Economics Stack Exchange Archive

Marginal cost and benefit

I’m stuck with a little exercise here, and I think you can help me. So suppose that the community of a village consists of three people with the following marginal benefits for streetlights provided in quantity $x$:

Tom: $MB = 1500 -5x $for$ x<300, MB = 0 $elsewhere

Sally: $MB = 2100 -7x $for$ x<300, MB = 0 $elsewhere

Huck: $MB = 2100 -5x $for$ x<420, MB = 0 $elsewhere

Now, if the marginal (and average) cost of streetlights is $$3150$, what is the optimal number of lights to install in the city? I thought that at $x=0$, the total cost will be smallest, and the benefit of each person will be largest. so this would be the optimal number. But it seems inconsistent with the rest of the exercise. So what do I have to do?

Answer 1108

Some questions to help you proceed to your answer:

  1. What’s the population (i.e. summed across all three residents) marginal benefit when x=1?
  2. What’s the population marginal benefit when x=2?
  3. So how much is the population marginal benefit changing with each extra streetlight?
  4. Up to what number of street lights will that change in population marginal benefit hold?
  5. Given that above rate of change in population marginal benefit, when does population marginal benfit switch from being greater than the marginal cost, to less than the marginal cost?
  6. How does that last criterion relate to the optimal number of streetlights?

Answer 1112

Another way to get started on the problem outside of EnergyNumbers excellent pointers is to think of the problem if there was only 1 resident. Try to figure out how many street lights Tom would want installed. Then see if you can generalize the answer to multiple residents.

Edit: I forgot to mention if you do do the thought experiment of having only 1 resident, you have to only consider them paying for 1/3 of the streetlight.

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