The structure of the model
The demographic-health sub-model
Person-years lived during year 
are classified by 5-year age group (indexed by 
) and by sex (indexed by 
). Within each age-sex-group, person-years are further subdivided by health status, as shown in Appendix Table 2.
| Non-disabled | Disabled | Total | |
|---|---|---|---|
| Survivor |
 |
 |
 |
| Decedent |
 |
 |
 |
| Total |
 |
 |
 |
The categories are related as follows:
(12) 
,
(13) 
,
(14) 
,
(15) 
,
(16) 
.
The expenditure sub-model
The notation for costs per person-year lived is set out in Appendix Table 3.
| Non-disabled | Disabled | Total | |
|---|---|---|---|
| Survivor |
 |
 |
 |
| Decedent |
 |
 |
 |
| Total |
 |
 |
 |
The costs per person-year lived are population-weighted averages, and are related to each other as follows:
(17) 
,
(18) 
,
(19) 
,
(20) 
,
(21) 
.
During any year t, all cost weights grow at the same rate, 
. Let 
be the launch year for the projections. When projecting into the future, we use the same value of g for all t, so that
(22) 
.
When back-casting, we allow to vary from year to year. The back-casting equivalent of Equation 22 is
(23) 
.
We can now derive Equation 1 in Section 4.2. We present the details only for back-casting, where 
, since we only carry out the decompositions on the historical data. The details for projections are very similar.
Let 
index all age-sex-groups and all health statuses, and let 
be total population. Then
(24) 
(25) 
(26) 
.
Equation 26 shows total expenditure as the product of three terms. The first term is simply population size. The second term, 
, is what we call the “ageing and health” effect. The third term, 
, is the “coverage and price” effect.
Equation 1 can be derived from Equation 26 by applying the general rule that if 
, then
, where 
is the instantaneous rate of change for 
. (To verify this rule, take logs of 
and differentiate.)
