Abstract Subsidized insurance premiums are present in nearly all public and some private insurance systems. Such subsidies are usually implemented to increase participation in the insurance program and to decrease the effects of adverse selection in the market. It has been argued that the increased demand for insurance due to subsidies also increases the market inefficiencies stemming from moral hazard in the market. While this argument is intuitive and some empirical evidence exists for it, it ignores the wealth effects of premium subsidies and their effect on moral hazard. We argue that such effects can be dominating for the majority of the insured, particularly in insurance markets where most insured do not have a choice between different insurance policies, as is sometimes the case, e.g. in health insurance. Our theoretical model shows that wealth effects do influence the moral hazard in a given insurance market and that the influence depends on contract design. The results offer policy implications for premium subsidies in public insurance systems calling for a differential treatment of premium subsidies in insurance systems with different premium schedules. One example for this is a different effect of premium subsidies in health insurance and long term care insurance. Introduction Insurance markets are often thought of as being competitively prized in a risk-based fashion. However, there are certain markets in which either the government or the insurance company itself subsidizes at least some of the policies such that they will be prized below actuarially fair value. Probably the most commonly cited example for this is the U.S. health insurance market. Through tax exemptions of employer sponsored health insurance and the corresponding insurance policies for the self-employed, it is estimated that the U.S. state and federal governments waived tax revenues of about $260 billion in 2009 alone (Gruber, 2011) . Introduction Subsidies are typically introduced to stimulate demand for a certain type of insurance. This is done, for instance, if it is judged to be socially desirable to have a large proportion of the population covered against a certain risk. Additionally, it has been argued that subsidies are a possible way to counter the effects of adverse selection (Glauber, 2004) . However, when discussing subsidies for insurance schemes, moral hazard has seldom been considered. Feldstein and Friedman (1977) were the first to consider the effect of premium subsidies on moral hazard in the health care market. They make the wide-scale introduction of tax exemptions for employer covered health insurance responsible for the increase in health care spending in the United States by arguing that the subsidies lead to increased insurance coverage which, through moral hazard, increased the medical expenditures in the country. Empirical evidence exists that subsidies make some people increase their coverage (Finkelstein, 2002; Goda, 2011) and that the increased coverage can lead to higher medical spending (Cutler and Zeckhauser, 2000; Finkelstein et al., 2012) . Introduction However, it is also apparent that many people do not react towards subsidies. This can be the case because the choice of insurance coverage is discrete and the subsidy is not sufficient to make the insured choose the next higher level of coverage. Another explanation, particularly relevant in health insurance, is that the people are not actually able to choose their level of coverage (Finkelstein, 2002) . In the case that insurance coverage does not change for the individual, premium subsidies will only influence the behavior of the insured through wealth effects. While Feldstein and Friedman (1977) explicitly exclude the consideration of such effects from their model, we focus our examination on them. We use a theoretical model to analyze the influence of premium subsidies on moral hazard if insurance coverage does not change. We thus amend the literature by considering the behavior of individuals that do not switch their coverage due to premium subsidies and by studying an additional aspect of subsidy induced behavior change for those who do. Introduction In our first analysis, we use a simple one period model based on Shavell (1979) . We model premium subsidies as a relative deduction of premium payment and look at their influence on the effort exercised by the insured to reduce the loss probability. This way, we show that premium subsidies increase the problematic effects of moral hazard. Since the analysis is based on Shavell (1979) , it models moral hazard as a change in loss prevention. To reflect the specific conditions in the health insurance market, we also consider the influence of premium subsidies on loss reduction instead of loss prevention. The analysis shows that the effects are similar in both models. Introduction When studying moral hazard, the design of the insurance contract is essential for the behavior of individuals. Prior studies have emphasized the importance of co-payments by the insured (Pauly, 1968; Einav et al., 2011) . We, however, focus on the premium payment plan, i.e. the question under which circumstances the insured has to pay a premium. In models of insurance demand, premium payments are usually non-contingent in the sense that whether or not a loss occurs, the insured is always required to pay the insurance premium. Introduction 1 While this is a common design feature in health insurance or property and casualty insurance, not all insurance policies feature non-contingent premiums. Certain policies do not require a premium payment once a loss has occurred. Long term care insurance or disability insurance are two possible examples for insurance systems in which such contingent premiums are common. In a second part of the paper we extend our analysis to this kind of contracts. Additionally, since virtually all examples for contingent premiums concern intertemporal decision problems, we reflect this in the model and use a two period set-up. We show that in contingent premium contracts the effect of premium subsidies is reversed and subsidies actually encourage loss prevention. Thus, contract design is essential for analyzing the wealth effects of premium subsidies. The rest of