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**ã R. Larry Reynolds (Boise State University)**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. Principles of Microeconomics**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 Principles of Microeconomics**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 Principles of Microeconomics**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” Principles of Microeconomics**Equations [cont . . .]**• An equation is a statement about a relationship between two or more variables • Y = fi (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” Principles of Microeconomics**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] Principles of Microeconomics**Y>0**The X axis [horizontal] +3 +2 +1 X<0 X > 0 -3 -2 -1 +1 +2 +3 -1 -2 -3 The Y axis [vertical] Y<0 Equations -- Graphs [Cartesian system] The North East Quadrant (NE), where X > 0, Y > 0 {both X and Y are positive numbers} (X,Y) where X<0, Y>0 (X,Y) where X>0 and Y<0 (X,Y) where X<0 and Y<0 (Left click mouse to add material) Principles of Microeconomics**6**5 3 2 1 1 2 3 4 5 6 When the values of the independent and dependent variables are positive, we use the North East quadrant (Left click mouse to add material) Go to the right {+3} units and up {+5} units! (X, Y) (3, 5) Right {+1} one and up {+6} six (1,6) (5, 1) (2.5, 3.2) Right 5 and up 1 to the right 2.5 units and up 3.2 units Principles of Microeconomics**A**6 5 4 3 2 1 B 1 2 3 4 5 6 Given the relationship, Y = 6 - 2X, (Left click mouse to add material) when X = 0 then Y = 6 [this is Y-intercept] Y sets of (X, Y) A line that slopes from upper left to lower right represents an inverse or negative relationship, when the value of X increases, Y decreases! (0, 6) when X = 1 then Y = 4 (1, 4) (2, 2) (3, 0) When X = 2, then Y = 2 The relationship for all positive values of X and Y can be illustrated by the line AB When X = 3, Y = 0, [this is X-intercept] X Principles of Microeconomics**Y**6 5 4 3 2 1 X 1 2 3 4 5 6 (Left click mouse to add material) 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? Principles of Microeconomics**Y**6 5 rise +2 4 3 1 2 run +1 slope = + 2 rise +1 1 run +2 X 1 2 3 4 5 6 -1 For a relationship, Y = 1 + 2X When X=0, Y=1 (0,1) When X = 1, Y = 3 slope = +2 (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 ! for a relationship Y = -1 + .5X This function shows that for a 1 unit increase in X, Y increases one half unit (Left click mouse to add material) Principles of Microeconomics**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)? Principles of Microeconomics**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. Principles of Microeconomics**Relationship [cont . . . ]**• Q = fp(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 Principles of Microeconomics**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 Principles of Microeconomics**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! Principles of Microeconomics**55**50 45 40 35 30 At P=$30, Q = 70 25 20 15 10 At a price of $10, the the quantity is $5 Demand 70 170 40 80 120 160 200 240 280 For the relationship, Q = 220 - 5P, the relationship can be graphed ... PRICE When the price is $44, 0 unit will be bought; at a price of $0, 220 units will be bought. 44 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! QUANTITY Principles of Microeconomics (Left click mouse to add material)**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 Principles of Microeconomics**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 Principles of Microeconomics**Y**6 5 4 DY= -1 3 2 1 DX = 2 X 1 2 3 4 5 6 rise slope is run Slope of a Line Y = 6 -.5X as the value of X increases from 2 to 4, the value of Y decreases from 5 to 4 DY is the rise [or change in Y caused by DX]{in this case, -1} so, slope is -1/2 or -.5 DX is the run {+2}, Principles of Microeconomics**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] Principles of Microeconomics**(Left click mouse to add material)**Y 6 Shifts right 5 4 shifts left 3 an increase in the function would represent an increase in the intercept [from 6 to a larger number] A decrease in the function would be Y’ = 4 - .5X 2 1 X 1 2 3 4 5 6 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] Given the function Y = 6 - .5X, Principles