1. Private Insurance: Suppose that there are two types of individuals: sickly and healthy Sickly get sick with probability (15 and Healthy get sick with probability qH with (13 > qH . The population is divided equally between both types. Each indi- vidual has income W (regardless of health or type). If sick, both type incur medical cost (1 to get better. The expected utility of an individual of type '1? : S, H is: EU = (1 *WUCW in) +q.:U(W idem +32%) The benet received when sick is such that him;- : p;- where m;- is the price per unit of insurance. The rst order condition that determine the optimal level of premium p; to be bought by this individual is: dEU/dpi- = (l qU'UV pi) + qi(l + l/i?1,;)U'(W d +pI-(l+ l/m) =0 The insurers expected prots are : EP = p" thH +315- qghs where the price charged for each unit of benet paid: 7a,; = pi/bi for 1' = 8,11. Assume that U[{:) = In. a, hence U'(rr) = l/(:, W = 100,000, (3 = 30,000, rm 2 .596 and q}; = "%. (a) (5 points) Derive the demand for insurance for each individual type as a function of the price of insurance it faces. (1)) (5 points) Show that if insurance companies and individuals can observe the types and there is perfect competition among insurers so that E!" : 0, each in- dividual type buys full insurance, that is bl- = d for all i = 8,11. (c) (5 points) Assume that there is information assymetry; individuals do know their type but insurance companies cannot observe the different types and have to charge the same insurance price m to everyone. There is perfect competition among insurers so that EP 2 0. Use the latter condition to derive a relation between the price of insurance and the amount sold to . (d) (5 points) Set up the conditions that you would use to nd the equilibrium price level of insurance and set up an equation that would allow you to nd it (no need to try to solve it)