Inflation Targeting as a Feedback Control System

 This post examines the process of inflation targeting by the Fed as a classic feedback control system. I think looking at it this way illuminates the differences between the neo-classical schools (at least the ones which accept the endogeneity of money in the short term, such as the Market Monetarists) and the post-Keynesian schools regarding the long term endogeneity of money (the post-Keynesians accept both the short term and long term endogeneity of money).

Figure 1: Fed and economy as a feedback control system tracking a targeted inflation rate

Figure 1 above illustrates the concept, and in particular how the actual measured inflation rate is subtracted from the target inflation rate (treated as an independent input variable here) to create an error signal, which the Fed's (central bank's) control law attempts to drive to zero. This is depicted this way in keeping with classic feedback control system theory diagrams, where an error signal is typically an input into the controller, which in turn produces in input signal to the "plant" (the system which the controller is attempting to control). The controller's outputs are manipulated in such as way (according to the control law) so as to minimize this error. In this case the federal funds rate (FFR) and Fed open market operations (OMOs: i.e. the buying and selling of Tsy debt and other assets by the Fed) are the signals which minimize the error. A more detailed depiction showing how the FFR and the OMOs work within another inner feedback loop, with the actual overnight rate achieved being fed back to create an inner loop error signal is depicted in figure 2 and discussed later in this post. Suffice it to say for now that the OMOs are adjusted to drive this inner loop error signal to zero.

One could argue that only the OMOs are important inputs into the economy, because the inner (unshown in figure 1) control loop already incorporates the FFR information into the OMOs, however, I show both the FFR and the OMOs as inputs into the economy because the targeted FFR not only determines the OMOs in this inner loop, but it acts as an important input all on its own (i.e. if the banks knows ahead of time what the targeted FFR is, they're unlikely to "fight" with the Fed ... they know the Fed can use the unlimited OMOs at its disposal to hit that targeted rate).

Also not shown in Figure 1 are the dependent state variables present inside the box "The Rest of the Economy" on the right hand side. One (several?) of those variable could be called "The Money Stock." Since the "loop is closed" on this control system (i.e. some of the economy's outputs are fed back to the input), it's difficult to say which of the dependent state variables in the economy determine the others. Post-Keynesians tend to put many factors determining the private sector's "Desire to Borrow at Terms Offered" (another state variable) as independent inputs, while I think the neo-classicals tend to think of this desire as wholly determined by the FFR or other dependent state variables. At least that's one possibility. Another possibility is that both schools accept that the desire to borrow is wholly determined by other state variables but that in an open loop system (where we cut the feedback signals) they disagree about which states determine other states (i.e. which states are more downstream than other states).

Figure 2 is a detailed view of the control law and two of the feedback loops shown in figure 1. The view here is narrower and focuses just on the control laws and the primary two feedback paths, especially the inner path associated with the FFR (the target or reference signal to track in the inner loop) and its associated control law (shown as the box with the conditional if-then statement inside). The differencing junction to the left in figure 2 is for the outer control loop and is the same as the differencing junction in figure 1: it likewise produces an inflation targeting error signal, although in figure 2 it's given the subscript "I" to distinguish it from the now explicit error signal produced by the rightmost differencing junction which represents the inner loop's FFR error signal (and  likewise distinguished by having an "R" subscript and being lower case). The inner FFR targeting loop in figure 2 can be assumed to be one of potentially a number of inner loops with feedback signals represented by the identifier "Other Feedback Outputs" in figure 1 (all depicted there without differencing junctions or control laws). Other outputs (feedback and otherwise) and fed-forward inputs illustrated in figure 1 are not shown in figure 2. The outer inflation targeting loop in figure 2 operates as described above for figure 1, tracking the targeted inflation rate with a large sample time period (low sample rate), in this case every six weeks, representing the time between Fed (or central bank) meetings to set a new target FFR. However, figure 2 makes explicit that a Taylor Rule is the outer loop control law and thus utilized to determine the new FFR target for the inner loop. Any similar rule (or the judgement of the board) could be substituted for this outer loop Taylor Rule control law.

The inner loop in figure 2 operates from the control law as depicted when there is not an abundance of excess reserves (ER) in the banking system (e.g. as it did in the USA in the three or so decades prior to 2008). The loop operates to track the targeted FFR provided by the outer loop, and fixed for six week intervals between estimates of the economy's actual inflation rate (the feedback signal for the outer loop). Note that the central bank offers discount window funds to the banks in the banking system at this targeted FFR, although there are disadvantages to using the discount window and most banks avoid doing so if possible. These disadvantages include a negative perception of the borrowing bank from other banks (and the central bank), penalty fees, and heavy collateral requirements. Instead the central bank prefers to ensure that the inter-bank funds market tracks the FFR by adjusting the liquidity of this market through OMOs.

The actual rate experienced in the market differs from the explicit FFR of the discount window and is referred to as the actual FFR and it indicated by the label "FFR*" in figure 2. FFR* is the measured true overnight rate for reserves in the inter-bank market and the measurement sample period for this signal is no more than 24 hours. Thus the inner FFR tracking loop operates with a much higher sample rate than the outer interest rate tracking loop. In terms of classical control systems, the plant for the inner loop mainly consists of the banking system which is a sub-component of the overall "rest of the economy" which serves as the plant for the outer loop and overall control system. Of course this is a bit of an oversimplification since the rest of the economy influences the banking system and thus the FFR*. The FFR* signal is fed back to a  produce an error signal (FFR - FFR*) which is an input into the inner loop's control law. When the reserve level in the banking system is too high the FFR* drops below the target FFR producing a positive error signal, which causes the central bank to engage in open market sales (OMSs): selling assets (Tsy debt & mortgage backed securities (MBSs)) on the open market for reserves, which removes reserves from the banking system and causes the FFR* to rise. Conversely when the reserve level is too low, this causes the FFR* to rise above target, the error signal is then negative, and the central bank engages in open market purchases (OMPs): purchasing these same kinds of financial assets on the open market with reserves, thus injecting reserves into the banking system and driving down the FFR*. The movement of reserves in both cases is indicated by the supplemental green arrows in figure 2. OMOs take place on a daily basis in response to rapid changes in the FFR*. Additionally, just the fact that the FFR target is publicly known helps to cause the system to converge more quickly as discussed above. Also note that OMOs here are often accomplished by the CB by using what's known as "repos" (OMSs) and "reverse repos," (OMPs) which are repurchase agreements: one party sells an asset at a discount with the agreement to repurchase it again for full price from the counterparty at a later date. A reverse repo is simply a repo from the asset purchaser's point of view.
 

Figure 2: Detail of inner and outer loops: especially inner FFR targeting loop for ER = 0 (pre-2008)

Note that with an abundance of ER in the banking system (e.g. due to quantitative easing (QE) post-2008 in the USA), figure 2 is no longer a good description of how tracking of the FFR is implemented by the Fed. With ER much greater than zero (ER >> 0), then the conditional statement serving as the inner loop's control law is ignored and the "switch" in figure 2 is essentially frozen in the up position (as pictured), and the CB always does more OMPs than OMSs which tends to drive the FFR* down as low as it can go. When the CB pays interest on reserves (IOR) this lower bound is the IOR rate itself. So in order for the CB to track the FFR with ER >> 0, it announces the target FFR (as always), sets the discount window rate to this same value (as always), but most importantly, sets the IOR rate to be equal to the target FFR. Scott Fullwiler, Cullen Roche, and Steve Randy Waldman all have good posts up about why the FFR = IOR when ER >> 0.

Update: Sept. 28, 2013: Here's some other people discussing "optimal control" in this context.