This PDF is a selection from an out-of-print volume from the National Bureau of Economic Research

Volume Title: Annals of Economic and Social Measurement, Volume 5, number 2

Volume Author/Editor: Sanford V. Berg, editor

Volume Publisher: NBER

Volume URL: http://www.nber.org/books/aesm76-2

Publication Date: April 1976

Chapter Title: Applications of Control Theory to Macroeconomics

Chapter Author: David Kendrick

Chapter URL: http://www.nber.org/chapters/c10438

Chapter pages in book: (p. 171 - 190) Annals of Economic and Social Measurement, 5/2, 1976

APPLICATIONS OF CONTROL THEORY TO MACROECONOMICS

BY DAVID KFNDRICK*

A survey of applications of control theory to macroeconomics is presented. control theory has been applied to about fifty different macroeconomic models containing anywhere froni one to inure than three hundred equations, and including models of the economies of the United States, canada, United Kingdom. West Germany, France, Belgium. Australia and the Netherlands. A wide range of control theory methods has been applied to these models. Deterministic methods for both quadratic - linear and general nonlinear models have been used. Uncertainty has been introduced in the form of an additive noise to the systems equations and in the form of uncertainty about parameter values, and the models have been solved either with closed loop policies and/or open loop optimal feedback policies. Also, adaptive control procedures have begun to he used on the smaller models. in addition there have been a Jew applications of decentralized control techniques and of differential games.

In the past decade, a number of engineers and economists have asked the question: "If modern control theory can improve the guidance of airplanes and spacecraft, can it also help in the control of inflation and unemployment?" Some of the results already available are displayed in Figure 1 in which inflation rates are plotted against unemployment rates. The origin of each arrow in the figure is the average inflation and unemployment rate experienced by the economy during the period studied. The authors of the study and the period covered appear above each arrow. The head of the arrow indicates the average inflation and unemployment rate obtained in a representative optimal control solution calculated by the authors. For the sake of comparison, the slope of the Phillips curve from the St. Louis FRB model [as displayed in Norman and Weatherby (74)] is also plotted. 'I'he location of the Phillips curve on the plot is arbitrary but its slope is as reported. If the controls solution provided unanibigu- ous improvements over the actual path followed by the economy then the arrows would point toward their origin. Instead, the solutions show a movement toward less unemployment and more inflation. However, the nature of the trade-off is important. On the one hand, it appears from Figure 1 that the optimal control solution could have brought substantial improvements at certain times,e.g., the Eisenhower years (see both the Friedman and Fair results), but in other periods, e.g., the Kennedy years, (Garbade and Pindyck results) the slopes are only slightly lower than for the Phillips curve. On the other hand, the results show that if an administration indeed prefers lower unemployment even at the cost of somewhat higher inflation rates, that result could be obtained using control methods. Or does it? In fact, the authors of the studies cited above viewed their work as only a first step in the direction of providing an answer because their control solutions do not take adequate account of uncertainty and decentralization.

This research was supported by the National Science Foundation undcrgrant Soc 72-05254.

171 S

References 1 Fair (75b) 7 2: Athans, et al (75) 3. Pindyck (73a) 4. Friedman (72) Fair:&c. 5: Garbade (75a) 6 69

C 5

C., 5, 0.

ccm4 c 0 C3

2f Wet

Q49

3 4 5 6 Unemployment Rate Percent

Figure 1Determinist Ic Control Results

In the actualeconomy there is uncertainty about theparameters that represent behavioralresponses in the economy, the of current endogenous state of the economy (values variables), future values ofexogenous variables and shocks to the system, and the formof the model that In the control solutions governs responses in theeconomy. shown, only a very limitedpart of that uncertainty is taken account of. Also, decision-makingpowers in the actual President, the Congress, economy are shared by the and the Federal ReserveBoard; in results shown, decision maker isassumed. a single

1. THE RANGEor CONTROL ThEORY APpUCATI0NSTO MACROFC'ONOMICS In the last twentyyears there have been some 39 different approximately 60 applicationsto macroeconometric models. Theyare listed in Appendix A in order of their size,and the names of those the model name,' who used each modelare listed under Appendix B providesa similar listing for theoretical The diversity of sizesand the range of models. readily apparent. countries for which theyhave been used are

