question archive Three residents of a community are to decide the size of their police force
Three residents of a community are to decide the size of their police force
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Three residents of a community are to decide the size of their police force. Each policeman costs $24,000. The marginal benefit of an additional policeman for each individual at a given number of existing policemen is as follows. What is the socially optimal number of policemen? If the number of policemen is decided by majority voting, who is the medium voter? If each consumer is subject to a head tax, how many policemen will be hired?
The socially optimal number of policemen (policeman's salary = $24,000)
| Number of policemen (z) | Marginal benefit for consumer A | Marginal benefit for consumer B | Marginal benefit for consumer C | Marginal social benefit |
| 1 | $19,000 | $16,000 | $13,000 | $48,000 |
| 2 | $17,000 | $14,000 | $11,000 | $12,000 |
| 3 | $15,000 | $12,000 | $9,000 | $42,000 |
| 4 | $13,000 | $10,000 | $7,000 | $36,000 |
| 5 | $11,000 | $8,000 | $5,000 | $30,000 |
| 6 | $9,000 | $6,000 | $3,000 | $24,000 |
| 7 | $7,000 | $4,000 | $1,000 | $18,000 |
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What is the socially optimal number of policemen?
The socially optimal number of policeman is 6 as that is when the salary for the police is equal to the marginal social benefit.
If the number of policemen is decided by majority voting, who is the medium voter?
Consumer B is the medium voter since his marginal benefit is exactly halfway between the other two consumers and thus he has the average and median marginal benefit of society.
If each consumer is subject to a head tax, how many policemen will be hired?
If a head tax was paid for police, there would be 5 policemen hired. This is because the size of the tax per person is $8,000 and the medium voter above demand 5 police at such as value.
