2. fall ‘ 97Principles of MicroeconomicsSlide 1 This is a PowerPoint presentation on fundamental math tools that are useful in principles of economics. A left mouse click or the enter key will add an element to a slide or move you to the next slide. The backspace key will take you back one element or slide. The escape key will get you out of the presentation.  R. Larry Reynolds (Boise State University)

3. fall ‘ 97Principles of MicroeconomicsSlide 2 Math Review ·Mathematics is a very precise language that is useful to express the relationships between related variables ·Economics is the study of the relationships between resources and the alternative outputs ·Therefore, math is a useful tool to express economic relationships

4. fall ‘ 97Principles of MicroeconomicsSlide 3 Relationships ·A relationship between two or more variables can be expressed as an equation, table or graph ·equations & graphs are “continuous” ·tables contain “discrete” information ·tables are less complete than equations ·it is more difficult to see patterns in tabular data than it is with a graph -- economists prefer equations and graphs

5. fall ‘ 97Principles of MicroeconomicsSlide 4 Equations ·a relationship between two variables can be expressed as an equation ·the value of the “dependent variable” is determined by the equation and the value of the “independent variable.” ·the value of the independent variable is determined outside the equation, i.e. it is “exogenous”

6. fall ‘ 97Principles of MicroeconomicsSlide 5 Equations [cont...] ·An equation is a statement about a relationship between two or more variables ·Y = f i (X) says the value of Y is determined by the value of X; Y is a “function of X.” ·Y is the dependent variable ·X is the independent variable ·A linear relationship may be specified: Y = a  mX [the function will graph as a straight line] ·When X = 0, then Y is “ a” ·for every 1 unit change in X, Y changes by “  m”

7. fall ‘ 97Principles of MicroeconomicsSlide 6 Y = 6 - 2X ·The relationship between Y and X is determined; for each value of X there is one and only one value of Y [function] ·Substitute a value of X into the equation to determine the value of Y ·Values of X and Y may be positive or negative, for many uses in economics the values are positive [we use the NE quadrant]

8. fall ‘ 97Principles of MicroeconomicsSlide 7 Equations -- Graphs [Cartesian system] The X axis [horizontal] The Y axis [vertical] The North East Quadrant (NE), where X > 0, Y > 0 {both X and Y are positive numbers} X > 0 +1+2+3 Y>0 +1 +2 +3 (X,Y) where X 0 X<0 -3-2 (X,Y) where X<0 and Y<0 Y<0 -2 -3 (X,Y) where X>0 and Y<0 (Left click mouse to add material)

9. fall ‘ 97Principles of MicroeconomicsSlide 8 When the values of the independent and dependent variables are positive, we use the North East quadrant 123456 1 2 3 5 6 (X, Y) (3, 5) Go to the right {+3} units and up {+5} units! (1,6) Right {+1} one and up {+6} six (5, 1) Right 5 and up 1 (2.5, 3.2) to the right 2.5 units and up 3.2 units (Left click mouse to add material)

10. fall ‘ 97Principles of MicroeconomicsSlide 9 Given the relationship, Y = 6 - 2X, 123456 1 2 3 4 5 6 sets of (X, Y) (0, 6) when X = 0 then Y = 6 [this is Y-intercept] (1, 4) when X = 1 then Y = 4 (2, 2) When X = 2, then Y = 2 (3, 0) When X = 3, Y = 0, [this is X-intercept] The relationship for all positive values of X and Y can be illustrated by the line AB A B A line that slopes from upper left to lower right represents an inverse or negative relationship, when the value of X increases, Y decreases! X Y (Left click mouse to add material)

11. fall ‘ 97Principles of MicroeconomicsSlide 10 123456 1 2 3 4 5 6 Y X Given a relationship, Y = 6 -.5X (0,6) (1,5.5) (2, 5) (4,4) (6,3) For every one unit increase in the value of X, Y decreases by one half unit. The slope of this function is -.5! The Y-intercept is 6. What is the X-intercept? (Left click mouse to add material)

