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# ps6fall2000 - 14.03 Fall 1999 Problem Set#6 Due in Class#22...

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14.03 Fall 1999 Problem Set #6. Due in Class #22 1. The government of Rothschildia (a small, densely populated island nation off of the coast of Cambridge) is considering implementing a voluntary national health insurance plan. Everyone in Rothschildia is risk averse with VNM utility function U ( W ) = W 0.5 . Each citizen has a wealth of 1 Stiglitz (the local curency) but should he or she become ill, s/he must spend her entire wealth of 1 Stiglitz on health care. (Since the cure is immediate and complete, the only disutility of illness is this 1 Stiglitz cost.) The only respect in which Rothschildians differ from one another is that each has a different ex ante probability of becoming ill, p i . Illness probability is distributed uniformly among citizens on the [0,1] interval, meaning that on average one-half of the population becomes sick in a year, but that each person has a different probability, p i , ranging from zero to one where all values of p i are equally likely. If p i = 0 , person (i) is certain not to become ill, if p i = 1 , person (i) is certain to become ill, if p i = .5 , person (i) has a 50% probability of becoming ill, etc. Moreover, each citizen knows his own individual illness probability but the government only knows about the distribution of probabilities. Finally, for this problem, you should assume that each citizen lives only 1 year but that a new generation is born every year (each also knowing his or her p i ). The government, of course, persists from year to year. You should also bear in mind the following facts about uniform probability distributions : 1. If a variable is distributed uniformly on the [0,1] interval, the mean (i.e., expected) value of the variable greater than or equal to a given cutoff, L , is ( L + 1) / 2 . Similarly, the expected value below a given cutoff, U , is ( U + 0)/ 2 = U / 2 . Hence, the expected value of observations ½ is ¾, and the expected value of observations £ ½ is ¼. 2. If you want to calculate the expected value of a function of a random variable , you must integrate that function over the probability distribution of the random variable which is specified by its ‘probability density function’ (PDF). The probability density function of a U[0,1] (where U stands for uniform) variable could not be simpler. It’s: f ( x ) = 1 Meaning that all values between 0 and 1 are equally likely (and, moreover, the sum of probabilities of all possible values – the integral of f ( x ) on [0,1] – is 1).

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ps6fall2000 - 14.03 Fall 1999 Problem Set#6 Due in Class#22...

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