H5N1 sudden colleague of late pushed me to write downwards a actually uncomplicated model that tin plow over notice clarify the intuition of how raising involvement rates mightiness raise, rather than lower, inflation. Here is an answer.
(This follows the last shipping service on the question, which links to a paper. Warning: this shipping service uses mathjax as well as has graphs. If you lot don't run into them, come upwards dorsum to the original. I conduct maintain to hitting shift-reload twice to run into math inwards Safari. )
I'll operate the measure intertemporal-substitution relation, that higher existent involvement rates cause you lot to postpone consumption, \[ c_t = E_t c_{t+1} - \sigma(i_t - E_t \pi_{t+1}) \] I'll distich it hither alongside the simplest possible Phillips curve, that inflation is higher when output is higher. \[ \pi_t = \kappa c_t \] I'll likewise assume that people know most the involvement charge per unit of measurement ascent ahead of time, thence \(\pi_{t+1}=E_t\pi_{t+1}\).
Now substitute \(\pi_t\) for \(c_t\), \[ \pi_t = \pi_{t+1} - \sigma \kappa(i_t - \pi_{t+1})\] So the solution is \[ E_t \pi_{t+1} = \frac{1}{1+\sigma\kappa} \pi_t + \frac{\sigma \kappa}{1+\sigma\kappa}i_t \]
Inflation is stable. You tin plow over notice solve this backwards to \[ \pi_{t} = \frac{\sigma \kappa}{1+\sigma\kappa} \sum_{j=0}^\infty \left( \frac{1}{1+\sigma\kappa}\right)^j i_{t-j} \]
Here is a plot of what happens when the Fed raises nominal involvement rates, using \(\sigma=1, \kappa=1\):
When involvement rates rise, inflation rises steadily.
Now, intuition. (In economic science intuition describes equations. If you lot conduct maintain intuition but can't quite come upwards up alongside the equations, you lot conduct maintain a hunch non a result.) During the fourth dimension of high existent involvement rates -- when the nominal charge per unit of measurement has risen, but inflation has non yet caught upwards -- consumption must grow faster.
People eat less ahead of the fourth dimension of high existent involvement rates, thence they conduct maintain to a greater extent than savings, as well as earn to a greater extent than involvement on those savings. Afterwards, they tin plow over notice eat more. Since to a greater extent than consumption pushes upwards prices, giving to a greater extent than inflation, inflation must likewise ascent during the catamenia of high consumption growth.
One agency to human face at this is that consumption as well as inflation was depressed earlier the rise, because people knew the ascent was going to happen. In that sense, higher involvement rates create lower consumption, but rational expectations reverses the arrow of time: higher futurity involvement rates lower consumption as well as inflation today.
(The example of a surprise ascent inwards involvement rates is a fleck to a greater extent than subtle. It's possible inwards that example that \(\pi_t\) as well as \(c_t\) saltation downwards unexpectedly at fourth dimension \(t\) when \(i_t\) jumps up. Analyzing that case, similar all the other complications, takes a newspaper non a weblog post. The indicate hither was to present a uncomplicated model that illustrates the possibility of a neo-Fisherian result, non to debate that the trial is general. My skeptical colleauge wanted to run into how it's fifty-fifty possible.)
I actually similar that the Phillips bend hither is thence completely quondam fashioned. This is Phillips' Phillips curve, alongside a permanent inflation-output tradeoff. That fact shows squarely where the neo-Fisherian trial comes from. The forward-looking intertemporal-substitution IS equation is the fundamental ingredient.