When the modeland the application are in the same article, a single listing is given. 172 2.l)ETERMINISTICREsuE:rs The deeiniiiistic control problem may be written as: find [u,] to minimize

subject to (2) x,41 =f(x,, u,) x given where (1) is the criterion function, (2) are the systems equations, the x's are state variables and the u's are control variables. For macroeconometric models the states arc typically levels of consumption, investment, employment, price indices, etc. and the controls are government expenditure, taxes, and the money supply. So the system equations (2) consist of the reduced form of a macroeconometric model and the criterion function represent preference about rates of inflation and unemployment. Many of the applications have used linearized systems equation and quadra- tic criterion functions and have written the problem as one of finding [u,J'l1' to minimize

+ >{(x, - .,)' %V,(x, - i,) + (u1 - ñ,)'A,(u, t=1 subject to

(4) x,+ = Ax, + Bu, X()given where x = state vector u = control vector i and ü = desired values for states and control respectively W. A = penalty weights on deviations of state and controls respectively from their desired paths. Studies of this type are listed in the quadratic-linear column of Table 1. The nonlinear models of the form (1l(2) are listed in the second column. Also, many of the second group of studies begin with (2) in implicit function form, i.e.,

(5) g(x,+,,x,, u,)=O. In fact since (5) may contain as many as two to three hundred equations, its solution is an important part of the nonlinear optimal control algorithms. Certain themes recur frequently in these studies: the importance of proper timing and coordination of fiscal and monetary policy,2 the importance of

2 See for example Pindyck (73a)p. 140, Wall and Weslcott (75) p. 16, and Craine, Havenner, and Tinsley (75) p. 12. TABLE I Dtt I15 MIN ISTI( Sir t)l FS

No. of Equations in Econometric Mode! Quadratic-Linear General Nonlinear 1-3 tlustin (53) Chcng & Wan (72) Phillips (59) Holt (62) Shupp (75)

3--9 Bogaard & Theil (59) Shupp (72) Theil (64) Heaie & Summers (7.4) Sandblom (70) Sandhloni (74) Thalberg (7!a, 7th) Njorrnari & Weatherhy (74) Paryani (72> Healey & MedinaI75) You (75)

10-25 Erickson. Leondcs, & Norton (70) Fair (74) tHo and Norton (72) Gupta, Meyer, Raines & Pindyck (72a) Tarn (75k Erickson & Norton (73) Kaul (75)

25-80 van Eijk & Sandee (59, l.ivsey (71> (74) Theil (65) Norman & Norman (73) Friedman (72) Fitzgerald, Johnston & Oudet (75) Bayes (73) Friedman & Ilowrcv (73) Holbrook (74a) Rouzier (74) Craine. Havenner Tinsley (75) 89-300 Fischer & tithe (75) Holbrook (73) (74b) Woodsjdc (73) Ando, Norman. Patash (75) Athans etal. (75) Fair (75a, 75h)

* Theoreticalmodels. t Linear-I inear. ± Classical rather than optimalcontrol.

carefully choosing thecriterion functIon,3 the when the length of the substantial alterations in the results planning horizon (i.e.,of N) is changed,4 and the importance of choosing thesolution procedure carefully models. when solving nonlinear

3Livsey(71)p. 542. 4Garbade (75a)p. 180 and Athans etal. (75). 5Ando, Norman, and Palash (75), Fair (74)and Holbrook (74a). 174 3. Srocris'ric SiU[)IES The stochastic colitiOl piobleiti lot teduced form models may be written ut a quadratic-linear form as: find[u,]ito minimize

3 = )N) WN(XN - XN) N-- I + (x1 - .1)'VV(x - i) + (u - ü,)'A1(u, - f I subject to systems equation

