A reprint from

American Scientist

the magazine of Sigma Xi, The Scientific Research Society

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Computing Science

Everything Is Under Control Brian Hayes

I

n 1949, faculty and students at the London School of Economics gathered to observe a demonstration. At the front of the room was a sevenfoot-tall contraption assembled out of plastic pipes, tanks, valves and other plumbing hardware. The device, later dubbed the MONIAC, was a hydraulic analog computer for modeling the flow of money through a national economy. When the machine was powered up, colored water gurgled through the transparent tubes and sloshed into reservoirs. Various streams represented consumption, investment, taxes, savings, imports and exports. Crankwheels and adjustable cams allowed the water levels and flows to be regulated—the hydraulic equivalent of setting interest rates or tax policies. This was real trickle­-down economics! The MONIAC attracted much attention, and it lives on in folklore. Later generations of students called it the “pink lemonade national income machine.” Punch magazine tried to satirize the device, but their cartoon was really no more outlandish than the construction drawings for the machine itself. There are tales of leaks; according to one source, the machine couldn’t cope with inflation, which caused red fluid to squirt out through a hole in one of the cylinders. And then there’s the story about the Chancellor of the Exchequer and the Governor of the Bank of England; when they were given a turn at the controls, the results showed “why the U.K. economy was in the state it was.” This is all good fun, but the MONIAC was not just a toy or a joke. It embodied a style of thinking about economic problems that may still be Brian Hayes is senior writer for American Scientist. Additional material related to the “Computing Science” column appears in Hayes’s blog at ­http:// bit-player.org. Address: 211 Dacian Avenue, Durham, NC 27701. Internet: [email protected] 186

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Can control theory save the economy from going down the tubes? worth revisiting, especially at a time when real economies are leaking liquid assets at an alarming rate. Engineering the Economy

The principal architect (and plumber) of the MONIAC was A. W. H. Phillips, a New Zealander who had been an electrical engineer before he turned to economics. It’s easy to see the influence of his engineering background. A hydraulic simulation of the economy makes sense only if you believe that the circulation of money through a society obeys definite, mathematical laws, like those that govern real fluids and other physical systems. And the crankwheels and cams on the MONIAC imply that the behavior of an economy is not only predictable but also controllable. If we twiddle the knobs and nudge the levers in just the right way, all the streams will flow smoothly and the various basins where wealth accumulates will never run dry or overflow. This notion of engineering an economy was and is controversial. Adam Smith and other classical economists had argued that markets are self­correcting; meddling with them can only impair their efficiency. By the 1930s, however, the British economist John Maynard Keynes was making a case for a specific kind of intervention by governments and central banks: They could and should act to stabilize economies, he said—to smooth out cycles of boom and bust. Phillips was one of Keynes’s many followers.

The basic idea in Keynesian economic policy is to counteract any oscillatory tendencies. When an economy overheats, with business activity growing at an unsustainable pace, the central bank raises interest rates and thereby restricts the money supply. At the same time, governments raise taxes or reduce spending, which also cools the economy. Conversely, when business slumps, the aim is to spur growth by lowering interest rates and by letting the government run a deficit, spending more than it takes in through taxes. Keynes has gone in and out of fashion, but even many of his detractors now accept the idea that controlling wild excursions of the business cycle is an appropriate policy goal. In the current economic downturn, it is taken for granted that governments will do their best to speed recovery and mitigate damage. In the U.S., both the recently departed Republican administration and the new Democratic one have enacted huge “stimulus” plans, and the Federal Reserve has cut interest rates to near zero. Everyone waits anxiously to see how well these measures will work. The economics profession, naturally, has much to say about these matters, but there is another intellectual tradition that may also offer useful counsel: control theory, the branch of applied mathematics and engineering that deals with feedback systems. Devising a scheme to suppress oscillations, like those seen in the business cycle, is a common task for control theorists. The theory also identifies certain unfortunate situations where attempts to impose control can actually make matters worse, destabilizing a system that might otherwise have found its own equilibrium. Control Freaks

On first acquaintance, the idea of feedback control seems straightforward enough. Consider the design of a cruisecontrol system for an automobile. A

