HW3,SP2000sols

HW3,SP2000sols - EEP101/ECON 125 Spring 00 Prof D Zilberman...

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EEP101/ECON 125 Spring 00 Prof D. Zilberman GSI’s: Malick/McGregor/St-Pierre Suggested Solutions to Problem Set 3 1. There are two graphs for this exercise. One refers to the two types of individual demands. The other refers to the aggregate demand for the public good and the marginal costs of providing that public good. (a) The aggregate marginal benefit curve (AMB) is the vertical sum of all the individual marginal benefit curves. Thus, 10 10 (200 10 ) (80 10 ) 280 20 8 H L AMB MB MB AMB Q Q AMB Q for Q = × + × = - + - = - This marginal benefit curve is valid only for Q 8 because the “low” value businesses aren’t willing to pay any additional positive amount for more than 8 lights. That is, there are no marginal benefits to this group from the ninth and subsequent lights, which we take into account in deriving the aggregate marginal benefits. The “high” value businesses are still willing to pay to have additional lights after 8 have already been provided. Therefore, the aggregate marginal benefits are still positive above 8 lights. The region of the aggregate marginal benefit curve above 8 lights is depicted as 10 0 10 200 10 8 H AMB MB AMB Q for Q = × + × = - The two segments of the aggregate marginal benefit curve are: 280 20 8 200 10 8 AMB Q for Q AMB Q for Q = - = - The individual business owner’s demand curve and the aggregate marginal benefit curve are depicted as follows
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100 10 20 MB ($) AMB Q MC Aggregate Marginal Benefit and Marginal Cost 200 300 Q* 8 10 20 10 20 MB ($) MB H Q MB L Individual Marginal Benefits
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The most a “high” value business owner is willing to pay for 2 streetlights is 38 2 40 2 20 ) ( 2 0 2 2 0 = - = - = Q Q dQ Q MB H This is equivalent to the area under the MB H curve from zero to two: 38 2 ) 18 20 ( 2 1 = × + The maximum willingness to pay for the “low” value business for two streetlights is 14 2 16 2 8 ) ( 2 0 2 2 0 = - = - = Q Q dQ Q MB L (b) Social welfare is maximized when the marginal cost of providing one more light equals the benefit of one more light, i.e., where marginal cost equals marginal benefit. Thus, MC = 100 + 10Q = 280 - 20Q = AMB. Solving for Q gives Q* = 6. Since this is less than eight, we know that we have used the relevant region of the aggregate marginal benefit curve. The total cost of providing Q* is the integral (area) under the marginal cost curve between 0 and 6. Total cost = 780 2 360 600 2 10 100 6 0 2 = + = + Q Q . (c) Streetlights can be considered a pure public good, that is, a good that is both non-rival and non-excludable. Since this good is non-excludable, it is difficult (and indeed wasteful of resources) to exclude consumers, and it is possible for a non-paying business to free-ride. If this were provided in competitive markets, less than the socially optimal quantity would be provided.
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One way that city government can finance the provision of streetlights is to require all 20 businesses to pay a portion of the cost of providing the public good. In other words, the government would provide Q* and levies a uniform tax on each business where the uniform tax is simply 39 20 780 *) ( = = n Q TC (d)
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