Optimal Premium Subsidy and Its Impact on Individual Choice for Insurance Coverage

The purpose of this article is to analyze whether a government premium subsidy is desirable by using a game theoretic approach. From our analysis, the following main results are derived. When policyholder’s effort for lowering accident probability is not considered, government conducts premium subsidy when the fraction of policyholders is small. In contrast, when policyholder’s effort is considered, whether government conducts premium subsidy is ambiguous even if the fraction of policyholders is small. Journal of Business & Financial Affairs J o u r n a l o f B usi ness & Finncial Af f a i r s ISSN: 2167-0234 Citation: Okura M, Jiang YY (2017) Optimal Premium Subsidy and Its Impact on Individual Choice for Insurance Coverage. J Bus Fin Aff 6: 260. doi: 10.4172/2167-0234.1000260


Introduction
The importance of sharing risks via the insurance market has been addressed by many previous studies [1][2][3]. However, it is always difficult for an individual to get full insurance coverage at a reasonable cost due to the failure of private insurance market. In general, factors such as market imperfections and existence of informational asymmetries between an individual and insurer which generate problems such as moral hazard and/or adverse selection are responsible for market failure.
Insurer would prefer to provide an incomplete coverage or conduct some monitoring activities in order to avoid incurring possible losses caused by individual's opportunistic behavior [4][5][6]. In the latter case, a full insurance coverage would probably become more expensive due to increase in transaction cost resulted from monitoring activities in relation to policyholder's behavior. In some cases, an insurance coverage itself would become unaffordable or unavailable if asymmetric information problems are significant enough that complicated and specialized insurance contracts associated with high costs become indispensable [7].
Previous studies have also shown that an individual may favor a partial coverage or no coverage at all if she does not have sufficient knowledge and/or information regarding certain risks and/or insurance coverage [8][9][10][11]. Moreover, Anderson [12] and Kaplow [13] suggest that an ex post government relief program (e.g., grants, low-interest loans) against financial losses undermines an individual's willingness to pay to be fully insured, which is known as a problem of charity hazard [14][15][16]. Although government relief can be efficient if insurance markets suffer more inefficiency from market imperfections other than moral hazard [17].
Given the presence of market failures and, in some cases, the resulting absence of markets for insurance, the overall distribution of risks would presumably be sub-optimal and if the total cost of individual's risk-bearing is consequently significant, then government intervention may be desirable [18]. More specifically, government intervention in certain markets as a supplier of insurance (e.g., public health care) can be cost-effective because all uncertainty will be borne publicly and the total of the costs of risk-bearing goes to zero as the population of taxpayers becomes large [7,19].
Government can also intervene in insurance markets by providing a government-sponsored reinsurance program, which has been relatively effective in dealing with natural disaster hazards [20]. Government can From our analysis, the following main results are derived. When policyholder's effort of lowering the probability of an accident is not considered in case the fraction of policyholders is small, premium subsidy conducted by government is desirable under which an individual chooses to purchase full coverage while the social welfare is maximized which consists of the welfare of policyholders, nonpolicyholders and insurance firm. In contrast, when policyholder's effort is considered, whether government should conduct premium subsidy is ambiguous even if the fraction of policyholders is small.
The remainder of this article is organized as follows. Section 2 builds a model and explains game structure of this model. Section 3 develops the model to derive the condition in which government conducts premium subsidy when policyholder does not conduct the effort. Section 4 provides a further analysis by introducing effort for lowering accident probability and investigates the effect of effort. The major findings of this article are summarized in Section 5.