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minimal version measures the current speed of the car, compares it with the desired speed, then adjusts the throttle by an amount proportional to the difference. If the car slows somewhat— perhaps on an upgrade—the controller senses the discrepancy and opens the throttle wider, so that the car regains some of the lost speed. But there is more to control theory than this simple proportional-feedback mechanism. A drawback of pure proportional control is that the car never quite attains the requested speed; as the error diminishes, so does the feedback signal, and the system settles into a state with some nonzero offset from the correct velocity. The offset can be eliminated by another form of feedback, based not on the error itself but on the integral of the error with respect to time. In effect, the integral measures the cumulative error, which keeps growing if the speed differs even slightly from the set point. Thus integral control ensures that over the long term the net error approaches zero and the average speed converges on the set-point speed. Yet integral control has drawbacks of its own. Suppose the car cannot maintain a commanded speed of 60 on an upgrade; an integral controller might compensate by going 80 on the other side of the hill, which could get you a speeding ticket. More generally, integral control has a tendency to overshoot and oscillate around the set point. A remedy is to add still another form of feedback, based on the time derivative of the error signal. Derivative feedback opposes rapid changes in speed and thus tends to damp out oscillations. Running Hot and Cold

Proportional, integral and derivative control (together known as PID) are basic tools of control theory. In designing a control system, an engineer sets the “gain” of each type of feedback—the amount of correction applied for a given error magnitude. High gains yield a sensitive controller that promptly detects and corrects any disturbance. But a controller that responds too vigorously risks destabilizing the system, magnifying departures from the set point rather than suppressing them. The hazard of controller-induced instability is most acute when there are delays, or time lags, built into the feedback circuit. The nature of this problem is familiar in everyday life. You step into the shower and find that www.americanscientist.org

Colored water in transparent plastic pipes modeled the flow of money through an economy in a hydraulic computer built in 1949 by the British economist A. W. H. Phillips. Fluid pumped to the top of the main circuit represented income; it streamed back down as consumption, with amounts diverted into taxes and savings. The flows were regulated by feedback devices, such as the one depicted in the diagram at right: A float in a reservoir operated a valve, which controlled the rate at which the reservoir filled. The drawing above, which shows an Americanized version of the Phillips computer, is in the James Meade Archive of the London School of Economics.

the water is too cool, so you twist the temperature-control valve counterclockwise. Nothing happens for a few seconds, and so you turn the valve a

cam valve

float reservoir

little more. When the hot water finally makes its way to the shower head, you find you’ve gone too far. You dial the valve back a little, but the water con-

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tinues to get hotter, so you turn the control further clockwise. Soon, you’re shivering. The temperature oscillations can keep growing until the shower is alternately emitting the hottest and the coldest water available. (In this situation the average temperature might be just right, but no one would count that a success of the control system.) Cruise control and a shower valve are examples of control systems that regulate a single variable, such as speed or temperature. An aircraft autopilot, in contrast, might have to maintain a constant altitude and heading as well as controlling motion around the roll, pitch and yaw axes. All of these variables are coupled; a change in one affects others. Similarly, a controller for a distilling column in an oil refinery

might need to regulate temperature, pressure and several flow rates. Again, the variables cannot be considered separately; turning up the heat alters pressures and flows. Solving such multivariable control problems was difficult and tedious with early design methods, which are now characterized as classical control theory. Those methods assess the performance of any given control law but leave to the intuition of the engineer the task of choosing which laws to test. Beginning in the 1960s, modern control theory introduced a new computer-intensive methodology that not only evaluates given laws but also searches for the best attainable laws under stated constraints. This collection of techniques, known as optimal control, identifies the

gains kp

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income Concepts from control theory underlie an economic model devised by Phillips in 1954. At the bottom of the block diagram is a feedback loop representing the operation of the economy itself: Spending by consumers results in demand for production, which after a time lag generates income, which in turn is spent to create still more demand. The rest of the diagram shows a hypothetical control circuit meant to stabilize the economy. The difference between actual income and a set point produces an error signal, e, that is transformed to become a negative feedback applied to production. Three forms of control feedback are included, namely signals proportional to e, proportional to the integral of e and proportional to the time derivative of e. proportional + integral + derivative control

change in economic output

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Feedback control prevents fluctuations in output in the Phillips model. The graph shows the economy’s response to an abrupt downward shift in demand. Without a stabilizing controller, production falls off and remains at a new, lower equilibrium. With proportional control only, output remains below the original level. The combination of proportional and integral control returns to the original level but then overshoots and oscillates. Including derivative control suppresses the oscillations. But a real economy is more challenging than this simplistic model. 188

