On Paulo and Bobby, English and Math
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 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:
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?
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?
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