Friday, October 13, 2023

Heterogeneous Agent Fiscal Theory

Today, I'll add an entry to my occasional reviews of interesting academic papers. The paper: "Price Level and Inflation Dynamics in Heterogeneous Agent Economies," by Greg Kaplan, Georgios Nikolakoudis and Gianluca Violante. 

One of the many reasons I am excited about this paper is that it unites fiscal theory of the price level with heterogeneous agent economics. And it shows how heterogeneity matters. There has been a lot of work on "heterogeneous agent new-Keynesian" models (HANK). This paper inaugurates heterogeneous agent fiscal theory models. Let's call them HAFT. 

The paper has a beautifully stripped down model. Prices are flexible, and the price level is set by fiscal theory. People face uninsurable income shocks, however, and a borrowing limit. So they save an extra amount in order to self-insure against bad times. Government bonds are the only asset in the model, so this extra saving pushes down the interest rate, discount rate, and government service debt cost. The model has a time-zero shock and then no aggregate uncertainty. 

This is exactly the right place to start. In the end, of course, we want fiscal theory, heterogeneous agents, and sticky prices to add inflation dynamics. And on top of that, whatever DSGE smorgasbord is important to the issues at hand; production side, international trade, multiple real assets, financial fractions, and more. But the genius of a great paper is to start with the minimal model. 

Part II effects of fiscal shocks. 

I am most excited by part II, the effects of fiscal shocks. This goes straight to important policy questions. 

Note: This figure plots impulse responses to a targeted and untargeted helicopter drop, aggregated at the quarterly frequency. The helicopter drop is a one-time issuance of 16% of total government nominal debt outstanding at = 0. Only households in the bottom 60% of the wealth distribution receive the issuance in the targeted experiment (dashed red line). The orange line plots dynamics in the representative agent (RA) model. The dashed black line plots the initial steady state. Source: Kaplan et al. Figure 7

At time 0, the government drops $5 trillion of extra debt on people, with no plans to pay it back. The interest rate does not change. What happens? In the representative agent economy, the price level jumps,  just enough to inflate away outstanding debt by $5 trillion. 

(In this simulation, inflation subsequent to the price level jump is just set by the central bank, via an interest rate target. So the rising price level line of the representative agent (orange) benchmark is not that interesting. It's not a conventional impulse response showing the change after the shock; it's the actual path after the shock. The difference between colored heterogeneous agent lines and the orange representative agent line is the important part.) 

Punchline: In the heterogeneous agent economies, the price level jumps a good deal more. And if transfers are targeted to the bottom of the wealth distribution, the price level jumps more still. It matters who gets the money. 

This is the first step on an important policy question. Why was the 2020-2021 stimulus so much more inflationary than, say 2008? I have a lot of stories ("fiscal histories," FTPL), one of which is a vague sense that printing money and sending people checks has more effect than borrowing in treasury markets and spending the results. This graph makes that sense precise. Sending people checks, especially people who are on the edge, does generate more inflation. 

In the end, whether government debt is inflationary or not comes down to whether people treat the asset as a good savings vehicle, and hang on to it, or try to spend it, thereby driving up prices. Sending checks to people likely to spend it gives more inflation. 

As you can see, the model also introduces some dynamics, where in this simple setup (flexible prices) the RA model just gives a price level jump. To understand those dynamics, and more intuition of the model, look at the response of real debt and the real interest rate 

The greater inflation means that the same increase in nominal debt is a lesser increase in real debt. Now, the crucial feature of the model steps in: due to self-insurance, there is essentially a liquidity value of debt. If you have less debt, the marginal value of higher; people bid down the real interest rate in an attempt to get more debt. But the higher real rate means the real value of debt rises, and as the debt rises, the real interest rate falls. 

