Question

A consumer with a utility function U = W 1 / 2 (square root of W , wealth) has an initial wealth of $50,000, the cost of illness is $25,000, with the probability of illness p = 0.25. 

a. Calculate an actuarially fair health insurance premium for this consumer. 

b. Illustrate the consumer's utility and expected utility on a graph. Indicate pure premium, different wealth amounts, etc. 

c. Can you tell how much extra this consumer will be willing to pay for health insurance on top of the actuarially fair/pure premium?

Answer 1

U = √W

A) actuarially fair insurance premium

= total loss in bad state * probability of bad state

,= 25,000*.25

= $ 6,250

( Illness is the bad state )

B) utility function is concave in wealth, hence individual is risk averse

Graph

C) now the Maximum Willingness to pay for the full insurance =

Intital wealth - certainty equivalent (CE) of the gamble

So EU = .75*√50,000 + .25*√25,000

= 207.23

Now for CE:

√CE = EU

CE = (207.23)² = 42,944.27

(Good state probability = 1-.25 = .75

Good state wealth = 50,000)

so Maximum WTP = 50,000 - 42,944.27

= 7055.73

so extra WTP = 7055.73 - 6250

= $ 805.73