The optimization problem outlined above imposes a structure on the assignment of services that leads to the largest reduction in nursing home use possible, given the amount of resources spent on community-based care. The optimization was conducted separately for each NLTCCD site, so that resources, while they could be reallocated among the individuals at a given geographical site, could not be reallocated between sites. This was to prevent the program from optimizing by "moving" budget resources to sites where community services are relatively cheap and nursing home beds relatively scarce or expensive. For our purposes, we wanted the optimization to be driven principally by the risk-related characteristics of individuals, relative local service prices, and relative service effectiveness in altering risk, rather than by variations across local economics and regulatory environments.
The structure imposed on service assignment by this optimization has a variety of technical characteristics that merit being made making explicit, since they define the conditions for efficiency being sought by the optimization algorithm. We note initially that because the expected nursing home cost per month at each site is taken to be the same for all individuals at that site (it is the site-specific average observed cost), minimizing nursing home use at a site is equivalent to minimizing expected nursing home expenditures. Thus:
When service assignment is optimal, the marginal reduction in nursing home use stemming from the last dollar spent on any two services offered to the same person must be equal. This characteristic of optimal assignment can be understood if one considers the cost of providing a given service to a given individual. The cost of providing the incremental hour of service is the hourly price of the service. The value of the reduction in nursing home use stemming from this service provision is the marginal impact on nursing home use of that hour of community service, multiplied by the expected cost of nursing home care (which we estimate as the observed average cost in each site or community). If one divides the value of the nursing home reduction by the price of the service, one obtains a measure of the dollar reduction in nursing home use per dollar of service provided. At the optimum, any two services provided to the same individual must result in the same dollar reduction in nursing home expenditures per dollar of service provided. If the two services did not result in the same cost savings per dollar spent, additional savings could be achieved by providing less of the relatively ineffective service and more of the other service. Therefore, in the optimal service assignment, the marginal value of the nursing home reduction stemming from the last dollar spent on any two services offered to the same person must be equal.
When service assignment is optimal, the marginal value of the reduction in nursing home use stemming from the last dollar spent on the same service offered to any two people must be equal. If the two marginal values are not equal, a dollar spent on providing the service to one person has a larger expected value, in terms of reduced nursing home expenditures, than does a dollar spent on providing that service to the other person. In this case, additional savings could be achieved by providing more of the service to the person for whom it has a larger deterrence effect, reducing the level of service provided to the person for whom the effect is relatively weak.
The optimal assignment need not result in all or any services being provided to a given client. When services are assigned strategically to reduce nursing home expenditures, resources must be directed toward the clients for whom the greatest reduction in nursing home use is expected as a result of providing those resources. With a limited budget, this targeting of services means that not every client will receive each, or any, community service. Only those clients for whom a service will have a relatively large deterrence effect will receive it.
The optimal assignment need not result in provision of all services. The effectiveness of each service in deterring nursing home use varies from service to service, and from individual to individual. The transition probability functions, in that they describe the technology available to reduce nursing home use through community-based services, provide a measure of each service's effectiveness in deterring nursing home use. These measures of effectiveness must then be weighed against the cost of each service, for it may or may not be the case that relatively inexpensive services are effective on a per-dollar basis. Optimizing within a limited budget, only those services that produce the largest reduction per dollar spent will be provided, regardless of their price.
Turning now to our model, consider first the i term in expression (2). As noted, this is the steady-state proportion of time that an individual with the characteristics of individual i would, conditional on survival, be expected to spend in nursing home care. The arithmetic mean of these terms is itself easily shown to be the steady-state proportion of time (total exposure) that the sample can be expected to spend in nursing home residence. It is this measure of total nursing home use that we seek to minimize through controlling the assignment of CLTC services in the sample, subject to the constraint that total expenditures for CLTC services not increase. We may state the problem formally as:
where i indexes individuals, and j indexes community service types.
Expression (3) is the objective function to be minimized, giving the steady-state proportion of total exposure the population is expected to spend in nursing home care. Expression (4) simply notes that is a function of a vector of exogenous characteristics (Xi) for the ith individual and her levels (hours per month) of services (Sij) where j indexes the different community service types. These service levels are the model's endogenous or "control" variables, in that they are taken to be under the control of the optimization algorithm.
The budget constraint (5) requires some elaboration. Consider first the left-hand side of the inequality; the bound, B. This bound is defined as
This bound is an estimate of total community service expenditures as actually observed in the study sample. The optimization may not exceed this aggregate expenditure level. In detail, i here again indexes individuals in the sample and j indexes the community service categories, while the superscript o indicates these were the values actually observed at the 6-month survey. The pj terms are the hourly charges (prices) for services. This bound is thus the sum of the price-weighted service quantities (i.e., nominal community service expenditures), weighted by the term gi (1-i). This latter term is a factor that discounts estimated community expenditures by correcting for the probability that the individual was actually in a true community service setting--not in a hospital or nursing home. The term gi is the individual probability of not being in the hospital, conditional on not being in a nursing home. (The manner in which this probability was estimated is described in Appendix B.) The term oi is simply that given by expression (2); that is, it is the steady-state risk for nursing home use for this individual given the services she was actually using at the time of the 6-month survey in the demonstration (Sijo).
Because one minus oi is the (unconditional) probability of being found outside of a nursing home given observed services, it follows by the joint probability law that the probability of being neither in a nursing home nor a hospital (and thus actually presumed to be using community program resources) is given by gi (1-oi). Observe that if this discount factor were not applied in expression (5), the implied budget bound would be that for the case in which all individuals spent the entire budget period using community resources. Because of hospital and nursing home use, however, estimated actual expenditures are less, as reflected in the discounting.
From a prospective standpoint, the undiscounted total expenditure bound may be thought of as what a program "offers" or "promises," while the discounted total is what it actually expects to spend, and hence determines its effective budget. An analogy might be the practice of commercial airlines, which promise more capacity than they actually have, counting on a certain proportion of passengers not to actually claim a seat.
The right-hand side of expression (5) has the same form and meaning, except that now community services (and the subsequent proportion of time spent in a nursing home) are taken to be endogenous (subject to the control of the optimization program), rather than to have the values observed in the survey. The quantity gi is taken to be exogenous (fixed); that is, we assume that hospital use is not affected by community service reallocation. This assumption is more realistic with respect to the pre-DRG era (1984) when the survey was conducted that it would be today.
The next constraint in the minimization problem, equation (6), restricts total community-based service expenditures for any given sample member in a month to be less than or equal to the average cost of nursing home care in their area for the same period (C). This bound is imposed partly for technical reasons--it greatly assists convergence in what is computationally a very complex NLP problem. But it is also substantively reasonable in that while the objective of the analysis is minimizing nursing home use (or, equivalently, expenditures), it would seem inappropriate in a budget-constrained context to spend more in the community than it would cost to maintain the client in a nursing home. As a practical matter, this constraint was not active in the final solutions anyway. The final constraint simply restricts the program to assign only non-negative values of community services to clients.