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L(m,p) 5 oi
[ f
i
(mi
) 2 pmi
],
where p is the Lagrangemultiplier chosen arbitrarily
(for the moment), and maximize L(.) over choices
of m (unconstrained). The outcome of this process
generates some ‘‘demands’’ for oil that may or may
not satisfy the resource constraint. If too much is
demanded, then p is raised in the next stage and vice
versa. The process iterates until the constraint is just
met. This process of simple steps repeated recur-
sively is just what a computer likes to do, which is
why these type of procedures are efficient computa-
tionally. Of course, the process may never exactly
converge, but this turns out not to be a problem in
practice.
It is clear that multiplier p is playing the role of a
price. At the maximization stage, each firm maxi-
mizes ‘‘profit’’ where cost is computed using p as the
input price. Furthermore, the adjustment stage is
just like a market equilibration process where p is
adjusted to equate supply and demand. Thus, it is
legitimate to think about the planning problem as
one of determining the correct shadow price of oil
and then choosing the allocation as private firms
would, facing that price. Note that (under standard
convexity assumptions) the outcome here is the
same as we would get by solving simultaneously for
m and p in the equation system:
›L
›mi
(m,p) 5
›f
i
›mi
(mi
) 2 p 5 0, all i, oi
mi
5 Z
This is what one normally does in an analytic
version of the Lagrangian method.
This logic generalizes to any constrained optimiza-
tion problem. Such problems can always be solved
by assigning shadow prices (multipliers) to each
constraint, maximizing the Lagrangian (generalized
social profit) and adjusting prices to meet the con-
straints. So thinking in terms of prices is helpful in
solving optimization problems even when the as-
sumptions of competition do not hold and public
intervention is essential. We next sketch how this
logic can be applied to analyze issues of dynamic
environmental asset management.
EXHAUSTIBLE RESOURCES
Suppose we think about ‘‘oil in the ground’’ as an
asset; our problem is to spread out the use of this
asset over time in an efficient (optimal) manner.
Now that we are in themacro world ofmanaging an
entire stock, we must expect that shadow prices will
be endogenously determined and will vary system-
atically over the life of the asset. At any moment of