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EconS 503 – Advanced Microeconomics II
1
Adverse Selection
Handout on Two-part tariffs (Second-degree price discrimination)
1. Introduction
Consider a setting where an uninformed firm is attempting to sell an item to a privately informed
customer. The firm’s profit function is 𝐹𝐹 − 𝑐𝑐𝑐𝑐, where 𝑐𝑐 > 0 represents the firm’s marginal costs, and F is
the fee paid from the customer to the firm in exchange for q units of the good (price for the package of
units, rather than a unit price). The customer’s utility function is 𝑢𝑢(𝑐𝑐, 𝑇𝑇, 𝜃𝜃) = 𝜃𝜃 ∙ 𝑣𝑣(𝑐𝑐) − 𝐹𝐹, where 𝑢𝑢′ > 0
and 𝑢𝑢′′ < 0. Parameter 𝜃𝜃 is privately observed by the consumer, and takes on either 𝜃𝜃
𝐿𝐿
with probability 𝛽𝛽
or 𝜃𝜃
𝐻𝐻
with probability 1 − 𝛽𝛽, where 𝜃𝜃
𝐻𝐻
𝐿𝐿
2. Complete Information
[
nd
Stage] For a given pair of fee 𝑇𝑇
𝑖𝑖
and quantity 𝑐𝑐
𝑖𝑖
𝑖𝑖
𝑖𝑖
), consumers with valuation 𝜃𝜃
𝑖𝑖
purchase the
good if and only if 𝜃𝜃
𝑖𝑖
𝑖𝑖
𝑖𝑖
[
st
Stage] Observing 𝜃𝜃
𝑖𝑖
(as we are in the complete-information version) and anticipating the buyers
decision rule in the second stage, 𝜃𝜃 𝑖𝑖
𝑖𝑖
𝑖𝑖
≥ 0 , the firm solves the PMP
max
𝑇𝑇
𝑖𝑖
,𝑞𝑞
𝑖𝑖
𝑖𝑖
𝑖𝑖
subject to 𝜃𝜃
𝑖𝑖
𝑖𝑖
𝑖𝑖
The participation constraint (P.C.) must bind. Otherwise 𝑇𝑇 𝑖𝑖
can be further increased, thus increasing
profits. Hence, 𝜃𝜃 𝑖𝑖
𝑖𝑖
𝑖𝑖
, which simplifies the above problem to the following unconstrained
maximization problem
max
𝑞𝑞
𝑖𝑖
𝑖𝑖
𝑖𝑖
𝑖𝑖
Taking F.O.C with respect to 𝑐𝑐 𝑖𝑖
𝑖𝑖
′
𝑖𝑖
) − 𝑐𝑐 ≤ 0 (= 0 in interior solutions)
Hence, under complete information, 𝑐𝑐 𝑖𝑖
is increased until the point in which the consumer’s marginal
utility of additional units coincides with the firm’s marginal cost. As we next show, when the firm is
uninformed about the customer’s type, this result doesn’t necesarily arise.
1
Felix Munoz-Garcia, Associate Professor, School of Economic Sciences, Washington State University, Pullman,
WA 99164-6210, fmunoz@wsu.edu.
MV = MC
3. Incomplete Information
The firm cannot observe the realization of 𝜃𝜃. The firm could offer contracts of the form (𝑇𝑇(𝑐𝑐), 𝑐𝑐), with
function 𝑇𝑇(𝑐𝑐) being as general as you can imagine.
For simplicity, let’s consider three types of contracts:
- Linear pricing: 𝑇𝑇(𝑐𝑐) = 𝑝𝑝 ∙ 𝑐𝑐 customers pay 𝑝𝑝 for every unit they buy.
