A Really Unproblematic Neo-Fisherian Model
Influenza A virus subtype H5N1 abrupt colleague of late pushed me to write downward a actually uncomplicated model that tin clarify the intuition of how raising involvement rates mightiness raise, rather than lower, inflation. Here is an answer.
(This follows the last transportation on the question, which links to a paper. Warning: this transportation uses mathjax together with has graphs. If you lot don't run across them, come upwards dorsum to the original. I receive got to hitting shift-reload twice to run across math inward Safari. )
I'll purpose the touchstone 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 span it hither amongst the simplest possible Phillips curve, that inflation is higher when output is higher. \[ \pi_t = \kappa c_t \] I'll too assume that people know nearly the involvement charge per unit of measurement rising ahead of time, hence \(\pi_{t+1}=E_t\pi_{t+1}\).
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 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 receive got intuition but can't quite come upwards up amongst the equations, you lot receive got 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, hence they receive got to a greater extent than savings, together with earn to a greater extent than involvement on those savings. Afterwards, they tin eat more. Since to a greater extent than consumption pushes upwards prices, giving to a greater extent than inflation, inflation must too rising during the menses of high consumption growth.
One means to aspect at this is that consumption together with inflation was depressed earlier the rise, because people knew the rising was going to happen. In that sense, higher involvement rates create lower consumption, but rational expectations reverses the arrow of time: higher time to come involvement rates lower consumption together with inflation today.
(The representative of a surprise rising inward involvement rates is a fleck to a greater extent than subtle. It's possible inward that representative that \(\pi_t\) together with \(c_t\) fountain downward unexpectedly at fourth dimension \(t\) when \(i_t\) jumps up. Analyzing that case, similar all the other complications, takes a newspaper non a spider web log post. The indicate hither was to exhibit a uncomplicated model that illustrates the possibility of a neo-Fisherian result, non to combat that the effect is general. My skeptical colleauge wanted to run across how it's fifty-fifty possible.)
I actually similar that the Phillips bend hither is hence completely onetime fashioned. This is Phillips' Phillips curve, amongst a permanent inflation-output tradeoff. That fact shows squarely where the neo-Fisherian effect comes from. The forward-looking intertemporal-substitution IS equation is the cardinal ingredient.
Model 2:
You mightiness object that amongst this static Phillips curve, at that spot is a permanent inflation-output tradeoff. Maybe we're getting the permanent rising inward inflation from the permanent rising inward output? No, but let's run across it. Here's the same model amongst an accelerationist Phillips curve, amongst piece of cake 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, amongst \(\lambda=0.9\).
As you lot tin see, nosotros even hence receive got a completely positive response. Inflation ends upwards moving i for i amongst the charge per unit of measurement change. Consumption booms together with hence piece of cake reverts to zero. The words are actually nearly the same.
The positive consumption response does non last amongst to a greater extent than realistic or amend grounded Phillips curves. With the touchstone forwards looking novel Keynesian Phillips bend inflation looks nearly the same, but output goes downward throughout the episode: you lot larn stagflation.
The absolutely simplest model is, of course, only \[i_t = r + E_t \pi_{t+1}\]. Then if the Fed raises
the nominal involvement rate, inflation must follow. But my challenge was to piece out the marketplace position forces
that force inflation up. I'm less able to say the corresponding storey inward real uncomplicated terms.
(This follows the last transportation on the question, which links to a paper. Warning: this transportation uses mathjax together with has graphs. If you lot don't run across them, come upwards dorsum to the original. I receive got to hitting shift-reload twice to run across math inward Safari. )
I'll purpose the touchstone 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 span it hither amongst the simplest possible Phillips curve, that inflation is higher when output is higher. \[ \pi_t = \kappa c_t \] I'll too assume that people know nearly the involvement charge per unit of measurement rising ahead of time, hence \(\pi_{t+1}=E_t\pi_{t+1}\).
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 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 receive got intuition but can't quite come upwards up amongst the equations, you lot receive got 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, hence they receive got to a greater extent than savings, together with earn to a greater extent than involvement on those savings. Afterwards, they tin eat more. Since to a greater extent than consumption pushes upwards prices, giving to a greater extent than inflation, inflation must too rising during the menses of high consumption growth.
One means to aspect at this is that consumption together with inflation was depressed earlier the rise, because people knew the rising was going to happen. In that sense, higher involvement rates create lower consumption, but rational expectations reverses the arrow of time: higher time to come involvement rates lower consumption together with inflation today.
(The representative of a surprise rising inward involvement rates is a fleck to a greater extent than subtle. It's possible inward that representative that \(\pi_t\) together with \(c_t\) fountain downward unexpectedly at fourth dimension \(t\) when \(i_t\) jumps up. Analyzing that case, similar all the other complications, takes a newspaper non a spider web log post. The indicate hither was to exhibit a uncomplicated model that illustrates the possibility of a neo-Fisherian result, non to combat that the effect is general. My skeptical colleauge wanted to run across how it's fifty-fifty possible.)
I actually similar that the Phillips bend hither is hence completely onetime fashioned. This is Phillips' Phillips curve, amongst a permanent inflation-output tradeoff. That fact shows squarely where the neo-Fisherian effect comes from. The forward-looking intertemporal-substitution IS equation is the cardinal ingredient.
Model 2:
You mightiness object that amongst this static Phillips curve, at that spot is a permanent inflation-output tradeoff. Maybe we're getting the permanent rising inward inflation from the permanent rising inward output? No, but let's run across it. Here's the same model amongst an accelerationist Phillips curve, amongst piece of cake 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, amongst \(\lambda=0.9\).
As you lot tin see, nosotros even hence receive got a completely positive response. Inflation ends upwards moving i for i amongst the charge per unit of measurement change. Consumption booms together with hence piece of cake reverts to zero. The words are actually nearly the same.
The positive consumption response does non last amongst to a greater extent than realistic or amend grounded Phillips curves. With the touchstone forwards looking novel Keynesian Phillips bend inflation looks nearly the same, but output goes downward throughout the episode: you lot larn stagflation.
The absolutely simplest model is, of course, only \[i_t = r + E_t \pi_{t+1}\]. Then if the Fed raises
the nominal involvement rate, inflation must follow. But my challenge was to piece out the marketplace position forces
that force inflation up. I'm less able to say the corresponding storey inward real uncomplicated terms.
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