classic example of begging the question, which technically means smuggling your conclusion into your premises.

The RGDP tracking story as I understand it rests on 3 pillars:

1) Stable NGDP is inherently a good thing because it represents a stable "tightness". This is plausible, but needs some filling out to escape being a tautology.

2) NGDP is a good proxy for employment (relative to some natural rate)

3) Within some operating envelope RGDP and inflation are in a sort of equilibrium and stable ngdp allows this equilibrium to assert itself. Outside the envelope it becomes classically deflationary on the high inflation side, and stimulating through the expectations channel on the low. In fact it is supposed that the expectations associated with this as a credible policy will act to keep NGDP within a narrow range through time arbitrage, since credibility makes expectations of a profit from such arbitrage rational.

OTOH there's the story where one of a pair of economists says "Look a $20 bill lying on the sidewalk", and the other says "Nonsense, if it were real, someone would have picked it up already."

I expect Nick can improve on this. I'm not an NGDP guru.

[Nick Rowe]

Peter N: Yep. The ^ is supposed to display over the variable. But I can't figure out how to do that in TypePad.

There's a whole literature on index number theory which talks about how to decompose NGDP into RGDP and P, and the limits and problems in doing so. It teaches you how to add production of apples and production of oranges, and what can go wrong. I have ignored that literature here. (All economists are familiar with the basics of it, and a few (not me) get more deeply into it.)

[Sergei]

Peter N, you are able to express much better what I mean than I am :)

"My guess is that is within limits, and we don't know what the limits are, but this is just a guess."

Yes, the typical scope of limits is the YoY result or similar. And what I am thinking about is rather DoD where D means decade. The structure does not change fast and YoY changes can often be neglected. They however accumulate and have profound effects over the longer term as the economy changes and adapts. But adapts to what?

My concluding qualitative assessment, which I tried to express above, is that the structure of economy continuously adapts (invisible hands or whatever) in a such way as to (again might be sloppy so please try to think beyond the words) minimize its cost of hitting the exogenously set policy targets.

Looking back into the past we had inflation caring/targeting central banks and over the period of 2-3 decades the economic structure evolved into the form which allowed it to arbitrage such inflation focused central banks away. And so we have economic structure which depends on asset bubbles to function. It worked "fine" until we failed to inflate another asset bubble.

How can I be assured that either this line of thinking is wrong or that we stay away from such bad equilibrium?

[Min]

"If we observe NGDP and RGDP both suddenly fall, as they did in many countries in 2008, we cannot say from that fact alone that RGDP would not have fallen as much as it did if the central bank had held NGDP constant (growing at trend). But anyone who denies that assertion would have to believe that P would have increased (relative to trend) by the same amount that RGDP fell (relative to trend) if the central bank had held NGDP constant (growing at trend). Is that belief plausible?"

When RGDP falls and inflation increases, don't we call that stagflation?