To understand why this is the equilibrium, it's worth looking at the debt accumulation equation, \[ \frac{db}{dt} = r_t (b_t; g_t) b_t - s_t. \]\(b_t\) is the real value of nominal debt, \(r_t=i_t-\pi_t\) is the real interest rate, and \(s_t\) is the real primary surplus. Higher real rates (debt service costs) raise debt. Higher primary surpluses pay down debt. Crucially -- the whole point of the paper -- the interest rate depends on how much debt is outstanding and on the distribution of wealth \(g_t\). (\(g_t\) is a whole distribution.)  More debt means a higher interest rate. More debt does a better job of satisfying self-insurance motives. Then the marginal value of debt is lower, so people don't try to save as much, and the interest rate rises. It works a lot like money demand,

Now, if the transfer were proportional to current wealth, nothing would change, the price level would jump just like the RA (orange) line.  But it isn't; in both cases more-constrained people get more money. The liquidity constraints are less binding, they're willing to save more. For given aggregate debt the real interest rate will rise. So the orange line with no change in real debt is no longer a steady state. We must have, initially \(db/dt>0.\) Once debt rises and the distribution of wealth mixes, we go back to the old steady state, so real debt rises less initially, so it can continue to rise. And to do that, we need a larger price level jump. Whew. (I hope I got that right. Intuition is hard!)  

In a previous post on heterogeneous agent models, I asked whether HA matters for aggregates, or whether it is just about distributional consequences of unchanged aggregate dynamics. Here is a great example in which HA matters for aggregates, both for the size and for the dynamics of the effects. 

Here's a second cool simulation. What if, rather than a lump-sum helicopter drop with no change in surpluses, the government just starts running permanent primary deficits? 

Note: Impulse response to a permanent expansion in primary deficits. The dotted orange line shows the effects of a reduction in surplus in the Representative Agent model. The blue line labelled “Lump Sum” illustrates the dynamics following an expansion of lump sum transfers. The dashed red line labelled “Tax Rate” plots dynamics following a tax cut. The orange line plots dynamics in the representative agent (RA) model. The dashed black line plots the initial steady state. Source: Kaplan et. al. Figure 8.

In the RA model, a decline in surpluses is exactly the same thing as a rise in debt. You get the initial price jump, and then the same inflation following the interest rate target. Not so the HA models! Perpetual deficits are different from a jump in debt with no change in deficit. 

Again, real debt and the real rate help to understand the intuition. The real amount of debt is permanently lower. That means people are more starved for buffer stock assets, and bid down the real interest rate. The nominal rate is fixed, by assumption in this simulation, so a lower real rate means more inflation. 

For policy, this is an important result. With flexible prices, RA fiscal theory only gives a one-time price level jump in response  to unexpected fiscal shocks. It does not give steady inflation in response to steady deficits. Here we do have steady inflation in response to steady deficits! It also shows an instance of the general "discount rates matter" theorem. Granted, here, the central bank could lower inflation by just lowering the nominal rate target but we know that's not so easy when we add realisms to the model. 

To see just why this is the equilibrium, and why surpluses are different than debt, again go back to the debt accumulation equation, \[ \frac{db}{dt} = r_t (b_t, g_t) b_t - s_t. \] In the RA model, the price level jumps so that \(b_t\) jumps down, and then with smaller \(s_t\), \(r b_t - s_t\) is unchanged with a constant \(r\). But in the HA model, the lower value of \(b\) means less liquidity value of debt, and people try to save, bidding down the interest rate. We need to work down the debt demand curve, driving down the real interest costs \(r\) until they partially pay for some of the deficits. There is a sense in which "financial repression" (artificially low interest rates) via perpetual inflation help to pay for perpetual deficits. Wow!

Part I r<g

The first theory part of the paper is also interesting. (Though these are really two papers stapled together, since as I see it the theory in the first part is not at all necessary for the simulations.) Here, Kaplan, Nikolakoudis and Violante take on the r<g question clearly. No, r<g does not doom fiscal theory! I was so enthused by this that I wrote up a little note "fiscal theory with negative interest rates" here. Detailed algebra of my points below are in that note,  (An essay r<g and also a r<g chapter in FTPL explains the related issue, why it's a mistake to use averages from our real economy to calibrate perfect foresight models. Yes, we can observe \(E(r)<E(g)\) yet present values converge.) 

