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INSURANCE

INSURANCE. Insurance Reduces Risk. What is risk? Something to do with variability of returns? Does insurance have variable returns? Insurance’s variability reduces overall portfolio variability. The Expected Value Concept. Actuarial Fairness.

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INSURANCE

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  1. INSURANCE

  2. Insurance Reduces Risk • What is risk? • Something to do with variability of returns? • Does insurance have variable returns? • Insurance’s variability reduces overall portfolio variability

  3. The Expected Value Concept

  4. Actuarial Fairness • Game/insurance with fee/premium equal to expected value of outcomes

  5. Risk Preferences • A RISK NEUTRAL person would pay as much as $.50 for a coin toss paying $1 heads, $0 tails • A RISK LOVING person would pay more than $.50 • A RISK AVERSE person would pay, at most, less than $.50

  6. Risk Aversion and Declining Marginal Utility • Gain from winning a dollar less than loss from losing a dollar

  7. A Risk Averse Utility Function Total Utility 200 140 Utility 20 10 0 Wealth ($000)

  8. Total and Marginal Utility TU Utility Wealth 0 Utility MU 0 Wealth

  9. Expected Utility of Wealth

  10. Certainty Equivalent of Expected Value • Utility if wealth actually equaled expected value of wealth

  11. An Example • Wealth = $20,000 if well with probability .95, $10,000 if ill with probability .05 • EU = .95 X (utility of $20,000) + .05 x (utility of $10,000) • EU = (.95 x 200) + (.05 x 140) • EU = 190 + 7 = 197 • EV = (.95 x $20,000) + (.05 x $10,000) • EV = $19,000 + $500 = $19,500 • CE = 199

  12. The Graph of the Example B D 200 199 FE shows max willingness to pay for insurance 197 E C F A Utility 140 0 17 19.5 10 20 Wealth ($000)

  13. Buying Insurance • Suppose our consumer is offered the opportunity to insure against this loss for $500 • Paying the premium means income will be $19,500 ($20,000 minus the $500 premium) no matter what happens • Utility at $19,500 (point D) exceeds expected utility (point C) • utility of 199 versus 197 • Willing to pay up to $3,000 • Distance FE • Any amount less than $3,000 gives more utility than the expected utility of $20,00 with probability .95 and $10,000 with probability .05

  14. The Demand for Insurance • How much insurance will an individual buy? • Notation: • p = the probability of illness • W = initial wealth • L = financial loss because of illness • Without insurance: • EU = p x (utility of net wealth ill) + (1-p) x (utility of net wealth well) • EU = p U(W - L) + (1-p) U(W)

  15. With insurance: Wealth ill = Wealth (W) - Loss (L) - insurance premium (q) + payment from insurance (q), where  is the premium rate and q is the coverage

  16. Wealth ill = W - L - q + q • = W - L - (1 - )q • Wealth well = Wealth (W) - premium (q) • Wealth well = W - q • Thus, Expected Utility = p U(W - L + (1 - )q) [1] + (1 - p) U(W - q) [2] • Expression [1] is related to the benefit of insurance and [2] is related to the cost • Individual will buy coverage (q) where MB = MC

  17. Marginal Benefit and Marginal Cost • MB related to expression [1] • As coverage (q) increases, the marginal utility of the extra money falls • MC related to expression [2] • As q increases, wealth when well decreases, so utility forgone rises

  18. Graph of MB and MC Utils Marginal cost (in utils) MC A Marginal benefit (in utils) MB 0 q* Coverage purchased

  19. What Happens if Premium Rate () Rises? • MB shifts down to MB1 • MC shifts up to MC1 • Equilibrium moves from A to B • Less coverage purchased

  20. Premium Rises Utils Marginal cost (in utils) MC MC1 B A MB MB1 0 q1 q2 Coverage purchased

  21. What Happens if Expected Loss (L) Increases? • MB shifts up to MB2 because at lower wealth, MU of any additional q is greater • L not in cost expression so MC does not change • Equilibrium moves from A to C • More coverage purchased

  22. Expected Loss Increases Utils Marginal cost (in utils) MC C A MB2 MB 0 q2 q1 Coverage purchased

  23. What happens if Wealth (W) Increases? • At any level of coverage (q), both the marginal utility of q when ill (MB) and the marginal utility of wealth forgone when well (MC) fall • Equilibrium moves from A to D • Affect on coverage purchased ambiguous • coverage goes up in following graph • would have gone down if fall in MB were larger or fall in MC smaller

  24. Wealth Increases Utils Marginal cost (in utils) MC1 MC2 A MB1 MB2 D 0 q1 q2 Coverage purchased

  25. The Supply of Insurance • What determines the premium rate ()? • As a point of departure, assume perfect competition • in long run, perfectly competitive firms earn zero profit • what  results in zero profit? • If representative customer is well, insurer earns q dollars

