Dominick Salvatore Schaums Outline of Microeconomics, 4th edition Schaums Outline Series 2006

CHAP. 3] THE MEASUREMENT OF ELASTICITIES 51
3.8 (a) Show that when QD y ¼ 600/P y (a rectangular hyperbola), the total expenditures on commodity Y
remain unchanged as P y falls. (b) From (a), derive the value of e along the hyperbola. (c) Verify (b) by
finding e geometrically at P y ¼ $4 and at P y ¼ $2.
(a) Table 3.11
Point
(1)
P y ($)
(3)
(2) Total
Q y Expenditures ($)
A 6 100 600
B 5 120 600
C 4 200 600
D 3 200 600
F 2 300 600
G 1 600 600
(b) Since QD y ¼ $600
P y
Fig. 3-9
(QD y )(P y ) ¼ $600 regardless of P y . Thus, for any given percentage fall in P y , QD y will increase by the same
percentage. Because the percentage changes in QD y and P y are always equal, e ¼ 1 at every point on the
rectangular hyperbola, D y .
(c) See Fig. 3-9.
At point C,
At point F;
e ¼ ML
OM ¼ 150
150 ¼ 1
e ¼ LH
OL ¼ 300
300 ¼ 1
3.9 Table 3.12 gives two demand schedules. Using only the total expenditure criterion, determine if these
demand curves are elastic or inelastic.
Table 3.12
P ($) 6 5 4 3 2 1
Q x 100 110 120 150 200 300
Q z 100 150 225 325 500 1100
Since total expenditures on commodity X fail continuously as P x falls [see column (3) of Table 3.13], e , 1
throughout the observed range of D x . Total expenditures on commodity Z rise continuously as P z falls [see column
(5) of the table], so e . 1 throughout the observed range of D z ,D x ,D z , and D y (from Problem 3.8) are sketched in
Fig. 3-10.