Does Laplace Have Le Place in Economics?

You may have only heard (and applied) Laplace Transform perhaps in your Mathematics subject or maybe in your Computer Networks subject, but did you know that it could be possible to apply this theory on certain stochastic and deterministic economic problems?

“In a deterministic economic process the value of each function under consideration is known in advance at every future point of time.” These functions can be the cost per unit time period, the number of demanded units per unit time period, etc. For example, the maximized discounted value of a future cash flow existing during a given time interval, finite or infinite, would take a very simple form in Laplace terms[1]. The Laplace expression would be more involved if we are only to study the discounted value from a finite time interval [0, T] . The derived equations using the Laplace transform such as this one:

130-1[1]

would be of great use to variety of problems such as feedback controlled-production problems, inventory problems, etc.

“In a stochastic economic process, the functions studied possess some property of giving uncertain outcomes.” This uncertainty stems from stochastic variables which have values that are unknown in advance. We also take note here the probability of these stochastic variables taking values within a certain interval which is the probability density function. If this probability function is time invariant, the process is stationary. After series of derivation and under the assumption that “using the expected value as suitable measure of the process result”, stochastic economic processes can be treated like deterministic processes.”[1] An example of a problem would be computing the expected present value of a season’s variational revenue flow using this derived equation:

130-2[1]

For the a detailed derivation and usage of Laplace transform in equation form, the source material is provided.

Source:

[1] http://ctr.maths.lu.se/matematiklth/courses/FMAF05/media/material/grub67.pdf

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