Believe those who are seeking the truth. Doubt those who find it. Andre Gide


Tuesday, November 18, 2014

Japan: Some Perspective

So Japan is in recession.  And it's all so unexpected. Ring the alarm bells!

Well, hold on for a moment. Take a look at the following diagram, which tracks the Japanese real GDP per capita since 1995 (normalized to equal 100 in that year). I also decompose the GDP into its expenditure components: private consumption, government consumption, private investment, and government investment (I ignore net exports). The GDP numbers go up to the 3rd quarter, the other series go up to only the 2nd quarter.



In terms of what we should have expected, I think it's fair to say that most economists would have predicted the qualitative nature of the observed dynamic in response to an anticipated tax hike. That is, we'd expect people to substitute economic activity intertemporally--front loading activity ahead of the tax hike, then curtailing it just after. And qualitatively, that's exactly what we see in the graph above. But does the drop off in real per capita GDP really deserve all the attention it's getting? I don't think so. The fact that the economy was a little weaker in the 3rd quarter than expected (the two consecutive quarters of GDP contraction is what justified labeling the event a "recession") is not really something to justify wringing one's hand over. Not yet, at least.

By the way, if you're interested in reading more about the Koizumi boom era, see my earlier post here: Another look at the Koizumi boom.

Saturday, November 15, 2014

Roger Farmer on labor market clearing.

While I'm a huge fan of Roger Farmer's work, I think he gets this one a little wrong:  Repeat After Me: The Quantity of Labor Demanded is Not Always Equal to the Quantity Supplied. I am, however, sympathetic to the substantive part of his message. Let me explain.

The idea of "supply" and "demand" is rooted in Marshall's scissors (a partial equilibrium concept). The supply and demand framework is an extremely useful and powerful way of organizing our thinking on a great many matters. And it is easy to understand. (I have a pet theory that if you really want to have an idea take hold, you have to be able to represent it in the form of a cross. The Marshallian cross. The Keynesian cross. Maybe even the Christian cross.)

The Marshallian perspective is one in which commodities are traded on impersonal markets--anonymous agents trading corn and human labor alike in sequences of spot trades. Everything that you would ever need to buy or sell is available (absence intervention) at a market-clearing price. The idea that you may want to seek out and form long-lasting relationships with potential trading partners (and that such relationships are difficult to form) plays no role in the exchange process--an abstraction that is evidently useful in some cases, but not in others.

I think what Roger means to say is that (repeat after me) the abstraction of anonymity, when describing the exchange for labor services, is a bad one. And on this, I would wholeheartedly agree (I've discussed some of these issues in an earlier post here).

Once one takes seriously the notion of relationship formation, as is done in the labor market search literature, then the whole concept of "supply and demand" analysis goes out the window. That's because these well-defined supply and demand schedules do not exist in decentralized search environments. Wage rates are determined through bargaining protocols, not S = D. To say, as Roger does, that demand does not always equal supply, presupposes the existence of Marshall's scissors in the first place (or,  more generally, of a complete set of Arrow-Debreu markets).

And in any case, how can we know whether labor markets do not "clear?" The existence of unemployment? I don't think so. The neoclassical model is one in which all trade occurs in centralized locations. In the context of the labor market, workers are assumed to know the location of their best job opportunity. In particular, there is no need to search (the defining characteristic of unemployment according to standard labor force surveys). The model is very good at explaining the employment and non-employment decision, or how many hours to work and leisure over a given time frame. The model is not designed to explain search. Hence it is not designed to explain unemployment. (There is even a sense in which the neoclassical model can explain "involuntary" employment and non-employment. What is "involuntary" are the parameters that describe an individuals' skill, aptitude, etc. Given a set of unfortunate attributes, a person may (reluctantly) choose to work or not. Think of the working poor, or those who are compelled to exit the labor market because of an illness.)

Having said this, there is nothing inherent in the neoclassical model which says that labor market outcomes are always ideal. A defining characteristic of Rogers' work has been the existence of multiple equilibria. It is quite possible for competitive labor markets to settle on sub-optimal outcomes where all markets clear. See Roger's paper here, for example.

The notion that supply might not equal demand may not have anything to do with understanding macroeconomic phenomena like unemployment. I think this important to understand because if we phrase things the way Roger does, people accustomed to thinking of the world through the lens of Marshall's scissors are automatically going to look for ways in which the price mechanism fails (sticky wages, for example). And then, once the only plausible inefficiency is so (wrongly) identified, the policy implication follows immediately: the government needs to tax/subsidize/control wage rates. In fact, the correct policy action may take a very different form (e.g., skills retraining programs, transportation subsidies, job finding centers, etc.)

