A NOTE ON RELATIONSHIP BETWEEN DEMAND ELASTICITY AND TOTAL REVENUEAuthor(s): Moon H. KangSource: Review of Social Economy, Vol. 24, No. 2 (September, 1966), pp. 176-178Published by: Taylor & Francis, Ltd.Stable URL: http://www.jstor.org/stable/29767825 .

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TEACHING AIDS A NOTE ON RELATIONSHIP BETWEEN DEMAND

ELASTICITY AND TOTAL REVENUE

By Moon H. Kang

University of Missouri at Rolla

From my experience in teaching the course of Principles of Eco? nomics I feel that there seems to be a need for a simple and effective

explanation of the relationship between demand elasticity and total revenue or expenditure. For many beginning students do not feel quite convinced with the commonly used explanation in the texts: with an elastic demand a price decline results in an increase in total revenue because the percentage increase in quantity demanded is greater than the percentage decrease in price, and a price rise brings about a decrease in total revenue because the percentage decrease in quantity demanded is

greater than the percentage increase in price; with an inelastic demand a price decline produces a decline in total revenue because the percentage increase in quantity demanded is smaller than the percentage decrease in

price, and a price rise brings about an increase in total revenue; with an

unitary demand elasticity a price change causes no change in total revenue. I believe the students' difficulty is because they have not been

prepared for a ready grasp of such a quantitative relationship. A mathematical snowing in general terms seems to be desirable to

facilitate their understanding. But the level of mathematics would have to be low enough to be commensurate with the elementary economics

{i.e., Principles of Economics). I have looked over many texts on

Principles, but I have failed to find any suitable mathematical exposition in general terms. The extent of my review has been: Paul A. Samuelson,

Economics, An Introductory Analysis, 6th edition, 1964; George L. Bach, Economics, An Introduction to Analysis and Policy, 4th edition, 1963; Campbell R. McConnell, Economics, Principles, Problems, and Policies, 2nd edition, 1963; Lowell C. Harris, The American Economy, Principles, Practices and Policies, 1962; Lloyd G. Reynolds, Economics, A General Introduction, 1963; Armen A. Alchian and William R. Allen, University Economics, 1964; Daniel Hamberg, Principles of a Growing Economy, 1961; W. Nelson Peach, Principles of Economics, 1960; Royall Brandis, Economics, Principles and Policy, revised edition, 1963.

I also have checked some of the texts for the intermediate level and the beginning graduate level of price theory. These were: Kenneth E. Boulding, Economic Analysis, 3rd edition, 1955; George J. Stigler, The

Theory of Price, revised edition, 1952; John F. Due and Robert W. Clower, Intermediate Economic Analysis, 4th edition, 1961; Harrison W. Carter and William P. Snavely, Intermediate Economic Analysis, 1961; Sidney Weintraub, Intermediate Price Theory, 1964; Joe S. Bain, Pricing, Distribution, and Employment, revised edition, 1953; H. H. Liebhafsky, The Nature of Price Theory, 1963; M. M. Bober,

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TEACHING AIDS 177

Intermediate Price and Income Theory, revised edition, 1962; George Malanos, Intermediate Economic Theory, 1962; James M. Henderson and Richard E. Quandt, Microeconomic Theory, A Mathematical Ap? proach, 1958. Of these ten only two, Weintraub's book and the one co authored by Henderson and Quandt, show some attempt to prove the

relationship in general terms by use of mathematics. Henderson and

Quandt prove positive, negative and constant marginal revenue for three different elasticities of demand by a mathematical analysis in the level of calculus.1 Calculus seems to be too high a level of mathematics for

elementary economics. Weintraub tries to prove the relationship with

elementary algebra, as shown below :2

MR = (P ? AP) (Q + AQ) ? PQ (P and Q denote price and

quantity demanded respectively) MR = PAQ

? QAP

? APAQ Dropping APAQ as being of an extremely small order,

MR = PAQ -QAP PAQ > PAQ .

Thus, MR depends on whether-? = 1 and ?-is the for QAP < QAP

mula for demand elasticity. The Weintraub's treatment does not seem to be very convincing to

the beginning students because they are puzzled why APAQ is dropped. It seems that a very elementary algebraic manipulation can be made in order to prove the relationship to the satisfaction of the beginning students without going to the level of calculas. Using the concept of arc

elasticity of demand we can prove it in general terms in a rather simple manner as shown below:

Suppose price declined from Pi to P2 and in response the quantity demanded increased from Qi to Q2. Let a and b represent the percentage change in price and the percentage change in quantity demanded respec? tively. Then,

2(Pi-P2) (1) b = 2(02-Qi)

(Q* + Qi) (2) a

Pi + P2

(3)

(4) From (3) and (4)

P1Q1 - P2Q2 = P1Q1 - P1Q1

a b P1Q1

- P2Q2 = 4PiQi

(2 +a) (2-b) (5)

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178 REVIEW OF SOCIAL ECONOMY

In the right side of the equation (5) Pi, Qi, and the denominator

portion (2 + a) (2 ?

b), are positive. Then the difference of the total revenues (i.e., PiQi

? P2Q2) is dependent on the numerical values of

a and b. Then,

(i) if a < b, the difference (i.e., P1Q1 ?

P2Q2) becomes negative. This means that with a decline in price the total revenue in? creases if the demand is elastic since the demand elasticity is

b measured by ?.

a

(ii) if a>b, the difference becomes positive. This means that with a decline of price the total revenue decreases if the demand is inelastic.

(iii) if a = b, the difference becomes zero. This means that with a decline in price the total revenue remains constant if the demand elasticity is unitary.

The same relationship can similarly be proved under a condition of

price rise.

1 James M. Henderson and Richard E. Quandt, Microeconomic Theory, A

Mathematical Approach (New York: McGraw-Hill Book Company, Inc., 1958), pp. 167-168.

2 Sidney Weintraub, Intermediate Price Theory (New York: Chilton Com?

pany, 1964), p. 18 and footnote No. 8.

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