# The dynamic demand for bank discount window borrowing with non-negativity constraints.

The Dynamic Demand for Bank Discount Window Borrowing with Non-Negativity ConstraintsI. Introduction

It is widely believed that the Federal Reserve (Fed) continues to employ the borrowed reserve operating procedure originally implemented in October 1982. As discussed in Sternlight et al. [1986] and Roley [1986], the Fed maintains a desired degree of reserve restraint consistent with an initial forecast of borrowed reserves. Since open market strategies rely critically on these initial borrowing assumptions, effective operation of this procedure implies the existence of a dependable Discount Window borrowings demand function. To this end, Marvin Goodfriend's [1983] study of borrowed reserve demand arguably offers a significant breakthrough in modeling bank behavior toward borrowing from the Fed.

Goodfriend extends previous work by including implicit costs which depend upon past as well as present borrowing. Consistent with documented Fed attitudes toward Discount Window non-price rationing [see e.g., Mengle, 1986], this cost function indicates that the Fed applies increasing discouragement to repeated as well as current requests. Consequently, the optimization problem becomes a dynamic program. Rationally formed expectations of future interest rate spreads between the Federal Funds rate and the Discount rate play a crucial role in deciding the magnitude of bank borrowing from the Fed. The Goodfriend model has since been employed to explore other facets of money market behavior [Goodfriend et al., 1986; Cosimano, 1987].

This paper extends Goodfriend's [1983] analysis by adding to his formulation another critical aspect of Discount Window borrowing: most banks borrow infrequently, if at all, from the Fed. This behavior implies a nontrivial zero Discount Window borrowing decision for the individual bank at any date. In this way the study follows Blinder [1986], who criticizes rational expectations based optimizing models for failing to consider meaningful corner solutions and aggregation problems. Numerous works, such as Judd and Scadding [1982], Bryant [1983], Foote [1985], provide strong empirical evidence concerning the importance of the zero borrowing decision in the demand for borrowed reserves.

The criterion for zero current period borrowing is derived in section II. This condition reflects the bank's reluctance to borrow over consecutive periods due to implied surveillance costs. A second extension arises in the solution for positive present period borrowing. In section III, it is shown that the bank reacts differently to changes in present and expected future spreads depending upon whether or not it borrowed from the Discount Window in the previous period.

The model demonstrates that the bank employs a two stage process in each period to determine its borrowing from the Fed. First, the bank reviews information on the current interest rate spread as well as its past borrowing and future borrowing plans. The bank uses this information to decide whether or not to borrow from the Fed. In the second stage, if the bank chooses to borrow from the Discount Window, it weighs this information along with expected future spreads to decide on the amount of borrowing. The bank will then pursue a strategy for borrowing according to its recent borrowing experience, reacting differently to expected future spreads if it used the Discount Window previously. Concluding remarks are given in section IV.

II. The Zero Discount Window Borrowing

Decision

Goodfriend [1983] assumes that the risk neutral individual bank chooses a path of borrowing by maximizing total discounted profit, defined as the net benefit of borrowing from the Fed. The optimization problem under uncertainty at date t = q is: where the exogenous variables, ft and dt, refer to the Federal Funds rate and the Discount rate at date t; Bt denotes date t Discount Window borrowing; b, co and c1 are parameters pertaining to the rate of time discount and Fed non-price rationing; and Eq denotes the mathematical expectation conditional upon information available at date q.

Assuming Bq-1 is known, Goodfriend [1983] obtains optimal Discount Window borrowing at date q by differentiating (1) with respect to Bq. Since the bank may choose not to borrow from the Fed, the analysis proceeds according to the KuhnTucker maximization conditions given by: where Sq = fq - dq denotes the date q interest rate spread.

