L8Functions - Introduction to Microeconomics Lecture 8...

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1 Introduction to Microeconomics Lecture 8 Maths: Functions Ian Crawford Outline z Function notation and some common functions z Linear, polynomial z Inverse functions z Composite functions z Exponential and logarithmic functions z Functions of several variables z Homogeneous Functions, and Returns to Scale Functions z y = 5 x – 8 A linear function z y depends on x. We often write y ( x ) to emphasize dependence z C(q) = 3 + 2 q 2 Total cost, C , depends on output, q z For general functions it is common to use f ( x ) in place of y : x is called the argument . z 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 z Ideally we should specify the domain – the set of inputs – and the range (the set of outputs). Usually in economics these are fairly obvious. z 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 z 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 z A monotonic function is either increasing or decreasing Graph of y = 5 x – 8 for x 0 30 40 50 -20 -10 0 10 20 0 2 4 6 8 10 12 x y = 5x - NB: sometimes we are interested in negative values of x and y Graph of C = 3 + 2 q 2 for q 0 150 200 250 0 50 100 0 2 4 6 8 10 12 q C Polynomials and related functions z A polynomial function takes the form f ( x ) = a n x n + a n -1 x n -1 + …+ a 0 z A linear function is a polynomial of degree 1 z A quadratic function is a polynomial of degree 2 z E.g. the cost function: C = 10 + q + 3 q 2 z A useful function is f ( x ) = x n for x > 0 z If n is an integer this is a polynomial of degree n z n can also be negative or a fraction z Q = p -2 is a demand function with n = 2 z f ( x ) = x 0.3 may represent a “production function” with x
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