We now assume that the parameter is not known with certainty. Instead, we represent it as a sum of its mean value and a random term distributed normally with zero mean value and standard deviation ():

, (9)

. (10)

By substituting the decomposition (9) into Eq. (8) we get

(11)

Where

, (12)

. (13)

It follows immediately from Eq. (11) that the mean value of capital at time is equal to

(14)

Where

. (15)

By substituting Eqs. (10) and (13) into Eq. (15) and taking the resultant integral we get

. (16)

It follows then from Eqs. (12), (14), and (16) that

. (17)

We now turn to calculation of standard deviation of (). It follows from Eq. (11) that the variance of is equal to

(18)

Where

(19)

is the variance of . The second term in the r.h.s. of Eq. (19) (i.e. ) can be obtained by squaring Eq. (16). Regarding the first term in the r.h.s. of Eq. (19) (i.e. ), we note from Eqs. (13) and (15) that it is provided by the same formula as (i.e. by Eq. (16)) if in the latter the parameter is replaced by :

. (20)

Finally,

, (21)

and, from Eqs. (12), (17)-(18), and (21)

. (22)

Numerical examples and discussion

To provide several numerical results, we adopt the following values of model parameters: 0.02 year-1 (i.e. 2 percent per annum); 0.05 year-1 (i.e. 5 percent per annum); 0.2(°C)-1 (i.e. the depreciation rate doubles if mean temperature rises by 5°C); 0.03°C/year (that corresponds to 3°C warming per century); 0.01°C/year.

As follows from Eq. (8), the (time-dependent) growth rate

(23)

, (24)

so it decreases linearly in time and eventually becomes negative. At the same time, according to Eq. (17), the time-dependent growth rate of mean value of capital in the case of uncertainty

(25)

is equal to

, (26)

and it increases at large times. The graphs of and are shown on Fig.

According to Eq. (8), in case of no uncertainty the time dependence of capital itself is given by a Gaussian curve: it first increases, reaches its maximum at and then starts decreasing, dropping to the initial value at and ultimately converging to zero at infinite time. Here

, (27)

and for numerical values of parameters specified above is equal to 133 years. The graph of normalized capital is shown on Fig. 2.

Finally, the graph of normalized mean value of capital in case of uncertainty is shown on Fig. 3. As seen on the figure, it rapidly increases at large times.

So why the solution in case of no uncertainty is so different in the long run from the average solution in case of uncertainty? The reason is in that the values of a random variable from a `symmetric' couple (, ) contribute asymmetrically (`unevenly') to the resultant moments: lower-than-average temperature growth rates contribute more than higher-than-average ones. This means that the resultant average solutions are shifted from high-end scenarios to low-end ones when climate change matters less, and therefore the average solution in case of uncertainty is overly much more optimistic in the long run that the solution in case of no uncertainty.

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