polgbs21_fin.dvi
MGT 611 – Policy Modeling
Fall 2021
Final Exam
Due: via Canvas before class December 13
This exam is open book, open notes, open web site, but closed
mouth! Do not talk to your classmates or other students at all
about this exam. You are to work entirely on your own. If you
have any questions, please contact either the professor or the TA.
Make sure to show all of your work! Good luck!
1. Terror Queue of the Heart (20 points, 5 points each part,
with apologies to Bonnie Tyler)
In the United States, the annual estimated all-cause number of deaths for those
with Coronary Heart Disease (CHD), including both known cases and those un-
diagnosed, approximately equals 670,000. There are an estimated 17.6 million
known CHD patients. It is also known that those newly afflicted with CHD
average 46.3 years until either diagnosis or death, whichever comes first.
(a) Given these data, what is the mean number of undiagnosed persons with
CHD?
(b) Suppose that the all-cause per capita mortality rate is 3.5 times higher
for undiagnosed persons with CHD than the per capital mortality rate for known
CHD patients. What are the per capita all-cause mortality rates for undiagnosed
versus diagnosed persons with CHD?
(c) What fraction of those newly afflicted with CHD will be diagnosed?
(d) What fraction of all persons living with CHD have been diagnosed?
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2. I Hear The Whistle But I Can’t Go (25 points awarded
as shown, with apologies to Steely Dan)
Imagine a scenario in which an average of 100 new undocumented immigrants
arrive to the United States each year from the fictional country of Eyfo (so we are
only considering a small portion of the total inflow of undocumented immigrants).
80% of these new arrivals are “movers” (short-term residents), while the remaining
20% of new arrivals are “stayers” (longer-term settlers). Suppose further that at
the end of each year in the United States, a mover returns to Eyfo with probability
25%, while a stayer returns to Eyfo after each year with probability 2%.
(a) (2 points) What is the average length of time spent in the United States
per visit by a mover? What is the average for a stayer?
(b) (2 points) A new undocumented immigrant from Eyfo has just arrived in
the United States. What is the probability that this immigrant returns to Eyfo
after one year?
(c) (4 points) Now suppose that a new undocumented immigrant from Eyfo
arrives to the United States at time 0. What is the probability that this new
immigrant returns to Eyfo after spending exactly years in the United States?
(d) (6 points) What is the conditional probability that a new undocumented
immigrant from Eyfo who arrived to the US at time 0 returns to Eyfo after spend-
ing exactly years in the United States, given that this immigrant spends at least
years in the United States?
(e) (6 points) Suppose that this process has continued long enough to reach
an equilibrium (steady state). Recall that 80% of new arrivals from Eyfo are
movers. In equilibrium, what is the total number of undocumented immigrants
in the United States who originated from Eyfo? Of all of the undocumented
immigrants from Eyfo residing in the United States, what fraction are movers?
And, over all newly arriving undocumented immigrants arriving to the United
States from Eyfo, what is the expected duration of stay?
(f) (5 points) Consider the first 20 years of the immigration from Eyfo. At the
end of this 20 year period, researchers conduct a survey in Eyfo of undocumented
immigrants who have returned to Eyfo. Note that only people who both left
Eyfo for the United States and returned within these 20 years can possibly be
included in the survey. Assuming that the researchers are able to obtain a random
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sample from the population of returnees to Eyfo over these first 20 years, what
fraction of the survey respondents are movers? (BIG HINT: after the first year
of immigration, only those who had arrived in the United States from Eyfo at
the start of the first year could return to Eyfo at the end of the first year of
immigration; after the second year of immigration, only those who had arrived
in the United States at the start of the first year from Eyfo and spent two years
in the US, plus those who arrived in the US at the start of the second year from
Eyfo and spent one year in the US, could return to Eyfo at the end of the second
year of immigration; …).
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3. Friends of Friends (15 points, 5 points each part)
Consider four kids in a kindergarten class named, oh, Hillary, Donald, Gary and
Jill. Hillary has three friends, Donald has one friend, and Gary and Jill each have
two friends.
(a) Let denote the number of friends of a randomly selected kid from
the group of four above. What is (), that is, what is the expected number
of friends when sampling from the four kids? Also, what is (2), the mean
squared number of friends?
