This appendix presents a simple simulation to calculate the cost of rare diseases under the current insurance system and other insurance systems.

Consider a business person, who is assumed to be male, and works for 40 years from 21 to 60 years old. After retirement (with no retirement allowance), he lives on a pension for 20 years until age 80 with the annual pension of *I* million yen. It is assumed that necessary expenses during working age would be higher than after retirement. Such expenses would include costs of raising children and mortgage or rental payments. To reflect this and to simplify the analysis, we assume that only *I* million yen is at his disposal so that from age 21 to 80, the subject consumes as if his income were flat at *I* million yen.

Suppose that a person has an annual utility function with a constant relative risk aversion. Then the utility function

*u*(

*x*) of this person can be represented as:

$$ u(x) = \frac{x^{1-\sigma }}{1-\sigma }, \ \ (\sigma \ne 1), $$

where

*x* is the amount of consumption in million yen beyond the necessary expenses mentioned above, and

\(\sigma =-\frac{u''(x) x }{u'(x)}\) is the constant rate of relative risk aversion that measures this person’s risk attitude. Furthermore, this person tries to maximize the discounted sum of utility:

$$ U(x_{21}, x_{22}, x_{23}, \ldots , x_{80})=u(x_{21})+\delta u(x_{22})+\delta ^2 u(x_{23})+ \cdots + \delta ^{59} u(x_{80}). $$

subject to the budget constraint (with no liquidity constraint) given by

$$\begin{aligned}&x_{21}+\delta x_{22}+\delta ^2 x_{23}+ \cdots + \delta ^{59} x_{80}\\&\quad \le \textit{LI} \equiv I+I \delta + I \delta ^2+ \cdots + I \delta ^{59} = \frac{I(1-\delta ^{60})}{1-\delta }, \end{aligned}$$

where

\(\textit{LI}\) is the lifetime income, or the discounted sum of total income. It is verified that the best consumption stream of this person is

\((I,I,\ldots , I)\), i.e., the one in which he consumes

*I* million yen every year. The total utility

\(U_N\) is

$$\begin{aligned} U_N=u(I)\frac{(1-\delta ^{60})}{1-\delta }=\frac{I^{1-\sigma }}{1-\sigma }\frac{(1-\delta ^{60})}{1-\delta }, \end{aligned}$$

where, in notation,

\(U_N\) stands for utility with no disease.

Suppose that there is a possibility that the subject suffers from a rare disease at the age of 40. As a result of ideal treatment, this person lives to age 80. The probability of this event (suffering from a rare disease) is

\(p\%\). The subject does not consider this possibility until he actually suffers from the disease. With an annual cost of treatment of

*C* million yen per year, the total utility becomes

$$\begin{aligned} U_D&=u(I)\frac{(1-\delta ^{20})}{1-\delta }+u(I-C)\frac{(\delta ^{20}-\delta ^{60})}{1-\delta }\\&\quad =\frac{I^{1-\sigma }}{1-\sigma }\frac{(1-\delta ^{20})}{1-\delta }+\frac{(I-C)^{1-\sigma }}{1-\sigma } \frac{(\delta ^{20}-\delta ^{60})}{1-\delta }, \end{aligned}$$

where

\(U_D\) stands for utility with a disease.

Then the expected utility is

$$ \textit{EU}=(1-p)U_N+\textit{pU}_D. $$

It is clear that the optimal policy toward rare diseases is to be fully insured. The question that remains is the magnitude of the welfare loss measured in terms of expenses if the current policy is maintained. This calculation requires quantification of the certainty equivalence (CE) of the situation with the possibility of disease without full insurance:

$$ \textit{CE}=\left[ (1-\sigma ) \frac{1-\delta }{1-\delta ^{60}}{} \textit{EU} \right] ^{\frac{1}{1-\sigma }}. $$

Thus, an individual welfare cost measured in terms of income caused by an incomplete insurance system (relative to a complete insurance system) is given by

$$ \textit{LI}-\textit{CE}. $$

Then the welfare cost evaluated at the age of 40 becomes

$$ (\textit{LI}-\textit{CE})/\delta ^{20}. $$

This value is listed in Tables

3.1 and

3.2.

Table 3.1 Lifetime loss, simulation 1: \(I=2\), \(C=0.37\), \(p=0.067\)

Explanation of the choices of variables are as follows. The income is a hypothetical one, but it is assumed that the utility function takes the value beyond a reference point. For example, if a person earns 5 million yen per year with some necessary expenses, such as education costs and housing loan, of 3 million yen, then the person’s utility comes from the remaining 2 million yen. The rates of relative risk aversion are roughly the lower bound and the upper bound taken from empirical researches (see, e.g., Shimono 2000). The discount factor is small, reflecting the recent low interest rate in Japan. The probability of suffering from a rare disease is taken from the fraction of patients with rare diseases as reported by WHO.

^{5}Table 3.2 Lifetime loss, simulation 1: \(I=1\), \(C=0.37\), \(p=0.067\)