L8Functions - Introduction to Microeconomics Lecture 8...

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Introduction to Microeconomics Lecture 8 Maths: Functions Ian Crawford
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Outline Function notation and some common functions Linear, polynomial Inverse functions Composite functions Exponential and logarithmic functions Functions of several variables Homogeneous Functions, and Returns to Scale
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Functions y = 5 x – 8 A linear function y depends on x. We often write y ( x ) to emphasize dependence C(q) = 3 + 2 q 2 Total cost, C , depends on output, q For general functions it is common to use f ( x ) in place of y : x is called the argument . The key feature of a function is that for each value of x , the input, there is a unique value of f ( x ), the output Ideally we should specify the domain – the set of inputs – and the range (the set of outputs). Usually in economics these are fairly obvious. If the graph of f ( x ) is upward-sloping then f ( x ) is an increasing function (or a “ monotonic increasing function ”), e.g. many supply functions If the graph of f ( x ) is downward-sloping the f ( x ) is a decreasing function (or a “ monotonic decreasing function ”), e.g. most demand functions A monotonic function is either increasing or decreasing
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Graph of y = 5 x – 8 for x 0 -20 -10 0 10 20 30 40 50 0 2 4 6 8 10 12 x y = 5x - 8 NB: sometimes we are interested in negative values of x and y
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Graph of C = 3 + 2 q 2 for q 0 0 50 100 150 200 250 0 2 4 6 8 10 12 q C
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Polynomials and related functions A polynomial function takes the form f ( x ) = a n x n + a n -1 x n -1 + …+ a 0 A linear function is a polynomial of degree 1 A quadratic function is a polynomial of degree 2 E.g. the cost function: C = 10 + q + 3 q 2 A useful function is f ( x ) = x n for x > 0 If n is an integer this is a polynomial of degree n n can also be negative or a fraction
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This note was uploaded on 04/12/2010 for the course ECON DEAM taught by Professor Vines during the Spring '10 term at Oxford University.

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L8Functions - Introduction to Microeconomics Lecture 8...

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