12. fall ‘ 97Principles of MicroeconomicsSlide 11 123456 1 2 3 4 5 6 Y X For a relationship, Y = 1 + 2X When X=0, Y=1 (0,1) When X = 1, Y = 3 (1,3) When X = 2, Y = 5 (2,5) This function illustrates a positive relationship between X and Y. For every one unit increase in X, Y increases by 2 ! run +1 rise +2 slope = +2 for a relationship Y = -1 +.5X This function shows that for a 1 unit increase in X, Y increases one half unit run +2 rise +1 slope = + 1212 (Left click mouse to add material)

13. fall ‘ 97Principles of MicroeconomicsSlide 12 Problem ·Graph the equation: Y = 9 - 3X ·What is the Y intercept? The slope? ·What is the X intercept? Is this a positive (direct) relationship or negative (inverse)? ·Graph the equation Y = -5 + 2X ·What is the Y intercept? The slope? ·What is the X intercept? Is this a positive (direct) relationship or negative (inverse)?

14. fall ‘ 97Principles of MicroeconomicsSlide 13 Equations in Economics ·The quantity [Q] of a good that a person will buy is determined partly by the price [P] of the good. [Note that there are other factors that determine Q.] ·Q is a function of P, given a Price the quantity of goods purchased is determined. Q = fp (P) ·A function is relationship between two sets in which there is one and only one element in the second set determined by each element in the first set.

15. fall ‘ 97Principles of MicroeconomicsSlide 14 Relationship [cont... ] ·Q = f p (P) {Q is a function of P} ·Example: Q = 220 - 5P ·If P = 0, then Q = 220 ·If P = 1, then Q = 215 ·for each one unit increase in the value of P, the value of Q decreases by 5

16. fall ‘ 97Principles of MicroeconomicsSlide 15 Q = 220 - 5P ·This is an inverse or negative relationship ·as the value of P increases, the value of Q decreases ·the “Y intercept” is 220, this is the value of Q when; P = 0 ·the “X intercept” is 44, this is the value of P when Q = 0 ·This is a “linear function,” i.e. a straight line ·The “slope” of the function is -5 ·for every 1 unit change in P, Q changes by 5 in the opposite direction

17. fall ‘ 97Principles of MicroeconomicsSlide 16 The equation provides the information to construct a table. However, it is not possible to make a table to include every possible value of P. The table contains “discrete” data and does not show all possible values!

18. fall ‘ 97Principles of MicroeconomicsSlide 17 For the relationship, Q = 220 - 5P, the relationship can be graphed... $5 10 15 20 25 30 35 40 45 50 55 PRICE QUANTITY 44 4080120160200240280 When the price is $44, 0 unit will be bought; at a price of $0, 220 units will be bought. Demand Notice that we have drawn the graph “backwards,” P{independent} variable is placed on the Y-axis. This is done because we eventually want to put supply on the same graph and one or the other must be reversed! Sorry! 70 At P=$30, Q = 70 170 At a price of $10, the the quantity is (Left click mouse to add material)

19. fall ‘ 97Principles of MicroeconomicsSlide 18 Slopes and Shifts ·Economists are interested in how one variable {the independent} “causes” changes in another variable {the dependent} ·this is measured by the slope of the function ·Economists are also interested in changes in the relationship between the variables ·this is measured by “shifts” of the function

20. fall ‘ 97Principles of MicroeconomicsSlide 19 Slope of a function or “line” ·The slope measures the change in the dependent variable that will be “caused” by a change in the independent variable ·When, Y = a  m X; m is the slope

21. fall ‘ 97Principles of MicroeconomicsSlide 20 123456 1 2 3 4 5 6 Y X Y = 6 -.5X as the value of X increases from 2 to 4,  X = 2 the value of Y decreases from 5 to 4  Y= -1  X is the run {+2},  Y is the rise [or change in Y caused by  X] {in this case, -1} slope is rise run so, slope is -1/2 or -.5 Slope of a Line

22. fall ‘ 97Principles of MicroeconomicsSlide 21 Shifts of function ·When the relationship between two variables changes, the function or line “shifts” ·This shift is caused by a change in some variable not included in the equation ·[the equation is a polynomial] ·A shift of the function will change the intercepts [and in some cases the slope]

