The Influence of Premium Subsidies on Moral Hazard in Insurance Contracts

Transcription

1 The Influence of Premium Subsidies on Moral Hazard in Insurance Contracts Johannes Jaspersen, Andreas Richter Munich Risk and Insurance Center ARIA 2013 Annual Meeting, August 6th Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 1/31

2 Table of Contents Motivation Prior Literature Model Non-Contingent Premiums Contingent Premiums Discussion Conclusion Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 2/31

3 Premium Subsidies in Insurance Markets Insurance markets are often competitively prized. However, especially in public insurance systems, subsidies exist: National Flood Insurance Program (NFIP): About $2.2 Billion between 1968 and 2005 (Kunreuther and Michel-Kerjan, 2009) Federal Crop Insurance Commission (FCIC): About $7.3 Billion in 2011 (Federal Crop Insurance Corporation / Risk Management Agency, 2011) Employer Sponsored Health Insurance (ESI): Over $725 Billion predicted between 2011 and 2015 (Joint Committee on Taxation, 2012) Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 3/31

4 Premium Subsidies and Insured s Behavior Premium Subsidies imply two different changes in behavior. Subsidies will, ceteris paribus, increase the demand for insurance. Increased coverage changes behavior Demand Effect Subsidies will, ceteris paribus, lower the premium which influences the wealth and the composition of the insurance contract Wealth Effect. Prior studies on premium subsidies and moral hazard have excluded wealth effects, we focus on them. Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 4/31

5 Wealth Effect - Significant? One could argue that wealth effects are secondary to demand effects. However, consider: 1. Discrete choice between coverage levels 2. Only one coverage level 3. Mandatory insurance policies Generally, if insured do not change their coverage, only the wealth effect applies. Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 5/31

6 Contribution We consider the influence of the wealth effect of subsidies on moral hazard. We offer a closed form solution for two different insurance contracts: Contracts with premium payments contingent on no loss occurring Contracts with non-contingent premium payments We show the wealth effect to be dependent on the contract type and thus provide specific recommendations for policy makers. Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 6/31

7 Table of Contents Motivation Prior Literature Model Non-Contingent Premiums Contingent Premiums Discussion Conclusion Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 7/31

8 Moral Hazard We define moral hazard as a change in behavior of the insured that is induced by an insurance contract. Similar to Shavell (1979) insured can take costly effort to reduce the loss probability. This is in contrast to much of the literature on moral hazard in health insurance (Arrow, 1963; Coulson et al., 1995). We thus also model moral hazard as loss reduction. Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 8/31

9 Prior Literature Some studies regarding subsidies and moral hazard exist for health insurance: Feldstein and Friedman (1977) use simulation to show significant increase in medical spending due to subsidized ESIs. Coulson et al. (1995) show that sponsored medigap insurance increases drug consumption. Jack and Sheiner (1997) give a close form solution for demand effects and show that additional subsidies can lessen some of the drawbacks. Stabile (2001) provides evidence for public moral hazard due to subsidized ESIs. Considerations exclude wealth effects. Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 9/31

10 Table of Contents Motivation Prior Literature Model Non-Contingent Premiums Contingent Premiums Discussion Conclusion Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 10/31

11 Model Set-Up We use a simple two-state model: x L with probability p > 0 or x H with probability 1 p whereas x L < x H. The insured has a von Neuman/Morgenstern utility function U( ) such that U ( ) > 0 and U ( ) < 0. The insured can influence p(e) with e [0, [: Effort has diminishing marginal benefit: p (e) < 0 and p (e) > 0. Effort causes costs c(e) with c (e) > 0 and c (e) > 0. Cost function is utility separable. Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 11/31

12 Non-Contingent Premiums The insured can take up coverage from the insurer. In exchange for a premium π, the insured will gain a fixed indeminty I in case of the low wealth state. Less than full insurance is purchased such that x H π > x L π + I. The insured takes up insurance (or not), determines his effort, the state of nature is determined and the insurance contract is executed. Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 12/31

