Optimal Economic Stabilization Policy: An Extended Framework

This paper outlines an optimization framework which extends the familiar Tinbergen-Theil model in two ways. First, a "piecewise quadratic" replaces the standard quadratic objective function. Second, the time horizon of the optimization becomes, within the context of economic stabilization problems, endogenous to the optimization process itself. The purpose of both extensions is to escape the conceptual re-strictiveness of the Tinbergen-Theil structure while preserving the practical convenience of that model for applied policy work. The paper also describes a solution algorithm incorporating these two extensions, and it presents the results of a sample computational application based on the 1957-58 recession.

Cal, convenient framework for planning quantitative aggregate policy to cope with such fluctuations, this is the familiar dynamic optimization of a quadratic criterion function subject to linear equality constraints.Nevertheless, the conceptual restrictiveness of this framework prevents it from being widely useful in actual applications to policy formulation.The quadratic criterion function, for example, does not seem to offer a very good representation of policy makers' preferences; and the necessity of an arbitrary choice of time horizon prevents using the framework to answer several significant policy questions.
At the same time, the more generalized mathematical programming2 and systems-control3 literatures have, on the whole, had too general a presentation to be of great use in economic policy applications.The principles enunciated within the various divisions of optimization research are of great relevance and importance in such problems, but the difficulty of adapting these principles remains a significant obstacle to their application to economic policy problems and in particular to economic stabilization.Much "bridge" work remains to be done to select that part of the generality which is of sufficient potential value to keep, as well as to render it readily usable in an operational and computational sense.This paper is, at least in part, an attempt along such lines.
This paper outlines an optimization framework which extends the Tinbergen-Theil model in two ways.First, a "piecewise quadratic" replaces the standard quadratic criterion function.Second, the time horizon of the optimization becomes, within the context of economic stabilization problems, endogenous to the optimization process itself.The object of both extensions is to escape the conceptual restrictiveness of the Tinbergen-Theil structure while preserving its practical convenience for applied policy work.To focus clearly on these two extensions, this paper deals with a deterministic system in which the potential impact of policy actions on the economy is known with certainty.4 Section I discusses the piecewise quadratic criterion function.Section II discusses the endogenous time horizon and the associated concept of the "policy interval."Section III outlines the solution algorithm for the optimization and analyzes the interaction of these two extensions.Section IV presents the results of a sample computational application of the algorithm.Section V briefly summarizes the optimization methods offered.For expositional simplicity (involving no loss of generality), treat a and b as null vectors, C as a null matrix, and A and B as nonnegative diagonal matrices.Then for each variable xi or y?, there is some desired value or zero-penalty point xi* or yi*, and from the matrices A and B some penalty rate aii or biL.The penalty attached to a given realization for any variable is then - Xi where xi -- xi, or 2 bit342, where

Yi
In applications to economic policy, however, the quadratic function is not a very satisfactory representation of preferences likely to be pursued by policy makers.Exact desired values yi* and xi* for given policy targets or instruments in any period may not exist in these real preferences; often, policy makers see certain variables more as constraints, in the sense of bearing an implicit loss only for values outside some range.An even more unrealistic aspect of the quadratic function is the requirement that deviations of a target or instrument variable from its desired value bear the same loss regardless of the direction of the deviation.
The piecewise quadratic function offers a more general framework for policy optimization.Specifically, it is in general asymmetrical and only convex (as opposed to strictly convex).In reality, it is three distinct functions, welded in such a way as to preserve those properties of the more restrictive quadratic form which are essential to the optimization process.The generalization achieved by this function relaxes the principal unattractive requirements associated with the quadratic form, and its inclusion within the framework of the Tinbergen-Theil approach therefore renders that approach more realistic and useful for examining economic policy problems.In addition to its generality and flexibility, which permit a more realistic representation of economic policy preferences, the piecewise quadratic criterion function has at least two computational advantages.One concerns its interaction with the extension to the Tinbergen-Theil approach developed in Section II below; Section III discusses this interaction explicitly.
