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The “radical” suggestion is that natural resources
whose rate of growth is slower than the market rate of interest should be
valued at their time to replacement.

Probably the most widely cited example of a renewable
resource that is damaged by our current valuation method is Old Growth Timber.
In the United States National Forests, access to the trees for clear-cutting is
given to timber companies on an auction basis.The bid price ultimately depends
on the market value of the board footage obtained from the trees. Older trees
are obviously more valuable than young trees and are cut first, so the average
age of the forest reduces over time. When European settlers came to the
northwestern states and began to harvest timber, the average age of our
temperate rainforests was 1500 years. Today the average age is about 400 years.
The forest as an asset has lost value over time, and that is a loss of equity
to successive generations when we account the ecological value of the forest in
sequestering carbon, preventing mudslides, moderating the microclimate of the
region, and providing recreational facility. There is some empirical evidence
that the bid price per stump conforms to Hotelling’s predictions, rising over
time at a rate approximate to the rate of interest [1]. However, the
administrative costs of holding the bid and building roads for the timber
trucks results in a loss that must be borne by taxpayers. This serves to keep
the price of board footage low so that a chronic externality is created in
which buyers of raw lumber (at Home Depot for example) are subsidized by
taxpayers, and homeowners in the forested regions suffer higher insurance
premiums while farmers suffer higher production costs due to the loss of
protection against mudslides and weather variations. There are also issues of
habitat loss and water purity that complicate the issue but valuing all of
these effects is beyond our scope.

To simplify matters we will consider only the carbon
sequestration value of the trees, and we will use the Climate Leadership
Council proposed carbon dividend of $40 per ton of carbon dioxide [2]. One
40-year old tree sequesters approximately 1 ton of carbon dioxide, accumulating
about 48 pounds each year, or about $0.96 in carbon value each year. [3] The
most rapid accumulation occurs in the early years of the tree’s life, but
purely for simplicity we will assume uniform sequestration in every year of the
tree’s life.

There are several compound interest formulas we can use to
talk about the value of the tree as a function of time. The first is the simple
formula for a perpetuity, also known as ‘capitalizing’ the asset. We simply
take the annual benefit and divide by the interest rate. So an asset that
yields $100 in revenue each year when the market rate of interest is 2% should
have a market value of $5,000:

Perpetuity: $100/0.2 = $5000

What that means is that a buyer of the asset would earn a
competitive rate of interest if he/she paid $5000 to obtain the asset today.
This is a ‘present value formula’ — it tells us how much we should pay today
for an asset that will earn revenues in the future.

The other formula is the ‘annuity formula’ to determine the
end value of a stream of benefits to be paid over a discrete number of years.
This is a ‘future value’ formula, it tells us how much the asset will be worth
in the future if its annual benefit is not cashed out but left to accumulate.
This is the value that should be of interest to future generations. The formula
is a little bit complicated:

Future value of an annuity = Annual Benefit x [(1 + r)t –
1]/r, where r is the interest rate and t is the number of years. Usually we
would look up this interest factor in a
table, or use a financial calculator or a website that does the calculation.

Future value of an annuity that pays $100/yr for 20 years: $100 x 24.297 = $2429.70
where the 24.297 interest factor was obtained from a table.

OK, after that long introduction, we are ready to do some
very simple calculations, ignoring inflation and interest rate fluctuations.

Question 1: Suppose we are looking at a 40 year-old tree in
an eastern U.S. National Forest. It is sequestering each year approximately 1
ton of carbon dioxide valued at $40/ton. The competitive interest rate is 2%.
Using the perpetuity formula, what is the present value of the tree if it is
not cut down but continues to exist and to sequester one ton of carbon each
year indefinitely into the future? This is the present value of the tree as a
living asset to the generation living right now.

Question 2: Now suppose that we cut down that same tree,
replant a new tree and must wait 40 years for the carbon sequestration benefit
to be replaced. It will accumulate $0.96 in value each year.
(a) Using the annuity formula, with an interest rate of 2%,
what would be the accumulated carbon value of the new tree in 40 years? The
annuity interest factor is 60.402; this interest factor should be multiplied by
the $0.96 annual benefit to get the accumulated future value of the new tree.
(b) If the original tree had been left standing, it would
have sequestered $40 of carbon every year for each of those 40 years, plus an
additional $0.96 each year as the tree grew. Using the annuity formula again
(interest factor of 60.402) what would have been the accumulated value of
sequestered carbon for the original one tonover the 40-year period? This is one
possible measure of the loss of intergenerational equity. (It holds when we
assume uniform sequestration each year so that the value added by one year of
growth is the same whether we allow the original tree to live or cut it down
and replant a new tree.)

Question 3: Now suppose that we left the original tree
standing and are valuing the tree at some moment 40 years into the future. It
is now sequestering 2 tons of carbon each year at a value of $40/ton.
(a) Using the perpetuity formula at 2%, what is the present
asset value of the tree to the generation living 40 years from now?
(b) Using the rule of 72 and back-calculating from the fact
that the present value of the tree (to two different, successive generations)
doubled in 40 years, what is the approximate interest rate yielded by the tree
when it is allowed to live? (Divide 72
by the 40-yr doubling period to get the interest rate)

Your answer to 3b might be considered the ‘social cost’ of
cutting down the tree, greatly simplified of course and measured only in carbon
value. But when we calculate the private return on investment from timber as
product, this valuation gives us a rate that should be deducted when
considering the returns to the economy as a whole. So if the private return is
3%, the social return is only 1.2%.

[1] Livernois, John, Henry Thille, and Xianqiang Zhang, “A
Test of the Hotelling Rule Using Old Growth Timber Data,” Faculty Working Paper
2003-4, University of Guelph, Ontario, Canada (June 2003), pp. 1-25

[2] Baker, James A. III et al, “The Conservative Cost
for Carbon Dividends, Climate Leadership Council, www.clcouncil.org

[3]
https://www.ncsu.edu/project/treesofstrength/treefact.htm

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