The Demand of Insurance

Lecture 2 The Demand of Insurance Xuezhu Shi Email: shixuezhu@outlook.com School of Insurance and Economics, UIBE Oct, 2020 . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE The Demand of Insurance The demand for insurance: demand for cover We are only going to learn the basic theory model behind the demand of insurance The basis of the approach: the state of the world . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE The state of the world The state of the world: corresponding to an amount of the loss incurred by the insurance buyer. Possible states incurs no loss an additional state for each possible loss losses: any value [0, Lm ], here is a continuum of possible states of the world Simple case incurs no loss incurs loss L . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Simple case: setting Insurance buyer’s wealth in each state of the world, W, state contingent wealth Endowments W0 W0 if no loss occurs W0 − L if lose L, L > 0 Buy insurance Get compensation C, depends on L pay a premium P W0 − P: no loss W0 − P − L + C : in the loss state . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Simple case: setting Maximisation of the expected utility The probability of the loss is π The expected value of wealth: W̄ = (1 − π)W0 + π(W0 − L) = W0 − πL πL is the expected value of income loss L < W0 . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Simple case: setting Insurance cover C at premium rate p, p ∈ [0, 1] The premium amount is P = pC C≥0 p is constant and independent of C . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE The Model of the Demand for Cover The buyer solves the problem max C≥0 Ū = (1 − π)U(W0 − P) + πU(W0 − L − P + C) subject to P = pC After substitution Ū(C) = (1 − π)U(W0 − pC) + πU(W0 − L − pC + C) . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE The Model of the Demand for Cover First order condition Ū′ = −p(1 − π)U′ (W0 − pC∗ ) + (1 − p)πU′ (W0 − L + (1 − p)C∗ ) where C∗ ≥ 0 and it comes from FOC=0 Second order condition ′′ ′′ ′′ Ū = p2 (1 − π)U (W0 − pC) + (1 − p)2 πU (W0 − L + (1 − p)C) < 0 where the sign follows because of the strict concavity of the utility function at all C ≥ 0 U′ (C∗ ) = 0 is both necessary and sufficient for optimal cover C∗ ≥ 0. . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE The Model of the Demand for Cover Implied conditions from FOC Optimal Cover is zero C∗ = 0 → p 1−p ≥ π U′ (W0 −L+(1−p)C∗ ) 1−π U′ (W0 −pC∗ ) p 1−p = π U′ (W0 −L+(1−p)C∗ ) 1−π U′ (W0 −pC∗ ) Optimal Cover is positive C∗ > 0 → U′ (W0 − pC∗ ) = π(1−p) ′ p(1−π) U (W0 − L + (1 − p)C∗ ) . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE The Model of the Demand for Cover fair premium When p = π, U′ (W0 − pC∗ ) = U′ (W0 − L + (1 − p)C∗ ). p = π is the case of fair premium p > π is the case of positive loading p < π is the case of positive loading . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE The Model of the Demand for Cover p = π ⇐⇒ U′ (W0 − pC∗ ) = U′ (W0 − L + (1 − p)C∗ ) ⇐⇒ C∗ = L; the buyer chooses full cover p > π ⇐⇒ U′ (W0 − pC∗ ) < U′ (W0 − L + (1 − p)C∗ ) ⇐⇒ C∗ < L; the buyer chooses partial cover p < π ⇐⇒ U′ (W0 − pC∗ ) > U′ (W0 − L + (1 − p)C∗ ) ⇐⇒ C∗ > L; the buyer chooses more than full cover taking zero cover implies U′ (W0 − L) > U′ (W0 ) and p must be sufficiently greater than π for the case to be possible . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE The Model of the Demand for Cover Figure: Optimal choice of cover . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE The Model of the Demand for Cover From U(C, P) = (1 − π)u(W0 − p) + πU(W0 − P − L + C), partial derivatives are: ′ UP = −[(1 − π)U′ (W0 − P) + πU′ (W0 − P − L + C)] ′ UC = πU′ (W0 − P − L + C); from the Implicit Function Theorem, we have that the slope is dP UC =− >0 dC UP IF C∗ = L, then dP dC =π . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE The Model of the Demand for State-Contingent Wealth The state contingent wealth values W1 and W2 W1 = W0 − pC and W2 = W0 − L + (1 − p)C The buyer’s expected utility is now written as Ū(W1 , W2 ) = (1 − π)U(W1 ) + πU(W2 ) Slope of the indifference curve ′ U 1 dW2 (1 − π) U′ (W1 ) = W = − ′ dW1 π U′ (W2 ) U W2 . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE The Model of the Demand for State-Contingent Wealth Figure: Indifference curve . