In the present day, I am going to add an entry to my occasional evaluations of fascinating tutorial papers. The paper: “Value Stage and Inflation Dynamics in Heterogeneous Agent Economies,” by Greg Kaplan, Georgios Nikolakoudis and Gianluca Violante.

One of many many causes I’m enthusiastic about this paper is that it unites fiscal idea of the worth degree with heterogeneous agent economics. And it exhibits how heterogeneity issues. There was a variety of work on “heterogeneous agent new-Keynesian” fashions (HANK). This paper inaugurates heterogeneous agent fiscal idea fashions. Let’s name them HAFT.

The paper has a superbly stripped down mannequin. Costs are versatile, and the worth degree is ready by fiscal idea. Folks face uninsurable revenue shocks, nevertheless, and a borrowing restrict. So that they save an additional quantity in an effort to self-insure towards dangerous occasions. Authorities bonds are the one asset within the mannequin, so this further saving pushes down the rate of interest, low cost charge, and authorities service debt price. The mannequin has a time-zero shock after which no mixture uncertainty.

That is precisely the suitable place to start out. Ultimately, after all, we would like fiscal idea, heterogeneous brokers, and sticky costs so as to add inflation dynamics. And on prime of that, no matter DSGE smorgasbord is vital to the problems at hand; manufacturing aspect, worldwide commerce, a number of actual belongings, monetary fractions, and extra. However the genius of an important paper is to start out with the minimal mannequin.

**Half II results of fiscal shocks.**

I’m most excited by half II, the results of fiscal shocks. This goes straight to vital coverage questions.

At time 0, the federal government drops $5 trillion of additional debt on individuals, with no plans to pay it again. The rate of interest doesn’t change. What occurs? Within the consultant agent economic system, the worth degree jumps, simply sufficient to inflate away excellent debt by $5 trillion.

(On this simulation, inflation subsequent to the worth degree leap is simply set by the central financial institution, by way of an rate of interest goal. So the rising worth degree line of the consultant agent (orange) benchmark isn’t that fascinating. It isn’t a traditional impulse response displaying the change after the shock; it is the precise path after the shock. The distinction between coloured heterogeneous agent strains and the orange consultant agent line is the vital half.)

Punchline: Within the heterogeneous agent economies, the worth degree jumps a very good deal extra. And if transfers are focused to the underside of the wealth distribution, the worth degree jumps extra nonetheless. It issues who will get the cash.

This is step one on an vital coverage query. Why was the 2020-2021 stimulus a lot extra inflationary than, say 2008? I’ve a variety of tales (“fiscal histories,” FTPL), one in every of which is a imprecise sense that printing cash and sending individuals checks has extra impact than borrowing in treasury markets and spending the outcomes. This graph makes that sense exact. Sending individuals checks, particularly people who find themselves on the sting, *does* generate extra inflation.

Ultimately, whether or not authorities debt is inflationary or not comes down as to whether individuals deal with the asset as a very good financial savings automobile, and dangle on to it, or attempt to spend it, thereby driving up costs. Sending checks to individuals prone to spend it provides extra inflation.

As you possibly can see, the mannequin additionally introduces some dynamics, the place on this easy setup (versatile costs) the RA mannequin simply provides a worth degree leap. To grasp these dynamics, and extra instinct of the mannequin, have a look at the response of actual debt and the true rate of interest

The higher inflation signifies that the identical improve in nominal debt is a lesser improve in actual debt. Now, the essential characteristic of the mannequin steps in: on account of self-insurance, there’s primarily a liquidity worth of debt. In case you have much less debt, the marginal worth of upper; individuals bid down the true rate of interest in an try to get extra debt. However the larger actual charge means the true worth of debt rises, and because the debt rises, the true rate of interest falls.

To grasp why that is the equilibrium, it is price trying on the debt accumulation equation, [ frac{db}{dt} = r_t (b_t; g_t) b_t – s_t. ](b_t) is the true worth of nominal debt, (r_t=i_t-pi_t) is the true rate of interest, and (s_t) is the true main surplus. Larger actual charges (debt service prices) increase debt. Larger main surpluses pay down debt. Crucially — the entire level of the paper — the rate of interest is determined by how a lot debt is excellent and on the distribution of wealth (g_t). ((g_t) is a complete distribution.) Extra debt means the next rate of interest. Extra debt does a greater job of satisfying self-insurance motives. Then the marginal worth of debt is decrease, so individuals do not attempt to save as a lot, and the rate of interest rises. It really works rather a lot like cash demand,

Now, if the switch have been proportional to present wealth, nothing would change, the worth degree would leap similar to the RA (orange) line. But it surely is not; in each circumstances more-constrained individuals get more cash. The liquidity constraints are much less binding, they’re keen to save lots of extra. For given mixture debt the true rate of interest will rise. So the orange line with no change in actual debt is not a gradual state. We will need to have, initially (db/dt>0.) As soon as debt rises and the distribution of wealth mixes, we return to the previous regular state, so actual debt rises much less initially, so it will probably proceed to rise. And to try this, we want a bigger worth degree leap. Whew. (I hope I received that proper. Instinct is difficult!)