- Nonlinear pricing (single two-part tariff for all types of customers)
- Nonlinear pricing (two two-part tariffs one for each type of customer)
3.1. Linear pricing, 𝑻𝑻(𝒒𝒒) = 𝒑𝒑 ∙ 𝒒𝒒
[
nd
Stage] Every customer with type 𝜃𝜃
𝑖𝑖
pays a price p per unit of q purchased, thus obtaining a utility
𝑖𝑖
− 𝑝𝑝𝑐𝑐 for all 𝑖𝑖 = {𝐻𝐻, 𝐿𝐿}
In order to maximize his utility (for every given p ), he increases q until
𝑖𝑖
1
Solving for q , we find 𝜃𝜃
𝑖𝑖
−Walrasian demand
𝑖𝑖
𝑖𝑖
Hence, 𝜃𝜃
𝑖𝑖
−customer’s utility is
𝑖𝑖
𝑖𝑖
𝑖𝑖
[
st
Stage] By backward induction, the monopolist anticipates the demand function 𝐷𝐷
𝑖𝑖
for 𝜃𝜃
𝑖𝑖
−type
buyer. Hence, the firm maximizes expected profits:
max
𝑝𝑝
(𝑝𝑝 − 𝑐𝑐) ∙ [𝛽𝛽 ∙ 𝐷𝐷
𝐿𝐿
𝐻𝐻
(𝑝𝑝)]
Let 𝐷𝐷(𝑝𝑝) ≡ 𝛽𝛽 ∙ 𝐷𝐷 𝐿𝐿
𝐻𝐻
(𝑝𝑝) denote the expected demand, which helps us simplify the above
program to
max
𝑝𝑝
Taking FOC with respect to p yields
′
Solving for p , we obtain a linear price of
𝑖𝑖
𝑖𝑖
Mathematically,
max
(𝐹𝐹,𝑝𝑝)
𝛽𝛽[𝐹𝐹 + (𝑝𝑝 − 𝑐𝑐) ⋅ 𝐷𝐷
𝐿𝐿
(𝑝𝑝)] + (1 − 𝛽𝛽)[𝐹𝐹 + (𝑝𝑝 − 𝑐𝑐) ⋅ 𝐷𝐷
𝐻𝐻
(𝑝𝑝)]
)[
𝐿𝐿
𝐻𝐻
𝐷𝐷(𝑝𝑝), i.e., expected demand
]
subject to 𝐹𝐹 ≤ 𝑆𝑆
𝑖𝑖
(𝑝𝑝) for all 𝑖𝑖 = {𝐻𝐻, 𝐿𝐿}
However, the seller can increase 𝐹𝐹 until 𝐹𝐹 = 𝑆𝑆 𝐿𝐿
(𝑝𝑝). Raising it any further would lead the low-type
customers to reject the purchase. Plugging 𝐹𝐹 = 𝑆𝑆 𝐿𝐿
(𝑝𝑝) into the above problem helps us obtain an
unconstrained PMP (with only one choice variable, 𝑝𝑝), as follows
max
𝑝𝑝
𝐿𝐿
Taking first-order conditions with respect to 𝑝𝑝 yields
𝐿𝐿
′
′
Solving for 𝑝𝑝 and rearranging, we obtain a price of the single two part tariff, 𝑝𝑝
𝑆𝑆𝑇𝑇𝐿𝐿𝑇𝑇
, of
𝑆𝑆𝑇𝑇𝐿𝐿𝑇𝑇
′
𝑝𝑝
𝐿𝐿𝐿𝐿
, price under
linear pricing
𝐿𝐿
′
Where the last term is positive since 𝑆𝑆
𝐿𝐿
′
(𝑝𝑝) < 0 and 𝐷𝐷
′
Remark : 𝑆𝑆
𝑖𝑖
′
(𝑝𝑝) can be found by applying the Envelope Theorem on
𝑖𝑖
𝑖𝑖
[
𝑖𝑖
)]
𝑖𝑖
In particular, second-order effects are absent, so that 𝐷𝐷
𝑖𝑖
(𝑝𝑝) is unaffected by a price change. As a
consequence
𝑖𝑖
′
𝑖𝑖
𝑖𝑖
Hence, prices in each setting are ranked as follows:
𝑆𝑆𝑇𝑇𝐿𝐿𝑇𝑇
𝐿𝐿𝐿𝐿
𝑐𝑐 (price under perfect competition)
The firm then sets a single two-part tariff
𝐿𝐿
𝑆𝑆𝑇𝑇𝐿𝐿𝑇𝑇
𝑆𝑆𝑇𝑇𝐿𝐿𝑇𝑇
Practice : Considering a demand function 𝐷𝐷 𝑖𝑖
𝑖𝑖
− 𝑝𝑝, where 𝜃𝜃
𝑖𝑖
= {1,2} and 𝛽𝛽 =
1
2
, find the profit-
maximizing two-part tariff.
