Econ 300_Summer 2009_Slides 1-Functions[1]

Econ 300_Summer 2009_Slides 1-Functions[1] - unction...

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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! Cant 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 Cobb-Douglas 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 Cobb-Douglas 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 Cobb-Douglas 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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Econ 300_Summer 2009_Slides 1-Functions[1] - unction...

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