Moral Hazard

Lecture 5 Moral Hazard Xuezhu Shi Email: shixuezhu@outlook.com School of Insurance and Economics, UIBE Nov, 2020 . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE Moral Hazard Ex-ante Safety belts are an insurance device People who installed the ”safety belt insurance” felt less inclined to drive safely Examples: Health insurance may induce people to be less careful when playing dangerous sports having a property insurance will make one think whether it is really necessary to take care of your premises with crop insurance, a farmer may work less hard to cultivate her fields . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE Moral Hazard The second meaning Insurance literature If people who hold health insurance consume more health services than would be optimal Ex-post moral hazard The behaviour occurs after the loss has occurred . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE Single Period Contracts Under Moral Hazard The Basic Model: Ex-ante moral hazard Two states of the world model: One state with no loss, the other where the loss occurs. The probability of the damage is not exogenous: depends on the effort (e) The expected utility of an individual with Ex-ante moral hazard in the insurance contact is: E[U] = (1 − π(e))U(W − P) + π(e)U(W − L + I) − c(e) I = C − P, I is the net compensation ′ π(e) s the probability that a loss occurs, which satisfies π (e) < 0, more effort leads to a lower probability of an accident ′ Cost of effort c(e) and c (e) > 0 . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE Single Period Contracts Under Moral Hazard The Basic Model: Ex-ante moral hazard The agent has the choice between two effort levels: e1 (”lazy”) and e2 > e1 (”hard working”) The ”first best” solution: when the efforts is observable. The premium would be P = π(e)L Either effort level e1 or effort level e2 is optimal, depending on which of the two expressions is larger u(W − π1 L) − c(e1 ) >=< u(W − π2 L) − c(e2 ) where πi = π(ei ), i = 1, 2 . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE Single Period Contracts Under Moral Hazard The Basic Model: Ex-ante moral hazard The ”second best” case: effort is not observable Suppose that also under asymmetric information the higher effort should be implemented. The contract must be designed such that the agent will work hard The optimisation problem: max(1 − π2 )u(W − P) + π2 u(W − L + I) − c(e2 ) P,I s.t PC :(1 − π2 )P − π2 I ≥ Π IC :(1 − π2 )u(W − P) + π2 u(W − L + I) − c(e2 ) ≥ (1 − π1 )u(W − P) + π1 u(W − L + I) − c(e1 ) . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE The Basic Model: Ex-ante moral hazard The ”second best” case Participation constraint (PC): the insurance company must obtain at least profit Π to agree to trade with the agent If Π = 0: the competitive market situation Because ex-ante, both parties have the same information. If Π is large enough, the solution to the monopoly problem will be obtained. IC is the incentive compatibility constraint As effort is not observable by the insurer, the contract must be such that it is in the interest of the insured to put in effort e2 rather than e1 . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE The Basic Model: Ex-ante moral hazard The ”second best” case Both constraints have to be binding. The Lagrangian function is given by: L = (1 − π2 )u(W − P) + π2 u(W − L + I) − c(e2 ) + λ1 [(1 − π2 )P − π2 I − Π] + λ2 [(1 − π2 )u(W − P) + π2 u(W − L + I) − c(e2 ) − (1 − π1 )u(W − P) − π1 u(W − L + I) + c(e1 )] . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE The Basic Model: Ex-ante moral hazard The ”second best” case The FOCs w.r.t P and I dL ′ = −(1 − π2 )u (W − P) + λ1 (1 − π2 ) dP ′ ′ − λ2 [(1 − π2 )u (W − P) − (1 − π1 )u (W − P)] = 0 dL ′ = π2 u (W − L + I) + λ1 π2 dI ′ ′ + λ2 [π2 u (W − L + I) − π1 u (W − L + I)] = 0 . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE The Basic Model: Ex-ante moral hazard The ”second best” case ′ ′ ′ ′ Denote u (W − P) = u1 and u (W − L + I) = u2 , then the FOCs 1 1 λ2 (1 − π2 ) − (1 − π1 ) 1 λ2 (1 − π1 ) ]= ) + [ + (1 − ′ = λ λ (1 − π ) λ λ (1 − π2 ) u1 1 1 2 1 1 1 1 λ2 π2 − π1 1 λ2 π1 + [ ]= + (1 − ) ′ = λ1 λ1 π2 λ1 λ1 π2 u2 . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE The Basic Model: Ex-ante moral hazard The ”second best” case Interpretation: One over the marginal utility is equal to a constant plus another constant times an expression An expression depends positively on the change in probability for different effort levels for that state and negatively on the probability of that state ′ ′ As π2 < π1 , then u1 < u2 , which implies P < L − I → there is less than full insurance. To implement higher effort levels, the agent must not obtain full insurance . