Rational Functions

A function such as x - 1

in which y is expressed as a ratio of two polynomials in the variable x, is known as a rational function. According to this definition, any polynomial function moist itself be a rational function, because it can always be expressed as a ratio to 1, and 1 is a constant function. A special rational function that has interesting applications in economics is the function a v = - or xy = a x which plots as a rectangular hyperbola, as in Fig. 2.8J. Since the product of the two variables is always a fixed constant in this case, this function may be used to represent that special demand curve—with price P and quantity Q on the two axes—for which the total f In the several equations just cited, the last coefficient (af) is always assumed to be nonzero; otherwise the function would degenerate into a lower-degree polynomial.

FIGURE 2,8

Linear

Linear

FIGURE 2,8

Quadratic

Quadratic

Rectangular Hyperbola Economics

expenditure 's constant at all levels of price. (Such a demand curve is the one with a unitary elasticity at each point on the curve.) Another application is to the average fixed cost (AFC) curve. With AFC on one axis and output Q on the other, the AFC curve must be rectangular-hyperbolic because AFC x Q{— total fixed cost) is a fixed constant.

The rectangular hyperbola drawn from xy = a never meets the axes, even if extended indefinitely upward and to the right. Rather, the curve approaches the axes asymptotically: as y becomes very large, the curve will come ever closer to the y axis but never actually reach it, and similarly for the x axis. The axes constitute the asymptotes of this function.

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  • Stuart Hansen
    How are rational functions used in economics?
    8 years ago

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