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control law that comes closest to satisfying a given criterion. A number of further variations have grown out of optimal control. Robust control finds laws that deliver reasonable performance even if the real system differs somewhat from the mathematical model that represents it. Stochastic control tolerates noise or errors in the measurements of the system’s state. Adaptive control applies the feedback principle to the control laws themselves, allowing the controller to continue working as the system evolves. An Invisible Hand on the Controls

Even without any external controls, economic systems are laced with multiple feedback loops. Adam Smith’s price mechanism is the best-known example: When demand exceeds supply, prices rise, thereby curtailing demand and allowing prices to fall back toward their original level. This is a negativefeedback loop, which has a stabilizing influence. Other loops introduce positive feedback, as with the inflationary spiral: Rising prices bring demands for higher wages, which lead to still higher prices. Smith believed that built-in feedback mechanisms are the best possible regulator of an economy; left alone, the system will find its own equilibrium, balancing production and consumption. Keynes agreed that economies tend toward equilibrium, but he pointed out that the same economy might wind up at many different points of equilibrium. A thriving economy has a high level of production balanced by a high level of consumption; for a depressed economy, supply and demand remain in balance, but both are at lower levels. The aim of Keynesian economic policy is to nudge us from a recessionary equilibrium toward a more prosperous one. Keynes did not formulate his ideas in the vocabulary of control theory, but some of his followers did. Phillips took this approach not only in the design of the MONIAC but also in later essays that explore PID control laws for economic variables. At about the same time, Arnold Tustin, a British engineer, published a treatise on “the problem of economic stabilisation from the point of view of control-system engineering.” These works and a few others from the same era include block diagrams and stability graphs that would be perfectly at home in a text on industrial control problems.

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2.5

lag = 4

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To create a control system for an economy, a first step is to identify economic equivalents of engineering concepts such as sensors and actuators. Filling the role of economic sensors are measurements of business activity, such as employment levels, trade balances and statistics on income, savings and spending. The actuators are monetary and fiscal policies. Monetary policy is how a central bank controls the supply of money. (Printing currency is a minor part of this process; the major part is regulating terms of credit.) Fiscal policy has to do with government revenues and expenditures, and especially the balance between these amounts. Because the government is typically the largest single participant in the market, a government surplus or deficit can have a decisive effect on the overall demand for goods and services. As control tasks go, regulating a national economy looks to be a major challenge. One notable difficulty is the wide range of time scales. Some economic events play out in a matter of hours or days (the 1987 stock market crash, the collapse of Lehman Brothers). At the other end of the spectrum, major trends can last for a decade or more (the Great Depression). Data from stock markets and commodity exchanges flow in minute by minute, but other statistics (retail sales, unemployment, corporate earnings) are calculated on a weekly, monthly or quarterly basis. On the actuator side of the control process, monetary policy is usually revised every month or two, but most aspects of fiscal policy are

system state (arbitrary units)

Economic Sensors and Actuators

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lag = 6

2.0 system state (arbitrary units)

With the advent of optimal control a decade later, there was another surge of interest in control theory as a tool for solving economic problems. In the early 1970s a series of joint conferences brought economists together with control engineers, and dozens of publications explored the synergies of the two disciplines. Furthermore, the Federal Reserve began experimenting with control-theory methods in the suite of statistical and mathematical tools that inform its policy deliberations. A 1974 review article by Michael Athans of MIT (a control theorist) and David Kendrick of the University of Texas at Austin (an economist) remarked on the atmosphere of excitement in a new interdisciplinary area that seemed to have bright prospects.