23. fall ‘ 97Principles of MicroeconomicsSlide 22 123456 1 2 3 4 5 6 Y X Given the function Y = 6 -.5X, A decrease in the function would be Y’ = 4 -.5X shifts left Shifts right an increase in the function would represent an increase in the intercept [from 6 to a larger number] the function shifts and its slope also changes Just the slope changes {in this case, an increase in the absolute value of.5 to -1.8} Y” = 6 - 1.8X [x intercept = 3.3] (Left click mouse to add material)

24. fall ‘ 97Principles of MicroeconomicsSlide 23 Shifts in functions ·In Principles of Economics most functions are graphed in 2-dimensions, this means we have 2 variables. [The dependent and independent] ·Most dependent variables are determined by several or many variables, this requires polynomials to express the relationships ·a change in one of these variables which is not shown on a 2-D graph causes the function to “shift”

25. fall ‘ 97Principles of MicroeconomicsSlide 24 Slope and Production ·The output of a good is determined by the amounts of inputs and technology used in production ·example of a case where land is fixed and fertilizer is added to the production of tomatoes. ·with no fertilizer some tomatoes, too much fertilizer and it destroys tomatoes

26. fall ‘ 97Principles of MicroeconomicsSlide 25 FERTILIZER tons of tomatoes 1 2 3 4 5 6 7 8 9 10 11 12 With no fertilizer we get 3 tons of tomatoes With 1 unit of Fertilizer [F], we get 6 tons The increase in tomatoes [  T] “caused” by  F is +3, this is the slope With 2 units of F, the output of T increases to 8 With the 3rd unit of F, T increases to 9 The maximum output of T possible with all inputs and existing technology is 10 units with 6 units of F use of more F causes the tomatoes to “burn” and output declines TP f 123456789 (Left click mouse to add material)

27. fall ‘ 97Principles of MicroeconomicsSlide 26 Slope and Marginal Product ·Since the output of tomatoes [T] is a function of Fertilizer [F], the other inputs and technology we are able to graph the total product of Fertilizer [TP f ] ·From the TP f, we can calculate the marginal product of fertilizer [MP f ] ·MP f is the  TP f “caused” by the   F

28. fall ‘ 97Principles of MicroeconomicsSlide 27 1 2 3 4 5 6 7 8 9 10 11 12 TP f Fertilizer [F]Tomatoes [T] 03 1 6  TP f = +3,  F = +1; +3/+1 = 3 [slope = +3] run=1 rise = +3 rise/run =+3 +3 3 MP f [slope] 3 {technically, this is between 0 and the first unit of F} 28  TP f = +2,  F = +1; +2/+1 = 2 Given: T = f (F,... ), MP f = [  TP f /  F] 39  TP f = +1,  F = +1; +1/+1 = 1 1 610  TP f = +1,  F = +3; +1/+3 .33 [ this is an approximation because  F>1].33 89  TP f = -1,  F = +2; -1/+2 = -.5 -.5 [a negative slope!] 123456789 2

29. fall ‘ 97Principles of MicroeconomicsSlide 28 Given a functional relationship such as: Q = 220 - 5P, we can express the equation for P as a function of Q Think of an equation as a “balance scale,” what you do to one side of the equation you must do to the other in order to maintain balance Q = 220 - 5P subtract 220 from both sides -220 -220 + Q = -5P divide every term in both sides by -5 44 - 1 5 Q = 1P or, P = 44 -.2 Q The equation P = 44 -.2Q is the same as Q = 220 -.5P (Left click mouse to add material)

30. fall ‘ 97Principles of MicroeconomicsSlide 29 How do economists estimate relationships ? ·Humans behavioral relationships are: ·modeled on the basis of theories ·models are verified through empirical observations and statistical methods ·The relationships are estimates that represent populations {or distributions} not specific individuals or elements

31. fall ‘ 97Principles of MicroeconomicsSlide 30 An Example ·Hypothesis: the amount of good X [Q] that Susan purchases is determined by the price of the good [P x ], Susans’s income [Y], prices of other related goods [P r ] and Susan’s preferences. ·Q = f i (P x, Y, P r, preferences,...) ·[... indicates there are other variables that are not included in the equation]

32. fall ‘ 97Principles of MicroeconomicsSlide 31 Model of Relationship ·Q = f i (P x, Y, P r, preferences,...) acts a a model to represent the relationships of each independent variable to Q [dependent variable] ·For simplicity, the relationship is described as “linear.” If the relationship were believed not to be linear, with a bit more effort we might construct a “nonlinear model.”

