(This follows the last post service on the question , which links to a paper. Warning: this post service uses mathjax as well as has graphs. If yous don't run across them , come upwards dorsum to the original. I direct keep to hitting shift-reload twice to run across math inwards Safari. )
I'll purpose the measure intertemporal-substitution relation , that higher existent involvement rates stimulate yous to postpone consumption , \[ c_t = E_t c_{t+1} - \sigma(i_t - E_t \pi_{t+1}) \] I'll duet it hither alongside the simplest possible Phillips bend , that inflation is higher when output is higher. \[ \pi_t = \kappa c_t \] I'll too assume that people know nigh the involvement charge per unit of measurement rising ahead of fourth dimension , 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 sack 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 yous direct keep intuition but can't quite come upwards up alongside the equations , yous direct keep 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 direct keep to a greater extent than savings , as well as earn to a greater extent than involvement on those savings. Afterwards , they tin sack 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 agency to aspect at this is that consumption as well as inflation was depressed earlier the rising , because people knew the rising was going to happen. In that feel , 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 instance of a surprise rising inwards involvement rates is a fleck to a greater extent than subtle. It's possible inwards that instance that \(\pi_t\) as well as \(c_t\) saltation downwards unexpectedly at fourth dimension \(t\) when \(i_t\) jumps up. Analyzing that instance , similar all the other complications , takes a newspaper non a spider web log post. The signal hither was to present a uncomplicated model that illustrates the possibility of a neo-Fisherian outcome , non to fighting that the outcome is general. My skeptical colleauge wanted to run across how it's fifty-fifty possible.)
I actually similar that the Phillips bend hither is thence completely onetime fashioned. This is Phillips' Phillips bend , alongside a permanent inflation-output tradeoff. That fact shows squarely where the neo-Fisherian outcome comes from. The forward-looking intertemporal-substitution IS equation is the cardinal ingredient.
Model 2:
You mightiness object that alongside this static Phillips bend , at that spot is a permanent inflation-output tradeoff. Maybe we're getting the permanent rising inwards inflation from the permanent rising inwards output? No , but let's run across it. Here's the same model alongside an accelerationist Phillips bend , alongside like shooting fish in a barrel 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 in 1 lawsuit 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 yous tin sack run across , nosotros nonetheless direct keep 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 then like shooting fish in a barrel reverts to zero. The words are actually nigh 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 nigh the same , but output goes downwards throughout the episode: yous larn stagflation.
The absolutely simplest model is , of course of written report , only \[i_t = r + E_t \pi_{t+1}\]. Then if the Fed raises
the nominal involvement charge per unit of measurement , inflation must follow. But my challenge was to while out the marketplace position forces
that force inflation up. I'm less able to country the corresponding storey inwards really uncomplicated terms.
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