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).
So, for example, if