SAN JOSÉ STATE UNIVERSITY
ECONOMICS DEPARTMENT
Thayer Watkins

## The Relationship Between Private Profitability and Net Social Benefit

A project will generally have some direct benefits and costs that result from the goods and services it produces and the resources it uses up. These goods and services and resources generally will involve market transactions. A project may also have external benefits and costs. These are the good things and the bad things that result from the project and are imposed upon society rather than resulting from market transactions. Thus the net scocial benefit of a project can be expressed as:

### The Benefits and Costs of the Direct Effects of a Project

For simplicity assume the project produces only one output, milk, and uses only one input, labor.

Let Pw/o denote the price of milk without the project and Pw the price with the project. The output and consumption of milk without the project is denoted as Qw/o and with the project as Qw. The output of the project is denoted as Qp and hence

#### Qw = Qw/o + Qp.

The gross benefit of an increase in consumption of milk from Qw/o to Qw is the area under the demand curve for milk from Qw/o to Qw. For a straight line demand function this is equal to the area of the trapezoid shown and thus:

#### Gross Social Benefit = (1/2)(Pw/o + Pw)Qp

The gross social cost of the increased resource use due to the project is the area under the supply curve for the resource over the range of resource use from what it would be without the project to what it would be with the project.

The use of labor without the project is denoted as Lw/o and with the project as Lw. The labor input of the project is denoted as Lp and hence

#### Lw = Lw/o + Lp.

Let Ww/o denote the wage rate for labor without the project and Ww the wage rate with the project. For a straight line supply function for labor this is equal to the area of the trapezoid shown and thus:

#### Gross Social Cost = (1/2)(Ww/o + Ww)Lp

The revenue that accrues to the project is equal to the price of milk with the project, Pw, times the output of the project, Qw. The cost of labor for the project is the wage rate with the project, Ww, times the labor used by the project, Lw. Thus the profit of the project is given by:

#### Private Profit = PwQp - WwLp

The net social benefit of the project can therefore be expressed in terms of private profit as follows:

#### Net Social Benefit = Private Profitability + (1/2)(Pw/o-Pw)Qp+ (1/2)(Ww/o-Ww)Lp- Net Costs of Externalities

In the absence of externalities the net social benefit of a project will be greater than the private profitability. Thus if a project is profitable necessarily it will be also socially worthwhile.

If there are some net external costs the project may still be worthwhile if these are not large enough to outweigh the net social benefits of the the direct effects of the project.

It is possible that a project may have a negative profitability and still be socially worthwhile. It is matter of whether the other terms are sufficiently large to outweigh the negative profitability.