1. Lecturer: Finn R. Førsund
ECON 4910 Spring Environmental Economics Lecture 8, The RAINS model Memorandum No 37/99
Lecturer: Finn R. Førsund
Briefly mentioned in Kolstad, Section 10.5 in Perman et al. RAINS is an European model
Important example of an environmental model being used for real policy decisions
Integrates an atmospheric transportation model, critical loads of pollutants in the environment, and purification cost functions at country level
RAINS

2. The cost function in RAINS
The emitting unit is a country, i
Purification costs for different sectors are agggregated in merit order
The emissions, eio, for a future year (2010) are based on projections of energy use and economic activities of key sectors
Costs change with projected emissions
Cost function are end-of-pipe, economic activities constant, power, transport, agriculture, chemicals
Cost function updated to the future year, stock of purification capital updated according to reinvestment cycles plus applying BAT,
Minimum feasible emission reflects both activity level and known BAT technology today
May simplify cost function to c(eo-ei)
RAINS

3. Purification measures with constant average cost
Marginal costs 3 2
Cost function in RAINS are piecewise linear, average cost equal to marginal cost for a certain purification technology, and the same for all sectors using this technology irrespective of country, but a certain adjustment due to differences in wage cost levels.
Measures may be choice of fuel types, like low sulphur coal, coke, oil, end-of-pipe like flue gas desulphurisation, limestone injection, and technology change like fluidised bed combustion 1
Purification r
max 1
RAINS

4. Merit order of purification measures
Marginal costs 3 2 1
Starting at eo and reducing emissions, measure 1 is chosen first, etc.
Sequence of measures no unique, some more expensive measures may exclude others,
cost function may depend on level of reduction
Emissions e
min e o
RAINS

5. The environmental block
Simplification to one pollutant, SO2
Deposition of SO2 into a grid cell from all sources (countries)
bj = background deposition
Average Accumulated Exceedance
RAINS

6. The optimisation problem
Minimise total purification costs subject to environmental targets for each grid cell and limits on emissions
AAE target determined as explained previously, negotiation about the x-factor
RAINS

7. The Lagrangian
Converting a minimisation problem to a maximisation problem
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8. First order conditions
The interpretation of the shadow price λj:
The increase in total cost if the environmental target AAE is made more ambitious,
a reduction of AAE* increases costs
an increase in AAE* decreases cost
NB! Only equality sign here in the first-order condition because emission is restricted to be between two positive numbers
The shadow price is zero if costs are not changed when environmental target is changed, may happen due to downwind configurations, receptors “protected” by targets for up-wind receptors.
RAINS

9. First order conditions, cont.
The interpretation of the environmental term
The “value” of the marginal increase in average accumulated exceedances in all receptors affected by a unit increase in emissions from country i expressed by positive transport coefficients
The physical change in AAE due to marginal transport from source i is valued to the shadow price om the environmental constraint, i.e. the cost of a marginal change in the target. This is not evaluation of environmental services proper.
RAINS

10. First order conditions, cont.
Both the shadow price on the upper and on the lower constraint on emissions cannot be positive at the same time.
If the constraints are not binding both shadow prices μi,γi will be zero. This is the interior solution
Upper constraint on emission is binding:
Marginal purification costs is greater than the shadow ”value” of the environmental effect at maximal emission
Upper constraint binding: insert upper constraint for ei
RAINS

11. First order conditions, cont.
Lower constraint on emission is binding:
Marginal purification cost is lower than the shadow ”value” of the environmental effect at minimal emission
Interior solution:
Marginal purification costs equal to total shadow-priced evaluation of contribution to accumulated exceedance
RAINS

12. Illustration of optimal solution
Marginal costs
S(emin) γ c’ c’
c’* c’ μ
S(.) is shadow-priced value of change in AAE
Three general cases
Emission on upper constraint, shadow value of AAE < c’
Emission on lower constraint, shadow value AAE > c’
Interior solution, shadow value AAE* = c’*
S(eo)
Emissions eo
emin e*
RAINS

13. Economic interpretations
Marginal purification costs differ between countries due to individual transfer coefficients
non-uniform reductions of pollutants
Shadow price for each receptor shows system marginal costs of deposition constraint
May have non-feasible solution due to fixed initial emissions and limited purification possibilities
maximal purification, reducing future projected emissions, or changing target loads
Chapter 9 in Kolstad
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14. Ambient tradable permits
Permits issued for each receptor, sources have to acquire pollution permits for each receptor affected by source
Finding the total target load for each receptor
Total ambient pollution quota for each receptor
Target for average accumulated exceedance can be transformed to target for totla deposition load, when issuing permits background deposition has to be taken into consideration
RAINS

15. Ambient tradable permits, cont.
Pollution quotas can be grandfathered, calculation of quotas ensuring that environmental targets are reached complicated
Source problem given price tj for pollution permit (dropping emission constraints)
RAINS

16. Ambient tradable permits, cont.
First-order condition for interior solution
Setting the price of pollution permits
The equilibrium price of pollutant permits in receptor j should reflect the shadow price on the receptor j specific constraint, multiplied with the marginal impact on AAEj
Realism: Over 5000 receptors, huge organisational task, feasible with country as source? Hardly, implying this market system is really not of practical interest
RAINS