By how much must carbon dioxide emissions be reduced to have a significant effect on atmospheric CO

Prior questions are whether increased atmospheric CO

Similarly, the effect of fossil fuel emissions on atmospheric CO

Using data, for example from the Earth Policy Institute, it is a relatively simple but instructive task to pursue and extend these investigations and to consider their implications in the light of policy objectives.

First, Harding's results are replicated here, using similar data from 1871 to 2008. The equation, with estimated coefficients, is as follows:

C

(0.18) (0.0014) (standard errors in parentheses)

where C

**Figure 1. Atmospheric CO _{2}
(ppm)
vs
cumulative emissions (gt of carbon)
**

Note that, as with many scientific publications, carbon dioxide
is
measured here in terms of gigatonnes of
carbon, not CO_{2}. The results,
plotted against time, are shown in Figure 2.

**Figure 2. Atmospheric CO _{2}
(ppm)
- cumulative emissions equation**

If the relationship is estimated in different units,
with both variables in the same units, the same basic
result is obtained, but the emissions coefficient can be more
usefully interpreted. The result is:

C =
.215653E+07
+ .546196 E^{ct} R^{2}
= .996 (Equation 2)

(1332.8) (0.0029) (standard errors in parentheses)

where C is atmospheric CO(1332.8) (0.0029) (standard errors in parentheses)

However a criticism of all these results can be mounted because
both
concentration and cumulative emissions contain strong time trends so
that the estimated relationship contains a spurious correlation. A
better estimate may be obtained by comparing changes in concentration
with annual emissions, i.e. by taking first differences in the
variables. If this is done, with the latter equation, the results are
as follows:

(139.4) (0.011) (standard errors in parentheses)

where C_{t} - C_{t-1}
is change in atmospheric CO_{2} in
million tones of CO_{2} and E_{t}
is annual CO_{2} emissions
from fossil fuels. A graph of these
results is shown in Figure 3. In this case a slope
coefficient of .48 is obtained
for the relative magnitude of additions and emissions. That is, in this
first-difference formulation, atmospheric increases are slightly less
than half of emissions whereas in the levels formulation they were
slightly more than half. The explanation for this difference is that
the levels formulation gives relatively more weight to the later period
observations, when the atmospheric increase is proportionally greater.
It
implies that emissions are increasingly being retained in the
atmosphere.

Figure 3.
Atmospheric CO_{2}
increases vs annual emissions (million
tonnes CO_{2})

The difference equation can be rearranged to dynamically
recompute the
level of atmospheric CO_{2} as
follows:

C_{t}
= C_{t-1 + }698.557
+ .484044 E_{t}
(Equation 4)

**Figure 4. Cumulative
CO _{2} increases - e**

The
inflow and
outflow of carbon dioxide from the atmosphere can be thought of as a
"carbon budget". An estimate of the
global carbon budget has been provided by the Global
Carbon Project for the years 1959-2008. The data are provided in
accordance with the following identity:

C_{t} - C_{t-1}
= E^{f}_{t} + E^{l}_{t}
- A^{o}_{t}
- A^{l}_{t}
- R_{t}
(Equation 5)

where C_{t}
- C_{t-1} is change in atmospheric CO_{2},
E^{f}_{t} is annual CO_{2}
emissions from fossil fuels consumption, E^{l}_{t}
is annual emissions from land use changes, A^{o}_{t}_{
}is absorption by ocean sinks, A^{l}_{t}_{
}is absorption by land sinks, and R_{t}_{
}is the residual. A graph of the data in Equation 5 is
shown in
Figure 5.

**Figure 5.
Carbon
budget (gigatonnes of carbon)
**

All variables are subject to measurement error, and as can be
seen from the graph, some are quite volatile, particularly land
absorption. This volatility is reflected in the atmospheric estimates.
The residual is the difference between atmospheric changes and
sources minus sinks. Since the atmospheric estimates and fossil fuel
emissions are subject to the least measurement error, it seems
reasonable to simplify the analysis by using these to form an estimate
of net absorption. The equation then becomes:

where "net absorption", A^{n}_{t},
represents (A^{o}_{t}
+ A^{l}_{t}
+ R_{t} - E^{l}_{t}), and
may be
calculated as E^{f}_{t}
- (C_{t} - C_{t-1}). As can be
seen from Figure 5, land use emissions and ocean absorption
approximately cancel out, so that net absorption is comprised mainly of
land absorption and the residual. The
resulting data for implied net absorption are shown in Figure 6,
together with the other variables of Equation 6.

The graph shows a clear upward trend in absorption, with absorption, over this time period, being generally slightly less than half of emissions, and atmospheric changes slightly more than half. A further advantage of this formulation, in terms of net absorption, is that for emissions and concentration, much longer time series are available. See for example the Earth Policy Institute. A graph of this data. in units of million tonnes of CO

It seems reasonable to assume that, other things being equal, the amount of absorption may be proportional to atmospheric CO

The linear equation is as follows:

A

(1387.63) (0.00059) (standard errors in parentheses)

where A_{t} is (net) absorption
of CO_{2} in
million tonnes and C_{t}
is atmospheric CO_{2}, also
in million tonnes. The coefficient of 0.02 indicates that approximately
2 percent of atmospheric CO_{2}
is absorbed each year.

The quadratic equation is

A_{t}
= -120729
+ .081122 C_{t}
+
-.117436E-07 C_{t}^{2}
R^{2} = .930
(Equation 8)

(18587.2) (.015393)
(.316572E-08)
(standard
errors in parentheses)

where in this case, C_{t}^{2}
is atmospheric CO_{2}
squared. The estimation results of these
equations are shown in Figure 8. While both these equations
may provide similar results over the
historical period, it may be expected that CO2EmAbConc.htmthey would behave quite
differently in forecasting.

**Figure 8.
Absorption of CO _{2}
vs atmospheric **

When plotted against time, both
equations show
a similar degree of precision, as shown in Figure 9, where atmospheric
CO2 is calculated in ppm for comparison with Figure 2.

E

emissions, absorption and atmospheric CO

Figure 10. Projected CO

Not surprisingly, the linear and quadratic absorption equations differ markedly as atmospheric CO

For comparison, a projection of concentration based on the emissions only equation, as in Figure 4 (converted to ppm) is also shown in Figure 11. As can be seen, to achieve stability in atmospheric concentration by 2050 requires both that emissions are reduced and that absorption remains in a linear relationship with concentration. Given limitation in fossil fuel resources, the rate of increase in emissions will inevitably decline. Whether absorption can be maintained is a crucial question. There is no intrinsic reason why absorption should follow a quadratic path, as this is only a mathematical construct. However it is doubtful that the linear trajectory can be maintained, so the eventual outcome may be somewhere between the two. Exactly where will be a critical question in the latter part of this century.

What does this mean for temperature? As calculated previously, a one-degree rise in temperature may be expected with a 100 ppm rise in CO

The objective of stabilising temperature must be considered in relation to both emissions and absorption. Emissions will be reduced by depletion of fossil fuel resources and by higher energy resource prices, no matter what emissions mitigation policies are adopted, or not adopted. Land use policies that are conducive to maintaining absorption are a critical ingredient of a solution. The only other alternative will be some other form of geo-engineering.

John L Perkins is a Senior Economist at the National Institute of Economic and Industry Research, Melbourne, Australia.