These slides review basic math tools used in our economics course: the concept of multi-variate relations and their mathematical representation as functions, bivariate functions, linear functions, etc.
Multi-variate relationships In economics as in other sciences, we are often interested in sorting out causes and eﬀects: “What causes prices to drop?” “What causes unemployment to increase?” These are typical questions in economics. However, ascertaining cause-eﬀect relationships is particularly diﬃcult in economics, because controlled experiments tend to be either very expensive or impracticable. Usually, economic events of interest result from multiple causes. However, if many of them are minor and if – to some extent – they oﬀset one another, then it is possible for economists to ﬁnd simpler yet powerful explanations, i.e. explanations that isolate a few yet important causes, especially those that – aside from being important – are under the control or conscious inﬂuence of individuals, organizations, or governments.
Multi-variate functions Speaking in math terms, we say that most (if not all) economic phenomena are the result of (functional) relationships between multiple variables. This equation denotes a multi-variate relationship or, more precisely, a multi-variate function: y = f (x1 , x2 , . . . , xn ) This equation says that variable y , a.k.a. the dependent variable, “is a function of” (in plain words, “its value depends on the value of”) the variables x1 , x2 , . . . , xn , a.k.a. the independent variables, where n can be any positive whole number, and f (.) is descriptive of the type of relationship among the variables.
Multi-variate functions The math does not say that the independent variables “cause” the dependent variable (or, more precisely, that changes in the independent variables cause the dependent variable to change). It only says that there is some deﬁnite “relationship” between y and the x’s, so that when the xs change, y also changes according to f (.). For all we know, the xs may cause y , or y may cause some or all the xs, or both y and the xs may be caused by some unidentiﬁed variable. Conventionally, y denotes the eﬀect and the xs denote causes. But economists must ascertain causality independently from the math. The math is used only to get a better sense of the logic involved.
Bivariate functions Since dealing with multi-variate functions mathematically gets very complicated very quickly, in this course we simply assume that there is only one independent variable of interest. All other independent variables are assumed constant, so we focus on one x at a time. This is not a terrible sacriﬁce. It is easier for us to grasp things with only one moving part at a time, while the other parts are assumed ﬁxed. In logic, when we imagine that everything else remains constant to focus on the relationship between only two variables, we are making the caeterius paribus assumption. This assumption eﬀectively reduces multi-variate functions to bivariate functions: y = f (x) Again, y is the dependent variable and x the (single) independent variable. The equation says that as x varies, y varies in accordance with rule f (.).
Bivariate functions Not that the caeteris paribus assumption requires that the xs are indeed (relatively speaking) independent causes – i.e. largely independent from one another. If at least some of xs are not suﬃciently independent, but they vary together, then the “everything-else-constant” assumption may lead to serious nonsense. In any case, bivariate functions are very convenient to work with, since we may represent them graphically on a plane (a.k.a. the Cartesian plane) – a ﬂat space divided by two perpendicular lines (axes), a vertical and a horizontal one. That way we can get a visual understanding of the relationship between the two variables. Conventionally, we plot the values of x on the horizontal axis and y on the vertical axis. Each axis is a number line and each pair of values (x, y ) corresponds to a point in the plane: a point’s projection on the horizontal axis is the value x and its projection on the vertical axis is the value y .
Identity function We now study the simplest type of bivariate functions. The simplest (but also somewhat trivial) example of a bivariate function is the identity function: y =x This says that every time x takes a value (e.g., x = 20), then y takes that same value (i.e., y = x = 20). Etc. Assigning values to x and using those values to determine the corresponding values of y as given by the function is called evaluating the function. By evaluating bivariate functions, we can generate data tables with two columns, one column with the values of x and the other one with the values of y . To plot the graph, we need at least two values of x, which by using the equation y = x, yield two corresponding values for y : e.g. x0 = −2, y0 = −2 and x1 = 2, y1 = 2.
