Friday, September 10, 2021

Inflation in the shadow of debt

(Note: This post uses mathjax equations. If you see garbled latex code, come to the original source.) 

The effect of monetary policy on inflation depends crucially on fiscal policy.

In standard new-Keynesian models, of the type used throughout the Fed, ECB, and similar institutions, for the central bank to reduce inflation by raising interest rates, there must be a contemporaneous fiscal tightening. If fiscal policy does not tighten, the Fed will not lower inflation by raising interest rates.

The warning for today is obvious: Fiscal policy is on a tear, and not about to tighten any time soon no matter what central banks do. An interest rate rise might not, then, provoke the expected decline in inflation.

Here is a very stripped down model to show the point. \begin{align*} x_t & = E_t x_{t+1} - \sigma(i_t - E_t \pi_{t+1}) \\ \pi_t & = \beta E_t \pi_{t+1} + \kappa x_t \\ i_t &= \phi \pi_t + u_t \\ \Delta E_{t+1}\pi_{t+1} & = - \sum_{j=0}^\infty \rho^j \Delta E_{t+1} \tilde{s}_{t+1+j} + \sum_{j=1}^\infty \rho^j \Delta E_{t+1}(i_{t+j}-\pi_{t+1+j}) \end{align*} The first two equations are the IS and Phillips curves of a standard new-Keynesian model. The third equation is the monetary policy rule.

The fourth equation stems from the condition that the value of debt equals the present value of surpluses. This condition is also a part of the standard new-Keynesian model. We're not doing fiscal theory here. Fiscal policy is assumed to be "passive:" Surpluses adjust to whatever inflation results from monetary policy. For example, if monetary policy induces a big deflation, that raises the real value of nominal debt, so real primary surpluses must raise to pay the now larger value of the debt. Since it just determines surpluses given everything else, this equation is often omitted, or relegated to a footnote, but it is there. Today, we just look at the surpluses. Without them, the Fed's monetary policy cannot produce the inflation path it desires.

Notation: \(\Delta E_{t+1} \equiv E_{t+1}-E_t\), \(\rho\) is a constant of approximation slightly less than or equal to one, \(\tilde{s}\) is the real primary surplus relative to debt. For example, \(\tilde{s}=0.01\) means the surplus is 1% of the value of debt, or 1% of GDP at current 100% debt to GDP. The last term captures a discount rate effect. If real interest rates are higher, that lowers the present value of surpluses. Equivalently, higher real interest rates raise the interest costs in the deficit, requiring still higher primary surpluses to pay off debt. (Reference: Equation (4.23) of Fiscal Theory of the Price Level.) \(x\) is the output gap, \(\pi\) is inflation, \(s\) is the real primary surplus, \(i\) is the interest rate, and the Greek letters are parameters. 

Now, suppose the Fed raises interest rates \(\{i_t\}\) following a standard AR(1). with coefficient \(\eta = 0.6\). However, there are multiple \(\{u_t\}\) which produce the same path for \(\{i_t\}\), each of which produces a different inflation path \(\{\pi_t\}\). Each of them also produces a different fiscal response \(\{s_t\}\). So, let's look for given (AR(1)) interest rate \(\{i_t\}\) path at the different possible inflation \(\{\pi_t\}\) paths, their associated monetary policy disturbance \(\{u_t\}\) and their associated fiscal underpinnings.

The top left panel shows a standard result. The interest rate in blue rises, and then returns following an AR(1). Here, the 1% interest rate rise causes a 1% inflation decline, shown in red. I use \(\eta=0.6, \sigma = 1, \kappa = 0.25, \beta = 0.95, \phi = 1.2 \) The monetary policy disturbance \(u_t\), dashed magenta.  is even larger than the actual inflation rise, but \( i_t = \phi \pi_t + u_t\) and  the disinflation in \(\pi_t\) bring the interest rate to a lower value. 

Now, let's calculate the implied "passive" surplus response. I use \(\rho=1\). With a 1% disinflation, the present value of surpluses must rise by 1%. However, the real interest rate rises substantially and persistently. From a present value point of view, that higher discount rate devalues government debt, an inflationary force.  From an ex-post point of view the higher real rates lead to years of higher debt service costs. Viewed either way, the constant-discounted sum of surpluses must rise by even more than one percent. In this case, the sum of surpluses must rise by 3.55, meaning 3.55 percent of debt or 3.55 percent of GDP at 100% debt to GDP ratio, or about $700 billion dollars. 

