The replacement cost method assigns a value to a cleaning technology only if this technology is less costly than other abatement measures for reaching certain emission targets in a cost effective solution. The cost effective solutions are, in turn, determined by a numerical model for minimizing total costs of achieving carbon and nutrient emission targets in a certain time and space. The construction of such a model is thus crucial for the determination of the value of forest ecosystem services in terms of carbon and nutrient sequestration. In the following, we therefore first provide a more intuitive presentation of the replacement cost method, which is followed by a description of the structure of the cost effectiveness model that is used to derive conditions for a positive value of forest carbon and nutrient sequestration and the magnitude of this value.

#### 2.1. Conceptual Framework of the Replacement Cost Method

The value of the services of land use in terms of carbon and nutrient sequestration is determined by their cost in relation to the costs of other abatement measures in a cost effective achievement of certain emission targets. The higher the cost of other abatement measures, the larger is the value of the ecosystem service. This simple principle for determining value is illustrated in

Figure 1 for carbon sequestration.

**Figure 1.**
Illustration of the calculation of the value of land use as a carbon sink in a cost effectiveness framework. SEK: Swedish crown; K^{T}: carbon emission reduction target.

**Figure 1.**
Illustration of the calculation of the value of land use as a carbon sink in a cost effectiveness framework. SEK: Swedish crown; K^{T}: carbon emission reduction target.

The horizontal axis illustrates carbon emission reductions, and K^{T} is the target to be achieved. The vertical axis shows the cleaning cost for different emission reduction levels. The curve C illustrates the minimum costs for achieving different emission reduction targets when the carbon sink provided by forests is not included as an abatement option in the cleaning program, and C^{K} illustrates the minimum costs when the carbon sink is included. Each point on C and C^{K} respectively reflects the allocation of all abatement measures that reach the target at minimum cost.

The value of carbon sequestration at the target

K^{T} is now determined by the difference in total minimum costs with and without carbon sequestration, which corresponds to the distance

C-C^{K} in

Figure 1. The value of the carbon sink is then determined by its construction cost and the abatement costs of other measures. The larger the difference between the abatement costs of other measures and that of the carbon sink, the higher the value of the carbon sink. This is, in turn, determined by the stringency in the carbon target since the costs of all abatement measures increase at higher emission reduction levels.

The value of land use for nutrient sequestration is calculated in the same way as for carbon sequestration. However, the value of both these sequestration options depends on whether carbon and nutrient targets are managed separately or simultaneously. When managed separately without consideration of the other pollutant target, the full cost of land use will be borne by the reduction in the pollutant in question. Under simultaneous management, land use measures will have a cost advantage compared with measures affecting only one pollutant and will therefore be implemented to a larger extent than under separate target management.

#### 2.2. Description of the Dynamic Cost Minimizing Model

The model used for calculating the value illustrated in

Figure 1 builds on the dynamic model developed by [

10], and adds water emission targets. Cost effectiveness is then defined as the allocation of abatement measures over time and at different locations, which reaches predetermined targets in a specific time at minimum costs. The location of carbon emission reductions and sequestration do not affect the impact on the atmosphere. This is in contrast to nutrient abatement and sequestration, where the location matters for the impact on the coastal waters because of the retention of nutrients from the source to the waters. We therefore divide the region into

m = 1,..,n drainage basins.

In each time period

t, the net business as usual (BAU) emission of pollutant

p, where

p is carbon dioxide equivalents (CO

_{2}e), nitrogen (N), and phosphorus (P), in drainage basin

m is determined by the emission of sources minus existing sequestration,

${X}_{t}^{pmBAU}$. These emissions can be reduced by decreases in the pollution at the emission sources,

${a}^{pmf}\text{}{A}_{t}^{pmf}$ where

f = 1,..,l are the different types of sources such as energy production, agriculture, and transport, and

a^{pm} is the impact on the target. This is unity for CO

_{2}e reductions since location of source does not matter for the impact, but 0 <

a^{pm} ≤ 1 for nutrient since some sources are located in upstream drainage basins where the effect is below unity. For emission sources located at coastal waters, such as sewage treatment plants, the discharge reduction at the source is the same as the load reduction to the sea, which implies that

a^{pm} = 1. Annual reductions in CO

_{2}e are also obtained by existing renewable technologies,

${A}_{t}^{pmr}$, where

r = 1,..,q technologies. These are obtained from investments,

${I}_{t}^{mr}$, which have a maximum technological life time, and are then regarded as capital investments subject to depreciation according to

where

a^{pmr} is the reduction in pollutant

p of one unit abatement by technology

r, and 0 <

δ^{r} < 1 is the annual depreciation rate of technology

r. Abatement by means of a renewable technology

r in time

t is thus determined by the accumulated number of investments prior to

t, and investment in period

t.

