# Inflation in the shadow of debt

(Note: This article uses the mathjax equation. If you see garbled Latex code, come Original source.)

The impact of monetary policy on inflation mainly depends on fiscal policy.

In the standard New Keynesian model used by the Federal Reserve, the European Central Bank and similar institutions, if the central bank is to raise interest rates to reduce inflation, it must simultaneously implement fiscal austerity. If fiscal policy is not tightened, the Fed will not raise interest rates to reduce inflation.

Today’s warning is obvious: fiscal policy is tearing apart, and no matter what the central bank does, it will not tighten in the short term. Therefore, raising interest rates may not trigger the expected decline in inflation.

This is a very condensed model to illustrate this 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 the standard new Keynesian model. The third equation is the monetary policy rules.

The fourth equation stems from the condition that the value of debt equals the present value of surplus. This condition is also part of the standard New Keynesian model. We are not studying fiscal theory here. Fiscal policy is assumed to be “passive”: the surplus is adjusted for any inflation caused by monetary policy. For example, if monetary policy triggers severe deflation, it will increase the real value of nominal debt, so the real basic surplus must be raised to pay for the larger debt value now. Since it only determines surplus given all other conditions, this equation is usually omitted or downgraded to a footnote, but it does exist. Today, we only look at surplus. Without them, the Fed’s monetary policy cannot produce the inflation path it wants.

Symbol: (Delta E_{t+1} equiv E_{t+1}-E_t), (rho) is an approximate constant, slightly less than or equal to 1, (tilde{s } ) Is the actual basic surplus relative to debt. For example, (tilde{s}=0.01) means that the surplus is 1% of the debt value, or 1% of GDP when the current debt-to-GDP ratio is 100%. The last item reflects the discount rate effect. If the real interest rate is higher, it will reduce the present value of the surplus. Similarly, a higher real interest rate increases the interest cost of the deficit, which requires a higher basic surplus to repay debt. (Reference: Formula (4.23) The fiscal theory of price levels.) (x) is the output gap, (pi) is inflation, (s) is the actual primary surplus, (i) is the interest rate, and the Greek letter is the parameter.

Now, suppose the Fed raises the interest rate according to the standard AR(1)({i_t}). The coefficient (eta = 0.6). However, there are multiple ({u_t}) that generate the same path for ({i_t}), and each will generate a different expansion path({pi_t}). Each of them also produces a different fiscal response({s_t}). Therefore, let us look for a given (AR(1)) interest rate({i_t}) path in different possible inflation paths({pi_t}), and their related monetary policy interference ({ u_t}) and its related financial basis.

The upper left panel displays the standard results. The blue interest rate rises and then returns according to AR(1). Here, a 1% increase in interest rates leads to a 1% decrease in inflation, shown in red. I use (eta=0.6, sigma = 1, kappa = 0.25, beta = 0.95, phi = 1.2 ) monetary policy disturbance(u_t), the dotted line is magenta. It is even greater than the increase in actual inflation, but the deflation of ( i_t = phi pi_t + u_t) and (pi_t) reduces the interest rate to a lower value.

Now, let us calculate the implied “passive” earnings response. I use (rho=1). With 1% deflation, the present value of the surplus must increase by 1%. However, real interest rates continue to rise sharply. From the perspective of present value, a higher discount rate will depreciate government debt, which is an inflationary force. From a post-engagement perspective, higher real interest rates have led to higher debt service costs for many years. Regardless of the angle of view, the sum of fixed discounted surpluses must increase by more than 1%. In this case, the sum of the surplus must increase by 3.55, which means that debt is 3.55% of GDP or 3.55% of GDP, and the debt-to-GDP ratio is 100%, or about $700 billion.

What if Congress saw this and just laughed? Well, the Fed must do something else. The panel in the upper right corner has different disturbance processes({u_t}). This interference produces exactly the same interest rate path, as shown in blue. But it will produce half of the initial deflation, -0.5%. Three years later, deflation also turned into slight inflation. As deflation decreases, the demand for greater value of government debt will decrease, so the total surplus can only increase by 2.23%.

The bottom left corner shows that inflation has not fallen at all, although the path of interest rates is exactly the same. This happens in different interference ({u_t}) as shown in the picture. Finally, in the lower right corner, this rate hike may lead to 0.5% inflation. In this case, fiscal policy will produce a slight deficit. No change in surplus or deficit occurs between 0% and 0.5% inflation.

To reiterate, in all four cases, the observable interest rate path is exactly the same. In the new Keynesian model, the difference lies in the dynamic path of monetary policy disturbances. Different potential interferences will produce different inflation results and different requirements for “passive” fiscal policy authorities. Of course (I cannot extricate myself here) For a fiscal theorist, ({u_t}) business is meaningless. Congress choosing to match the Fed’s deflation with its own deflation will create a deflationary path. If Congress does not do this, we will have an inflation path.

Regardless of the angle of view, in a completely standard New Keynesian model, the impact of rising interest rates depends to a large extent on fiscal policy. If fiscal policy does not agree to tighten as interest rates rise, rising interest rates will not produce lower inflation.

Hat tip: This point was raised in a discussion with Eric Leeper on his 2021 Jackson Hole paper on fiscal-monetary interactions.

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calculate. In order to generate the chart, I wrote the monetary policy rules in different forms[

i_t = i^ast_t + phi ( pi_t – pi^ast_t)

]

[

i^ast_t = eta i^ast_{t-1} + varepsilon_t

]

Then I can directly specify the interest rate AR(1) in (i^ast_t) and the initial inflation in (pi^ast_t). These forms are equivalent. In fact, I constructed ( u_t = i^ast_t-phi pi^ast_t ) to draw it.

I use the analysis solution of inflation, according to the interest rate path derived from 26.4 of the 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;

Signal = 1;

kap = 0.25;

Sum = 0.6;

Stake = 0.95;

? = 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);

Time = (0:1:T-1)’;

Pit = zero point (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 index = 4: T;

Pit (indx) = sig*kap/(lam1-lam2)*…

(and ^ (indx-3) / (1-lam2 * and) + lam1i * (and ^ (indx-3) -lam1i ^ (indx-3)) / (and-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,:));

Stem = rterm-pim(2,:);

Display(‘r’);

Display (rterm);

disp(‘s’);

disp (stem);

If 0;% all together

number;

C = color scale;

hold on

Plot (Tim, Pim,’-r’,’line width’, 2);

plot(tim,um,’–m’,’linewidth’,2);

Plot (Tim, it,’-b’,’line width’, 2);

Plot (team, 0*team,’-k’)

axis([ 0 6 -inf inf])

end;

Number;% 4 Panel view

For index = 1:4;

Subgraph (2,2, index);

hold on;

plot(tim,pim(:,indx),’-r’,’linewidth’,2);

If index == 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’,’line width’, 2);

Plot (team, 0*team,’-k’)

title([‘Sigma s=” num2str(sterm(indx),”%4.2f’)],’Font size’,16)

axis([ 0 6 -1 1.5])

end

If eta == 0.6

Print-dpng nk_fiscal_1.png

end