Mathematics for Economics/Dynamics

Economic dynamics

Related rates are a common paradigm of mathematical modeling. A pair of differential equations with rates expressed as a 2x2 matrix calls for a solution to a matrix differential equation. w:Giancarlo Gandolfo develops the model in pages 237 to 45 of Economic Dynamics (1997)[1980]. A different development profiles the matrix space M(2, R) as a real 4-algebra with planar sub-algebras:

• Abstract Algebra/2x2 real matrices

The related rates matrix differential equation finds solution according to the specific plane in M(2, R) that contains the matrix. The solution bifurcates on whether the matrix involves an imaginary unit or a hyperbolic unit. The first is periodic, the second is not.

• Abstract Algebra/2x2 real matrices#Differential equations

These differential equations model deterministic systems while stochastic terms are admitted in the social sciences. The Markov chain is a stochastic model that uses a matrix of probabilities for transitions between states of the mode. The columns sum to one. Markov chains are probability models with long-term limiting states. w:J. Laurie Snell and w:John G. Kemeny brought these models into classrooms.

Time series

The dynamic nature of economics is evident in time series studies. These are based on lists of economic data recorded over fixed time intervals. For instance, each nation has an annual gross domestic product. These studies may show leading indicators such as demographic series of youth entering the labor force. The idea that one series may be correlated with another is made precise by the correlation coefficient. Say two time series determine two points in n-space, where n is the list length. Then the list of zeroes and these two points determine a plane in n-space.

Definition: $R=\cos \Theta$ is the correlation coefficient of two time series with Θ being the angle at 0 between the points represented by the time series.

Definition: The series pairs with R near zero are correlated. Those with R near π are anti-correlated. With R near π/2 are un-correlated or orthogonal.

Time series analysis involves considerations of autocorrelation, such as the year-over-year figures of series subject to seasonal variation. The autocorrelation study contracts the length n of lists. For instance, the one period difference $t_{i}-t_{i-1}$ will have only n − 1 values with both i and i – 1 in the index set. When n is large, the loss is not significant.

Dynamic behavior is separated into trend and periodic behavior, with account taken for shocks. Whatever the source of the shock, if it is exterior to the series milieu, associated data points are called into question.

Dynamic factors may bounce around, vigorously at times, inducing vertigo in humans. The practice of smoothing of time series is called the moving average. These smoothing operations, using neighboring data points in a formula, reduce the jittery flight of a series.

A common mathematical approach to time series deals with the backshift or lag operator $BX_{i}=X_{i-1}.$ With 1 as identity operator, $(1-B)X_{i}=X_{i}-X_{i-1},$ so the binomial 1 – B represents the first-difference series. The mean of the first-difference indicates the steps of a trend line, arithmetic series, co-initial and co-terminal with the original. One may subtract the trend from the original to obtain a series with zero trend.

Consider now the forecast formula $t_{i}=-t_{i-1}-t_{i-2}.$ Start with any two data values and compute a third. Using the formula again returns the initial value. In fact, this formula generates a three-step series, repeating. The formula is equivalent to the trinomial 1 + B + B² set to zero. The repetitive nature of the associated series can be considered as the product $1-B^{3}=(1-B)(1+B+B^{2})$ where the two factors correspond to either a constant series or a three-step.