Business Cycle Models

Business cycle theories try to describe and explain the observablefluctuations in major economic variables like the national product, the employment rate, or the inflation rate. There are several patterns in different time scales. Gabisch and Lorenz (1989) introduce three classes of patterns: the Kondratieff, Juglar and Kitchin cycles.

Kondratieffcycles with a period of between 40 and 60 years are usually considered to be caused by technological innovations and subsequent structural change. Schumpeter distinguished three main cycle periods: 1790—1850 (steam power), 1850—1890 (rail-roads) and 1890—1930 (automobile). World War II started a new period but it is not yet easy to compare the postwar period with the previous periods. Juglar cycles with periods between 7 and 11 years were observed by Juglar in an investigation of French, British, and US banking figures, interest rates and prices. These patterns seemed to correspond with a life cycle of investment goods generating fluctuations in the GNP, inflation rate and employment. After World War II the average length of these cycles decreased to approximately 5 to 7 years for most Western countries. Kitchin concen-trated on short waves with a length of between 2 and 4 years in a study of British and US bank clearings and wholesale prices. There seemed to be great influence of exogenous shocks.

This classification does not cover all relevant fluctuations, and the real motion to observe is the superposition of several basic cycles. It follows that it may be empirically difficult to assign an observed cycle to one of the three different patterns mentioned above. It is also difficult to isolate a basic cycle from otherfluctuations. A recent phase

of research efforts includes two tasks. First the characterization of cyclical phenomena in the form of stylized facts (for the method see Danthine and Donaldson (1993)). The second task is the construction of models of economy capable of replicating these facts.

Gabisch and Lorenz (1989) list the broadly accepted stylized facts of business cycles with wavelengths between 4 and 8 years as follows:²

i) Observed values of economic variables can be divided into leaders, laggers, and coinciders. Aggregate output movements and profits are coinciders; different but related sectors move in conformity. Stock exchange rates, the value of sold shares, and employment in hours per week in the manufacturing industries are leaders, while factory payrolls and medium-term money market rates are laggers.

ii) Business profits show much greater amplitudes than other series.

iii) The production of durable goods exhibits greater amplitudes than the production of non-durable ones does.

iv) Agricultural production and the exploitation of natural resources to-gether with the associated prices exhibit greater than average devia-tions from the scheme of leading, lagging and coinciding series.

v) Different economic variables can move in different directions: the aggregate price level and short-term interest rates are procyclical;

long-term interest rates move only slightly in the same direction as the coinciding series. Monetary aggregates are procyclical.

vi) Trade cycle statistics correspond with leading, lagging, or coinciding series in many countries. The conformity is higher in small countries.

vii) Labor productivity is procyclical.

viii) Real wages do not unambiguously move anticyclically.

ix) Observed cycles are asymmetric: the velocity of an upswing is typi-cally slower than that of a downswing.

x) The amplitude of observed cycles decreased in the postwar period.

Several business cycle theories have been proposed but none of them can produce all these facts. We shall now describe some methods of constructing business cycle theories. The short list presented in this section is based on Gabisch-Lorenz (1987)

2The list is a modified version of the one proposed by Lucas (1977).

and Puu (1989). The main dichotomy between models is the source of the cyclic behaviour.

5.2.1 Shock-dependent models

Shock-dependent models explain how systems tend to a stable state after a disturbance.

So the source of the cyclic behaviour is in the environment. The model shows the natural period for the system, and explains the effects of past shocks.

The classical discrete-time linear shock dependent business cycle model is Samuel-son’s (1939) multiplier accelerator model (described below). It leads to a second-order linear differential equation.

Hicks (1950) introduced a modification of Samuelson’s model, which still is linear.

For his linear model Hicks made a nonlinear extension adding the ceiling and the floor for the growth curve. The model is fascinating but essentially verbal and its parameters are hard to estimate.

Kalecki published a series of papers (1935—1943) of models based on mixed difference-differential equations. The early linear version of the Kalecki model exhibits a dynamics very similar to the multiplier-accelerator models of Samuelson and Hicks. A typical behaviour of the Kalecki model is damped oscillations and the model needs exogenous shocks to exhibit permanent oscillations. In later versions Kalecki made the model essentially non-linear. The non-linear version is intuitively appealing, but it does not generate cycles in all cases and it lacks a rigorous mathematical treatment.

The idea of exogenous shock is very natural and real. For example the political business cycles can drive the damped system so that the cyclic behaviour is permanent.

Still some governments have actively tried to stabilize the economy, often without success. So it seems that the national economy has some kind of endogenous procyclic nature.

5.2.2 Shock-independent models

Kaldor’s (1940) business cycle model is one of thefirst attempts to study the effect of non-linearities in economics. Kaldor’s model is an elegant representation of a complex

system. Kaldor supposes that investments are a non-linear function of real income.

Savings are also supposed to be a non-linear function of real income. Fast dynamics leads the state to an equilibrium where savings are equal to investments. Slow dynam-ics leads to a slow-shifting motion in the savings- and investment-curves so that the equilibrium starts to move in the phase space. The movement of the equilibrium is cyclical and may even have catastrophic jumps. Kaldor’s model is very appealing, flex-ible and modern. Still it has been criticized for its ad hoc assumptions (see references in Gabisch-Lorenz page 129).

Goodwin (1951) presented a paper that covered several versions of the nonlinear accelerator-type models. The simplest form of Goodwins model is based on the as-sumption that there exist levels of real capital stock where the real investments are turned off or on. So the net investments exhibit alternate behaviour and the capi-tal stock has subsequent increasing and decreasing phases. The simple model is very crude. Goodwin also made a piecewise linear (but globally nonlinear) modification of the investment function of Samuelson’s classical model. Goodwin was one of the earliest users of computer simulation.

5.2.3 The models using the Poincar´ e-Bendixson theorem

The ideas of non-linear models have been generalized by several authors by using the Poincar´e-Bendixson theorem, by which one can study the existence of the limit cycle.

Good textbooks about the Poincar´e-Bendixson theorem is Hirsch and Smale (1974) and Verhulst (1990). The economic applications are presented in Gabisch and Lorenz (1987) and Puu (1989).

Chang and Smyth have (1971) reformulated Kaldor’s model (see Gabisch and Lorenz page 132). Chang and Smyth’s version makes clear the requirements for the existence of the limit cycle. It makes some essential improvements on Kaldor’s original model.

Some recent works use the Li´enard-van der Pol type models (Ichimura (1955), Schinasi (1985) (see Gabisch and Lorenz pages 158—161), Puu (1989) ). These models usually make some symmetry assumptions and thus they have not been so attractive

in economic application (as in physical application where the symmetry often arises from the system itself). One major problem in using the Poincar´e-Bendixson theorem is that the theorem asserts the existence of the cycle, but says little of the amplitude or time period of the solution. Puu (1989) and Verhulst (1990) describe the so-called Poincar´e-Lindstedt method by which one can approximate the periodic solution of the non-linear system. The method is laborious and needs an explicit and simple form for the nonlinearity in the model.

Puu makes a modification of Samuelson’s model, assuming that the investment function is odd and nonlinear. He uses the simplest possible form, namely the third-order polynomial. This is not very realistic, but makes the mathematics easy. Still the approximations of the amplitude and time period by the Poincar´e-Lindstedt method are laborious tasks.

With the tools derived in the previous chapters we can extend the work of Puu.

There are no easy formulas, but the majorant principles make it possible to get reliable bounds for the amplitude by simulating majorant and minorant systems. The estimate of the time period (T ≥2π in rescaled time) is not accurate in all cases, but it is useful and accurate in many cases. Above all, it is general and reliable.