Structural estimation of a principal-agent model: moral hazard in medical insurance.

Despite the importance of principal-agent models in the development of modern economic theory, there are few estimations of these models. I recover the estimates of a principal-agent model and obtain an approximation to the optimal contract. The results show that out-of-pocket payments follow a concave profile with respect to costs of treatment. I estimate the welfare loss due to moral hazard, taking into account income effects. I also propose a new measure of moral hazard based on the conditional correlation between contractible and noncontractible variables.

1. Introduction

* Contract theory has been extremely important in the development of modern economic theory during the last thirty years. However, the increasing sophistication of the theory has not gone hand-in-hand with empirical validation of the models, as Salanie (1997) points out. Chiappori and Salanie (2003) offer an up-to-date perspective on the literature that has tried to link econometrics and contract theory. Most of the existing works have used a reduced-form approach. (1)

This article's main contribution is to estimate the parameters of a principal-agent model with moral hazard. This allows me to use the principal-agent paradigm when solving for the optimal contract. This presents two main advantages. First, principal-agent models have developed in the last thirty years as a rigorous framework for studying the moral hazard concept. For my purposes, the optimal contract can be obtained directly from first principles, so I do not need to make further assumptions about the first-best level of utilization. Second, this approach requires the analyst to make a clear distinction between contractible and noncontractible variables. The relation between these variables provides important information for deriving the optimal contract.

I concentrate on the problem of health care insurance. Moral hazard in the use of medical services has been one of the most recurrent issues in health economics; early references on the topic are Arrow (1963), Pauly (1968), and Zeckhauser (1970). Moral hazard arises because health shocks are not contractible and consequently contracts are not complete. It might then be optimal for insurers to give incentives so the consumer will not seek expensive treatments for minor health shocks.

In the health care setting, it is natural to think that the noncontractible variable is the health shock, while the contractible variable is treatment cost. This will be important when deriving the optimal contract, since cost can indicate the severity of the health shock. This will also be the basis for our proposal of a new measure of moral hazard: the correlation between health shocks and treatment costs.

Previous articles have tried to estimate optimal health care insurance contracts (Feldstein, 1973; Feldman and Dowd, 1991; Buchanan et al., 1991; Newhouse et al., 1993; Manning and Marquis, 1996). However, their methodology is based on optimal taxation rather than asymmetric information theory. (2) Medical insurance may distort the consumption of health care services, since it lowers the marginal price of consumption. In this respect, the problem of optimal taxation is similar to optimal health insurance. As Besley (1988) points out, however, there is a crucial difference between them: the insurance problem is against a background of incomplete markets. The previous approaches are based on comparing the welfare loss of a given insurance contract to the situation of no insurance. Consequently, they assume that the first-best level of health care services corresponds to the one where there is no insurance. Ma and Riordan (1997, 2002) have shown that this assumption does not hold true in the presence of income effects. In fact, the implementation of the first best requires the consumer to be responsible for only a fraction of treatment costs, because the marginal valuation of income rises once the consumer pays her out-of-pocket portion. (3,4) In this article I can deal with this issue because I obtain the optimal contract directly from first principles as the solution to the principal-agent problem. (5)

This article also differs from previous literature in the way I model health care consumption. In previous approaches, the individual decision is over the amount of monetary resources dedicated to health care. Hence, utility would be a function of the amount of dollars spent in an illness episode. Though this is a simplifying assumption, it is undesirable because it assumes that the larger the health care costs, the higher the utility. It is preferable to disentangle quantity consumed from the cost of producing it, since the individual will derive utility from quantity but not from the cost of production. In my model, the individual decides whether or not to have treatment against an illness spell with some level of severity. The costs of treatment are given to the individual as a technological relation. My approach, though more complicated from an econometric point of view, allows me to disentangle quantity from costs. An important advantage of this approach is that I can exploit the stochastic relation between costs and health shocks when solving for the optimal contract. If treatment costs are strongly correlated with health shocks, then the problem of moral hazard will be alleviated because the insurer can infer the value of the noncontractible variables from the contractible ones. (6) This informational relation will be my basis for proposing a new measure of moral hazard using the correlation between health shocks and treatment costs. This measure is based on the informational content that contractible variables (treatment costs) have over noncontractible ones (health shocks). The previous literature has used elasticities of health expenditures with respect to copayments as a measure of moral hazard. (7) view my measure of moral hazard as a complement rather than a substitute to the traditional elasticity measure. My measure checks for the support condition that is commonly assumed in theoretical models but has not previously been examined in empirical research. (8) My measure is especially valuable if nonlinear contracts are allowed, as is commonly the case in health insurance contracts. See Cutler (2002) and Cutler and Zeckhauser (2000) for examples of nonlinear health insurance contracts in the United States.

I use data from the RAND Health Insurance Experiment (HIE) that randomly assigned individuals to insurance contracts. This is a significant advantage: in particular, I do not need to model the individual's choice of insurance contracts. Moreover, the randomization will also be important for the identification of the model.

The article is organized as follows. Section 2 describes the theoretical model and some of its implications. Section 3 describes the data. Section 4 discusses the econometric strategy used to estimate the theoretical model. Section 5 gives the results of a descriptive analysis. Section 6 discusses the estimates of the structural parameters and evaluates the suitability of the model. Section 7 sets up and solves the principal-agent problems and discusses my measure of moral hazard. Section 8 concludes. Finally, the Appendix contains details of the computation of the log-likelihood function.

2. The demand model

* Individual decision problem. This section is devoted to modelling individual decisions about whether or not to be treated when suffering an illness spell. This is the basis for the estimation of the parameters of the principal-agent model. In my setup, the consumer faces a specific insurance contract that will influence her decision.

My model draws on Ma and Riordan (1997,2002). Their model is well suited for my purpose, as they consider income effects and separate quantity from treatment costs. The main difference between their model and mine is that I allow treatment costs to be random and stochastically related to illness severity (health penalty). From an empirical point of view, this is important for obtaining my measure of moral hazard. (9)

In the model, the individual decides whether or not to be treated but does not decide the cost of treatment. In fact, Keeler and Rolph (1988) and Newhouse et al. (1993) found that insurance contracts mainly influence the decision whether or not to seek treatment against an illness episode, rather than the treatment costs. This is expected given the informational asymmetry between doctor and patient. I shall also assume that the doctor chooses treatment costs independently of the individual's insurance contract and income. This corresponds to the situation where the medical guideline that the doctor follows does not take into account individual economic characteristics but gives the most cost-effective treatment. Consequently, I shall assume that treatment costs come from a given technological relation. I emphasize one important hypothesis in my model: The individual is rational and compares benefits and costs when she decides whether or not to seek treatment. This might be a strong assumption when one is dealing with severe illnesses for which the individual lacks experience and can hardly value the benefits and costs. Furthermore, the treatment decision in the case of very severe illnesses might depend on long-term effects that would severely complicate the model. In the empirical application I shall restrict the type of illness spells studied to make behavior more likely to conform to modelling assumptions.