NGDP Targeting in an OLG Model (Another Try)

I would like to follow up on my earlier post: NGDP Targeting in an OLG Model. The purpose of that post was to evaluate the desirability of an NGDP target policy within the context of an explicit (mathematical) macroeconomic model. Josh Hendrickson does a good job of explaining the motivation behind my approach here (and btw, thank you for the kind words, Josh.) Josh goes on to provide a list of reasons for why an NGDP target is a good policy prescription, but he does not really address the point I was trying to make with my simple model. And so, let me try again, this time in less technical terms.

First, let me describe the model economy I employed in my earlier post. The economy is populated by different types of people. At any point in time, there are people with relatively large wealth positions and high consumption propensities--and there are people with relatively small wealth positions who have high saving propensities. This is not a "representative agent" model economy: people are different--and these differences matter.

There are two types of assets in the economy, that I label "capital" and "money" (or government debt). I model capital as physical capital, but it should be clear that one may substitute any form of private investment in its place, including human capital investment, or recruiting investment (as would be the case for a labor-market search model). Capital investment is just a metaphor for any activity involving a sacrifice today for an uncertain return reward in the future. In the model, peoples' perceptions of this future reward (whether such perceptions are rational or not) are a key driver of investment demand (and hence, aggregate demand). Does this sound crazy? (I don't think so.)

The government money/debt plays an important role in this model economy. In particular, the competitive equilibrium turns out to be inefficient without it (there is a dynamic inefficiency). Of course, this does not mean that the government can print paper haphazardly. From a social welfare perspective, it will want to manage the supply of its paper in a particular way (that I will describe below).

There is a "sticky" nominal price in the economy: the nominal interest rate on government paper cannot be made contingent on any contemporaneous information (in particular, the expectations shocks that afflict investment demand). I imagine that one could also include nominal private debt (like mortgage debt), but it would not affect the qualitative results I report below. All other prices are flexible.

Now, let me describe how this economy behaves over time, assuming a "passive" government policy of keep the nominal supply of debt fixed.

There are two types of shocks: (1) a news-shock that affects investment demand (and so looks like an AD shock), and (2) a productivity shock that affects the ex post return on capital (and so looks like an AS shock).

Good news creates a rational optimism: investors revise upward their forecast over future returns to capital investment. Agents "dump" money/bonds and substitute in private securities. The money dump results in a surprise jump in the price-level. The real value of outstanding government debt declines. There is a redistribution of wealth from bondholders to investors.

The opposite happens when the news is bad (a productivity slowdown?). A sequence of bad news shocks results in a deflation. Capital spending contracts, and with it, future GDP. There is a redistribution of purchasing power away from investors toward bondholders that further depresses investment spending.

I claim that qualitatively, this model generates dynamics that most people would have a hard time distinguishing from the data. 

The optimal policy here turns out to be a price-level target (PLT). The role of the PLT here is to prevent variation in the real value of nominal debt that is not indexed to the price-level. Ex ante, agents want to avoid transfers of wealth stemming from uninsurable price-level shocks interacting with nominal debt.

So, if the news is bad, the government should increase the supply of money to accommodate the increase in demand for money (via the asset substitution induced by bad news over the expected return to investment). But if the news truly is fundamentally bad, the future real GDP should decline, and along with it, the NGDP should decline as well (it's decline is stemmed in part by maintaining the price level target). Stabilizing the NGDP in this context would mean increasing the price-level so high as to create a transfer of wealth from creditors to debtors (instead of debtors to creditors) --something these agents would have wanted to prevent ex ante if nominal debt could have been indexed to the price-level.

This was the gist of my argument. I was just curious to see what NGDP targeting advocates thought of it. What is missing in my model? Are frictions other than nominal debt required? I have a hard time seeing how the presence of sticky nominal wages or prices are going to alter my conclusion here. But who knows, maybe someone can tell me?

Note: If the expectation shocks I describe above are not rational (e.g., possibly psychological "animal spirits"), then obviously there is a role for NGDP (and RGDP) targeting. However, I don't really hear Scott Sumner and others making this claim (or do they?). 

European Bond Yields

It is a fascinating picture (h/t Frances Wooley at WCI):

Frances asks "what were they thinking?" It seems clear enough that with the introduction of the Euro, bond traders came to view the debt of several European sovereigns as very close substitutes--a perception that seems to have vanished since the beginning of the financial crisis.

