Economicae^©

an illustrated encyclopedia of economics

Famous Economists

Mathematics of Economics

		Mathematics of Economics
arc elasticity:	The calculation of arc elasticity avoids the problem of ambiguity about bases inherent in measures of point elasticity by taking the means of the variables being assessed for their relative responsiveness. For example, the arc formula for the price elasticity of demand (or supply) is calculated as: [(Q1-Q2) ∕ ((Q1+Q2) ∕ 2)] ∕ [(P1-P2) ∕ (P1+P2) ∕ 2)] See also elasticity, mid-point bases, and simplified arc formula for elasticity.
axis:	An axis is one of the intersecting 90° lines used to measure how variables are related in a Cartesian coordinate system. The x-axis in a two-dimensional Cartesian coordinate system measures the variable laid out horizontally and the Y-axis measures the variable laid out vertically. The plural of axis is axes. See the Cartesian coordinate system figure for more discussion.
cardinal measurement:	A variable is cardinally measurable if a given interval between measures has a consistent meaning, i.e., if the measure of the variable corresponds to fixed-interval points along a line. For example, height, output, and income are cardinally measurable. Open the cardinal vs. ordinal file for more discussion, or see also ordinal measurement.
Cartesian coordinate:	Cartesian coordinates are an ordered set of numbers (x,y) that identifies how variables may be related graphically along a horizontal x-axis and a vertical y-axis. Open the figure for the Cartesian coordinate system for more discussion.
causation:	The relationship whereby one variable (the independent variable) is not only highly correlated with another variable (the dependent variable), but actually causes changes in the independent variable. In 2003, C.W.J. Granger [1934- ] was awarded a Nobel Prize, in part because of his development of time-series statistical methods for ascertaining causation.
confidence interval:	In statistical models (including econometric models), a confidence interval identifies a range of values in which the unknown population parameter (μ) is likely to be found with a certain percentage of confidence. The interval is calculated from a set of sample data. Confidence intervals are usually calculated at the 90% or 95% or 98% or 99% levels. The statistics used to calculate a confidence interval are: (a) the sample mean (M), (b) the value of Ζ (your chosen percentage level of confidence), and (c) the standard error of the mean (σM). Solving, the confidence interval can be expressed as M – Ζ σM ≤ μ ≤ M + Ζ σM.
determinant:	A determinant is a variable that influences or affects another variable.
determinate:	A system of equations is said to be determinate if the equations yield a solution.
dimensionless:	A function is dimensionless when the units of measure do not alter results. The price elasticity of demand, for example, is dimensionless because whether the quantity demanded is measured in pounds, grams, or tons has no effect on the calculation, nor does whether the price is measured in US cents, US dollars, or euros. See also zero degree homogeneity.
discounted present value:	See present value.
elasticity:	Elasticity is a measure of the sensitivity of one variable relative to some other variable. See also arc elasticity, income elasticity of demand, price elasticity of demand, and price elasticity of supply.
flow variable:	A flow variable is meaningful only if measured over a period of time. Income and production are examples. See also stock variables.
functions:	Functions are relationships among variables. Functions that relate two variables may be positive (e.g., how caloric intake and weight are related), negative (e.g., how marijuana smoking affects income), or variable (e.g., how physical strength varies with age). For examples of economic functions, see demand function or supply function.
Gini coefficient:	A Gini coefficient is a summary measure of how unequally distributed one variable is related to another and is a number between 0 and 1, where perfect equality has a Gini coefficient of zero, and absolute inequality yields a Gini coefficient of 1. The Gini coefficient is calculated from the relative areas under a Lorenz curve. Open the Lorenz curve link for a graph and more discussion
Herfindahl-Hirschman Index:	The Herfindahl-Hirschman Index (HHI) is the sum of the squares of the market shares of the firms in an industry. HHIs are now used as a guideline for antitrust actions.
homogeneity of degree n:	A function is homogeneous of degree n if multiplication of all elements of the functions by a constant scalar α yields an increase in the value of the function by a^n. Thus: a^n ¦(x1, x2,..., xn) = ¦(ax1, ax2, .... , axn). All homogeneous functions are also homothetic, in that all their level sets are uniform radial expansions of each other.
homogeneity of degree one:	A function is linearly homogeneous (homogeneous of degree one) if multiplication of all elements of the functions by a constant scalar α yields an increase in the value of the function by a. Thus: a¦(x1, x2,..., xn) = ¦(ax1, ax2, .... , axn). See also linearly homogeneous function.
homogeneity of degree zero:	A function is homogeneous of degree zero if multiplication of all elements of the functions by a constant scalar α yields an increase in the value of the function by a^0 = 1. Thus: a^0¦(x1, x2, ... , xn) = ¦(ax1, ax2, .... , axn), and the function is unaffected if the variables x1, x2, ... , xn are all simultaneously changed by any constant factor a because a^0 = 1.
horizontal summation:	The process of summing variables along the x-axis (e.g., the various quantities of a good that people will buy or sell) for each value measured along the y-axis (e.g., each possible price for that good).
index numbers:	Index numbers are summary measure used to make relative comparisons of a specific variable between time periods.
indifference curve:	A line connecting the various combinations of two goods that yield the same total utility; the consumer is indifferent among the various bundles of goods along an indifference curve.
