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 Etct+1 (expected future consumption).
The young are endowed with y units of output and they possess an investment technology such that kt units of output invested at date t yields zt+1f(kt) 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 nt that is useful for forecasting future productivity zt+1. Let z(nt) = E[zt+1 | nt] and assume that z(nt) is increasing in nt. I call nt "news", higher realizations of nt "good news," and lower realizations of nt "bad news." Assume that nt 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 Rt denote the gross nominal interest rate paid on the outstanding stock of government money/debt Mt-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 RtMt-1 is not indexed to the price-level pt; 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: ptkt + mt = pty - ptTt
when old: pt+1ct+1 = pt+1zt+1f(kt) + Rt+1mt
So the young "work" to produce output y, pay taxes ptTt, investment in capital kt, and money mt (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 qt = mt/pt (real money balances) and Πt+1 = pt+1/pt (the gross inflation rate). The two equations above may now be combined and expressed as follows:
ct+1 = zt+1f(y - qt - Tt ) + (Rt+1 / Πt+1)qt
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(nt)f ' ( y - qt - Tt ) = Rt+1 E[1 /Πt+1 | nt]
The equation above implicitly defines the aggregate demand for investment kt = y - qt. 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:
ptGt + (Rt - 1)Mt-1 = (Mt - Mt-1) + ptTt
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 Gt = 0 for all t.
Let Mt = μtMt-1 and rearrange the GBC as follows:
(Rt - μt )Mt-1 /pt = Tt
Notice here that a surprise increase in the price-level reduces the real burden of the debt.
Finally, impose the market-clearing conditions:
ptM t= qt for all t
which implies:
Πt+1 = μt+1qt/qt+1
Combine this with the FOC above to form:
(*) z(nt)f ' (y - qt -Tt ) qt = Rt+1 E[ qt+1 / μt+1 | nt]
Finally, we have: NGDPt = pt[ y + ztf(kt-1) ]
A Benchmark Policy
Set Rt = μt = 1 for all t, so that Tt = 0 for all t.
Notice that since nt is i.i.d., we have E[ qt+1 | nt] = Q (some constant). Consequently, condition (*) may be written as:
z(nt)f ' (y - qt)qt = Q
Proposition 1: qt is a decreasing function of nt.
The proof follows from the strict concavity of f(.), the fact that z(.) is increasing in nt , 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 nt), 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 kt, and an increase in the demand for government securities qt. From the market-clearing condition, pt = M / qt. 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., Rt = R > μt = μ = 1.
From the GBC above, (Rt - μt )Mt-1 /pt = Tt, 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 Tt 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.
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 Etct+1 (expected future consumption).
The young are endowed with y units of output and they possess an investment technology such that kt units of output invested at date t yields zt+1f(kt) 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 nt that is useful for forecasting future productivity zt+1. Let z(nt) = E[zt+1 | nt] and assume that z(nt) is increasing in nt. I call nt "news", higher realizations of nt "good news," and lower realizations of nt "bad news." Assume that nt 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 Rt denote the gross nominal interest rate paid on the outstanding stock of government money/debt Mt-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 RtMt-1 is not indexed to the price-level pt; 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: ptkt + mt = pty - ptTt
when old: pt+1ct+1 = pt+1zt+1f(kt) + Rt+1mt
So the young "work" to produce output y, pay taxes ptTt, investment in capital kt, and money mt (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 qt = mt/pt (real money balances) and Πt+1 = pt+1/pt (the gross inflation rate). The two equations above may now be combined and expressed as follows:
ct+1 = zt+1f(y - qt - Tt ) + (Rt+1 / Πt+1)qt
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(nt)f ' ( y - qt - Tt ) = Rt+1 E[1 /Πt+1 | nt]
The equation above implicitly defines the aggregate demand for investment kt = y - qt. 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:
ptGt + (Rt - 1)Mt-1 = (Mt - Mt-1) + ptTt
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 Gt = 0 for all t.
Let Mt = μtMt-1 and rearrange the GBC as follows:
(Rt - μt )Mt-1 /pt = Tt
Notice here that a surprise increase in the price-level reduces the real burden of the debt.
Finally, impose the market-clearing conditions:
ptM t= qt for all t
which implies:
Πt+1 = μt+1qt/qt+1
Combine this with the FOC above to form:
(*) z(nt)f ' (y - qt -Tt ) qt = Rt+1 E[ qt+1 / μt+1 | nt]
Finally, we have: NGDPt = pt[ y + ztf(kt-1) ]
A Benchmark Policy
Set Rt = μt = 1 for all t, so that Tt = 0 for all t.
Notice that since nt is i.i.d., we have E[ qt+1 | nt] = Q (some constant). Consequently, condition (*) may be written as:
z(nt)f ' (y - qt)qt = Q
Proposition 1: qt is a decreasing function of nt.
The proof follows from the strict concavity of f(.), the fact that z(.) is increasing in nt , 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 nt), 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 kt, and an increase in the demand for government securities qt. From the market-clearing condition, pt = M / qt. 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., Rt = R > μt = μ = 1.
From the GBC above, (Rt - μt )Mt-1 /pt = Tt, 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 Tt 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.
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