The third abatement option is the creation of the carbon and nutrient sink,

${L}_{t}^{ms}$, which can be formed by afforestation. However, it may take some time before the full potential of the sink is achieved. This growth depends on biomass growth and soil processes, which, in turn, are determined by type of land use change. Since there are few data on these processes, we assign a simple function where the rate of growth, δ

^{s}, is constant over time and carbon and nutrient sequestration achieve a maximum. Recall that the pollutant sink in each time period is subject to stochastic weather conditions. We therefore assign the carbon and nutrient sink in a given period

t, ${A}_{t}^{pms}$, as dependent of land use changes in prior periods,

${L}_{\tau}^{ms}$, and an additive stochastic parameter,

${\epsilon}_{t}^{pms}$, which is written as

where

q^{pms} is the maximum sequestration per unit of land area of pollutant

p. The larger the

t − τ value, the higher the

${A}_{t}^{pms}$ for a given

${L}_{\tau}^{ms}$. Similar to renewable technologies, total pollutant sink in a municipality is then determined by the accumulated carbon sink additions. The difference is that sink capacity is increasing and capacities of renewable technologies are decreasing over time.

Total net emissions of each pollutant,

${X}_{t}^{p}$ are then the sum of emissions in BAU minus emission reduction obtained by the three classes of measures in each drainage basin:

The exercise of each abatement measure is subject to a cost, which is described by the cost functions ${C}^{pmf}\left({A}_{t}^{pmf}\right)$, ${C}^{pmr}\left({I}_{t}^{pmr}\right)$ and ,${C}^{pms}\left({L}_{t}^{pms}\right)$ which are assumed to be increasing and convex in their arguments, i.e., costs are increasing at a higher rate in the use of the measure.

Additional assumptions are that abatement in each period is subject to restrictions where only part,

b^{f}, of the BAU emissions can be reduced, part of total emissions from fossil fuel or nutrient use can be replaced by a specific technology energy,

b^{r}, in each period, and that part of agriculture land under BAU,

b^{s}, can be converted into forests in each period. The reasons for these restrictions are that drastic reductions in emissions and land use are difficult to implement in a very short time period. The capacity restrictions are then written as

The decision maker at the Stockholm-Mälar region is now assumed to apply a safety-first approach in reaching targets on total emission where they formulate a minimum probability or reliability level,

α^{p}, of achieving the maximum emission target,

${\overline{T}}_{t}^{p}$, for each pollutant, which is written as

This can be expressed in terms of mean emissions or leaching,

${\mu}_{{T}^{p}}^{p}$, risk aversion,

${\varphi}^{{a}^{p}}$, and variance in loads

${\sigma}_{{T}^{p}}^{p}$ as [

22]

where

${\mu}_{t}^{p}=E\left[{X}_{t}^{p}\right]$ with

E as expectation operator;

${\varphi}^{{a}^{p}}$ shows the choice of α

^{p} as the acceptable deviation of the load from the mean. The level of

${\varphi}^{{a}^{p}}$ depends on the shape of the probability distribution, which is discussed in

Section 3. The left hand side of Equation (8) thus shows that reliability in achieving the target is obtained at a cost, which increases with reliability concern or the probability of achieving the target,

i.e., in

${\varphi}^{{a}^{p}}$, and in

${\sigma}_{t}^{p}$.

Recall from

Section 2.1 that the value of

${L}_{\tau}^{ms}$ is calculated as the difference in cost for achieving the targets in Equation (7) with and without the inclusion of

${L}_{\tau}^{ms}$ as an abatement option. Given Equations (1)–(8), the decision maker is then assumed to choose among available options,

${A}_{t}^{pmf}$ and

${I}_{t}^{mr}$, or

${A}_{t}^{pmf}$,

${I}_{t}^{mr}$, and

${L}_{\tau}^{ms}$, in order to minimize total cost in present terms for achieving the carbon emission and nutrient leaching targets. When all options are available, this is written as

s.t. Equations (1)–(9)

where ${\rho}^{t}=\frac{1}{{(1+i)}^{t}}$ with i as the discount rate.