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE The Basic Model: Ex-ante moral hazard The ”second best” case The problem is not solved yet Calculated how the contract would look if the higher effort level is implemented Not sure whether it is optimal Need to choose effort level . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE Model with many effort levels Suppose the possible effort levels are e ∈ E, where E is some discrete or continuous set. The problem, which has to be solved, is the following max (1 − π(e))u(W − P) + π(e)u(W − L + I) − c(e) e∈E,P,I s.t. PC : (1 − π(e))P − π(e)I ≥ Π IC : e = arg max[(1 − π(ẽ))u(W − P) + πẽu(W − L + I) − c(ẽ)] ẽ∈E . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE Model with many effort levels Deal with an argmax function: in the textbook, we go with the continuous effort levels (Mirrlees, 1971) With assumptions on the second derivative of the cost and probability function are positive, then we will have an interior solution ′ ′ IC : −π (e)[u(W − P) − u(W − L + I)] − c (e) = 0 ′ ′ As π (e) < 0 and c (e)0 −→ u(W − P) > u(W − L + I) −→ partial insurance Second order condition: IC : −π ” (e)[u(W − P) − u(W − L + I)] − c” (e) < 0 This holds for any partial insurance contract if π ” (e) > 0 and c” (e) > 0, larger effort becomes more and more costly, and less and less productive . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE Model with many effort levels The IC above already shows that to implement any effort level larger than emin , partial insurance is necessary More extensive insurance is desired as the agent is risk averse, while less insurance gives the agent more incentives to avoid the accident The optimal effort level is actually higher under asymmetric than under symmetric information: In a second best world, the agent may either work less hard or harder than in the first best world. . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE Model with continuous loss Two cases Loss-prevention: the agent can influence the probability of a loss Loss-reduction: by exercising effort, the insured influences the size of the loss . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE Model with continuous loss Loss-prevention Suppose losses are random with a distribution function F(L) and density f(L), defined on L ∈ [L, L̄] Denote by I(L) the (net) indemnity as a function of the size of the loss, then the optimisation problem becomes: max(1 − π(e))u(W − P) + π(e) e,P,I ∫ L̄ L u(W − L + I(L))f(L)dL − c(e) s.t. PC : (1 − π(e))P − π(e) ∫ L̄ L IC : e = arg max(1 − π(e))u(W − P) + π(e) e I(L)f(L)dL ≥ Π ∫ L̄ L. . Xuezhu Shi . u(W − L + I(L))f(L)dL − c(e) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE Model with continuous loss Loss-prevention ′ FOIC : π (e)[−u(W − P) + ∫ L̄ L ′ u(W − L + I(L))f(L)dL] − c (e) = 0 and SOIC : π (e)[−u(W − P) + ” ′ ∫ L̄ L u(W − L + I(L))f(L)dL] − c” (e) < 0 ′ if π (e) < 0 and c (e) < 0. We also assume C(L) ≥ 0 or I(L) ≥ P. . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE Model with continuous loss Loss-prevention At the optimum: ′ ′ ′ π(e)u (W − L + I(L))f(L) − λ1 π(e)f(L) + λ2 π (e)u (W − L + I(L))f(L) ≤ 0 I(L) = P if this inequation is strictly lower than zero. When I(L) > P, ′ 1 1 λ2 π (e) = + ′ u (W − L + I(L)) λ1 λ1 π(e) One over the marginal utility is equal to some constant plus a term which depends on the change in probability divided by the probability . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE Model with continuous loss Loss-prevention ′ u (W − L + I(L)) is independent of L and I(L) + P = C(L) cannot be negative If the agent can only influence the probability of an accident, then the optimal insurance contract has a deductible: C(L) = max[L − D, 0]. To make an insurer work hard, she needs to be punished in case a loss occurs, but rewarded for no loss This is achieved by giving the insured an income of W − P in the no-loss state, but W − P − D in the loss state, if loss exceeds D. . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE Model with continuous loss Loss-reduction . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE Model with continuous loss Loss-reduction In case of loss-reduction, the optimal contract depends on the exact characteristics of the environment. Under some assumptions on the distribution function and on the feasibility of contracts, the optimal contract is closer to a coinsurance contract than to an insurance policy with a deductible. . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22/22 Lecture 5 Moral Email:Hazard shixuezhu@outlook.com School of Insurance and Economics, UIBE