(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.**********

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

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