(b) Now imagine that each of the four kids writes their own name plus the
names of each of their friends on little squares of paper, their own name and one
friend’s name per paper square, and hands the paper squares in to the teacher (who
receives eight paper squares in total). The teacher places all of the squares into
a hat, mixes them up, and draws one out at random. What are the probabilities
that the teacher picks one of Hillary’s, Donald’s, Gary’s or Jill’s friends?
(c) Let denote the number of friends of a kid from the orginal group of
four as sampled at random from the teacher’s hat (remember — each paper square
has the name of one of the four kids in the class as well as the name of one of
their friends). What is ( )? Answer first using your result from part (b), then
answer again using your result from part (a).
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4. The Federal Housing Agency (10 points)
The Federal Housing Agency (FHA) guarantees mortgages or trust deeds for qual-
ifying individuals purchasing residences. Although most insurees make their pay-
ments on time, a certain percentage are always overdue. Some debtors never pay,
and their residences must be subjected to foreclosure proceedings. The FHA’s
experience has been that when a mortgage is two or more payments behind, the
residence generally will have to be foreclosed. At the beginning of each month, the
mortgage officer reviews each account and classifies it as paid, current, overdue,
or a foreclosure. Current accounts are those being paid on time, while the overdue
category refers to an account that is one payment behind. The table entries below
report the probability that an account dollar in a particular category last month
will be in a particular category this month. A loan officer wants to know the
probabilities that current and overdue accounts eventually end up as paid or as
foreclosures. What are these probabilities? (HINT: East Rock)
This Month’s Account Status
Last Month’s
Account Status
Paid Current Overdue Foreclosure
Paid 1 0 0 0
Current 2 6 2 0
Overdue 1 3 4 2
Foreclosure 0 0 0 1
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5. Repeat Screening for HIV Infection (20 points, 5 points
each part)
Throughout the history of the HIV/AIDS epidemic, certain populations have been
subjected to repeat HIV testing. These include, but are not limited to, commercial
sex workers, prisoners, and persons in military service. The reasons for such
testing are varied: commercial sex workers are typically screened because they are
potentially at high risk for the transmission and acquisition of HIV, while persons
in military service are considered part of a “walking blood bank.” In some sense,
persons in the population at large who engage in high risk behavior and regularly
have themselves tested also constitute a repeatedly screened population.
This problem considers one aspect of this issue: how often should persons be
tested? Let us agree that undetected infections are costly, and identify a “cost of
infection” per person-year of undetected infection in the population. Let us also
agree that testing is costly, at a charge of per test. For simplicity, we will assume
that HIV testing is perfectly sensitive and specific: if the test says HIV+, then the
person tested is truly infected, while an HIV- result means that the person is free
of infection. Let the incidence rate of new infections (per uninfected person per
unit time) remain constant and equal to , and let the time in between successive
HIV tests (the “screening interval”) be denoted by .
(a) Recall that each person in this population is screened once every time
units. Suppose an individual has just become infected. On average, how much
time will pass from the moment of infection until the next screening test (when
the infection is detected)? (BIG HINT: the timing of infection is completely
independent of the timing of HIV tests, the latter happening once every time
units without fail!)
(b) Suppose that there are persons in the population, and that the number
of infected persons is sufficiently small relative to the population size that the
aggregate rate of new infections in the population can be written as per unit
time. What is the expected number of undetected infected persons in the popula-
tion? (ANOTHER BIG HINT: it obviously has something to do with the length
of time that a newly infected person remains undetected!!)
(c) Now, if it costs per test, and there are persons in the population each
getting tested once every time units, what is the aggregate cost of testing per
unit time? (THE HINTS KEEP COMING: if a person is tested once every time
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units, then what is the testing rate for that person? There are persons in the
population, so what is the total number of tests per unit time in the population?)
(d) In the State of Nevada, commercial sex work is legal and regulated in cer-
tain counties. The HIV incidence rate among these sex workers has been estimated
as 4 new HIV infections per thousand sex workers per year (that is, = 0004).
The state uses standard enzyme immunoassay tests (EIAs) to detect HIV anti-
body; these cost $5 per test. Finally, suppose that the public health department
has, using a variety of arguments, decided that the cost of undetected infection
equals $360,000 per year of undetected HIV infection in a commercial sex worker.
Total public health costs can be thought of as the sum of the total cost of unde-
tected HIV infection and the total cost of screening. The former cost is simply
equal to times the number of undetected infected persons in the population,
while the latter cost you found in part (c) above. Given the data values provided
for , and , what is the optimal screening interval? That is, what numerical
value of minimizes the total public health costs?
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