Some basic mathematics∗
Luc Hens
October 1, 2014
Equation of a straight line
In the theory of international trade we will develop some formal models. This
document outlines some of the mathematics used in the models. I expect that
you know the formulas in this section by heart.
Suppose you start somewhere on a straight line and move horizontally to the
right over a certain distance. This distance is called the run. Unless the line
was horizontal, you’ll be off the line. You can get back on the line by moving
vertically. The vertical distance is called the rise (if you had to move up, the rise
is a positive number; if you had to move down, the rise is a negative number).
The ratio of the rise over the run is called the slope (or gradient) of the line:
slope =
rise
run
(1)
For a straight line, the slope will be the same regardless of where you are on
the line and how much you moved to the right.
The equation of any straight line in the x-y plane can be solved for y and
be written in the following form:
y = mx + a
(2)
The coefficient of x, m, is the slope of the line. The term a is called the
y-intercept of the line.
Suppose a straight line goes through a point A with coordinates (x1 , y1 ) and
a point and B with coordinates (x2 , y2 ). The equation of that line can be found
as follows:
y2 − y1
y − y1 =
(x − x1 )
(3)
x2 − x1
Memorize equations (1), (2), and (3). You will need them throughout this
course.
Self test 1
Solve 4x + 2y = 16 for y to obtain an equation of the form y = mx + a. What
is the slope? What is the y-intercept? Graph the line (on scale). Identify the
slope in the graph (choose +1 as the run). Identify the y-intercept in the graph.
(The answer is at the end of this document.)
∗ Handout
for Chapter 3 in Krugman et al. (2015).
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Self test 2
Find the equation for the straight line trough A(1, 5) and B(5, 2). What is the
slope? What is the y-intercept? Graph the points and the line (on scale). (The
answer is at the end of this document.)
Production Possibility Frontier
Now let us apply these formulas to the model of Krugman et al. (2015, chapter
3). We saw that Home’s production possibility frontier goes throught the points
with coordinates (0, L/aLW ) and (L/aLC , 0) (Krugman et al., 2015, figure 3-1
p. 59). With the help of equation (3) we can compute the equation of the
production possibility frontier. The variable on the x-axis is QC and the variable
on the y-axis is QW ; moreover, we know that:
x1
=
0
y1
= L/aLW
x2
= L/aLC
y2
=
0
Plugging these values into equation (3) yields:
QW − L/aLW =
0 − L/aLW
(QC − 0)
L/aLC − 0
Expanding the right-hand side yields:
QW − L/aLW = −
aLC
QC
aLW
Adding L/aLW to both sides yields:
aLC
QC + L/aLW
QW = −
aLW
(4)
This is an equation of the form y = mx + a (2), with x = QC and y = QW .
Hence the equation represents a straight line with
aLC
−
as the slope and
aLW
L/aLW
as the y-intercept
As aLC /aLW is the opportunity cost of cheese (Krugman et al., 2015, p. 60), we
find that (omitting the minus sign) the slope of the production possibility
frontier is the opportunity cost of the good on the horizontal axis.
Finally, multiplying both sides of equation (4) by aLW and re-arranging
terms yields:
aLC QC + aLW QW = L
(5)
The variable QC measures the quantity (in pounds) of cheese produced, and
the unit labor requirement aLC measures how many hours of labor are required
to produce one pound of cheese. Hence, aLC QC is the number of hours of
labor employed to produce cheese. Similarly, aLW QW is the number of hours of
labor employed to produce wine. In sum, this form of the production possibility
frontier equation shows how the total labor supply (L) is allocated between the
cheese sector and the wine sector.
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Answers to self tests
1. The equation is y = −2x + 8. The slope is −1. The y-intercept is 8.
2. The equation is y = − 34 x + 5 34 . The slope is − 34 . The y-intercept is 5 43 .
References
Krugman, P. R., Obstfeld, M., and Melitz, M. J. (2015). International Economics: Theory and Policy. Pearson Education, Harlow, 10th edition.
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