Are sunk costs sunk, is salvage value salvageable? A paradox in engineering economics analysis

Salvage value is defined as “The estimated value of an asset at the end of its useful life.”
Sunk cost is defined as “Cost already incurred which cannot be recovered regardless of future events.”
It is often said in economics that “sunk costs are sunk”, meaning they should not be considered a cost in economic analysis, because the money has already been spent.
Now consider two cases
In case 1, we have a road project that costs $10.00 today, and at the end of 10 years has some economic value remaining, let’s say a salvage value of $5.00, which when discounted back to the present is $1.93 (at 10% interest). This value is the residual value of the road. Thus, the total present cost of the project $10.00 – $1.93 = $8.07. Clearly the road cannot be moved. However, its presence makes it easier to build future roads … the land has been acquired and graded, some useful material for aggregate is on-site perhaps, and can be thought of as the amount that it reduces the cost of future generations to build the road. Alternatively, the land could be sold for development if the road is no longer needed, or turned into a park.
Assume the present value of the benefit of the road is $10.00. The benefit/cost ratio is $10.00 over $8.07 or 1.23. If we treat the salvage value as a benefit rather than cost, the benefit is $10.00 + $1.93 = $11.93 and the cost is $10, and the B/C is 1.193.
In 10 years time, the community decides to replace the old worn out road with a new road. This is a new project. The salvage value from the previous project is now the sunk cost of the current project (after all the road is there and could not be moved, and so does not cost the current project anything to exploit). So the cost of the project in 10 years time would be $10.00 – $5.00 = $5.00. Discounting that to the present is $1.93.
The benefit in 10 years time is also $10.00, but the cost in 10 years time was $5.00, and the benefit/cost ratio they perceive is $10.00/$5.00 = 2.00
Aggregating the two projects
the benefits are $10 + $3.86 = $13.86
the costs are $8.07 + $1.93 = $10.00
the collective benefit/cost ratio is 1.386
the NPV is benefits – costs = $3.86
One might argue the salvage value is a benefit, rather than a cost reduction. In that case
the benefits are $10.00 + $1.93 + $3.86 = $15.79
the costs are $10.00 + $1.93 = $11.93
the collective benefit/cost ratio is 1.32
the NPV remains $3.86
Case 2 is an identical road, but now the community has a 20 year time horizon to start.
The initial cost is $10, and the cost in 10 years time is $5.00 (discounted to $1.93). The benefits are $10 now and $10 in 10 years time (discounted to $3.86). There is no salvage value at the end of the first period, nor sunk costs at the end of the second period.
What is the benefit cost ratio?
the costs are $11.93
the benefits are still $13.86
the benefit/cost ratio is 1.16
the NPV is $1.93.
If you are the community, which will you invest in?
Case 1 has an initial B/C of 1.23 (or 1.193), Case 2 has a B/C of 1.16. But the real benefits and real costs of the roads are identical.
The salvage value in this example is, like so much in economics (think Pareto optimality), an accounting fiction. In this case no transaction takes place to realize that salvage value. On the other hand, excluding the salvage value over-estimates the net cost of the project, as it ignores potential future uses of the project.
Time horizons on projects must be comparable to correctly assess relative B/C ratio, yet not all projects do have the same benefit/cost ratio.
This “paradox” was first noted to me by Mark Snyder. I don’t know how widely it is known or understood, but it does affect analysis.