Mathematics for Economics/Hyperbolas

Hyperbolas in Economics

Introduction
The approach to mathematical economics in this chapter is geometric. Direct and inverse proportion correspond to constants that determine rays and hyperbola branches in the first quadrant. Entry into this environment of mathematics is illustrated with economic topics of wages, partitioning, stocks and market dynamics, mentioned superficially. Then students are expected to come to grips with hyperbolic geometry.

Johann von Thunen brought hyperbolas into economics in 1875 when he used geometric mean of two values to determine a third. The extreme minimum wage, allowing the worker a bare survival, was much less than the value produced when equipped with the employer’s tools, evidence of capitalization. Von Thunen solved the problem of deciding on the wage paid, by multiplying the two values, then extracting the square root. This algebraic procedure has been called taking the geometric mean of the two original values, and von Thunen called it the natural wage. When a natural wage is set, the plot of two original values giving that result is a hyperbola.

The equi-partition function p = S/n for n receivers, where S is the total supply $$S = \sum_1^n p .$$ When supply S is rationed equally, then the equi-partition function provides the rationed quantity p. With n a natural number, as in social partitioning, the points (1, S), (2, S/2), ... (k, S/k) lie in the Cartesian plane. The points are connected with the smooth curve y = p(n) = S/n when n takes real values between the integers.

A company has x thousand shares held publicly, and today the shares are selling at k$. In terms of market capitalization, the product $ x k 000 represents the company value in the market for shares. As k varies from day to day, so company value is in flux. Sometimes management finds k intolerably low and company funds are used to repurchase some shares, thus reducing x. The coordinates (x, k) where market capitalization is constant form a hyperbola.

Supply and demand are spoken of as inversely related. Price of an item is a commercial factor, with price and demand related as follows: high price suppresses demand and cheap price augments demand. Supply is known by the vendor, and vendor sets the price, which in case of scarcity can be very high.

The points (x, y) such that x>0 and y>0 is called the first quadrant Q of the Cartesian plane. The hyperbola branch in Q can be seen at pages 31, 49, 144, and 173 of W. Stanley Jevons The Theory of Political Economy, 5th edition, 1957, a book originally published in 1871.

Economic behavior observed in Q can be informed of differential geometry associated with it: that Q is a two-dimensional manifold with (0,0) as the only boundary point. In fact, Q can be read as a copy of Lobachevski's hyperbolic plane when the radial and hyperbolic angle coordinates are introduced.

The radial coordinate determining a hyperbola is wage level in the first example, the supply S in the second example, and company value in the third. In retail, total sales receipts forms the radial coordinate. The hyperbolic angle is an area of the corresponding hyperbolic sector to the unit radius. Whereas the conventional circular angle is less than 360 degrees or 2 &pi; radians, the hyperbolic angle is unbounded. Thus, a zero point is required to anchor its measurement. According to tradition, the point (1,1) on the hyperbola xy= 1 is the zero point. The hyperbolic angle then determined by (x, 1/x) is log x, the natural logarithm with base e = 2.718...

For hyperbolic metric geometry, the hyperbolic angle determines a horizontal axis, with vertical given by the square root of the radial coordinate in Q. These new coordinates in the upper half-plane H are given a metric, a positive distance function satisfying the triangle inequality. Tracking distances via H for movement in Q, over days, weeks, months or years, provides consistent measurement for long and short term perspective.

One of the measures of maturity in an individual or community is willingness to delay rewards until work is done. The psychological subject is delayed gratification. The study in economics is called discounting, a factor of consumer behavior where payoff in the future is tested against a lesser payoff earlier. An encouraging result of economic studies shows people prefer hyperbolic discounting over exponential discounting, which is encouraging since reasonable people use this pattern of deferred gratification, a sign of the proverbial economic man, something sometimes doubted in other contexts.