the paper is structured as follows. In the next section, we review the literature on premium subsidies and make some considerations regarding insurance contracts and moral hazard. Section three introduces our extension of the model by Shavell (1979) and shows our results regarding the wealth effects of premium subsidies on moral hazard for loss prevention and loss reduction. In section four we investigate the influence of premium subsidies on moral hazard in a two period setting with contingent premiums. Section five summarizes and discusses the findings and offers some implications for public policy. The last section concludes and considers some possible directions for further research. Literature Review Many public insurance systems include some kind of premium subsidy for at least parts of the insured population. In insurance markets such as the market for employer sponsored health insurance or the federal crop insurance system in the U.S., policies are deliberately subsidized by the government in order to increase participation. Subsidies are usually administered through one of two different channels: direct subsidization through pricing and tax exemptions. Public programs such as the Federal Crop Insurance Commission (FCIC) issue policies that are deliberately priced below the actuarially fair value and do not include additional loadings for administrative expenditures or capital costs. In the FCIC, the relative rate of subsidization is tied to the coverage level of the insured but not to the risk. The National Flood Insurance Program (NFIP) on the other hand does not subsidize all policies but only those for older houses which have a substantial risk of being damaged by a flood. Since unsubsidized coverage for these houses could not be afforded by most owners, the NFIP ties the relative subsidization rate to risk exposure instead of coverage level. Literature Review The second possible channel to administer subsidies is through tax exemptions of premium payments to private insurance policies. This is, for example, done in the U.S. and Canada for health insurance and in the U.S. long term care insurance market. Tax exemptions keep the relative rate of subsidization constant in both coverage level and risk exposure, but tie it to the insured's marginal tax rate. Since most taxation systems have variable marginal tax rates in income, the relative rate of subsidization will thus be dependent on the income of the insured. Literature Review Some subsidies are relatively small in size. Kunreuther and Michel-Kerjan (2009) estimate the accumulated losses of the NFIP between 1968 and 2005 to be about $2.2 billion. However, other insurance systems are subsidized with considerably larger amounts. The FCIC paid out total premium subsidies of $7.3 billion and incurred a loss of $11.3 billion in the fiscal year of 2011 alone (Federal Crop Insurance Corporation / Risk Management Agency, 2011) . The substitution of health insurance dwarfs all other subsidies in the U.S. by comparison. The Joint Committee on Taxation estimates that solely the exclusion of employer contributions for health care, health insurance premiums, and long-term care insurance premiums from taxation will lead to tax expenditures just by the federal government of $725 billion between 2011 and 2015 (Joint Committee on Taxation, 2012). This poses the largest tax expenditure of the U.S. federal government (Gruber, 2011) . Gruber and Poterba (1996) estimate an average subsidy of employer sponsored health insurance of 31.8% of the initial premium through tax deductions. While this is a rather large number, Gruber (2011) also states that the actual amount of the subsidy varies by income of the insured. The increasing marginal income tax of the United States leads to more highly subsidized premiums for high income individuals than for those with lower income. Literature Review Subsidies for some policyholders can even appear in public insurance systems which are self financing in nature. In systems such as the U.S. unemployment insurance system (Anderson and Meyer, 1993) or the German health insurance system (Buchner and Wasem, 2003) , the pricing of the insurance policies does not adequately reflect the risk of the individual policyholder. Thus, even if the entire system is self-financing, some of the insured can buy policies below the actuarially fair price. For those insured that can profit from such cross subsidies, the effect is essentially the same as that of subsidies administered to all insured. Literature Review Several studies have examined the effect of subsidies on insurance demand. Most of the studies have focused on health insurance. They usually take advantage of the fact that the subsidy towards the insurance policy depends on the marginal tax rate of the insured. Cutler and Zeckhauser (2000) survey the literature regarding the United States and find estimates of demand elasticities for employer provided health care between -0.14 and -1.5. More recent literature validates these estimates (Royalty, 2000; Gruber and Lettau, 2004) . Finkelstein (2002) employs a difference-in-difference estimation and takes advantage of a change in Canadian tax legislation to show that the findings are robust in the international context. Often cited as most trusted for their accuracy are the estimates from the RAND Health Insurance Experiment (Manning et al., 1987; Newhouse and Rand Corporation Insurance Experiment Group, 1993) which deem the demand elasticity to be at -0.2. Literature Review Studies in other insurance markets have also found evidence for an increase in demand due to subsidies. Glauber (2004) states that the introduction of heavily subsidized crop insurance policies significantly increased the participation in the program. Goda (2011) shows that the introduction of tax subsidies for long term care insurance increases demand. However, some evidence to the contrary also exists. In his survey study, Kunreuther (1976) states that the demand for flood insurance in the U.S. is small even though the policies are (at the time of the study) heavily subsidized. Literature Review In this study, we are concerned with the influence of premium subsidies on moral hazard. Moral hazard is a broadly defined term. In our model