What if Congress looks at that and just laughs? Well, the Fed must do something else. The top right panel has a different disturbance process \(\{u_t\}\). This disturbance produces exactly the same path of interest rates, shown in blue. But it produces half as much initial deflation, -0.5%. The disinflation also turns to slight inflation after 3 years. With less disinflation, there is less need to produce a larger value of government debt, so the sum of surpluses must only rise by 2.23%. 

The bottom left shows a case that inflation does not decline at all, though again the path of interest rates is exactly the same. This occurs with a different disturbance \(\{u_t\}\) as shown. Finally, in the bottom right, it is possible that this interest rate rise produces 0.5% inflation. In this case, fiscal policy produces a slight deficit. The case of no change in surplus or deficit occurs between 0% and 0.5% inflation. 

To reiterate the point, the observable path of interest rates is exactly the same in all four cases. In a new-Keynesian model, the difference is the dynamic path of the monetary policy disturbance. Different underlying disturbances then produce the different inflation outcomes, and the different requirements for the "passive" fiscal policy authorities. Of course (I can't help myself here) to a fiscal theorist the \(\{u_t\}\) business is meaningless. Congress' choice to match the Fed's tightening with its own tightening produces the deflationary path, and if Congress does not do so, we get an inflationary path. 

Looked at either way, in a totally standard new-Keynesian model, the effects of an interest rate rise depend crucially on fiscal policy. If fiscal policy does not agree to tighten along with an interest rate rise, the interest rate rise will not produce lower inflation. 

Hat tip: This point emerged out of discussions with Eric Leeper on his 2021 Jackson Hole paper on fiscal-monetary interactions.  

The next post, an essay at Project Syndicate, provides larger context. 

**********

Calculations. To produce the plots I write the monetary policy rule in a different form \[ i_t = i^\ast_t + \phi ( \pi_t - \pi^\ast_t) \] \[ i^\ast_t = \eta i^\ast_{t-1} + \varepsilon_t \] Then I can specify directly the interest rate AR(1) in \(i^\ast_t\), and the initial inflation in \(\pi^\ast_t\).  These forms are equivalent. Indeed, I construct \( u_t = i^\ast_t - \phi \pi^\ast_t \) in order to plot it. 

I use the analytical solutions for inflation given an interest rate path derived 26.4 of Fiscal Theory, \[ \pi_{t+1}=\frac{\sigma\kappa}{\lambda_{1}-\lambda_{2}}\left[ i_{t}+\sum _{j=1}^{\infty}\lambda_{1}^{-j}i_{t-j}+\sum_{j=1}^{\infty}\lambda_{2}% ^{j}E_{t+1}i_{t+j}\right] +\sum_{j=0}^{\infty}\lambda_{1}^{-j}\delta_{t+1-j}. \] \[ \lambda_{1,\ 2}=\frac{\left( 1+\beta+\sigma\kappa\right) \pm\sqrt{\left( 1+\beta+\sigma\kappa\right) ^{2}-4\beta}}{2}, \]

Matlab code: T = 50;
sig = 1;
kap = 0.25;
eta = 0.6;
bet = 0.95;
phi = 1.2;
pi1 = [-1 -0.5 0 0.5];

lam1 = ((1+bet+sig*kap)+ ((1+bet+sig*kap)^2-4*bet)^0.5)/2;
lam2 = ((1+bet+sig*kap)- ((1+bet+sig*kap)^2-4*bet)^0.5)/2;
lam1i = lam1^(-1);

delt = pi1 - sig*kap/(lam1-lam2)*lam2/(1-lam2*eta);

tim = (0:1:T-1)';

pit = zeros(T,1);
pit(2) = sig*kap/(lam1-lam2)*lam2/(1-lam2*eta) ; % t=1
pit(3) = sig*kap/(lam1-lam2)*(1/(1-lam2*eta)) ;
for indx = 4:T;
pit(indx) = sig*kap/(lam1-lam2)*...
(eta^(indx-3)/(1-lam2*eta) + lam1i*(eta^(indx-3)-lam1i^(indx-3))/(eta-lam1i) );
end;

pim = [pit*(1+0*pi1) + [0*delt;(lam1i.^((0:T-2)')).*delt]];
it = [0; eta.^(0:1:T-2)'];
um = it*(1+0*pi1) - phi*pim;
rterm = sum(it(2:end-1,:)-pim(3:end,:));
sterm = rterm-pim(2,:);
disp('r');
disp(rterm);
disp('s');
disp(sterm);

