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Unformatted text preview: unction Function • mapping from each x in X to some y in Y A mapping from each x in X to some y in Y f:X g111 Y omain is X; Range is 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 it a function? 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? = 10  x y 10 2 x 10 8 6 4 2 1 2 3 4 5 y = 2x 2 8 6 4 2 g16 2 g16 1 1 2 2 2 2 4 y x x g32 g16 g16 8 6 4 2 g16 2 g16 1 1 2 3 g16 2 g16 4 2 25 y x g32 g16 5 4 3 2 1 1 2 3 4 5 Multivariate functions (many variables) • = f(x,y) z f(x,y) • y = f(x 1 ,x 2 ,…,x n ) 1 2 n i n y x x x x g32 g32 g32 g14 g14 g14 g166 g34 1 i n 1 2 1 i n i y x x x x g32 g32 g32 g152 g152 g152 g150 g34 Consumption as a function of income and wealth: C = 300 + .6I + .02 W 1/2 1/2 1 2 3 y x x g32 2 1/2 Draw for various y in 2 dimensions CobbDouglas 1/2 1/2 1 2 3 y x x g32 function 2 y x With y fixed, x 2 = f(x 1 ) 2 1 / 9 x y x g32 CobbDouglas in economics onsumer preferences 1/2 1/2 1 2 3 y x x g32 1/2 1/2 1 2 3 y x x g32 Consumer preferences • y = utility • pears x 1 = pears • x 2 = cheese difference curve • Indifference curve CobbDouglas in economics irm production 1/2 1/2 1 2 3 y x x g32 1/2 1/2 1 2 3 y x x g32 Firm production • y = output quantity • capital x 1 = capital • x 2 = labor oquant • Isoquant roperties of functions Properties of functions • xtreme values: maximum and minimum Extreme values: maximum and minimum • Limits and continuity • Monotonicity: increasing, decreasing • Concavity and convexity odel of rational behavior Model of rational behavior • eople have preferences 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 xtreme Values Extreme Values Global maximum ocal Local maximum Local minimum it a continuous function? Is it a continuous function?...
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 Spring '08
 Hayes
 Derivative, Monotonic function, Convex function, average rate, 0E

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