Lecture Notes Companion 2—Elasticity concepts
A. Price Elasticity and Total Expenditure/Revenue
P
Linear Demand Curve
50
Total revenue:
(a) P = $25, Q = 15: TR = PQ = $25*15 = $375
(b) P = $20, Q = 20: TR = PQ = $20*20 = $400
maximum revenue
(c) P = $15, Q = 25: TR = PQ = $15*25 = $375
40
η
Total revenue (TR) is maximized at the price which corresponds
to the point of unit elasticity on the demand curve. Thus:
> 1 (elastic)
30
25
(a)
20
η
inelastic good
elastic good
= 1 (unit elastic)
If the price . . .
rises
falls
TR rises TR falls
TR falls
TR rises
(b)
15
(c)
η
< 1 (inelastic)
10
The general rule is:
• TR rises anytime the price moves toward the unit-elastic price.
• TR falls anytime the price moves away from the unit-elastic price.
Note: TR = P × Q means that TR is drawn as a rectangle (recall that
the area of a rectangle is A = L × W)
0
0
5
10
15
20
25
30
35
40
Qd
Puzzlers:
1.
Since firms that sell price-inelastic goods can increase total revenue just by
raising their price, and since most firms’ goods are price-inelastic, then why
don’t they just raise their prices?
TR ($)
400
300
2.
200
100
0
5
10
15
20
TR rises
25
30
35
40
Qd
TR rises
TR falls
Gasoline is highly price-inelastic. Does it follow that Chevron gasoline is
highly price inelastic? Explain.
C. Graphical Calculation of Price Elasticity
P
12
11
10
9
8
7
6
5
4
3
2
1
Using Marshall’s method, the (absolute value of) elasticity
of demand at point X,
i.e., at P = $4 and Qd = 4, is
D
t
a
XT ÷ Xt
X
= √80 ÷ √20
=
Where XT and Xt are derived using the Pythagorean Theorem as
D′
(XT)2 = (XM)2 + (MT)2 =
M
2.0
T
O
1 2 3 4 5 6 7 8 9 10 11 12
Giving
Q
“The elasticity of demand can be best traced
in the demand curve with the aid of the
following rule. Let a straight line touching
the curve at any point X meet OQ in T and
OP in t, then the measure of elasticity at the
point X is the ratio of XT to Xt . . . Another
way of looking at the same result is this:
--the elasticity at the point X is measured by
the ratio of . . . MT to MO (XM being drawn
perpendicular to OM) . . .
--adapted from Alfred Marshall, Principles of
Economics, 8th ed. (1920), pp. 86-87
4 2 + 82
=
16 + 64 = 80
XT = √80
And
(Xt)2 = (Xa)2 + (at)2
Giving Xt = √20
=
42 + 2 2
=
16 + 4 = 20
Alternatively (and more simply), the elasticity can be found as
MT ÷ MO
=
8 ÷ 4
=
2.0
Verifying this, η = ΔQ/ΔP × P1/Q1 = -2 × 4/4 = -2.0
(Remember, we’ll ignore the negative sign.)