The better question, as Frances points out, is why were they thinking that? Actually, the "they" in this question should probably be replaced with a more uncomfortable "we." Yes, what were we thinking--if we were thinking anything at all. (If you were thinking otherwise, I presume you were holding significant short positions throughout the episode?)

I remember what I was thinking when I was teaching my section on International Monetary Systems years ago. After discussing the benefits of a common currency (or multilateral fixed exchange rate agreement), I'd turn to the evidence and discuss why the experiment seems to have worked in some cases and not others. A recurring theme for success appeared to be (among other things) some notion of "fiscal coordination" among potentially disparate regions of the union. I came to view this conclusion as a "duh, kinda obvious" sort of lesson that any future monetary union would surely respect and deal with accordingly.

Oops. So maybe I was being unduly naive in this respect. But is it reasonable to suppose that agents managing large bond portfolios were equally naive?

Anyway, if you have an interesting take on the picture above, please comment below.

In the meantime, you may be interested in this piece by my colleagues Fernando Martin and Chris Waller: Sovereign Debt: A Modern Greek Tragedy, and by my colleague Silvio Contessi: An Application of Conventional Sovereign Debt Sustainability Analysis to the Current Debt Crisis.

And here's a piece by Ferguson and Roubini in Der Spiegel: This Time, Europe Really is on the Brink.

Interesting times...

NGDP targeting in an OLG model

I'm still trying to work through this NGDP targeting idea. A lot of people graciously replied to my earlier query here, including David Beckworth here (David links up to others who have also contributed their thoughts.)

So much material. So little time. I find myself reading, and then re-reading these replies, trying to absorb the arguments. As I continue to do so, I thought that I'd reciprocate with a gift of my own; something that people strongly in favor of NGDP targeting can mull over and reflect upon. I am going to approach things a little differently here, however. I want to present my argument within the context of a formal economic model, where the assumptions are laid bare. Along the way, I'll try to present the economic intuition as best I can.

Let me consider a simple OLG model. Before I begin I should like to say that if you have something against the OLG model relative to standard macro models, you should read Michael Woodford (1986). Woodford shows that the dynamics of debt-constrained economies can look a lot like OLG dynamics. So I could use the Woodford model in what I am about to say, but I stick to the OLG model because it is simpler and the economic intuition is the same.

An OLG Model

There is a constant population of 2-period-lived overlapping generations (and an initial old generation). All agents care only for consumption when old; in particular, the preferences for a date t agent are E_tc_t+1 (expected future consumption).

The young are endowed with y units of output and they possess an investment technology such that k_t units of output invested at date t yields z_t+1f(k_t) units of output at date t+1. Capital depreciates fully after it is used in production. The future productivity of capital is a random variable. There is another random variable n_t that is useful for forecasting future productivity z_t+1. Let z(n_t) = E[z_t+1 | n_t] and assume that z(n_t) is increasing in n_t. I call n_t "news", higher realizations of n_t "good news," and lower realizations of n_t "bad news." Assume that n_t is an i.i.d. process.

Note that because there is no growth in this economy, the "natural" real rate of interest is zero.

There is a second asset in this economy in the form of interest-bearing government money/debt (I make no distinction here between money and bonds). Let R_t denote the gross nominal interest rate paid on the outstanding stock of government money/debt M_t-1.

Nominal debt is an important consideration for the arguments in favor of an NGDP target and so, I make the assumption here. In particular, I assume that the nominal burden of the debt R_tM_t-1 is not indexed to the price-level p_t; see also, Champ and Freeman (1990). Because agents differ at a point in time with respect to their wealth portfolios, a surprise change in the price-level will induce unexpected wealth transfers.

The budget constraints for a representative young agent are given by:

when young: p_tk_t + m_t = p_ty - p_tT_t
when old: p_t+1c_t+1 = p_t+1z_t+1f(k_t) + R_t+1m_t

So the young "work" to produce output y, pay taxes p_tT_t,  investment in capital k_t, and money m_t (from the old). When the young become old, they consume out of the returns from capital and money/bond investments (that is, they consume the returns to their capital and sell their money to the new generation of young for goods and services).