I'll give the basic idea here. To keep it simple, think about the question what happens with a negative real interest rate \(r<0\), a constant surplus \(s\) in an economy with no growth, and perfect foresight. You might think we're in trouble: \[b_t = \frac{B_t}{P_t} = \int e^{-r\tau} s d\tau = \frac{s}{r}.\]A negative interest rate makes present values blow up, no? Well, what about a permanently negative surplus \(s<0\) financed by a permanently negative interest cost \(r<0\)? That sounds fine in flow terms, but it's really weird as a present value, no? 

Yes, it is weird. Debt accumulates at \[\frac{db_t}{dt} = r_t b_t - s_t.\] If \(r>0\), \(s>0\), then the real value of debt is generically explosive for any initial debt but \(b_0=s/r\). Because of the transversality condition ruling out real explosions, the initial price level jumps so \(b_0=B_0/P_0=s/r\). But if \(r<0\), \(s<0\), then debt is stable. For any \(b_0\), debt converges, the transversality condition is satisfied. We lose fiscal price level determination. No, you can't take a present value of a negative cashflow stream with a negative discount rate and get a sensible present value. 

But \(r\) is not constant. The more debt, the higher the interest rate. So  \[\frac{db_t}{dt} = r(b_t) b_t - s_t.\] Linearizing around the steady state \(b=s/r\), \[\frac{db_t}{dt} = \left[r_t  + \frac{dr(b_t)}{db}\right]b_t - s.\] So even if \(r<0\), if more debt raises the interest rate enough, if \(dr(b)/db\) is large enough, dynamics are locally and it turns out globally unstable even with \(r<0\). Fiscal theory still works! 

You can work out an easy example with bonds in utility, \(\int e^{-\rho t}[u(c_t) + \theta v(b_t)]dt\), and simplifying further log utility \(u(c) + \theta \log(b)\). In this case \(r = \rho - \theta v'(b) = \rho - \theta/b\) (see the note 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 \(r<0\) part still gives stable dynamics and multiple equilibria. But if \(\theta>-s\), then dynamics are again explosive for all but \(b=s/r\) and fiscal theory works anyway. 

This is a powerful result. We usually think that in perfect foresight models, \(r>g\), \(r>0\) here, and consequently positive vs negative primary surpluses \(s>0\) vs. \(s<0\) is an important dividing line. I don't know how many fiscal theory critiques I have heard that say a) it doesn't work because r<g so present values explode b) it doesn't work because primary surpluses are always slightly negative. 

This is all wrong. The analysis, as in this example, shows is that fiscal theory can work fine, and doesn't even notice, a transition from \(r>0\) to \(r<0\), from \(s>0\) to \(s<0\). Financing a steady small negative primary surplus with a steady small negative interest rate, or \(r<g\) is seamless. 

The crucial question in this example is \(s<-\theta\). At this boundary, there is no equilibrium any more. You can finance only so much primary deficit by financial repression, i.e. squeezing down the amount of debt so its liquidity value is high, pushing down the interest costs of debt. 

The paper staples these two exercises together, and calibrates the above simulations to \(s<0\) and \(r<g\). But I bet they would look almost exactly the same with \(s>0\) and \(r>g\). \(r<g\) is not essential to the fiscal simulations.* 

The paper analyzes self-insurance against idiosyncratic shocks as the cause of a liquidity value of debt. That's interesting, and allows the authors to calibrate the liquidity value against microeconomic observations on just how much people suffer such shocks and want to insure against them. The Part I simulations are just that, heterogeneous agents in action. But this theoretical point is much broader, and applies to any economic force that pushes up the real interest rate as the volume of debt rises. Bonds in utility, here and in the paper's appendix, work. They are a common stand in for the usefulness of government bonds in financial transactions. And in that case, it's easier to extend the analysis to a capital stock, real estate, foreign borrowing and lending, gold bars, crypto, and other means of self-insuring against shocks. Standard ``crowding out'' stories by which higher debt raises interest rates work. (Blachard's r<g work has a lot of such stories.) The ``segmented markets'' stories underlying faith in QE give a rising b(r). So the general principle is robust to many different kinds of models. 