1.5

lag = 6 lag = 4 lag = 3

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Time lags in a control system can have a disastrous effect on stability. The graphs show the response of a hypothetical system governed by a simple controller when the set point is abruptly changed from 0 to 1 (in some arbitrary units). In the upper graph the gain of the controller is 0.3; in other words, if the measured state of the system differs from the set point by an amount x, the controller applies a correction of –0.3x. The four curves differ in the time lag between when the state is measured and when the correction is applied. With a lag of one month the system settles smoothly to a new steady state; a lag of three months causes some overshoot, and a lag of four months induces damped oscillations. When the lag reaches six months, the system is unstable, with oscillations that grow in amplitude. In the lower graph the lags are the same but the gain has been reduced by half, to 0.15. Now the system is stable, but it takes many months to readjust after a change in the set point.

set annually. Because of all these time lags, regulators can never really know the current state of the economy they are trying to control; they have to act on the basis of measurements made over an interval that extends months into the past, and their actions will not produce their full effects for months into the future. Another issue is that economic control is a multivariable problem, even though policymakers often speak as if there were a single dial, like a thermostat, that governs the overall pace of

business activity. Control actions have different effects on lenders vs. borrowers, workers vs. investors, tenants vs. landowners, importers vs. exporters. A boon to farmers may be a burden to food buyers. The effects on various sectors of the economy cannot be adjusted independently; all the variables interact, much like temperature and pressure in an industrial process. Still another question is whether governments and central banks actually have the power to tame the business cycle. No algorithm can keep a car up

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to speed if the engine lacks the oomph to climb a hill. Likewise, agencies trying to correct a severe economic downturn may simply lack the resources to restore prosperity. In the case of monetary policy, it is often pointed out that

12

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A model of the U.S. economy created by Ray C. Fair of Yale University could be viewed as an updated version of the MONIAC, without the gurgling and splashing. The model consists of 127 equations, defining relations among quantities such as industrial production, consumer spending and interest rates. For the period up to the end of 2008, the coefficients of the equations are fitted to empirical data; beyond that date (purple region of graphs) the model runs in a forecasting mode. The first two variables plotted above—gross domestic product and the unemployment rate— are measures of economic health. The other two variables—the interest rate on short-term Treasury notes and the government surplus or deficit—are instruments of monetary and fiscal policy often employed to counteract adverse trends. The model is available online at http://fairmodel.econ.yale.edu/main2.htm. 190

interest rates are a blade with only one edge: You can raise them as high as you wish, but you cannot lower them below zero. (On the other hand, bailout packages for banks are tantamount to a negative interest rate: The banks are being paid to borrow.) Among all these impediments to effective control, the most worrisome are the time lags between measuring the state of the economy and applying corrections. Such delays, if combined with overly zealous control actions, bring a risk of controller-induced oscillations—of getting scalded or frozen in the shower. We might still be trying to stimulate a sluggish economy when it is already on the verge of overheating; then in the next phase of the cycle we might overreact in the other direction. Control theory offers well-defined criteria for deciding whether a system will eventually settle down to a steady state or will enter some runaway regime of exponential growth or unbounded oscillations. Control engineers employ those methods to calculate stability margins—a measure of how closely the system approaches the edge of instability. As far as I know, such stability analyses are not routinely performed for monetary and fiscal controls applied to major economies. Perhaps it is not possible to do so given our imperfect knowledge of the state of the economy; if that’s the case, however, then we also cannot know how the economy will respond to our policies.

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Viewing economics through the lens of control theory amounts to treating the economy as a machine. The machine may be an unwieldy one—something like a heavily laden supertanker, slow to start and stop, difficult to steer—but it still obeys deterministic laws of motion. In particular, a machine never tries to second-guess or outwit the controller. But a human society is not a machine. In 1976 Robert E. Lucas, Jr., of the University of Chicago presented a caustic critique of mathematical modeling as a tool for setting economic policy. Models, he said, predict the effect of policy changes without acknowledging that rational agents will alter their behavior under the new policies in ways that invalidate the assumptions on which the models were built. For example, if everyone knows (or can infer) the rule by which the Federal Reserve sets interest rates, borrowers and

