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M. E. Belfiori
The central element in the social optimum is the social cost of carbon. The social
cost of carbon is the cost that climate change imposes on individuals and the economy.
Formally, it is given by the following expression
μ
∗
t =
∞
j=0 [β (1 − δ)]
j u
(C
∗
t+ j )x
(S
∗
t+ j )F t+ j
u (C
∗
t )
(11)
Equation (11) states that the social cost of climate change is equal to the present
value of the output losses associated with a marginal increase in carbon emissions.
Emissions remain in the atmosphere for 300 years, on average, and some are everlasting (Archer 2005). Thus, damages will happen over time, and the social cost of
climate change is equal to the present value of the future losses.
Efficient Energy Use
Another critical question concerns the efficient use of alternative energy sources.
How much should the economy rely on renewable versus non-renewable energy?
There is a simple rule that governs the optimal use of non-renewable energy.
This is the Hotelling rule (Hotelling 1931), and it is implied in Eq. (12). It says that
the marginal benefit from oil extraction must be the same at any moment in time.
If benefits are high in the future, it is better to keep oil underground and extract it
later. The opposite holds if future benefits are low. In this case, it is better to extract
more oil today. Therefore, an acceleration in oil extraction may be observed if oil
companies foresee a drop in future benefits.
The benefit of oil extraction is that oil allows the production of consumption goods.
This is the first term on the right-hand side of Eq. (12). Also, carbon emissions arise
from oil extraction, and these costs must be net out from the benefits.
βu
(C t+1 )[F
e 1 ,t+1 − μ
∗
t+1 ] = u
(C t )[F
e 1 ,t − μ
∗
t ]
(12)
On the other hand, if there is a deferment of oil extraction, higher consumption
will result in the future, and there will also be a delay in the occurrence of climate
damages. This is the left-hand side of Eq. (12). Overall, the Hotelling rule solves the
optimal depletion rate of global oil reserves.
The efficient use of renewable energy requires that the marginal rate of substitution between consumption and leisure (and work effort) equals the marginal rate of
transformation. It is the standard condition for economic efficiency. It balances out
the economic benefits and costs of producing renewable energy. Also, the efficient
use of resources requires that the marginal productivity of factor inputs is the same
across productive sectors. For renewables, in particular, this implies that
F
e 2 ,t f
n,t = F
n,t
(13)
M. E. Belfiori
The central element in the social optimum is the social cost of carbon. The social
cost of carbon is the cost that climate change imposes on individuals and the economy.
Formally, it is given by the following expression
μ
∗
t =
∞
j=0 [β (1 − δ)]
j u
(C
∗
t+ j )x
(S
∗
t+ j )F t+ j
u (C
∗
t )
(11)
Equation (11) states that the social cost of climate change is equal to the present
value of the output losses associated with a marginal increase in carbon emissions.
Emissions remain in the atmosphere for 300 years, on average, and some are everlasting (Archer 2005). Thus, damages will happen over time, and the social cost of
climate change is equal to the present value of the future losses.
Efficient Energy Use
Another critical question concerns the efficient use of alternative energy sources.
How much should the economy rely on renewable versus non-renewable energy?
There is a simple rule that governs the optimal use of non-renewable energy.
This is the Hotelling rule (Hotelling 1931), and it is implied in Eq. (12). It says that
the marginal benefit from oil extraction must be the same at any moment in time.
If benefits are high in the future, it is better to keep oil underground and extract it
later. The opposite holds if future benefits are low. In this case, it is better to extract
more oil today. Therefore, an acceleration in oil extraction may be observed if oil
companies foresee a drop in future benefits.
The benefit of oil extraction is that oil allows the production of consumption goods.
This is the first term on the right-hand side of Eq. (12). Also, carbon emissions arise
from oil extraction, and these costs must be net out from the benefits.
βu
(C t+1 )[F
e 1 ,t+1 − μ
∗
t+1 ] = u
(C t )[F
e 1 ,t − μ
∗
t ]
(12)
On the other hand, if there is a deferment of oil extraction, higher consumption
will result in the future, and there will also be a delay in the occurrence of climate
damages. This is the left-hand side of Eq. (12). Overall, the Hotelling rule solves the
optimal depletion rate of global oil reserves.
The efficient use of renewable energy requires that the marginal rate of substitution between consumption and leisure (and work effort) equals the marginal rate of
transformation. It is the standard condition for economic efficiency. It balances out
the economic benefits and costs of producing renewable energy. Also, the efficient
use of resources requires that the marginal productivity of factor inputs is the same
across productive sectors. For renewables, in particular, this implies that
F
e 2 ,t f
n,t = F
n,t
(13)