My note explores one issue the paper does not, and it's an important one in asset pricing. OK, I see how dynamics are locally unstable, but how do you take a present value when r<0? If we write the steady state \[b_t = \int_{\tau=0}^\infty e^{-r \tau}s d\tau = \int_{\tau=0}^T e^{-r \tau}s d\tau + e^{-rT}b_{t+T}= (1-e^{-rT})\frac{s}{r} + e^{-rT}b,\]and with \(r<0\) and \(s<0\), the integral and final term of the present value formula each explode to infinity. It seems you really can't discount with a negative rate. 

The answer is: don't integrate forward \[\frac{db_t}{dt}=r b_t - s \]to the nonsense \[ b_t = \int e^{-r \tau} s d\tau.\]Instead, integrate forward \[\frac{db_t}{dt} = \rho b_t - \theta - s\]to \[b_t = \int e^{-\rho \tau} (s + \theta)dt =  \int e^{-\rho \tau} \frac{u'(c_t+\tau)}{u'(c_t)}(s + \theta)dt.\]In the last equation I put consumption (\(c_t=1\) in the model) for clarity. 

  • Discount the flow value of liquidity benefits at the consumer's intertemporal marginal rate of substitution. Do not use liquidity to produce an altered discount rate.  

This is another deep, and frequently violated point. Our discount factor tricks do not work in infinite-horizon models. \(1=E(R_{t+1}^{-1}R_{t+1})\) works just as well as \(1 = E\left[\beta u'(c_{t+1})/u'(c_t)\right] r_{t+1}\) in a finite horizon model, but you can't always use \(m_{t+1}=R_{t+1}^{-1}\) in infinite period models. The integrals blow up, as in the example. 

This is a good thesis topic for a theoretically minded researcher. It's something about Hilbert spaces. Though I wrote the discount factor book, I don't know how to extend discount factor tricks to infinite periods. As far as I can tell, nobody else does either. It's not in Duffie's book. 

In the meantime, if you use discount factor tricks like affine models -- anything but the proper SDF -- to discount an infinite cashflow, and you find "puzzles," and "bubbles," you're on thin ice. There are lots of papers making this mistake. 

A minor criticism: The paper doesn't show nuts and bolts of how to calculate a HAFT model, even in the simplest example. Note by contrast how trivial it is to calculate a bonds in utility model that gets most of the same results. Give us a recipe book for calculating textbook examples, please!

Obviously this is a first step. As FTPL quickly adds sticky prices to get reasonable inflation dynamics, so should HAFT. For FTPL (or FTMP, fiscal theory of monetary policy; i.e. adding interest rate targets), adding sticky prices made the story much more realistic: We get a year or two of steady inflation eating away at bond values, rather than a price level jump. I can't wait to see HAFT with sticky prices. For all the other requests for generalization: you just found your thesis topic. 

Send typos, especially in equations. 


*Greg wrote, and pointed out this isn't exactly right. "In the standard r>g, s>0 case, an increase desire to hold real assets (such as more income risk) leads to a lower real rate and higher real debt - the standard "secular stagnation” story. With r<g, s<0, an increased desire to hold real assets leads to higher real rates and higher debt." To understand this comment, you have to look at the supply and demand graph in the paper, or in my note. The "supply" of debt in the steady state \(b = s/r/), plotted with \(r\) as a function of \(b\) flips sign from a declining curve to a rising curve when \(s\) and \(r\) change sign. The "demand" \( r(b)) is upward sloping. So when demand shifts out, \(b\) rises, but \(r\) falls when \(r>0\) and rises when \(r<0\). With positive interest rates, you produce a greater amount of real debt, for the same surplus, with a lower real interest rate. With negative interest rates and a negative surplus, you produce more debt with a less negative real rate. Hmm. The \(r<g\) region is still a little weird. There is also the possibility of multiple equilibria, like the New-Keynesian zero bound equilibria; see the paper and note.  