In addition, 𝑐𝑐 𝐻𝐻
𝐻𝐻
𝑆𝑆𝑇𝑇𝐿𝐿𝑇𝑇
𝐿𝐿
𝑆𝑆𝑇𝑇𝐿𝐿𝑇𝑇
𝐿𝐿
. We can depict this two-part tariff in the (𝐹𝐹, 𝑝𝑝)-
quadrant, as follows.
p
STPT
F =S L
(p
STPT
)
S
L
(p
STPT
) +p
STPT
q
L
q
H
q
F
Graphical representation of the indifference curves using the same (𝐹𝐹, 𝑝𝑝)-quadrant:
q
F
θ i
-type
indifference curve
Same utility from:
-LowF and lowq
-HighF and highq
q
F
IC
i
IC
i
Increasing utility
For compactness, the literature refers to the former conditions as “participation constraints,” as they
guarantee the participation of all types of customers; whereas the latter conditions are referred to as
“incentive compatibility” conditions. In particular, the participation constraints in this context are
𝐿𝐿
𝐿𝐿
𝐿𝐿
𝐿𝐿
𝐻𝐻
𝐻𝐻
𝐻𝐻
𝐻𝐻
while the incentive compatibility conditions are
𝐿𝐿
𝐿𝐿
𝐿𝐿
𝐿𝐿
𝐻𝐻
𝐻𝐻
𝐿𝐿
𝐻𝐻
𝐻𝐻
𝐻𝐻
𝐻𝐻
𝐿𝐿
𝐿𝐿
𝐻𝐻
We can rearrange the above four inequalities and insert them as constraints into the monopolist’s profit
maximization problem, as follows:
max
𝐹𝐹 𝐿𝐿
,𝑞𝑞 𝐿𝐿
,𝐹𝐹 𝐻𝐻
,𝑞𝑞 𝐻𝐻
[
𝐻𝐻
𝐻𝐻
] + (
− 𝑝𝑝)[𝐹𝐹
𝐿𝐿
𝐿𝐿
]
subject to
𝐿𝐿
𝐿𝐿
𝐿𝐿
𝐻𝐻
𝐻𝐻
𝐻𝐻
𝐿𝐿
[𝑢𝑢(𝑐𝑐
𝐿𝐿
𝐻𝐻
)] + 𝐹𝐹
𝐻𝐻
𝐿𝐿
𝐻𝐻
[𝑢𝑢(𝑐𝑐
𝐻𝐻
𝐿𝐿
)] + 𝐹𝐹
𝐿𝐿
𝐻𝐻
Since both 𝑃𝑃𝐶𝐶 𝐻𝐻
and 𝐼𝐼𝐶𝐶
𝐻𝐻
are now expressed in terms of the fee 𝐹𝐹
𝐻𝐻
, we can easily see that the monopolist
increases 𝐹𝐹 𝐻𝐻
until such fee coincides with the lowest of 𝜃𝜃
𝐻𝐻
𝐻𝐻
and 𝜃𝜃
𝐻𝐻
[
𝐻𝐻
𝐿𝐿
)] +
𝐿𝐿
, as
depicted in figure 1, for all 𝑖𝑖 = {𝐿𝐿, 𝐻𝐻}. Otherwise, one (or both) constraints will be violated, leading the
high-demand customer to not participate (and/or select the two-part tariff meant for the low-demand
customer). We examine this result more closely in the next discussion.
PC
i
is
binding
IC
i
is
binding
Maximal F
i
that achives participation and self-selection
F
i
F
i
i
u ( q
i
i
u ( q
i
i
[ u ( q
i
) - u ( q
j
)] + F
j
i
[ u ( q
i
) - u ( q
j
)] + F
j
Maximal F
i
that achives participation and self-selection
Figure 1 PC condition binds (upper panel) and IC condition binds (lower panel)
High-demand customer. Let us first focus on the high-demand consumer and show that 𝐼𝐼𝐶𝐶
𝐻𝐻
is binding,
(the lower panel of figure 1 arises for this type of customer).
