The Internet Economy

Econ 407

Differential Equations and Their Applications to Economics

An equation which has a   derivative(s) as one of its component is known as a differential equation. A linear differential equation is made up of additive terms while a nonlinear differential equation will have additive as well as multiplicative components. In differential equations, the order of the derivative term(s) determine the order of the equation. Thus, an equation which has only first derivatives is referred to as first-order differential equation. An important assumption made about differential equations is that the coefficients of all the independent variables including the derivative term(s) are constant over the interval of interest.

Dynamic models that are influenced by time can often be described by a system of differential equations. To find the underlying structure of such a system, one must first attempt to approximate a solution by first solving the differential equation devised to describe the system. A solution to a differential equation results in an equation with no derivative terms of any order. Once a solution is obtained, the underlying model can be approximated by mathematical manipulations using a computer and simulation techniques.

An Example in Economics:

Suppose the marginal cost function is adequately described by the following linear equation:

MC= 10q= d(TC)/d(q).

We can solve this equation by integrating it once. The result will be:

TC= 5q^2 +c1, where c1 is a constant and can be assumed to be the fixed cost.

Normally, a differential equations is solved by finding a function that satisfies the differential equation. A trajectory is then determined by starting the solution with a particular initial conditions. For example, if we want to predict, what is involved in expanding production from zero to a million unit, we need to find the fixed cost, figure out the marginal cost, and then integrate the differential equation starting from the assumed initial conditions.