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,
II
A DYNAMIC OPTIMIZATION MODEL OF
DEPLETABLE RESOURCES
by
Eduardo M. Modiano
and
Jeremy F. Shapiro
OR 082-78
August 1978
Supported in part by the U.S. Army Research Office (Durham) under contract
No. DAAG29-76-C-0064.
r
Abstract
A dynamic optimization model is presented to describe the normative
behavior of a supplier holding a depletable resource.
The supplier sells
the resource to an economic sector so as to maximize the present value of
his IioldIings over a finite planning horizon.
are discussed briefly.
A number of extensions
An implemented version of the model to the dynamic
allocation of coal supply to the U.S. energy sector is also presented.
w
A Dynamic Optimization Model of Depletable Resources
by
Eduardo M. Modiano
and
Jeremy F. Shapiro
1, Introduction
The model developed in this paper considers the normative behavior
of a supplier holding a single scarce primary resource who sells the
resource to an economic sector so as to maximize the present value of
his net income over a given, finite planning horizon.
The gross revenue
he receives in each period for the resource supplied to the economic sector is derived from the cost savings realized by the sector in using the
resource to meet end-use demands.
The model is a simplified abstraction
of reality but it has provided interesting numerical results when applied
to the dynamic allocation of coal supply to the U.S. energy sector.
More-
over, the simplified model permits us to clearly develop some new ideas
about modeling the depletion of scarce resources and indicates extensions
to richer, more complex models using these ideas.
Although some extensions
of the model will be discussed briefly in this paper, the details will be
presented in later papers.
The main contributions of this paper are derived from the integration
of mathematical programming models and methods with the economic theory
of depletable resources.
The mathematical programming models permit the
incorporation of large quantities of data about the economic systems under
investigation; specifically, the economic sector which uses the depletable
1 Modiano
(1978) gives complete discussions of all of the extensions to be
discussed in this paper.
2
resource and the supplier who produces or extracts the resource.
More-
over, mathematical programming permits the explicit optimization of these
systems according to well defined criteria.
An important side benefit of the mathematical programming analysis
is a derived demand curve in each period for the depletable resource as
the result of minimizing the production, conversion and distribution
costs incurred by the economic sector in meeting end-use demands by consumers.
This cost minimization over time permits substitution of other,
more efficient resources for the depletable resource and the introduction
of new technologies to replace those dependent on the depletable resource.
The derived demand curves for each period are used by the supplier to
optimize his supply decisions over the entire planning horizon.
Mathe-
matical programming decomposition methods permit us to systematically
sample these demand curves at the levels of interest to the supplier.
Thus, it is not necessary to generate the entire derived demand curve for
the depletable resource in each period, a task which could be quite substantial for large models.
Section two contains a statement of our basic model of depletable
resources.
Sections three and four give more detail about the two compo-
nents of this model:
the supplier's problem and the sectoral problem.
Section five reports on the results of the model applied to the supply of
coal to the U.S. energy sector.
3
2.
Statement of the Model
Our basic model consists of an economic sector that is the sole
user of a depletable resource and a unique supplier of this resource.
We describe first the behavior of the supplier.
In order to decide on
1
2
T
a supply schedule r , r ,..., r over a specified planning horizon of
T periods, the supplier maximizes the net present value of profits resulting from the selling decisions.
The value of the holdings of the R
units of the depletable resource VT(R) from period 1 through T is obtained by solving the supplier's problem:
T
VT(R)
=
t t-1l
t
gt(
r,rt)}
j=l
t
max
t-lt(rt)
t=l
T
+ aT
(R -
rt)
t=l
T
S.t.
r
(1)
< R
t=l
t
rt >
where
t
=
1,2,...,T
(<l) is a discount factor,
t is the total revenue function, gt
is the extraction cost function and
is the salvage function.
The
revenue function denotes the supplier's revenue in period t for an amount
t
r
supplied of the depletable resource.
It is assumed to be concave, con-
tinuous, non-decreasing and satisfying Bt(0) = 0.
