I was once told by an English professor that Joseph Conrad preferred to write in English (his third language) because sentence meanings in that language often had a wonderful ambiguity that added an artistic flair to his prose.
 
Well, I'm not sure if that story is true. But I do know that it is easy to misinterpret what people mean when they try to communicate their economic theories in "plain" English. That is why academic economists, when speaking among themselves, prefer to communicate in a much more precise language--math.

For those among you who do not understand this language, I'm sorry. I'll do my best to translate into English as I go along. What I want to do here is provide a formal (mathematical) framework to evaluate the discussion on Ricardian equivalence these past few days (see my previous two posts).

Before I get started, I want to make a few things clear. I was not trying to defend Lucas' claim that G fully crowds out private spending. I am not a Republican (I am a Canadian). I agree with some of things that Krugman says (just take a look at some recent posts). I am annoyed that Krugman repeatedly attacks Lucas for "not understanding his own theory." Not only was that was a low blow, but that sort of talk just promotes a division that I do not think exists in the profession. Moreover, and more to the point of what motivated my original post, in delivering his low blow, Krugman presented his own muddled view of the role that Ricardian equivalence played in Lucas' argument.

So let me try to clear things up. Note that I do not speak for Lucas here. What follows is one possible interpretation of what he had in mind. More precisely, it is what came to my mind when I was trying to interpret the content of his speech.

The model I have in mind is a simple overlapping generations (OLG) economy. People live for two periods; they are "young" and then "old." The population is constant. For simplicity, the young do not care for consumption. Instead, everybody wants to postpone consumption to old age (this is not a critical assumption).

The young are endowed with a unit of labor that produces output y (the young supply this labor inelastically, so we may treat y as an endowment). The young also possess an storage technology; k units of investment today yields F(k,g) units of output tomorrow, where g denotes government investment spending. I assume that output F(k,g) is increasing in both k (private investment) and g (public investment). For simplicity, assume that all capital depreciates fully after it is used in production.

Consider the following two specifications of F(k,g):

PF1: F(k,g) = f(k+g)
PF2: F(k,g) = A(g)f(k)

In PF1, private and public capital are perfect substitutes in production. What this implies is that an increase in g lowers the marginal product of (the return to) private capital spending. In PF2, private and public investment are complements. What this implies is that an increase in g increases the marginal product of (the return to) private capital spending.

I believe, though I am  not sure, that Lucas had in mind specification PF1. At least, this is an assumption that is consistent with his conclusions. He would have come to a different conclusion if he believed PF2. Note: the choice of PF1 vs PF2 has nothing to do with Ricardian equivalence. 

Let me continue to describe my model economy. There is a government security that earns a gross real rate of return R. In the present economic climate, with nominal interest rates close to zero, R<1 is the inverse of the gross rate of inflation. I treat R here as a policy parameter.

The budget constraints for a young agent in this economy are given by:

k + m = y - t
c = F(k,g) + Rm - T

So here, a young person must take his after tax income (y-t) and make a portfolio allocation choice: how much to invest in private capital k and how much in government money/bonds m. In old age, the agent gets to consume the proceeds of his investments, minus his  future tax obligation T.

Next, we have to specify the government budget constraint. I consider two extreme cases.

GBC1: g = t + T/R
GBC2: g = (1-R)m

Under GBC1, I am assuming that the burden of financing g falls entirely on the young. This assumption (together with my use of lump-sum taxes) is going to generate a Ricardian equivalence result: the young are not going to care whether they are taxed now or later for g. (Note: Ricardian equivalence would not hold if I assume instead that the burden of finance falls on both the young and old--that is, if I assume that current g is financed by the current young and current old--in contrast, here I assume current g is financed by current young and future old).

Under GBC2, I assume that g is financed entirely through money creation (seigniorage revenue).

Finally, I consider two experiments:

E1: a permanent increase in g
E2: a temporary increase in g

OK, now let's investigate some of the properties of this simple model and see how it can be used to make sense of things.

Analysis

Case 1: PF1, GBC1, either E1 or E2

The key equation is the one that equates the marginal product of private capital investment to its opportunity cost:

[1] f'(k+g) = R

Result: An increase in g fully crowds out k (so future GDP remains unchanged). This is independent of whether the young are taxed now or later.

Does this conclusion rely on Ricardian equivalence? Well, yes and no (assuming distortionary tax finance would imply that an increase in g would decrease future GDP). Consider the next case.