of Microeconomics**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” Principles of Microeconomics**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 Principles of Microeconomics**12**11 10 TPf 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 The maximum output of T possible with all inputs and existing technology is 10 units with 6 units of F tons of tomatoes With the 3rd unit of F, T increases to 9 With 2 units of F, the output of T increases to 8 With 1 unit of Fertilizer [F], we get 6 tons The increase in tomatoes [DT] “caused” by DF is +3, this is the slope With no fertilizer we get 3 tons of tomatoes use of more F causes the tomatoes to “burn” and output declines (Left click mouse to add material) FERTILIZER Principles of Microeconomics**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 [TPf] • From the TPf, we can calculate the marginal product of fertilizer [MPf] • MPf is the DTPf “caused” by theDF Principles of Microeconomics**12**11 10 TPf 9 8 7 6 3 5 rise = +3 4 +3 3 2 run=1 1 1 2 3 4 5 6 7 8 9 Given: T = f (F, . . . ), MPf = [DTPf/DF] DTPf = +1, DF = +3; +1/+3 @ .33 [this is an approximation because DF>1] DTPf = -1, DF = +2; -1/+2 = -.5 Fertilizer [F] Tomatoes [T] 0 3 MPf [slope] 3 {technically, this is between 0 and the first unit of F} 1 6 2 8 2 3 9 1 6 10 .33 rise/run =+3 8 9 -.5[a negative slope!] DTPf = +3, DF = +1; +3/+1 = 3 [slope = +3] DTPf = +2, DF = +1; +2/+1 = 2 DTPf = +1, DF = +1; +1/+1 = 1 Principles of Microeconomics**-220**-220 -5 -5 -5 1 44 - Q = 1P 5 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 + Q = -5P divide every term in both sidesby -5 or,P = 44 - .2 Q The equation P = 44 - .2Q is the same as Q = 220 - .5P (Left click mouse to add material) Principles of Microeconomics**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 Principles of Microeconomics**An Example**• Hypothesis: the amount of good X [Q] that Susan purchases is determined by the price of the good [Px], Susans’s income [Y], prices of other related goods [Pr] and Susan’s preferences. • Q = fi (Px,Y, Pr, preferences, . . .) • [. . . indicates there are other variables that are not included in the equation] Principles of Microeconomics**Model of Relationship**• Q = fi (Px,Y, Pr, 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.” Principles of Microeconomics**Empirical verification**• To test the model, we would like to observe Susan’s buying pattern. • If Px,Y, Pr 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 Px has changed. Y, Pr and preferences have remained unchanged over the period in which we observe Susan’s purchases Principles of Microeconomics**Price of good X**18 16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20 22 Quantity per week During a 5 week period, Susan was observed making the following purchases Data from these observations can be plotted on the graph Clearly there is a pattern, however it is not a perfect relationship. Through statistical inference we can estimate some general characteristics about the relationship Principles of Microeconomics**Given the observed data about Susan’s purchases:**Price of good X 18 16 ( Q= 10, P= $15) 14 (15, 11) 12 (20,10) 10 8 (22,7) 6 (22,6) 4 2 2 4 6 8 10 12 14 16 18 20 22 24 26 Quantity per week We can estimate a line that minimizes the square of the difference that each point [that represents two variables] lies off the estimated line. P = 23 - .75Q may be written Q = 30.667- 1.333P 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 Principles of Microeconomics**Given the observed data about Susan’s purchases:**Price of good X 18 16 14 12 10 P = 10 8 P = 6 6 4 Q = 22.67 2 Q = 17.37 2 4 6 8 10 12 14 16 18 20 22 24 26 Quantity per week 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] 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 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 Principles of Microeconomics**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 Principles of Microeconomics**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 = DY = Y1 - Y2 and run = DX = X1 - X2, Principles of Microeconomics**DerivativeThere 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: Principles of Microeconomics**dY**dX dY dX 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” Principles of Microeconomics