Monday, November 10, 2014

A dirty little secret


Shhh...I told you *nothing!* 
There's been a lot of talk lately about the so-called "Neo-Fisherite" proposition that higher nominal interest rates beget higher inflation rates (and vice-versa for lower nominal interest rates). I thought I'd weigh in here with my own 2 cents worth on the controversy.

Let's start with something that most people find uncontroversial, the Fisher equation:

[FE]  R(t) = r(t) + Π (t+1)

where R is the gross nominal interest rate, r is the gross real interest rate, an Π is the expected gross inflation rate (all variables logged).

I like to think of the Fisher equation as a no-abitrage condition, where r represents the real rate of return on (say) a Treasury Inflation Protected Security (TIPS) and (R - Π) represents the expected real rate of return on a nominal Treasury. If the two securities share similar risk and liquidity characteristics, then we'd expected the Fisher equation to hold. If it did not hold, a nimble bond trader would be able to make riskless profits. Nobody believes that such opportunities exist for any measurable length of time.

Let me assume that the real interest rate is fixed (the gist of the argument holds even if we relax this assumption). In this case, the Fisher equation tells us that higher nominal interest rates must be associated with higher inflation expectations (and ultimately, higher inflation, if expectations are rational). But association is not the same thing as causation. And the root of the controversy seems to lie in the causal assumptions embedded in the Neo-Fisherite view.

The conventional (Monetarist) view is that (for a "stable" demand for real money balances), an increase in the money growth rate leads to an increase in inflation expectations, which leads bond holders to demand a higher nominal interest rate as compensation for the inflation tax. The unconventional (Neo-Fisherite) view is that lowering the nominal interest leads to...well, it leads to...a lower inflation rate...because that's what the Fisher equation tells us. Hmm, no kidding?
 
The lack of a good explanation for the economics underlying the causal link between R and Π is what leads commentators like Nick Rowe to tear at his beard. But the lack of clarity on this dimension by a some writers does not mean that a good explanation cannot be found. And indeed, I think Nick gets it just about right here. The reconciliation I seek is based on what Eric Leeper has labeled a dirty little secret; namely, that "for monetary policy to successfully control inflation, fiscal policy must behave in a particular, circumscribed manner." (Pg. 14. Leeper goes on to note that both Milton Friedman and James Tobin were explicit about this necessity.)

The starting point for answering the question of how a policy affects the economy is to be very clear what one means by policy. Most people do not get this very important point: a policy is not just an action, it is a set of rules. And because monetary and fiscal policy are tied together through a consolidated government budget constraint, a monetary policy is not completely specified without a corresponding (and consistent) fiscal policy.

When Monetarists claim that increasing the rate of money growth leads to inflation, they assert that this will be so regardless of how the fiscal authority behaves. Implicitly, the fiscal authority is assumed to (passively) follow a set of rules: i.e., use the new money to cut taxes (via helicopter drops), finance government spending, or pay interest on money. It really doesn't matter which. (For some push back on this view, see Price Stability: Is a Tough Central Banker Enough? by Lawrence Christiano and Terry Fitzgerald.)

When Neo-Fisherites claim that increasing the nominal interest rate leads to inflation, the fiscal authority is also implicitly assumed to follow a specific set of rules that passively adjust to be consistent with the central bank's policy. At the end of the day, the fiscal authority must increase the rate of growth of its nominal debt (for a strictly positive nominal interest rate and a constant money-to-bond ratio, the supply of money must be rising at this same rate.) At the same time, this higher rate of debt-issue is used to finance a higher primary budget deficit (just think helicopter drops again).

Well, putting things this way makes it seem like there's no substantive difference between the two views. Personally, I think this is more-or-less correct, and I believe that Nick Rowe might agree with me. I hestitate a bit, however, because there may be some hard-core "Neo-Wicksellians" out there that try to understand the interest rate - inflation dynamic without any reference to fiscal policy and nominal aggregates. (Not sure if this paper falls in this class, but I plan to read it soon and comment on it: The Perils of Nominal Targets, by Roc Armenter).

