Problem Set 3 - Solution Hints

Transcription

1 ETH Zurich D-MTEC Chair of Risk & Insurance Economics (Prof. Mimra) Exercise Class Spring 2016 Anastasia Sycheva Contact: Office Hour: on appointment Zürichbergstrasse 18 / ZUE, Room F2 Problem Set 3 - Solution Hints 1. Demand for Insurance a.) The farmer has the initial endowment of state contingent wealths W 0 in good state and W 0 L in bad state. This corresponds to the point (W 0, W 0 L) in the two-states-of-the-world diagram. Indifference curves of the farmer show combinations of wealth in the two states of the world which yield the same utility ( u = u ), and for a specified level of u are given by all sets of W 1, W 2 that satisfy: u = (1 ) u(w 1 ) + u(w 2 ) The Budget Constraint (BC), describing all combinations of W 1 and W 2 that the individual can achieve by buying insurance given it s resources, is derived from: W 1 = W 0 pc W 2 = W 0 L + (1 p)c W 2 = W 0 pl p (1 p)w 1 + pw 2 = W 0 pl 1 p W 1 p }{{} slope of the BC The Certainty Equivalent (CE) for the farmer is the riskless amount of wealth that has the same utility as the state contingent wealths W 0 and W 0 L. Thus it is given by the coordinates of the intersection of the indifference curve (IC) through (W 0, W 0 L) with W 2 = W 1 line (certainty line) (see Figure 1). 1

2 The risk premium r is given as E(W ) CE. Without insurance, E(W ) = (1 )W 0 + (W 0 L) = W 0 L. In the BC set = p and find the coordinates of intersection of BC with certainty line: (1 )W 1 + W 1 = W 0 L W 1 = W 2 = W 0 L = E(w) 1 Introduction The resulting risk premium is depicted in Figure 1. w IC w 0 L r CE w 0 L w 0 w 1 Figure 1: Figure1 Figure 1: Task a) initial endowment, certainty equivalent (CE) and risk premium of the farmer b.) The insurance lines (budget constraints) for an actuarially fair premium = p and actuarially unfair premium p > are depicted in Figure 2. From the lecture we know that the optimal cover (C ) is a solution to the following maximization problem: max C [(1 )u(w 0 pc) + u(w 0 L pc + C)] with the First Order Conditions given by: u (W 0 pc ) = (1 p) p(1 ) u (W 0 L + (1 p)c ) Risk aversion of the farmer implies u (W ) < 0. We differentiate between 3 cases: p = u (W 0 pc ) = u (W 0 L + (1 p)c ) W 0 pc = W 0 L + (1 p)c C = L p > u (W 0 pc ) < u (W 0 L + (1 p)c ) 2

3 W 0 pc > W 0 L + (1 p)c C < L p < u (W 0 pc ) > u (W 0 L + (1 p)c ) W 0 pc < W 0 L + (1 p)c C > L Thus the farmer buys full insurance (C = L) if the premium is actuarially fair 1 Introduction (p = ). w p p w 0 L CE w 0 pc p> w 0 w 1 w 0 pc p= Figure 1: Figure1 Figure 2: Task b) insurance line (budget constraint) for an actuarially fair premium = p and an actuarially unfair premium p > c.) The insurance policy selected by the farmer is given by the tangency point between the Indifference Curve (IC) and Budget Constraint (BC). It corresponds to point a with coordinates (W 0 pc, W 0 L+(1 p)c ) on Figure 3. Through point a draw a line, parallel to the certainty line. For point a, b, d, e in Figure 3 it holds that: bd = ad = W 0 L + (1 p)c W 0 + L = (1 p)c de = W 0 W 0 + pc = pc be = ad + de = (1 p)c + pc = C Thus in Figure 3 the premium amount P = pc corresponds to de and cover C to be 3