In a earlier submit on heterogeneous agent fashions, I requested whether or not HA issues for aggregates, or whether or not it’s nearly distributional penalties of unchanged mixture dynamics. Right here is a superb instance during which HA issues for aggregates, each for the dimensions and for the dynamics of the results.

This is a second cool simulation. What if, slightly than a lump-sum helicopter drop with no change in surpluses, the federal government simply begins working everlasting main deficits?

Within the RA mannequin, a decline in surpluses is precisely the identical factor as an increase in debt. You get the preliminary worth leap, after which the identical inflation following the rate of interest goal. Not so the HA fashions! Perpetual deficits are totally different from a leap in debt with no change in deficit.

Once more, actual debt and the true charge assist to grasp the instinct. The true quantity of debt is completely decrease. Meaning individuals are extra starved for buffer inventory belongings, and bid down the true rate of interest. The nominal charge is fastened, by assumption on this simulation, so a decrease actual charge means extra inflation.

For coverage, this is a vital outcome. With versatile costs, RA fiscal idea solely provides a one-time worth degree leap in response to sudden fiscal shocks. It doesn’t give regular inflation in response to regular deficits. *Right here we do have regular inflation in response to regular deficits!* It additionally exhibits an occasion of the final “low cost charges matter” theorem. Granted, right here, the central financial institution might decrease inflation by simply reducing the nominal charge goal however we all know that is not really easy after we add realisms to the mannequin.

To see simply why that is the equilibrium, and why surpluses are totally different than debt, once more return to the debt accumulation equation, [ frac{db}{dt} = r_t (b_t, g_t) b_t – s_t. ] Within the RA mannequin, the worth degree jumps in order that (b_t) jumps down, after which with smaller (s_t), (r b_t – s_t) is unchanged with a continuing (r). However within the HA mannequin, the decrease worth of (b) means much less liquidity worth of debt, and folks attempt to save, bidding down the rate of interest. We have to work down the debt demand curve, driving down the true curiosity prices (r) till they partially pay for among the deficits. There’s a sense during which “monetary repression” (artificially low rates of interest) by way of perpetual inflation assist to pay for perpetual deficits. Wow!

**Half I r<g**

The primary idea a part of the paper can be fascinating. (Although these are actually two papers stapled collectively, since as I see it the speculation within the first half is in no way mandatory for the simulations.) Right here, Kaplan, Nikolakoudis and Violante tackle the r<g query clearly. No, r<g doesn’t doom fiscal idea! I used to be so enthused by this that I wrote up just a little word “fiscal idea with adverse rates of interest” right here. Detailed algebra of my factors under are in that word, (An essay r<g and likewise a r<g chapter in FTPL explains the associated challenge, why it is a mistake to make use of averages from our actual economic system to calibrate good foresight fashions. Sure, we will observe (E(r)<E(g)) but current values converge.)

I am going to give the fundamental thought right here. To maintain it easy, take into consideration the query what occurs with a adverse actual rate of interest (r<0), a continuing surplus (s) in an economic system with no progress, and excellent foresight. You would possibly assume we’re in hassle: [b_t = frac{B_t}{P_t} = int e^{-rtau} s dtau = frac{s}{r}.]A adverse rate of interest makes current values blow up, no? Properly, what a few completely adverse surplus (s<0) financed by a completely adverse curiosity price (r<0)? That sounds high-quality in circulation phrases, nevertheless it’s actually bizarre as a gift worth, no?

Sure, it’s bizarre. Debt accumulates at [frac{db_t}{dt} = r_t b_t – s_t.] If (r>0), (s>0), then the true worth of debt is generically explosive for any preliminary debt however (b_0=s/r). Due to the transversality situation ruling out actual explosions, the preliminary worth degree jumps so (b_0=B_0/P_0=s/r). But when (r<0), (s<0), then debt is steady. For any (b_0), debt converges, the transversality situation is glad. We lose fiscal worth degree willpower. No, you possibly can’t take a gift worth of a adverse cashflow stream with a adverse low cost charge and get a wise current worth.

