Consider a world in which there are two types of people (robust and frail) who work for a single employer, MU. Both types of people have utility function given by

U(W)=1 - exp(-W/500),

where "W" is wealth. Wealth is a stock, and income (Y) is a flow into that stock. Since

W=W_t-1 +DY_t that is, income is simply a flow into the stock of wealth,don't get caught up in

whether there is a Y or a W in the utility function. The function "exp" is the exponential function wheref(x)=exp(x)= e^x. It is the inverse of the natural logarithm, so,ln[exp(x)]=x. Further,

f(x)=exp(x)(d/dx) . To save you the math, I assure you that in this case that the function is concave inW; that is,U(W)>0andU(W)<0. During any given year, the probability of falling ill isprobust=0.1andpfrail=0.2. If a person falls ill, the cost of treatment is $200. MU is

deciding whether to offer health insurance to their workers. Each worker pays MU a premium at the beginning of the year, and if the worker falls ill, the insurer will pay the worker some amount of coverage. When they decide whether to buy insurance, workers know if they are robust or frail (though they do not know if they will fall ill during the year), but neither MU nor MU's insurance company can observe any worker's health status. Workers will, of course,

misrepresent their type if it is in their interest to do so. The firm decides what insurance plan (coverage and premium) to offer workers, and workers decide whether to purchase the plan or not by maximizing their expected utility. All workers have $1,000 in initial wealth.

(a)Are these people risk-averse with respect to wealth?

(b)MU first proposes contract A. The contract offers full coverage of the damages (i.e., an individual receives $200 if he falls ill). What is the maximum premium MU could price the contract, so that it remains attractive for robust people?

(c)MU decides to price the contract A at a premium of $23. Would frail people want to buy this contract?

(d)Thinking about the insurance company, and hence MU, would they want to offer this contract (what are their expected profits?)?

(e)If only frail workers buy the contract, what is the minimum premium MU could charge for a full insurance contract to just break even (think about profits again)? Would frail people want to buy this contract? Would robust people want to buy it?