The responsiveness of consumers to changes in the price of a good must be con- sidered by managers of price-setting firms when making pricing and output deci- sions. The price elasticity of demand gives managers essential information about how total revenue will be affected by a change in price. As it turns out, an equally important concept for pricing and output decisions is marginal revenue. Marginal revenue (MR) is the addition to total revenue attributable to selling one additional unit of output:
MR 5 DTRyDQ
Because marginal revenue measures the rate of change in total revenue as quan- tity changes, MR is the slope of the TR curve. Marginal revenue is related to price elasticity because marginal revenue, like price elasticity, involves changes in total revenue caused by movements along a demand curve.
Marginal Revenue and Demand
As noted, marginal revenue is related to the way changes in price and output af- fect total revenue along a demand curve. To see the relation between marginal revenue and price, consider the following numerical example. The demand sched- ule for a product is presented in columns 1 and 2 of Table 6.3. Price times quantity gives the total revenue obtainable at each level of sales, shown in column 3.
Marginal revenue, shown in column 4, indicates the change in total revenue from an additional unit of sales. Note that marginal revenue equals price only
marginal revenue (MR) The addition to total revenue attributable to selling one additional unit of output; the slope of total revenue.
F I G U R E 6.3 Constant Elasticity
of Demand 80
60
40
20
33.33 66.67
0
Price (dollars)
1,000 2,000 3,000
Quantity V: Ev = –1.5
U: Eu = –1.5 D: Q = 100,000P –1.5
for the first unit sold. For the first unit sold, total revenue is the demand price for 1 unit. The first unit sold adds $4—the price of the first unit—to total revenue, and the marginal revenue of the first unit sold equals $4; that is, MR 5 P for the first unit. If 2 units are sold, the second unit should contribute $3.50 (the price of the second unit) to total revenue. But total revenue for 2 units is only $7, indicating that the second unit adds only $3 (5 $7 2 $4) to total revenue. Thus the marginal revenue of the second unit is not equal to price, as it was for the first unit. Indeed, examining columns 2 and 4 in Table 6.3 indicates that MR , P for all but the first unit sold.
Marginal revenue is less than price (MR , P) for all but the first unit sold be- cause price must be lowered in order to sell more units. Not only is price lowered on the marginal (additional) unit sold, but price is also lowered for all the inframar- ginal units sold. The inframarginal units are those units that could have been sold at a higher price had the firm not lowered price to sell the marginal unit. Marginal revenue for any output level can be expressed as
MR 5 Price 2 Revenue lost by lowering price
on the inframarginal units
The second unit of output sells for $3.50. By itself, the second unit contributes
$3.50 to total revenue. But marginal revenue is not equal to $3.50 for the second unit because to sell the second unit, price on the first unit is lowered from $4 to
$3.50. In other words, the first unit is an inframarginal unit, and the $0.50 lost on the first unit must be subtracted from the price. The net effect on total revenue of selling the second unit is $3 (5 $3.50 2 $0.50), the same value as shown in column 4 of Table 6.3.
If the firm is currently selling 2 units and wishes to sell 3 units, it must lower price from $3.50 to $3.10. The third unit increases total revenue by its price, $3.10.
To sell the third unit, the firm must lower price on the 2 units that could have been sold for $3.50 if only 2 units were offered for sale. The revenue lost on the 2 inframarginal units is $0.80 (5 $0.40 3 2). Thus the marginal revenue of the third
inframarginal units Units of output that could have been sold at a higher price had a firm not lowered its price to sell the marginal unit.
(1) Unit sales
(2) Price
(3)
Total revenue
(4)
Marginal revenue (DTR/DQ)
0 $4.50 $ 0 —
1 4.00 4.00 $4.00
2 3.50 7.00 3.00
3 3.10 9.30 2.30
4 2.80 11.20 1.90
5 2.40 12.00 0.80
6 2.00 12.00 0
7 1.50 10.50 21.50
T A B L E 6.3
Demand and Marginal Revenue
unit is $2.30 (5 $3.10 2 $0.80), and marginal revenue is less than the price of the third unit.