The Model
Suppose that there are weakly risk-averse identical (potential) individuals, a risk-neutral insurance firm, and the government. One type of individuals has two types of initial assets. The first type is liquid assets such as salary and bonuses, which is denoted by The second type is fixed assets such as a house and vehicle, which is denoted by 0 F w > . In contrast, another type of individuals has only liquid assets. Assume that a fixed asset has the potential to incur damage, and the individual considers purchasing insurance for covering the damage. In contrast, it is assumed that there is no possibility of losing the amount of the liquid asset. After purchasing insurance, the policyholder chooses whether the effort of lowering the probability of an accident is conducted in "effort case" section. Because that effort is not observable by others, the policyholder chooses the effort only if it is profitable. i π represents the accident probability and { } 0,1 i ∈ where "0" and "1" indicate the cases of no effort and effort, respectively. Thus, it is natural that Effort can lower the accident probability, but the policyholder has to incur the cost involved. Effort cost is assumed to be constant and represented by 0 k > . The following four-stage game is considered.
In the first stage, the government decides both the subsidy rate, which is denoted by , and the tax rate, which is denoted by The government subsidizes a portion of the insurance premium from tax revenue. Thus, the policyholder actually pays the insurance premium. To apply the premium subsidy, the government imposes a tax on each liquid asset w L such as income tax.
( ] 0,1 β ∈ denotes the fraction of policyholders involved. Then, the government chooses both the subsidy rate and the tax rate to satisfy the following equation which specifies that the government revenue and expenditure are equal: where 0 i p > represents the insurance premium. Then, the government chooses the subsidy rate in accordance with the following equation: In the second stage, the insurance firm chooses the insurance premium. For simplicity of the model, the amount of damage is assumed to be equal to w F . Thus, when the policyholder purchases full insurance, the insurance money is equal to w F . In the third stage, the policyholder chooses the insurance coverage rate, which is represented In the fourth stage, the policyholder chooses whether to conduct the effort.
In this situation, the expected utility for the policyholder, which is denoted by EU i , can be written as follows: By using the computation in Asai and Okura [31], the certainty equivalent, which is denoted by CE i , can be derived as follows: where r ≥ 0 represents the policyholder's degree of absolute risk aversion.

No effort case
In this section, we consider the case in which policyholder's effort is excluded. To analyze this game, we temporarily remove the fourth stage from the original game and a three-stage game is considered by backward induction. Because policyholder does not conduct the effort, subscript of all variables in this section is surely zero.
We first investigate the third stage. The optimal insurance coverage rate has to satisfy the following first-order condition: where the asterisk represents the equilibrium value. From equation (7), we show Further, consider the second stage. The expected profit of the insurance firm, denoted by 0 EΠ , is: Then, substituting equations (2) and (8) into equation (9), we have The optimal insurance premium has to satisfy the following firstorder condition: From equation (11), the optimal insurance premium can be derived as follows: Also, substituting equation (12) into equation (8), the optimal insurance coverage rate can be rewritten as: From equation (13), we find the larger the tax rate (and the subsidy rate), the larger the optimal insurance coverage rate. However, there is an upper limit for the tax rate because 0 1 α ≤ has to be satisfied. Let 0 t be the upper limit of the tax rate. The government can choose a tax rate beyond 0 t . However, even if such a tax rate is chosen, the policyholder will not change the insurance coverage rate. Thus, it is not desirable for the government to choose a tax rate beyond 0 t . Then, we have ( ) In contrast, 0 1 2 α * = is realized when the tax rate (and the subsidy rate) is zero. Thus, the premium subsidy offered by the government always raises the optimal insurance coverage rate. Substituting equation (12) into equation (2), the subsidy rate is: Further, substituting equations (12), (13), and (15) into equations (6), and (10), the certainty equivalent of the policyholder and the expected profit of the insurance firm are derived as: Lastly, we investigate the first stage. In this stage, the government actually chooses both the premium subsidy and the tax rates. However, both rates are related in equation (15), and the problem at this stage can be reduced to derive the optimal tax rate. It means that if government chooses strictly positive tax rate 0 0 t * > , government chooses strictly positive premium subsidy 0 0 s * > and vice versa. It is assumed that the government chooses the tax rate for maximizing social welfare, which consists of the welfare of policyholders, non-policyholders, and the insurance firm. Then, the social welfare, denoted by i SW , is shown as: where [ ] 0,1 γ ∈ represents the weight of consumers' welfare used to calculate the social welfare.
Substituting equations (16) and (17) into equation (18), we obtain Equation (19) is the quadratic and convex functions of t 0 , and so the value to minimize social welfare is uniquely determined. This value is denoted by 0 t , and the first-order condition can be shown as: From equation (20), we have: From equations (14) and (21), the following relationships are indicated: From equations (22) and (23), the following three cases are possible.
Case 1: From equation (22), we find 0 0 t ≤ . Then, the social welfare function in equation (19) Also define . Then, f 0 is calculated as: From equation (26), the following relationship can be confirmed: By combining the above three cases, the following two possible results are derived. The above results show that whether the premium subsidy is desirable depends on the weight of consumers' welfare γ and the fraction of policyholders β. If γ and/or β are relatively small, then the government introduces a premium subsidy and full insurance is realized. In contrast, if γ and/or β are relatively large, then the government does not offer a premium subsidy and partial insurance is realized.