If the view I expressed above is correct, then it suggests that just limiting attention to (say) the dynamics of the Fed's balance sheet is not very informative without reference to the perceived stance of fiscal policy and how it interacts with monetary policy. Macroeconomists have of course known this for a long time but have, for various reasons, downplayed the interplay for stretches of time (e.g., during the Great Moderation). Maybe it's time to be explicit again. Let's help Nick keep his beard.
 

Monday, October 20, 2014

What's holding back female employment?

Almost four years ago, I asked whether the U.S. was in for a labor market slump similar to the slump experience in Canada during the 1990's. Evidently, the answer turned out to be yes.

How is the U.S. faring relative to Canada back then? American prime-age males seem to be tracking their Canadian counterparts, both in terms of employment-to-population ratios and in labor force participation rates. American females, on the other hand, appear to be lagging behind their Canadian counterparts. Let me show you some data.

Let's begin by looking at the employment ratio for prime-age males:


As you can see, the sharp drop and subsequent recovery dynamic for prime-age males is remarkably similar across these two countries and time periods. (The initial E-P ratio was about 87% for both countries; see here).

Here is what their labor force participation rates look like:


Again, the recovery dynamic looks almost identical (The initial part rate for Canada was 93%, for the US about 91%; see here).

Alright, now let's take a look at the same statistics for prime-age females. First, the employment ratios:


These dynamics look quite a bit different. The main effect of the recession in Canada was to slow down the growth rate in the employment ratio. In the U.S., the effect has been to reduce the employment ratio, with only a very weak sign of recovering in the past year.

Here is what the labor force participation rate dynamics look like:


Again, two very different recovery dynamics.

A colleague of mine suggested that state-level layoffs in education and government may explain a good part of the lackluster recovery dynamic for U.S. females. This is certainly worth looking into. However, if we take a look at the following diagram, we see that the discrepancy appears to have happened much earlier -- around 1997, in fact.


It seems unlikely to me that the divergence between Canadian and American prime-age females is driven by cyclical considerations (although, a small part of the recent gap may be). Work incentives are likely to have changed, although what these changes were, I do not yet know. In any case, I doubt that monetary policy is a tool that can be used to close this gap. I can think of plenty fiscal interventions that might help, however.

Addendum Oct. 22, 2014

My colleague, Maria Canon, points me to the following paper by Sharon Cohany and Emy Sok Trends in labor force participation of married mothers of infants, as well as this interesting set of slides by Jennifer Hunt: Female labor force participation: slack and reform.

And here's a real doozy "Universal Child Care, Maternal Labor Supply, and Family Well-Being" by Michael Baker, Jonathan Gruber, and Kevin Milligan (JPE 2008). From the abstract:
We analyze the introduction of highly subsidized, universally accessible child care in Quebec, addressing the impact on child care utilization, maternal labor supply, and family well-being. We find strong evidence of a shift into new child care use, although some crowding out of existing arrangements is evident. Maternal labor supply increases significantly. Finally, the evidence suggests that children are worse off by measures ranging from aggression to motor and social skills to illness. We also uncover evidence that the new child care program led to more hostile, less consistent parenting, worse parental health, and lower-quality parental relationships.

Monday, September 8, 2014

Who's Afraid of Deflation?

Everyone knows that deflation is bad. Bad, bad, bad. Why is it bad? Well, we learned it in school. We learned it from the pundits on the news. The Great Depression. Japan. What, are you crazy? It's bad. Here, let Ed Castranova explain it to you (Wildcat Currency, pp.160-61):

Deflation means that all prices are falling and the currency is gaining in value. Why is this a disaster? ... If you hold paper money and see that it is actually gaining in value, it may occur to you that you can increase your purchasing power--make a profit--by not spending it...But if many people hold on to their money, this can dramatically reduce real economic activity and growth...

In this post, I want to report some data that may lead people to question this common narrative. Note, I am not saying that there is no element of truth in the interpretation (maybe there is, maybe there isn't). And I do not want to question the likely bad effects that come about owing to a large unexpected deflation (or inflation).  What I want to question is whether a period of prolonged moderate (and presumably expected) deflation is necessarily associated with periods of depressed economic activity. Most people certainly seem to think so. But why?

The first example I want to show you is for the postbellum United States (source):


Following the end of the U.S. civil war, the price-level (GDP deflator) fell steadily for 35 years. In 1900, it was close to 50% of its 1865 value. In the meantime, real per capita GDP grew by 85%. That's an average annual growth rate of about 1.8% in real per capita income. The average annual rate of deflation was about 2%. I wonder how many people are aware of this "disaster?"

