# MANAGED HEALTH CARE: Shadow Price

The shadow prices implied by individuals’ predictions and a risk adjustment policy are contained in Table 4. Three information assumptions are combined with three risk adjustment policies to produce nine sets of profit-maximizing shadow prices. The q for the “other” category is normalized to 1.00 in all cases, so each entry in the table needs to be read as the shadow price relative to this numeraire.

Begin with the first three columns of results, computed for the assumption that individuals can forecast health costs based only on their own age and sex. The very first column shows the consequences of no risk adjustment with this informational assumption. Individuals cannot forecast very well at all, so the incentives plans have to distort are small, even with no risk adjustment. All estimated q’s are close to 1.00 with the exception of birth-related expenditures. Birth-related expenses are more predictable with age and sex, so plans will have an incentive to ration these services more tightly. Interestingly, the two risk-adjustment systems tested, ADGs and HCCs, each exacerbate the distortion in birth expenses.Apparently the risk adjusted payments under these systems are relatively less correlated with the predictable expenses, and magnify plans’ incentives to discriminate against these categories of expenditures.

The second set of three columns reports q’s when individuals can predict based on 25 percent of the information contained in prior spending. Specifically, predicted spending for each person was figured as a weighted average of the prediction based on age-sex and the prediction based on age, sex, and prior year spending in that category. When we say “can predict based on 25% of the information contained in prior spending,” we mean, operationally, that the weight on the prior spending prediction is 25%, and the weight on the age-sex prediction is 75%.

# MANAGED HEALTH CARE: Measuring Shadow Prices 4

Equation (12) is a logit regression. Equation (13) is a linear regression that estimates the impact of the Xs on the square root of the level of spending on each service for individuals with positive spending on that service. We chose the square root transformation to deal with skewness in the distribution of spending rather than the more common logarithmic transformation because the smearing estimator for the square root model is less sensitive to heteroskedasticity than the log transformation.11 The difficulties in retransformation in the context of the two-part model have been treated in detail by Manning (1998) and Mullahy (1998). In those papers, it is shown how sensitive expected spending estimates can be to distributional properties such as heteroskedasticity this.

The use of a transformation to account for skewness in the spending data necessitates use of the “smearing” estimator to retransform the predicted values of spending to the expected levels of spending consistent with the original distributions of spending (Duan et al., 1983). Since this application calls for predicting 1993 spending using 1992 data and coefficients from the two part model of 1992 spending on 1991 right side variables, the smearing factor is taken from the error term of the 1991-1992 regressions. Since we use a square root transformation, the smearing factor is additive as opposed to the multiplicative form in the case of the logarithmic transformation. The resulting empirical analysis consists of a set of 18 regressions for each of the two informational assumptions we make.

# MANAGED HEALTH CARE: Measuring Shadow Prices 3

Expected Spending: The variable mis is the expected level of spending by each individual for each category of service. Estimating expected spending requires assumptions about the information available to individuals. The literature reflects a wide range of conceptions of what consumers might know about their health risks. Newhouse et al. (1989) suggest that individuals know all the information contained in measurable aspects of health status plus the time invariant-person specific component of the unobserved factors contributing to variation in health care spending.

# MANAGED HEALTH CARE: Measuring Shadow Prices 2

Treatments for some conditions are likely to be expensive, some much less so. Together, the seven conditions examined account for about 46% of all spending. Some treatments for included conditions are arguably quite predictable, such as birth-related spending, while others might be considered more random, such as injuries and poisonings. More info We classify all health care claims according to the primary diagnosis attached to the claim.

Patterns of Spending

Table 1 provides an overview of the patterns of utilization across the service types identified. By and large utilization of different service types is relatively stable over the three years observed. The most notable change is the reduction in the share of people with birth related spending. In 1991, 25.7% of the sample had birth-related spending compared to 16.7% in 1993. Birth of a child may have initiated a period of eligibility for some of these women, accounting for the elevated rate in the first year.

# MANAGED HEALTH CARE: Measuring Shadow Prices

In this section we illustrate how to use our measure. As we noted in the introduction, the data we will use are from an “unmanaged” plan, so the findings can not be regarded as definitive. Our purpose here is to illustrate how to use presently available data to calculate the distortion index. At the same time, the elements that feed into incentives to distort, such as predictability of various services, and correlation among use in various categories of service, are likely to be largely common to managed and unmanaged patterns of care. We believe our findings are therefore of some interest in themselves.

The empirical building blocks for estimation of shadow prices are the expected spending of individuals by service class and the correlation of expected spending across services and under differing information assumptions (see equation (11)). Our estimation strategy is aimed at obtaining estimates of future spending, conditional on the information assumptions, which minimize the forecast error. The performance of a number of estimation strategies for health care spending data have been assessed over the past fifteen years.

# MANAGED HEALTH CARE: Uncertainty 2

To investigate which shadow prices are set high relative to other shadow prices, we use (10′) to construct a ratio of qs to qs. where s’ is some other service. We simplify by abstracting from individual differences in enrollment response by assuming that O’j = Ф\ This amounts to saying that an increase in the value of plan i increases the likelihood of joining for all individuals equally. Equation (10′) can now be used to write the ratio of two shadow prices, q and q\ Note that the Ф’ term cancels out of this expression:

# MANAGED HEALTH CARE: Uncertainty

So far we have assumed that each individual i knows with certainty his valuation of each of the s services vis(mis), and, hence, given some q, the dollar amount of the different services that will be provided to him upon joining the plan. In order to make our model more realistic and to prepare for empirical application, we shall now allow for each individual to be uncertain about his future demands for the different services. Let us suppose that each individual has a set of prior beliefs about his possible health care demands, and that the plan shares these beliefs review.

# MANAGED HEALTH CARE: Profit Maximization 3

To see what (6) implies for various services, we make some substitutions. The change in the probability of joining can be written as the product of two derivatives:

for every i. Note that the assumption that for every shadow price qs the elasticity of demand for service s is the same for all individual s does not imply, of course, that all individuals have the same demand curve for that service. It only implies that demand curves of different individuals, for a certain services, are “horizontal multiplications” of some “basic” demand function for the service. Individuals will differ in their relative demands. One interpretation of this assumption, as in Glazer and McGuire (1998), is that given someone is sick, a common function describes valuation of a service, but people differ in the probability that they become ill.

# MANAGED HEALTH CARE: Profit Maximization 2

Let q s be the service-specific shadow price the plan sets determining access to care for service s. A patient with a benefit function for service s of vis(«) will receive a quantity of services, mis determined by:

Let the amount of spending determined by the equation above be denoted by mis (q s). Note that (3) is simply a demand function, relating the quantity of services to the (shadow) price in a managed care plan. See Figure 1.