Question

Problem 2: Maximizing Net Benefits

There are important trade-offs involved in granting "Wild and Scenic River Status" to portions of a river. How much of this public good, a free-flowing river, should be protected from further development? As an analyst in the Office of Policy Analysis of the U.S. Department of the Interior, you are called upon to make a recommendation. Each year, 1,000 people benefit from the river's various services. A contingent valuation survey carried out by your office has estimated that each individual beneficiary has the same demand function for river preservation,

Q = 75 - (0.25)(P)

where P is the price-per-mile which persons are willing to pay (per year) for Q miles of river preserved. You find that the marginal (opportunity) cost of preservation is $60,000 per mile per year. ($60,000 for every one mile)

[Hint: You need to derive the market (aggregate) demand curve for a public good.]

a)How many miles of the river would be preserved in an efficient allocation?

b) What is the magnitude of the total (gross), annual benefits associated with this (efficient allocation) policy?

c)What are the total, annual costs of the policy?

d) What is the magnitude of the total (annual) consumers' surplus?

e)How large are net, annual benefits?

f) If it turns out that the marginal cost of preservation is only $20,000 per mile per year, how many miles of the river would be preserved in an efficient allocation?

g) Now assume substitute sites are available to beneficiaries, so their demands are substantially more elastic: their individual demand functions for river preservation are Q = 75 - (0.75)(P) In this case, with the original marginal costs of preservation of $60,000 per mile per year, how many miles of the river would be preserved in an efficient allocation?

Answer 1

Answer (a)

Inverse demand function: Individual demand curves expressed in terms of P to aggregate the demand curve vertically:

Q = 75 - (0.25)(P)

0.25P = 75-Q

P = 300 - 4Q

As stated that 1000 people would be benefited from the river's various services, so multiplying the right hand side, in above equation, to derive aggregate marginal benefit curve:

P = 300,000 - 4000Q

Now, we need to find the efficient level of public goods that ideally should be provided, by equating marginal cost to price (MC=P). Marginal cost here is $60,000 per mile per year:

60,000 = 300,000 - 4000Q

4000Q = 300,000 - 60,000

4000Q = 240,000

Q = 60 miles of river per year

Answer (b):

Total (Gross) annual benefits would be equivalent to the area under demand curve upto Q = 60

Total (Gross) Annual Benefits = 60 * 60,000 + 0.5 * 60 * (300,000 - 60,000)

= 3,600,000 + 7,200,000

= 10,800,000

Answer (c):

Total (Annual) cost of the policy: Area under the Marginal Cost curve upto Q = 60 miles

Total (Annual) cost = 60 * 60,000

= 3,600,000

Answer (d):

Magnitude of Total (Annual) Consumer Surplus: Marginal Cost of $60,000 is not borne directly by the users, hence, not impacting the consumer surplus:

= 60 * 60,000 + 0.5 * 60 * (300,000 - 60,000)

= 3,600,000 + 7,200,000

= 10,800,000

Answer (e):

Net Annual Benefits can be calculated by:

= Total (Gross) Annual Benefits - Total Annual Costs

= 10,800,000 - 3,600,000

= 7,200,000

Answer (f):

If the marginal cost of preservation is only $20,000 per mile per year and demand remaining unchanged, efficient preservation would be:

20,000 = 300,000 - 4000Q

4000Q = 300,000 - 20,000

4000Q = 280,000

Q = 70 miles of river per year

Answer (g):

In the scenario of demand being substantially more elastic, owing to the availability of substitutes, inverse demand curve is used to calculate market aggregate demand curve:

Q = 75 - 0.5P

0.5P = 75 - Q

P = 150 - 2Q

Multiplying right hand side by 1000 (1000 individuals getting benefited from river's various services)

P = 150,000 - 2000Q

Marginal cost is given as $60,000 per mile per year so the efficient level can be calculated as:

60,000 = 150,000 - 2000Q

2000Q = 90,000

Q = 45 river miles per year