We considers the dynamic incentive remuneration design for the functional recovery care by applying the dynamic principal-agent theory, where the long-term care provider (simply, provider) is formulated as the agent and the long-term care insurer (simply, insurer) as the principal. The insurer designs a dynamic incentive remuneration strategy to encourage the provider to provide the functional recovery care. In particular, the insurer pays the remuneration to the provider based on the condition improvement of a person requiring long-term care at the end of a contract period. In addition, an optimal functional recovery care effort level is recommended to the provider.

1. Dynamic Incentive Remuneration Design for Functional Recovery Care
⾃⽴⽀援介護を促進するインセンティブ型報酬制度に関する研究
Masaru Unno
NTT Communications Corporation, Japan

2. Masaru Unno Dynamic Incentive Remuneration Design for Functional Recovery Care
Japan is one of longevity countries in the world
12017/12/26
(Source)
World Population Prospects 2017,
United Nations
The proportion of elderly in Japan becomes the highest in the world in the early 2000s, and the current proportion is 27%.
Super-aging society
21%
Now
Japan

3. Masaru Unno
The proportion of the elderly(65+) rises to 40% in the future
2
elderly
(65+)
38%
(Source) National Institute of Population and
Social Security Research, Japan
In the future, as the total population decreases, the proportion of the elderly will continue to increase.
In 2060, 1 in 2.5 people will be 65 years old and over.
(10,000 persons)

4. Masaru Unno
Number of Persons Requiring Long-Term Care Needed
3
(1000 person)
6,077
2,182
2,582
3,029
3,484
3,874
4,108
4,348 4,408 4,548
4,690
4,870
5,076
5,330
5,643
5,859
The number of persons requiring long-term care is increasing at a rate of 7% per year from 2000 to 2015.
7% per year

5. Masaru Unno
Japan's long-term care insurance system (overview)
4
Insured person
Insurer
Long-term care
service providers
municipality
Long-term care services
Payment of long-term
care service expenses
Persons certified
long-term care
need
(aged 40+)
(mainly 65+)
Insurance
fund
State, Prefecture
The insurance benefit of the long-term care insurance system comes from the insurance premiums paid by the insured
persons (50%) and the fiscal expenditure of public sector (state : 25%, prefecture : 12.5%, municipality : 12.5%).
(Insurance benefits)
Premiums of
insured person
50%
Public
expenditure
37.5%
Public
expenditure
12.5%
12.5%25%
Co-payment of long-term care service expenses

6. Masaru Unno
Long-term care expenses is increasing continuously
5
Insurance benefit of long-term care service increases continuously due to aging.
There is a possibility that the long-term care insurance system can not be maintained in the future.
(100 million yen) National budget
= 96 trillion yen
(cf.)

7. Masaru Unno
Functional Recovery Care
6
In this situation, “ functional recovery care ” has become more important.
ü Functional recovery care provides medical and welfare services so that a person can carry out
a daily life independently after receiving the functional recovery care.
ü It has been reported that the condition of a person requiring long-term care is improved by
receiving the functional recovery care.

8. Masaru Unno
The effect of Functional Recovery Care (image)
7
Level of long-term
care need
start
0 𝑇
Cumulative
Improvement
time
Normal long-term care
Functional Recovery Care
∆𝑡
end
good
worse
condition
The condition of a person requiring long-term care is expected to be improved by receiving the functional
recovery care.

9. Masaru Unno
Functional Recovery Care
8
It is expected that the amount of the insurance benefit can be reduced by the functional recovery care.
The long-term care service provider has little motivation to provide the functional recovery care.
But
Providerʼs revenue will decrease as the persons requiring long-term care recover from the care under
the current long-term care insurance system.
Because

10. Masaru Unno
Motivation
9
To solve this problem,
We consider to introduce a new long-term care remuneration strategy such that the providerʼs remuneration
is directly related to the condition improvement of a person requiring long-term care.
In particular
We consider a dynamic incentive remuneration design by applying the principal-agent theory of Holmstrom
and Milgrom (1987)．
(The insurer is a principal and the care provider is an agent.)