33. fall ‘ 97Principles of MicroeconomicsSlide 32 Empirical verification ·To test the model, we would like to observe Susan’s buying pattern. ·If P x,Y, P r and preferences were all changing at the same time, we would use a multivariate analysis called “multiple regression.” For simplicity we have been lucky enough to find a period where only P x has changed. Y, P r and preferences have remained unchanged over the period in which we observe Susan’s purchases

34. fall ‘ 97Principles of MicroeconomicsSlide 33 Quantity per week Price of good X 2 4 6 8 10 12 14 16 18 Data from these observations can be plotted on the graph During a 5 week period, Susan was observed making the following purchases Clearly there is a pattern, however it is not a perfect relationship. Through statistical inference we can estimate some general characteristics about the relationship 2468 10121416182022

35. fall ‘ 97Principles of MicroeconomicsSlide 34 2 4 6 8 10 12 14 16 18 20 22 24 26 Quantity per week Price of good X 2 4 6 8 10 12 14 16 18 Given the observed data about Susan’s purchases: ( Q= 10, P= $15) (15, 11) (20,10) (22,7) (22,6) We can estimate a line that minimizes the square of the difference that each point [that represents two variables] lies off the estimated line. No single point may lie the line, but the line is an estimate of the relationship P = 23 -.75Q is our estimate of the relationship between the price and the quantity that Susan purchases each week, ceteris paribus or all other things equal P = 23 -.75Q may be written Q = 30.667- 1.333P

36. fall ‘ 97Principles of MicroeconomicsSlide 35 2 4 6 8 10 12 14 16 18 20 22 24 26 Quantity per week Price of good X 2 4 6 8 10 12 14 16 18 Given the observed data about Susan’s purchases: and our estimated function: P = 23 -.75Q or Q = 30.67 - 1.33P, we would predict that at a price of $10 Susan would purchase about 17.37 units, [Q = 30.67 - 1.33 P, P = 10 so Q =17.37] Q = 17.37 P = 10 We observed that Susan bought 20 units when the price was $10 so estimate is off by a small amount [-2.63 units] At a price of $6 our equation predicts that 22.67 units will be purchased P = 6 Q = 22.67 Since we observed that she purchased 22, we are off by.67 units our estimates are not perfect, but they give an approximation of the relationship

37. fall ‘ 97Principles of MicroeconomicsSlide 36 Statistical Estimates ·The estimates are not “perfect” but they provide reasonable estimates ·There are many statistical tools that measure the confidence that we have in out predictions ·these include such things as correlation, coefficient of determination, standard errors, t-scores and F-ratios

38. fall ‘ 97Principles of MicroeconomicsSlide 37 Slope & Calculus ·In economics we are interested in how a change in one variable changes another ·How a change in price changes sales. How a change in an input changes output. How a change in output changes cost. etc. ·The rate of change is measured by the slope of the functional relationship ·by subtraction the slope was calculated as rise over run where rise =  Y = Y 1 - Y 2 and run =  X = X 1 - X 2,

39. fall ‘ 97Principles of MicroeconomicsSlide 38 Derivative There are still more slides on this topic ·When we have a nonlinear function, a simple derivative can be used to calculate the slope of the tangent to the function at any value of the independent variable ·The notation for a derivative is written:

40. fall ‘ 97Principles of MicroeconomicsSlide 39 Summary ·a derivative is the slope of a tangent at a point on a function · is the rate of change, it measures the change in Y caused by a change in X as the change in X approaches 0 ·in economics jargon, [ the slope or rate of change] is the “marginal” dY dX dY dX