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I know if the Fed continues to increase the Fed Fund Rate, inflation eventually goes down. But what if the Fed stops hiking the current Fed Fund Rate, how does it affect current inflation rate if the current Fed Fund Rate stays constant? In other words, does prolonging the current Fed Fund Rate (no change for a longer period of time) have an increasing, constant, or decreasing effect on the current inflation rate?

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Ceteris paribus, if interest rate is kept constant also inflation will be kept constant. The relationship between Fed's interest rate and inflation is typically modeled as (see Romer Advanced Macroeconomics pp 520-523):

$$\pi_t = \alpha + \beta \Delta i_{Fed} +\gamma \pi_t^C + \epsilon_t$$

If everything else stays constant, and Fed's federal funds rate (Fed's interest rate) also does not change, then inflation stays as it was given the value of $\alpha$, people's inflation expectations $\pi_t^C$ and exogenous shocks and there is no further change in inflation unless something changes. There are more ways how to model this relationship, the model above is one of the simplest ones, but regardless of that, in generally used models no change in interest rate will produce no change in inflation rate if nothing else changes (e.g. previous exogenous shocks to inflation dying down etc).

However, note in real life things are not being constant so inflation could come down on its own even if interest rate is kept constant. For example in terms of models above $\pi^C$ could drop or there could be negative exogenous shocks $\epsilon_t$ or even parameters $\alpha$ and $\gamma$ could have changed, but the drop in inflation would not be result of interest rates being kept same it would be result of it 'naturally' dying out due to adjustment in other factors.

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  • $\begingroup$ Would you please define what each of the factors represent in Romer's Advanced Macroeconomics? Generally, what are some exogenous shocks that can increase or decrease inflation rates? Thanks. $\endgroup$ Feb 20, 2023 at 17:19
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    $\begingroup$ @HelloDarkWorld $\pi$ is the inflation rate, $i$ Fed's interest rate, $\pi^C$ is expected inflation of consumers or people $\epsilon$ are some random shocks and $\alpha$, $\beta$ and $\gamma$ are just some parameters. Exogenous shock such as for example supply chain issues or shocks to GDP in general (e.g. sudden recession or strong expansion) $\endgroup$
    – 1muflon1
    Feb 20, 2023 at 17:41
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Summary:

  • If Fed makes the mistake of sticking to an overly restrictive rate for an extended period of time, inflation will go below the target and the economy will be hurt more than necessary.
  • If the mistake was to stick to a rate that was insufficiently restrictive, inflation may stay above target or even rise and ultimately also hurt the economy.

If the Fed sets a policy rate, it's because it thinks that rate is appropriate to have medium-term inflation close to the target. Specifically, if inflation is above target, the appropriate policy rate is restrictive; that is, it's set at a level high enough to bring inflation down in the medium term, according to the Fed's judgement.

Now, if, subsequently, inflation is indeed coming down as expected, a new, lower policy rate may be appropriate. But if the Fed still maintains the rate at its previous levels, its policy is overly restrictive and would hurt the economy more than needed to bring inflation back to target, and inflation would undershoot the target. Conversely, if the Fed has set the terminal rate not sufficiently restrictive, this would allow second-round effects to materialize, and inflation would stay high or increase even further, also ultimately hurting the economy. Because monetary policy affects the economy and inflation with a lag, judgments have to be made ahead of time. Monetary policy is a tricky business, and mistakes can be made.

Formally: It is often assumed that central banks set policy according to a so-called Taylor rule (of which there are now many varieties): $$ i_t = \pi_t + r_t^* + a_\pi ( \pi_t - \pi_t^* ) + a_y ( y_t - \bar y_t ) $$ where $i$ is the policy rate, $\pi$ the inflation rate, $\pi^*$ the inflation target rate, $y_t-\bar{y}_t$ is the output gap (measuring how far GDP is from potential GDP), and the $a$s are coefficients. Importantly, $r^*$ is the neutral real interest rate - at this rate, inflation remains at its target rate if inflation is at target and GDP is at potential, and ignoring outside factors.

If the actual policy rate is significantly different from $i$, the Fed makes a mistake (provided that it's a good rule). Mistakes may occur because the Fed misjudges or miscalculates some of the right-hand side elements that are not directly observed, such as $r^*$ or potential output $\bar{y}$.

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