  26. If customer gets ill, insurer loses q - q, or (1 - )q dollars • Either way, insurer incurs loading cost t • cost of servicing transactions • Exp profit = (1 – p)q – p(1 - )q - t

  27. Assume perfect competition (zero profit) • Then (1 – p)q – p(1 - )q - t = 0 • Or q - pq – pq + pq - t = 0 • Or q - pq - t = 0 • Or  = p + t/q • Thus, premium rate () equals the probability of illness plus loading cost as a proportion of coverage

  28. E.g., if the probability of illness is .05 and loading costs are 10% of coverage, then the premium will be $.15 for every dollar of coverage • If more is charged, other insurers will take all the business • if less is charged, profits will be lost • If loading costs are zero, insurance will be actuarially fair:  = p

  29. Optimal Coverage (q) • What amount of insurance (q) will consumer choose? • Recall that EU = pU(W – L + (1 - )q)+ (1 – p)U(W - q) • To find max EU, take derivative of EU with respect to q and set equal to zero: p(1 - )MU(W – L + (1 - )q) – (1 – p)MU(W - q) = 0 where MU(. . .) refers to marginal utility (i.e., derivative of U)

  30. MU(W – L + (1 - )q) = XMU(W - q) where X = [(1 – p)]/[p(1 - )]

  31. If  = p (actuarially fair), then X = 1 and MU(W - L + (1 - )q) = MU(W - q) • This can only happen if W - L + (1 - )q = W - q or q = L • So if  = p, the consumer will fully insure

  32. Recall, though, that under perfect competition, insurers will set  = p + t/q • So if there are loading costs, consumers will not fully insure • To see exactly why, return to the MB = MC relationship

  33. p(1 - )MU(W - L + (1 - )q) = (1 - p)MU(W - q) • Recall that under perfect competition (1 - p) = p(1 - ) + t/q • Substitute this expression for (1 - p) into equation above to get • p(1 - )MU(W - L + (1 - )q) = [p(1 - ) + t/q]MU(W - q)

  34. divide through by p(1 - ) to get MU(W – L + (1 - )q) = [p(1 - ) + t/q]/[p(1 - )]MU(W - q) Now [p(1 - ) + t/q]/[p(1 - )] = 1 + [t/q]/[p(1 - )] = 1 + t/[qp(1 - )] So, MU(W – L + (1 - )q) = (1 + Z)MU(W - q) where Z = t/[qp(1 - )]

  35. If loading costs (t) are zero, then consumer fully insures (q = L) • If t > 0, then Z > 0 and MC shifted up by (1 + Z) • Thus, q < L (i.e., the consumer underinsures)

  36. PHEW!!! Here’s the Bottom Line • If loading costs (t) are zero, perfect competition forces insurers to charge actuarially fair premiums • If premiums are actuarially fair, consumers will fully insure • If there are loading costs, premiums will be higher and consumers will buy less than full insurance

  37. Moral Hazard • Analysis assumes, so far, that the loss (L) is fixed • What if L is not fixed? • Say L is affected by the health care price faced by consumer?

  38. Illustration of Moral Hazard Inelastic Demand Price-Sensitive Demand $ $ D D P1 P1 Health care Health care Q1 Q1 Q2

  39. If insurer charges premium based on L = P1Q1, it will lose money because loss will actually be P1Q2 • If insurer charges premium based on L = P1Q2, consumer may not buy since premium may exceed what he would pay for health care in absence of insurance

  40. Testable Hypotheses • There will be more complete coverage the less elastic is demand • Insurance develops first for those services that are less elastic • Cross sectional data support first hypothesis • Time-series data support second

  41. The Effect of Deductibles • No effect if deductible is small, allowing consumer to buy health care at zero marginal price once deductible is paid • Causes consumer to self-insure if deductible is so large that the gain from being able to buy at zero marginal price less than deductible

  42. Coinsurance • Insurance requiring consumer to pay a percentage of the loss • E.g., a 20% coinsurance requires consumer to pay 20% of the cost of his consumption of health care • What does coinsurance do to demand?

  43. Illustration of Coinsurance D (100% coinsurance) $ D (< 100% coinsurance) A B P1 S P2 C 0 Health Care Q1 Q2

  44. Insurance causes demand to swivel out causing more health care to be demanded • The lower is coinsurance, the less elastic is demand • At 0% coinsurance, demand becomes totally inelastic

  45. Welfare Loss • If no insurance, demand reflects all benefits (assuming no externalities) • Insurance causes welfare loss because market demand does not reflect benefits of health care

  46. Illustration of Welfare Loss $ D (100%) D (20%) S Deadweight Loss 0 Q1 Q2 Health Care

  47. Thus, insurance results in over-allocation to insured forms of health care at expense of non-insured forms (good nutrition, exercise) and also at expense of non-health care goods and services

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