Updated June 30, 1998

# TOTAL REVENUE AND ELASTICITY

Figure 1a Demand Curve1
Figure 1b Total Revenue Curve

Figure 1a shows a downward-sloping straight line demand curve in which price is shown on the vertical axis and quantity demanded is shown on the horizontal axis. The midpoint of the demand curve is of special interest because, as will be demonstrated later, that is the point at which the curve is unit-elastic (elasticity = 1) and the point at which total revenue (Figure 1b) is maximized. (2)

Directly beneath the demand curve graph is the total revenue curve. Both graphs share the same horizontal axis measuring quantity demanded. The law of the downward-sloping straight line demand curve is that the total revenue curve will be a parabola whose high point ("vertex") is the point at which total revenue is maximized.

Whereas two points determine a straight line, three points determine the parabola and can be found as follows. The quantity-value for which the demand curve crosses the vertical axis has the highest possible price but nothing is sold at such a high price: the point (0,0) is on the total revenue (TR) curve.

The demand curve's maximum quantity (Q) value has, unfortunately, a zero price (P) and thus produces a TR = 0. A vertical line dropped from the horizontal axis intercept of the demand curve will produce a second point on the parabola at which TR = 0.

The third point of the total revenue curve is derived as follows. The symmetry of the parabola implies that the Q-value for the vertex of the parabola will be at the midpoint of the two Q-axis intercepts of the total revenue curve, a point that is vertically beneath the midpoint of the demand curve.

ELASTICITY

The algebraic formula for elasticity (e) is

e = - %DQ divided by %DP

Since %DP and %DQ have opposite signs (when price goes up, demand goes down and vice versa), elasticity is always positive.

The simplest graphical formula for elasticity is

e = (B/Q) - 1

where B is the horizontal axis intercept of the demand curve.

Each point on the demand curve has a different elasticity, ranging from zero (when Q = B) on up to values approaching plus infinity as Q approaches a value of zero. 3

Looking at the formula for elasticity: if Q is less than B/2, then the elasticity is greater than one; if Q is more than B/2, then elasticity is less than one; and if Q = B/2, then the elasticity is equal to one and you're at the midpoint of the demand curve. Thus the unit-elasticity point is also the midpoint of the demand curve. If, from that midpoint you drop a vertical line down to the parabola, TR, you have the point of maximum total revenue.

IMPLICATIONS

An analysis of Figure 1 suggests that a revenue-maximizing firm which is not operating at the midpoint of its demand curve should raise price if demand is inelastic (e less than one) and should lower price if demand is elastic (e greater than one). If demand is inelastic, the customers are "unresponsive" ("Oh, you raised price? Why I hadn't even noticed!"). But if demand is elastic, customers are very responsive and will take notice of even a nickel increase. Hence you should raise price only when demand is inelastic which means you are to the right of the demand curve midpoint and to the right of TR Max.

TWO ALGEBRAIC EXAMPLES

First, some background. The simplest formula for the equation of a straight line can be used if the two axis intercepts are known. If we call the vertical axis intercept (X,Y) = (Q,P) = (0, A) and the horizontal axis intercept (X,Y) = (Q,P) = (B,0), then the "Double Intercept" formula is

Y/A + X/B = 1

where A is the vertical axis intercept and B is the horizontal axis intercept. In English, the Double Intercept formula is

"Y over its intercept plus X over its intercept equals one"

Other basic facts are that total revenue is price times quantity and that marginal revenue is the derivative of total revenue.

Example #1

The vertical axis intercept of the demand curve is (Q,P) = (0,200) and the horizontal axis interecept is (Q,P) = (40,0). Find the following:

• equation of the demand curve in the form P = mX + b (slope-intercept form)
• equation of TR curve in form TR = f(Q)
• equation of marginal revenue (MR) curve
• elasticity when
• Q = 0
• Q = 20
• Q = 40
• TR Max (numerical value)
Solution

P/200 + Q/40 = 1
P + 5Q = 200
P = -5Q + 200 Demand Curve in Slope-Intercept Form
TR = (P)(Q) = (-5Q + 200)(Q)
TR = -5Q2 + 200Q Total Revenue Curve
MR = derivative of total revenue = derivative of (-5Q2 + 200Q )
MR = -10Q + 200 Marginal Revenue Curve
When Q = 0, both the demand curve and marginal revenue curve are the same point in Fig 1a
When MR = 0, then Q= 20, which is the midpoint of the interval [0,40] on the horizontal axis.
Elasticities

• when Q = 0, e is undefined, but as Q approaches zero e approaches plus infinity
• when Q = 20, e = 1 "unit elasticity"
• when Q = 40, e = 0
TR Max = PQ = (20)(100) = 2,000