However (r) isn’t fixed. The extra debt, the upper the rate of interest. So [frac{db_t}{dt} = r(b_t) b_t – s_t.] Linearizing across the regular state (b=s/r), [frac{db_t}{dt} = left[r_t + frac{dr(b_t)}{db}right]b_t – s.] So even when (r<0), if extra debt raises the rate of interest sufficient, if (dr(b)/db) is massive sufficient, dynamics are regionally and it seems globally unstable even with (r<0). Fiscal idea nonetheless works!

You may work out a simple instance with bonds in utility, (int e^{-rho t}[u(c_t) + theta v(b_t)]dt), and simplifying additional log utility (u(c) + theta log(b)). On this case (r = rho – theta v'(b) = rho – theta/b) (see the word for derivation), so debt evolves as [frac{db}{dt} = left[rho – frac{theta}{b_t}right]b_t – s = rho b_t – theta – s.]Now the (r<0) half nonetheless provides steady dynamics and a number of equilibria. But when (theta>-s), then dynamics are once more explosive for all however (b=s/r) and financial idea works anyway.

This can be a highly effective outcome. We normally assume that in good foresight fashions, (r>g), (r>0) right here, and consequently optimistic vs adverse main surpluses (s>0) vs. (s<0) is a vital dividing line. I do not know what number of fiscal idea critiques I’ve heard that say a) it does not work as a result of r<g so current values explode b) it does not work as a result of main surpluses are at all times barely adverse.

That is all flawed. The evaluation, as on this instance, exhibits is that fiscal idea can work high-quality, and does not even discover, a transition from (r>0) to (r<0), from (s>0) to (s<0). Financing a gradual small adverse main surplus with a gradual small adverse rate of interest, or (r<g) is seamless.

The essential query on this instance is (s<-theta). At this boundary, there is no such thing as a equilibrium any extra. You may finance solely a lot main deficit by monetary repression, i.e. squeezing down the quantity of debt so its liquidity worth is excessive, pushing down the curiosity prices of debt.

The paper staples these two workout routines collectively, and calibrates the above simulations to (s<0) and (r<g). However I wager they might look nearly precisely the identical with (s>0) and (r>g). (r<g) isn’t important to the fiscal simulations.*

The paper analyzes self-insurance towards idiosyncratic shocks as the reason for a liquidity worth of debt. That is fascinating, and permits the authors to calibrate the liquidity worth towards microeconomic observations on simply how a lot individuals endure such shocks and wish to insure towards them. The Half I simulations are simply that, heterogeneous brokers in motion. However this theoretical level is far broader, and applies to any financial pressure that pushes up the true rate of interest as the quantity of debt rises. Bonds in utility, right here and within the paper’s appendix, work. They’re a standard stand in for the usefulness of presidency bonds in monetary transactions. And in that case, it is simpler to increase the evaluation to a capital inventory, actual property, overseas borrowing and lending, gold bars, crypto, and different technique of self-insuring towards shocks. Customary “crowding out” tales by which larger debt raises rates of interest work. (Blachard’s r<g work has a variety of such tales.) The “segmented markets” tales underlying religion in QE give a rising b(r). So the final precept is powerful to many various sorts of fashions.

My word explores one challenge the paper doesn’t, and it is an vital one in asset pricing. OK, I see how dynamics are regionally unstable, however how do you’re taking a gift worth when r<0? If we write the regular state [b_t = int_{tau=0}^infty e^{-r tau}s dtau = int_{tau=0}^T e^{-r tau}s dtau + e^{-rT}b_{t+T}= (1-e^{-rT})frac{s}{r} + e^{-rT}b,]and with (r<0) and (s<0), the integral and ultimate time period of the current worth components every explode to infinity. It appears you actually cannot low cost with a adverse charge.

The reply is: do not combine ahead [frac{db_t}{dt}=r b_t – s ]to the nonsense [ b_t = int e^{-r tau} s dtau.]As an alternative, combine ahead [frac{db_t}{dt} = rho b_t – theta – s]to [b_t = int e^{-rho tau} (s + theta)dt = int e^{-rho tau} frac{u'(c_t+tau)}{u'(c_t)}(s + theta)dt.]Within the final equation I put consumption ((c_t=1) within the mannequin) for readability.

- Low cost the
*circulation*worth of liquidity advantages on the shopper’s intertemporal marginal charge of substitution. Don’t use liquidity to provide an altered low cost charge.

That is one other deep, and ceaselessly violated level. *Our low cost issue methods don’t work in infinite-horizon fashions*. (1=E(R_{t+1}^{-1}R_{t+1})) works simply in addition to (1 = Eleft[beta u'(c_{t+1})/u'(c_t)right] r_{t+1}) in a finite horizon mannequin, however you possibly can’t at all times use (m_{t+1}=R_{t+1}^{-1}) in infinite interval fashions. The integrals blow up, as within the instance.