It is now easy to see why P 5 MR for the first unit sold. For the first unit sold, price is not lowered on any inframarginal units. Because price must fall in order to sell additional units, marginal revenue must be less than price at every other level of sales (output).
As shown in column 4, marginal revenue declines for each additional unit sold.
Notice that it is positive for each of the first 5 units sold. However, marginal rev- enue is 0 for the sixth unit sold, and it becomes negative thereafter. That is, the seventh unit sold actually causes total revenue to decline. Marginal revenue is positive when the effect of lowering price on the inframarginal units is less than the revenue contributed by the added sales at the lower price. Marginal revenue is negative when the effect of lowering price on the inframarginal units is greater than the revenue contributed by the added sales at the lower price.
Relation Marginal revenue must be less than price for all units sold after the first, because the price must be lowered to sell more units. When marginal revenue is positive, total revenue increases when quan- tity increases. When marginal revenue is negative, total revenue decreases when quantity increases. Mar- ginal revenue is equal to 0 when total revenue is maximized.
Figure 6.4 shows graphically the relations among demand, marginal revenue, and total revenue for the demand schedule in Table 6.3. As noted, MR is below price (in Panel A) at every level of output except the first. When total revenue (in Panel B) begins to decrease, marginal revenue becomes negative. Demand and marginal revenue are both negatively sloped.
Sometimes the interval over which marginal revenue is measured is greater than one unit of output. After all, managers don’t necessarily increase output by just one unit at a time. Suppose in Table 6.3 that we want to compute marginal rev- enue when output increases from 2 units to 5 units. Over the interval, the change in total revenue is $5 (5 $12 2 $7), and the change in output is 3 units. Marginal revenue is $1.67 (5 DTRyDQ 5 $5y3) per unit change in output; that is, each of the 3 units contributes (on average) $1.67 to total revenue. As a general rule, whenever the interval over which marginal revenue is being measured is more than a single unit, divide DTR by DQ to obtain the marginal revenue for each of the units of output in the interval.
As mentioned in Chapter 2 and as you will see in Chapter 7, linear demand equations are frequently employed for purposes of empirical demand estimation and demand forecasting. The relation between a linear demand equation and its marginal revenue function is no different from that set forth in the preceding rela- tion. The case of a linear demand is special because the relation between demand and marginal revenue has some additional properties that do not hold for nonlin- ear demand curves.
When demand is linear, marginal revenue is linear and lies halfway between demand and the vertical (price) axis. This implies that marginal revenue must be twice as steep as demand, and demand and marginal revenue share the same
intercept on the vertical axis. We can explain these additional properties and show how to apply them by returning to the simplified linear demand function (Q 5 a 1 bP 1 cM 1 dPR) examined earlier in this chapter (and in Chapter 2). Again we hold the values of income and the price of the related good R constant at the spe- cific values __M and __P R, respectively. This produces the linear demand equation Q 5 a9 1 bP, where a9 5 a 1 c __M 1 d __P R. Next, we find the inverse demand equation by solving for P 5 f(Q) as explained in Chapter 2 (you may wish to review Technical Problem 2 in Chapter 2)
P 5 2 a__ 9 b 1 1 __ b Q 5 A 1 BQ
where A 5 2a9yb and B 5 1yb. Since a9 is always positive and b is always nega- tive (by the law of demand), it follows that A is always positive and B is always negative: A . 0 and B , 0. Using the values of A and B from inverse demand, the equation for marginal revenue is MR 5 A 1 2BQ. Thus marginal revenue is linear, has the same vertical intercept as inverse demand (A), and is twice as steep as in- verse demand (DMR/DQ 5 2B).