Identity function With that, we have enough information to plot the graph of this function, which is a straight line that goes through those points. In this case (the identity function), the line also goes through the origin (i.e. the point where x = 0 and y = 0). Here’s a data table with a few selected values of x and, therefore, y : y ($) x ($) -2 -2 0 0 2 2
Identity function And here’s the graph of the identity function y = x:
Proportional function A bit less simple is the proportional function: y = bx. Here b means a given or constant number. For example: y = 3x In this case, b = 3. That is, y is always the triple of x. Thus, if x = 10, then y = 3 × 10 = 30.
Linear functions A more realistic example of a proportional function is currency conversion at a given exchange rate. Suppose that today’s U.S. dollar-Mexican peso exchange rate is S(USD/MXN) = 10. In algebraic form: y = 10 x where x is the number of U.S. dollars and y the number of Mexican pesos. Determine the equivalent in Mexican pesos of x = 327 U.S. dollars: y = 10 × 327 = 3, 270 In words, 327 U.S. dollars are equivalent to 3,270 Mexican pesos in today’s foreign exchange market.
Linear functions Note that when b = 1 the proportional function “degenerates” into the identity function. In other words, the identity function is the proportional function in the particular case when b = 1. The constant b is called the slope, and it indicates the scale at which y expands or shrinks as x changes. Graphically, b determines the inclination (slope) of the linear graph representing y = bx. The slope b also indicates the change in y when x changes in one unit: ∆y y1 − y0 b= = ∆x x1 − x0
Linear functions To show that b = ∆y /∆x, let x0 = 0, then y0 = b × 0 = 0. Now, let x1 = 1, then y1 = b × 1 = b. Clearly: ∆x = x1 − x0 = 1 − 0 = 1 ∆y = y1 − y0 = b − 0 = b ∆y b b= = =b ∆x 1
Linear functions To double check, alternatively, let x0 = 10, then y0 = b × 10 = 10b. Now, let x1 = 20, then y1 = b × 20 = 20b. Note that: ∆x = x1 − x0 = 20 − 10 = 10 ∆y = y1 − y0 = 20b − 10b = 10b ∆y 10b b= = =b ∆x 10
Linear functions A linear function has the following algebraic form: y = a + bx Here a and b are both given or constant numbers. Clearly, the proportional function is a linear function when a = 0. We already know that b is the slope, which indicates the change in y when x changes in one unit: ∆y b= ∆x
Linear functions As noted above, b determines how steep or shallow the linear graph is. On the other hand, the constant a is called the vertical intercept or, simply, the intercept, because it determines the location of the graph in the plane. More speciﬁcally, a determines the point at which the linear graph crosses the vertical axis. When a = 0 (the proportional case), the line crosses the vertical axis at the origin, i.e. when y = 0. In the more general case, a can be positive or negative. If a > 0, then the linear graph crosses the vertical axis above the origin (on the positive region of y ). If a < 0, then the linear graph crosses the vertical axis below the origin (on the negative region of y ).
Linear functions An example is the formula to convert degrees from the Celsius temperature scale into degrees in the Farenheit scale: 9 y = 32 + x 5 where x means a temperature in the Celsius scale and y means its equivalent in the Farenheit scale. Convert from Celsius water’s “freezing point” (x = 0) into Farenheit: y = 32 + [(9/5) × 0] = 32 + 0 = 32 The water starts to freeze at 32◦ F. Note that a = 32 immediately gives us this information. Convert Celsius water’s “boiling point” (x = 100) into Farenheit: y = 32 + [(9/5) × 100] = 32 + 180 = 212 The water starts to boil at 212◦ F.
Linear relationships Note that if b > 0 (positive slope), then the change in y associated with the unit change in x is positive. In other words, there is a positive or direct relationship between x and y . If b < 0 (negative slope), then the change in y associated with the unit change in x is negative. That is, there is a negative or inverse relationship between x and y . If b = ∞, then even a very tiny change in x sends y through the roof: the graph is a vertical line. If b = 0, then no matter how much x changes, y does not change at all: the graph is a ﬂat or horizontal line.