Example #2

The vertical axis intercept of the demand curve is (Q,P) = (0,300) and the horizontal axis intercept is (Q,P) = (100,0). Find the following:

• equation of the demand curve in the form P = mX + b (slope-intercept form)
• equation of TR curve in form TR = f(Q)
• equation of marginal revenue (MR) curve
• elasticity when
• Q = 0
• Q = 50
• Q = 100
• TR Max (numerical value)
Solution
 P = -3Q + 300 TR = -3Q2 + 300Q MR = -6Q + 300 elasticities:undefined, but approaching + infinity e = 1e = 0 TR Max = 7500

TOTAL REVENUE, TOTAL RETURNS, AND MOMENTUM

MOMENTUM

There is a level of sales, such as sales for a given year, and there is a percentage change in sales obtained by looking at sales in two different time periods. Much more information is obtained from percentage change numbers than from levels numbers. Momentum is a measure of whether sales are, as an example, increasing at a increasing rate (good news) or increasing at a decreasing rate (bad news). If you remember your calculus days, a curve is concave up if the second derivative is positive in which case the slope of the curve is increasing by either becoming more positive or less negative.

EXAMPLES:

 Year EX#3 EX#4 EX#5 EX#6 1990 100 100 100 100 1991 105 110 105 110 1992 96 121 89 115 1993 90 130 87 105 1994 120 150 80 125 1995 125 160 82 120 1996 131 170 78 127 1997 140 172 85 115 1998 156 180 96 108

Example #3 (see data in above table, 1990-1998)
Calculate the following and form a conclusion:
(a) %D(1990-98)
(b) %D(1990-94)
(c) %D(1994-98)

Solution

%D(1990-98) = 156/100 - 1 = 56%
%D(1990-94) = 120/100 - 1 = 20%
%D(1994-98) = 156/120 - 1 = 30%
Conclusion: Positive Momentum -- increasing at an increasing rate

Solution to Example #4:
%D(1990-98) = 80%
%D(1990-94) = 50%
%D(1994-98) = 20%
Conclusion: Negative Momentum -- increasing at a decreasing rate

Solution to Example #5
%D(1990-98) = -4%
%D(1990-94) = -20%
%D(1994-98) = +20%
Conclusion: Favorable Momentum Reversal

Solution to Example #6
%D(1990-98) = 8%
%D(1990-94) = 25%
%D(1994-98) = -13.6%
Conclusion: Unfavorable Momentum Reversal

TOTAL REVENUE

Total revenue is also known as sales or dollar sales. We can distinguish between domestic sales of a U. S. company, TRDOM, and sales by the same company in the rest-of-the-world, TRROW. Total revenue is price (P) times quantity (Q). Quantity sold is frequently referred to as unit sales or volume of sales. Since the firm's financial statement is in dollars, both types of TR will need to be measured in dollars. Assume a two-country world in which the "ROW" is Germany (G).

Domestic Sales:

TRDOM = (P)(Q)
tup(TRDOM) = tup(P)tup(Q)
1 + %DTRDOM = (1 + %D P)(1 + %DQ)

Sales In ROW measured in dollars:

TRROW = (TRG)(VODM) = [(PG)(Q)](VODM)
tup(TRROW) = tup(TRG)tup(VODM) = tup(PG)tup(Q)tup(VODM)
1 + %DTRROW = (1 + %DTRG)(1 + AODM) = (1 + %DPG)(1 + %DQ)(1 + AODM)

Example #7

Total revenue went up 5% and unit sales were up 12%. Find the percentage change in price.

Solution:
1 + .05 = (1 + %DP)(1 + .12)
%D P = -6.25%

Example #8

News item (May 28, 1998). IBM Chairperson Louis Gerstner Jr. told industry analysts that revenue could grow at double-digit rates, in constant currency terms. Assume, for simplicity, that IBM sells only in Germany and that sales converted into dollars were up 5% and the value of the deutschemark fell 10%. Find the percentage increase in sales in constant currency terms. Hint: sales in constant currency terms means sales as if there had been no change in the exchange rate, so you need to find what happened to sales in Germany itself -- sales in DM's.
Solution

1 + .05 = (1 + %DTRG)(1 - .10)
1 + %DTRG = 1.16667
%DTRG = 16.667%
Sales in constant currency terms grew over 16%, which is growth at a double digit rate.

Example #9

Same data as Example #8 except that the value of the deutschemark rose 12%. Find the percentage change in constant currency terms.

Solution:

1 + .05 = (1 + %DTRG)(1 + .12)
1 + %DTRG = .9375
%DTRG = -6.25%
Sales in constant currency terms fell 6.25% -- sales in Germany itself fell 6.25, but sales in dollars were up due to the favorable impact of falling dollar or, equivalently, the rising deutschemark.

