Functions Professor Peter Cramton Economics 300Function • A mapping from each x in X to some y in Y f:XY Domain is X; Range is Y • y = f(x) • Shows how y depends on x • Each x maps into one y – If I know x, I can determine yIs it a function? x y No! Can’t say what y is knowing xExamples with one variable: univariate functions What economic relationships might these functions describe?y = 10 - 2 x 1234524681021122468y = 2x22112342246822 2 4y x x  1234512345225yxMultivariate functions (many variables) • z = f(x,y) • y = f(x1,x2,…,xn) 121niniy x x x x    121niniy x x x x    Consumption as a function of income and wealth: C = 300 + .6I + .02 W1/2 1/2123y x x1/2 1/2123y x xDraw for various y in 2 dimensions Cobb-Douglas function 221/9x y xWith y fixed, x2 = f(x1)Cobb-Douglas in economics 1/2 1/2123y x xConsumer preferences • y = utility • x1 = pears • x2 = cheese • Indifference curve 1/2 1/2123y x xCobb-Douglas in economics 1/2 1/2123y x xFirm production • y = output quantity • x1 = capital • x2 = labor • Isoquant 1/2 1/2123y x xProperties of functions • Extreme values: maximum and minimum • Limits and continuity • Monotonicity: increasing, decreasing • Concavity and convexityModel of rational behavior • People have preferences • People seek to make optimal decisions – Maximize utility – Maximize profit – Minimize cost • Use optimization to find extreme values – Maxima and minima of functionsExtreme Values Local maximum Local minimum Global maximumIs it a continuous function? x y Yes! Can draw it without lifting pen.Is it a continuous function? No! No! No!Limits • Left-hand limit – Value of function as approach a from left • Right-hand limit – Value of function as approach a from right • Limit – Value of function as approach a in any direction – Requires lim ( )xafxlim ( )xafxlim ( )xafxlim ( ) lim ( )x a x af x f xContinuity • f(x) is continuous at a if • f(x) is a continuous function if it is continuous at all points x in domain X lim ( ) lim ( ) ( )x a x af x f x f aIs it a continuous function? No! No! No! f(3) not defined lim ( ) ( )xaf x f alim ( ) lim ( )x a x af x f xMonotonicity • If for all a,b in X with a < b, f(a)  f(b) then f is increasing f(a)  f(b) then f is decreasing f(a) < f(b) then f is strictly increasing f(a) > f(b) then f is strictly decreasing • f is monotonic if it is increasing or decreasing • f is strictly monotonic if it is strictly increasing or strictly decreasingMonotonic functionsStrictly monotonic functions have inversesAverage rate of change • How quickly does f(x) change as x goes from a to b? ( ) ( )y f b f ax b a0 1 1 3 4 2 8 5 6 7 2 3 4 5 6 7 8 “run” ( = 2) “rise” ( = 1) Slope = rise / run (In this case = 1/2) Average rate of change is constant for a linear function ( ) ( )slopey f b f ax b aAverage rate of change is slope of secant line ( ) ( )y f b f ax b a22113360260yx22113330130yxB: x from 0 to 6 A: x from 0 to 3Convexity • Concave • ConvexWhy economists care? • Diminishing marginal utility 510151234Why economists care? • Increasing marginal costs 123451015Equivalent definitions of convex • f is convex if average rate of change is increasing • f is convex if f is at or below all secant lines • f is convex if for all a,b in X and  in (0,1), ( (1 ) ) ( ) (1 ) ( )f a b f a f b       A convex function 1233()f a b1233( ) ( )f a f b()fa()fbStrictly convex • f is strictly convex if average rate of change is strictly increasing • f is strictly convex if f is below all secant lines • f is strictly convex if for all ab in X and  in (0,1), ( (1 ) ) ( ) (1 ) ( )f a b f a f b       Concave • f is concave if average rate of change is decreasing • f is concave if f is at or above all secant lines • f is concave if for all a,b in X and  in (0,1), ( (1 ) ) ( ) (1 ) ( )f a b f a f b       Strictly concave • f is strictly concave if average rate of change is strictly decreasing • f is strictly concave if f is above all secant lines • f is strictly concave if for all ab in X and  in (0,1), ( (1 ) ) ( ) (1 ) ( )f a b f a f b       Some LogicNecessary and sufficient conditions All are equivalent • If P then Q • P  Q (P implies Q) • P only if Q • P is sufficient for Q • Q is necessary for P P dog Q  animalNecessary and sufficient conditions All are equivalent • If P then Q & if Q then P • P  Q & Q  P • P  Q (P equivalent to Q) • P  Q (P if and only if Q) • P is necessary and sufficient for Q • Q is necessary and sufficient for P P  f has constant rate of change Q  f is linearIs a line convex or concave? • Linear  convex • Linear  concave • Therefore, linear  convex & concave • Does convex & concave  linear? • Concave  • Convex  • Both  • Therefore, linear  convex & concave ( (1 ) ) ( ) (1 ) ( )( (1 ) ) ( ) (1 ) ( )( (1 ) ) ( ) (1 ) ( )f a b f a f bf a b f a f bf a b f a f b                     Useful functions • Power functions • Polynomial functions • Exponential functions ()pf x kx20 1 2()nnf x a a x a x a x    ()xf x kbPower function (even exponent) ()pf x kxPower function (odd exponent) ()pf x kxPower function (negative exponent) ()pf x kxRules of Exponents • x0 = 1 • x1 = x • x-1 = 1 / x • (xa ) b = (xb ) a = xab • xa x b = xa+b • xa / x b = xa – b • xa y a = (xy)a • xa / y a = (x/y)a …