This can be a good thesis matter for a theoretically minded researcher. It is one thing about Hilbert areas. Although I wrote the low cost issue guide, I do not know the way to lengthen low cost issue methods to infinite intervals. So far as I can inform, no one else does both. It isn’t in Duffie’s guide.

Within the meantime, for those who use low cost issue methods like affine fashions — something however the correct SDF — to low cost an infinite cashflow, and you discover “puzzles,” and “bubbles,” you are on skinny ice. There are many papers making this error.

A minor criticism: The paper does not present nuts and bolts of the way to *calculate* a HAFT mannequin, even within the easiest instance. Notice in contrast how trivial it’s to calculate a bonds in utility mannequin that will get many of the similar outcomes. Give us a recipe guide for calculating textbook examples, please!

Clearly it is a first step. As FTPL shortly provides sticky costs to get affordable inflation dynamics, so ought to HAFT. For FTPL (or FTMP, fiscal idea of financial coverage; i.e. including rate of interest targets), including sticky costs made the story rather more real looking: We get a yr or two of regular inflation consuming away at bond values, slightly than a worth degree leap. I am unable to wait to see HAFT with sticky costs. For all the opposite requests for generalization: you simply discovered your thesis matter.

Ship typos, particularly in equations.

*Updates*

*Greg wrote, and identified this is not precisely proper. “In the usual r>g, s>0 case, a rise want to carry actual belongings (akin to extra revenue danger) results in a decrease actual charge and better actual debt – the usual “secular stagnation” story. With r<g, s<0, an elevated want to carry actual belongings results in larger actual charges and better debt.” To grasp this remark, it’s a must to have a look at the availability and demand graph within the paper, or in my word. The “provide” of debt within the regular state (b = s/r/), plotted with (r) as a perform of (b) flips signal from a declining curve to a rising curve when (s) and (r) change signal. The “demand” ( r(b)) is upward sloping. So when demand shifts out, (b) rises, however (r) falls when (r>0) and rises when (r<0). With optimistic rates of interest, you produce a higher quantity of actual debt, for a similar surplus, with a decrease actual rate of interest. With adverse rates of interest and a adverse surplus, you produce extra debt with a much less adverse actual charge. Hmm. The (r<g) area continues to be just a little bizarre. There’s additionally the opportunity of a number of equilibria, just like the New-Keynesian zero certain equilibria; see the paper and word.

Erzo Luttmer has a associated HAFT paper, “Everlasting Major Deficits, Idiosyncratic Lengthy-Run Danger, and Development.” It is calibrated in rather more element, and likewise extra detailed on the r<g and long term deficit questions. It consists of fiscal idea (p. 14) however doesn’t appear centrally targeted on inflation. I have never learn it but, nevertheless it’s vital for those who’re getting in to those points.

I nonetheless regard r<g as a technical nuisance. In many of the circumstances right here, it doesn’t relieve the federal government of the necessity to repay money owed, it doesn’t result in a Magic Cash Tree, and it doesn’t undermine fiscal worth degree willpower. I’m nonetheless not a fan of OLG fashions, which delicately want the economic system really to go on for infinite progress. I am not completely persuaded HA is first-order vital for getting mixture inflation dynamics proper. The Phillips curve nonetheless looks as if the largest rotten timber within the ship to me. However these points are technical and complicated, and I may very well be flawed. Consideration is proscribed, so it’s a must to place your bets on this enterprise; however thankfully you possibly can nonetheless learn after different individuals work it out!

Noah Kwicklis at UCLA has a really fascinating associated paper “Switch Funds, Sacrifice Ratios, and Inflation in a Fiscal Concept HANK“

I numerically remedy a calibrated Heterogeneous Agent New-Keynesian (HANK) mannequin that options nominal rigidities, incomplete markets, hand-to-mouth households, nominal long-term authorities debt, and energetic fiscal coverage with a passive financial coverage rule to research the implications of the fiscal idea of the worth degree (FTPL) in a setting with wealth and revenue inequality. In mannequin simulations, the full cumulative inflation generated by a fiscal helicopter drop is basically decided by the dimensions of the preliminary stimulus and is comparatively insensitive to the preliminary distribution of the funds. In distinction, the full actual GDP and employment response relies upon rather more strongly on the stability sheets of the switch recipients, such that funds to and from households with few belongings and excessive marginal propensities to devour (MPCs) transfer mixture output rather more strongly than funds to or from households with low MPCs….