Example #10

News item (Dec. 9, 1997).Coca Cola stock fell 4% due to concern about the continued weakness of foreign currencies against the dollar. The strength of the dollar has been of concern to Coca Cola investors since the company derives 80% of its profit from overseas. Suppose sales in DM's were 1000DM and the VODM was \$2/1DM in the initial period and that sales in the subsequent period were still 1000DM but the new exchange rate was \$1.50/1DM; in other words, the VODM fell and the VOD rose. Find the percentage decrease in the VODM, percentage increase in the VOD, and the percentage change in the dollar value of sales in Germany.

Solution
%DVODM = -25%
percentage decrease in VODM = 25%
%DVOD = 33.333% = percentage increase in VOD
percentage change in dollar sales = -25% -- dollar sales fell by the same percentage amount as did the VODM. Implication: a strong dollar -- a rising dollar -- is bad news for companies with a lot of business overseas. This is the negative impact of a strong dollar.

Example #11

News item (July 22, 1997). A T & T's Business Services unit posted a strong 15% growth rate in traffic volume, triggering a moderate 4.1 % increase in revenue to \$5.6 billion. Find the following:
(a) percentage change in average price charged
(b) elasticity. Hint: use e = - %DQ/%DP

Solution

(a) 1 + %DTR = (1 + %D P)(1 + %DQ)
1 + .041 = (1 + %DP)(1 + .15)
%DP = -9.478%
(b) e= -(.15)/(-.09478) = 1.58

TOTAL RETURNS FROM STOCKS

The "internal rate of return" requires that investment earnings be reinvested so that compound interest takes place. Assume that all dividends received are used to buy additional shares of stock . The rate of return can be calculated by finding two simple percentage changes: the percentage change in the quantity of stock held and the percentage change in the price of the stock. The initial value [call it TR for Total Returns where TR = (P)(Q)] is the initial price times the initial quantity -- P0 times Q0 and the ending value after one time period is P1 times Q1:
TR = (P)(Q)
tup(TR) = tup(P)tup(Q)
1 + %DTR = (1 + %DP)(1 +%DQ)
The %DTR is the percentage change in the value of your investment and is thus the rate of return ROR. The percentage change in price is the capital gain yield, CGY. The percentage change in quantity represents the dividend yield , DY. The formula then becomes
1 + ROR = (1 + CGY)(1 + DY)
and
ROR = CGY + DY + (CGY)(DY)

For investment by a U.S. resident in the German stock market, the formula will be

1 + ROR = (1 + CGY)(1 + DY)(1 + AODM)

Example #12

You buy 100 shares of stock at \$20 a share. Ten years later (or one time period later) the price is \$40 a share and you have 240 shares. Find the following:

(a) ROR
(b) DY
(c)CGY

Solution:

(a)The initial value of the investment is (\$20 per share)(100 shares) = \$2,000. The ending value is (\$40)(240) = \$9600. The percentage change in the value of the investment is (9600/2000) - 1 = 3.80 = 380%, so ROR = 380%.
(b) The dividend yield DY is (ending # of shares/beginning # of shares) - 1 = 1.40 = 140%
(c) The capital gain yield CGY is (ending price)/(beginning price) - 1 = 2 - 1 = 1 = 100%
1 + ROR = (1 + CGY)(1 + DY) = (2)(2.4) = 4.8 and ROR = 4.8 - 1 = 3.80 = 380%

Example #13

You invest \$1000 in the German stock market and the initial value of the dollar is 2DM/\$1 so your investment in deutschemarks is 2000DM. You buy 50 shares at 40DM a share. The ending value is 75 shares valued at 90 DM a share. The ending va lue of the dollar is 1.25DM/\$1. Find the following:

(a) CGY
(b) DY
(c) AODM
(d) ROR

Solution:

(a) CGY = 90/40 - 1 = 1.25 = 125%
(b) DY = (75/50) - 1 = .50 = 50%
(c) initial VODM = \$ .50/1DM and the ending VODM is \$ .80/1DM so the percentage change in the VODM, or AODM, is (.8/.5) - 1 = 1.6 - 1 = .60 = 60%
(d) 1 = ROR = (1 + 1.25)(1 + .50)(1 + .60)= 5.4 and ROR = 5.4 - 1 = 4.40 = 440%

1. Thanks very much to Jeffrey Lau for the graph!

2. The marginal revenue (MR) curve starts where the demand curve does but drops twice as fast and thus intersects the horizontal axis at a point directly below the midpoint of the demand curve
3. The curve (not shown) with e on the vertical axis and Q on the horizontal axis is a rectangular hyperbola with center at (Q,e) = (0,-1).

.

Page created June 15, 1998