11. Masaru Unno
Framework
10
Insured person
Insurer
Long-term care
service providers
municipality
Premiums
Long-term care services
Payment of long-term
care service expenses
Persons certified
long-term care
need
Insurance
fund
State, Prefecture
Public
expenditure
Public
expenditure
(Insurance benefits)
Service usage
decreases
condition
improved
Under the new remuneration strategy, the insurer pays the remuneration to the provider based on the
condition improvement of a person requiring long-term care.
+ Functional Recovery Care
Reduce
expenses
New Incentive
Remuneration

12. Masaru Unno
Problem Formulation
11
The cumulative condition improvement X(t) of a person requiring care at time t depends on
l a stochastic fluctuation of condition of a person : 𝑍 𝑡 (standard Brownian motion)
l the functional recovery care effort made by provider : 𝜇 𝑡
𝑔 𝜇 𝑡
l the deterioration of the condition under normal care : −𝛿 (constant)
d𝑋 𝑡 = 𝑔 𝜇 𝑡 d𝑡 + 𝜎d𝑍 𝑡
cumulative condition
improvement
function of ,𝜇 𝑡
𝑔 0 = −𝛿
standard Brownian motion
constant

13. Masaru Unno
Problem Formulation
12
𝔼 Γ 𝑋 𝑇 − 6 ℎ 𝜇 𝑡 d𝑡
8
9
The provider incurs the cost .ℎ 𝜇 𝑡
The insurer pays the remuneration to the provider.Γ 𝑋 𝑇
l is determined based on X(T) of the cumulative condition improvement at T (the end time of
the contract period).
Γ 𝑋 𝑇
When the provider makes functional recovery care effort, the expected revenue of the
provider is
𝜇 𝑡 , 0 ≤ 𝑡 ≤ 𝑇

14. Masaru Unno
Problem Formulation
13
𝔼 𝑢 Γ 𝑋 𝑇 − 6 ℎ 𝜇 𝑡 d𝑡
8
9
= 𝔼 −exp −𝜌 Γ 𝑋 𝑇 − 6 ℎ 𝜇 𝑡 d𝑡
8
9
The provider is risk averse and has the exponential utility function of the revenue :
risk sensitivity of the provider (constant)

15. Masaru Unno
Problem Formulation
14
𝔼 6 𝜆 d𝑋 𝑡
8
9
− Γ 𝑋 𝑇 = 𝔼 6 𝜆𝑔 𝜇 𝑡 d𝑡
8
9
− Γ 𝑋 𝑇
The insurer will be able to reduce the amount of insurance benefit by ( is constant).
The total expected revenue of the insurer is
𝜆d𝑋 𝑡 𝜆
The reduction of the insurance benefit can be deemed to be the revenue of the insurer.

16. Masaru Unno
Insurer's Problem
15
𝔼 6 𝜆𝑔 𝜇 𝑡 d𝑡
8
9
− Γ 𝑋 𝑇
𝜇 𝑡 ∈ arg max
HI J
𝔼 𝑢 Γ 𝑋 𝑇 − 6 ℎ 𝜇 𝑡 d𝑡
8
9
Assumption :
The insurer's problem :
maximize
subject to
: incentive compatibility constraint
: participation constraint
will determine the level of the functional recovery care effort to maximize its utility,
under the condition of the functional recovery care remuneration.
l The provider
l The insurer will design a remuneration strategy , knowing the behavior of the provider.Γ 𝑋 𝑇
𝔼 Γ 𝑋 𝑇 − 6 ℎ 𝜇 𝑡 d𝑡
8
9
≥ 0,

17. Masaru Unno
Remuneration under Incentive Compatibility Constraint
16
Γ 𝑋 𝑇 = 6
ℎ′ 𝜇 𝑡
𝑔′ 𝜇 𝑡
d𝑋 𝑡
8
9
− 6
ℎU 𝜇 𝑡
𝑔U 𝜇 𝑡
𝑔 𝜇 𝑠 d𝑡
8
9
+ 6
1
2
𝜌𝜎Y
ℎ′ 𝜇 𝑡
𝑔′ 𝜇 𝑡
Y
d𝑡
8
9
+ 6 ℎ 𝜇 𝑡 d𝑡
8
9
Proposition 1
The stochastic process , which satisfies the incentive compatibility constraint, is
implemented only if
𝜇 𝑠 , 0 ≤ 𝑡 ≤ 𝑇
We arrive at the following Proposition:

18. Masaru Unno
Provider's Optimal Functional Recovery Care Effort
17
d
d𝜇
𝜆𝑔 𝜇 𝑡 −
1
2
𝜌𝜎Y
ℎU 𝜇 𝑡
𝑔U 𝜇 𝑡
Y
− ℎ 𝜇 𝑡 = 0
Proposition 2
The optimal functional recovery care effort of the provider, which satisfies the incentive compatibility
constraint and maximizes the total expected revenue of the insurer, is implemented only if the following
condition is satisfied :
From Proposition 1, the solution of the insurer's optimization problem can be obtained :