and for the sake of the discussion, we define moral hazard as a change in behavior of the insured that is induced by an insurance contract. Behavior is modeled similar to Shavell (1979) as costly effort taken by the insured to reduce the expected loss, so called loss prevention. Models with loss prevention analyze ex-ante moral hazard. While this is a common approach in modeling moral hazard, it is in contrast to much of the literature on moral hazard in health insurance (Arrow, 1963; Coulson et al., 1995) . We thus also model the specific case for loss reduction, which can be argued to be more relevant for the health insurance case. Literature Review Nevertheless, the relevance of loss prevention in many insurance markets cannot be doubted. Provisions such as sprinkler systems or lightning rods clearly decrease the likelihood of damages to a house in case of fire or storm. Similarly, good work effort will decrease the probability of unemployment. Even though, amongst others, Cutler and Zeckhauser (2000) state that ex-ante moral hazard is not necessarily applicable for health insurance, certain examples still come to mind. A healthy lifestyle generally leads to less health problems and thus ex-ante moral hazard is at least a possibility in health insurance. As an example, anecdotal evidence suggests that in countries such as Switzerland, in which few people have comprehensive dental insurance, most white collar workers brush their teeth after lunch at the office to avoid dental problems. Hence, both ex-ante and ex-post moral hazard have relevance for health insurance. Literature Review There is empirical evidence that moral hazard exists in insurance markets (Chiappori et al., 1998; Stabile, 2001) and it has been noted that the differentiation between ex-ante and ex-post moral hazard is important in this respect (Abbring et al., 2008) . In health insurance, the existence of ex-post moral hazard is well documented. Finkelstein et al. (2012) find in a study similar to the RAND Experiment that individuals from the lower income sector increase their annual utilization of medical care by 25% or $778 if they obtain health insurance. Two aspects of their results are particularly interesting. The first is that the costs were mostly generated by treatments about which the insured has some control. In contrast, the utilization of ER related medical care did not change. This implies that the insurance does influence the spending behavior of insured, while there is no effect in situations in which the health situation is so critical that it requires an ER visit and the insured has no possibility of denying medical care. The second interesting aspect of the study is that utilization of preventive care also increased due to insurance. Such preventive care could lower the medical spending in the long run. The authors acknowledge that their study has a short time horizon and that such an effect could indeed occur. Based on considerations of price elasticities for medical care, multiple studies have estimated the welfare loss due to moral hazard in health insurance in the united states. Based on the calculations by Feldstein (1973) , Feldman and Dowd (1991) estimate the welfare loss due to health insurance to be between $33.4 billion and $109.3 billion in 1984. Converted to 2012 dollars, this would correspond to a welfare loss between $74.4 billion and $243.4 billion. 2 It is obvious that subsidized insurance will amplify this effect, because more people will be insured. 2 In contrast to the influence of subsidies on demand, the influence of premium subsidies on moral hazard is not researched as well. Generally, subsidies can influence the market inefficiencies due to moral hazard in two ways: subsidies can increase the insurance coverage level which in turn can lead to an increase in moral hazard. We call this demand effects. The second influence is a change in the composition of the insurance contract. Subsidies reduce the premium and thus increase the wealth of the insured for any given insurance coverage. We call this wealth effects. Which of the two effects dominates depends, among other factors, on the structure of the insurance market. 2 It could be argued that demand effects will be more significant economically than wealth effects. Feldstein and Friedman (1977) exclude wealth effects from their consideration entirely. However, while this is valid for some cases, in other cases wealth effects will be more important than demand effects. Some cases will not show any demand effects at all and thus only wealth effects will jeffreyayala@example.net. Three specific market settings come to mind: 2 1. If insured can only choose between a few different coverage levels and a subsidy is 2 Calculations are based on an annual inflation rate of 2.9%. 2 introduced to the market, it might not give the insured a sufficiently large incentive to switch to the next higher coverage level. In this case, the insured will remain on the existing coverage level and subsidies will only have consequences for the individuals' wealth and not for their demand. 2 2. A similar argument applies in case only one level of coverage exists. In this situation, the insured can only decide to purchase insurance or not. Those who are already insured cannot increase their coverage due to the subsidy and only the wealth effect influences their behavior. 2 3. If insurance coverage is mandatory, a subsidy will not change the insurance demand at all and will only affect the wealth of the insured. 