# The “index of public safety” in a city

1. The “index of public safety” in a city is given by one minus the crime rate. For example, and
index of 0.90, or 90%, means that 90% of the population will not be a crime victim during a given
year. The public safety index, denoted S, depends on the number of police in a city as well as its
population. The relationship is
S = 13.2(P.20)(n–.30),
Where P is the number of police and n is the city population.
a) Using the formula, identify and interpret the elasticity of congestion for public safety.
b) Suppose both P and n double. Does the level of public safety rise or fall? Based on your
answer, if n is expect to double, what must happen to the level of police to maintain a
constant level of public safety (must the police force more or less than double)?
c) Consider a city of 100,000 people with a police force of 60 officers. Compute the public
safety index for this city, and interpret it.
d) Suppose the city population grows to 500,000 but that the police force size remains at
2. Compute the new index of public safety, and compare it to the level in part (c).
e) Suppose the larger city wishes to achieve the same level of public safety as it enjoyed
when population was 100,000. How much must the police force be expanded to achieve
this goal? Compare the proportional increase in P to the proportional increase in
population.
f) Suppose the cost of a policeman is measured in thousands of dollars, with the cost equal
to 30. Compute the per capita cost (total cost divided by population) of the original
police force. Then compute the per capita cost of the police force from part (e),
remembering that the city is now larger. What is the proportional change in the cost?
3. Consider a city with three consumers: 1,2, and 3. The city provides park land for the enjoyment
of its residents. Parks are a public good, and the amount of park land (which is measured in acres)
is denoted by z. The demands for park land for the three consumers are as follows:
D1 = 40 – z
D2 = 30 – z
D3 = 20 – z
These formulas give the height of each consumer’s demand curve at a given level of z. Note that
each demand curve cuts the horizontal axis, eventually becoming negative. For the problem to
work out right, you must use this feature of the curves in deriving DΣ. In other words, don’t assume
that the curves become horizontal once they hit the axis.
a) The height of the DΣ curve at a given z is just the sum of the heights of the individual
demands at that z. Using this fact, compute the expression that gives the height up to the
DΣ curve at each z.
b) The cost of park land per acre, denoted c, is 9 (like the demand intercepts, you can think
of this cost as measured in thousands of dollars). Given the cost of park land, compute
the socially optimal number of acres of park land in the city.
c) Compute the level of social surplus at the optimal z. Remember that this is just the area
of a surplus triangle.
d) Suppose there are two other communities, each with 3 consumers, just like the given
community. Compute total social surplus in the three communities, assuming each
chooses the same amount of park acres as the first community.
e) Now suppose the population is reorganized into 3 homogeneous communities. The first
has 3 type-1 consumers (i.e., high demanders). The second has 3 type-2 consumers
(medium demanders), and the third has 3 type-3 consumers (low demanders). Repeat
parts (a), (b), and (c) for each community, finding the DΣ curve, the optimal number of
park acres, and social surplus in each community.
f) Compute total social surplus by summing the social surplus results from part (e) across
communities. How does the answer compare to social surplus from part (d)? Based on
communities?
4. Suppose that the cost function for a particular public good is given by
C(z,n) = (40n – 12n2 + n3
)z
In this case, the unit cost of z is 40n – 12n2 + n3
, a function of n.
a) Using the above formula for the unit cost of z, derive the formula for the unit cost of z
per capita. This is just the unit cost of z divided by n.
b) The best community size is where the unit cost of z on a per capita basis is as small as
possible. Using the results of part (a), compute this per capita cost for n =
1,2,3,…,8,9,10. What community size minimizes unit cost per capita? Multiply your
unit cost per capita by the relevant n to get total unit cost in the optimal-size community
(i.e., the c value for use with the DΣ curve below).
c) Suppose that all consumers in the economy are identical, and each consumer’s demand
for z is given by D = 20 – z. Using the results of part (b), compute the DΣ curve for an
optimal-size community (this is a community of the size found in part (b)). This is done
by adding up as many individual D’s as there are people in the optimal community.
d) Using the unit cost figure from part (b), find the optimal level of z in the optimal-size
community. Also, compute social surplus in the optimal-size community.
e) Suppose the economy contains 18 people. How many optimal-size communities can be
created out of this population? Using the results of part (d), what is total social surplus
in the economy?
f) Now suppose that instead of being divided into optimal-size communities, the
population is divided into 2 communities of size 9. Using the previous results, find the
unit cost of z per capita in these communities. Then find the optimal level of z in each
community, as well as social surplus in each community.
g) Compute social surplus in the whole economy when there are two 9-person
communities. Compare you answer to that from part (e). How big is the loss from nonoptimal community sizes?
5. Suppose that the cost function for a particular public good is given by
C(z,n) = (8n – n2 + .05n3
)z
In this case, the unit cost of z per capita is 8 – n + .05n2
.
a) Compute unit cost per capita for n = 1,2,…,10,11,12. At what value of n is this cost
minimized? Compute c (the total cost per unit of z) for n = 5 and n = 10. As before,
this is done simply by multiplying your unit cost per capita values by the relevant n.
Suppose the economy contains 5 high demanders and 5 low demanders. The demand curves for z
are given by D1 = 6 – z and D2 = 20 – z for the low and high demanders, respectively. Suppose the
economy is organized into two homogeneous communities each with population 5, one for the low
demanders and one for the high demanders.
b) Using the results of part (a), compute the socially-optimal z level in the low-demand
community. Also compute the level of social surplus in that community at the
optimum.
c) Repeat part (b) for the high-demand community. Add the social surplus levels from
both communities to get overall social surplus in the homogeneous case.
Now suppose the economy is organized into one mixed community of population 10, containing 5
low demanders and 5 high demanders.
d) Using the results of part (a), compute the socially-optimal z level in the mixed
community. Also compute the level of social surplus in the mixed community, which
gives the overall surplus level since this is only community.
e) Comparing your answers to parts (c) and (d), decide whether the mixed or
homogeneous arrangement is superior. Given an intuitive explanation for your
conclusion.
f) Now suppose that the demand rises for the low demanders, with D1 = 10 – z giving the
demand curve. Repeat part (b) using this new demand curve. Using your result along
with part (c), recompute overall social surplus in the homogeneous case.
g) Using the assumptions of part (f), compute the socially-optimal z level in the mixed
community of size 10, and compute social surplus.
h) Comparing your answers to parts (f) and (g), decide whether the mixed or
homogeneous arrangement is superior.
i) Give a thorough intuitive explanation of why different conclusions are reached in parts
(e) and (h).