Case 2: PF2, GBC1, either E1 or E2

The key equation now takes the form:

[2] A(g)f'(k) = R

Result: an increase in g stimulates k (so future GDP increases). This is independent of whether the young are taxed now or later.

This is the sense in which I believe Lucas' remarks have nothing to do with Ricardian equivalence (it has to do with his belief of PF1 over PF2). And indeed, what he literally says is "and taxing them later is not going to help." That is, it might even hurt--which can only be true if one departs from Ricardian equivalence (e.g., by assuming that the future tax hit will be distortionary). Again...words, words, words...we need an explicit model to decipher and evaluate what he really meant.

Aside: I often hear people say things like "Well, yes, if the increase in g is permanent, then it will fully crowd out. But this does not hold if the increase in g is temporary." My reply to this is: you are wrong. Take a look at the model above. It is possible for a permanent increase in g to increase GDP permanently. In particular, Cases 1 and 2 remain valid whether or not the increase in g is temporary or permanent (they hold for E1 and E2).

Case 3: PF1, GB2, E1

The key equation here is again given by [1]. A permanent increase in g is financed here by an inflation tax. Increasing g obviously requires increasing inflation (lowering R, the real return on government money). But if R is lowered, then condition [1] implies that k+g increases. That is, individuals substitute out of money and into capital (private or public). Consequently, if the government increases g permanently and finances it with money creation, output expands. (Note: this result need not be welfare improving. Do not confuse GDP with  economic welfare).

Case 4: PF1, GB2, E2

OK, so here we have a one-time increase in g financed by a one-time increase in the money supply. I think that this is what Lucas likely had in mind when he claimed that a money-financed increase in g stimulates.

The analysis here becomes a little more complicated because we have to do "out of steady state" analysis. Let me instead give you the intuition.

It is known that for OLG models, that money is not generally neutral (despite the fact that prices are completely flexible--indeed, I think that price flexibility is critical for the  non-neutrality result). In this model, a one-time increase in the money supply to finance a temporary increase in g will cause a surprise jump in the price level, which has the effect of reducing the purchasing power of the money brought into the period by the old. (If you are an Austrian, you will complain that the old have had their savings stolen by the surprise inflation policy). The effect is to divert purchasing power away from the old (who want to consume) toward the young (who would rather invest). This money-financed increase in g will stimulate; which is consistent with what Lucas said. Moreover, the result relies on a failure of Ricardian equivalence. (In a model with an infinitely-lived representative agent, the money-financed increase in g would have no effect at all, given PF1).

Conclusion

A reader of mine provided me with this quote (apparently, from Brad DeLong):

I learned this from Andy Abel and Olivier Blanchard before my eyes first opened: increases in government purchases are ineffective only if (a) "Ricardian Equivalence holds and (b) what the government buys (and distributes to households) is exactly what households would buy for themselves. RE by itself doesn't do it."

I think this is a nice way to summarize things. (Keep in mind that "ineffective" in the quote above means "no effect--whether good or bad.")

In conclusion, Lucas' remarks need not be interpreted as his theory relying on RE. Indeed, as I hope to have made clear above, his remarks, when taken together, require a departure from RE. The key assumption he makes, in my view (who really knows?) is the part (b) in DeLong's quote (my PF1). That part has nothing to do with RE.

Happy New Year, everyone!

Postscript Dec. 31, 2011
An economist that I admire once said this:

"...just talking plausibly about economics is not the same as having a real understanding; for that you need crisp, tightly argued models."

In case you missed it, Krugman takes a nice shot at me here: I Like Math. I like the cartoon! Moreover, I agree with what he says: "If you resort to math to justify what looks like a very foolish claim, and you can't find a way to express that justification in plain English, then something is wrong."

By "foolish," I presume he means "logically invalid" and not "empirically implausible." For those who speak the language of math and are familiar with OLG models, I have shown that there is a logic to the Lucas view as expressed in that speech. (I don't personally believe that the view is empirically plausible, but that is beside the point of my original post). I have shown that the logic implies a departure from RE; contrary to Krugman's claim. I have tried to express this logic in plain language here and here. And in keeping with the sentiment of the quote above (yes, by PK), I tried to re-express the logic in mathematical form to complement what I said earlier. If I have failed in any way, it is in my ability to communicate the idea in "plain" English. I am not as talented as Krugman in this regard. The logic of my argument, however, remains sound.