Erzo Luttmer has a related HAFT paper, "Permanent Primary Deficits, Idiosyncratic Long-Run Risk, and Growth." It's calibrated in much more detail, and also more detailed on the r<g and long run deficit questions. It includes fiscal theory (p. 14) but does not seem centrally focused on inflation. I haven't read it yet, but it's important if you're getting in to these issues. 

I still regard r<g as a technical nuisance. In most of the cases here, it does not relieve the government of the need to repay debts, it does not lead to a Magic Money Tree, and it does not undermine fiscal price level determination. I am still not a fan of OLG models, which delicately need the economy truly to go on for infinite growth. I'm not totally persuaded HA is first-order important for getting aggregate inflation dynamics right. The Phillips curve still seems like the biggest rotten timber in the ship to me. But these issues are technical and complex, and I could be wrong. Attention is limited, so you have to place your bets in this business; but fortunately you can still read after other people work it out!  

Noah Kwicklis at UCLA has a very interesting related paper "Transfer Payments, Sacrifice Ratios, and Inflation in a Fiscal Theory HANK"

I numerically solve a calibrated Heterogeneous Agent New-Keynesian (HANK) model that features nominal rigidities, incomplete markets, hand-to-mouth households, nominal long-term government debt, and active fiscal policy with a passive monetary policy rule to analyze the implications of the fiscal theory of the price level (FTPL) in a setting with wealth and income inequality. In model simulations, the total cumulative inflation generated by a fiscal helicopter drop is largely determined by the size of the initial stimulus and is relatively insensitive to the initial distribution of the payments. In contrast, the total real GDP and employment response depends much more strongly on the balance sheets of the transfer recipients, such that payments to and from households with few assets and high marginal propensities to consume (MPCs) move aggregate output much more strongly than payments to or from households with low MPCs....



  1. Seems to me there's a welfare cost of inflation calculation to be done here. In Woodford the cost is symmetric around the zero optimal value because it's derived from pricing frictions causing us to end up producing and consuming a slightly suboptimal basket of goods. The liquidity value of debt sounds like it's saying that a small positive inflation is more costly than just the pricing distortions would imply.

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  3. "Why was the 2020-2021 stimulus so much more inflationary than, say 2008?"

    One answer: Sterilization and paying IoR. That kept a lid on inflation and allowed banks' balance sheets to heal. Too much money in consumer's wallets requires an asbestos wallet to keep demand-pull inflation from spiraling out of control.

  4. Wow, so happy to see this! Finally an emphasis on levels and simply equilibria that can be tested.

    Here, I would suggest the following, to add some science:

    1. Analyze the history between fiscal deficits/surpluses, and put a predictive model that measures the correspondance, over a long time, say 30 years, between fiscal deficits in $s, and changes in M1, M2, and debt levels in $. Obviously, the amount of money printed should equal the amount of money outstanding in one form or another.

    2. Analyze how the quantity of debt outstanding corresponds with interest rates. There should be a clear relationship, with the interest rate going down as the quantity of debt in public hands decreases.

    3. Analyze how M1, M2 etc. correspond with price levels. We should see, over the long term that price levels vary linearly with money outstanding.

    Put these three connections together, and we can define the long-run relationship between interest rates, inflation, fiscal policy, quantity of debt, M1 M2 etc. These long-term relationships should be a solid bedrock to rest macroeconomic policy on.

    Of course the remaining question is what happens in the short term. For that, we have this lovely heterogenious approach. Once we have the long-term model worked out, we can always refine the timing of the changes we'll see based on what people do what with the money in the short term. Ideally, this can then inform policy when it comes to crises so we know not just how much to print or save, but who to give or take from


  5. I'm the PhD student with the FTPL-HANK working paper -- thanks for linking to it in the edits. It's interesting, I saw the first graph that you posted and read it rather differently. My sense is that the three lines (pertaining to a targeted vs un-targeted transfer in HAFT, vs a transfer in the RA model) are still fairly close together, and the first two converge rapidly to the RA line rapidly over time. No matter where the fiscal helicopter drop goes, the price level rises by roughly the same amount by the time 30 quarters have passed.