The second computational advantage arises in the context of the distinction between equality and inequality constraints.The Theil model optimizes the function (1) subject to a set of linear equality constraints (2); this model, unlike linear programming or general mathematical programming techniques, does not admit inequality constraints.The piecewise quadratic function facilitates incorporating inequality constraints while at the same time staying within the operationally convenient Theil framework.This inequality constraint capability is a further aspect of the piecewise quadratic function's more realistic representation of policy preferences.7 which does not utilize the full flexibility of the piecewise quadratic form.Defining xix = X or xil =o, for example, specifies a function which assigns a loss to deviations of xi on one side only of a given value.The discussion below treats the analogous case for ai = ?? or aii = o or both.
7 It is possible to argue that, since piecewise quadratic criteria involve more individual parameters than do standard quadratics, the information required would be more difficult to extract from policy makers.Such an argument seems not to be the case.Precisely because the piecewise quadratic function is capable of representing policy preferences more realistically, the required information should be easier to ex- 8 This technique is the "penalty method" of nonlinear programming with constraints in Zangwill (1969), chap.12.The basic idea is to approximate the feasible region from the outside.6) and ( 7).The former is strictly convex for a, > 0 and has a minimum at x* --a2j2ai, where w1(x*) -a -a3 (a2) 2/4a1.The latter is convex for bu, bl > 0, having a flat zero range for yE M(y).The curves as drawn imply bl > a, > bu.
Since convexity (even when not strict) is preserved in the addition operation (Managasarian 1969, chap.9), w(x,y) in equation ( 5) is itself convex.Using a linear relation y Rx + s, for scalar R and s, permits plotting equation ( 5) for values of x.Three cases emerge, depending upon which of the three sets of values of y contains (Rx* + s). Figure 3 plots w(x,y The fundamental dynamic concept of the endogenous time horizon approach to optimization is the "policy interval," defined as that period of time during which specific stabilization policy is in effect.
This concept rests upon the notion of a stable economy, which in the short run strays from its "normal" path.Given such a deviation, the goal of the policy authorities is to return the economy to this path while minimizing specified costs associated with being away from the path.The discussion of zero loss ranges in Section I leads easily to the idea of identifying this long-run path as a set of ranges of acceptable values of key variables, rather than as a single set of required point values for each period in time.
The policy-interval concept also assumes the existence of some "normal" economic policy which pertains as long as the economy stays within the acceptable bounds of the long-run path.In cases of deviation, the policy authorities pursue objectives associated with returning to this path by implementing specific stabilization policy actions-hence the name "policy interval."When the economy has in fact returned to the long-run path, the stabilization element of economic policy terminates and the policy reverts to the appropriate long-run norm.
This intuitive and descriptive definition of the policy interval raises at least three questions.What determines whether the economy has deviated sufficiently from the long-run norm to identify the deviation as a policy interval?Once a policy interval has begun, what determines whether the economy has returned sufficiently to the long-run norm to end the policy 9Theil (1964), p. 154.Theil offers the alternative approach of an infinite horizon, solved using infinite band matrices (chap.5), as well as a moving horizon, which is the truncated case of an infinite horizon (pp.154-61).The concept of the endogenous time horizon developed here is more closely related to Theil's finite horizon base, but, as the discussion indicates, it has more flexibility.
10See, for example, the discussion of two-point boundary value problems in Bellman and Dreyfus (1962).

interval? What goals do the particular stabilization policy actions within the policy interval pursue?
In response to these questions, these three properties more precisely identify a given policy interval: initial conditions, terminal conditions, and criterion function.These three elements of the optimization enter the analysis in the following manner.
The initial conditions specify the conditions which the economy must satisfy for a policy interval to begin.According to these initial conditions, the policy interval itself begins at that point in time when a particular economic variable (or set of variables) strays outside the given limits of the long-run acceptable range.Examples may be the unemployment rate or the rate of price increase rising too high, or the growth rate of real output falling too low.Beginning in the period which first satisfies the initial conditions for a policy interval, the authorities undertake specific stabilization policy actions.
