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.

1. Basic math tools:
Relations between variables
juliohuato@gmail.com
September 5, 2011

2. Topics
Multi-variate functions
Bivariate functions
Linear functions
Nonlinear functions

3. 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.

4. 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.

5. 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.

6. 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 (.).

7. 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 .

8. 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.

9. 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

10. Identity function
And here’s the graph of the identity function y = x:

11. 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.

12. 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.

13. 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

14. 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

15. 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

16. 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

17. 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 ).

18. 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.

19. 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.

20. 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

21. 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

22. 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.

23. 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/.

24. 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 .

25. 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.

26. 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.

27. 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.

28. 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.

29. 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 ).