By substituting Bq = 0 into (2), the following criterion for zero present period borrowing is obtained:

The bank will not use the Discount Window at date q when the marginal cost of obtaining Federal Funds falls short of the minimum marginal cost of borrowing from the Fed. Among other things, this condition implies that the bank will not borrow from the Fed when the Discount rate exceeds the Federal Funds rate. Zero Discount Window borrowing at date q also becomes more likely if the bank has borrowed in the previous period or anticipates using the Discount Window in the immediate future. The bank may not wish to incur large marginal implicit cost due to its past borrowing from the Fed. The bank may also desire to prevent additional future non-price rationing.

Since Bq > 0 implies that (2) holds with equality, positive borrowing reverses the inequality of (3). Dutkowsky [1984] demonstrates that the nontrivial corner solution produces a switching regression model for borrowed reserve demand. One regime describes zero or frictional borrowing. The zero borrowing condition defines the criterion for switching between regimes.

III. Discount Window Behavior Under

Positive Borrowing

If, at date q, the bank observes that the interest rate spread exceeds the minimum marginal cost of Discount Window borrowing, then the bank will decide to borrow from the Discount Window, i.e., Bq will be nonzero. The bank will then borrow to the point which the marginal costs of borrowing from the two sources are equal.

In this way, the bank determines date q Discount Window borrowing by solving the Euler equation:

Goodfriend [1983] solves this equation first by assuming perfect foresight, and then distributes the conditional expectations operator throughout the solution. Since the linear-quadratic optimization problem permits certainty equivalence, this method is equivalent to solving the Euler equation under uncertainty [Sargent, 1979, p. 338].

Under suitable regularity conditions, particularly if co > (1 + b)c1 and Bq - 1 > 0, Goodfriend's [1983] borrowing function is: where:

Goodfriend [1983, p. 349] assumes that agents determine expectations of the interest rate spread according to:

The process in (7) describes an interest rate spread that asymptotically converges after some previous displacement from its long-run value o. This formulation is consistent with rational expectations only if the interest rate spread is exogenous to the individual bank. Otherwise, expectations of the interest rate spread would be determined by the variable's semi-reduced form equation.

Substituting (7) into (5) yields the following reduced form solution for Discount Window borrowing:

The above solution holds only under positive values of Bq - 1, Bq, and EqBq + 1 for all dates t = q. This result indicates that the Goodfriend function is valid only if the bank used the Discount Window in the previous period, borrows from the Fed at the current time, and plans to borrow in the following period. This persistent borrowing appears unlikely for most banks, especially given Fed attitudes concerning the Discount Window, well-documented by Goodfriend [1983].

This behavior indicates that the Euler equation (4) at date q should be reexamined. Equality of this equation implies that the bank has already decided to borrow from the Fed based upon favorable conditions described in (3). Under rationality, if the bank assigns a probability to positive Discount Window borrowing outcomes at date q + 1 given date q information, then EqBq + 1 is positive. It is assumed that, at any date, future states in which the bank borrows from the Discount Window exist with positive probability.

However, the bank may not have borrowed from the Fed at date q - 1. Furthermore, if the bank has chosen to borrow, it knows this information and will utilize it rationally in planning the magnitude of date q borrowing. If the bank used the Discount Window at date q - 1, then Goodfriend's [1983] solution for date q borrowing applies. On the other hand, zero previous borrowing requires solving (4) with Bq - 1 set to zero.

Under certainty equivalence and the assumed regularity conditions, date q borrowing in this case is: Substituting the expectations process (7) into (9) yields the reduced form:

Obtaining the complete reduced form model for Discount Window borrowing requires expressing the switching criterion (3) in terms of observed variables. Since the condition arises from zero current period Discount Window borrowing, the bank will use (10) to determine date q + 1 borrowing if it decides to borrow from the Fed. Alternatively, the bank may choose not to borrow from the Discount Window at date q + 1.