If the time period used in the analysis is short, or if the relevant datareporting machinery entails long delays, problems may arise with the role of the initial conditions as specified above.Specifically, although the value of a variable in a given period may satisfy the appropriate initial conditions for a policy interval, this fact may not become apparent until some time later.Such cases require a reformulation of the role of the initial conditions to incorporate either a forecasting procedure or a lagged policy response.
The terminal conditions specify the conditions which the economy must satisfy for a policy interval in progress to come to an end, that is, for the authorities to revert to the long-run policy actions which pertain in the absence of specific stabilization efforts.The policy interval itself ends, that is, specific stabilization policy terminates, at that point in time when all relevant economic variables first satisfy the applicable terminal conditions.That particular period marks the time horizon of the optimization procedure which determines the optimal stabilization policy during the policy interval." A necessary part of the terminal conditions is the return to acceptability of the particular variable or variables which initiated the policy interval via the initial conditions, but the terminal conditions may involve other variables as well.In a policy interval initiated by an excessive unemployment rate, for example, typical terminal conditions may permit the policy interval to end only when both the unemployment rate and the rate of price inflation are within acceptable bounds; hence, a policy which returns unemployment to its normal level, but only at the expense of inducing a price inflation, is not sufficient.
More generally formulated terminal conditions are also possible.In the example above of terminal conditions based on values of the unemployment rate and the rate of price inflation, satisfaction of specified conditions in one period only may for some applications be insufficient.In such cases, the terminal conditions may require values of the former variable in a given range for K consecutive periods and values of the latter in a given range for L consecutive periods, where in general K 7 L.An even more general extension of the terminal conditions is the requirement that the specific stabilization policies undertaken during the policy interval not induce particular undesirable effects after the policy interval's conclusion.'2 The criterion function sets forth in a mathematical framework those economic goals which the authorities pursue during the policy interval.These goals may be, but are not necessarily, related to long-run goals associated with the long-run normal path of the economy.Just as it is necessary for the specific variable or variables in the initial conditions to appear in the terminal conditions, all variables in the terminal conditions must necessarily appear in the criterion function.The criterion function may, analogously, have additional arguments.It may be desirable, for example, to optimize some function of the balance-of-payments surplus during the policy interval, without necessarily making such a surplus a specific constraint by including it in the terminal conditions.
As one simple illustration of the mechanics of the policy-interval concept and the associated endogenous time horizon optimization, assume that the normal long-run growth rate of real output, dXldt, is sufficiently in excess of r to warrant identifying r as the "minimum acceptable rate" of growth.The initial condition for a "recessionary" policy interval is then that dXldt falls below r.This initial condition means that an observed dXldt < r calls for stabilization policy actions; for dXldt > r, policy maintains its long-run course which is independent of any immediate stabilization needs.
Once a policy interval has begun, it is possible to define a "minimum acceptable level" of real output by projecting forward, at the "minimum acceptable rate" r, the last observation of real output before the beginning of the policy interval.One possible terminal condition for this policy interval may then be that real output attains or exceeds this "minimum desired level."'3 Figure 4 shows paths for actual and minimum desired real output as functions of time.Through period t -0, real output grows at dX/dt > r, and so the minimum desired path is unnecessary.In period t -1, how- ever, dX/dt < r, which satisfies the initial condition.The minimum de- sired path is then an extrapolation at growth rate r of the value of real output in period t 0. The effect of the specific stabilization policy actions, together with the inherent long-run stability of the economy, returns actual output in figure 4  Specifying intervals of stabilization policy in this way, as well as using the associated terminal condition and endogenous time-horizon techniques, is not a universally applicable procedure.It is, however, suitable for the application of optimization analysis to short-term stabilization, defined, as above, as the restoration of the economy to its "normal, long-run acceptable" path after a deviation from that path.
In this context, it is important to distinguish the problem of stabilizing the economy, during deviations from a long-run path, from the problem of directing the economy from one long-run path to a different one.'4Methods designed for the former problem may not suffice for the latter.In particular, in the long-run case the time horizon chosen becomes the dominant element in the solution; and the policy interval itself emerges as an artificially contrived deviation from the original long-run path,15 perhaps unmaintainable without an infinite time horizon to policy.