13 Actuarially Fair Premiums - Asymmetric Information Insurer cannot observe insured s effort. Insurer will anticipate insured s effort and set price accordingly. The insured has the following maximization problem: max EU = (1 p(e))u(x H π) + p(e)u(x L π + I ) c(e) e The frst order condition reads: p (e) [U(x L π + I ) U(x H π)] = c (e) Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 13/31

14 Premium Subsidies I Premium subsidies lead to negative expected value for the insurer. We model this by a subsidy parameter λ 1 such that the budget constraint reads: λπ p(e)i = 0 λ is modeled as a relative subsidy. If λ > 1, only 1 of the expected indemnity need to be covered λ by the premium income. All results hold for absolute subsidies, too. Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 14/31

15 Premium Subsidies II Considering the effect of the premium subsidies on effort: e λ = p (e) π [U (x λ H π) U (x L π + I )] p (e) [U(x L π + I ) U(x H π)] c (e) < 0 (1) Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 15/31

16 Premium Subsidies III λ has a negative effect on the effort exercised by the insured. With non-contingent premiums, subsidies will increase the effects of moral hazard. Premium subsidies increase the wealth regardless of the state of nature. Thus, subsidies decrease the marginal utility of additional wealth, while marginal costs of effort are unaffected. This leads to less effort by the insured. Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 16/31

17 Loss Reduction I It has been argued (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. We thus look at loss reduction in non-contingent premium contracts, as well. The model changes: Size of loss is randomly determined according to f (L, e) with support ]L, L[ Effort e [0, [ is still exercised at utility separable costs c(e), F (L,e) such that e > 0 and 2 F (L,e) < 0. e 2 Insurance coverage is given by the coinsurance rate δ. Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 17/31

18 Loss Reduction II With the insurer s budget constraint λπ p L (1 δ)lf (L, e)dl = 0, the L insured s maximization problem is: [ ] L max EU = (1 p)u(x H π) + p U(x H π δl)f (L, e)dl c(e) e We arrive at: δ L L and e λ = U (x H π δl) L L π F (L, e) dl = c (e) e λ U (x H π δl) < 0 L L U (x H π δl) 2 F (L,e) e dl c (e) 2 L F (L,e) e dl Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 18/31

19 Table of Contents Motivation Prior Literature Model Non-Contingent Premiums Contingent Premiums Discussion Conclusion Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 19/31

20 Motivation The explanation given above shows that the premium payment plan is crucial for the influence of premium subsidies on moral hazard. If premiums are not paid in every state of the world, the results might change. Example: long-term care insurance. Insured buys policy and pays premiums as long as no claim is made. If insured requires long-term care, he seizes to pay premiums and the insurance company indemnifies. Similar structures of insurance contracts can, for example, be seen in disability insurance or unemployment insurance. Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 20/31

21 Changes Since virtually all examples are intertemporal decisions, contingent premiums are modeled in a two period set-up. In period 1, the insured has wealth x1 with certainty and pays the insurance premium α. Wealth in period 2 either is x2l with probability p > 0 or x 2H otherwise. Premium α is still to be paid in the high wealth state. In the low wealth state, the insured will gain a payment β. No full insurance is possible: x2h α > x 2L + β. Cost and benefit of effort are modeled as above. Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 21/31

22 Effort and Premium Subsidies - Asymmetric Information I In the asymmetric information case, the insurer s budget constraint equals λ(2 p(e))α p(e)β = 0. The insured s maximization problem is: max EU = U(x 1 α)+(1 p(e))u(x 2H α)+p(e)u(x 2L +β) c(e) e which renders the first order condition: p (e) [U(x 2L + β) U(x 2H α)] = c (e) Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 22/31