O.K., well maybe that was just long ago. Sure. Let's take a look at some more recent data from the United States, the United Kingdom, and Japan. The sample period begins in 2009 (the trough of the Great Recession) and ends in late 2013. Here is what the price level dynamic looks like since 2009:


Over this five year period, the price level is up about 7% in the United States and about 11% in the United Kingdom. As for Japan, well, we all know about the Japanese deflation problem. Over the same period of time, the price level in Japan fell by almost 7%.

Now, I want you to try to guess what the recovery dynamic--measured in real per capita GDP--looks like for each of these countries. Surely, the U.K. must be performing relatively well, Japan relatively poorly, and the U.S. somewhere in the middle?

You would be correct in supposing that the U.S. is somewhere in the middle:


But you would have mixed up the U.K. with Japan. Since the trough of the past recession, Japanese real per capita GDP is up 15% (as of the end of 2013)--roughly 3% annual growth rate. Is deflation really so bad? Maybe the Japanese would like the U.K. style inflation instead? I don't get it.

I have some more evidence to contradict the notion of deflation discouraging spending (transactions). The evidence pertains to Bitcoin and the data is available here: Blockchain.

Many people are aware of the massive increase in the purchasing power of Bitcoin over the past couple of years (i.e., a massive deflationary episode). As is well-known, the protocol is designed such that the total supply of bitcoins will never exceed 21M units. In the meantime, this virtual currency and payment system continues to see its popularity and use grow.


One might think that given the prospect of continued long run deflation--i.e, price appreciation (it's hard to believe that holders of bitcoin are thinking anything else)--that people would generally be induced to hoard and not spend their bitcoins. And yet, available data seems to suggest that this may not be the case:


Maybe deflation is not so bad after all?  Let's hope so, because we may all have to start getting used to the idea!

Additional readings:
[1] Good vs. Bad Deflation: Lessons from the Gold Standard Era (Michael Bordo and Angela Redish).

[2] Deflation and Depression: Is There an Empirical Link? (Andy Atkeson and Pat Kehoe).

[3] The Postbellum Deflation and its Lessons for Today (David Beckworth).

Friday, July 25, 2014

Debt: The First 5000 Years

Ah, the airport bookstore. As monetary theorist and history buff, I could not resist this tantalizing title: Debt: The First 5000 Years. The book is authored by anthropologist David Graeber, a leading figure in the Occupy Wall Street movement. But what grabbed me was the summary on the back cover, which states (among other things) that every economics textbook is wrong in the way it explains the emergence of money, which goes something like this: "Once upon a time, there was barter. It was difficult. So people invented money." [p28].

I think we (economists) have to score one for the anthropologists here. I remember being taught that story and it took me some time to figure out it was wrong. What makes barter difficult? We are taught that the difficulty stems from a "lack of coincidence of wants." Consider, for example, an island populated by three people, Adam, Betty and Charlie. Adam wants breakfast, Betty wants lunch, Charlie wants dinner. Adam can deliver dinner, Betty can deliver breakfast, and Charlie can deliver lunch. There are no bilateral gains to trade (no voluntary trade would occur between any arbitrary pairing of individuals). And yet, there are clearly multilateral gains to trade.


The solution, we are told, is to introduce a monetary object and endow it to Adam, who may then purchase his breakfast from Betty with cash. Betty then uses her money to buy lunch from Charlie. Charlie then uses his money to buy dinner from Adam, and so on.

As anthropologists have pointed out for a long time, there is really little evidence of trade taking this form in primitive communities (see: Famous Myths of Fiat Money, by Dror Goldberg). Instead, these societies operated as "gift giving" economies, or informal credit systems. The principle should be familiar to all of us: it is reflected in the way we trade favors with friends, family, and other members of social networks to which we belong.

What then, explains monetary exchange (really, the coexistence of money and credit)? According to Kiyotaki and Moore, Evil is the Root of All Money. "Evil" here is interpreted as the existence of untrustworthy (noncooperative) people. Untrustworthy individuals readily accept gifts from the community, but cannot be trusted to fulfill their implicit obligation to reciprocate in-kind when an opportunity to do so arises. However, we know from game theory that a system of "cooperative" exchange might still be sustained if untrustworthy people can be compelled to behave properly, say, by the threat of punishment for noncompliant behavior (e.g., ostracism from the community).