2 The first case is probably the most common setting for an insurance market. Almost no insurance provider offers a choice of coverage levels on a continuous scale. The fewer policies are offered to the insureds, the less likely it will be for the subsidy to influence the demand of the insured and the more important will be the wealth effect in comparison to the demand effect. Case two is to some extend the extreme case of case one. It is relatively common in public insurance programs such as unemployment insurance. The relevance of wealth effects in comparison to demand effects in this case can further be demonstrated by the results of Gruber and Washington (2005) who find a very small price elasticity of insurance take-up if tax subsidies are introduced. Insurance coverage can be factually mandatory as is the case for most European public insurance systems. Germany, for example, requires mandatory health care coverage, unemployment insurance, annuity coverage and automobile liability coverage of every citizen. If such a mandatory insurance scheme is combined with a single level of coverage, any subsidy applied to the contract will solely have a wealth effect on the insured's behavior. Insurance coverage can also be circumstantially mandatory even if it is not directly required by law. In some areas of the U.S. every household who takes up a mortgage from a federally backed or regulated lender is required to buy flood insurance coverage from the NFIP (Michel-Kerjan and Kousky, 2010). As Finkelstein (2002) mentions, many employees also do not have a choice whether to take up their employer's health plan or not. Save from changing employment, this also poses a mandatory insurance scheme, particularly if the employees are not offered a choice between different health plans. 2 A different argument against considering demand effects of moral hazard is presented by de Meza (1983) and Nyman (1999) . The argument is based on the fact that health insurance redistributes wealth from the healthy to the sick. This redistribution enables the sick to consume more medical care and other commodities in an ill state than would be possible without health insurance. For extremely expensive procedures, health insurance might be the only possibility to gain access to medical jeffreyayala@example.net. As such, the authors conclude that part of the increased utilization of medical care due to health insurance is not caused by moral hazard but rather by the redistribution inherent in any insurance market. Incorporating such considerations in calculations like those made by Feldman and Dowd (1991) , would greatly diminish the estimates of welfare loss due to moral hazard in health insurance. 2 Despite these considerations, the prior literature on the influence of premium subsidies on moral hazard solely focused on the demand effect. Feldstein and Friedman (1977) explore both the issue of increasing insurance demand and rising cost of third party services (see also Nell et al. 2009 ) in a simulation model of the health care market. They conclude that the impact is highly economically significant and might be responsible for some of the increased health care spending in the United States. An argument similar to that of Feldstein and Friedman (1977) is proposed in Pauly's (1986) review of price elasticities for health insurance and medical care. He cites evidence that subsidized insurance premiums will increase insurance demand, lead to overinsurance and ultimately to overconsumption of medical care. Such an argument has been repeated by multiple authors since (Feldstein, 1995) . Jack and Sheiner (1997) offer a closed form solution supporting this result and argue that under certain conditions subsidies to out-of-pocket medical spending can offset the effect of premium subsidies at least to a certain degree. This entire line of research has contributed significantly to the understanding of the economic effects of premium subsidies. However, the argumentation does not differentiate between ex-ante and ex-post moral hazard and assumes that insurance coverage only influences moral hazard through demand effects. 2 In the perfect information case, the effect of insurance pricing on loss prevention and loss reduction through both demand effects and wealth effects has been investigated before. In a model with non-contingent premiums, Ehrlich and Becker (1972) find that increasing the price for insurance will decrease the effort for loss prevention. This implies that a subsidy would increase loss prevention. The result can intuitively be explained on the account that in the model of Ehrlich and Becker (1972) , insurance and loss prevention can be complements. Thus, if insurance becomes more expensive, the effectiveness of loss prevention can decrease. However, since asymmetric information changes the interaction of insurance and loss protection, the results of the model will not necessarily carry over toward the asymmetric information case. 2 It can be seen that even though premium subsidies and moral hazard have been discussed in many respects, the discussion has been focused on demand effects. Thus, even though wealth effects of subsidies on ex-ante moral hazard under asymmetric information can be economically significant, they have not yet been investigated empirically or theoretically. We offer a theoretical foundation for such a discussion in the next sections. Premium Subsidies and Moral Hazard with Non-Contingent Premiums Basic Model and Loss Prevention For our analysis, we use a simple model with two states of the world. The insured is assumed to maximize the expected utility of his wealth which is subject to randomness. He is expected to have a von Neumann/Morgenstern utility function U (·) showing risk aversion such that U (·) > 0 and U (·) < 0. The wealth takes on the value x L with probability p > 0 and the value x H with probability (1−p), whereas x L < x H . The insured can exercise effort e ∈ [0, ∞[ that will influence the probability. The effort has a diminishing marginal benefit in terms of loss probability reduction such that p (e) < 0 and p (e) > 0. Exercising effort causes costs c(e) for the insured which are increasing convexly such that c (e) > 0 and c (e) > 0. We assume the cost function to be measured in utility units such that the costs are separable from the insured's utility function. Basic Model and Loss Prevention Effort is measured in utility units to emphasize the fact that most effort in the situations under scrutiny here is non monetary in nature. Consider, for example, health insurance. One substantial driver of health risk are unhealthy eating habits. While it could be argued that there is a surcharge associated with healthy eating, most people choose not to eat healthily not because of the price of healthy nutrition, but because they rather have the taste for hamburgers and french fries than for salad. This becomes even more apparent when considering the amount of food consumed. Eating less is cheaper