INSURANCE

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## INSURANCE

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**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**Actuarial Fairness**• Game/insurance with fee/premium equal to expected value of outcomes**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**Risk Aversion and Declining Marginal Utility**• Gain from winning a dollar less than loss from losing a dollar**A Risk Averse Utility Function**Total Utility 200 140 Utility 20 10 0 Wealth ($000)**Total and Marginal Utility**TU Utility Wealth 0 Utility MU 0 Wealth**Certainty Equivalent of Expected Value**• Utility if wealth actually equaled expected value of wealth**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**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)**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**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)**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**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**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**Graph of MB and MC**Utils Marginal cost (in utils) MC A Marginal benefit (in utils) MB 0 q* Coverage purchased**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**Premium Rises**Utils Marginal cost (in utils) MC MC1 B A MB MB1 0 q1 q2 Coverage purchased**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**Expected Loss Increases**Utils Marginal cost (in utils) MC C A MB2 MB 0 q2 q1 Coverage purchased**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**Wealth Increases**Utils Marginal cost (in utils) MC1 MC2 A MB1 MB2 D 0 q1 q2 Coverage purchased**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**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**Assume perfect competition (zero profit)**• Then (1 – p)q – p(1 - )q - t = 0 • Or q - pq – pq + pq - 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**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**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)**MU(W – L + (1 - )q) = XMU(W - q)**where X = [(1 – p)]/[p(1 - )]**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**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**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)**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 - )]**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)**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**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?**Illustration of Moral Hazard**Inelastic Demand Price-Sensitive Demand $ $ D D P1 P1 Health care Health care Q1 Q1 Q2**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**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**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**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?**Illustration of Coinsurance**D (100% coinsurance) $ D (< 100% coinsurance) A B P1 S P2 C 0 Health Care Q1 Q2**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**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**Illustration of Welfare Loss**$ D (100%) D (20%) S Deadweight Loss 0 Q1 Q2 Health Care**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