the Aggregate supply is given by the Lucas Supply Curve -

$y_t = ȳ + b(P_t - E_{t-1}P_t) + μ_t$

where $μ_t$ is stochastic supply shock (following standard normal error properties).

Aggregate demand equation is given by -

$y_t = m_t - P_t + v_t$

where $v_t$ is stochastic demand shock (following standard normal error properties).

Monetary authorities follows the policy rule -

$m_t = \bar{m} + m_{t-1} - cy_{t-1} + dμ_t +fv_{t-1}$

For the system, after solving for $y_t$ under the assumption of rational expectations (ie. Agents incorporate monetary policy changes into their decisions), I get it as a function of demand and supply shock.

$y_t$ = $b/(1+b) v_t$ + $(1+bd)/(1+b) μ_t$

Here, though $y_t$ is a function of policy parameter 'd' but $μ_t$, being supply shock of the current period equally random to both public and monetary authorities, is unanticipated part of money supply.

PIP argues that any anticipated changes in money supply cannot affect real variables. Since, $μ_t$ is unanticipated so by this regard PIP must hold. But since the policy parameter 'd' enters the output decision so monetary policy do have some influence over real variable. Then does it mean PIP doesn't hold.

I am confused between the two arguments. It will be helpful if someone can explain which one of the two is right? (Whether PIP holds or not and why?)

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    $\begingroup$ Done! Thanks for the correction $\endgroup$ – Elina Gilbert Jul 21 '20 at 22:00
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    $\begingroup$ I have zero economics background so don't feel too comfortable trying to answer. I think is that this is an example of PIP not holding because shocks are not "intended". They can be negative or positive. PIP holds when the government purposely tries to change something but the "actors" anticipate the change so nothing actually happens. There's an example at the beginning of Shaw's "Introduction to RE" book. There may also be an example in the paper "a child's guide to RE". I'm gonna see if I can find that paper when get a chance. If someone can chime in here to help us out, it's appreciated. $\endgroup$ – mark leeds Jul 22 '20 at 17:33
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    $\begingroup$ Elina: One more thing. Note that the fact that $d$ multiplies the supply shock still doesn't make it intended or anticipatory. It's still a "surprise" if you will so PIP doesn't hold and doesn't need to hold. PIP applies only when there are expected changes. $\endgroup$ – mark leeds Jul 22 '20 at 17:36
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    $\begingroup$ Hi Elina: I think we're on the same page but I'm not sure if "PIP holds" is the right way to say it because, since $y_t$ will respond to a shock, that means that supply-demand does have un-antipicated behavior. It's the anticipated policy that it doesn't respond to. So, I guess you're right that PIP still holds in the sense that policy isn't changing supply-demand. Like I said, hopefully someone else can confirm or respond or correct because RE is still a little fuzzy to me. I'm self taught and the road hasn't been an easy one. Still, I think we are in agreement. $\endgroup$ – mark leeds Jul 23 '20 at 4:33
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    $\begingroup$ Hi Elina: I'm too tired to read it carefully right now but there's a very nice example at the end of this in appendix B ( that I read a while back ) that tells me that what you said in your last comment is correct. thanks for the refresher on RE. It tells me that I did an okay job teaching myself. pdfs.semanticscholar.org/e99e/… $\endgroup$ – mark leeds Jul 23 '20 at 4:45

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