Demand  and  Supply  of  Insurance

EC426: Class question 3, topic 7 David Tan

Question  3 a)  Consider the graphical representation in Einav et al.

(2010). For a given demand function, how would differences in incomes affect the cost curve related to that demand function?

b)  What would happen if people with higher income have better health? How does this affect the welfare cost of adverse selection?

Notation Two Contracts: •  H = High coverage insurance contract •  L = Low coverage insurance contract •  p = Relative price of contract H to contract L

o  Can assume contract L is free and has zero coverage o  Price is endogenous

•  Consumers make a discrete decision, either buy H or L Consumer characteristics: •  Population distributed by G(ζ) •  ζ = vector of consumer characteristics

o  Einav et al. don’t specify the nature of this vector but it can describe for example the consumer’s ex ante risk perception or preferences

Utility •  vH(ζi, p) = Utility of buying contract H

o  Strictly decreasing in p

•  vL(ζi) = Utility of buying L o  vH (p=0) > vL

•  Let v = vH(ζ) – vL (ζ) = π(ζ) + r(ζ) o  This is essentially the relative willingness to pay for insurance contract H

•  π(ζ) = Cost of insurance to the insurer à the risk term

•  r(ζ) = Net-value of insurance à the preference term o  Assumes CARA and normal risk x o  r(ζ) = γ/2 var (x)

Demand  and  Supply  of  insurance

Demand for insurance à individuals buy insurance contract H IFF: vH(ζi, p) > vL(ζi) à v(ζ) ≥ p •  D(p) = 1 – Fv(p)

o  Fv(p) = CDF of values in the population for a given price o  Dependent on the distribution G(ζ) o  Einav et al. à Essentially represents the highest price p that an individual is

willing to pay for contract H

Supply of insurance •  Perfect competition, with no. of firms ≥ 2 •  Firms cannot price discriminate. Must charge one

price for all.

Cost curves: •  π(ζ) = Cost of insurance to the insurer (1) AC(p) = E(π(ζ) | v(ζ) ≥ p). (2) MC(p) = E(π(ζ) | v(ζ) = p). *MC (p(q)) = E(π) + [cov(π,v)/var (v)].[p(q) – E(v)] •  Costs are determined by individuals who

endogenously choose H.

For a given demand function, how would differences in incomes affect the cost curve related to that demand function?

•  Income differences will affect the cost curves through ζ – the consumer characteristics vector

•  Introducing income differences is essentially adding more heterogeneity.

Base case: no income differences, i.e. other consumer characteristics influence demand and costs for insurance

Case 1: income differences •  Assume income differences affect preferences for

insurance but are unrelated to risk. •  Income differences affect v through the r term (the

preferences term) and introduces more variance in v (var (v) increases).

•  Risks stay the same for each individual in comparison to the base case.

MC (p(q)) = E(π) + [cov(π,v)/var (v)].[p(q) – E(v)] •  If var (v) increases then the MC curve gets flatter. •  Causes the AC curve to get flatter as well

What would happen if people with higher income have better health? How does this affect the welfare cost of adverse selection?

Case 2: income differences AND a negative correlation between income and health risk AND health is a normal good. •  Now higher income means better health and thus lower

health risk. •  Health is a normal good: Assume that income is

positively correlated with the r term. i.e. higher income means increased preferences for insurance.

cov(π,v) = var(π) + cov(π,r): •  Cost of insurance becomes negatively correlated with

the preference term r. o  i.e. as income increases for an individual à he gets healthier à risk falls à cost

of insurance falls.

•  Depending on how strong the effect is, the cov(π,v) could become negative è advantageous selection.

•  The MC curve becomes upward sloping o  People who have the highest willingness to pay have the lowest expected cost

to the insurer.

The Welfare Costs: •  Advantageous selection still incurs a welfare cost

but through overinsurance. •  Welfare cost = EDC •  The inframarginal customers that are acquired by

reducing price have relatively good risks(low expected costs) à Gives insurance providers incentives to reduce price.