lenders will anticipate any changes in rates, adjusting their behavior in ways that tend to neutralize the effect of the policy change. It’s as if the supertanker, with a mind of its own, could change the shape of the rudder in order to resist commands from the helm. The Lucas critique attacks not just control theory and mathematical modeling but any reasoned strategy for managing an economy. The most extreme form of the thesis holds that if a policy can be predicted, it will be undermined and rendered ineffective. Thus the attempt to regulate the economy becomes a futile spiral in which people adjust to new policies, the policies are adjusted to account for those adjustments, and so on. Within the context of control theory, however, the kind of circularity cited by Lucas is not anything out of the ordinary. In any system with closed-loop feedback control—animate or inanimate—the controller affects the state of the plant, which in turn affects the state of the controller, which affects the plant, and so on. The circularity is an essential part of the design and does not lead to undefined behavior or an endless round of readjustments. Assuming that stability criteria are satisfied, the combined system of controller and plant converges on some definite and predictable equilibrium state. Another branch of mathematics reaches a similar conclusion from somewhat different premises. In game theory, a contest like the one between a regulator and a population of economic agents generally has a fixed point (called a Nash equilibrium), where neither party can gain by making further changes in strategy. This idea has developed into a theory of “policy games,” which strive to identify those economic models that remain controllable even when the players don’t wish to be controlled. From a more pragmatic point of view, it’s not clear that real-world economic agents are as strongly motivated to “game the system” as Lucas supposed. Ray C. Fair of Yale University, using a model of the U.S. economy based on decades of empirical data, tested variations of the model in which agents could look ahead and base their behavior on predictions of future regulatory policies. The results suggest that such activity is not common in the real economy. Another series of studies by Glenn D. Rudebusch of the Federal Re-

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serve Bank of San Francisco reached a similar conclusion. Whatever the merits of the Lucas critique, it had the collateral effect of dampening enthusiasm for applications of control theory in macroeconomics. Research in the area did not end entirely, but the undertaking lost momentum, and control theory has never again been fully in the mainstream of economic thought. Nor has it become a common tool of those who put policy into practice. The Great Moderation

With or without control theory, governments and central banks have not given up on attempts to smooth out business cycles. On the contrary, every major recession and inflationary episode in recent years has been met with a “counter-­cyclical” response, regardless of the political ideology of those in power. Feedback principles have probably helped to frame some of those countercyclical actions. In 1993 John B. Taylor of Stanford University proposed mathematical rules for setting centralbank interest rates in response to inflation and economic growth—rules that may well have influenced Federal Reserve decisions over the past decade. The Taylor rules constitute a rudimentary feedback mechanism. On the other hand, they do not exploit the full power of modern control theory. In particular, Taylor’s formulas ignore the question of controller-induced instability, and they take no account of the uncertainties that enter into stochastic and robust control methods. Until recently, the economic strategies of governments and banks seemed to be working rather well. The peaks and troughs of the business cycle have been shallow, with only mild inflation. By 2002, economists were calling our era the Great Moderation. But confidence in a perpetually purring economy has been swept away by the financial crisis of the past year. Governments have taken extraordinary measures in an effort to stabilize the system. One would like to believe that those interventions were guided by scientific principles of some kind— that the timing and the magnitude of the stimulus plans were calibrated to produce the fastest possible recovery without the kind of overcorrection that might bring oscillations. Needless to say, the cost of miscalculation is high. Ben S. Bernanke, a www.americanscientist.org

