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PROFIT-MAXIMIZING QUANTITY-DEPENDENT PRICING
Here we will show how to find the profit-maximizing two-part tariff for Clearvoice,
the wireless telephone monopolist of Section 18.4, as well as the profit-maximizing
menu of two-part tariffs. We’ll assume, as in Section 18.4, that there are two
types of consumer: high-demand consumers, each of whose demand function is
QH 5 100 2 100PH , and low-demand consumers, each of whose demand function
is QL 5 50 2 100PL , where PH and PL are the per-minute prices in dollars in the two
service plans. These are the demand functions shown in Figures 18.9–18.12. The corresponding inverse demand functions are PH 5 1 2 0.01QH and PL 5 0.5 2 0.01QL .
We’ll assume in most of the discussion that there are 400 low-demand consumers
and 100 high-demand consumers.
TWO-PART TARIFFS
To find the most profitable two-part tariff, we need to compare the best tariff that
induces only high-demand consumers to buy with the best tariff that induces both
types of consumers to buy. As we saw in Section 18.4, if Clearvoice sells only to highdemand consumers, it does best by setting the per-minute price equal to the marginal
cost of 10 cents per minute. The fixed fee can then be set equal to a high-demand
consumer’s surplus at that price, which is $40.50 (see Worked-Out Problem 18.1,
page 632). Since there are 100 high-demand consumers, profit is $4,050.
Now let’s find the best two-part tariff when Clearvoice sells to both types of
consumers. Notice that if Clearvoice decides on the number of minutes to sell to
each low-demand consumer, QL , its decision determines both the per-minute charge
and the fixed fee (which equals a low-demand consumer’s surplus at that price).
To find the best two- part tariff, then, we can examine the effect of marginally
increasing the quantity that the low-demand consumer buys, QL , by one minute.
At the profit-maximizing quantity, the marginal revenue from this change should
equal its marginal cost (according to the No Marginal Improvement Principle from
Chapter 3).
Marginally increasing the quantity that a low-demand consumer buys by one
minute changes Clearvoice’s revenue in three ways:
1. It changes the revenue Clearvoice receives from a low-demand consumer through
the sale of minutes. If Clearvoice sells one additional minute to a low-demand
consumer, the marginal revenue effect from the sale of minutes to that consumer is
MRL 5 P 1 (DP/DQL)QL
5 (0.5 2 0.01QL) 2 0.01QL
(1)
5 0.5 2 0.02QL
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2. It changes the revenue Clearvoice receives from a high-demand consumer through
the sale of minutes. Since the change in the per-minute price is ∆P 5 20.01∆QL ,
and ∆QH 5 2100∆P , the change in QH is ∆QH 5 2100(20.01 ∆QL) 5 QL . That
is, the high-demand quantity changes by the same amount as the low-demand
quantity, namely one minute. So the marginal revenue from the sale of minutes
to a high-demand consumer is
MRH 5 P 1 (DP/DQH)QH
5 (1 2 0.01QH) 2 0.01QH
5 1 2 0.02QH
(2)
Since at any per-minute price we have QH 5 QL 1 50, we can substitute for QH in
expression (2) and write this marginal revenue as a function instead of QL:
MRH 5 1 2 0.02(QL 1 50)
5 20.02QL
(3)
3. It changes the fixed fee, F, that Clearvoice can charge without losing the lowdemand consumer. The extra surplus the low-demand consumer enjoys when
the price falls by ∆P is approximately 2(∆P 3 QL), which is the low-demand
consumer’s saving from the price decrease. (Draw a graph showing the lowdemand consumer’s surplus before and after the price change and examine
the difference; it is approximately a rectangle with a height of ∆P and a width
of QL.) So an increase of one minute in the number of minutes bought by a lowdemand consumer, which requires a decrease in the per-minute price equal to
∆P 5 20.01, raises the fixed fee the monopolist can charge by
DF 5 0.01QL
(4)
The overall marginal revenue from this change is therefore
MR 5 500(DF ) 1 400(MRL) 1 100(MRH )
5 500(0.01QL) 1 400(0.5 2 0.02QL) 1 100(20.02QL)
(5)
5 200 2 5QL
How does this change affect costs? With the change, each consumer buys one additional minute. Thus, 500 additional minutes are sold. The marginal cost is therefore
MC 5 $50. So we can find the best per-minute price by equating marginal revenue
with marginal cost:
200 2 5QL 5 50
Solving for QL , we find that QL 5 30. From the low-demand consumer’s inverse
demand function, this implies that the profit-maximizing per-minute price is 20 cents.
The fixed fee equals the low-demand consumer’s surplus when facing a per-minute
price of 20 cents, which is $4.50 (see Figure 18.10, page 645).
Now let’s calculate Clearvoice’s profit under this two-part tariff. Low-demand
consumers buy 30 minutes and high-demand consumers buy 80 minutes. So the total
number of minutes sold is (400 3 30) 1 (100 3 80) 5 20,000. Profit from the sale
of those minutes is therefore $2,000 since Clearvoice makes a 10-cent profit on each
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minute it sells. All 500 consumers also pay the fixed fee, yielding another $2,250.