if 0; % all together
figure;
C = colororder;
hold on
plot(tim,pim,'-r','linewidth',2);
plot(tim,um,'--m','linewidth',2);
plot(tim,it,'-b','linewidth',2);
plot(tim,0*tim,'-k')
axis([ 0 6 -inf inf])
end;

figure; % 4 panel plot
for indx = 1:4;
subplot(2,2,indx);
hold on;
plot(tim,pim(:,indx),'-r','linewidth',2);
if indx == 1;
text(1.8,-0.7,'\pi','color','r','fontsize',18)
text(1,0.7,'i','color','b','fontsize',18);
text(2.4,1,'u','color','m','fontsize',18)
end
plot(tim,um(:,indx),'--m','linewidth',2);
plot(tim,it,'-b','linewidth',2);
plot(tim,0*tim,'-k')
title(['\Sigma s = ' num2str(sterm(indx),'%4.2f')],'fontsize',16)
axis([ 0 6 -1 1.5])
end
if eta == 0.6
print -dpng nk_fiscal_1.png
end

20 comments:

  1. If I respond, will it just get me banned for being uninformed, irresponsible, or motivated by unworthy interests?

    First, where is your key to the variables? Aren't you willfully contributing to ambiguity by listing equations without defining the variables in the same place?

    Second, aren't you completely leaving out finance? Isn't government debt priced by markets without regard to unobservable surpluses? Don't banks create money to trade government debt, so that the debt takes on a market value that has far more to do with funding plumbing than this notional "surplus" you assume and never bother to actually measure?

    How can you leave out finance? Don't financial flows measurably eclipse real flows by at least an order of magnitude?

    When you include a properly scaled financial sector as in https://www.bis.org/publ/work890.htm (a BIS DSGE model), do you find that real goods flows are too small to influence prices?

    Can you respond to Fischer Black in "Noise"?

    "I think that the price level and rate of inflation are literally indeterminate. They are whatever people think they will be. They are determined by expectations, but expectations follow no rational rules. If people believe that certain changes in the money stock will cause changes in the rate of inflation, that may well happen, because their expectations will be built into their long term contracts."

    ReplyDelete
    Replies
    1. John has a tendency to be expert level. The key is that he told you what each equation was so you can find it in most advance macro textbooks (or in earlier posts). The IS curve relates output (xt) to expected output and an intertemporal elasticity of substitution (sigma) discounted real interest rate ( nominal rate (i) less expected inflation (pi)). The new keynesian phillips curve relates inflation (pi) to expected inflation and output (xt) with k being the slope of the phillips curve under stable inflation expectations i think. Next he gives the monetary policy or taylor rule which describes central bank behavior setting the interest rate (i) as a function of expected inflation and random shocks to the economy (u). The parameter phi is usually taken to be greater than one such that the fed tends to overreact and thereby induce stability. The final equation is John’s labor of love which is one of several forms of an equilibrium condition relating today’s change in inflation rate to the discounted stream (or present value) of changes in future government surpluses (s) and discounted stream of changes in real interest rates. As simple as can be really ;)

      Delete
    2. Based on the code provided at the end of the post, which variables do you need defined? You can certainly quibble with the assumptions, but the terms and their values seem straightforward.

      Delete
    3. RSM,

      "I think that the price level and rate of inflation are literally indeterminate. They are whatever people think they will be. They are determined by expectations, but expectations follow no rational rules."

      Inflation is in large part determined by productivity defined as Real GDP / Total Debt Outstanding. How much real growth can you get out of an economy for a given amount of total debt?

      The inflation expectations leads to higher inflation argument falls flat and is based on some bad logic.

      If I were expecting inflation in the price of hammers, would I as a carpenter run out and buy a couple hundred hammers above and beyond what I ever intend to use? There is a utilitarian aspect to consider.

      If I were expecting inflation in the price of new bread, would I go out and purchase a couple of hundred loaves of bread recognizing that most the bread will turn to mold before I get a chance to eat it? There is a depreciation aspect to consider.

      If I were expecting inflation in price of garbage collection or electricity, would I accelerate or decelerate my generation of garbage and use of electricity? There is a conservation factor to consider.