23 Effort and Premium Subsidies - Asymmetric Information II The influence of premium subsidies on effort is given by: e λ = p (e) αu (x λ 2H α) p (e) [U(x 2L + β) U(x 2H α)] c (e) > 0 The sign of α is not obvious, since there is no submodularity. λ One can, however, show graphically that the influence must be negative. Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 23/31

24 Effort and Premium Subsidies - Asymmetric Information III The wealth effect of subsides can decrease moral hazard. Subsidies decrease the premium and increase the utility in the high wealth state. Insured has more incentive to be in the high wealth state. Increased incentives lead to increased effort. The design of the premium payment plan is crucial. Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 24/31

25 Table of Contents Motivation Prior Literature Model Non-Contingent Premiums Contingent Premiums Discussion Conclusion Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 25/31

26 Contingent and Non-Contingent Premiums The effect of premium subsidies depends on the contract design. Both types of contracts are relevant: Contingent Premiums Non-Contingent Premiums Payment of only if no loss occurs regardless of state Premium Influence of decrease moral hazard increase moral hazard Subsidies Examples - unemployment insurance - most P&C insurance - long-term care insurance - health insurance - disability insurance Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 26/31

27 Discussion and Implications Thus, when subsidies are debated, considerations of moral hazard should include type of contract. Some direct implications: Subsidies of ESIs in health insurance will lead to moral hazard through wealth effects and demand effects. This is not the case for long-term care insurance. As long as the participation constraint holds or insurance is mandatory, the results hold for premium loadings as well as subsidies. Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 27/31

28 Table of Contents Motivation Prior Literature Model Non-Contingent Premiums Contingent Premiums Discussion Conclusion Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 28/31

29 Conclusion Premium subsidies influence moral hazard in the market. Prior studies have focused on demand effects. Demand effects and wealth effects need not be equal. In certain markets, demand effects might not appear. We show wealth effects of subsidies to be dependent on contract type: With non-contingent premiums, moral hazard will be increased, whether loss prevention or loss reduction is of concern. With contingent premiums, moral hazard is reduced. Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 29/31

30 References I Arrow, Kenneth J., Uncertainty and the Welfare Economics of Medical Care, The American Economic Review, 1963, 53 (5), Coulson, N. Edward, Joseph V. Terza, Cheryl A. Neslusan, and Bruce C. Stuart, Estimating the Moral-Hazard Effect of Supplemental Medical Insurance in the Demand for Prescription Drugs by the Elderly, The American Economic Review, 1995, 85 (2), Cutler, David M. and Richard J. Zeckhauser, The Anatomy of Health Insurance, in Anthony J. Culyer and Joseph P. Newhouse, eds.,, Vol. 1, Part A of Handbook of Health Economics, Elsevier, 2000, pp Federal Crop Insurance Corporation / Risk Management Agency, Financial Statements for Fiscal Years 2011 and 2010, Audit Report, United States Department of Agriculture Feldstein, Martin and Bernard Friedman, Tax Subsidies, the Rational Demand for Insurance and the Health Care Crisis, Journal of Public Economics, 1977, 7 (2), Jack, William and Louise Sheiner, Welfare-Improving Health Expenditure Subsidies, The American Economic Review, 1997, 87 (1), Joint Committee on Taxation, Estimates of Federal Tax Expenditures dor Fiscal Years , Report, United States Congress Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 30/31

31 References II Kunreuther, H. and E. Michel-Kerjan, At War With the Weather: Managing Large-Scale Risks in a New Era of Catastrophes, Cambridge: Mit Press, Shavell, Steven, On Moral Hazard and Insurance, The Quarterly Journal of Economics, 1979, 93 (4), Stabile, Mark, Private Insurance Subsidies and Public Health Care Markets: Evidence from Canada, The Canadian Journal of Economics / Revue canadienne d Economique, 2001, 34 (4), Johannes Jaspersen, Andreas Richter Premium Subsidies and Moral Hazard 31/31