Difference Equations

Difference Equations

LOGISTIC GROWTH

AGE STRUCTURE OF A POPULATION

ARMA MODEL OF TIME SERIES

BUSINESS CYCLES

BIBLIOGRAPHY

Difference equations or, rather, systems of these are mathematical models of some systems in the real world that are believed to change their states at discrete and equidistant points of time. An example of such a system is a university that accepts students twice a year. The state of such a university would be described in terms of student numbers—which are constant for half a year and change at two discrete points in the course of a year (based on the number of students that pass their exams and the number of students accepted). Continuous systems, on the other hand, change their states at any time. Systems in the real world can usually be described in terms of both continuous and discrete time. When, however, the most important state changes occur only within a very small part of a time span (such as within just a week per semester in a university), difference equations are the method of choice. The same preference applies when processes are modeled that are only measured at regular time intervals (once a year, once a month)—as is common, since most statistics are only published at discrete times (even though individual events such as births, deaths, or unemployment occur more or less continuously).

Difference equation models connect the future state of a system to the current state or even past states of the same system, in a way comparable to the manner in which differential equations allow the calculation of future states of a system from its current state. The application of difference equations supposes, however, that the processes within the modeled system are discrete in time, whereas in the case of differential equations processes are continuous in time.

For a number of real systems, the use of difference equations seems appropriate—for instance, in the case of populations of animals with non-overlapping generations. Here, one is only (or at least mainly) interested in the size x_t+1 of a given population next year when x_t, the size of the population in a given year t, is known; x_{t+ 1} is then expected to be some function of x_t. More generally speaking, t is the parameter of a process {x_t, t∊T } where T is an enumerable set, and the general form of a (first-order) difference equation is

x_{t + 1} = f (x_t ).

Difference equations of higher order are also possible; a second-order difference equation has the general form

x_{t + 1} = f (x_t, x _{t - 1})

and is often transformed into a system of difference equations, such that

x_{t + 1} = f (x _t, y _t); y _{t + 1} = x _t.

LOGISTIC GROWTH

One of the simplest cases of a difference equation in one variable—which also displays some interesting behavior—is the so-called logistic or Verhulst equation, which in its time-discrete version has the form

x _{t + 1} = rx _t(1 - x _t/k ).

One of the interpretations of this equation is that it describes a population in a habitat with carrying capacity K, whose size in the next generation is proportional to a growth constant (sometimes called the Malthusian parameter) r, to the current population size x_t and to the proportion of the habitat that, in some way, is so far unused (1 - x_t/K ). The equation has two stationary solutions, namely x_st0 = 0 and x_st1 = K(r - 1) /r. For r ≤ 1, only x_st0 = 0 is stable, and the population dies out, whereas for 1 < r < 3, x _st1 = K(r- 1)/r is stable, and the population will stabilize at this size, and for r ≥ 3, x_st1 is again unstable, and the system displays some interesting chaotic or fractal behavior (Schuster 1984, pp. 31-46).

AGE STRUCTURE OF A POPULATION

Another example also comes from mathematical demography, and here a system of difference equations is used to project the age distribution in a given population characterized by age-dependent death rates (which may differ between males and females) and birth rates that depend on the age of the mother. If one writes, for instance, m_a (t ) and f_a(t) for the number of males and females, respectively, in the a -th year of age, and and for the age-dependent death rates of males and females, then the death part of the system of difference equations is easily formulated:

for all a > 1

and for a = 1 the difference equation is

m ₁(t + 1) = γΣ _a=15⁴⁵β_a f_a (t ) and f ₁(t + 1) = (1-γ ) Σ _a=15⁴⁵ β_af_a(t )

where γ is the proportion of male births and β_a the age-dependent fertility rate. This system of difference equations yields a series of age pyramids.

ARMA MODEL OF TIME SERIES

Time series are also often modeled with the help of difference equations. The so-called ARMA model (Box and Jenkins 1970) is of this type, where the current value of a time series is modeled as a sum of autoregressive (AR) terms and a (weighted) moving average (MA) of random shocks:

x_t = α ₁x _{t - 1} + α₂x_{t - 2} + … + α_px_{t - p} + ε_t + β ₁ε _{t - 1} + β₂ε_{t - 2} + … + β_qε_{t - q}.

BUSINESS CYCLES

Finally, differential equations with delay are a mixture of difference and differential equations. One major example is Michał Kalecki’s model of business cycles, which can be formulated as follows:

K(t) = α/זּK(t) - (α/זּ + δ)K(t-זּ)

where K and K represent capital and capital growth respectively, and т is the delay between the decision to invest and the realization of this decision, while α and γ are two parameters that weight the influence of profit and of capital on the decision. Generally speaking, this model shows oscillating behavior for a wide range of values of α, γ and т, where these oscillations can be of constant amplitude, damped or negatively damped.

As in other cases, the Kalecki equation can be rewritten into a system of difference and differential equations:

K(t) = α/זּK(t) - (α/זּ + δ)L(t-זּ)

L(t)=K(t-זּ)

where the first is a differential equation and the second is a difference equation. Kalecki’s approach shows that with relatively simple assumptions about the investment behavior in an economy, business cycles can be modeled. The idea is that there is always a delay between an investment decision and the realization of the capital investment, and Kalecki’s model relates business cycles to just this delay. К and K are aggregated variables that in principle can be measured at very short time intervals and thus can be modeled in terms of differential equations, whereas т is a considerably longer period of time.

SEE ALSO Comparative Dynamics; Differential Equations; Kalecki, Michł; System Analysis

BIBLIOGRAPHY

Box, George E. P., and Gwilym M. Jenkins. 1970. Time Series Analysis: Forecasting and Control. San Francisco: Holden-Day.

Kalecki, Michł. 1935. A Macrodynamic Theory of Business Cycles. Econometrica 3 (3): 327–344.

Pressat, Roland. 1972. Demographic Analysis. Methods, Results, Applications. Trans. Judah Matras. New York: Aldine-Atherton.

Schuster, Heinz Georg. 1984. Deterministic Chaos: An Introduction. Weinheim, Germany: Physik-Verlag.

Verhulst, Pierre-François. 1847. Deuxième mémoire sur la loi d’accroissement de la population. Nouveaux mémoires de l’Academie Royale des Sciences et Belles-Lettres de Bruxelles 20: 1–32.

Klaus G. Troitzsch