Alfie, Bill, And Coco Each Value Police Protection

Alfie Bill and Coco each value police protection differently Alf

Alfie, Bill, and Coco each value police protection differently Alf

Alfie, Bill, and Coco each value police protection differently. They have individual demand functions: Alfie's demand is Q = 55 – 5 P, Bill's demand is Q = 80 – 4 P, and Coco's demand is Q = 100 – 10 P. The marginal cost of providing police protection is $13.5. To find the socially optimal level of police provision, demand functions are rewritten to express price as a function of quantity. This yields:

  • Alfie: PA = 11 – 0.2 Q
  • Bill: PB = 20 – 0.25 Q
  • Coco: PC = 10 – 0.1 Q

The total marginal social benefit (MSB) of providing the police protection is the sum of individual willingness to pay at each quantity level, which is:

PA + PB + PC = 41 – 0.55 Q

Setting this equal to the marginal cost (MC = 13.5) to find the socially optimal quantity:

41 – 0.55 Q = 13.5

Solving for Q:

Q = (41 – 13.5) / 0.55 = 27.5 / 0.55 ≈ 50

At Q = 50, individual marginal benefits are:

  • Alfie: 11 – 0.2(50) = 1
  • Bill: 20 – 0.25(50) = 7.5
  • Coco: 10 – 0.1(50) = 5

The share of the tax burden under Lindahl pricing corresponds to each individual's marginal benefit over the total marginal cost:

  • Alfie's share: 1 / 13.5 ≈ 7.4%
  • Bill's share: 7.5 / 13.5 ≈ 55.6%
  • Coco's share: 5 / 13.5 ≈ 37%

This implies that each person pays a proportionate share of the tax corresponding to their marginal benefit at the optimal provision level.

In summary, the socially optimal level of police protection is approximately Q = 50 units, with individual contributions to the Lindahl tax burden ranging from about 7.4% for Alfie, 55.6% for Bill, and 37% for Coco, based on their marginal valuations of police services and the marginal cost of provision.

Sample Paper For Above instruction

Determining the socially optimal level of public goods, such as police protection, requires aggregating individual valuations and equating the total marginal social benefit (MSB) to the marginal cost (MC) of provision. In this case, the diverse valuations of Alfie, Bill, and Coco exemplify the challenges in providing public goods optimally. Each has their demand function for police protection, indicating their maximum willingness to pay at different levels of Q, the quantity of police protection provided.

By transforming the demand functions into price as a function of quantity (P = a – bQ), we can express each individual’s willingness to pay at any given quantity. For Alfie, PA = 11 – 0.2 Q; for Bill, PB = 20 – 0.25 Q; and for Coco, PC = 10 – 0.1 Q. The total marginal social benefit at each quantity level is the sum of these individual willingness to pay functions, yielding:

MSB = PA + PB + PC = (11 – 0.2 Q) + (20 – 0.25 Q) + (10 – 0.1 Q) = 41 – 0.55 Q.

To find the optimal quantity of protection, this total MSB must equal the marginal cost of provision, which is $13.50. Setting the equations equal:

41 – 0.55 Q = 13.5

Solving for Q yields:

Q = (41 – 13.5) / 0.55 ≈ 50

At this optimal level, individual marginal benefits are computed by substituting Q = 50 back into each person's demand function:

  • Alfie: 11 – 0.2 * 50 = 1
  • Bill: 20 – 0.25 * 50 = 7.5
  • Coco: 10 – 0.1 * 50 = 5

These values represent the individual marginal benefits or the Lindahl prices that each individual would be willing to pay at the optimal Q. The proportion of the total tax burden each would bear corresponds to their share of the total marginal benefit relative to the total marginal cost:

  • Alfie: 1 / 13.5 ≈ 7.4%
  • Bill: 7.5 / 13.5 ≈ 55.6%
  • Coco: 5 / 13.5 ≈ 37%

This distribution respects individual valuations, aligning individual contributions with their marginal benefits. Such an approach ensures efficient resource allocation and equitable burden sharing, crucial concepts in public economics, especially with regard to public goods where individual preferences significantly influence optimal provisioning levels.

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