Question

The city of Ventura would like to build a seawall to protect the city from the threat of tsunamis. Each additional inches of height further protects the city and the 100 residents are each willing to pay $11 per inch of seawall height, regardless of how many inches are provided. The cost of building a wall that is i inches high is given by c(i) = 5i^2.

What is the Pareto Optimal height for the seawall?

Answer 1

Answer: 110 inches

Step-by-step explanation:

The Pareto Optimal point is the level at which complete efficiency has been reached. This means that at this point, any other strategy will do more harm than good.

In Production, the optimal efficiency is reached where Marginal Cost equals Marginal Revenue. Same here.

The Pareto Optimal height is the point where the Marginal Revenue (MR) equals the Marginal Cost (MC).

The Marginal Revenue in this scenario is the $11 that each resident will pay.

There are 100 residents so the total is,

= 11 * 100

= $1,100

The Marginal Cost is given by 5i². This can be differentiated for easier calculations.

= (d/dx) nx²

= 2nx

So,

= (d/dx) 5i²

= 10i

Equating MR = MC

1,100 = 10i

i = 1,100/10

i = 110 inches

The Pareto Optimal height for the seawall is 110 inches.