= A (0)x, + B(O1)u + , and measurement equations

where 0 = a vector of unknown parameters, = systems noise. q = measurement noise, and yobservation vector. Here it is assumed that the state vector cannot be observed directly but rather only through a noisy observer (8). The unknown parameters in A and B are stacked up in the vector 0. Three sources of uncertainty are included here: additive noise, inthe reduced form of the macroeconometric model, additive noises,,,in (8), and uncertainty about the parameter values, 0. Studies that consider systems noise,, only are listed in thefirst column of Table 2. If the problem is quadratic linear, the certainty equivalence theorem of Simon (56) and Theil (57) is applied and the problem is solved as a deterministic- quadratic linear control problem.6 Garbade (75a, 75b) discusses a method to be used with nonlinear models and additive systems noise. The second column of Table 2 contains studies that treat A and B as stochastic. For example the Cooper and Fischer (75) study examines the question of whether it is better to have a fixed growth rate rule for the money supply or to have a discretionary policy. The stochastic parameters are those of a lag distribu- tion. Thus they address the Friedman question of whether or not a constant growth rate rule is better when the lags in the economy are long and uncertain. The more general case of unknown parameters is discussed by Chow (73a). Many of the methods used here are akin to those which engineers call open loop optimal feedback (OLOF). in these, the control may be cautious because of uncertainty about parameter values. in the studies listed under "Dual" in Table 2, the parameters are unknown but it is assumed that they can be learned over time. The control has the "dual" purposes of achieving the desired targets and learning the parameters. However, this is in a sense a false dichotomy since the single goal of meeting the targets is the essential one and only that learning done in early periods helps in meeting the targets in later periods. Four different adaptive control methods have been applied to mac- roeconometric models, Prescott (67). MacRae (72, 75). Abel (75) and Chow

6The WallWestcott method does not use the certainty-equivalence theorem. Also their model is linear in percent changes.

175 TABLE 2 Sioci i tsi'ic Sit DI

No. ot Equations in Econometoc Additive Model Parameter Noise Uncertainty Dual 1-3 Fisher (62) Zellner (66) (71) Zeilner & Geiset (68) MacRae & MacRae *Henderson &Turnovsky Abel (75) (70) (72) Chow (75) Chow (73) *Turnovsky (73. 74a,74b, 75a) 3-9 Chow (72b) Burger. Kalish & Babb *Karekcn Muench,Wallace(71) Prescott (67) (71) (73) Bowman & Laporte (72) Brim & Hester (74) Kendrick (73) Phelps & Taylor (75) Aoki (74a, 75a) tSargent & Wallace (75) Cooper & Fischer (75) Shupp (75) 10-25 Bray (74) (75a) Kendrick & Majors Pindyck & Roberts (74) (74) Upadhyay (75) Wall & Westcott Walsh & Cruz (75) (74. 75)

25-80 Garbade (75a, 75b)

80-300 Gordon (74)

* Theoretical models.

(75a), and Upadhyay(75). It is other method not yet clear whichof these methods still untried)will prove (or some to be superiorin applications roeconometric models.So far noneof the to mac- macroeconomicmodels have applications ofadaptive controlto the updating included theerrors in measurement, of macroeconometrictime series i. However, reported eachquarter are indeed would indicatethat the first data help in noisy; therefore,the use of this understanding theuncertainty which procedure could One of theattractive aspects surroundsmacroeconomic policy. not only estimates of adaptive of the x and control is that itcontinually updates and°°, the parameter as well. Thus, vector 0, but alsotheir convariances, of the policymakers learn economy associated not only theexpected performance Uncertainty. with differentpolicy measures but also thedegree of

4. DECENTRALIZATIONSTUDIES Though by macroeconomic policyat least in the decentralizationin decision U.S. is definitely efforts to making, there characterized model this have so firbeen relatively few McFadden (69) phenomenon.Those involving and Aoki(75c), which decentralized controlare contain threeto nine equations. The 176 models involving conflicting objectives are Kydland (73, 76) and Myoken (75a), which also contain three to nine equations, and Pau (73) and Pindyck (76), which contain ten to twenty-five equations. The model in Myoken is theoretical only.

TABLE 3 DECENTRALIZATIONSTuDIEs

No.of Equations in Econometric Model Decentralized Control Conflicting Objectives

1-3

3-9 McFadden (69) Kydland (73) (76) Aoki (75c) *Myoken (75a)

10-25 Pau (73) Pindyck (76)

25-80

* Theoretical.