#### 2.3. Determinants of the Value of Carbon and Nutrient Sink in Forests

The determinants of the magnitude of the value of forests as pollutant sinks can be found by solving for the decision problem in Equation (9). We solve for the cost effective solution by formulating the Lagrange expression,

L, which is written as

where

${\lambda}_{{T}^{p}}\le 0$ is the Lagrange multiplier for the targets, and

${\gamma}^{pmf}\le 0$,

${\gamma}^{pmr}\le 0$, and

${\gamma}^{pms}\le 0$ are those for the different capacity constraints on abatement measures. The Lagrange multipliers have an interesting interpretation; they provide information on the change in total discounted minimum cost for a relaxation of the constraint. For example, a relaxation of the carbon emission target in 2050 by one unit will decrease the cost corresponding to

${\lambda}_{T}^{p}\le 0$ where

p is CO

_{2}e.

We derive the conditions for a positive value from the first-order condition of a cost-effective solution,

i.e., the marginal abatement cost for obtaining a unit reduction in the targets shall be equal for the dynamic and spatial allocation for all measures and equal to

${\lambda}_{{T}^{p}}^{p}$. For ease of exposition but without loss of generality, we assume interior solutions where the capacity constraints are not binding. The first-order conditions for the cost effective solution are then written as

All three conditions state that the marginal abatement cost in the present value, i.e., the discount factor times the marginal cost, on the left hand side shall equal the weighted sum of marginal impacts on the different targets on the right hand side. The Lagrange multipliers, ${\lambda}_{{T}^{p}}^{p}$, constitute the weights that reflect the marginal cost of changing the respective target by one unit. A measure has a cost advantage for a relatively low marginal abatement cost and high weighted impact.

Starting by comparing the cost of abatement of sources and investments in renewable energy and nutrient cleaning, i.e., Equations (11) and (12), we can see that investment has a marginal cost advantage for a low depreciation rate, ${\delta}^{pr}$, and long time period, T^{p} − t, when the marginal decrease in investment, a^{pmr}, acts. Next we compare the marginal cost of reaching the target by investment in cleaning technologies with increased forest area, i.e., Equations (12) and (13). Disregarding uncertainty for the moment, we can then see that the larger the pollutant sink per area of land, a^{pms}q^{pms}, the higher the impact on the respective target. This is also the case for the accumulated sink over time, $\left(1-{(1-{\delta}^{pms})}^{{T}^{p}-t}\right)$which is increasing because of the growth of the planted forest trees. A comparative advantage with pollutant sink compared with investment in renewable energy and nutrient cleaning is then the growth over time in the sink capacity, instead of a decline, which needs replacement of new technology when the old is worn out. This cost advantage is counteracted when we consider the uncertainty term, $\frac{{\phi}^{ap}}{2}\frac{\partial {\sigma}_{t}^{pms}}{\partial {L}_{t}^{ms}}$, which instead reduces the marginal impact when uncertainty in the pollutant sink is increasing in the area of afforestation and, hence, increases the marginal cost for reaching the target.

We can also note that the marginal impact increases for all three types of measures when more than one target is affected. A marginal increase in the forest area, which affects all three targets, then has a cost advantage when the other measures affect only one or two targets. The relative advantage of growth in sink per unit of land is then enhanced. On the other hand, the disadvantage of the uncertainty discount is also increased. When the advantages exceed the disadvantages, the value of the services is also determined by simultaneous or separate management of the targets. In practice, each pollutant target is treated separately, at least in Sweden. This means that the advantages of multifunctional measures are not fully utilized since their cost advantages of impacts of several targets are not accounted for.

Based on this simple dynamic model with uncertainty in carbon and nutrient sinks, we can thus conclude that investment in forest ecosystem services has an advantage over other abatement and investment measures when

- -
the cost of forest plantations is relatively low

- -
the weighted marginal impact of forest on carbon and nutrient sequestration is high

- -
the annual growth in sink capacity and depreciation of investment in other technologies are large

- -
the uncertainty discount of the pollutant sink is low

- -
the management of all targets occurs simultaneously instead of separately

As will be shown in the following, these factors can have a considerable effect on the calculated value of afforestation in the Stockholm-Mälar region.