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE The Model of the Demand for State-Contingent Wealth W1 = W0 − pC and W2 = W0 − L + (1 − p)C leads to The budget constraint (1 − p)[W0 − W1 ] + p[W0 − L − W2 ] = 0 or (1 − p)W1 + pW2 = W0 − pL. max Ū(W1 , W2 ) = (1 − π)U(W1 ) + πU(W2 ) subject to (1 − p)W1 + pW2 = W0 − pL . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE The Model of the Demand for State-Contingent Wealth First order conditions using Lagrangian method ′ ŪW1 = (1 − π)U′ (W∗1 ) − λ(1 − p) = 0, ′ ŪW2 = πU′ (W∗2 ) − λp = 0, ′ Ūλ = (1 − π)W∗1 + πW∗2 = W0 − pL; The first two can be expressed as ′ 1−p (1 − π) U (W∗1 ) = π U′ (W∗2 ) p Equality of the marginal rate of substitution with the price ratio or tangency of an indifference curve with budget line . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE The Model of the Demand for State-Contingent Wealth Rewrite the previous equation ′ U (W∗1 ) = (1 − p)π ′ ∗ U (W2 ) p(1 − π) p = π ⇐⇒ U′ (W∗1 ) = U′ (W∗2 ) ⇐⇒ W∗1 = W∗2 ; the buyer chooses full cover p > π ⇐⇒ U′ (W∗1 ) < U′ (W∗2 ) ⇐⇒ W∗1 > W∗2 ; the buyer chooses partial cover p < π ⇐⇒ U′ (W∗1 ) > U′ (W∗2 ) ⇐⇒ W∗1 < W∗2 ; the buyer chooses more than full cover . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE The Model of the Demand for State-Contingent Wealth The expected value or fair odds line is (1 − π)W1 + πW2 = W̄ = W0 − πL . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Comparative Statics Recall that FOC is ′ Ūc = −p(1 − π)U′ (W0 − pC∗ ) + (1 − p)πU′ (W0 − L + (1 − p)C∗ ) According to the Implicit Function Theorem: ŪCW0 ∂C∗ =− ; ∂W0 ŪCC and ∂C∗ ŪCL =− ; ∂L ŪCC ŪCp ∂C∗ =− ∂p ŪCC ŪCπ ∂C∗ =− ∂π ŪCC . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Comparative Statics The effect of a change in wealth Recall ŪCC (second order derivation w.r.t C) is ′′ ′′ ŪCC = p2 (1 − π)U (W0 − pC∗ ) + (1 − p)2 πU (W0 − L + (1 − p)C∗ ) < 0 and the second order derivation w.r.t C and W0 is ′′ ′′ ŪCW0 = −p(1 − π)U (W0 − pC∗ ) + (1 − p)πU (W0 − L + (1 − p)C∗ ) < 0 . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Comparative Statics The effect of a change in wealth The first case: when p = π and C∗ = L ŪCW0 ∂C∗ =− =0 ∂W0 ŪCC Full cover is bought and L stays unchanged. A change in wealth has no effects on the demand of insurance . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Comparative Statics The effect of a change in wealth The second case: when p > π and C∗ < L ŪCW0 ∂C∗ =− ∂W0 ŪCC ? 0 ?: can be >, <, = ′′ (W) depending on the absolute risk aversion A(W) = − UU′ (W) The ŪCW0 can be rewritten as: ′′ ŪCW0 = −p(1 − π)U (W∗2 )[(A(W∗1 ) − A(W∗2 ))] . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Comparative Statics The effect of a change in wealth W∗1 > W∗2 and (A(W∗1 ) > A(W∗2 )) =⇒ insurance is a normal good W∗1 > W∗2 and (A(W∗1 ) < A(W∗2 )) =⇒ insurance is a inferior good (commonly expected) . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Comparative Statics The effect of a change in loss The second order derivation w.r.t C and L is ′′ ŪCL = −(1 − p)πU (W0 − L + (1 − p)C∗ ) > 0 so ∂C∗ ŪCL =− >0 ∂L ŪCC Given risk averse, increase in the loss would increase the demand of insurance . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Comparative Statics The effect of a change in premium rate The second order derivation w.r.t C and p is ′ ′ ′′ ′′ ŪCp = −[(1−π)U (W∗1 )+πU (W∗2 )]+[p(1−π)U (W∗1 )−(1−p)πU (W∗2 )]C∗ so ′ ′ ŪCp ŪCW0 ∂C∗ (1 − π)U (W∗1 ) + πU (W∗2 ) =− = + C∗ ∂p ŪCC ŪCC ŪCC The first term: substitution effect and must be negative The second term: wealth/income effect, can be either positive or negative depends on ŪCW0 . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Comparative Statics The effect of a change in premium rate Intuition: Normal good: the wealth effect is negative −→ the demand falls as p rises Inferior good: the wealth effect is positive −→ the demand depends on which effects is larger . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Comparative Statics The effect of a change in loss probability The second order derivation w.r.t C and π is ′ ′ ŪCπ = pU (W∗1 ) + (1 − p)U (W∗2 ) > 0 so ∂C∗ ŪCπ =− >0 ∂π ŪCC Given risk averse, increase in the loss probability would increase the demand of insurance In the real life, π increase leads to increase in p . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Multiple Loss States and Deductibles Two ways of insurance Coinsurance: a fixed proportion of the loss is paid in each state Deductibles: Nothing is paid for losses below a specified value, called the deductible, while, when losses exceed this value, the insured receives an amount equal to the loss minus the deductible Deductibles are more commonly observed. Example showed below . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Coinsurance and Deductibles The two-state model by assuming now that the possible loss lies in some given interval L ∈ [0, Lm ], Lm < W0 . The probability of L = f(L) Under proportional coinsurance we have cover: C = αL, α ∈ [0, 1] α = 0 −→ no insurance α = 1 −→ full insurance Under deductibles,D means deductible. We have cover: C = 0 for L ≤ D; C = L − D for L > D; D = Lm −→ no insurance D = 0 −→ full insurance . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Coinsurance and Deductibles Figure: Cover as a function of the loss . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Coinsurance and Deductibles The buyer’s state-contingent wealth: Coinsurance: Wα = W0 − L − P + C = W0 − (1 − α)L − P Deductibles: WD = W0 − L − P + C = W0 − L − P + max(0, L − D) Important: When L = D, ŴD = W0 − D − P Under a deductible, wealth cannot fall below ŴD , however high the loss. . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Coinsurance and Deductibles Figure: Wealth under coinsurance and deductible . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Insurance Demand with State Dependent Utility Recall that at fair premium, the insurance buyer will always want to equalize marginal utilities of wealth across states. π = p −→ U′ (W1 ) = U′ (W2 ) Three senses of the full insurance: choice of cover that equalizes marginal utilities of wealth across states choice of cover that equalizes total utilities of wealth across states choice of cover that equalizes wealth across states . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Insurance Demand with State Dependent Utility Utility is state independent and the premium is fair, three senses coincide Under state dependent utility: marginal utilities will be equalized, but it remains an open question whether incomes and total utilities are equalized Find an economically meaningful way of relating the state dependent utility functions to each other . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Cook and Graham (1977) At every wealth level W, assume there is an amount of income ω(W) that satisfies U1 (W − ω(W)) = U2 (W) ω(W): the consumer’s maximal willingness to pay to be in the ”good” state 1 rather than the ”bad” state 2. Any given level of W in state 2, ω(W) gives the reduction in this income level required to yield an equal level of utility in state 1 . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Cook and Graham (1977) Figure: State dependent. utility . . . . . . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Cook and Graham (1977) Differentiate: U1 (W − ω(W)) = U2 (W) ′ ′ ′ We get U1 (W − ω(W))[1 − ω (W)] = U2 (W) ′ ω (W) = 1 − ′ U2 (W) ′ U1 (W−ω(W)) Willingness to pay changes as wealth varies Determined by the slopes of the utility functions at equal utility values . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Cook and Graham (1977) ′ Assume ω (W) ≥ 0 Willingness to pay to be healthy rather than sick at least not to fall with wealth ′ ′ ′ ′ ′ ′ ′ ′ ′ ω (W) = 0 −→ U2 (W) = U1 (W − ω(W)) −→ U2 (W) > U1 (W) ′ ω (W) > 0 −→ U2 (W) < U1 (W − ω(W)) −→ U2 (W) >=< U1 (W) . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Cook and Graham (1977) ′ ′ ′ If ω (W) = 0, then U2 (W∗2 ) = U1 (W∗2 − ω(W∗2 )). ′ ′ Recall U2 (W2 ) = U1 (W1 ) given fair premium then W∗1 = W∗2 − ω(W∗2 ) . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Cook and Graham (1977) ′ If ω (W1 ) > 0 ′ ′ U2 (W) = U1 (W) at each W, marginal utilities are state independent optimum is at α, wealth fully insuranced ′ ′ U2 (W) > U1 (W) at a given wealth, increasing wealth increases utility more in the bad state than in the good optimum is at β, more than full wealth insuranced ′ ′ U2 (W) < U1 (W) at a given wealth, increasing wealth increases utility more in the good state than in the bad optimum is at γ, more than less wealth insuranced . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE Cook and Graham (1977) Figure: Equilibrium possibilities . . . . . . . Xuezhu Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42/42 Lecture 2 The Demand Email: of shixuezhu@outlook.com Insurance School of Insurance and Economics, UIBE