The punishment/reward system that implicitly exists in gift-giving societies requires (to the extent that some community members are untrustworthy) a communal monitoring of individual behavior. In small communities, "everybody knows everything about everyone" and so this is arguably why "communistic" societies can be sustained in small groups. It also suggests why the arrangement breaks down for larger groups. The virtual communal data bank -- a distributed network of computer brains -- is simply not capable of recording all the information necessary to support an informal credit system in a large population. In a large population, people can remain anonymous. We necessarily become strangers to most people. And its tough to trust a stranger (a person you are not likely ever to meet again).

Nevertheless, multilateral gains to trade may still exist even among strangers. And if credit is difficult, or impossible, then the solution is money (see: The Technological Role of Fiat Money, by Narayana Kocherlakota). According to this theory, money serves as a substitute for the missing communal memory. Contributions to society are now measured not by virtual credits in the collective mind of the community; instead, they are recorded by money balances (this assumes, of course, that money, like virtual credit, is difficult to counterfeit/steal).

So, in a nutshell, economic theory suggests that we use informal credit arrangements to govern exchange among people we know (family, friends, colleagues, etc.) and we use money to facilitate exchange with "strangers." The emergence of money then seems tied to the emergence of strangers. An obvious explanation for this is population growth (and the associated rise of large urban areas).

One thing I learned from Graeber is that the relative importance of money and credit seems to have waxed and waned over time. Money (in particular, coinage) emerged around 800BC and remained significant until about 600AD, an era associated with many great empires, and the associated need to pay transient professional armies. With the collapse of the great empires, new states emerged, increasingly under the regulation of religious authorities. Coinage declined in importance, with credit systems taking over (600AD-1450AD). This latter observation is consistent with the general decline of urban areas in western Europe, but Graeber points to many other factors as well. Monetary exchange waxes once again with the age of the "great capitalist empires" (1450-1971AD).

My comments above only scratch the surface of the book's much broader thesis concerning the moral nature of debt. The presentation is not as clean as it could be, the analysis is sloppy in several places, and the conclusion is rather weak but, heck, it's still a very interesting read. If nothing else, it encouraged me to interpret various aspects of history in ways that I am not accustomed to.

Alas, every gain comes at a price (beyond the $22 cash I paid for his book). His opening chapter, in particular, is so annoying that it almost led me to abort the enterprise. In the book, and in the many interviews he gives, he relays the following story (source):
And one of the things that really fascinated me was the moral power of the idea of debt. I would tell stories to people, very sympathetic people, liberal lawyers, well-meaning do-gooder types, and you’d tell these stories about horrible things. You know, in Madagascar, for example, the IMF came in with these policies, you have to cut the budgets because, god knows, we can’t reduce the interest payments you owe to Citibank, they owed all this money. And they had to do things like get rid of mosquito eradication programs, as a result that malaria returned to parts of the country where it had been wiped out for a hundred years and tens of thousands of people died and you had dead babies being buried and weeping mothers. I was there, I saw this sort of thing. You described this to people and the reaction would be, well, that’s terrible, but surely people have to pay their debts. You’re not suggesting they cancel it or default, that would be outrageous. And one of the things that really fascinated me was the moral power of the idea of debt.
I'm not completely sure, but if I was to relay this story to the average person I know, I would hardly expect them to say "well, that's terrible, but surely people have to pay their debts!" I'm pretty sure that most of the people I know would have replied "that's $^%& outrageous!" But then, maybe I don't know too many "sympathetic" people, liberal lawyers and well-meaning do-gooder types.

Moreover, I'm pretty sure that a significant majority of the people I know would have questioned the claim that the IMF kills African babies. After all, we are not speaking here of a paragon of good government.
Since Madagascar gained independence from France in 1960, the island's political transitions have been marked by numerous popular protests, several disputed elections, an impeachment, two military coups and one assassination. The island's recurrent political crises are often prolonged, with detrimental effects on the local economy, international relations and Malagasy living standards. (source)
Of course, malaria was for a long time a big problem on the African continent (see here) and elsewhere. But the disease was practically wiped out with the use of the pesticide DDT (see here). The use of DDT was then banned, owing to pressure from "well-meaning do-gooder" environmental groups. [Evidently, the ban was primarily for agricultural use, and only sometimes in vector control]. Now, according to this source:
In the 1980’s Madagascar stopped using DDT and immediately had an epidemic of malaria, resulting in the death of more than 100,000 people.
Hmm. And according to this source:
A strong malaria epidemic with a high mortality rate occurred on the Madagascar Highlands in 1986-88. Vector control and free access to antimalaria drugs controlled the disease.
This latter source also mentions the lack of immunity and a shortage of medicaments as factors contributing to the mortality rate. Is Graeber suggesting that the shortage of medicaments was the consequence of IMF imposed austerity measures on Madagascar's government and the desire to service Citibank debt? It seems an unlikely story (although, it's not easy to find details). According to this data from the World Malaria Report, almost all the resources for fighting malaria in Madagascar originates from international aid organizations, like USAID and The Global Fund. Did the IMF prevent these agencies from doing their good work?