This distinction serves to emphasize the importance of the identification of the "normal, long-run acceptable" path or state of the economy and fashion; it is important to note that this terminal condition is only one of many possible formulations.
14 The long-run problem is that of Ramsey (1928) and the subsequent literature of optimal growth. 15 The necessity that K be nonsingular, an important point below, follows from the second-order minimum condition that K be positive definite.The optimal y follows from substituting x into (2).Hence, the Level I problem yields an optimal solution (,9y) and associated'8 optimal x and y.
It is important to note the dimensions of the vectors and matrices in this solution.The vector x is a stacked vector; hence, for the case of m instrument variables, the vector x has mT elements, where T is the fixed time horizon for the Level I problem.Similarly, for the case of n target variables, vectors y and s have mT elements.All other vectors and matrices are conformable.
The Level II problem solves a series of Level I problems, each with different A and/or B matrices and a different s vector.
On the first subiteration,19 the diagonal elements aii assume, for those xi 18 In Theil's standard quadratic case, the optimal x and y follow from the definitions x = xx* and yy -y*.For a piecewise quadratic criterion function with nontrivial zero loss sets, the optimal Z follow from decision rule (4), and similarly for the optimal 'Y. 19The repeated solutions of the Level I problem within a given Level II problem are called "subiterations"; "iterations" are the repeated solutions of the Level II problem within a given Level III problem.A second restriction is that any variable which is a part of the terminal conditions in the Level III problem must be treated in the piecewise quadratic mode in the Level II problem.Suppose, for example, that in the example of Section II, the only terminal condition is the restoration of real output to the minimum desired level.In the absence of piecewise quadratic treatment, those elements yi* which correspond to successive periods' values of real output assume, as fixed values, the successive minimum desired levels.Then a problem arises in that the Level I problem penalizes not only values of real income below the minimum desired level, but any value above this level as well; this effect retards satisfaction of the terminal condition, with distorting effects for the Level III problem.In general, piecewise quadratic treatment of the terminal-condition values is necessary to provide a zero-penalty region on the appropriate side of the relevant (xi*, yi*).Hence, piecewise quadratic optimization, although it is an entirely independent generalization of the Tinbergen-Theil approach, is a necessary precursor to the application of endogenous time horizon optimization to economic stabilization policy.
A third major problem, left unanswered in the exposition of the Level III problem above, is the possibility that the optimal (x, y) may satisfy the terminal conditions for no reasonable value of T.22 A terminal condition involving 2 percent unemployment and zero price inflation, for example, is unlikely to be satisfied in the context of any reasonable model of the economy.It is this problem which effectively restricts the applicability of the policy interval concept and the associated algorithm to short-term stabilization policy, as the discussion of Section II indicates.

IV. A Computational Example
The  A compromise solution to this problem is to select real output as the primary indicator of a recessionary policy interval.A subjective view of the historical period suggests 1957:111 as an arbitrary initial quarter of the policy interval, and a terminal condition which requires that real output (X) equal or exceed its minimum acceptable level, specified as a 3 percent per annum projection of the level of real output in 1957:JI.Table 1 shows the quarterly values of this path.
The criterion function assigns a loss to any value of real output below the relevant minimum acceptable level.There is no reason to assign losses to all values above this level, however, and so it is necessary to treat real output in a piecewise quadratic function.Hence, table 1 labels the projection from the observed $453.2 billion in 1957:1I as Xt', because these values divide the sets L(Xt) and M(Xt).
Choosing the corresponding Xt(, to divide the sets M(Xt) and U(Xt), is an entirely arbitrary matter.Table 1 shows Xt"' values which are a projection, at the same 3 percent annual rate used for the Xt1, from a base level $10 billion above the actual value of real output in 1957:11.Since the sample exercise presented here treats a recessionary policy interval, the Xt" values would be relevant only in the case of extreme overshooting.This situation does not in fact arise, so the Xt" values are largely superfluous; the sets M(Xt) could just as well be unbounded above.
