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ECON 4910, Lecture notes 3
Waste accumulation
Perman et al. (2003), Chapter 16 Stock pollution problems
Haavelmo (1971) introduced the accumulation of waste as an analogy to the concept of entropy in physics (second law of thermo dynamics). Emissions, Mt, are accumulated into the stock of pollutants, At:
(1)
The accumulated waste may decay in Nature due to natural processes. Assuming a “radioactive” decay with a constant coefficient yields the net accumulation :
(2)
where is the decay coefficient.
To make the most simple model we connect a benefit function, B(.) to the emissions:
(3)
and a damage function, D(.), to the accumulated waste:
(4)
Both functions may be measured in money.
The optimal planning problem with infinite horizon
(5)
The current value Hamiltonian:
(6)
The static first order condition:
(7)
The dynamic first order condition:
(8)
Interpretations
Rearranging (7) yields:
(9)
The shadow price on the stock of waste is negative; more waste reduces the objective function. At each point in time we have a balance between marginal benefit of emission and the damage of the accompanying increase in the stock of emissions from t to infinity of the unit of emission at time tmeasured by the shadow price on the stock of waste.
Inserting (9) into (8) yields:
(10)
The shadow price increases (decreases) when marginal damage of the stock is higher(lower) than the “gross interest” on the marginal benefit of emission. The gross interest rate is the sum of the interest rate and the decay coefficient, both working on the stock of waste.
Steady state
In steady state the stock of waste and the shadow price on the stock are constant. Inserting (9) into (10) we then have:
(11)
From the growth equation (2) for waste we have (dropping time subscript t):
(12)
In steady state the flow of emissions is restricted to the amount of decay taking place.
Eq. (11) gives us an interpretation of the shadow price on the stock of waste in steady state:
(13)
The right hand side is the present value of the marginal damage created from present time t to infinity of a unit of emission with the gross rate (effective rate) as the discounting factor. Finally, this present value of damages shall be equal to the marginal benefit in steady state:
(14)
We now have two equations to determine the variables A and M in steady state; (12) and (14). Substituting for the accumulated waste in the equilibrium condition (14) yields an equation in M:
(15)
Given the value of parameters and the functional forms we can then solve for the steady state M and then find A (from (12)).
To elucidate the nature of the steady state solution further we may differentiate the damage function in steady state, , w.r.t M:
(16)
Rearranging (15) and using (16) yield:
(17)
Special cases of parameter values
Case A:
There is no discounting of benefits and damages, and (17) collapses to:
(18)
This is the standard “static” condition of marginal damage equal to marginal benefit; steady state is “as if” we are solving a static model of flow pollution. The model is:
(19)
But there is also a stock interpretation. Using the steady state relation between current emission and the stock we can write:
(20)
The expression in the middle is the present value of the marginal stock damage where the decay rate acts as the discount factor.
Case B: and case D:
When α = 0 there is no decay and there cannot be a steady state unless emissions are zero. This is the original Haavelmo disaster case.
Case C:
Eq. (17) is then relevant. Comparing case A and C we see that the steady state level of emissions is increased with positive discounting, i.e. the steady state stock is larger; future damages are discounted more strongly.