APRIA 2014 Annual Conference Optimal Premium Subsidy and Its Impact on Individual Choice for Insurance Coverage Mahito Okura YingYing Jiang APRIA 2014 Annual Conference July 27-30, 2014

1. Introduction Market failure in insurance market caused by asymmetric information. Moral hazard Adverse selection Government intervention can alleviate problems in asymmetric information. Compulsory insurance Government-sponsored reinsurance program Financial Subsidy (e.g. premium subsidy)

1. Introduction Government intervention may generate new problems such as moral hazard and/or adverse selection. Thus, whether government intervention is desirable (more efficient) has been controversial.

1. Introduction This study focuses on the case of a premium subsidy by using a game theoretical framework. The main result of this study: When policyholder’s effort for lowering accident probability is not considered, government conducts premium subsidy when the fraction of policyholders is small. When policyholder’s effort for lowering accident probability is considered, whether government conducts premium subsidy is ambiguous even if the fraction of policyholders is small.

2. The model In the market: (Potential) policyholders (identical and weakly risk-averse) Nonpolicyholders Insurance firm (risk-neutral) Government Policyholders have following two types of initial assets: : liquid assets (e.g. salary and bonuses) that do not have the potential to incur damage. : fixed assets (e.g. house and vehicle) that have the potential to incur damage. Nonpolicyholders have only liquid assets (thus, they do not need to purchase insurance). Four-stage game is considered.

2. The model Fourth stage: Policyholder chooses whether to conduct the effort for lowering accident probability. Accident probability: and 0: no effort case 1: effort case Thus, Effort cost:

2. The model First stage: Government decides both the subsidy rate ( ) and tax rate ( ). Government subsidizes a portion of the insurance premium from tax revenue. Government revenue (tax) and expenditure (subsidy) must be equal (government zero budget constraint). Government imposes a tax on each liquid asset (e.g. income tax) to all individuals (not only policyholders but also nonpolicyholders). : a fraction of policyholders

2. The model Government zero budget constraint: : insurance premium

2. The model Second stage: Insurance firm chooses the insurance premium . Insurance money is in the case of full insurance. It is assumed that the amount of damage is equal to the amount of fixed assets .

2. The model Third stage: Fourth stage: Policyholder chooses the insurance coverage rate. : insurance coverage rate Fourth stage: Policyholder chooses whether to conduct the effort for lowering accident probability.

2. The model Policyholder’s expected utility ( ): Policyholder’s certainty equivalent ( ): :policyholder’s degree of absolute risk aversion

3. No effort case Third stage: Consider the case in which policyholder’s effort is excluded (thus, subscript of all variables is “0”). Temporarily remove the fourth stage from the original game and a three-stage game is analyzed by backward induction. Third stage: Optimal insurance coverage rate:

3. No effort case Second stage: Optimal insurance premium: Optimal insurance coverage rate: : upper limit of the tax rate:

3. No effort case First Stage: Certainty equivalent of the policyholder: Expected profit of the insurance firm: If government chooses strictly positive tax rate , it means government chooses strictly positive premium subsidy rate and vice versa.

3. No effort case Government chooses the tax rate for maximizing social welfare. Social welfare ( ): : weight of consumers’ welfare used to calculate the social welfare Social welfare function:

3. No effort case Social welfare function is the quadratic and convex function of . : the value to minimize social welfare

3. No effort case The following relationships are indicated: From the above two equations, the following three cases are possible: Case 1: Case 2: Case 3:

3. No effort case Case 1: Social welfare function is a monotone increasing function of in the range of . Thus, Government chooses . Policyholder chooses full insurance.

3. No effort case Case 2: Social welfare is maximized in either or in the range of . The following relationship is satisfied: Then, If , then and full insurance is realized. If , then and partial insurance ( ) is realized.

3. No effort case Case 3: Social welfare function is a monotone decreasing function of in the range of . Thus, Government chooses Policyholder chooses partial insurance ( )

3. No effort case Combining three cases: Implications: If , then and full insurance is realized. If , then and partial insurance is realized. Implications: If and/or are relatively small, then government conducts a premium subsidy. If and/or are relatively small, then government does not conduct a premium subsidy.

4. Effort case Condition in which policyholder chooses the effort in the fourth stage: Including policyholder’s effort may change the condition whether government conducts premium subsidy. First case: Policyholder never chooses to conduct the effort because full insurance is chosen. Then, is chosen.

4. Effort case Government may want to lower to the following tax rate. : maximum tax rate in which the policyholder conducts the effort and it can be derived as follows: : insurance coverage rate in the case of is always satisfied

4. Effort case Second case: In this situation, policyholder chooses . Assume that the policyholder decides to conduct the effort in the case of . Compare two situations and .

4. Effort case Results: Define that In the case of If , then and are realized If , then and are realized

4. Effort case Two implications: In the case of ( ) If , then and are realized. If , then and are realized. If , then and are realized. Two implications: The smaller , the larger and .

4. Effort case Unlike no effort case, whether government chooses premium subsidy is ambiguous even if is small. becomes critical value whether government conducts premium subsidy. is a function of and is neither monotone increasing nor decreasing function of . This result shows whether premium subsidy conducted by government is surely desirable depends on whether policyholder’s effort is considered when is small.

5. Concluding remarks The main result of this study: When policyholder’s effort for lowering accident probability is not considered, government conducts premium subsidy when the fraction of policyholders is small. When policyholder’s effort for lowering accident probability is considered, whether government conducts premium subsidy is ambiguous even if the fraction of policyholders is small.