19. ( are constants)
Masaru Unno
Numerical Simulation
18
𝑔 𝜇 = 𝑎𝜇 − 𝛿
ℎ 𝜇 𝑡 =
𝑏
2
𝜇 Y + 𝑐𝜇 + 𝑘
𝑎, 𝑏, 𝑐, 𝑘, 𝛿, 𝜆, 𝜌, 𝜎
Identifying the model :
: the effect of the functional recovery care effort to the
condition improvement
: providerʼs cost
d𝑋 𝑡 = 𝑎𝜇 − 𝛿 d𝑡 + 𝜎d𝑍 𝑡 : dynamics of the cumulative condition
improvement
: If the contract period T is not very long, the effect of the
functional recovery care effort is considered to be a linear
approximation.

20. Masaru Unno
Numerical Simulation
19
where
𝜇∗ =
𝜆𝑎b
𝑏 𝑎Y + 𝜌𝜎Y 𝑏
−
𝑐
𝑏
Γ 𝑋 𝑇 = 𝛼𝑋 𝑇 + 𝛽𝑇,
𝛼 =
𝜆𝑎Y
𝑎Y + 𝜌𝜎Y 𝑏
𝛽 =
1
2
𝜌𝜎Y −
𝑎Y
2𝑏
𝜆𝑎Y
𝑎Y + 𝜌𝜎Y 𝑏
Y
+
𝑎𝑐
𝑏
+ 𝛿
𝜆𝑎Y
𝑎Y + 𝜌𝜎Y 𝑏
−
𝑐Y
2𝑏
+ 𝑘
𝛼, 𝛽 are constants
The optimal care effort and the optimal incentive remuneration strategy :𝜇∗
: optimal care effort
: optimal remuneration strategy
constant

21. Masaru Unno
Numerical Simulation
20
𝑎 = 0.3, 𝑏 = 480, 𝑐 = 280, 𝑘 = 0, 𝛿 = 0.14, 𝜆 = 27545, 𝜌 = 0.001, 𝜎 = 1.1
Furthermore, define the parameters as follows :
The optimal remuneration strategy :
Γ 𝑋 𝑇 = 3696𝑋 𝑇 + 8065𝑇
𝜇∗ = 1.73
𝑇 = 52 (week)
𝑔 𝜇 = 𝑎𝜇 − 𝛿
expected value of 𝑋 𝑇 = 19.7
expected value of Γ 𝑋 𝑇
= 492, 014
expected revenue of the insurer = 49,320

22. Masaru Unno
Numerical Simulation
21
0 10 20 30 40 50
−1001020304050
week (t)
CareLevel(X)
A 1 17
µ∗
= 1.73
µ = 0.66
µ = 2.64
(a) µ 1/2
(b) µ 2
Sample paths of cumulative condition improvement (100 times)
𝑋 𝑇
36.9
1.3
average = 19.6
consistent with the
expected value 19.7

23. Masaru Unno
Numerical Simulation
22
Histgram of Remuneration
Remuneration (Γ(T))
Frequency
420000 440000 460000 480000 500000 520000 540000 560000
051015202530
Histgram of Revenue of Insurer
Revenue of Insurer (Π)
Frequency
−4e+05 −2e+05 0e+00 2e+05 4e+05
0510152025
Insurer's Revenue Remuneration
The simulation histograms of the insurer's revenue and the remuneration for 100 times
average = 48,964 average = 491,959
Γ 𝑋 𝑇
consistent with the
expected value 492, 014
consistent with the
expected value 49, 320

24. Masaru Unno
Numerical Simulation
23
0.0 0.5 1.0 1.5 2.0 2.5 3.0
390000400000410000420000430000
Amount of Service Provision (µ)
RevenueofCareProvider(V)
µ*=1.73
Finally, the provider will maximize its revenue by making the optimal functional recovery care 𝜇∗
𝜇∗ = 1.73

25. Masaru Unno
Conclusion
24
We have derived the necessary conditions for the optimal incentive remuneration strategy to satisfy.
If the new remuneration strategy to promote the functional recovery care is realized, it is expected that the
national and local governments will be able to suppress the insurance benefit.
(and maintain the long-term care insurance system)
In this paper, it is assumed that the dynamics of the condition improvement of a person requiring long-term
care is obeyed by Brownian motion.
Further empirical studies are required to verify the dynamics and the relationship between the functional
recovery care effort and the effectiveness of the proposed optimal incentive remuneration strategy.
The condition improvement is also required to be analyzed quantitatively.

26. Masaru Unno 25
Thank you for your attention.