The extraction cost
function is assumed convex, continuous, increasing and satisfying
gt(.,O) =
.
It is a function of how much is extracted in period t as
well as of the cumulative past extractions.
The salvage function is
assumed to be concave, continuous, increasing and satisfying
T+l ()
=
The convex, increasing shape of the curve reflects increasing marginal
costs resulting from increasing effort necessary to extract deeper
deposits [e.g. (Zimmerman, 1977)].
O.
4
It gives a valuation for the resource stock left at the end of the
planning horizon.
The objective in the supplier's problem (1) is concave due to the
concavity and convexity assumptions about the revenue, salvage and
extraction cost functions.
t
If
and gt are differentiable, the Kuhn-
Tucker conditions are necessary and sufficient conditions for optimality.
Namely, they require the existence of (rl,...,r ,X) satisfying
t
T
t
a
drt
drt
i=t+l
i-t
i
dg i _
-r
drt
1-t +
t+l
T-t+l dB t+
drt
< 0
-
(2.a)
t = 1,2,...,T
T
drt
a
it+l
drt
i-t
4
lt
AC -X
drt
T t
Z r < R,
+
T-t+l d
) rt = O
)r
=0
drt
t = 1,2,...,T
(2.b)
^t
r
>0
(2.c)
t = 1,2,...,T
t=l
^
T
(
^
r
- R) = O0
(2.d)
t=l
>0
(2.e)
In our development below, the revenue function
t will be derived
from a dynamic linear programming model implying it is concave, continuous
but not everywhere differentiable.
Technical difficulties due to non-
5
differentiability can be overcome by the suitable use of suhgradients,
which exist everywhere, in place of the gradients in these conditions. 3
In order to interpret (2.a) and (2.b) it is insightful to verify
that for rt > 0, 2(b) can be rewritten as
dt
d
drt
-
t
drt
T
=
i-t d
i=t+l
i
drt
+
-t
t
T-t+
1-
T+1
d(3)
drt
The left-hand side of (3) is the difference between marginal revenue and
marginal immediate cost.
The right-hand side can be regarded as a mea-
sure of total user's cost, including the sacrifice of future profit due
to present sales.
The sacrifice of future profits due to present sales
is composed of three items.
The first term reflects the discounted effects
of present extractions in elevating the cost of future extractions.
The
second term is pure scarcity rent, whereas the last term reflects the
sacrifice in salvage value.
Therefore (3) can be restated as a require-
ment that marginal revenue equals marginal total costs (immediate plus total
user's costs).
The conditions (2.c) are the constraints on the supplier's problem
(1).
The shadow price of the depletable resource constraint A, which can
be interpreted as the marginal present value of incremental reserves, is
non-negative by (2.e).
By the complementary slackness condition (2.d),
it vanishes when the resource is not fully utilized over the planning
horizon T.
3
See, for example Grinold (1972) or Shapiro (1978, chapter 6).
4
Cummings (1969), Gordon (1976) and Scott (1967) derive user's cost terms
for different types of models of depletable resources derived from Hotelling's classic model (1931). These models are different, for example,
in that they do not incorporate a sectoral model using the depletable
resource.
The depletable resource is used by a single economic
sector for
which vectors of end-use demands dt are given for T periods. The demand
is met by production transforming the primary supplies into end-use
goods.
Demand met through conversion and transmission of primary
supplies in the energy sector is an example.
We assume the existence
of a single depletable resource among the primary supplies available
to the economic sector.
Denoting by r the amount of the depletable
resource available to the sector for production of end-use goods, a
model of the economic sector can be developed to calculate in each
period the cost function,
4t(r) = minimum cost of meeting the
demand for end-use goods in
(4)
period t when r is the quantity
of depletable resource available in period t.
In this paper, we will use a linear programming representation of
the economic sector to calculate the cost function.