than eating more and would be the more healthy option for many people. Thus, dietary restrictions are mostly a question of discipline, which is, by definition, non monetary effort. Basic Model and Loss Prevention Non-contingent premiums imply that the premium payment is independent of whether or not a loss occurs. As stated above, such contracts are common in health insurance and property and casualty contracts. They are also prevalent in most commercial insurance lines such as the crop insurance policies offered by the FCIC. The premium is typically paid at the beginning of the period or payments are spread out over it. In case of a loss within the time period, the insurer will make an indemnity payment which does not influence the payment plan for the premium. We investigate this type of contract using a model set-up in which the insured can take up insurance for a fixed indemnity of I and a premium of π to be paid regardless of the state of nature. The principle offers only one possible insurance contract to the insured. We simplify to this in order to investigate the isolated wealth effect of premium subsidies on moral hazard. As is common in most insurance markets, the insured will only be able to buy less than full coverage such that x H − π > x L − π + I will always be fulfilled. Basic Model and Loss Prevention In the model the insurer offers a fixed insurance level I at premium π. The insured decides to take up the insurance or not and contingent on that choice determines his effort. Consecutively, the state of nature is determined and the insurance contract is executed. Basic Model and Loss Prevention To identify the first best solution of the problem under perfect information we assume that the premium π charged by the insurer is dependent on the effort exercised by the insured and observed by the insurer. The insurer charges actuarially fair premiums and thus the contract will be subject to a budget constraint such that the expected value of the insurance contract is zero, i.e. that the expected premium payment equals the expected indemnity: Basic Model and Loss Prevention π(e) − p(e)I = 0 Basic Model and Loss Prevention We assume the insurer to incurr zero profits. This could be due to multiple reasons. If the insurance market is competitive, expected profit will be zero in equilibrium. Basic Model and Loss Prevention 3 However, it could also be the case that the insurer is the government and that the government does not have a profit motive when pricing insurance policies. In the case that subsidies are introduced to the market, the insurance policies and thus the budget constraints will be modeled such that the expected profit for each policy will be negative. This will of course not mean that the expected profit of the insurance companies has to be negative. The model does not make specific assumptions on whether the insurer is reimbursed for the subsidy by the government such that expected profits will again be zero or the insurer in fact incurs a loss (if, for example, the insurer is the government). This is a natural way to model most insurance markets which are subsidized. If, for example, US crop insurance is considered, private insurance companies in a competitive market sell policies for which the premium income does not cover the expected loss of the insurer. They get reimbursed for the calculated loss from the FCIC. In the case of health insurance, the insurers that offer health care plans actually do so in a competitive market and the tax exemption of premium payments make the policies appear to the insureds as if they were priced with a calculated loss for the insurance company (which is again carried by the government through the tax expenditure). As such, the model accurately describes the perception from the point of view of the insured, whose behavior is what is of interest in this study. Basic Model and Loss Prevention Given this price setting mechanism and the concavity of U (·) the insured will always take up insurance 4 since the participation constraint will always be fulfilled: Basic Model and Loss Prevention In case the insured takes up insurance he will anticipate the price-setting mechanism and chose e as to maximize expected utility, whereas expected utility is given as: Basic Model and Loss Prevention For ease of expression, we abbreviate U (x H − π) as U H and U (x L − π + I) as U L . The first order condition is given as 5 : Basic Model and Loss Prevention It can be seen that the marginal benefit of administering effort is twofold. First, it decreases the probability of the loss state (first term on the LHS) and secondly it increases the wealth because the premium decreases with increased effort (second term on the LHS). When looking at the second-best solution, effort is no longer observable by the insurer. In case of asymmetric information, the insurer cannot observe the insured's effort. In this case the insurer will assume the insured to exercise effort such that her utility is maximized. This renders the following maximization problem: Basic Model and Loss Prevention which renders the first order condition to be 6 : Basic Model and Loss Prevention Comparing the solution from (2) to the first-best solution in (1) shows that the marginal 4 Fairly priced insurance in this sense leads to a mean preserving contraction of the lottery that determines the wealth of the insured. 5 Recalling that x H − π > x L − π + I and noting that the premium is concavely decreasing in effort, the second order condition holds as can be seen by: Basic Model and Loss Prevention benefit due to premium reduction is removed. With less marginal benefit but the same marginal cost function, there will be less effort exercised. At any given premium π, the insured will now exercise less effort and have a higher chance of the low wealth state, than in the perfect information case. This is the effect of moral hazard. Premium subsidies price the insurance contract at a rate such that it will have a negative expected value for the insurer. In order to model this, we use a loading factor λ that implies subsidies if λ > 1. The budget constraint of the insurer now is: Basic Model and Loss Prevention It can be seen from this equation that λ is modeled as a relative subsidy. 