To summarize, the policy interval in this exercise begins in 1957:III and terminates when the optimal value of real output Xt in any period satisfies Xt e M(Xt) {XtIXt' < Xt < Xtu}.The exercise focuses on five variables: three target variables-real output, the gross national product price deflator, and the total unemployment rate; and two instrument variables -total government purchases and the central bank discount rate.26Table interval approach suggests both the difficulty and thie nonnecessity of specifying such conditions precisely in historical exercises.
26 It is possible to argue that the central bank discount rate is not an exogenous tool of monetary policy but, rather, an endogenous variable which follows market rates.Even under this view, there remain two related rationales for using the discount  27 Any such use of actual values implies some assumption that expenditure policy at the time corresponded to social needs evaluated independently of any need for antirecessionary fiscal policy.
-2 This paper presents no arguments to defend the particular criterion function The piecewise quadratic weights for X assign bjjl 50 for Xt f L(Xt) and bj" -5 for Xt e U(Xt), reflecting the priority of the recessionary policy interval.The quadratic weight29 for P is ba-3.0X 106.The quadratic weight for UN is bit 25.The piecewise quadratic weights for G assign aiil -1.5 for Gt e L(Gt) and aiu 1 for Gt eU(Gt).The piecewise quadratic weights for ID assign air' 200 for IDt e L(IDt), reflecting the effective imposition of the constraint ID > 3.0, and aciu 20 for IDt U(IDt).
The algorithm converges to T-6 regardless of the initial arbitrary T. On the final iteration, three subiterations are required for the piecewise quadratic convergence.The criterion function, evaluated for the actually observed x and y over the interval, has value 5.34 X 104; evaluated for the optimal x and 9 over the interval, its value is 9.83 X 102.Table 3 shows the optimal x and y.
It is clear from a comparison of tables 2 and 3 that the main effect of the optimization is to raise government purchases and, in so doing, to raise real output.The optimal ?t are consistently above the actual Xt; the difference exceeds $25 billion in 1958:I, the trough quarter of the actual AA recession.In 1958:IV, Xt > Xtf to yield time horizon T -6 for the policy interval.
The increase in X in the optimal solution is possible only at the cost of additional price inflation, however, and the optimal Pt are somewhat above the actual Pt, with the difference growing to 0.006 by the end of the policy interval.The narrowness of this margin indicates the unresponsiveness of prices in the Wharton model.The performance of the optimal UNt is superior to that of the actual UNt.
The force moving the economy from the historical base to the optimal path is fiscal policy, operating through massive government purchases.The weights chosen, and none is intended.The object of this exercise is simply to illustrate the operation of the piecewise quadratic and endogenous time-horizon optimization methods.
29 Because of the different units of measurement of the various x and y, the a and b values may be deceptive; bii = 3.0 X 106 for the price index does not represent a very large weight.A optimal Gt are greater than the actual Gt by more than $10 billion in the two quarters of and immediately following the trough of the actual recession.The large a'il placed on the discount rate effectively prevents that variable from falling significantly below its IDt1 boundary, thereby providing a good example of the use of the piecewise quadratic criterion function to impose an inequality constraint on the optimization.

V. Summary
The piecewise quadratic criterion function generalizes the standard quadratic so as to facilitate a much more reasonable representation of economic policy preferences; its asymmetrical property also provides a ready method for imposing inequality constraints upon the optimization.Endogenous time-horizon optimization averts the arbitrariness of the selection of the time horizon in previous optimization methods, thereby permitting the optimization to deal with questions involving the desired speed of economic recovery; it also facilitates dealing with terminal conditions in the sense of dynamic programming.These two methods are independent in motivation but not in operation, in that the solution algorithm presented makes the piecewise quadratic criterion function a necessary precursor to endogenous time-horizon optimization.The combined effect of these two extensions to the Tinbergen-Theil model is to provide a framework which is not only sufficiently broad and flexible to treat many of the basic problems of formulating quantitative economic stabilization policy, but is also sufficiently straightforward in its operational and computational aspects to render such a treatment easily accessible for applied policy work.