Linear functions Let, x = Income and y = Consumption spending, and consider the following data on selected levels of x and y : Income ($) Consumption ($) 0 50 100 100 200 150 300 200 400 250
Linear functions A simple visual inspection of the data shows that there is a linear relationship between x and y . From one row to the next, x increases in 100 and, as a result, y increases in 50. By taking data from any couple of rows, we can then determine the slope of this relationship. Let us take the ﬁrst and the last row: x0 = 0, y0 = 50, x1 = 400, y1 = 250. Therefore: ∆x = x1 − x0 = 400 − 0 = 400 and ∆y = y1 − y0 = 250 − 50 = 200. The slope is then: ∆y 200 b= = = .5 ∆x 400 With this information, we know that the equation representative of the linear relationship between Income and Consumption spending has the following form: y = a + .5 x
Linear functions However, we still don’t know the value of a, the vertical intercept. We need to determine a to completely pin down our linear equation. To determine a, we need information from any row in the data table. Let’s use the third row: y = a + .5 x, i.e. 150 = a + .5 × 200 = a + 100. This is a simple linear equation. To solve, subtract 100 from each side of the equation: 150 − 100 = a = 50 We got it! a = 50. So, we know that the linear equation representative of the data in the table is: y = 50 + .5 x With this linear function, we are ready determine any level of Consumption spending whenever the level of Income is given.
Linear functions We can graph the points in the data table and then join them with a straight line. Or, alternatively, we can evaluate the linear equation y = 50 + .5 x twice and plot the resulting graph. A graphic calculator or a computer can do this. Or we can use this free online graph generator: http://rechneronline.de/function-graphs/.
Linear functions Sometimes, we may want to use symbols other than x and y . For example, let C = Consumption and Y = Income. Then: C = 50 + .5 Y . Note that y = C is the dependent variable, because the value of C depends on the value of Y , the independent variable. The intercept a = 50 indicates that C = 0 when or if Y = $50. Finally, the slope b = .5 indicates that when Y increases by $1 (or one dollar), C increases by $.5 (or 50 cents). Note that b = .5 > 0, which means that C increases when Y increases, i.e. there is a positive or direct relationship between C and Y .
Linear functions Note that the equation form or algebraic formulation conveys the same information and then more than the contained in a numerical data table. And it does so in a much more compact manner. Because algebra uses general symbols, rather than speciﬁc numbers or graphical objects, it is very powerful. A great mathematician said that, in math, we do not “understand” things: we just get used to them! So get used to the algebraic form of a linear functional relationship.
Nonlinear functions In our course, we will not use the algebraic or equation form for nonlinear relationships between two variables, x and y . For them, we will only use graphs and intuition. Usually, when dealing with nonlinear relationships, intercepts are of little or no interest. Most of the interest focuses on the varying slopes.
Nonlinear functions This graph shows a curve that is concave to the origin (the point in the plane where x = 0 and y = 0). Note that that, throughout, the slope of the curve is negative. When x is close to zero, the slope is a very small negative number (almost zero). Then, as x increases, the slope of the curve becomes increasingly negative and, when it hits the horizontal axis, it is very negative.
Nonlinear functions This following graph shows also a downward-sloping curve, but this is convex to the origin. When x is close to zero, the slope is very high (it tends to inﬁnity). Then, as x goes up, the curve’s slope becomes less and less negative. When it hits the horizontal axis, the slope of the curve is almost zero.
Nonlinear functions This ﬁnal graph shows a more complicated relationship between x and y . It is a curve that changes direction. When 0 < x < 1.5, 4.7 < x < 7.7 the slope of the curve is positive and for all other values of x in the graph, the slope is negative. Also note that the curve’s slope becomes zero at (x = 1.5, y = 4), (x = 4.7, y = 2), and (x = 7.7, y = 4).1 1 At these points the slope changes from positive to negative or vice versa. A ﬂat or horizontal tangent line can be drawn to touch them. These points are either maxima or minima (the plural of maximum and minimum, i.e. the highest and lowest values of y ).