Question

Britney and Paris are neighbors. During the winter, it is impossible for a snowplow to clear the street in front of Britney’s house without clearing the front of Paris’s. Britney’s marginal benefit from snowplowing services is 12 - Z, where Z is the number of times the street is plowed. Paris’s marginal benefit is 8 - 2Z. The marginal cost of getting the street plowed is $ 16. Sketch the two marginal benefit schedules and the aggregate marginal benefit schedule. Draw in the marginal cost schedule, and find the efficient level of provision for snowplowing services

Answer 1

Britney’s marginal benefit is MBBRITNEY = 12-Z, and Paris’s is MBPARIS = 8-2Z. The marginal benefit for society as a whole is the sum of the two marginal benefits, or MB=20-3Z (for Z ≤ 4), and is equal to Britney’s marginal benefit schedule afterwards (for Z>4). The marginal cost is constant at MC = 16. Setting MB=MC along the first segment gives 20-3Z = 16, or Z = 4/3, which is the efficient level of snowplowing. Note that if either Britney or Paris had to pay for the entire cost herself, no snowplowing would occur since the marginal cost of $16 exceeds either of their individual marginal benefits from the first unit ($12 or $8). Thus, this is clearly a situation when the private market does not work very well. Also note, however, that if the marginal cost were somewhat lower, (e.g., MC ≤ 8), then it is possible that Paris could credibly free ride, and Britney would provide the efficient allocation. This occurs because if Britney believes that Paris will free ride, Britney provides her optimal allocation, which occurs on the second segment of society’s MB curve, which is identical to Britney’s MB curve (note that Paris gets zero marginal benefit for Z > 4). Since Paris is completely satiated with this good at Z = 4, her threat to free ride is credit if Britney provides Z > 4. See the graph below.