Total profit is therefore $4,250 per month. Since that amount is greater than the
profit from selling only to high-demand consumers ($4,050), this plan is the profitmaximizing two-part tariff.
Finally, how does the optimal per-minute price when selling to both types of consumers depend on the proportion of low-demand versus high-demand consumers?
To see the answer, suppose there are N consumers in total, and a share sL of them
are low-demand consumers. So there are sLN low-demand consumers and (1 2 sL)N
high-demand ones. Now, the overall revenue effect of selling one additional minute
to a low-demand consumer is
MR 5 N(DF ) 1 sLN(MRL ) 1 (1 2 sL )N(MRH )
5 N 3 (0.01QL) 1 sL(0.5 2 0.02QL ) 1 (1 2 sL )(20.02QL ) 4
(6)
5 N 3 0.5sL 2 0.01QL 4
The extra cost of expanding QL by one minute is (0.10 3 N) since every consumer expands his consumption by one minute (recall that ∆QH 5 ∆QL ), there are
N total consumers, and providing a minute of service costs Clearvoice 10 cents. So
the marginal cost is MC 5 0.10N. Setting marginal revenue equal to marginal cost,
we have
N 3 0.5sL 2 0.01QL 4 5 0.10N
(7)
Dividing both sides of (7) by the total number of consumers N, we can rewrite (7) as
0.5sL 2 0.01QL 5 0.10
(8)
The solution is
QL 5
0.5sL 2 0.10
0.01
(9)
Notice that for any proportion of low-demand consumers less than one (sL , 1), QL
is less than 40, which—from the low-demand inverse demand function—implies that
the per-minute price is greater than the marginal cost of 10 cents per minute. Moreover, QL falls as the fraction of low types, sL , gets smaller, which means that the perminute price increases. Clearvoice’s most profitable two-part tariff is then found by
comparing the profit of this plan that makes sales to both types of consumers to the
profit from selling only to the high-demand consumers.
THE PROFIT-MAXIMIZING MENU OF TWO-PART TARIFFS
Now let’s find the most profitable menu of two-part tariffs.1 As in Section 18.4, we
know that the most profitable policy includes a two-part tariff plan intended for
high-demand consumers with a per-minute charge of 10 cents (equal to the marginal cost).
A high-demand consumer will buy 90 minutes under this plan. The profitmaximizing policy also includes a two-part tariff intended for low-demand consumers
1In
fact, the most profitable menu of two-part tariffs can be shown to be the most profitable of any type of sales policy in
this setting.
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that caps the number of minutes that can be purchased at the number a low-demand
consumer desires given that plan’s per-minute price. (The cap reduces the benefit a highdemand consumer derives if he chooses the low-demand plan, raising the amount the
high-demand consumer can be charged as a fixed fee.)
Clearvoice therefore need only determine the per-minute price in the low-demand
plan and the fixed fee for each plan. It can decide on these three items by thinking
about the number of minutes low-demand consumers should buy, which we’ll call
QL. Once QL has been set, all three items can be determined (see the discussion in
Section 18.4):
•
•
•
The low-demand plan’s per-minute price is the price that causes a low-demand
consumer to buy QL minutes of time. It is determined from a low-demand consumer’s
demand curve (or more precisely, the low-demand consumer’s inverse demand
function).
The low-demand plan’s fixed fee equals a low-demand consumer’s surplus at that
per-minute price.
The high-demand plan’s fixed fee is the fixed fee that makes a high-demand
consumer indifferent between the high-demand and low-demand plans, given
that the per-minute price in the high-demand plan is 10 cents.
For example, if Clearvoice wants the low-demand consumer to buy 30 minutes, then
(as we saw in Figure 18.12 and the third column of Table 18.3 on pages 651 and 653)
the low-demand plan’s per-minute price should be 20 cents, the low-demand plan’s
fixed fee should be $4.50, and the high-demand plan’s fixed fee should be $25.50.
Clearvoice’s profit, given 400 low-demand and 100 high-demand consumers, would
be $5,550 (see Table 18.3).
To find the most profitable pricing policy, we can look at the effects of increasing
the number of minutes the low-demand consumer buys, QL, by one minute. With the
profit-maximizing plan, the marginal revenue from this change must equal its marginal cost. This marginal increase in QL has three effects on revenue:
1. It changes the revenue Clearvoice receives from a low-demand consumer through
the sale of minutes. This is the same as the first effect we saw when finding the
most profitable two-part tariff. Thus,
MRL 5 0.5 2 0.02QL
(10)
2. It changes the fixed fee Clearvoice can charge without losing the low-demand
consumer. This is the same as the third effect we saw when finding the most
profitable two-part tariff. Letting FL be the fixed fee in the low-demand plan,
we have
DFL 5 0.01QL
(11)
3. It changes the fixed fee Clearvoice can charge a high-demand consumer. To
illustrate this effect, the red-striped area in Figure 1 shows the reduction in the
high-demand fixed fee when Clearvoice increases the number of minutes in
the low-demand plan from 30 to 31 minutes. That area is approximately equal
to the area of a rectangle whose height is the distance between the high- and
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0.70
0.69 0.60
0.50
0.40
0.30
0.20
P 5 0.19
0.15
1.00
0.90
Price ($/minute)
0.80
0.70
0.60
DH
0.50
0.40
The Change in the
High-Demand Plan’s Fixed
Fee. When the amount
sold in the low-demand plan
increases from 30 to 31 minutes
(corresponding to a reduction
in the per-minute price from
20 to 19 cents), the high-demand
plan’s fixed fee is reduced by the
red-striped area.