F I G U R E 6.4
Demand, Marginal Revenue, and Total Revenue
–0.50
Price and marginal revenue (dollars)
0
Quantity 2.00
1.50 2.50 4.00
1.00 0.50
–1.00 –1.50
5 4 3 2
1 Q
7 6 3.00
3.50
Panel A
D
MR
Total revenue (dollars)
Quantity 2.00
2 3 4 5 6 7 Q
4.00 6.00 8.00 10.00 12.00
TR
Panel B
0 1
Relation When inverse demand is linear, P 5 A 1 BQ (A . 0, B , 0), marginal revenue is also linear, intersects the vertical (price) axis at the same point demand does, and is twice as steep as the inverse de- mand function. The equation of the linear marginal revenue curve is MR 5 A 1 2BQ.
Figure 6.5 shows the linear inverse demand curve P 5 6 2 0.05Q. (Remember that B is negative because P and Q are inversely related.) The associated marginal revenue curve is also linear, intersects the price axis at $6, and is twice as steep as the demand curve. Because it is twice as steep, marginal revenue intersects the quantity axis at 60 units, which is half the output level for which demand inter- sects the quantity axis. The equation for marginal revenue has the same vertical intercept but twice the slope: MR 5 6 2 0.10Q.
Marginal Revenue and Price Elasticity
Using Figure 6.5, we now examine the relation of price elasticity to demand and marginal revenue. Recall that if total revenue increases when price falls and quan- tity rises, demand is elastic; if total revenue decreases when price falls and quan- tity rises, demand is inelastic. When marginal revenue is positive in Panel A, from a quantity of 0 to 60, total revenue increases as price declines in Panel B; thus de- mand is elastic over this range. Conversely, when marginal revenue is negative, at any quantity greater than 60, total revenue declines when price falls; thus demand must be inelastic over this range. Finally, if marginal revenue is 0, at a quantity of 60, total revenue does not change when quantity changes, so the price elasticity of demand is unitary at 60.
F I G U R E 6.5
Linear Demand, Marginal Revenue, and Elasticity (Q 5 120 2 20P )
Quantity
Price and marginal revenue (dollars)
3
60 120
E = 1
6 E > 1
E < 1
MR = 6 – 0.10 Q Inverse D: P = 6 – 0.05 Q
Panel A 4
2
0 40
Quantity
Total revenue (dollars)
180
120 TR = P 3Q = 6Q – 0.05Q2
Panel B E = 1
E > 1 E < 1
60 0
Except for marginal revenue being linear and twice as steep as demand, all the preceding relations hold for nonlinear demands. Thus, the following relation (also summarized in Table 6.4) holds for all demand curves:
Relation When MR is positive (negative), total revenue increases (decreases) as quantity increases, and demand is elastic (inelastic). When MR is equal to 0, the price elasticity of demand is unitary.
The relation among marginal revenue, price elasticity of demand, and price at any quantity can be expressed still more precisely. As shown in this chapter’s ap- pendix, the relation between marginal revenue, price, and price elasticity, for linear or curvilinear demands, is
MR 5 P ( 1 1 1 __ E )
where E is the price elasticity of demand and P is product price. When demand is elastic (|E| . 1), |1yE| is less than 1, 1 1 (1yE) is positive, and marginal revenue is positive. When demand is inelastic (|E| , 1), |1yE| is greater than 1, 1 1 (1/E) is negative, and marginal revenue is negative. In the case of unitary price elasticity (E 5 21), 1 1 (1yE) is 0, and marginal revenue is 0.
To illustrate the relation between MR, P, and E numerically, we calculate marginal revenue at 40 units of output for the demand curve shown in Panel A of Figure 6.5. At 40 units of output, the point elasticity of demand is equal to 22 [5 Py(P 2 A) 5 4/(4 2 6)]. Using the formula presented above, MR is equal to 2 [5 4(1 2 1y2)]. This is the same value for marginal revenue that is obtained by substituting Q 5 40 into the equation for marginal revenue: MR 5 6 2 0.1(40) 5 2.
Relation For any demand curve, when demand is elastic (|E| . 1), marginal revenue is positive. When demand is inelastic (|E| , 1), marginal revenue is negative. When demand is unitary elastic (|E| 5 1), mar- ginal revenue is 0. For all demand and marginal revenue curves:
MR 5 P ( 1 1 1 __ E )
where E is the price elasticity of demand.
Now try Technical Problems 12–14.