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ch02-functions (2)

ch02-functions (2) - Functions Professor Erkut Ozbay...

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Functions Professor Erkut Ozbay Economics 300
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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
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Is it a function? x y No! Can’t say what y is knowing x
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Examples with one variable: univariate functions
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y = 10 - 2 x 1 2 3 4 5
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2 1 1 2 2 y = 2x 2
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2 1 1 2 3 2 2 2 4 y x x = - -
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1 2 3 4 5 2 25 y x = -
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Multivariate functions (many variables) z = f(x,y) y = f(x 1 ,x 2 ,…,x n ) 1 2 1 n i n i y x x x x = = = + + + L 1 2 1 n i n i y x x x x = = = ��� L
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Consumption as a function of income and wealth: C = 300 + .6I + .02 W
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1/2 1/2 1 2 3 y x x =
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1/2 1/2 1 2 3 y x x = Draw for various y in 2 dimensions Cobb-Douglas function 2 2 1 / 9 x y x = With y fixed, x 2 = f(x 1 )
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Cobb-Douglas in economics 1/2 1/2 1 2 3 y x x = Consumer preferences y = utility • x 1 = pears • x 2 = cheese Indifference curve 1/2 1/2 1 2 3 y x x =
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Cobb-Douglas in economics 1/2 1/2 1 2 3 y x x = Firm production y = output quantity • x 1 = capital • x 2 = labor Isoquant 1/2 1/2 1 2 3 y x x =
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Properties of functions Extreme values: maximum and minimum Limits and continuity Monotonicity: increasing, decreasing Concavity and convexity
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Model of rational behavior People have preferences
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