student of the Great Depression, has argued that timid and misguided policies of the Federal Reserve were partly to blame for the length and severity of the 1930s depression. Bernanke is now chairman of the Fed, and he has presided over a monetary response that is anything but timid. Since September 2008 the Fed has injected well over a trillion dollars into the U.S. economy. Not all economists see timidity as the main peril. Athan­asios Orphanides of the Federal Reserve has argued that overaggressive corrections in the 1970s contributed to the “stagflation” of that decade. And still another faction questions whether monetary adjustments really have much effect at all. Through an ingenious computer simulation, Christopher A. Sims of Yale imposed the policies of the modern Fed on the economy of the 1930s, and vice versa. Swapping the strategies had little effect on the outcome. I don’t know what to make of all these seemingly contradictory observations, and I find it unnerving that in a matter of such consequence there is so little consensus on basic principles. If the designers of an airplane disagreed so vehemently about the engineering of the flight-control system, the airplane would not leave the ground until the conflict had been resolved. But grounding the economy is not an option. It was the economic catastrophe of the 1930s that led Keynes to propose a mechanism for moderating cycles of growth and contraction. Perhaps the current crisis will inspire a further examination of strategies for managing these fluctuations. The precepts of control theory suggest that one useful goal would be to reduce time lags in the feedback loop. On the sensor side, this would mean quicker collection and evaluation of basic business data, such as price levels and employment statistics. Speeding up the actuator side might be more problematic. Although the Fed and other central banks can act promptly on monetary policy, many aspects of fiscal policy are tied to the legislative calendar. The most promising approach may be greater reliance on “safety net” mechanisms such as unemployment compensation, which function automatically, without any need to enact new laws. Standing in the middle between sensors and actuators are the control algorithms. Finding the optimal control law—and deciding what measure of

economic wellbeing should be optimized—is the intellectually difficult and politically contentious part of the process. Maybe the answer is to power up the MONIAC again. We can let the cascade of colored water tell us how far to twist the hot and cold faucets. Bibliography Åström, Karl Johan, and Richard M. Murray. 2008. Feedback Systems: An Introduction for Scientists and Engineers. Princeton: Princeton University Press. Athans, Michael, and David Kendrick. 1974. Control theory and economics: A survey, forecast, and speculations. IEEE Transactions on Automatic Control 19:518–524. Barr, Nicholas. 2000. The history of the Phillips machine. In Leeson, 2000, pp. 89–114. Bernanke, Ben S. 2004. Essays on the Great Depression. Princeton: Princeton University Press. Bissell, Chris. 2007. Historical perspectives: The Moniac: A hydromechanical analog computer of the 1950s. IEEE Control Systems Magazine 27(1):69–74. Fair, Ray C. 2004. Estimating How the Macroeconomy Works. Cambridge, Mass.: Harvard University Press. See also http://fairmodel. econ.yale.edu/main2.htm Kendrick, David A. 1988. Feedback: A New Framework for Macroeconomic Policy. Boston: Kluwer Academic Publishers. Kendrick, David Andrew. 2005. Stochastic control for economic models: Past, present and the paths ahead. Journal of Economic Dynamics and Control 29:3–30. Leeson, Robert (editor). 2000. A. W. H. Phillips: Collected Works in Contemporary Perspective. Cambridge: Cambridge University Press. Lucas, Robert E., Jr. 1976. Econometric policy evaluation: A critique. Carnegie-Rochester Conference Series on Public Policy, Vol. 1, pp. 19–46. Orphanides, Athanasios, and John C. Williams. 2005. The decline of activist stabilization policy: Natural rate misperceptions, learning, and expectations. Journal of Economic Dynamics and Control 29:1927–1950. Phillips, A. W. 1954. Stabilisation policy in a closed economy. The Economic Journal 64(254):290–323. Reprinted in Leeson, 2000. Rudebusch, Glenn D. 2002. Assessing the Lucas critique in monetary policy models. Federal Reserve Bank of San Francisco, Working Paper 2002–02. http://www.frbsf.org/publications/economics/papers/2002/wp0202bk.pdf Sims, Christopher A. 1998. The role of interest rate policy in the generation and propagation of business cycles: What has changed since the ’30s? In Beyond Shocks: What Causes Business Cycles?, edited by Jeffrey C. Fuhrer and Scott Schuh. Boston: Federal Reserve Bank of Boston, Conference Series No. 42. Taylor, John B. 1993. Discretion versus policy rules in practice. Carnegie-Rochester Conference Series on Public Policy, Vol. 39, pp. 195–214. Tustin, Arnold. 1953. The Mechanism of Economic Systems: An Approach to the Problem of Economic Stabilization from the Point of View of Control-System Engineering. Cambridge, Mass.: Harvard University Press.

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