It turns out to be convenient to express things in "real" terms. To this end, define q_t = m_t/p_t (real money balances) and Π_t+1 = p_t+1/p_t (the gross inflation rate). The two equations above may now be combined and expressed as follows:

c_t+1 = z_t+1f(y - q_t  - T_t ) + (R_t+1 / Π_t+1)q_t

So, conditional on news n, a young person chooses his demand for real money balances q (and implicitly capital investment k) to maximize expected consumption (after-tax wealth, in this case). Since f(.) is increasing and strictly concave, the first-order condition describing money demand is:

z(n_t)f ' ( y - q_t  - T_t ) = R_t+1 E[1 /Π_t+1 | n_t]

The equation above implicitly defines the aggregate demand for investment k_t = y - q_t. The RHS is the expected real interest rate. An increase in the expected real interest rate reduces investment demand. A good news shock increases investment demand (for any given expected real interest rate). Notice how a news shock looks like an aggregate demand shock (the aggregate demand for output rises with no contemporaneous increase in output). If you want, you can think of  the equation above as defining an IS curve, with y pinned down by exogenous factors (labor market clearing, in a neoclassical model). In short, I think this is all pretty conventional.

I consolidate the monetary and fiscal authority, so that the government budget constraint (GBC) is given by:

p_tG_t + (R_t - 1)M_t-1 = (M_t - M_t-1) + p_tT_t

The LHS is the sum of government purchases plus (net) interest on the debt; the RHS is new debt plus net tax revenue. In what follows, I assume G_t  = 0 for all t.

Let M_t = μ_tM_t-1 and rearrange the GBC as follows:

 (R_t -  μ_t )M_t-1 /p_t  = T_t

Notice  here that a surprise increase in the price-level reduces the real burden of the debt.

Finally, impose the market-clearing conditions:

p_tM _t= q_t for all t

which implies:

Π_t+1  = μ_t+1q_t/q_t+1

Combine this with the FOC above to form:

(*) z(n_t)f ' (y - q_t -T_t ) q_t = R_t+1 E[ q_t+1  /  μ_t+1  | n_t]

Finally, we have: NGDP_t = p_t[ y + z_tf(k_t-1) ]

A Benchmark Policy

Set R_t = μ_t = 1 for all t, so that T_t = 0 for all t.

Notice that since n_t  is i.i.d., we have E[ q_t+1 | n_t] = Q (some constant). Consequently, condition (*) may be written as:

z(n_t)f ' (y - q_t)q_t = Q

Proposition 1: q_t  is a decreasing function of n_t.

The proof follows from the strict concavity of f(.), the fact that z(.) is increasing in  n_t , and that Q is a constant. The intuition is as follows: Good news raises the expected return to capital formation--the demand for capital rises, and the demand for government money/debt falls. This is a strait forward portfolio reallocation effect. If the news is "bad" (a decline in  n_t), then the demand for government securities rises -- this looks like a "flight to safety" event.

Consider a bad news event. There is a collapse in investment demand k_t, and an increase in the demand for government securities q_t. From the market-clearing condition, p_t = M / q_t. That is, the bad news event causes a surprise drop in the price-level (a sequence of such events would lead to a surprise deflation).

The surprise drop in the price-level leads to a surprise increase in the purchasing power of government securities. In this simple set up, the stock of government securities is held entirely by the old (the high propensity to consume agents) prior to the realization of the news shock. The young (the high propensity to invest agents) wish to acquire these securities as part of their wealth portfolios. The decline in the price level makes the real value of nominal government debt more expensive. In this way, bond holders are able to secure more labor power (y) from the young, so that fewer resources are now available for investment. The decline in capital spending leads to an expected decline in future NGDP (and RGDP). Note: I say expected because the future capital stock is lower; but future GDP may turn out to be higher or lower than expected depending on the realization of the productivity shock z.

Stabilizing the (expected) NGDP

It is possible here for the government to stabilize the expected NGDP path by conditioning the nominal interest rate on news (or, if the lower bound is a constraint, the same effect could be achieved by altering the expected inflation rate via money creation). The key is to stabilize capital spending; and the way to do this is to lower the nominal (hence real) interest rate on government securities. In this way, the decline in the price-level can be avoided. And NGDP remains elevated, despite the bad news, because capital spending is "subsidized" and the price-level remains stabilized.

