1
ECON 4910, Exercise 15-16 March 2004
A simple version of the RAINS model
A solution to the following constrained optimisation problem is per definition cost effective:
(1)
Symbols:
= purification costs of country i
= emissions from country i
= lowest emission that is feasible
= the maximal emission that is feasible (= the projected emission for a future year)
= the unit transfer coefficient showing how much per unit of emission from source i ends up in receptor j
=background deposition in receptor j
= the environmental target in receptor j (greater or equal to the critical load)
,, = the country cost function build on the merit order aggregation principle of starting purification with the cheapest purification measures and ending up with the most expensive
The Lagrangian for problem (1) (see Sydsæter, Strøm and Berck: Economists’mathematical manual, Eq. 15.14):
(2)
Necessary first order conditions:
(3)
The shadow prices on the upper and lower constraints on emissions from a source cannot both be positive at the same time. If we are at the upper boundary i will be positive and i zero, and vice versa at the lower boundary. For an interior solution both are zero. Equation (3) must then hold with equality: The marginal cost of emitting one unit more (> 0) will then be equal to the value of the marginal contribution to all receptors with positive transfer coefficients priced at the shadow value on the environmental constraint for receptor j:
(4)
The shadow value is positive for binding constraints (environmental target is just kept) and zero when the actual deposition is less than the target.
Ambient standards
Regard a country i as a decision-making unit with cost minimisation as a strategy. A supra-national authority sets quotas for depositions in each receptor equal to the target loads . Each country is allocated quotas free of charge for every receptor, and we have:
(5)
The countries can trade the ambient permits, but each country must have sufficient permits for each receptor that receives emissions from the country. A price, exogenous for each country, is set for trading a unit of the ambient permit.
The cost minimisation problem for a country is:
(6)
where is the amount of ambient permits the country has after trade.
Assumption: A country is rational and will not sit with more permits than necessary implying:
(7)
The Lagrangian for problem (6) is then:
(8)
The first order condition is:
(9)
Assuming an interior solution () we have:
(10)
Comparing (4) and (10) we see that there may exist prices such that the social planning solution (4) is realised. But it is very difficult to device a mechanism that will guide the system to the optimal social solution with zero prices for depositions in recipients with non-binding constraints. In the RAINS model we have over 600 receptors. It is quite challenging to administer and control that the rules are obeyed in so many markets.