4 1 Introduction w p p w 0 L + (1 p)c a w 0 L b d e w 0 pc w 0 w 1 Figure 1: Figure2 Figure 3: Task c) Premium amount P = pc and cover C for an optimal insurance policy (point a) with the premium rate p d.) If the farmer is offered an insurance policy with a fair premium rate and additional fixed payment k then he has the same FOC on the optimal amount of cover as a farmer with initial endowment (W 0 k, W 0 L k), a fair insurance premium and without fixed payment. Since = p in Task b) we have showed that the farmer will purchase full insurance C = L However if the k is two large than it might be the case that: EU(W 0, W 0 L) = (1 )u(w 0 ) + u(w 0 L) > u(w 0 L k) Therefore farmer s initial state contingent wealth gives him more utility than the one with insurance and he won t purchase any policy. The graphical illustration is provided in Figure 4 2. Demand for Insurance: Comparative Statics 1 In our setting the optimal cover C for the agent can be found as a solution of a maximization problem: max C [(1 )u(w 0 pc) + u(w 0 L pc + C)] Denote by F (C, W 0, p,, L) = p(1 )u (W 0 pc ) + (1 p)u (W 0 L + (1 p)c ) The First order conditions for the maximization problem can be rewritten as p(1 )u (W 0 pc ) = (1 p)u (W 0 L + (1 p)c ) 4

5 1 Introduction w w 0 L k (too large) k (sufficiently small) k (sufficiently small) w 0 w 1 Figure 1: Figure4 Figure 4: Task d) The optimal insurance policy for the case when the insurance company offers fair premium and additionally charges fixed payment k F (C, W 0, p,, L) = 0 Thus FOC can be interpreted as an implicit function F (C, W 0, p,, L) = 0 for C. The impact of various parameters on the optimal cover can be studies using the Implicit Function Theorem: a.) We first calculate F = F W 0 F C, p = F p F C = F, F C L = F L F C and note that since u < 0 we have: F = p2 (1 )u (W1 ) + (1 p) 2 u (W2 ) < 0 W 1 1 = W 0 pc, W2 = W 0 L + (1 p)c [ ] [ F Thus we can conclude that: sgn w 0 = sgn w 0 ]. [ ] sgn = sgn [ p(1 )u (W 0 pc ) + (1 p)u (W 0 L + (1 p)c )] [ p(1 )u (W1 ) = sgn + (1 ] p)u (W2 ) p(1 )u (W1 ) (1 p)u (W2 ) 5

6 In the step from (1) to (2), the two terms get divided by the two terms from the FOC, respectively. Since both terms are equal this operation does not change the sign of the expression. [ ] sgn [ ] = sgn u (w1) u (w1) + u (w1) u (w1) (1) = sgn [A(w 1) A(w 2)], (2) where A( ) is the Arrow-Pratt-Measure of absolute risk aversion (see Problem Set 2, Exercise 4). Since p > in Exercise 1 b.) we have showed that W 1 conclude: > W 2. Thus we can DARA-functions: A(W 1 ) < A(W 2 ) = sgn [A(W 1 ) A(W 2 )] < 0 = < 0 inferior good CARA-functions: A(W 1 ) = A(W 2 ) = sgn [A(W 1 ) A(W 2 )] = 0 = = 0 IARA-functions: A(W 1 ) > A(W 2 ) = sgn [A(W 1 ) A(W 2 )] > 0 = > 0 normal good b.) If the insurer charges premium amount P = k + λc with λ > 1 then we have: EU = (1 ) u(w 1 ) + u(w 2 ) = (1 ) u(w 0 k λc) + u(w 0 L k + (1 λ)c) Denote F (C, k, λ,, p, L, W 0 ) = ( λ) (1 ) u (W1 ) + (1 λ) u (W2 ) W 1 = W 0 k λc, W 2 = W 0 L k + (1 λ)c Thus the First Order Conditions on the optimal cover C yield: F (C, k, λ,, p, L, W 0 ) = 0 Using the Implicit Function Theorem we have F k = k We first calculate F and since u < 0: F F = (λ)2 (1 )u (W 1 ) + (1 λ) 2 u (W 2 ) < 0 6