    To my eye, it looks like the total cumulative inflation generated in all of the scenarios is essentially the same, after shorter-run dynamics related to precautionary savings and wealth effects have played out and real rates revert back to the natural rate in the long-term. I think your hunch that the heterogeneous agents don't add much to the FTPL's inflation dynamics is mostly correct. In either setting, people expect that the government will inflate away debt and price it accordingly, which in turn generates inflation that inflates away the debt, making their expectation rational. An upshot of this, though, is that the distribution of transfers is second-order for inflation when compared to the magnitude of the debt-financed stimulus spending. Nominal debt not backed by future surpluses remains the nominal anchor.

    (Of course, since MPC heterogeneity matters for the output response in HANK, fiscal theory + HANK seems like it suggests some pretty stark implications for the trade-off between output and inflation -- which is what my paper is all about).

    1. Noah: Thanks for writing. I agree with your intuition. The HAFT part seems to generate a bit more short run inflation (price level rise) by a transient change in the real rate of interest. That makes a lot of sense. I guess the larger output response then means HA is altering the Phillips curve; given the path of inflation you get more output response. Great paper!

  6. I suggest that you have made an error in the case where r = - a, and the constant a is positive.
    Start with the state transition equation, db(t)/dt = r∙b(t) - s(t) and make the substitution r = - a to find db(t)/dt = -a∙b(t) - s(t). Provided s(t) is bounded (the sign of s(t) is not material) the accumulated quantity of real bonds goes to zero as time goes to infinity, for
    b(T) = exp(-a∙(T - t))∙b(t) - ∫exp(-a∙(T - τ))∙s(τ)∙dτ , in which the integral evaluated between τ = t and τ = T. This is easily demonstrated by setting s(t) = 0 for all t. Then db(t)/dt = -a∙b(t) which has the solution
    b(t) = b(0)∙exp(-a∙t) , or translating the t-axis to conform with your notation
    b(T) = b(t)∙exp(-a∙(T - t)).
    Intuitively, this is the expected result. For if r < 0 for all time, then the real bond accumulation tends to zero by the Fisher equation i(t) = r(t) + π(t) which when re-arranged is r(t) = i(t) - π(t) < 0. The real (exchange) value of Treasury-issued debt diminishes as time proceeds to the right. But, the present value, b(t), is, nevertheless, given by b(T)∙exp(+a∙(T - t)).

    As interesting as the mathematics is, ultimately it is immaterial because economic conditions rarely persist for more than a generation. WWI (1914 - 1918) to 1929, 1931 to WWII (1939 - 1945), for one example. October 1987 to the 1996-7; 2001-2002 to 2007-2009; 2015 to 2020 &c. Earlier episodes can be found. In essence, although it is mathematically possible to write down an integral equation with integration limits of t = 0 to t = ∞, the equation is not to be considered real in the sense that ∞ will ever be reached with the model assumptions invariant under time. Pontryagin's Maximum Principal is not applicable to optimization if the future date is unbounded time. Likewise, in business practice, no decision-maker would seriously consider a model that relies on an infinite time horizon as a basis for real-world decision-making.

    1. That's the point, not a mistake. When r<0, and s<0 db_t/dt = r b_t + s goes geometrically to the steady state b=s/r, for any initial b_0. Thus, b_0 is not pinned down as the only non-explosive solution.

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  8. Do you think that Covid fiscal response has had a large impact going forward on our expectations around the variance of consumption? And hence dictate a lower savings insurance rate? Pre Covid, large risk shocks were met with high unemployment and longer growth shocks. (Hence the individual desire to save to consumption smooth. And also the existence of substantial risk premium to own assets that are negatively correlated with consumption). Covid demonstarted a new govt fiscal response - in essence it was the “great consumption smoothing”. We borrowed from the future to furlough workers today in a manner we probably couldn’t have imagined. (At the expense of higher inflation and borrowing costs). Going forward do households now risk assess future crisis differently? One might assume we need to save less for insurance against future consumption shocks because we price in a high probability of govt fiscal smoothing. Similarly we might expect risk premium more generally to compress as we reduce expectations of the variance of future consumption over different periods?


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