Proof. An indirect way to show that 𝐼𝐼𝐶𝐶 𝐻𝐻
binds, i.e., 𝐹𝐹
𝐻𝐻
𝐻𝐻
[𝑢𝑢(𝑐𝑐
𝐻𝐻
𝐿𝐿
)] + 𝐹𝐹
𝐿𝐿
, is to demonstrate that
𝐻𝐻
𝐻𝐻
𝐻𝐻
) (i.e., as depicted be in the lower panel of Figure 1). By contradiction, consider that
𝐻𝐻
𝐻𝐻
𝐻𝐻
). If this condition holds, then 𝐼𝐼𝐶𝐶
𝐻𝐻
can be rewritten as
𝐻𝐻
𝐻𝐻
𝐿𝐿
𝐿𝐿
𝐻𝐻
, which simplifies to 𝐹𝐹
𝐿𝐿
𝐻𝐻
𝐿𝐿
In addition, we can combine this result with the property that 𝜃𝜃 𝐻𝐻
𝐿𝐿
to obtain
𝐿𝐿
𝐻𝐻
𝐿𝐿
𝐿𝐿
𝐿𝐿
That is, 𝐹𝐹
𝐿𝐿
𝐿𝐿
𝐿𝐿
). This finding, however, violates the participation constraint of the low-demand
customer, 𝑃𝑃𝐶𝐶
𝐿𝐿
, indicating that we have reached a contradiction and, therefore, 𝐹𝐹
𝐻𝐻
𝐻𝐻
𝐻𝐻
) (i.e., 𝑃𝑃𝐶𝐶
𝐻𝐻
is not binding). Thus, 𝐼𝐼𝐶𝐶
𝐻𝐻
is binding but 𝑃𝑃𝐶𝐶
𝐻𝐻
is not, confirming that for the high-demand customer the
lower panel of Figure 1 applies (i.e., 𝐹𝐹 𝐻𝐻
𝐻𝐻
[𝑢𝑢(𝑐𝑐
𝐻𝐻
𝐿𝐿
)] + 𝐹𝐹
𝐿𝐿
). Q.E.D.
Low-demand customer. Let us now use a similar approach to show that the top panel of Figure 1 arises for
the low-demand customer (i.e., 𝑃𝑃𝐶𝐶
𝐿𝐿
binds since 𝐹𝐹
𝐿𝐿
𝐿𝐿
𝐿𝐿
Proof. Similarly as for high-demand customers, we can prove this result by instead showing that 𝐹𝐹 𝐿𝐿
𝐿𝐿
[𝑢𝑢(𝑐𝑐
𝐿𝐿
𝐻𝐻
)] + 𝐹𝐹
𝐻𝐻
holds. Proving this result by contradiction, assume that 𝐹𝐹
𝐿𝐿
𝐿𝐿
[𝑢𝑢(𝑐𝑐
𝐿𝐿
𝐻𝐻
)] + 𝐹𝐹
𝐻𝐻
. Plugging this expression into 𝐼𝐼𝐶𝐶
𝐻𝐻
(which binds, as shown in our discussion of the high-
demand customer), we obtain
𝐻𝐻
[
𝐻𝐻
𝐿𝐿
)] +
𝐿𝐿
[
𝐿𝐿
𝐻𝐻
)] +
𝐻𝐻
𝐻𝐻
This expression simplifies to
𝐻𝐻
[𝑢𝑢(𝑐𝑐
𝐻𝐻
𝐿𝐿
)] = 𝜃𝜃
𝐿𝐿
[𝑢𝑢(𝑐𝑐
𝐿𝐿
𝐻𝐻
)]
and ultimately reduces to 𝜃𝜃 𝐻𝐻
𝐿𝐿
, violating the initial assumption 𝜃𝜃
𝐻𝐻
𝐿𝐿
. Therefore, 𝐹𝐹
𝐿𝐿
𝐿𝐿
[
𝐿𝐿
𝐻𝐻
)] + 𝐹𝐹
𝐻𝐻
cannot hold, but instead 𝐹𝐹
𝐿𝐿
𝐿𝐿
[
𝐿𝐿
𝐻𝐻
)] + 𝐹𝐹
𝐻𝐻
must be true. As a
consequence, the top panel of Figure 1 applies for the low-demand customer, ultimately implying that
𝐿𝐿
binds while 𝐼𝐼𝐶𝐶
𝐿𝐿
does not. Q.E.D.
