Differential Equations in Economics

STUDY OF APLICATIONS OF DIFFERENCE/DIFFERENCIAL EQUATIONS IN ECONOMICS

1. AIM

The aim of this work is to understand the utilities of the differential equations to solve economic problems.  

1. INTRODUCTION

Both difference and differential equations became an essential tool for the economic analysis, since the computer usage made it possible.

In economics, as a rule, it’s studied how a variable changes over time to make a decision or to know, for instance, the spending, the benefits, or the risk that a financial product will have. To obtain these answers it’s necessary to study dynamic systems (which change over time and/or depending on other parameters) and its evolution. These systems are created through difference and differential equations.

Differential equations are expressions of how the studied variable changes depending on other factors (time, labour, investment amount…) and the variation of the function with respect to another factor (first order derivative or superior). We use differential equations when we know how does the dynamic system behave, each value of the factors that influenciate it (in most cases it includes time variable)

Difference equations and differential equations, respond to the same problems, but we have to take into account that difference equations are used in discrete functions (not continuous), while differential equations are used in continuous functions. This way, the studied phenomena should follow the same criteria.

Both difference and differential equations don’t have a unique solution, from initial conditions we get particular solutions.

The mathematical application to solve economic problems is always formed by three phases:

• Translate the economic information into mathematical language in order to able to obtain an economic model (it can be a differential equation, a linear system or any other mathematical expression)
• We have to operate from the model created, such that we can find a solution for the problem.
• We have to interpret and understand the obtained results in terms of economics.

Differential equations

A differential equation is an equation where the unknown is a function (x(t)), and the equation is set in terms of the relationship between the unknown function, its derivatives (x,x, etc. The dot means the derivative with respect to variable t) and others functions of the independent variable (usually t).

Examples of differential equations:

y'=x.cos⁡(x)

y''-2y=2x

δf(x,y)δx= -x.δf(x,y)δy

The equations that only depend on one independent variable are called ordinary differential equations (ODE), if only first order derivatives appear, the order of the differential equation consists on the greater derivative order of the equation. If the functions depends on more than one independent variable, they are called partial derivatives functions (PDF). In the previous examples, the two first are ODE while the second and the third are PDF.

Systems can be deterministic or stochastic. Randomness doesn’t play a role the future states of the system in the deterministics systems. In these systems, knowing the actual state and the behaviour of the other factors we can determine the future state without incurring any risk. In the stochastic systems, the state of the system is determined by the predictable actions of the process as for the random elements.

• Difference equations

Difference equations are expressions where we can see how the studied variable changes conditioned by the other factors (time, labour, investment amount) and the variation of the function with respect to one of the factors (first order derivative or superior)

Examples of difference equations:

xt+1=xt(1+xt)

C0(n)xn+C10(n)xn+1+…+Ck(n)xn-k=f(n)

The order of the equation depends on the number of terms that interfere in the expression, so that the first expression is the one of order one, while the second K.

The first expression is nonlinear, since the superior degree is two, while the second equation is linear.

Difference equations could also be deterministic or it can happen that a term follows an aleatory process.

• DEVELOPMENT

We are to present examples of application of difference and differential equations in economics and financials, the relation is a sample with the intention to give an idea of the extent of their use.