But is stabilizing the NGDP path a desirable policy?

Well, it depends on what one means by "desirable." If you objective is to stabilize NGDP, then the answer is "yes." In terms of maximizing the expected utility of the representative young agent, however, the answer appears to be "no;" at least, not in this case. (Welfare calculations in heterogeneous agents economies, like this one, can be complicated--as is the case in reality.)

The intuition is this. When the news is bad concerning the future return to investment, it is optimal for investment to contract (and for savings to flow into more stable return vehicles, like government securities). To put it in more colloquial terms: the real rate of return on capital spending sucks (at least, in expectation). In fact, the real return would be less than the population growth rate -- the natural rate of interest in this economy.

Animal Spirits?

Implicit in a lot of discussions about the desirability of stabilization policy is the idea that the business cycle is inefficient. One way in which they may be inefficient is if expectations are prone to fluctuate purely for "psychological" (exogenous) reasons. In the context of the model developed above, we might instead assume that "news events" are instead just "animal spirits" that move expectations around for no particular reason. Assuming that policymakers are somehow immune to such effects, it would indeed be desirable to stabilize NGDP in this model.

Is this what proponents of NGDP targeting have in mind?  I have no idea as they rarely, if ever, are explicit about what they assume are the driving forces of the business cycle. All I mostly ever hear is a "negative AD shock," whatever that is supposed to be. (The two examples above, rational pessimism and irrational pessimism, both lead to a reduction in AD in some sense, for example.)

An Alternative Policy

Let me modify policy in a minor way; i.e., R_t = R > μ_t = μ = 1.

From the GBC above,  (R_t -  μ_t )M_t-1 /p_t  = T_t, so that under this policy, the young are required to finance the carrying cost of the public debt.

It is easy to see that if the young must allocate more resources to service the debt, less resources will be available for capital spending. And a surprise decrease in the price-level now has two effects. First, there is the effect described above. Second, the real tax burden on the young must rise, if the government's nominal obligations are to be met.

Although I haven't fully worked it out, it seems to me that this second force constitutes a drag on capital spending that should be avoided, if possible. In particular, a better policy would apply the tax T_t to the old, instead of the young.

So in this case, it seems that some policy designed to support the price-level (hence NGDP) might be desirable. Although, once again, if the information that leads agents to reduce capital expenditure is the best information available, then one would not want to stabilize NGDP perfectly.

Conclusions

The model presented above is highly abstract. Nevertheless, I think that it captures some forces that may presently be at work in real world economies. Pessimistic expectations over the future return to investment (whether via a productivity slowdown, as documented here, or through the rational--or irrational--expectation of a higher tax rate on investments) will act as a drag on the economy, and make competing savings vehicles, like US treasuries, relatively more attractive. The effect is deflationary and, to the extent that nominal debt is not indexed, there will be redistributive consequences.

Even though the model delivers a plausible interpretation of some recent macroeconomic developments, a NGDP target is not an obvious solution. But of course, as I said, the model is highly abstract. It is likely missing some features of the real world that NGDP target proponents think are important. If this is the case, then I'd like to hear what they are, and how these elements might be embedded in the model above. If nothing else, it would be a contribution to the debate if we could just get straight what assumptions we are making when stating strong propositions concerning the desirability of this or that policy.


Postscript

There are still a lot of theoretical issues to resolve concerning the relative merits of different monetary policy rules, especially in the context of an open economy. One such paper that explores this question is: "What to Stabilize in the Open Economy" by Bencivenga, Huybens, and Smith (IER 2002). Among other things, the authors find a price-level target gives rise to an indeterminancy, and endogenous volatility driven by expectations.

Postscript June 15, 2012: Josh Hendrickson offers an extended comment here. Thanks to Josh for this; I will reply soon.

Feynman on the Scientific Method

A colleague of mine recently pointed me to this fantastic lecture by Richard Feynman (1918-1988) explaining his take on the scientific method.

What a treat it would have been to sit in on his lectures! And I confess to missing the old blackboard technology. Can't imagine him giving this lecture using Powerpoint, in particular.

Speaking of the cursed PPT presentation, ever wonder what Lincoln's Gettysburg Address may have looked like had he prepared his notes using Powerpoint? Take a look here.

Am off to look for that old box of chalk...