Effort case
It is easy to verify that 1 0 CE CE ≥ is the condition in which policyholder chooses the effort in the fourth stage of the game. However, introducing policyholder's effort may change the condition whether government conducts premium subsidy. For example, the government may want to lower the optimal tax rate in the case of ( ) 0 6 6 5 γ β ≤ ≤ + for lowering the optimal insurance coverage rate and conducting the effort. To investigate the effect of the effort, we reexamine two situations, ( ) 0 6 6 5 γ β ≤ ≤ + and ( ) First, consider the case where ( ) In this situation, the policyholder never chooses to conduct the effort because full insurance is chosen. Thus, the optimal tax rate is surely 0 t . Then, the government may be able to increase the social welfare by lowering the tax rate, because this leads to lowering the optimal insurance coverage rate, and then the policyholder voluntarily conducts the effort. Let 1 t  be the maximum tax rate at which the policyholder conducts the effort. Then, 1 t  can be derived by solving Of course, 1 t does not necessarily exist, because 1 t is never greater than = − that is the upper limit of the tax rate when policyholder conducts the effort. In the latter analysis, it is assumed that 1 t  always exists. Let 1 α  be the insurance coverage rate in the case of 1 t  . From same manner deriving equation (13), we find that Then, if 0 g ≥ is realized, the government chooses 1 t  instead of 0 t . There are two kinds of conditions to realize 0 g ≥ , as shown below.
In the case that ( )( In the case that The denominator on the right-hand side of equations (30) and (31) represents the change in the expected profit for the insurance firm, represented by In this situation, the change from 0 t to 1 t  improves the social welfare of non-policyholders, because the tax rate becomes lower and the expected profit for the insurance firm is increased because while the social welfare of policyholders is the same because 1 0 . CE CE = Consequently, it is desirable for all parties to choose the tax rate 1 t  .
In contrast, when (31) is the unique condition because there exists no γ that satisfies equation (30), since . However, we cannot know whether γ satisfies equation (31) and so whether the government chooses 0 t or 1 t  is ambiguous.
Next, we consider the case where ( ) SW be the social welfare when the tax rate is zero and policyholder conducts the effort. If we define Then, if 0 h ≥ is realized, the government chooses 1 t  instead of zero. From equation (32), the following condition for satisfying 0 h ≥ is derived: Because expected profit of the insurance firm is an increasing function of t i by equations (17) and (29), 1 1 E E Π ≥ Π is always satisfied. Thus, it is possible to realize 1 t  , but whether the government chooses zero or 1 t  is ambiguous.
The above analysis shows that there are three kinds of optimal tax rates 0, 1 t  , and 0 t and three insurance coverage rates, 1/2, 1 α  , and 1. To know which optimal tax rate and insurance coverage rate are realized, we define the following two variables, which are the right-hand side of equations (31) and (33): From the results in this analysis, we derive the following implications. First, the smaller γ, the larger i t * and i α * . Second, unlike no effort case, whether government conducts premium subsidy is ambiguous even if β is small. It is easy to verify that γ 2 becomes the critical value whether government conducts premium subsidy. However, the relationship between γ 2 and β is unclear because 1 t  is a function of β and 1 t  is neither monotone increasing nor decreasing function of β. This result shows whether premium subsidy conducted by government is surely desirable depends on whether policyholder's effort is considered when the fraction of policyholders is small.

Concluding Remarks
The main findings of this article are summarized as follows. When policyholder's effort for lowering accident probability is not considered, and the fraction of policyholders is small, government conducts premium subsidy which stimulates an individual to choose full insurance coverage while maximizes social welfare which consists of policyholders, non-policyholders and insurance firm. In contrast, when policyholder's effort is considered, it becomes ambiguous whether premium subsidy is desirable even in the case of small fraction of policyholders.
This result suggests that, as a means of fostering the growth and development of insurance industry, a premium subsidy policy by government will probably be more effective when the targets are underdeveloped or in the rudimentary stage at least when policyholder's effort is not necessarily considered. Concretely, a government grant will facilitate the dissemination of knowledge and/or information regarding insurance and increase its penetration rate. Meanwhile, achieving sustainable growth in the insurance sector will ultimately lead to an improvement in social and economic welfare.
This finding is consistent with prior studies that assert government intervention can be Pareto-improving in a market associated with imperfect and/or costly information [28,29]. It also helps to explain why agricultural insurance programs in most developing countries would be assisted by a subsidy from the government. As a matter of fact, it has also been empirically supported by some recent studies on China's Cooperative Medical Scheme, a government-subsidized voluntary health insurance program in rural regions [32,33].
However, our study is only focused on a short-term contract. A further analysis based on a long-term contract and/or a repeated contract should be considered. Moreover, the problem of moral hazard between the insurance firm and the government also deserves special attention, because insurance firms may be reluctant to pursue sound practices such as risk classification and/or loss assessment when they know the government will eventually pay the bill.