5. FUTURE RESEARCH The answer to the original question of this paper remains elusive. Efforts are now underway to include both uncertainty and decentralization, but only a bare beginning has been made. So the central direction of future research will be the application of methods of adaptive control and game theory to macroeconometric models of increasing size. Some other areas worth further research effort are listed below: The Federal Reserve Board can make monetary policy decisions fairly quickly. 1-lowever, fiscal decisions are made by the President, but must then go to Congress, and back to the President. No control theory application has yet taken account of the difference in timing between the policy-making actions. The measurement errors in (8) above have not yet been systematically included and should be. This should include not only the fact that macroeconomic time series are characterized by different degrees of uncertainty, but also a careful consideration in the timing of the availabil- ity of data. Related to point b, above, are the differences in the way data are collected: most data are quarterly, but some are daily, weekly, or monthly. The problem raised is how best to integrate monthly or weekly models with quarterly ones. 177 The response of agents to the announcement offeedback control needs lobe considered because the mere Policy announcement may changethe behavior of agents and thereby render the policysuhoptjtil land and Prescott (75). viz. Ky(l- Policy decisions about macroeconomicsare highly visible debated in the political arena. Consequently, and much policy modelsused in this field cannot be divorced from but rather must he enriched by thePolitical environment which surrounds these decisions.For an interesting pie see Fair (75b). exam

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Editor's Note: This material is from the book Frontiers of Quantitative Economics,Vol. Ill, ed. M. D. Intriligator, North-}-{olland Publishing Company, forthcoming, 1976.

