# 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{sigmakappa}{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+sigmakapparight) pmsqrt{left(
1+beta+sigmakapparight) ^{2}-4beta}}{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

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