AP_Krugman_Textbook

this by calculating the slope of the linear curve between points Aand Band between

points Cand Din panel (b) of Figure A.3.

= = 5

= = 5

Horizontal and Vertical Curves and Their Slopes

When a curve is horizontal, the value of yalong that curve never changes—it is con-

stant. Everywhere along the curve, the change in yis zero. Now, zero divided by any

number is zero. So regardless of the value of the change in x,the slope of a horizontal

curve is always zero.

If a curve is vertical, the value of xalong the curve never changes—it is constant. Every-

where along the curve, the change in xis zero. This means that the slope of a vertical line is

a ratio with zero in the denominator. A ratio with zero in the denominator is equal to in-

finity—that is, an infinitely large number. So the slope of a vertical line is equal to infinity.

A vertical or a horizontal curve has a special implication: it means that the x-variable

and the y-variable are unrelated. Two variables are unrelated when a change in one vari-

able (the independent variable) has no effect on the other variable (the dependent vari-

able). To put it a slightly different way, two variables are unrelated when the dependent

variable is constant regardless of the value of the independent variable. If, as is usual,

the y-variable is the dependent variable, the curve is horizontal. If the dependent vari-

able is the x-variable, the curve is vertical.

Δy

Δx

20

4

Δy

Δx

10

2

38 section I Basic Economic Concepts

5

30

25

20

15

10

5

0 10 15 20 25 30 35 40 45 x

A

y

(a) Negative Constant Slope

B

Δx = 10

Slope = –

Δy = –5 Slope = 5

12345678910

60

50

40

30

20

10

0 x

y

(b) Positive Constant Slope

Δx = 2

Δx = 4

Δy = 10

Δy = 20

A

B

C

D

Slope = 5

1

2

Calculating the Slope

Panels (a) and (b) show two linear curves. Between points Aand

Bon the curve in panel (a), the change in y(the rise) is −5 and

the change in x(the run) is 10. So the slope from Ato Bis

= = − = −0.5, where the negative sign indicates that the

curve is downward sloping. In panel (b), the curve has a slope from

Ato Bof = = 5. The slope from Cto Dis = = 5. The

slope is positive, indicating that the curve is upward sloping. Fur-

thermore, the slope between Aand Bis the same as the slope be-

tween Cand D,making this a linear curve. The slope of a linear

curve is constant: it is the same regardless of where it is calcu-

lated along the curve.

20

4

Δy

Δx

10

2

Δy

Δx

1

2

− 5

10

Δy

Δx

figure A.3