7 Thus we have that: sgn [ ] [ F k = sgn ] k [ ] sgn = sgn [(λ) (1 )u (W1 ) (1 λ) u (W2 )] (3) k [ (λ) (1 )u (W1 ) = sgn (λ)(1 ) u (w1) (1 λ) ] u (W2 ) (4) (1 λ) u (W2 ) In the step from (1) to (2), (1) the two terms get divided by the two terms from the FOC, respectively. [ ] sgn w 0 [ ] u (W1 ) = sgn u (W1 ) u (W1 ) u (W1 ) (5) = sgn [ A(W 1 ) + A(W 2 )], (6) Since the premium rate p = P C showed that W1 > W2. = k+λc C > λ > in Exercise 1) we have IARA-function implies: A(W 1 ) > A(W 2 ) = sgn [ A(W 1 ) + A(W 2 )] < 0 = k < 0 c.) Consider an individual with utility function u(w) = ln(w): i) The FOC could be written as follows: (1 )pu (W 0 pc )+(1 p)u (W 0 L+(1 p)c ) = (1 )pu (W 1 )+(1 p)u (W 2 ) = (1 )p (1 p) = + W 1 W 2 W 1 (1 p) = W 2 (1 )p ii) Using task i) the cover demand function C = C(W 0, p, L, ) is given by: [W 0 pc ](1 p) = [W 0 L + (1 p)c ](1 )p (1 p)pc [(1 ) + ] = (1 p)w 0 (1 )p[w 0 L] iii) The Effect of a Change in Wealth: C = p W p [W 0 L] = p (1 p)p < 0 Thus the optimal cover decreases with wealth and insurance is an inferior good. As we have shown in task a) in general sing of the partial derivative of cover demand function is determined by the properties of risk aversion A(W ) 7

8 The Effect of a Change in Loss L = 1 1 p > 0 As in the general case, if an individual is risk averse, an increase in loss increases the demand for cover other things being equal The Effect of a Change in Loss Probability = W 0 Lp (1 p)p > 0 Increase in loss probability increases demand for coverage. However in reality one would expect that the premium rate would change with the probability of loss thus there might be some ambiguity. The Effect of a Change in Premium Rate p = ( 1)(L W 0) (p 1) 2 W 0 p 2 < 0 The demand for cover decreases with premium rate. 3. Multiple Loss States and Deductible a.) Since both policies charge actuarially fair premiums it holds that: p P olicy1 = = 200, p P olicy2 = 1 ( )+1 = ( ) b.) The financial terms and possible outcomes for each policy are summarized in Table 1. Policy 1 Policy 2 Loss Reimbursement Premium Net Loss Table 1: Financial terms of different policies c.) To answer the question we compare the Net Losses under two policies (see Table 1. The distribution functions of Net Losses are illustrated in Figure 5). To determine the preferences of a risk averse agent we use the equivalence conditions from Problem Set 2, Exercise 2 b.) namely: 8

9 1.0 Policy 1 Policy Figure 5: Distribution of losses corresponding to different policies If xdf (x) = xdg(x) and for every t 0, can conclude that: F SOSD G t G(x)dx t F (x)dx one In task a.) we have showed that both policies yield the same expected values of losses. Since the areas in ed and blue in Figure 5 are equal, it holds that t t 0, F P olicy1(x)dx t F P olicy2(x)dx F P olicy2 SOSD F P olicy1. Thus the risk averse individual will prefer Policy 2 4. State Dependent Utility a. i) Analogous to task a.) Exercise 1 the agent s indifference curve consists of all combinations of state contingent wealths W 1 and W 2 that yield the same utility and for every fixed utility level u is given by: u = (1 )u 1 (W 1 ) + u 2 (W 2 ) For each fixed value of u the expression above can be viewed as an implicit function for W 2 with respect to W 1. Denote F (W 1, W 2 ) = (1 )u 1 (W 1 ) + u 2 (W 2 ) u. Using the Implicit Function Theorem we can calculate the slope of the indifference curve as follows: dw 2 = dw 1 df (W 1,W 2 ) dw 1 df (W 1,W 2 ) dw 2 = 1 u 1(W 1 ) u 2(W 2 ) The slope of the indifference curve in the point where it crosses the security line is given by 1 u 1 (W 1) u 2 (W 1). On the other hand the slope of the insurance line 9