For
FIG. 1.-Prototypical piecewise quadratic function FIG. 2.-Functions 6 (a) and 7 (b) Setting ad u oc imposes upon the optimization the constraint xi < xiu; setting ajil -oc imposes the corresponding constraint xi > xi'.In computational practice, infinite values are not manageable for aiju and air.Nevertheless, approximating infinity by values of aidu or a,,l larger than the other elements aij, bij, and cij by several orders of magnitude is an effective way to approximate the relevant inequality constraint in the solution to the piecewise quadratic optimization.8Figures 2 and 3 show how individual piecewise quadratic terms combine to form the piecewise quadratic function for a given problem.For purposes of illustration, they treat a problem in which x and y are scalars.The preference function contains no term in the cross-product of x and y and so is separable: W (X, Y)

Figure 2
Figure 2 plots functions (6) and (7).The former is strictly convex for a, > 0 and has a minimum at x* --a2j2ai, where w1(x*) -a -a3 FIG. 3.-Three cases for function 5 to a value above the minimum desired level in period t -T thereby satisfying the terminal condition.T is then the time horizon of the policy-interval optimization problem.Using the policy-interval concept in this way makes the time horizon an endogenous, simultaneously determined element of the optimization procedure.The addition of terminal conditions makes the optimization problem similar in nature to the two-point boundary value problem, as noted above.Viewed in this context, endogenous time-horizon optimization is a form of two-point boundary value problem with the terminal point not fixed.
FIG. 4.-Recessionary policy interval based on variable X which have piecewise quadratic treatment, either the a..u or the ail values; any number of arbitrary decision rules may suffice.The xi*, to be used , assume the values xqu to correspond to aiiu or xi to correspond to aiiI.For those xi which do not bear piecewise quadratic treatment, the aii and xi* values are straightforward.The bii and yi* values for the first subiteration follow from the same arbitrary decision rule; and the yi* and xi* vectors, together with the other information necessary for the Level I problem, suffice to derive the s vector 20 for the first subiteration.Using these A and B matrices and this s vector, the algorithm solves the Level I problem, yielding a set of optimal xi and Yi.For any subiteration other than the first, the algorithm adjusts the (aij, X*) and the (bi, yi*) according to decision rules (3) and (4), using the set of optimal (xi, 9i) from the previous subiteration.Having applied these decision rules, the algorithm uses the adjusted A and B matrices and the adjusted s vector to resolve the Level I problem.The Level II problem terminates on the first subiteration for which decision rules (3) and (4) call for no adjustment to be made in any (a*,, xi*) or (bie, yi*).Any further subiterations would simply reproduce the final solution.The Level III problem solves a series of Level II problems, each with a different trial time horizon.On the first iteration, the trial time horizon T is arbitrary.The algorithm then solves the Level II problem for fixed time horizon T. After solving the Level II problem, the algorithm checks the optimal (x, y) for satisfaction of the terminal conditions.There are three possibilities: If (xy ) first satisfies the terminal conditions in T -T the Level III problem terminates.This (x, y, T) is the final solution for the policy in- terval.If (x, y) first satisfies the terminal conditions in T < T, the algorithm resets T -T for the next iteration and again solves the Level II problem.If (x, y) fails to satisfy the terminal conditions in any period up to and including T, the algorithm resets T -T + 1 for the next iteration and again solves the Level II problem.At this point, several observations are in order about the interaction among the three levels of the algorithm: First, a qualification to the flexibility of the piecewise quadratic criterion function arises as a "nonredundancy" restriction on the number of the 20 The s vector varies from one subiteration to the next, because it effectively normalizes constraints (2) about the particular xi or x u being used for xi* and the particular yil or yiu being used for yi*.To be explicit, the algorithm involves rewriting constraints (2) as - R= ' Rx+' and using this "s" for s.Since 'x and y vary (for given x and y) according to decision rules (4), this G"s varies also.xi and yi which can simultaneously bear the full piecewise quadratic treatment.Any number may have asymmetrical terms with aiu 74 aii' or biju 74 biji so this restriction applies only to those terms with nontrivial sets M(x;) or M(y;), that is, with xi" 74 xi' or yiU 74 ybl.The source of this re- striction