Finally, let me point readers to Ken Rogoff's defense of IMF policies here. See also the article here, by Masood Ahmed.

Make no mistake, the malaria episode described by Graeber is a tragic story. People were dying and somewhere the resources existed that could have mitigated the losses (a continued program of DDT spraying would have prevented it altogether). Among other things, the government of Madagascar could have reallocated resources away from some expenditure (say, military, which is 1% of GDP according to CIA Factbook) toward medicaments. That it evidently chose not to is revealing. Does Graeber truly believe that a "debt jubilee" for governments of this nature would have prevented the episode in question? (Note: I am not against debt jubilees.)

Graeber has many useful and interesting things to say in his book. I personally find it annoying that a scholar and writer of such high caliber has to resort to stories like this to sell his ideas. But maybe that's just me. In any case, my recommendation is to read the book and filter out as much of the noise as you can. 

Wednesday, June 25, 2014

Excess reserves and inflation risk: A model

===================================================================
Note: The following is an edited version of my original post. Thanks to Nick Edmonds for pointing out an inconsistency in my earlier analysis. Nick's comment forced me to think through the properties of my model more carefully. In light of his observation, I have modified the original model to include capital investment. My earlier conclusions remain unchanged. 
===================================================================

I should have known better than to reason from accounting identities. But that's basically what I did in my last post and Nick Rowe called me out on it here. So I decided to go back and think through the exercise I had in mind using a simple model economy.

Consider a simple OLG model, with 2-period-lived agents. The young are endowed with output, y. Let N denote the number of young agents (normalize N=1). The young care only about consumption when they are old (hence, they save all their income y when young). Agents are risk-averse, with expected utility function E[u(c)]. There is a storage technology. If a young agent saves k units of output when young, he gets x*f(k) units of output in the next period, where x is a productivity parameter and f(.) is an increasing and strictly concave function (there are diminishing returns to capital accumulation). Assume that capital depreciates fully after it is used in production.

If x*f'(y) > 1, the economy is dynamically efficient. If x*f'(y) < 1, the economy is dynamically inefficient (and there is a welfare-enhancing role for government debt).

Now, imagine that there are two such economies, each in a separate location. Moreover, suppose that a known fraction 0 < s < 1 of young agents from each location migrate to the "foreign" location. The identity of who migrates is not known beforehand, so there is idiosyncratic risk, but no aggregate risk.

Next, assume that there are two other assets, money and bonds, both issued by the government supply (and endowed to the initial old). Let M be the supply of money, and let B denote the supply of bonds. Let D denote the total supply of nominal government debt:

[1] D = M + B

Money is a perpetuity that pays zero nominal interest. Bonds are one-period risk-free claims to money. (Once the bonds pay off, the government just re-issues a new bond offering B to suck cash back out of the system.) Assume that the government keeps D constant maintains a fixed bond/money ratio z = B/M, so that [1] can be written as:

[2] D = (1+z)*M

In what follows, I will keep D constant throughout and consider the effect of changing z (once and for all). Note, I am comparing steady-states here. Also, since D and M remain constant over time, and since there is no real growth in this economy, I anticipate that the steady state inflation rate will be equal to zero.

Let R denote the gross nominal interest rate (also the real interest rate, since inflation is zero). Assume that the government finances the carrying cost of its interest-bearing debt with a lump-sum tax,

[3] T = (R-1)*B

The difference between money and bonds is that bonds (or intermediated claims to bonds) cannot be transported across locations. Only money is transportable. The effect of this assumption is to impose a cash-in-advance constraint (CIA) on the young agents who move across locations. (Hence, we can interpret the relocation shock as an idiosyncratic liquidity shock).

Young agents are confronted with a portfolio allocation problem. Let P denote the price level. Since the young do not consume, they save their entire nominal income, P*y. Savings can be allocated to money, bonds, or capital,

[4] P*y = M + B + P*k

There is a trade off here: money is more liquid, but bonds and capital (generally) pay a higher return. The portfolio choice must be made before the young realize their liquidity shock.