      And that is just the spending decision. I could with elevated inflation expectations decide to pay down debts, I could decide facing elevated inflation expectations to invest in a productive enterprise, I could make numerous other decisions regarding how to allocate money that would not track with inflation expectations leading to higher demand or price inflation.

      Delete
    4. @LAL
      Do you have some reading recommendations?

      Delete
    5. You can find John’s book on the Fiscal Theory if the price level in his website. Along with his Time Series and Asset Pricing books they are my favorite. But specifically for new Keynesianism Romer’s Advanced Macroeconomics textbook, Woodford’s Interest and Prices (this I think is what John uses), or maybe Mankiw. Alternativley, you can get it from lots of papers: Smets-Wouters is what John’s colleague Harald Uhlig would teach undergrads at the univeristy of Chicago. Olivier Blanchard’s papers on monopolistic price setting and John Taylor and Calvo’s papers on staggered wage settings I would say form the core of the micro foundations. (Taylor’s papers again for monetary policy rules). Most papers by Mankiw are well written, and his New Keynesian Manifesto is heart warming.

      Delete
    6. 》 If I were expecting inflation in the price of hammers, would I as a carpenter run out and buy a couple hundred hammers above and beyond what I ever intend to use? There is a utilitarian aspect to consider.

      Why not buy out-the-money call options on hammers? If they go in-the-money can't you just sell the option and never even handle physical hammers?

      Why do you ignore finance completely, when financial stocks and flows eclipse their real counterparts by at least a factor of ten?

      Did you even look at the BIS DSGE model i cited?

      Delete
    7. "Why not buy out-the-money call options on hammers? If they go in-the-money can't you just sell the option and never even handle physical hammers?"

      Because I am a carpenter, not a financial contract trader. (I thought that point was already established).

      "Why do you ignore finance completely, when financial stocks and flows eclipse their real counterparts by at least a factor of ten?"

      I don't ignore finance and yes I looked at the model. My response was in regard to your Fisher Black reference:

      "I think that the price level and rate of inflation are literally indeterminate. They are whatever people think they will be. They are determined by expectations, but expectations follow no rational rules."

      Elevated inflation expectations are not a sufficient condition to drive inflation and the price level higher.

      If you can point to where inflation expectations are discussed in the paper and how they feed through to realized inflation, I would be happy to take a second look.

      Delete
  2. Also i forgot to plug John’s upcoming Fiscal Theory of the Price Level book which goes through all of this step by step in discrete and continuous time domains.

    ReplyDelete
  3. Just an observation: The fourth equation in the article resolves to read as follows,
    Eᵥ₊₁ {∑ρᵡ (πᵥ₊₁₊ᵪ + s ̃ᵥ₊₁₊ᵪ – ρ iᵥ₊₁₊ᵪ )} = Eᵥ {∑ρᵡ (πᵥ₊₁₊ᵪ + s ̃ᵥ₊₁₊ᵪ – ρ iᵥ₊₁₊ᵪ )}
    where the indices in the summands and in the expectations operator have their equivalent in the article, i.e., χ = j = 0, 1, 2, …, n, n+1, … ∞ ; and, v = t .

    This suggests that either the summand = 0, or the expectations do not change from period t to period t+1, suggesting that the fourth equation is, at best, an identity, at worst a nullity.

    The first three equations can be resolved by algebraic manipulation to obtain expressions for x(t), π(t), and i(t), solely in terms of E{x(t+1)}, E{π(t+1)}, and u(t), i.e.,

    x(t) = [ E{x(t+1)} + σ∙(1-ϕ∙β)∙E{π(t+1)} -σ∙u(t)]/(1-σ∙κ∙ϕ)

    π(t) = [ -κ∙E{x(t+1)} + (β -σ∙κ)∙E{π(t+1)} +σ∙κ∙u(t)]/(1-σ∙κ∙ϕ)

    i(t) = [ -κ∙ϕ∙E{x(t+1)} -ϕ∙(σ∙κ +β)∙E{π(t+1)} +u(t)]/(1-σ∙κ∙ϕ)

    Whereas, u(t) is a disturbance term in the interest rate policy equation, E{x(t+1)} and E{π(t+1)} are expectations of the next period's capacity shortfall, x(.), and inflation rate, π(.), and as such one would want to have some means of determining, in a mathematical sense, how the values of those expectations are developed, in order to determine x(t), π(t), and i(t) in the current period.