185 APPENDIX A Name (Date) NUMERICAL MACROEcONOMF.TzICMODELS USED IN CONTROL THEORYCountry APPLICATIONS Period- icity Estimation Period BehavioralEquation Identities EquationsSystems Criterio&' States Control 2. Chow1. Zellner (74) & Geisel (68) US A 1921-291953-72 1 (I Linear Quadratic Quadratic 1 2 4. Abel3. Shupp (75) (75a) (a) Abel (75) US 00 67111541-631V 7111 22 0 Log-Linear Linear QuadraticQuadratic 2 2 7.6. PhillipsGoodwin5. HoIt (62)(54) (51) (6) Chow (75a) US 0 632 1 Linear1.inearLii ear NoneQuadratic 9.8. St.Klein Louis (50) FRB Andersen & CarlsonThcilBogaard (64) (70) & Theil (59) US 0A 551-71111921-40 4 3 Linear Quadratic (h)(a) CooperBurger, &Kalish Fischer & (75) Norman & Wcathcrhy (74)Bahh (71) NonlinearNonhnear Quadratic 2 10. McFadden (69) (a)Ic) HealyMcFadden(f) Healy & Summers (69)& Medina (74i (75) Bowman & Laporle (72) 4 4 Linear Quadratic 11. Kmenta-Smith (73) (h) Kydland (73) (76) US 541-63lV 54 34 Linear Quadratic 12.13. Sandblom Chow (7) (70) (a)(a) Paryani Thalhcrg (72) (7 Ia. 71l) US 0A 1945-631921-40 4S5 5 LinearLinear Quadratic None 5 13. Chow (67) is A T9Z11948-63 -U APPENDIX A (Continued) Name (Date) NUMERICAL MACROECONOMETRIC MODELS USED IN CONTROL TfCountry IIaORY APPLICATiONS Period- icity' Estimation Period BehavioralEquation Identities EquationsSystems Criterion States Control KendrickChowPrescott (72b) (73) (67) (71) Linear QuadraticOuadratic 14. Shupp (72) FairYou (74) (75) US 8 1 NonlinearLinear Nonlinear Quadratic 16.15. PauPindyck (73) PiridyckKendrick (72a, & b;Majors 73a, b) (74) USDenmark 0A 551-671V 9 2 LinearNonlinear Nonlinear Quadratic 3028 8 25 3 17. Laidler (73) (a)(d) Gupta,Pindyck Meyer. (76) Raines Walsh & Cruz (76) linearLinear Quadratic game 2830 3 19.18. KaulDept. (75) of Finance Model (a) Ho & Norton (72) & Tarn (75) CanadaUS AQ 8 15 5 LinearNonlinear LinearNonlinear 11 3 20. Erickson, Leondes & Norton (70) EricksonErickson, &Leondes Norton &(73)Norton (70) US 0 5 11 Linear Quadratic 21. FRB Monthly Model Thompson,Perry (74) Pierce & NI 10 8 1.inear Quadratic 46 ID 22.23. Fair Livsey (70) (71) (a) FairPindyck (74) & Roberts (74 UKUS 0 571-661V 14 8 11 5 NonlineaiNonlinear Nonlinear NUMERICAL MACROECONOMETRIC MODELS USED IN CONTROL THEORY APPLICATIONS APPENDiX A (continued) 24. PREMI Wall & Westcott (74) Name (Date) UK Country Period-Icity Q Estimation Period BehavioralEquation Identities EquationsSystems Criterionb States Control (c)(b)(a) WallBray &(74)(75a) Westcott (75)(74) 551-7311 151311 1210 Linear in Percent QuadraticOuadratic 26.25. CentralRouzicr Planning(74) Bureau (a) van Eijk & Sandee(56) (56) Nether-Belgium lands A 1953-74 16 27 10 LinearNon-linear Change QuadraticLinearQuadratic 4 2 28.27. Bogaard Klein (69)& Barten & Norman (59) (69) (a)(a) Theil Norman (64) & Norman (73) Nether-US lands A 1929-64 12 27 28 LinearNonlinear Quadratic 4 5 30.29. GarbadeMINNIE (75a) Battenberg,(a)(b) GarbadeTheil Enzler(65) (75a, & b) US Q 471-691V 2131 4012 NonlinearLinear QuadraticNonlinear 54 4 65 31. Michigan Model Hymans(a) Craine, & 1-lavennerShapiro (73) & Havennerrinsley (75)(74) US 0 21 40 Nonlinear Quadratic 32. NIF-2 Higgins(a) Holbrook & Fitzgerald (74a) (72) Australia Q Nonlinear Ouadratic >61 3 (a) Fitzgerald, Johnson & Bayes (68) 26 41 NonlInear QuadraticPiecewIse 3 APPENDIX A (Continued) NUMERICAL MACROECONOMETRIC MODELS USED IN CONTROL THEORY APPUCATIONS Period- icitya Estimation Behavioral Systems Wharton(a)Evans Friedman Model& Klein (72) (68) Name (Date) US Country Q 481-64W Period Equation 47 Identities 29 LinearizedNonlinear Equations Piecewise Criterion' States Control (a)Boulle,STAR Oudet Bayer, (76) Mazier& Olive (74) France A 77 Nonlinear Quadratic 36. Athans ex at. (75) Fair (75a) FairFair (75) (76) US 0 19541 37 82 46 NonlinearLinear NonlinearQuadratic 31 10 38.37. FMS Krelle (74) Ando,(a) Fischer Modigliani & Uebe & (75) USW. Germany 0A 1955-7119731V 20() Nonlinear Quadratic 20 39. Bank of Canada RDX2 (h)(a) b PalashAndo, 'Nonlinear" Norman(75)NumberA =& annual;Raashe olsiates Q (72) quarterly: used in the M Ieedback.= monthly. Palash (75) here means nonquadratic and nonlinear, Canada 260 Nonlinear Quadratic 40. Data Resources, Inc. (a)Heliweil Gordon etal. (74) (71) HolbrookWoodside (73) (73) (74b) US 00 168 153 NonlinearNonlinear Quadratic a APPENDIX B Name (Date) THEOREtICAL MACROECONOMETRIC MODEI.S USED Country Pertod icity Estimation Period BehavioralEquation Identities IN CoNiRot.TutoRy APNICATI0NS EquationsSystems Criterion Stitcs &'ntrol Kareken.Henderson Muench & (72)Turnovsky & Wallace (73) S 0 3, Mundell (65) Myoken (75a) (a) Cheng & Wan (72) Linear QuadraticMinimum time 2 Phelps & aylor (75) 3 ()3 Quadratic ZellnerTurnovskyShuppSargent (76) (66)& Wallace (73) (74a) (75) (74b) (75a) 5 C) LinearLineirlinear Quadratic