- 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 - 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? - 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

your answer, are homogeneous communities superior to the original mixed

communities? - 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? - 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).

Sample Solution

n small before it became small. Moreover, if things only became smaller, and not larger, eventually everything would be miniscule. And if it was the other way around, where everything only became larger, and not smaller, everything would eventually be one thing, because everything would have joined together. If this were the case then we would notice that things only become smaller, shorter, or uglier, and never their opposites, or vice versa. Socrates shows that things do transition from two opposites, by referencing to observable examples. He contrasts this to death, and claims that there has to be a cycle of becoming alive and becoming dead, or else everything would become dead, or vice versa. The analogies that Socrates uses are applicable to every corporeal thing in the universe. Everything is either large or small, tall or short, etcetera. He claims that there is a process of becoming from its opposite (e.g. something becoming larger from being small), and that this process is cyclical. For if everythi>

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