# Mathematics for Economics

## Get a grip

The first quadrant of the Cartesian plane has slippery slopes called rectangular hyperbolas with the axes as asymptotes. Since requisitioning resources, and other topics in economic study, deal in inverse proportion, these trajectories are ubiquitous. Moving into the parameter spaces of trade, commerce, and promise, the hyperbola is an experienced slippery slope calling for traction. The first chapter of this book provides that traction with metric geometry. Use of hyperbolic coordinates yields parameters in the upper half-plane where a common model of hyperbolic geometry lives. In other words, this book promulgates economic motivations to use hyperbolic geometry and appropriate mappings. Although the quadrant appears to be bounded by the x-and-y-axes, in the hyperbolic topology it is an infinite surface with curvature negative-unity everywhere.

## Coding change

Trends and cycles are seen in econometrics and the second chapter considers methods of systemics for use with temporal data. A differential equation expresses a model by a function and its rate of change. A case where a pair of differential equation hold simultaneously leads to matrix analysis.

In business, comparison of this month’s total to last month gives the lag series of difference of totals. If

${\displaystyle Bx_{n}=x_{n}-x_{n-1},\quad {\text{then}}\quad B^{2}x_{n}=x_{n}-2x_{n-1}+x_{n-2}.}$

Considering a time series ${\displaystyle (x_{n})_{n=1}^{k}}$ of length k, polynomials in B can be evaluated on the time series. If one of them gives a value of zero, then xk can be expressed with of the other terms of that polynomial, yielding a predictive model.

## Pure mathematics used in econometrics

The following topics treated in other Wikibooks prepare one for technical economics:

• Calculus, including the Chain rule, higher order derivatives, Taylor series, optimization, implicit differentiation, antiderivatives, integra formulas, improper integrals, Leibniz's rule, differential equations,
• Linear Algebra, including bases, linear subspaces, systems of linear equations, scalar products, hyperplanes, eigenvalues, quadratic forms, semidefinite matrices, dominant diagonal.
• Multivariable calculus, including open and closed sets, convex sets, level surfaces, gradients, tangent planes, chain rule, Taylor's theorem, convex and concave functions