    If x(t), π(t), and i(t) are known quantities in the current period, then one would anticipate that the expectations E{x(t+1)} and E{π(t+1)} are resolved up to the stochastic disturbance term, u(t), typically. But, in point of fact, with three known quantities on the left-hand side of the equals signs, E{x(t+1)} and E{π(t+1)} and u(t) are completely determined. This suggests that the model is missing one or more terms that the fourth equation cannot provide (see above).

    The algebra is linear, and straight-forward, in the first four equations appearing in the article. The parameters σ, κ, and β are given constants. Is there any reason to believe that the parameters are independent of time, t, or of x(t), or π(t)? The fourth parameter, ϕ, is a gain (in an engineering
    control systems sense, and is under the control of the monetary policy planner--he can select any suitable value from the real number line, provided the selection does not result in instability, i.e., all of the poles of the system lie in the left-hand half of the complex frequency plane). The values for the four parameters σ, κ, β and ϕ, yield a positive determinant. The charts presented in the article demonstrate that the resulting system is stable.

    Given that the fourth equation is a cipher, it is difficult to draw an inference that the fiscal management has any bearing on the behaviour of x(t), or π(t) or i(t) in this pared-down NK model. If the determination of i(t) was not automatic (third equation), then one could conceive of a monetary system manager eye-balling the ratio of real surplus to debt ratio and making an adjustment, perhaps to the gain,ϕ, to trim either or both of x(t) and π(t), or the expectations E{x(t+1)} and E{π(t+1)}, via the interest rate, i(t). Just how that would work, in an monetary system model is not clear, but it would provide a more direct linkage between the fiscal side and the monetary policy side. For example, the monetary system controller, noting that the fiscal authority is over-spending and running up the debt which would be anticipated to raise inflation expectations and create supply-side shortages [ x(t) < 0, where x(t) = 1- Y(t)/Y*(t) , Y*(t) is the natural full capacity of the economy], would adjust the interest rate to immunize or "cool" the economy to dampen the run-up in inflation expectations. As noted above, the gain parameter ϕ might serve that purpose and become a function of real surplus to debt ratio, s ̃ᵥ₊₁₊ᵪ , e.g.., ϕ = f(s(t)) = ϕ₀ + ϕ₁∙s(t) + ϕ₂∙s²(t), a quadratic function, say, or some other similar suitable function that would replicate the human being in the control loop (see various engineering control systems models for examples).





    ReplyDelete
    Replies
    1. OEE,

      "For example, the monetary system controller, noting that the fiscal authority is over-spending and running up the debt which would be anticipated to raise inflation expectations and create supply-side shortages [ x(t) < 0, where x(t) = 1- Y(t)/Y*(t) , Y*(t) is the natural full capacity of the economy], would adjust the interest rate to immunize or cool the economy to dampen the run-up in inflation expectations."

      And that adjustment of interest rates upward would create a larger interest expenditure on government debt that could in turn be spent by the public.

      The Fed can't simultaneously remove bonds from circulation through open market operations (to reduce interest expenditures to the public) and raise interest rates
      to "cool" the economy.

      For raising interest rates to actually cool the economy, one or more of the following has to happen:

      1. Government switches to zero coupon bonds (or other zero coupon securities) where interest / returns are accrued versus readily spendable.

      2. Private sector credit formation must contract

      And that assumes that the Ponzi limit isn't hit (total interest expenditures by government exceeds total tax revenue). At that point, bond auctions would likely fail and all bets are off.

      Delete
    2. The U.S. Treasury issues U.S. federal government debt. I'm not aware of the Treasury issuing zero-coupon notes or bills, but it might have done. It certainly issues Treasury bills at a discount to face value.

      Typically, the FOMC buys T-bills and short maturity notes, not bonds. It is only with "Operation Twist" that the FOMC has waded in and purchased bonds and mortgage backed securities, and then only to lower long-term interest rates to stimulate the business economy and re-inflate commercial bank balance sheets during the 2009 recession and its after-math. The scenario that you have quoted does not apply to the situation in which the FOMC would be buying bonds and MBS.

      When the FOMC raises the interest rate on short term securities by raising the Fed Funds Rate and the Discount Rate and Overnight Repo/Reverse Repo and Interest on Excess Reserves, the FOMC is typically focussed on reeling in the "animal spirits" in the financial markets in the U.S.--the opposite of a situation that would see QE and Operation Twist employed.

