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# Inter Microecon ECON 3020

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This 4 page Class Notes was uploaded by Hallie Kuphal on Wednesday October 28, 2015. The Class Notes belongs to ECON 3020 at University of Wyoming taught by Abul Hussain in Fall. Since its upload, it has received 15 views. For similar materials see /class/230357/econ-3020-university-of-wyoming in Economcs at University of Wyoming.

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Date Created: 10/28/15

ECON 3020 INTERMEDIATE MICROECONOMICS HUSSAIN FALL 2008 MATH REVIEW I FUNCTIONS AND CALCULUS This set of reviewsnotes covers topics from the CALCULUS APPENDIX at the end of the textbook pages CA l through CA 30 You must read these notes in conjunction with the textbook calculus appendix and also the MATH REVIEW II notes FUNCTION A function is a form of relationship between two variables x and y such that for each x value there is one unique y value In other words each x value uniquely determines one y value A function can also be thought of as a mapping or a transformation from one variable x to another variable y A function f can then be taken as the rule of transformation from variable x to variable y And this rule of transformation can take various forms e g linear non linear exponential etc Some examples of functions include 1 y f X X 3 Identity function 2 y f X 0 3 Zero function 3 y f X J 3 Square root function 4 y f X mx c 3 Linear function 5 y f X 9X 3 Exponential function 6 C F 32 g 3 Temperature conversion equation For a function f x is known as the independent variable and y as the dependent variable The set of all possible values that x can take is known as the domain of the function and the resultant set of y values is known as the range of the function The above functions are also examples of explicit functions since variable y is always explicitly defined as function of variable x in each of the above functions functions 1 through 6 In cases of implicit functions on the other hand the functional relationship is not explicit Examples of an implicit function will be 7 gXy ozy 3 ij4 63 dy2 eyX0 8 gXy y 0 9 gXy y eX 0 In all of the functions above 7 through 9 the functional relationship between is not explicitly defined In case of functions 8 and 9 we can very easily transform them into explicit functions But what about function 7 Page 1 PROPERTIES OF FUNCTIONS Some of the important properties of functions include Monotonicity A monotonic function is one that is always increasing or decreasing The function y x is a monotomcally increasing function and the function y is a x monotonically decreasing function Graph these two functions for better understanding Continuity A function without any jumps kinks discontinuities or breaks is said to be continuous Think of a function which is smooth over all possible values of x as opposed to breaks or steps at some values of x Slope The slope of a function measures the degree of steepness of a function More importantly slope measures the ratio of change in the y variable per unit change in the x variable 7 the concept of slope is extremely relevant for economics The formula for calculating slopes is given by Rise yl yz 51 ape Run Xl XZ 1 Concavity and convexity Concavity and convexity measures the rate of change of the slope of a function Alternatively concavity and convexity measures the curvature of a function 7 again the concepts of concavity and convexity is fundamentally important in economics How do we calculate the rate of change of slope of a function 0 Homogeneity A function f x y is said to be homogeneous of degree n if fa xa y aquot fxy for any a gt 0 How do we check for degree of homogeneity MULTIVARIABLE FUNCTIONS Multi variable functions are functions that define a rule of transformation or mapping from many more than one variables to a single variable Some examples include 1 ZfXyXy 2 UUxyAnyZ 3 Area BaseHeight In all of the above examples functions 1 2 3 and 4 two variables together determine one unique value of a third variable Page 2 DERIVATIVES The derivative of a function f X at a specific point gives us the slope of that function at that specific point The derivative of function f X will generally be denoted by f 39 X or M dX 01f or a df X 0 If d gt 0 at any pomt then the function f X is increasing at that pomt X df X 0 If d lt 0 at any pomt then the function f X is decreasing at that pomt X df X 0 If d 0 at any pomt then the function f X is stationary at that pomt ie neither X increasing nor decreasing 0 Derivatives apply to single variable functions only 0 There are numerous rules for computing derivatives and these rules have been discussed in MATH REVIEW 11 PARTIAL DERIVATIVES In case of a function with several variables a partial derivative lets us know how the value of the function changes as we change only one variable while holding all the variables constant Consider a function y f Xz the total change in y due to changes in both x and z is the sum of individual changes due to x and z And we will have two partial derivatives is the partial derivative of y with respect to x 7 it is the change in y due to a small X d change in x while z remains constant This partial derivative is denoted by fX or simply 0 is the partial derivative of y with respect to z 7 it is the change in y due to a small 2 d change in z while x remains constant This partial derivative is denoted by fz or simply 3 Rules and formulas for computing partial derivatives are same as the ones for computing simple derivatives Page 3 OPTIMIZATION FINDING MAXIMUM OR MINIMUM Consider a single variable function y f X 2 0 The function reaches an optimum can be both a maximum or a minimum when the slope of the function is zero In other words dfX f 39 X d 0 3 This is known as the first order condition x 0 The value of x which satisfies the above condition is called the critical value or stationary value The critical value is commonly denoted by x 0 The first order condition test leads us to either maximum or minimum 7 how do we distinguish between a maximum and a minimum For this we use the second order condition 0 The set of second order conditions are given by f xgt0 3 Minimumatxx f xlt0 3 Maximum atxx Now consider a multi variable function y Xz 0 The first order condition is given by fX 0 and fl 0 0 The second order conditions evaluated at the optimum ie 232quot are given by Maximum at 262quot Minimum at 232quot fXXlt0andfzzlt0 fXXgt0andfzzgt0 fxxfwfxyfyxgt0 fxxfwfxyfyxgt0 0 The first and second order conditions are very often used in economics e g in utility maximization profit maximization and cost minimization problems FURTHER READING IF REQUIRED Chiang Alpha C 1984 Fundamental Methods of Mathematical Economics Third Edition Singapore McGraw Hill Book Company Page 4

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