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Functions Professor Peter Cramton Economics 300 Function 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 y Is it a function y x No Can t say what y is knowing x Examples with one variable univariate functions What economic relationships might these functions describe y 10 2 x 10 8 6 4 2 1 2 3 4 5 y 2x2 8 6 4 2 2 1 1 2 y 2x 2x 4 2 8 6 4 2 2 1 1 2 4 2 3 y 25 x 2 5 4 3 2 1 1 2 3 4 5 Multivariate functions many variables z f x y y f x1 x2 xn n y xi x1 x2 i 1 n y xi x1 x2 i 1 xn xn Consumption as a function of income and wealth C 300 6I 02 W y 3x 1 2 1 2 1 2 x Draw for various y in 2 dimensions y 3x 1 2 1 2 1 2 x Cobb Douglas function With y fixed x2 f x1 2 2 1 x y 9x Cobb Douglas in economics y 3x 1 2 1 2 1 2 y 3x11 2 x21 2 x Consumer preferences y utility x1 pears x2 cheese Indifference curve Cobb Douglas in economics y 3x 1 2 1 2 1 2 y 3x11 2 x21 2 x Firm production y output quantity x1 capital x2 labor Isoquant Properties of functions Extreme values maximum and minimum Limits and continuity Monotonicity increasing decreasing Concavity and convexity Model 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 functions Extreme Values Global maximum Local maximum Local minimum Is it a continuous function y x 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 lim f x x a Right hand limit lim f x Value of function as approach a from right x a Limit Value of function as approach a in any direction Requires lim f x x a lim f x x a lim f x x a Continuity f x is continuous at a if lim f x lim f x f a x a x a f x is a continuous function if it is continuous at all points x in domain X Is it a continuous function No lim f x f a x a No No f 3 lim f x lim f x not x a x a defined 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 decreasing Monotonic functions Strictly monotonic functions have inverses Average rate of change How quickly does f x change as x goes from a to b y f b f a x b a Average rate of change is constant for a linear function y f b f a slope x b a 8 7 Slope rise run In this case 1 2 6 5 4 rise 1 3 2 run 2 1 0 1 2 3 4 5 6 7 8 Average rate of change is slope of secant line y f b f a x b a A x from 0 to 3 3 0 y 1 x 3 0 1 3 2 1 3 2 B x from 0 to 6 6 0 y 2 x 6 0 1 3 2 1 3 2 Convexity Concave Convex Why economists care Diminishing marginal utility 4 3 2 1 5 10 15 Why economists care Increasing marginal costs 15 10 5 1 2 3 4 Equivalent 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 f a 1 b f a 1 f b A convex function 1 3 f a 23 f b f a f b f 13 a 23 b Strictly 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 f a 1 b f a 1 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 f a 1 b f a 1 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 f a 1 b f a 1 f b Some Logic Necessary 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 animal Necessary 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 linear Is a line convex or concave Linear convex Linear concave Therefore linear convex concave Does convex concave linear Concave f a 1 b f a 1 f b Convex f a 1 b f a 1 f b Both f a 1 b f a 1 f b Therefore linear convex concave Useful functions Power functions f x kx p Polynomial functions f x a0 a1 x a2 x 2 Exponential functions f x kb x an x n Power function even exponent f x kx p Power function odd exponent f x kx p Power function negative exponent f x kx p Rules of Exponents Rule Example 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 x1 a a x 20 1 21 2 2 1 1 2 21 3 23 1 8 22 23 25 32 23 22 21 2 22 32 62 36 42 22 4 2 2 4 9 1 2 9 3 Polynomial functions Linear 1st order f x a0 a1 …


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UMD ECON 300 - Functions

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