10 1 No Risk, No Fun W Slope > 1 W 2 W 0 L W 1 W 0 W 1 Figure 1: Figure1 Figure 6: Two-state-of-the-world-diagram for task a.i) - a.ii). The point (W1, W2 ) correspond to the state contingent wealths under an optimal insurance contract. Less than full insurance is purchased. under the assumption of a fair premium rate (p = ) is given by: 1. Using the assumption that : u 1(W ) > u 2(W ) 1 u 1(W 1 ) u 2(W 1 ) > 1 a. ii) The First Order Conditions on the optimal cover C for an agent with a state dependent utility can be derived as follows: EU = (1 )u 1 (W 0 C) + u 2 (W 0 L + (1 )C) EU = 0 (1 )u 1(W 1 )+(1 )u 2(W 2 ) = 0 u 1(W 1 ) = u 2(W 2 ) where W 1 = W 0 C, W 2 = W 0 L + (1 )C This implies that in the optimum, the individual equalizes marginal utilities across states, but not necessarily wealth or total utilities. Using the assumption u 1(W ) > u 2(W ) and the concavity of the utility functions (u 1 < 0, u 2 < 0) we conclude that: u 1(W 1 ) = u 2(W 2 ) W 1 > W 2 L > C Thus the agent will purchase less than full insurance of wealth 1 a. iii) The two-state-of-the-world-diagram is given in Figure 6 10

11 b. i) To infer the effect of the parameters a and b we first need to derive the First Order Conditions on the cover: EU = (1 )u 1 (W 0 pc) + (a + bu 1 (W 0 L + (1 p)c)) F (C, a, b) = p(1 )u 1(W 1 ) + b(1 p)u 1(W 2 ) = 0 The effect of the parameter a on the optimal insurance cover can be evaluated using the Implicit Function Theorem: Since F (a,c ) a a = a = 0, we get = 0. The fixed parameter a does not influence a the insurance decision. The intuition behind is as follows: with state-dependent utility, the optimum condition requires that marginal utilities are equalized, i.e., u 1(W 1 ) = u 2(W 2 ). Since a has no influence on the marginal utilities, it does not influence the insurance decision. We repeat the previous steps to look at the effect of parameter b on the demand for the cover C. b b b = b = (1 p)u 1(W 2 ) > 0 since the utility function is increasing in wealth. = p 2 (1 )u 1(W 1 ) + b(1 p) 2 u 1(W 2 ) < 0 since u (W ) < 0. Thus > 0. The parameter b has a positive effect on the insurance demand. The larger b, the larger is the marginal utility of wealth in state 2 for a given wealth level. Thus, the equalization of marginal utilities (the optimum condition) coincides with a higher wealth level in state 2 and thus more insurance. b. ii) The First Order Conditions from the previous subtask can be rewritten as follows: b 1 p p Full insurance of wealth requires: 1 = u 1(W 1 ) u 1(W 2 ) W 1 = W 2 u 1(W 1 ) u 1(W 2 ) = 1 b 1 p p 1 = 1 b = p 1 p 1 Since p > 1 p > 1 This implies that for an unfair premium rate the p 1 full insurance of wealth might still be optimal if it holds that b > 1. 11