Because there is idiosyncratic liquidity risk, the young can be made better off by pooling arrangement that we can interpret as a bank. The bank issues interest-bearing liabilities, redeemable for cash on demand. It uses these liabilities to finance its assets, M+B+P*k. Interest is  only paid on bank liabilities that are left to mature into the next period. (The demandable nature of the debt can be motivated by assuming that the idiosyncratic shock is private information. It is straightforward to show that truth-telling here in incentive-compatible.)

Let me describe how things work here. Consider one of the locations. It will consist of two types of old agents: domestics and foreigners. The old foreigners use cash to buy output from the domestic young agents. The old domestics use banknotes to purchase output from the young domestics (the portion of the banknotes that turn into cash as the bond matures). The remaining banknotes can be redeemed for a share of the output produced by the maturing capital project. The old domestic agents must also pay a lump-sum tax.

As for the young in a given location, they accumulate cash equal to the sales of output to the old. After paying their taxes, the old collectively have cash balances equal to D. The young deposit this cash in their bank. The bank holds some cash back as reserves M and uses the rest to purchase newly-issued bonds B. The bank also uses some of its banknotes to purchase output P*k from the young workers, which the bank invests. At the end of this operation, the bank has assets M+B+P*k and a corresponding set of (demandable) liabilities. The broad money supply in this model is equal to M1 = M+B+P*k. The nominal GDP is given by NGDP = P*y + P*x*f(k).

Formally, I model the bank as a coalition of young agents. The coalition maximizes the expected utility of a representative member:  (1-s)*u(c1) + s*u(c2), where c1 is consumption in the domestic location and c2 is consumption in the foreign location. The maximization above is constrained by condition [4] which, expressed in real terms, can be stated as:

[5] y = m + b + k

where m = M/P and b = B/P (real money and bond holdings, respectively).

In addition, there is a budget constraint:

[6] (1-s)*c1 + s*c2 = x*f(k) + R*b + m - t 

where t = T/P (see condition [3]).

Finally, there is the "cash-in-advance" (CIA) constraint:

[7] s*c2 <= m

Note: the CIA constraint represents the "cash reserves" the bank has to set aside to meet expected redemptions. Because there is no aggregate risk here, the aggregate withdrawal amount is perfectly forecastable. This constraint may or may not bind. It will bind if the nominal interest rate is positive (i.e., R > 1). More generally, it will bind if the rate of return on bonds exceeds the rate of return on reserves. If the constraint is slack, I will say that the bank is holding "excess reserves." (with apologies to Nick Rowe).

Optimality Conditions

Because bonds and capital are risk-free and equally illiquid, they must earn the same real rate of return:

[8] R = xf'(k)

The bank constructs its asset portfolio to equate the return-adjusted marginal utility of consumption across locations:

[9] R*u'(c1) = u'(c2)

Invoking the government budget constraint [3], the bank's budget constraint [6], reduces to:

[8] (1-s)*c1 + s*c2 = x*f(k) + b + m 

In equilibrium,

[9] m = M/P and b = B/P

We also have the bank's budget constraint [4]:

[10] y = m + b + k

Because the  monetary authority is targeting a bond/money ratio z, we can use [2] to rewrite the bank's budget constraints [8] and [10] as:

[11]  (1-s)*c1 + s*c2 = x*f(k) + (1+z)*m 

[12] y = (1+z)*m + k

Finally, we have the CIA constraint [7]. There are now two cases to consider.

Case 1: CIA constraint binds (R > 1).

This case occurs for high values of x. That is, when the expected return to capital spending is high. In this case, the CIA constraint [7] binds, so that s*c2 = m or, using [12],

[13] m = (y - k)/(1+z)

Condition [11] then becomes (1-s)*c1 = xf(k) + z*m. Again, using [12], we can rewrite this as:

[14] (1-s)*c1 = x*f(k) + A(z)*(y - k)

where A(z) = z/(1+z) is an increasing function of z. Combining [8], [9], [13] and [14], we are left with an expression that determines the equilibrium level of capital spending as a function of parameters:

[15] x*f'(k)*u'( [x*f(k) + A(z)*(y-k)]/(1-s) ) = u'( (y-k)/(s*(1+z)) )

Now, consider a "loosening" of monetary policy (a decline in the bond/money ratio, z). The direct impact of this shock is to decrease c1 and increase c2. How must k move to rebalance condition [15]? The answer is that capital spending must increase. Note that since [8] holds, the effect of this "quantitative easing" program is to cause the nominal (and real) interest rate to decline (the marginal product of capital is decreasing in the size of the capital stock).

