capital curve, by V
t
, to K2. If the resource is grandfathered so that F
t
= 0, this results in an
equilibrium with higher interest rates, depicted by E2. If there are publicly held assets F
t
, these increase the saving curve S1 to curve S2 or S3. If
future generations receive claims for an unspoiled environment, so that environmental damages are
compensated, this implies that the publicly held assets exceed the value of the partly spoiled
resource, F
t
\
V
t
, and thus, the equilibrium moves to E3. We will thus expect that grandfathering
raises the interest rate, whereas a trust fund re- duces the interest rate. This will become clear
from a comparison of scenarios I and II with III and IV, respectively.
4. Numerical illustration
Below, a number of figures are presented for the scenarios. Fig. 2 depicts the interest rate,
defined for the years between two periods starting at 2020. Fig. 3 and Fig. 4 present the flow vari-
ables CO
2
emission price and net CO
2
emission, each describing a flow during the periods 2000 –
2020, 2020 – 2040 and so forth. In these graphs, the periods are identified with the corresponding
central years, 2010, 2030, etc. Fig. 5 and Fig. 6 illustrate the state variables CO
2
concentration and global mean temperature, defined for the
years between two periods: 2000, 2020, etc. Fig. 2 shows the interest rate evolution for each
of the four scenarios. Three results require special attention. All three results confirm the inferences
drawn in Section 3 from Fig. 1. First, comparing the interest rates between scenarios I + II and
III + IV in 2020, we see that grandfathering the environmental resource implies initially an interest
rate of close to 2 percent per year higher than in the trust fund case. Secondly, the intergenera-
tional distribution of the property rights over the environmental resource has an even stronger im-
pact on the long-term value of the interest rate. Sharing the property rights with future genera-
tions, instead of grandfathering it, decreases the interest rate from 7.3 to 2.9 percent per year in
2200 scenario I vs. III, when the population is not assumed to age. Including the demographic
Fig. 2. Interest rate for the period 2000 – 2200.
change, we find that sharing the property rights decreases the interest rate from 2.5 to 0.5 percent
per year in 2200 scenario II vs. IV. Thirdly, comparing the scenarios with and without ageing,
one concludes that ageing decreases the future interest rate substantially, from 7.3 to 2.5 percent
in 2200 under grandfathering scenario I vs. II, and from 2.9 to 0.5 percent in 2200 scenario III
vs. IV if property rights are shared with future generations. The figure reveals the importance of
both demographic changes and environmental policies for the long-term evolution of the interest
rate. To compare our results with the results obtained with a typical dynastic model, we note
that in a dynastic model the assumed slow-down of per capita economic growth from 2 percent per
year in 2000 to zero in 2200 would cause a decrease in the interest rate of about 3 percent per
year: Dr:gDg, where g=1r=1.5 and Dg=2 percent per year. Except for the first scenario, all
other scenarios show a much stronger decrease in the interest rate than the levelling of economic
growth can account for. Overall, the figure unde- niably reveals that assumptions on demography
and environmental policies substantially affect the interest rate, contrary to what is suggested in the
literature Stephan et al., 1997; Manne, 1999.
Not considering ageing, grandfathering leads to low emission prices, reaching 100 UStC in 2200
see Fig. 3. If we include ageing, the interest rate decreases, and the carbon emission price slowly
increases from nearly zero in 2000 – 100 UStC in 2100 and 400 UStC in 2200. Scenario I can be
considered as being the closest to the business as usual, or benchmark, scenario. The CO
2
emissions
Fig. 3. CO
2
emission price for the period 2000 – 2200.
of scenario I because of the ageing of the popula- tion: the resulting increase in life-cycle savings
and the associated increase in the man-made capi- tal stock produces more carbon dioxide than in an
economy without ageing. In the 22nd century, however, the small effect of this demographic
change is dominated by the increasing difference in the carbon dioxide price of these two scenarios.
The steady rise in emission prices in scenario II outweighs the increase in emissions resulting from
economic growth. This leads to a rapid decrease of emissions in scenario II, whereas in scenario I
only a modest emission reduction takes place at the end of the 22nd century. In scenario II, cumu-
lative emissions from today up to the year 2100 amount to about 1450 GtC, which is comparable
with the corresponding IPCC IS92a scenario IPCC, 1992. By the year 2200, net emissions will
have attained the level of zero.
Altering the analysis from a ‘victims pay’ to a ‘polluters pay’ perspective, by the introduction of
a trust fund, substantially shifts upwards the car- bon dioxide emission price curve. In the case of
the ageing scenario, the increase is already visible in the first period, whereas it becomes apparent in
the second half of the 21st century for the no ageing case. The inclusion of ageing has a consid-
erable effect on the emission price evolution. If no ageing is considered, the trust fund emission price
remains neatly between the two grandfathering scenarios over the entire time lapse studied. This
applies as well to the emission levels, insofar as the 22nd century is considered. If ageing is in-
cluded in the calculations scenario IV, the emis- sion price increases rapidly to 400 UStC in less
than a century. Scenario IV does not limit the expansion of net emissions in the medium term
up to 2030, leaving present generations the pos- sibility to adapt and to develop alternative energy
sources.
After 2050
net emissions
decrease rapidly, and by 2080 a complete substitution of
fossil fuel energy carriers by carbon-free energy sources has taken place.
As can be seen from a comparison between Fig. 4 and Fig. 5, CO
2
concentration levels in the atmosphere lag behind CO
2
emission patterns. In turn, an inspection of Fig. 6, depicting the global
average temperature increase relative to the pre- in this scenario see Fig. 4 are roughly stable on
the time scale considered. An autonomous in- crease in energy efficiency AIEE and an au-
tonomous expansion of backstop technologies, assumed in the benchmark scenario see the para-
graph directly following Eq. 11, mainly explain this approximately stable emission level. The rela-
tively small reduction in emissions close to the year 2200 is induced by the expected emission
price increase around that date for this scenario. It should be understood, however, that these esti-
mates are dependent on the benchmark assump- tions made. Chakravorty et al. 1997, for
example, assume an autonomous decline of fossil fuel energy use before 2100 because of competitive
solar energy supply.
Let’s compare the two grandfathering scenarios in which the inclusion of ageing provokes a de-
crease in the interest rate. In the early periods, net emissions of scenario II exceed slightly the levels
Fig. 4. Net anthropogenic CO
2
emission for the period 2000 – 2200.
Fig. 5. Atmospheric CO
2
concentration for the period 2000 – 2200.
5. Conclusion