Summarizing, from the high-demand customer we have that 𝜃𝜃 𝐻𝐻
[
𝐻𝐻
𝐿𝐿
)] + 𝐹𝐹
𝐿𝐿
𝐻𝐻
whereas
from the low-demand customer we obtained that 𝜃𝜃 𝐿𝐿
𝐿𝐿
𝐿𝐿
. We can now plug this information about
𝐻𝐻
and 𝐹𝐹
𝐿𝐿
into the monopolist’s expected PMP, which now becomes an unconstrained maximization
problem, as follows:
max
𝑞𝑞
𝐿𝐿
,𝑞𝑞
𝐻𝐻
≥
𝑝𝑝[𝐹𝐹
𝐻𝐻
𝐻𝐻
] + (1 − 𝑝𝑝)[𝐹𝐹
𝐿𝐿
𝐿𝐿
]
Figure 2 Output for the low-demand customer
Summarizing, the amount offered to high-demand customers is socially efficient (recall that 𝜃𝜃
𝐻𝐻
′
𝐻𝐻
𝑐𝑐). In other words, 𝑐𝑐
𝐻𝐻
𝐻𝐻
𝑠𝑠𝑠𝑠
, and there is no output distortion for high-demand customers relative to
complete information allocations. That is a common finding in principal-agent models where the principal
(in this case the monopolist) cannot observe the private type of the agent (in this case the consumer). In
contrast, the output offered to low-demand customers entails a distortion relative to complete information,
𝐿𝐿
𝐿𝐿
𝑠𝑠𝑠𝑠
, as depicted in Figure 2. Furthermore, this output distortion 𝑐𝑐
𝐿𝐿
𝑠𝑠𝑠𝑠
𝐿𝐿
is increasing in term
𝑝𝑝
1−𝑝𝑝
𝐻𝐻
𝐿𝐿
). Specifically, it increases in the frequency of high-type buyers, 𝑝𝑝, and on the difference
between high- and low-type buyers, (𝜃𝜃
𝐻𝐻
𝐿𝐿
In addition, the fact that constraint 𝑃𝑃𝐶𝐶 𝐿𝐿
binds while 𝑃𝑃𝐶𝐶
𝐻𝐻
does not, entails that only the high-demand
customer retains a positive utility level, i.e., 𝜃𝜃 𝐻𝐻
𝐻𝐻
𝐻𝐻
- In other words, the firm’s lack of
information provides the high-demand customer with an “information rent.” Intuitively, this information
rent emerges from the seller’s attempt to reduce the incentives of the high-type customer to select the
contract meant for the low type. In particular, while the low-demand buyer pays a lower fee, the output
that he receives is sufficiently low to make it unattractive for the high-demand buyer, 𝑐𝑐 𝐿𝐿
𝐿𝐿
𝑠𝑠𝑠𝑠
. In other
words, the output distortion 𝑐𝑐 𝐿𝐿
𝑠𝑠𝑠𝑠
𝐿𝐿
that we described above stems from the seller’s purpose to reduce
the information rent of the high-type buyer.
Example. Consider a monopolist selling a textbook to two types of graduate students, low- and high-
demand, with utility function
𝑖𝑖
𝑖𝑖
𝑖𝑖
𝑖𝑖
𝑖𝑖
𝑖𝑖
2
𝑖𝑖
𝑖𝑖
c
q
L
q
L
SO
u' ( q
L
) [ θ
L
p
1 − p
𝐻𝐻
𝐿𝐿
)]
u' ( q
L
L
q
where 𝑖𝑖 = {𝐿𝐿, 𝐻𝐻} and 𝜃𝜃 𝐻𝐻
𝐿𝐿
. In this context, we obtain the direct demand function 𝑐𝑐
𝑖𝑖
𝑖𝑖
− 𝑝𝑝. In
addition, assume that the proportion of high-demand (low-demand) students is 𝛾𝛾 ( 1 − 𝛾𝛾, respectively).
The monopolist’s constant marginal cost is 𝑐𝑐 > 0, which satisfies 𝜃𝜃 𝑖𝑖
𝑐𝑐 for all 𝑖𝑖 = {𝐿𝐿, 𝐻𝐻}. Consider for
simplicity that 𝜃𝜃
𝐿𝐿
𝜃𝜃
𝐻𝐻
+𝑐𝑐
2
, which implies that each type of student would buy the textbook, both when the
firm practices uniform pricing and when it sets two-part tariffs (as we next show).