The bank's subjective probability at date q that it will borrow at all from the Fed at date q + 1, given zero current period borrowing and all other information available at date q, is denoted by pi. The parameter pi, to be estimated with the rest of the model, can be interpreted as the individual bank's implicit policy toward using the Discount Window in general. The bank determines its expected Discount Window borrowing at date q + 1, conditional upon date q information and zero borrowing, by using (10) to obtain: Substituting for EqSq + 1 using (7) yields:

After substituting (12) into (3) for EqBq + 1, collecting terms, and normalizing on the interest rate spread, the following reduced form zero borrowing criterion results:

The above condition extends Dutkowsky [1984] in a straightforward way. The bank decides whether to borrow from the Fed by comparing a linear combination of the interest rate spread and its past borrowing experience to an unknown switchpoint. As in (3), the bank will be less likely to borrow from the Fed if it borrowed in the previous period. The switchpoint is composed of parameters reflecting non-price rationing, expected future interest rate spread determination, and the bank's borrowing policy. The assumption in (8), that co>(1+b)c1, results in a positive switchpoint. Even with zero past borrowing, non-price rationing at the Discount Window implies that the Federal Funds rate has to exceed the discount rate by an acceptable magnitude before the bank chooses to borrow from the Fed.

The model composed of equations (8), (10) and zero borrowing falls into the class of deterministic switching regression models with three possible regimes for borrowing. Switching is determined by (13) and the bank's borrowing from the Fed in the previous period. If the bank undertakes Discount Window borrowing in the current period based upon conditions that reverse the inequality of (13), it determines the magnitude of borrowing according to whether it borrowed at date q - 1. If the bank used the Discount Window in the previous period, it borrows according to the behavior modeled by Goodfriend [1983] and portrayed in (8). Zero previous borrowing, with the bank deciding to borrow in the current period, is represented in (10).

By comparing the interest rate spread parameters between (8) and (10), it is observed that the bank will unambiguously react more strongly to changes in the spread at date q if it borrowed from the Fed at date q - 1. If the bank has a past borrowing record, small changes in the interest rate spread generate an amplified response in Discount Window borrowing. Zero borrowing takes place when (13) holds, regardless of the level of Bq - 1.

Estimation of the three regime system given individual bank borrowing data proceeds in a straightforward manner using maximum likelihood algorithms, such as those discussed in Goldfeld and Quandt [1972]. However, data on individual bank Discount Window borrowing are presently unavailable. Due to the highly sensitive and confidential relationship between the Fed and Discount Window borrowers, the authors' request for such data under the Freedom of Information Act was refused.

IV. Concluding Remarks

Goodfriend [1983] cites several Fed sources that express difficulties encountered in forecasting aggregate Discount Window borrowing. Goodfriend's model exposes two sources of prediction problems: the complexity of Discount Window nonprice rationing and the difficulties banks experience in forming accurate expectations of the interest rate spread.

The Goodfriend model augmented with non-negativity constraints moves toward resolution of these concerns. The extended model reveals that three borrowing regimes exist, with non-price rationing parameters indicating the regime in which the individual bank borrows. Furthermore, non-price rationing and spread forecasting parameters determine different borrowing reactions to the interest rate spread. From the Fed's standpoint, the existence of three borrowing regimes requires substantially different open market responses, depending upon the size of the spread and past and expected future borrowing from the Discount Window.

While the three regime model accurately portrays borrowing for the individual bank, it is incorrectly specified for use with aggregate data. Goldfeld and quandt [1976] demonstrate that aggregation alters the functional form of a switching regression model developed for the individual unit. Dutkowsky and Foote [1985] provide empirical evidence of aggregation effects in the static demand for Discount Window borrowing.

Derivation of the model for aggregate borrowing would extend Goldfeld and Quandt [1976], since the criteria for regime switching consits of linear combinations of several variables and unknown switching parameters. Resolution of the aggregation problem requires significant research far beyond the scope of this study. All of these factors point out the continued obstacles inherent in finding a reliable borrowed reserve demand function.

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Author: | Foote, William G.; Dutkowsky, Donald H. |
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Publication: | Atlantic Economic Journal |

Date: | Jun 1, 1989 |

Words: | 2189 |

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