But I think it is now time to stop. Let me end by alerting you to an interesting take on the matter by Henrik Jensen: The Krugman Multiplier is Too Big. (He includes a link to a video of the speech by Lucas.)

Postscript Jan. 2, 2012

I should have linked up to this classic paper by Neil Wallace earlier than this, but better late than never: A Modigliani-Miller Theorem for Open Market Operations. As macroeconomists know, there is a strong connection between RE and MM (they are essentially the same proposition applied in different contexts). The Wallace paper asserts that open market operations "matter" only to the extent that some or all of the assumptions that underlie RE/MM are violated. Lucas believes that monetary policy matters. Ergo, his arguments (whatever they might be) cannot be based on RE alone.

Postscript Jan. 09, 2012

Well, I'm sure this one is going to fly under the radar, but I feel the need to record it here. It seems that Brad DeLong agrees with me (h/t Mark Thoma); see here. (Well, he doesn't mention me by name, but his elaboration squares with what I have been trying to say all along.)

Yes indeed, one may question whether the mix of publicly provided goods and services substitutes more or less well with privately supplied goods and services. It matters for whether how a change in G is likely to impact the economy. Ultimately, it is an empirical question. And it has nothing to do with RE. Krugman was wrong to question Lucas' understanding of his own theory. Instead, he could have legitimately questioned Lucas' parameter estimates governing the substitutability of private and public expenditure. But really now, I suppose that would have been a lot less fun.

Postscript Jan. 11, 2012

Krugman is like your neighbor's annoying little puppy that just won't stop gnawing at your feet. Scott Sumner weighs in here: Nobel Prizes for Alchemy?

Ah, controversy. What a great way to end the year!

I want to comment on Mark Thoma's post today about the Ricardian Equivalence Theorem (RET). Linking up to the interview with Barro was a good idea, Mark. Everyone agrees that the theorem has nothing to say about the effectiveness of G, and Barro explains all of this splendidly. Moreover, everyone agrees that since the conditions needed to render the proposition valid are violated in reality, the proposition cannot possibly be expected to make a perfectly accurate prediction of how altering the timing of taxes (holding G fixed) is likely to impact the economy. I guess that this is about where our mutual agreement ends.

What is there left to argue about? It's the holidays--I'm sure we'll find something. Let's start with Mark's opening paragraph:

I haven't said much about the recent flare up over Ricardian equivalence. Why? The answer's simple, the empirical evidence does not support it. Why argue about something when we already know it fails to adequately explain the data? Making the Ricardian equivalence assumption might be okay as a first approximation for some questions--though I'd argue that it mostly isn't--but in any case the theory does not adequately capture economic behavior.

I'm not exactly sure which flare up he is talking about, but I suspect that I may be involved in it somewhere, owing to this post here: Does Krugman Understand the Ricardian Equivalence Theorem?

I want to clear up a few things regarding that post. First, I was not trying to defend Lucas' views on fiscal stimulus. Lucas's view on the matter (insofar as one can gather it from what was clearly an informal and off-the-cuff speech) appears to be that a money-financed increase in G is stimulative, while a tax-financed increase in G is not. Now, there may be several ways to criticize the "rationale" of his argument. But whatever criticism you pick, it most certainly cannot be centered on Lucas' alleged appeal to the Ricardian equivalence theorem. For crying out loud--the man is claiming that the method of financing matters for a given G. This can only be true if the Ricardian proposition fails to hold in reality.

Now, what of Mark's claim that the empirical evidence does not support the RET? Well, as I said above, given that we live in a world of distortionary taxation, borrowing constraints, finite planning horizons, etc., etc., it would indeed be remarkable if the predictions of RET held up exactly in the data.

But surely that is setting the bar a little too high (not one of us has a theory that can perfectly predict such outcomes). Rather, the question is whether or not the assumptions constitute sufficiently good approximations for the purpose at hand (i.e., for a given policy experiment). Indeed, in the interview posted by Mark, Barro states his view on the matter quite plainly:

As a first-order proposition, it is right that it matters little whether you pay for government spending with taxes today or taxes tomorrow...