Uniform pricing. Consider first that the monopolist does not practice price discrimination (i.e., it sets a
uniform price that induces both types of customers to purchase positive units). In this setting, the
monopolist sets a unique price p that solves the expected PMP
max
𝑝𝑝
𝛾𝛾[(𝑝𝑝 − 𝑐𝑐)(𝜃𝜃
𝐿𝐿
− 𝑝𝑝)] + (1 − 𝛾𝛾)[(𝑝𝑝 − 𝑐𝑐)(𝜃𝜃
𝐻𝐻
− 𝑝𝑝)],
where 𝑐𝑐 𝑖𝑖
𝑖𝑖
− 𝑝𝑝 for every type- i customer. Taking first-order conditions with respect to 𝑝𝑝 yields
𝐿𝐿
𝐻𝐻
And solving for p we obtain the uniform price
𝑈𝑈𝑈𝑈𝑖𝑖𝑈𝑈𝑠𝑠𝑈𝑈𝑈𝑈
𝐿𝐿
𝐻𝐻
which yields monopoly profits of
𝑈𝑈𝑈𝑈𝑖𝑖𝑈𝑈𝑠𝑠𝑈𝑈𝑈𝑈
[𝛾𝛾𝜃𝜃
𝐿𝐿
𝐻𝐻
− 𝑐𝑐]
2
Note that the monopolist could use a uniform price to only serve high-demand students. The price that
would maximize its profits in this case solves
max
𝑝𝑝
𝐻𝐻
thus ignoring the segment of low-demand students. Taking first-order conditions with respect to 𝑝𝑝 and
solving for 𝑝𝑝 yields 𝑝𝑝 𝐻𝐻
𝜃𝜃
𝐻𝐻
+𝑐𝑐
2
. In this context, monopoly profits become
𝑈𝑈𝑈𝑈𝑖𝑖𝑈𝑈𝑠𝑠𝑈𝑈𝑈𝑈−𝐻𝐻
𝐻𝐻
2
which are larger than those when serving both types of students (i.e., 𝜋𝜋
𝑈𝑈𝑈𝑈𝑖𝑖𝑈𝑈𝑠𝑠𝑈𝑈𝑈𝑈−𝐻𝐻
𝑈𝑈𝑈𝑈𝑖𝑖𝑈𝑈𝑠𝑠𝑈𝑈𝑈𝑈
) if the
proportion of low-demand customers, 𝛾𝛾, is sufficiently small, that is,
𝐻𝐻
𝐻𝐻
𝐿𝐿
𝐻𝐻
𝐿𝐿
2
Intuitively, the frequency of high-value customers is large, thus inducing the seller ignore low-value
customers to focus on high-value customers alone. For instance, parameter values 𝜃𝜃 𝐻𝐻
𝐿𝐿
and 𝛾𝛾 =
3
4
satisfy this condition since
𝐻𝐻
𝐻𝐻
𝐿𝐿
𝐻𝐻
𝐿𝐿
2
2
𝐿𝐿
16
7
, and fees of 𝐹𝐹
𝐻𝐻
444
49
≅ 9.06 and 𝐹𝐹
𝐿𝐿
96
49
≅ 1.95. As a consequence, expected profits from two-
part tariffs are
𝑇𝑇𝐿𝐿𝑇𝑇
[
𝐻𝐻
𝐻𝐻
] + (
)[
𝐿𝐿
𝐿𝐿
] =
In contrast, those under uniform pricing become
𝑈𝑈𝑈𝑈𝑖𝑖𝑈𝑈𝑠𝑠𝑈𝑈𝑈𝑈
[𝛾𝛾𝜃𝜃
𝐿𝐿
+(1−𝛾𝛾)𝜃𝜃
𝐻𝐻
−𝑐𝑐]
2
4
49
64
≅ 0.76, and
𝑈𝑈𝑈𝑈𝑖𝑖𝑈𝑈𝑠𝑠𝑈𝑈𝑈𝑈−𝐻𝐻
𝐻𝐻
2
Hence, practicing two-part tariffs is profit-enhancing for the monopolist since
𝑇𝑇𝐿𝐿𝑇𝑇
𝑈𝑈𝑈𝑈𝑖𝑖𝑈𝑈𝑠𝑠𝑈𝑈𝑈𝑈−𝐻𝐻
𝑈𝑈𝑈𝑈𝑖𝑖𝑈𝑈𝑠𝑠𝑈𝑈𝑈𝑈