is the requirement that matrix K, as defined in (9), be nonsingular for the solution of the Level I problem in (10).If a sufficient number of xi and ye bear piecewise quadratic treatment with nontrivial M(xi) and M(yi), and if a particular relationship among these variables maintains via equation (2), then, in the event that certain xi fall in M(xi) and certain y, fall in M(yi) simultaneously, matrix K will have one or more rows and columns consisting of zero vectors.More specifically, each column of matrix K corresponds to one xi, the value of a particular instrument variable in a particular time period.If the optimal xi falls in M(xi), then that xi depends only upon the effect of movements in xi on the target variables in the problem.If, in addition, however, all y, affected by xi via equation (2) fall in their respective M (vy), then the K matrix will apply only zero elements to values of xi.Hence, the value of Xi under such circumstances is indeterminate, as illustrated operationally by the singularity of K.The nonsingularity or "nonredundancy" restriction in the Level II problem is a prohibition barring any pattern of piecewise quadratic terms which could yield such a result in an associated Level I problem.2' 1957-58 recession in the United States may serve as a good example of a policy interval to illustrate the application of the methods developed above.The model of the economy used is the Wharton model (Evans and Klein 1968), linearized so as to reflect the behavior of the economy in the 1957-58 period.23The 1957-58 recession has been, to date, the most severe in the postwar period.Real output fell from a peak of $455.2 billion24 in 1957:III (third quarter of 1957) to $437.5 billion in 1958:1, for a decline of nearly 4 percent.Not until 1958:IV did real output regain its prerecession peak value.At the same time, the total unemployment rate rose from 4.0 percent in 1957:I to over 7.4 percent, the highest value observed in the postwar period, in 1958:III.To date 1958:II and 1958:III have been the only quarters in the postwar period to register a total unemployment rate above 7 percent.In the terminology of Section II, the 1957-58 experience constitutes an easily recognizable recessionary policy interval.Nevertheless, it is difficult, especially in the context of the purely expository aim of this section, to arrive at clearly satisfactory initial and terminal conditions for this policy interval.The unemployment rate first began to rise in 1957:1I, but the increase was not pronounced until 1957: IV, and the unemployment rate did not go above 5 percent until 1958:I.Real output reached a clear peak in 1957:111, but this peak represented a growth of less than 2 percent at an annual rate since the previous quarter, and the level of real output was lower in 1957:1I than in 1957:1.Hence, in the absence of a clear definition of the policy of the period-as seems always the case in historical exercises -no straightforward initial condition for the policy interval is obvious.25 22 This point concerns the convergence properties of the Level III problem.For a discussion of these convergence properties, as well as those of the Level II problem, see Friedman (1971), chaps.6 and 7. 23 The motivation and technique of the linearization are discussed in Friedman (1971), chaps. 2 and 3. 24 Values for output and government purchases, stated in this section, are seasonally adjusted quarterly values at annual rates.Price and unemployment values are also seasonally adjusted.Data are from Survey of Current Business; The National Income and Product Accounts of the United States, 1929-1965; and Federal Reserve Bulletin. 25The discussion in Section II of the function of initial conditions in the policy- The central bank discount rate (ID) also bears piecewise quadratic treatment, with IDt* IDth -IDt1 constant at 3.0 percent.Hence, the zero loss ranges M(IDt) exist only trivially as points.The piecewise quadratic treatment arises in the differential criterion function weighting pattern for sets L(IDt) and U(ID&).The criterion function in this exercise assigns loss values only to squared deviations of the target and instrument variables and not to cross-products of deviations; that is, the criterion function vectors a and b are null vectors, matrices A and B are diagonal matrices, and matrix C is a null matrix, as in the simplified exposition of Section I.The weighting patterns for all five variables are as follows.28rate as an instrument variable.First, even though it follows market rates, it bears an announcement effect which influences expectations by confirming Federal Reserve approval of financial market trends and intentions of maintaining them.Second, it may at times serve as a more general proxy for shifting monetary policy stances which are otherwise difficult to quantify.