Mathematıcal Economıcs (ENG) — Ünite 7 Soru-Cevap

Dynamic variables change with time. In
other words, they are function of time.

The solution of a difference equation reveals the exact
time path of a variable or a group of variables

Absolutely yes. Modern economics has already
evolved into dynamic models.

The mathematical tool studying the
motion of a variable or group of variables
along time/ periods is called a difference
equation whenever time t can only take
integer values, i.e., t is discrete.

Dynamic economic model: An
economic model in which at least
one variable is a function of time.

A difference function with more than one independent variable is called a partial difference equation.

A difference equation is autonomous if the independent variable enters in the difference equation
only through yt–j, j = 0, 1, 2, ... That is, no independent t or t is not an argument of at–j in the difference
equation (see constant coefficient versus variable coefficient argument below). Otherwise, it is called
non-autonomous. Hence, equation (1) is a non-autonomous (due to 0.7 . t ), while, for example,
yt = (1.02) . yt–1 – 0.5 . yt–2 + 0.2 is autonomous. As a hint, if you do not see explicit t in a difference
equation, it means it is an autonomous difference equation.

A difference equation is autonomous if the independent variable enters in the difference equation
only through yt–j, j = 0, 1, 2, ... That is, no independent t or t is not an argument of at–j in the difference
equation (see constant coefficient versus variable coefficient argument below). Otherwise, it is called
non-autonomous.

As a hint, if you do not see explicit t in a difference
equation, it means it is an autonomous difference equation.

The order of a difference equation is the difference between the highest and the lowest time subscripts
in a difference equation.

Two methods to solve linear difference equations may be heuristically called the informal method (solution by iteration) and the formal method (solution by using algebra). There is not a clear-cut distinction between the two approaches. The simplest way of solving a difference equation is via iteration,which fits very well with the first-order linear difference equations. A unique solution may be easily obtained if the problem also informs the reader about a starting period and a corresponding starting value for the dependent variable. Things get quickly cumbersome with the iteration method once the order of the difference equation is higher and/ or the difference equation is non-linear. Plus, the number of initial values required to identify a unique solution increase with the order. Therefore, the iteration method is not advised for higher-order difference equations.

The formal method to solve an ordinary linear difference equation is the second method. Although
we heuristically call it formal, literally speaking, it is not, à la formal methods available in differential
equations. The formal method requires one to solve the difference equation in two parts: complementary
solution and particular solution. The complementary solution is the solution to the homogenous part of
the difference equation, and the particular solution is the solution to the difference equation itself. The
latter will coincide with the long-run equilibrium (steady-state) value of the variable, if one exists and the
form capture deviations from the long-run value (during the transitional periods). The sum of the two
solutions defines the general solution

In order to find the complementary solution to a difference equation, the trial-and-error (guess-andverify)
method is often advised.

A backwardlooking
difference equation problem can be solved by starting
from the current time and moving back up until the starting
time or by starting from the initial value and moving forward
until the current time.

A forcing term is a function of time.

The difference between
the dependent variable and the forcing term is that the latter’s
time path is given to the equation.

The general form of a second-order
difference equation is
yt = a0 + a1 . yt–1 + a2 . yt–2

Second-order linear difference equation: A linear
difference equation in which the highest difference between
the highest and the lowest time subscripts is two.