Model 2:
You mightiness object that alongside this static Phillips curve, in that location is a permanent inflation-output tradeoff. Maybe we're getting the permanent ascent inwards inflation from the permanent ascent inwards output? No, but let's run into it. Here's the same model alongside an accelerationist Phillips curve, alongside slow adaptive expectations. Change the Phillips bend to \[ c_{t} = \kappa(\pi_{t}-\pi_{t-1}^{e}) \] \[ \pi_{t}^{e} = \lambda\pi_{t-1}^{e}+(1-\lambda)\pi_{t} \] or, equivalently, \[ \pi_{t}^{e}=(1-\lambda)\sum_{j=0}^{\infty}\lambda^{j}\pi_{t-j}. \]
Substituting out consumption again, \[ (\pi_{t}-\pi_{t-1}^{e})=(\pi_{t+1}-\pi_{t}^{e})-\sigma\kappa(i_{t}-\pi_{t+1}) \] \[ (1+\sigma\kappa)\pi_{t+1}=\pi_{t}+\pi_{t}^{e}-\pi_{t-1}^{e}+\sigma\kappa i_{t} \] \[ \pi_{t+1}=\frac{1}{1+\sigma\kappa}\left( \pi_{t}+\pi_{t}^{e}-\pi_{t-1} ^{e}\right) +\frac{\sigma\kappa}{1+\sigma\kappa}i_{t}. \] Explicitly, \[ (1+\sigma\kappa)\pi_{t+1}=\pi_{t}+\gamma(1-\lambda)\left[ \sum_{j=0}^{\infty }\lambda^{j}\Delta\pi_{t-j}\right] +\sigma\kappa i_{t} \]
Simulating this model, alongside \(\lambda=0.9\).
As you lot tin plow over notice see, nosotros nevertheless conduct maintain a completely positive response. Inflation ends upwards moving 1 for 1 alongside the charge per unit of measurement change. Consumption booms as well as and thence slow reverts to zero. The words are actually most the same.
The positive consumption response does non last alongside to a greater extent than realistic or improve grounded Phillips curves. With the measure forrard looking novel Keynesian Phillips bend inflation looks most the same, but output goes downwards throughout the episode: you lot larn stagflation.
The absolutely simplest model is, of course, simply \[i_t = r + E_t \pi_{t+1}\]. Then if the Fed raises
the nominal involvement rate, inflation must follow. But my challenge was to piece out the marketplace forces
that force inflation up. I'm less able to country the corresponding even out inwards really uncomplicated terms.
(This follows the last shipping service on the question, which links to a paper. Warning: this shipping service uses mathjax as well as has graphs. If you lot don't run into them, come upwards dorsum to the original. I conduct maintain to hitting shift-reload twice to run into math inwards Safari. )
I'll operate the measure intertemporal-substitution relation, that higher existent involvement rates cause you lot to postpone consumption, \[ c_t = E_t c_{t+1} - \sigma(i_t - E_t \pi_{t+1}) \] I'll distich it hither alongside the simplest possible Phillips curve, that inflation is higher when output is higher. \[ \pi_t = \kappa c_t \] I'll likewise assume that people know most the involvement charge per unit of measurement ascent ahead of time, thence \(\pi_{t+1}=E_t\pi_{t+1}\).
Now substitute \(\pi_t\) for \(c_t\), \[ \pi_t = \pi_{t+1} - \sigma \kappa(i_t - \pi_{t+1})\] So the solution is \[ E_t \pi_{t+1} = \frac{1}{1+\sigma\kappa} \pi_t + \frac{\sigma \kappa}{1+\sigma\kappa}i_t \]
Inflation is stable. You tin plow over notice solve this backwards to \[ \pi_{t} = \frac{\sigma \kappa}{1+\sigma\kappa} \sum_{j=0}^\infty \left( \frac{1}{1+\sigma\kappa}\right)^j i_{t-j} \]
Here is a plot of what happens when the Fed raises nominal involvement rates, using \(\sigma=1, \kappa=1\):
Now, intuition. (In economic science intuition describes equations. If you lot conduct maintain intuition but can't quite come upwards up alongside the equations, you lot conduct maintain a hunch non a result.) During the fourth dimension of high existent involvement rates -- when the nominal charge per unit of measurement has risen, but inflation has non yet caught upwards -- consumption must grow faster.