So, to Barro it seems that the empirical evidence broadly supports the proposition, at least, to a first-order approximation. If so, that is bad news for me, because I like to work with models where the proposition fails. It would, however, be good news for those promoting an increase in G in the face of large deficits (the size of the deficit should not factor into the debate, if the proposition holds true).

In any case, I'm not sure whether Mark's claim about the empirical evidence not supporting RET is entirely valid. I am reminded of a paper I once saw Emanuela Cardia present: Replicating Ricardian Equivalence Tests With Simulated Series. Here is the abstract:

This paper  replicates standard consumption function  tests of Ricardian equivalence  using series  generated from  a  model which nests Ricardian equivalence within a  non-Ricardian alternative (due  to finite  horizons and/or  distortionary taxation). I show that the estimates of the effects of taxation on consumption are not robust and that standard tests may have weaknesses which can lead to conflicting results, whether Ricardian equivalence holds or not. The simulations also show that no clear conclusions about Ricardian equivalence can be drawn from observing a low correlation between the current account and government budget deficits.

In short, I think that the empirical evidence may be somewhat more mixed than what Mark suggests.

At the end of the day, I think that the key lesson of the RET is not (for example) that "deficit financed tax cuts do not matter." Rather, the lesson should be that "such a policy is likely to be much less stimulative than you would expect if you were to base your thinking on a model that did not incorporate Ricardian forces."

Now who wants to argue with that?

Suppose that the government wants to acquire the resources necessary to implement a new expenditure program G = {g₁, g₂, g₃ ... }, where g_t denotes government purchases of goods and services at date t.

Let us take G as given. To begin their evaluation of G, macroeconomists ask the following two questions. First, what are the likely macroeconomic consequences of implementing program G? Second, does the answer to first question depend on how G is financed? (Financing is assumed to take the form of taxes, deficits, and money creation, or some combination thereof).

The sovereign debt crisis in Europe has garnered most of our attention as of late. But should Europe really be our main concern? For several months now, many economists (including myself) have been casting a nervous eye over to China. Paul Krugman summarizes these concerns nicely in his NYT article today: Will China Break? Mark Gongloff earlier asked the million dollar question here: China's Shadow Banking System: The Next Subprime?  Hmm...

P.S. And what's with these stories I keep hearing about China's missing bosses? (e.g., China's Vanishing Factory Bosses). Sounds ominous, if true. We truly do live in interesting times. 

Addendum: Dec. 13, 2011

As I have stressed in an earlier post, one should be careful in using these raw correlations to identify the source of disturbance; see: Interpreting the Beveridge Curve.

A reader points out that the Monster Employment Index (available since 2004) might be of some use for measuring regional employment opportunities. 

On October 9, 2010, I posted some regional vacancy-unemployment data for the United States; see: Beveridge Curves for 36 U.S. Cities.

My measure of vacancies was the Conference Board's help-wanted index (HWI).  A colleague of  mine (Silvio Contessi) pointed me to a paper by Regis Barnichon (EL 2010) that identifies a major flaw in this data series. Barnichon summarizes the problem here:

The traditional measure of vacancy posting is the Conference Board Help-Wanted Index (HWI) that measures the number of help-wanted advertisements in 51 major newspapers. However, since the mid-1990s, this “print” measure of vacancy posting has become increasingly unrepresentative as advertising over the internet has become more prevalent. Instead, economists increasingly rely on the Job Openings and Labor Turnover Survey (JOLTS) measure of job openings. However, this measure is only available since December 2000 and cannot be used to contrast current labor market situations with past experiences.

In this paper, I build a vacancy posting index that captures the behavior of total—“print” and “online”—help-wanted advertising, by combining the print HWI with the online Help-Wanted Index published by the Conference Board since 2005. 

Here is how Barnichon's correction looks for the aggregate data.


That is, the secular decline (blue line) in the original HWI series is estimated to be entirely the consequence of a substitution away from print to electronic forms of job advertising.

With this in mind, I asked my tireless research assistant (Constanza Liborio) to recalculate our regional Beveridge curves using Barnichon's correction (for those interested, I can email you a file describing the exact procedure employed).

The regional vacancy data was purchased from the Conference Board (their Help Wanted Online data series), so unfortunately, I cannot make it available to you without their permission. I have permission to display the data, however. Here is what we get.

What principles should govern the way a lender-of-last-resort (LLR) operates during a financial crisis? On this question, one is frequently referred to two key principles, attributed to Walter Bagehot in his book Lombard Street. The two principles are usually summarized as "lend freely and at a penalty rate." What does this mean?