Revised percentages
The time value of money is poignantly shown by inflation of prices for the necessities of life. As business costs rise, prices of goods and services reflect the monetary tide. More objectively, economists may express the change as a de facto currency devaluation. Wing units of hyperbolic angle follow the units of natural logarithm, so that the angle between (1,1) and (e,1/e) is one unit. One percent change occurs between 1 and 1.01, and loge (1.01) = .0099503 wings or 9.9503 milliwings. The first ten percentage points have these angular values: Increments from one percentile to the next decrease; the first percent has the largest angle. In contrast, for mechanical velocities measured in nanowings, each marginal increase is virtually the same as the previous step. (Compare Kinematics/Transformations) Naturally, the milliwings used here for economics are a million times the magnitude of the nanowings of mechanics.
 * 1%: 9.9503 milliwings   6%: 58.269 milliwings
 * 2%: 19.8026      "     7%: 67.659     "
 * 3%: 29.558     "      8%: 76.691      "
 * 4%: 39.2207    "     9%: 86.178       "
 * 5%: 48.790      "     10%: 95.310       "

The laws of economy apply on macro and micro scales as far a population goes. Thus the quadrant has an infinite density near the origin, and the level curves xy = constant express a reference but are not paths of least distance in Q, as developed below. The hyperbolic angle parametrization of Q complements the level curves and provides a portal to the higher geometry of Q: the rays from (0,0) represent constant slopes as well as constant hyperbolic angles. The half plane (u,v) with positive v, takes the geometric mean, square root of xy, to determine v. The hyperbolic angle of a point in Q corresponds to u in the half plane: u = log √(x/y).

The half plane is a standard model of hyperbolic geometry, often taken together with the disk model and complex numbers, where a linear fractional transformation makes the connection. The geodesics, or shortest-distance curves in the half plane are semicircles with center on the u-axis, or half lines { (x0, y) : y > 0 }, x0 fixed. A point (u,v) in the half plane corresponds to a point in Q given by x = v exp(u) and y = v exp(&minus;u). When the geodesics of the half plane are plotted in Q, an economic scenario of growth and decline is suggested. For instance, the production of a particular product increases as it finds a market, rises to its summit of favorability, then falls off as superseded by its market. Similarly, a corporation starts small, grows with success, as can be traced with historic points in Q. But note that the rays in Q converging at the origin correspond to vertical lines in the half plane that each have a different foot on the u-axis. This feature of the half plane's expression of phenomena in Q allows compatibility of macro and micro market analysis.

Metric geometry
In a tradition begun by Arthur Cayley, the boundary of the half-plane is called the absolute: $$\{(u,\ v): v > 0, \ \ u \in \Reals \} .$$ Semicircles used for geodesics have center on the absolute.

To take meaningful measurements in Q, reference to the half-plane brings a metric that is respected by motions. These are hyperbolic motions corresponding to whichever model of the hyperbolic plane is at hand, in this case the half-plane (and by bijective correspondence, a metric in Q).

The first motion is magnification or contraction about (0,0): (u,v) to (su, sv), where s>0. The second motion is translation along the absolute: (u, v) to (u + t, v). These motions make any semicircle with center on the absolute equivalent to the unit semicircle u u + v v = 1. Indeed, the center can be moved to (0,0) by translation, and the radius normalized by magnification or contraction.

A vertical interval from (u, a) to (u, b), where b > a, is taken to the interval from (su, sa) to (su, sb). The distance formula, or metric, for this interval is log b – log a = log (b/a), which is invariant under the motion. The second type of motion moves the half-plane to the left or right: (u, v) to (u + t, v). Looking a Q, such a motion corresponds to a hyperbolic rotation, a type of linear transformation that preserves difference of hyperbolic angles in Q.

The third hyperbolic motion is an involution and involves inversion in the unit semicircle in the half-plane. Under this inversion the ray (1, tan &theta;), 0 < &theta; < &pi;/2, which is tangent to the unit circle, is taken to the semicircle of radius ½ with center at (½, 0). Since $$1 + \tan^2 \theta = \sec^2 \theta ,$$ the hypotenuse of the right triangle on [0,1] to (1, tan &theta;) has length sec &theta;, so the reciprocal is cos &theta;, leading to the said semicircle. Now any interval on this semicircle corresponds to an interval on the vertical ray, hence a distance is defined on this semicircle. Invoke the second motion to bring the center to (0, 0), and the first motion to double the radius, resulting in the unit semicircle. Thus, any interval on the unit semicircle has a distance by correspondence with the one centered at (½, 0), radius ½. Apparently any semicircle with center on the absolute has a distance formula to apply to intervals.

This hyperbolic geometry features in differential geometry as a surface of negative curvature. It is a saddle surface as every point has an expansive feature not found in Euclidean geometry. The treatment given here with Cartesian geometry and standard trigonometry provides accessibility not requiring complex numbers.