People eat less ahead of the fourth dimension of high existent involvement rates, thence they conduct maintain to a greater extent than savings, as well as earn to a greater extent than involvement on those savings. Afterwards, they tin plow over notice eat more. Since to a greater extent than consumption pushes upwards prices, giving to a greater extent than inflation, inflation must likewise ascent during the catamenia of high consumption growth.
One agency to human face at this is that consumption as well as inflation was depressed earlier the rise, because people knew the ascent was going to happen. In that sense, higher involvement rates create lower consumption, but rational expectations reverses the arrow of time: higher futurity involvement rates lower consumption as well as inflation today.
(The example of a surprise ascent inwards involvement rates is a fleck to a greater extent than subtle. It's possible inwards that example that \(\pi_t\) as well as \(c_t\) saltation downwards unexpectedly at fourth dimension \(t\) when \(i_t\) jumps up. Analyzing that case, similar all the other complications, takes a newspaper non a weblog post. The indicate hither was to present a uncomplicated model that illustrates the possibility of a neo-Fisherian result, non to debate that the trial is general. My skeptical colleauge wanted to run into how it's fifty-fifty possible.)
I actually similar that the Phillips bend hither is thence completely quondam fashioned. This is Phillips' Phillips curve, alongside a permanent inflation-output tradeoff. That fact shows squarely where the neo-Fisherian trial comes from. The forward-looking intertemporal-substitution IS equation is the fundamental ingredient.
Model 2:
You mightiness object that alongside this static Phillips curve, in that location is a permanent inflation-output tradeoff. Maybe we're getting the permanent ascent inwards inflation from the permanent ascent inwards output? No, but let's run into it. Here's the same model alongside an accelerationist Phillips curve, alongside slow adaptive expectations. Change the Phillips bend to \[ c_{t} = \kappa(\pi_{t}-\pi_{t-1}^{e}) \] \[ \pi_{t}^{e} = \lambda\pi_{t-1}^{e}+(1-\lambda)\pi_{t} \] or, equivalently, \[ \pi_{t}^{e}=(1-\lambda)\sum_{j=0}^{\infty}\lambda^{j}\pi_{t-j}. \]
Substituting out consumption again, \[ (\pi_{t}-\pi_{t-1}^{e})=(\pi_{t+1}-\pi_{t}^{e})-\sigma\kappa(i_{t}-\pi_{t+1}) \] \[ (1+\sigma\kappa)\pi_{t+1}=\pi_{t}+\pi_{t}^{e}-\pi_{t-1}^{e}+\sigma\kappa i_{t} \] \[ \pi_{t+1}=\frac{1}{1+\sigma\kappa}\left( \pi_{t}+\pi_{t}^{e}-\pi_{t-1} ^{e}\right) +\frac{\sigma\kappa}{1+\sigma\kappa}i_{t}. \] Explicitly, \[ (1+\sigma\kappa)\pi_{t+1}=\pi_{t}+\gamma(1-\lambda)\left[ \sum_{j=0}^{\infty }\lambda^{j}\Delta\pi_{t-j}\right] +\sigma\kappa i_{t} \]
Simulating this model, alongside \(\lambda=0.9\).
As you lot tin plow over notice see, nosotros nevertheless conduct maintain a completely positive response. Inflation ends upwards moving 1 for 1 alongside the charge per unit of measurement change. Consumption booms as well as and thence slow reverts to zero. The words are actually most the same.
The positive consumption response does non last alongside to a greater extent than realistic or improve grounded Phillips curves. With the measure forrard looking novel Keynesian Phillips bend inflation looks most the same, but output goes downwards throughout the episode: you lot larn stagflation.
The absolutely simplest model is, of course, simply \[i_t = r + E_t \pi_{t+1}\]. Then if the Fed raises
the nominal involvement rate, inflation must follow. But my challenge was to piece out the marketplace forces
that force inflation up. I'm less able to country the corresponding even out inwards really uncomplicated terms.