      Private sector "credit formation" (bank loans and broker call loans) do contract but only when the bank prime rate exceeds a threshold and the FFR exceeds the 10-year T-note yield rate. Then, as we know from past experience, the business economy begins to contract, first in construction services, then in manufacturing, then in retail sales, and services, at which point the FOMC is frantically attempting to reverse the course of the business contraction. We have then entered into a recession; and, now, the FOMC frantic and endevours to drive the short-term interest rates down as the unemployment reaches a local maximum. Congress at this point is hopping mad at the FOMC for removing the "punch bowl" without Congress's permission, or advanced warning, &c.

      Ponzi schemes have no limit, except when the government pounces on the operator and hauls him off to gaol. In economics, the term is used in the sense of a constraint on the modeller's ability to create wealth out of thin air (Ponzi's objective). Thus, we see the modeller declare that the constraints he has imposed on the model guarantee adherence to the "no Ponzi scheme" principle. What the U.S. government does about it in real life remains an open question.

      Delete
    3. "Ponzi schemes have no limit, except when the government pounces on the operator and hauls him off to goal."

      Unlike the private Ponzi operator, the federal government is fairly transparent in publishing it's income and expenditure statements, so you will need to try a little harder than that.

      You really don't think bond auction participants also pay close attention to US government tax and spending policies?

      Delete
  4. John,

    "What if Congress looks at that and just laughs? Well, the Fed must do something else."

    Congress is just one of the three branches of US Government. There is also the Executive Branch (Presidency, IRS, Treasury Department) and the Judicial Branch (court system).

    Prior to Federal Reserve Act of 1913, economic policy on the national level was handled through the US Treasury Department.

    ReplyDelete
  5. "In standard new-Keynesian models, of the type used throughout the Fed, ECB, and similar institutions, for the central bank to reduce inflation by raising interest rates, there must be a contemporaneous fiscal tightening. If fiscal policy does not tighten, the Fed will not lower inflation by raising interest rates."

    Does the Fed really believe this? Do they operate under the assumption that no amount of the interest rate rising will do a thing to inflation if there is 0 Fiscal coordination?

    I know this makes a lot of logical sense, especially for countries like Zimbabwe. But I get sense that the mainstream views the US as a pretty special case where fiscal policy doesn't really matter currently.

    ReplyDelete
    Replies
    1. James,

      It is a little bit more complex than that.

      Enough interest rate raising will either:
      1. Force fiscal coordination
      Or 2. Cause government bond auctions to fail

      At 27 Trillion Dollars + in US debt, interest rates across all maturities of government debt need only rise to about 8% before all available tax revenue is used making interest payments (the Ponzi limit). After that point, government bond buyers are putting their own long positions at risk by buying any more government debt.

      The consideration is this - if excess government spending is the root cause of higher inflation, then does it really matter how that spending manifests itself whether that spending takes the form of government wages, entitlement benefits, or even government bond interest payments?

      Delete
    2. @Frank

      I think the it might, as Japan has shown. If domestic pension holders are just going to park that money in safes, while foreign bond holders will buy consumer good, then you might see two different inflation results.

      Delete
    3. James,

      But in Japan is that a cultural thing or is it a government policy that keeps it's interest payments from creating a self loop re-inforcing loop of higher inflation leading to higher interest payments leading to even higher inflation?

      What I mean is that there is more than one way to kill the inflation cat. Nixon implemented price controls back in the 1970's for instance. What inflation containment measures does the government and / or bank of Japan engage in beyond changes in interest rates?

      Does Japan institute automatic government spending cuts on other services when / if inflation ever becomes problematic?

      Delete
  6. On the latest version of the Fiscal Theory, abailable in John's website, the path 26.4 is something completely different. Which makes the last equation of the note, used to build the matlab code, kind of a cipher. And that has nothing to do with the "expert level" of the note.

    ReplyDelete
    Replies
    1. It took me a while to find. But derivation of the last equation of the note is actually equation (25.21) (not 26.4) in the Fiscal Theory draft available at John's website. It's on the book's appendix.

      Delete

Comments are welcome. Keep it short, polite, and on topic.

Thanks to a few abusers I am now moderating comments. I welcome thoughtful disagreement. I will block comments with insulting or abusive language. I'm also blocking totally inane comments. Try to make some sense. I am much more likely to allow critical comments if you have the honesty and courage to use your real name.