A Cost-Benefit Analysis

Consider a project that requires $5 million to build a dam and canal which will provide water to irrigate desert land that has no other use. The land is used to grow cucumbers. Suppose the demand function for cucumbers were known to be

q = 18 - 0.1p,
 

where q is in units of millions of pounds per year and p is the price of cucumbers in cents per pound.

Suppose that a project produces 0.5 million pounds per year. Without the project the quantity of cucumbers available is 6 million pounds per year. With the project then the quantity is 6.5 million pounds per year.

What is the gross benefit of the project? The gross benefit is the area under the demand curve from the level of consumption without the project to the level with the project. Since the demand function is linear we can calculate the area under the curve as the area of the trapezoid. This requires the level of the market price without the project and with the project.

The price of cucumbers without the project is found by solving for p; i.e.,

p_w/o = (18-6)/0.1 = 120 cents per pound
and
p_w = (18-6.5)/0.1 = 115 cents per pound.
 

The area of the trapezoid is then

(1/2)(120+115)(0.5 million) = 58.75 million cents per year
= $0.5875 million per year
= $587,500 per year.
 

Suppose the only input required to produce cucumbers, besides the the land and water which have no opportunity cost is labor and each 10,000 pounds of cucumbers requires 1 person-year of labor.

This means the project's 0.5 million pounds of cucumbers require 50 person-years of labor.

Suppose also that the supply function for labor in the area of the project is

L = -20,000 + 4w,
 

where L is person-years of labor supplied and w is the wage rate in dollars per person-years. Without the project the amount of labor utilized is 2000 person-years. With the project the labor demand is 2500 person-years. The annual wage rate without the project is then

w_w/o = (20,000 + 2000)/4 = $5,500 per year.
 

With the project it is

w_w = (20,000 + 2050)/4 = $5,512.50 per year.
 

The social cost of the labor for the project is the area under the supply curve from the level of demand for labor without the project to the level with the project. Since the supply schedule for labor is linear the social cost is the area of the trapezoid; i.e.,

(1/2)(5,500+5,512.50)(50) = $275,312.50
 

The annual net social benefit of the project is then

587,500 - 275,312.50 = 312,187.50
 

If the project provides this net benefit forever and the interest rate is 5 percent, the present value of the future net benefits is

(312,187.50)/0.05 = 6,243,750.
 

When we deduct the $5 million required to construct the project the net benefit of the project is

6,243,750 - 5,000,000 = 1,243,750.
 

There is an alternate way of tallying the costs and benefits of a project. It involves the consumers' and producers' surpluses, but some care must be taken to capture all of the consequences of the project. The previous analysis was in terms of the benefits of increased consumption of commodities less the costs of increased resource use. The situation may be depicted diagrammatically. The alternative is to total up the net gains of consumers, businesses, and resource owners; i.e.,

Usually what is above labeled "businesses' surplus" is referred to as producers' surplus. Shortly it will be shown why businesses' surplus is a more appropriate term.

First, let us compute consumers' surplus. It is the area of the trapezoid which is bounded by p_w/o and p_w above and below and the vertical axis on the left and the demand schedule on the right. Its value is equal to

(1/2)(q_w/o + q_w)( p_w/o - p_w),
 

which in the case of the project is

0.5(6.0+6.5)(120-115) = (6.25)(5cents)
= 31.25 million cents per year
= $312,500 per year.
 

The surplus for the resource owners is computed as

(1/2)(L_w/o + L_w)( w_w - w_w/o),
 

which in the case of the project is

0.5(2000+2050)(12.50) = (2025)(12.50)
= $25,312.50 per year
 

The gain for the businesses selling cumcumbers comes from the higher price of cucumbers, but the loss for businesses comes from the higher price they have to pay for labor. This is not just the cost for businesses involved in the cumcumber market, instead it includeds all businesses which use labor. Producers' surplus would tend to mislead one into counting only the effects on the producers of cucumbers. Without the project the revenue received by businesses from the sale of cucumber was 6.0(1.20) = $7.2 million. After the project the revenue received by businesses is 6.5(1.15) = $7.475. Thus the gain in revenue to businesses from the sale of cucumbers is$7.475-7.20 = $275,000 per year. But the project increased the payment for labor from 2000(5500) = $11 million to 2050(5512.50) = $11,300,625 per year, an increase of $300,625. Therefore there is a net loss to businesses; i.e. the net gain is negative and equal to:

275,000 - 300,625 = -25,625.
 

Now we can tally up the net social gain of the project:

ΔConsumers' surplus = $312,500.00
ΔResource Owners' surplus = $25,312.50
ΔBusinesses' surplus = -$25,625.00
Annual Net Social Benefit of Project = $312,187.50.
 

This is exactly the same as was computed by the other method.