7 By increasing it to a value larger than one, only 1 λ of the expected indemnity need to be covered by the premium income. This way, we model subsidies as they appear most often in practice. All of the examples mentioned above are relative subsidies in nature. In this model we assume the relative subsidy to be constant. In practice, the relative rate of subsidization is often tied to one or more parameters of the insurance contract, such as coverage level. The incentives set through such variable subsidization rates are outside the scope of this study, but offer a natural direction for further work. Basic Model and Loss Prevention Considering the effect of the premium subsidies on the effect of moral hazard in the case of asymmetric information, we use the first order condition rendered by (2) and apply the implicit function theorem to arrive at 8 : Basic Model and Loss Prevention λ has a negative effect on the effort exercised by the insured. Thus, with this contract design the wealth effect of premium subsidies will increase the market inefficiency due to moral hazard. Intuitively, this can be explained by the fact that premium subsidies influence the wealth independent of the state of nature. Due to the condition that x H − π > x L − π + I and the diminishing marginal utility of U (·), the low wealth state will have higher marginal utility than the high wealth state. Thus, the increase in wealth increases the attractiveness of the low-wealth state by more than it increases the attractiveness of the high-wealth state. Basic Model and Loss Prevention The insured now has relatively less incentive to be in the high-wealth state and will decrease 7 We model λ to be a relative subsidy since this is common in practice. However, the results also hold for an absolute subsidy. In that case the budget constraint would be λ + π − p(e)I = 0. The proof is available from the jeffreyayala@example.net. 8 The effect of subsidies on the premium in (4) is negative. From the effort to reach it. Thus, with non-contingent premiums, the wealth effect will amplify the demand effect and increase the malicious consequences of moral hazard in the insurance market. Moral Hazard and Premium Subsidies in Loss Reduction As stated above, health insurance is one of the major examples for the relevance of premium subsidies. It has been argued by different people (Arrow, 1963; Coulson et al., 1995; Cutler and Zeckhauser, 2000) that a lack of loss prevention is not the major driver of moral hazard in medical care markets. The way in which insured individuals deal with treatment costs once an illness has occurred is often much more problematic. Thus, a lack of effort to reduce losses once they occurred is a major driver of moral hazard in health insurance. In this section we deal with this issue by looking at the wealth effects of premium subsidies on loss reduction in non-contingent contracts. Moral Hazard and Premium Subsidies in Loss Reduction The model set-up has to be adjusted to fit the context. For simplicity, we define the low wealth state in terms of the high wealth state and a random loss: Moral Hazard and Premium Subsidies in Loss Reduction The probability that a loss occurs is p. If it occurs, the size of the loss is randomly determined according to a probability distribution Moral Hazard and Premium Subsidies in Loss Reduction The insured can exercise effort e ∈ [0, ∞[ at convexly increasing costs c(e) to influence this probability distribution. Let f (L, e) be twice continuously differentiable, effort then influences the probability distribution such that ∂F (L,e) ∂e > 0 and Moral Hazard and Premium Subsidies in Loss Reduction < 0. I.e. for any two effort levels e 1 < e 2 , F (L, e 1 ) first order stochastically dominates F (L, e 2 ), but the influence of e on the probability distribution is concave. It is assumed that for any effort level, the support of the probability distribution remains unchanged. Moral Hazard and Premium Subsidies in Loss Reduction Instead of an absolute indemnity, I, we model insurance as a coinsurance contract such that the indemnity now is a fraction of the loss determined by the coinsurance rate, δ. The insured bears a share of the loss equal to the coinsurance rate and the insurer indemnifies the rest. Moral Hazard and Premium Subsidies in Loss Reduction In the interest of brevity, we focus on the asymmetric information case in this extension. Moral Hazard and Premium Subsidies in Loss Reduction 9 The insurer cannot observe the insured's effort. In anticipation of utility maximizing behavior by the insured, the insurer will offer an insurance contract (δ, π), such that his Moral Hazard and Premium Subsidies in Loss Reduction (1 − δ)Lf (L, e)dL = 0 and the no full insurance or overinsurance condition δ > 0 is fulfilled. Moral Hazard and Premium Subsidies in Loss Reduction This renders the following maximization problem for the optimal level of effort by the 9 As before, the effort under symmetric information will be higher than under asymmetric information. Results are available from the jeffreyayala@example.net. insured: Moral Hazard and Premium Subsidies in Loss Reduction we arrive at the first order condition in (5) 10 . Moral Hazard and Premium Subsidies in Loss Reduction Using this and the implicit function theorem, we can derive the influence of premium subsidies on loss reduction with non-contingent premiums as 11 : Moral Hazard and Premium Subsidies in Loss Reduction We can see from (4) and (6) that the result for the wealth effects of premium subsidies on moral hazard are the same whether loss prevention or loss reduction are analyzed. The intuitive explanation for both results is similar, as well. Premium subsidies increase the wealth of the insured in both states of nature. Due to U H > U L and U (·) < 0, this will decrease the marginal benefit in the low-wealth state by more than in the high wealth state. Thus, spending effort to increase the utility in the low-wealth state becomes less attractive and less effort will be exercised. Moral Hazard and Premium Subsidies in Loss Reduction 10 Some conditions need to be fulfilled for the first order approach to be valid. Appendix A shows that they are fulfilled and gives the intuition for the derivation of (5). Moral Hazard and Premium Subsidies in Loss Reduction 11 The denominator in (6) can be seen to be negative. The implicit function theorem and (5) render: Moral Hazard and Premium Subsidies in Loss Reduction We now apply the implicit function theorem on the insurers budget constraint to identify the influence of the subsidy parameter on the premium: Moral Hazard and Premium Subsidies in Loss Reduction Integrating by parts and the assumption that the support of F (L, e) does not change in effort, this renders: Moral Hazard and Premium Subsidies in Loss Reduction Thus we can see that the influence of premium subsidies on effort is indeed negative as stated in equation (6). Moral Hazard and Premium Subsidies in Loss Reduction That the result for non-contingent premiums is valid for loss reduction as well as loss prevention shows that premium subsidies will increase moral hazard no matter what behavioral pattern is more important in a given insurance market. For the medical care market, which in part motivated the inclusion of loss reduction in this study, this shows that wealth effects of premium subsidies will aggravate their already problematic demand effects. Dependence on Contract Design -Loss Prevention and Contingent Premiums The intuitive explanation given above shows that the premium payment plan is crucial for the influence of premium subsidies on moral hazard. Especially the case in which premiums are not paid in every state of the world might change the result given above. A good example for an insurance contract in which the premium is not paid independently of the loss is longterm care insurance. Here the insured buys a policy and only pays premiums as long as no claim is made. If the insured requires long-term care, he seizes to pay premiums and the insurance company starts paying the indemnity. Similar structures of insurance contracts can, for example, be seen in disability insurance or unemployment insurance. Dependence on Contract Design -Loss Prevention and Contingent Premiums When considering examples for insurance contracts with contingent premiums, it becomes apparent, that all of them are intertemporal decision problems. To model contingent premiums as simple as possible, we assume a model with two periods. In the first period, the insured has wealth x H with certainty. His wealth in the second period either takes on the value x L with probability p > 0 or the value x H with probability (1 − p), whereas x L < x H . The insured can exercise effort that will influence the probability distribution of his wealth in the second period at convexly increasing costs c(e). To protect himself from the low wealth state in the second period, he is able to take up insurance with a contingent premium. In exchange for a premium α to be paid in the first period and in the high wealth state of the second period he will gain a payment β in case of the low wealth state in the second period. We assume the insurer to offer a fixed level of insurance β. All changes in the insurance contract, as could, for example, be induced by premium subsidies, will only influence α, while β will stay constant. As above, the insurer offers only one possible insurance contract and no full insurance or overinsurance is possible such that x H − α > x L + β will always be fulfilled. Dependence on Contract Design -Loss Prevention and Contingent Premiums The second period is added to the model to emphasize the intertemporal nature of insurance situations with contingent contracts. This becomes clear when taking unemployment insurance as an example. The insured enters a job in the first period and earns x H as a wage and pays α as a premium for unemployment insurance. In this period she exercises effort such as coming to work on time and working hard in order to stay employed in the second period. Once the uncertainty about the employment status is resolved, the insured enters the second period either still earning the wage x H and paying α to protect herself against unemployment in the future or earning her reservation income x L and gaining unemployment benefits β. This example makes clear that the model could be stretched for more than two periods, but the essential features of the decision situation of the insured are captured in this simplified version. Additionally, the example shows why in this model as well, effort is measured in utility cost and not in monetary units, since the examples for effort are non-monetary in nature. Dependence on Contract Design -Loss Prevention and Contingent Premiums For simplification we assume the insured not to have time preferences and no possibility of transferring wealth between the two periods. The insurer will offer an insurance contract, the insured will accept or decline it and contingent on that decision determine his effort. Based on the effort, the state of nature for the second period is determined and the insurance contract is executed. Dependence on Contract Design -Loss Prevention and Contingent Premiums The insured takes up insurance, anticipates the price-setting mechanism and chooses e as to maximize expected utility. The first best solution under perfect information thus implies the following maximization problem of the insured: Dependence on Contract Design -Loss Prevention and Contingent Premiums For convenience, we abbreviate U (x H −α) as U H and U (x L +β) as U L . The budget constraint of the insurer reads: Dependence on Contract Design -Loss Prevention and Contingent Premiums (2 − p(e))α(e) − p(e)β = 0 Dependence on Contract Design -Loss Prevention and Contingent Premiums Thus the first order condition is given as 13 : Dependence on Contract Design -Loss Prevention and Contingent Premiums As in (1), the case of non-contingent premiums, the marginal benefit of effort is twofold, the loss probability and the premium are decreased through effort. The second best solution is again characterized by non-observability of effort. The maximization problem is thus: Dependence on Contract Design -Loss Prevention and Contingent Premiums 12 As above, the fairly priced insurance policy leads to a mean preserving contraction of the lottery that determines the wealth of the insured and thus the participation constraint is always fulfilled 13 Noting