5 $0.50
DL
25
DL
Figure 1
DH
30 31
35
5 1 minute
0.30
0.20
MC
0.10
10 20 30 40 50 60 70 80 90 100
31
Minutes per month
low-demand consumers’ demand curves, $0.50 [5 (1 2 QL) 2 (0.5 2 QL)], and
whose width is one, the change in the number of low-demand minutes. So the
change in the high-demand plan’s fixed fee is
DFH 5 20.5
(12)
Putting these three effects together, the marginal revenue from this change is
MR 5 400(MRL ) 1 400(DFL ) 1 100(DFH )
5 400(0.5 2 0.02QL ) 1 400(0.01QL ) 2 100(0.5)
(13)
5 150 2 4QL
What is the effect of this change on costs? The 400 low-demand consumers will
each buy one more minute, but the high-demand consumers won’t change their purchase quantity: they will continue to buy 90 minutes. So the marginal cost is $40. The
profit-maximizing per-minute price for low-demand consumers equates marginal
revenue with marginal cost:
150 2 4QL 5 40
Solving for QL tells us that QL 5 27.5 minutes. From the low-demand consumer’s
inverse demand function, this implies that the per-minute price in the low-demand
plan is 22.50 cents per minute. The fixed fee in the low-demand plan should equal
the low-demand consumer’s surplus at a price of 22.50 cents per minute, which is
$3.78125 (5 0.5 3 0.275 3 27.5).
The high-demand fixed fee is the amount that makes a high-demand consumer
indifferent between the two plans. At a per-minute price of 10 cents, the high-demand
consumer has a surplus of $40.50 cents. His overall surplus from the high-demand plan
is therefore $40.50 cents less the high-demand plan’s fixed fee FH . If instead he chooses
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the low-demand plan, he will have a surplus of $13.75, which equals the 27.5 minutes
he will use times the $0.50 amount by which his willingness to pay for each of those
minutes exceeds that of a low-demand consumer. (Recall that a low-demand consumer
has a surplus of zero from this plan. Since a high-demand consumer uses the same number of minutes and pays the same charges, his surplus from the low-demand plan equals
the amount that his willingness to pay exceeds that of the low-demand consumer.) So
the fixed fee that makes him indifferent between the two plans satisfies
40.5 2 FH 5 13.75
Solving this formula tells us that the high-demand plan’s fixed fee should be $26.75.
In sum, the profit-maximizing menu offers two plans: one intended for highdemand consumers, with a monthly fee of $26.75 and a per-minute price of 10 cents,
and the other intended for low-demand consumers, with a monthly fee of $3.78125,
a per-minute price of 22.5 cents, and a cap at 27.5 minutes per month. Clearvoice
will earn $4,187.50 in fixed fees and approximately $1,375 in per-minute sales to lowdemand consumers. (It will earn nothing on sales of minutes to high-demand consumers, since their price equals its marginal cost.) So its total profit will be $5,562.50
per month, $1,312.50 more than the $4,250 it would earn from the most profitable
two-part tariff we identified above (roughly a 30 percent increase).
Finally, how does the optimal per-minute price in the low-demand plan depend
on the proportion of low-demand versus high-demand consumers? To see the answer,
suppose again that the share of low-demand consumers is sL . Now, the overall revenue effect of selling one additional minute to a low-demand consumer is
MR 5 N 3 sL(MRL ) 1 sL(DFL ) 1 (1 2 sL )(DFH ) 4
5 N 3 sL(0.5 2 0.02QL ) 1 sL(0.01QL ) 1 (1 2 sL )(20.5) 4
(14)
5 N 3 sL(1 2 0.01QL ) 2 0.5 4
The marginal cost of expanding QL by one minute is MC 5 (0.10 3 sLN ) since only
low-demand consumers change their consumption. Setting marginal revenue equal
to marginal cost, we have
N 3 sL(1 2 0.01QL ) 2 0.5 4 5 0.10sL N
(15)
Dividing both sides of (15) by the total number of consumers N, we can rewrite
(15) as
sL(1 2 0.01QL) 2 0.5 5 0.10sL
(16)
The solution is
QL 5 90 2
50
sL
(17)
Notice that for any proportion of low-demand consumers less than one (sL , 1), QL
is less than 40, which—from the low-demand inverse demand function—implies
that the per-minute price in the low-demand plan is greater than the marginal cost
of 10 cents. Moreover, QL falls as the fraction of low types, sL , gets smaller, which
means that the per-minute price in the low-demand plan increases.
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