inflection point:	In inflection point is a point in a continuous function where the second derivative changes sign.
isocost:	An isocost is a line connecting all identical levels of expenditures or costs. A consumer isocost is called a budget line.
isoquant:	An isoquant is a line on the surface of a production function that connects all identical levels of output.
Lerner index of monopoly power:	The Lerner index of monopoly power (LMP) is an estimate of market power proposed by Abba Lerner based on the proportion by which the price of output exceeds marginal cost. Market power is then measured as: (P ‑ MC) ∕ P.
linear relationship:	A linear relationship is describable by an equation Y = mX + b, where m is the slope of a straight line and b is the intercept of the line in a Cartesian space.
nonlinear relationship:	A nonlinear relationship is a relationship for which neither the slope nor the intercept of a tangent to a curve are constant.
optimization:	Optimization is the process of maximizing any constrained function. Most economists assume that human decisions entail optimization wherein a goal [e.g., satisfaction or profit] is maximized subject to the opportunity set – constraints on the opportunities available [e.g., an individual’s budget, the competitive environment facing a firm, and limited information and time are all constraints.] Optimization for the individual (satisfaction or utility) or firm (profit) occurs requires the marginal personal benefits and the marginal personal costs to be equated (MB=MC). Optimization for the society at large requires the marginal social benefits and marginal social costs of every activity to be equated (MSB=MSC). If MB ≠ MSB or if MC ≠ MSC, then private decisionmaking results in a market failure. See also law of equal marginal advantage.
ordinal measurement:	A variable is ordinally measurable if ranking is possible for values of the variable. For example, a gold medal reflects superior performance to a silver or bronze medal in the Olympics, or you may prefer French toast to waffles, and waffles to oat bran muffins. All variables that are cardinally measurable are also ordinally measurable, although the reverse may not be true. Open the cardinal vs. ordinal file for more discussion, or see also cardinal measurement.
payoff matrix:	In game theory, a payoff matrix is a table that matches sets of gains or losses when “players” choose from the options available to them. The payoff to any player from selecting a particular option depends on the option(s) selected by other players.
preference function:	A preference function is a foundation for explaining consumer decisions. Individuals are assumed capable of identifying which one of any two bundles (sets) of goods they prefer, or whether they are indifferent between the two bundles. See the link to preference functions for more discussion.
present value:	The present value of any asset is the value now of the income stream expected from the asset, discounted by the interest rate; the demand price of the asset.
random walk:	A variable follows a “random walk” when the direction or magnitude of change cannot be predicted from past patterns. The random walk hypothesis for financial assets is based in the theory of efficient markets.
rise ∕ run:	“Rise-over-run” is a shorthand formula for slope. Divide the rise (change in altitude) by the run (change in horizontal distance), and you’ve computed the slope of a function. See also slope.
risk:	Risk refers to the statistical distribution of alternative outcomes from an action, and is usually estimated by the variance, standard deviation, etc., of the possible outcomes (e.g., the probability of death plus the probability of living = 1.) If the probabilities of alternative outcomes are reasonably well known, a probability function can be constructed. In investment analysis, risk is commonly measured as the standard deviation of the expected return on total investment. Another example: An insurance actuary can estimate risks with reasonable precision if significant amounts of historical data are available, and then ascertain appropriate insurance premiums for bearing risk. See the link for risk and uncertainty for more discussion. See also certainty, uncertainty, and Knightian uncertainty.
simplified arc formula for elasticity:	Finding the simplified arc formula for elasticity entails simplifying the mathematical formula used to calculate arc elasticity. For example, the formula for the price elasticity of supply [(Q1-Q2) ∕ ((Q1+Q2) ∕ 2)] [(P1-P2) ∕ (P1+P2) ∕ 2)] simplifies to [(Q1-Q2) x (P1+P2)] [(Q1+Q2) x (P1-P2)]. See also elasticity, point elasticity, mid-point bases, and arc formula for elasticity.
Y-axis:	The Y-axis is the vertical line along which one variable is measured in a 2-dimensional system of Cartesian coordinates. See the Cartesian coordinate system figure for more discussion.
Y-intercept:	The Y-intercept is the value of b in the formula of linear equation Y = mX + b. In a 2-dimension Cartesian space, the slope of the straight line is m, and b is the value of the variable on the Y-axis when the variable on the X-axis is zero. See the Cartesian coordinate system figure for more discussion.
zero degree homogeneity:	A function has elements of zero degree homogeneity if multiplication of a set of the independent variables in the function by any scalar (constant) results in no change in the dependent variable. For example, the neoclassical conclusion that money is neutral in the long run is equivalent to an assertion that functions to maximize utility and profit are homogeneous of degree zero in monetary prices. If all monetary prices are multiplied by any positive number, the behavior that maximizes utility and “real” profit is assumed unaffected. See also homogeneity of degree zero.
zero sum game:	A zero sum game is any interaction in which benefits experienced by any party are precisely offset by losses to some of the other parties. Gambling is an example, in that any winnings by some players are necessarily equal to the losses of other players. (In the case of casino gambling, the casino and its employees could be considered “players.”)