Suppose the state is trying to decide how many miles of a very scenic river it should preserve. there are 100 people in the community, each of whom has an identical inverse demand function given by p=10-1.0q, where q is the number of miles preserved and p is the per-mile price he or she is willing to pay for q miles of preserved river.
(a)if the marginal cost of preservation is $500 per mile, how many miles would be preserved in an efficient allocation?
(b)how large is the economic surplus?

(a) Efficient Quantity of miles preserved Q = 1000/600 = 1.67miles

(b) Economic Surplus = = 139 dollars

Explanation:

(a) Efficient Quantity is obtained where Demand curve meets Supply Curve

(Aggregate) Demand Curve is given by P = 1000 - 100q

(That's adding 100 individual demand curve. For example, at the first mile preserved, each would be willing to pay 10 - 1 = 9 dollar, so 100 people would be willing to pay 900 dollar)

Supply Curve is given by P=500q

(For every additional mile preserved the the price increases 500 dollar, and the first mile would cost 500 dollar)

Demand and Supply Curves meet at equilibrium price, given by 1000 - 100q = 500q.

Solving for q we get 1.67

(b) Economic Surplus is given by the area under the Demand Curve but above the Price (the vertical dotted line). In other words,

Because the community is willing to pay for a+b+c to preserve the river at the efficient quantity but only have to pay b+c (Quantity x Price)

Efficient Price is obtained at efficient quantity. P = 1000 - 100*1.67 = 833

= 0.5*(1000-833)*1.67 = 139 dollars

Question answered