Microeconomics Insurance and Risk Aversion PDF

Preview

Summary

Microeconomics insurance and risk aversion topic covering summarised notes from different sources....

Description

Insurance and Risk Aversion  Assumption of risk neutrality (concerned only with expected yield from uncertainty) is not considered or appropriate now Attitudes towards risk:  Risky situation or prospect is one which has associated with it a set of outcomes or payoffs, where each outcome is associated with particular state of world which occurs with some positive probability  Sum of respective state-probabilities is unity, since one of the states must occur  E.g. Risk neutral: assume a choice between certain pay off of £5 or lottery where 50% chance of payoff of £10, and 50% chance of zero where expected return is £5  Individual preferring lottery ticket rather than to be given £5 is risk loving while an individual is risk averse if prefer certain payoff of £5 to lottery ticket  These figures show utility of wealth for an individual – zero origin represents individual’s existing wealth level











Figure 5.1 (i) shows constant marginal utility of wealth i.e. second derivative is zero. E(R) indicates expected wealth from lottery. EU(L) = U (£5) = U[E(R)]; expected utility from owning lottery = level of utility from certain payoff of £5 = utility gained from certain payoff equal to expected payoff from lottery – indifferent hence risk neutral Figure 5.2 (ii) shows diminishing marginal utility of wealth where second derivative of utility function is less than zero. E(R) of lottery is £5 and reading up shows U(£5) which is utility of certain, which equals U[E(R)]. Expected utility of lottery located vertically halfway between U(£0) and U(£10), reading up from E(R) on x-axis until point B, which read across gives EU(L), and is on the chord joining points for wealth levels of 0 and 10 This time, EU(L) does not coincide with U[E(R)] or U(£5), but lies below it. Individual would prefer payoff of £5 with certainty to lottery with expected payoff of £5 – risk averse Figure 5.3 (iii) shows increasing marginal utility of wealth where second derivative exceeds zero. Expected return again of lottery is £5 and reading up to point A shows across to see utility level associated with payoff of £5 for certain, U(£5) or U[E(R)] This individual would prefer lottery ticket to payoff of £5 with certainty – risk loving

Risk aversion and insurance:  Assume now throughout that individuals are risk averse and that leads them to demand insurance against risks, and in particular large risks i.e. car or house





People are prepared to pay small amounts for pleasure of watching lottery win or any bet; thus gambling viewed as a recreational activity involving repeated small costs hence risk loving However, large amounts and serious risks is assumed to be risk aversion

The demand for insurance:  Individual’s total wealth denoted as T, and item of property such as car have its worth denoted by C, and probability of theft is p (known to both individual and insurance company)

    

 

  

Figure 5.2 shows risk averse individual’s utility of wealth Situation where individual does not insure If car is not stolen, wealth will be T and utility U(T), while if car stolen, wealth will be (T-C) and utility U(T-C) Expected wealth, E(W) = p(T-C) + (1-p)T Expected utility, EU(N) = pU(T-C) + (1-p)U(T) where N indicates no insurance Risk averse individual hence expected utility from risky situation, EU(N) is less than utility, U[E(W)], would get from certain level of wealth equal to the expected wealth of E(W) – shown on diagram by point B being vertically below point A Certain level of wealth, S on x-axis, for certain would yield same level of utility as EU(N) i.e. point B to D Thus, if insurance company promised to replace car, or pay amount of compensation Y equal to C, if it was stolen, individual would be willing to pay a premium of X for this insurance contract as long as X was no greater than (T-S) since then would guarantee him income level of S or more When customer fully compensated for loss then this is full insurance case Once fully insured, individual becomes certain of wealth level; if no thef then wealth level is (T-X), and if thef then still (T-X) since compensated Since, U(S) is equal to EU(N), with an insured level of wealth greater than S he obtains more utility than his expected utility in uninsured state and so willing to purchase insurance





Individual purchasing full insurance is paying premium, X, in return for uncertain repayment from insurance company; actual repayment zero with prob (1-p), that is with car not stolen; and C with prob p, that is when car stolen A risk averse person is one who prefers level of wealth of Z with certainty to a risky prospect with a level of expected wealth equal to Z

The supply of insurance:  Risk averse person will insure against theft as long as X for compensation payment C is less than (T-S)  Assume insurance companies are risk neutral and market for insurance is competitive thus companies compete against each other for customers by reducing premium charged to level that yields them only normal profits  In equilibrium, assuming full insurance, premium X is given by: X = pC called either zero profit or competition constraint or fair odds constraint Insurance companies expected outgoings equal its receipts and expected profits equal 0 so long as above equation holds and premium=expected compensation pay  Insurance at this cost is said to be insurance at fair odds; If X>pC then unfair odds; if XS, the certain level of wealth which yields insuree level of utility equal to expected utility if uninsured, it is clear insuree gains increase in utility by purchasing full insurances at fair odds; utility after purchasing full insurance, UE(W) is greater than expected utility if uninsured, EU(N)  Therefore, premium pC paid is less than (T-S), which is amount could pay and remain indifferent between purchasing full insurance or not Full insurance at fair odds:  Regardless of whether compensation level Y is greater, equal or less than C, insurance is still fair as long as premium X equals expected value of compensation, pY; that is value of compensation times the probability of receiving it  It can be that: p = price of unit of compensation, Y = amount purchased and X = total cost of purchasing insurance given Y  Insurance offered at fair odds if price of unit of compensation is equal to probability of theft, and is offered at unfair odds if price>p, or favourable odds is price...