In "normal times," firms regularly borrow cash on a short-term basis (say, to meet payroll). These loans are usually collateralized with a host of assets (e.g., accounts receivable, property, securities). The dictum "lend freely" in this context means to extend cash loans freely against the collateral that is normally put up to secure short-term lending arrangements.

The rationale commonly offered for the LLR is that during a crisis, "perfectly good" collateral assets are either no longer accepted as security for short term loans, or that if they are, they are heavily discounted (e.g., a creditor will only lend 75 cents for every dollar in collateral, instead of the usual 99 cents). Whatever the ultimate cause, this type of "liquidity crisis" creates havoc in the payments system. This havoc can be avoided, or at least mitigated, by a LLR that stands ready to replace the "missing" lending activity. (Or so the story goes.)

Let's say that the normal discount rate on high grade collateral is 0 < d < 1. So if a creditor offers to lend $99 for every $100 in collateral, d = 0.01 (one percent discount). Let's suppose that during a crisis, the discount rate rises to c > d. (If c = 0.5, then there is a 50% discount or "haircut" on collateral.) One issue that the LLR must address is the discount rate it should offer on an emergency loan. Let me call this discount rate p.

Now, if the LLR sets p = c, then what is the point of having an LLR? So clearly, if the LLR is to influence lending activity in any manner, it must  set p < c. Opponents of LLR activity like to label p < c a "bailout." (This term is rarely defined precisely; it appears to be a label to attach to programs that one does not like.)

At the other extreme, the LLR could set p = d. In this case, the LLR is discounting collateral in the same way that the market does during "normal" times. If the LLR instead set p > d, it is charging a "penalty rate." (Note: I do not think that Bagehot ever used this term.) How should the LLR set this penalty rate and why? Here is Bagehot:

First. That these loans should only be made at a very high rate of interest. This will operate as a heavy fine on unreasonable timidity, and will prevent the greatest number of applications by persons who did not require it. The rate should be raised early in the panic, so that the fine may be paid early; that no one may borrow out of idle precaution without paying well for it; that the Banking reserve may be protected as far as possible. (emphasis, my own)

Well, O.K., so he does not appear to answer the question of what discount rate to apply; only that it should be "very high." But I am less interested here in the precise penalty rate as to the rationale for why a penalty rate is necessary. A colleague of mine (who appears to have done a great deal of reading in the area) suggests that the rationale was primarily to ensure that the Bank of England did not run out of reserves (an event that would have led to its failure, and the subsequent end of civilization in the minds of many at the time).

Of course, the Federal Reserve Bank of the United States does not face the prospect of running out of cash reserves, the way that the Bank of England did back then. This is because "cash" back then took the form of specie (gold and silver coin). "Cash" today takes the form of small denomination paper notes (and electronic digits in reserve accounts) that the Fed can issue "out of thin air." In light of our modern institutional structure, I wonder whether Bagehot, living in today's world, might not have dropped his "penalty rate" dictum?

Is there any good reason left for the penalty rate? Perhaps there is. But it is clearly of second-order importance during a financial crisis. First, lend freely. It is probably not the time to worry about this penalty rate or that penalty rate in the depths of a liquidity crisis. If an institution is deemed, after the fact, to have benefited "unfairly" at the expense of society during an emergency lending episode, then a "fee for service" (i.e., tax) could be applied after the crisis has passed.

P.S. Would be interested to hear from historians on this subject.

Postcript: January 30, 2012
I would like to thank Josh Hendrickson for sending me the link to this paper:

Turning Bagehot on his Head.
Abstract: Ever since Bagehot’s (1873) pioneering work, it is a widely accepted wisdom that in order to alleviate (ex ante) bank moral hazard, a lender of last resort should lend at penalty rates only. In a model in which banks are subject to shocks but can exert effort to affect the likelihood of these shocks, we show that the validity of this argument crucially relies on banks always remaining solvent. The reason is that when banks become insolvent, Bagehot’s prescription dictates to let them fail. Penalty rates charged when banks are illiquid (but solvent) then reduce banks’ incentives to avoid insolvency ex ante and thus increase bank moral hazard. We derive a condition which shows precisely when this effect on ex ante incentives outweighs the traditional one and show that it is fulfilled under plausible scenarios.