that the premium is concavely decreasing in effort, the second order condition holds as can be seen by: Dependence on Contract Design -Loss Prevention and Contingent Premiums which renders the first order condition: Dependence on Contract Design -Loss Prevention and Contingent Premiums Which compared to (7) shows no marginal benefit due to premium reduction. We use this situation in order to analyze the influence of premium subsidies on the effort exercised by the insured. We again introduce λ as a relative subsidy factor 14 . In the budget constraint this reads: Dependence on Contract Design -Loss Prevention and Contingent Premiums As above, we use (8) and apply the implicit function theorem to arrive at 15 : Dependence on Contract Design -Loss Prevention and Contingent Premiums The result shows that higher subsidies increase the effort exercised by the insured and reduce the problematic effects of moral hazard in insurance contracts with contingent premiums. This can be intuitively explained by the fact that by subsidizing the premium to be paid at a constant coverage level the wealth in the high-wealth state increases. This now increases the incentive for the insured to be in the high-wealth state and thus he will increase the exercised effort in order to have a higher probability of achieving that state. In contrast to the demand effect, the wealth effect in contingent premium contracts thus decreases the inefficiency due to moral hazard. Discussion The results from the model above highlight two factors of premium subsidies. Firstly, premium subsidies in insurance programs can change the behavior of individuals even if they do not change their insurance coverage. Secondly, the design of the insurance system and particularly the premium payment plan, will determine the influence premium subsidies will have on the insured's behavior. By focusing the discussion of premium subsidies and moral hazard on demand effects, it has implicitly been limited to those people that change their coverage due to the premium subsidies. The goal of virtually every policy measure to introduce or increase premium subsidies is to increase insurance take-up. A change in behavior is an obvious consequence of insurance coverage. Even though this change in behavior might be problematic, the increased insurance coverage was the initial goal of the subsidization. So any change in behavior must initially have been judged as an acceptable cost for the benefit of increased coverage. Nevertheless, subsidies do not only affect those people that will end up changing their coverage. Any person that is already insured will benefit from the subsidy even if they do not change their insurance jeffreyayala@example.net. Discussion This study shows that this benefit will change the behavior of the insured. By focusing on demand effects, public policy has thus been lead to focus on only part of the problem. It is important to consider all market participants when considering premium subsidies and not only those which will change their insurance coverage or start to participate in the insurance program for the first time. Depending on the structure of the insurance program and the available coverage levels, those people that do not change their coverage can be much more economically significant than those who do. This might change the outlook on some of the economic policies which are in consideration or already in force. Discussion The question whether demand effects of wealth effects are more important is directly tied to the design of the insurance system. In a market with many different accessible coverage levels, demand effects will most probably be more influential on the insured's behavior. However, with a decreasing number of options to choose from and increasing transactions costs associated with the change of one's coverage, wealth effects will gain in importance. This does not mean that both effects will always appear. In the extreme case of mandatory insurance coverage, only the wealth effect will appear if a subsidy is introduced. In other cases both changes in demand and changes in behavior due to a different composition of the insurance contract are possible. Nevertheless, relatively speaking one or the other effect will be more significant. Discussion The second design aspect shown to be important by this study, is the way premiums are paid by the insured. By subsidizing the premium, the wealth of the insured changes in all states of the world in which the insured pays a premium. Thus, if the premium payment is contingent, the benefit of the subsidy is only gained if no loss occurs. This way, the premium subsidy provides the insured with an incentive to actively prevent a loss. While this result is important for analyzing the effect of a subsidy on moral hazard, it also holds potential implications for redeveloping old insurance systems or designing new ones. Discussion Apart from the results regarding premium subsidies and moral hazard, the results of the study also hold for positive premium loadings. None of the results in the study are bound to λ ≥ 1. However, if the premium payments are loaded instead of subsidized (λ < 1), the participation constraint of the insured need not hold. In a case where it does not hold any more, the insured will not take up insurance and the moral hazard problem disappears. Nevertheless, for sufficiently risk averse decision makers or in the case the insurance coverage is mandatory, the results will hold and can be applied to premium loadings. Discussion The results of the above models have implications for public policy. It is apparent that when discussing subsidization for insurance, considerations are mostly focused on the tradeoff between cost of the subsidy and the potential positive effects of increased participation and decreased adverse selection (Glauber, 2004) . Neglecting moral hazard as a possible consequence of subsidized insurance policies is a common, but possibly dangerous mistake. When implementing premium subsidies, the moral hazard problem should always be considered. The study highlights, however, that the considerations must differ according to the design of the insurance scheme. For example,
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