What is the effect of this QE program on the price-level? To answer this, refer to condition [4], but rewritten in the following way:

 [16] P = D/(y - k)

This is something I did not appreciate when I wrote my first post on this subject. That is, notice that the equilibrium price-level depends not on the quantity of base money, but rather, on the total stock of nominal government debt. In my original model (without capital spending), a shift in the composition of the D has no price-level effect (I erroneously reported that it did). In the current set up, a QE program (holding D fixed) has the effect of lowering the interest rate and expanding real capital spending. The real demand for government total government debt D/P must decline, which is to say, the price-level must rise.

[ Note: as a modeling choice, I decided to endogenize investment here. But one might alternatively have endogenized y (through a labor-leisure choice). One might also have modeled a non-trivial saving decision by assuming that the young derive utility from consumption when young and old. ]

Case 2: CIA constraint is slack (R = 1).

This case occurs when x is sufficiently small -- i.e., when the expected productivity of capital spending is diminished.  In this case, the equilibrium quantity of real money balances is indeterminate. All that is determined is the equilibrium quantity of real government debt d = m + b. Conditions [11] and [12] become:

[17]  (1-s)*c1 + s*c2 = x*f(k) + d 

[18] y = d + k

Condition [15] becomes:

[19] u'( [x*f(y - d) + d]/(1-s) ) = u'( d/s )

Actually, even more simply, from condition [8] we have xf'(k) = 1, which pins down k (note that k is independent of z). The real value of D is then given by d = y - k. [Added July 10, 2014]. 

Condition [19] determines the equilibrium real value of total government debt. The composition of this debt (z) is irrelevant -- this is a classic "liquidity trap" scenario where swaps of two assets that are perfect substitutes have no real or nominal effect. The equilibrium price-level in this case is determined by:

[20] P = D/d

A massive QE program in case (a decline in z, keeping D constant) simply induces banks to increase their demand for base money one-for-one with the increase in the supply of base money. (Nice Rowe would say that these are not "excess" reserves in the sense that they are the level of reserves desired by banks. He is correct in saying this.)

The question I originally asked was: do these excess reserves (as I have defined them) pose an inflationary threat when the economy returns to "normal?"

Inflationary Risk

Let us think of  "returning to normal" as an increase in x (a return of optimism) which induces the interest rate to R >1. In this case, we are back to case 1, but with a lower value for z. So yes, as illustrated in case 1, if z is to remain at this lower level, the price-level will be higher than it would otherwise be. This is the sense in which there is inflationary risk associated with "excess reserves" (in this model, at least).

Of course, in the model, there is a simple adjustment to monetary policy that would prevent the price-level from rising excessively. The Fed could just raise z (reverse the QE program).

In reality, reversing QE might not be enough. In the model above, I assumed that bonds were of very short duration. In reality, the average duration of the Fed's balance has been extended to about 10 years. What this means is that if interest rates spike up, the Fed is likely to suffer a capital loss on its portfolio. The implication is that it may not have enough assets to buy back all the reserves necessary to keep the price-level in check.

Alternatively, the Fed could increase the interest it pays on reserves. But in this case too, the question is how the interest charges are to be financed? If there is full support from the Treasury, then there is no problem. But if not, then the Fed will (effectively) have to print money (it would book a deferred asset) to finance interest on money. The effect of such a policy would be inflationary.

Finally, how is this related to bank-lending and private money creation? Well, in this model, where banks are assumed to intermediate all assets, broad money is given by M1 = D + P*k. We can eliminate P in this expression by using [16]:

[21] M1 = [ 1 + k/(y-k) ]*D

So when R > 1, reducing z has the effect of increasing capital spending and increasing M1. In the model, young agents want to "borrow" banknotes to finance additional investment spending. But it is not the increase in M1 that causes the price-level to rise. Instead, it is the reduction in the real demand for total government debt that causes the price-level to rise.

Likewise, in the case where R = 1 and then the economy returns to normal, the price-level pressure is coming from the portfolio substitution activity of economic agents: people want to dump their money and bonds in order to finance additional capital spending. The price-level rises as the demand for government securities falls. The fact that M1 is rising is incidental to this process.