The specific model
is the LP-sectoral problem
s.t
(5.a)
+ fs
~t(r) = min ctx
pt t
t
Axt - s
t
Ax
O < xt,
< r
(5.b)
< 0
(5.c)
d
O< st <Mt
(',.d)
(5.e)
7
where the decision variables xt denote the levels of conversion, transmission and transportation activities and the decision variables st
denote the amounts purchased of the alternative primary supplies.
Unit
costs are given by the non-negative vectors ct and ft respectively.
The
matrices A1 and A2 give the technological coefficients for supply and
demand respectively and pt is a vector denoting the unit depletable
resource usage of the corresponding activities.
Since the depletable
resource is non-productible, pt > 0.
The objective (5.a) is the mini-
mization of costs in each period t.
Costs include both the operational
costs of the sector's activities as well as the expenditures on the alternative primary supplies' purchase.
The fact that the depletable resource
usage cannot exceed its availability is indicated by (5.b).
Constraints
(5.c) and (5.d) are the ordinary supply and demand restrictions.
Finally,
(5.e) represents the non-negativity constraints as well as the existence
of upper bounds on the utilization of alternative primary supplies.
The properties of linear programs ensures that the cost function ~t
is non-increasing in r and convex.
We further assume that
t(0 ) < + -,
or in other words, the demands in each period can be met (at finite cost)
without using any of the depletable resource.5
lim
Finally, we assume that
t(r) > - I, a condition that is ensured, for example, if c t > 0 and
r-c
ft > 0 in problem (5).
These assumptions permit us to define the cost
savings function
t(0) -
t(r) = cost savings in meeting the demand for enduse goods in period t when r is the quantity
of depletable resource available in period t.
5
The depletable resource is then inessential to the sector's operations
in the sense of DasGupta and Heal (1974).
(6)
8
It is useful for future reference to note further that
t is the
piecewise linear convex function given for all r > 0 by
qt(r)
maximum {at,k
k=l,...,Kt
=
where the scalars at,k and
_ t,kr} ,
tsk are derived from the extreme points of
the linear programming dual to problem (5).
k
t,k
'
The non-negative scalars
are the shadow prices on the constraints (5.b).
This form implies
that the cost savings function is piecewise linear and concave and given
by
t()
-
t(r) =
minimum {(4 t(o) _ at,k) + nt,kr}
k=l...,K t
.(7)
The derived demand curve of the sector for the depletable resource is shown
t,K are decreasing because
in figure 1 where the shadow prices
%t is a convex function.
assumed lim
r-eo
The final shadow price HtK = o because we have
t(r) > - a.
With the above assumptions on the basic model, we proceed to determine how an equilibrium is reached between the depletable resource supplier
and the economic sector in this hypothetical economy.
We assume that the
economic sector is willing to pay in period t for r units of the depletable
resource a quantity not exceeding the cost savings implied from a resource
availability of the amount r,
supply of r units less than
ing demand.
t(0) t(0) -
t(r).
Any quantity paid for a
t(r) lowers the sector's cost of meet-
The sector is indifferent when the cost of purchasing the
supply is precisely
t(O) -
t(r).
This permits us to def.ine the supp.lier's
revenue function:
St (r) _
t(0)
-
4t(r),
and to establish an equilibrium condition:
(8)
9
price
quantity rt
Figure 1
10
"For any non-negative resource
12
T
levels r ,r ,...,r satisfying
T
Z rt < R we say that the supt=l
plier's and the sectoral problems
(9)
are in equilibrium if these resource levels permit the supplier
to maximize his profit; namely,
t=l
j=l
t=l
Equilibrium between the economic sector and the unique supplier can
be reached through an iterative or tatonnement process depicted in Fig.
First, the supplier announces a supply schedule of the depletable re12
source, r ,r ,...,r
T
; then the economic sector reacts to it by announc-
t
ing for each period, at those levels, its cost of meeting demand Ot, or
equivalently, its cost-savings through
t as defined in (8), and its
shadow price eIt which can be viewed as bid prices for marginal depletable
resource units.
At this point, either equilibrium as defined in (9 ) has
been reached or, given the new information about his revenue function,
the supplier revises his supply schedule.
At each iteration prior to
equilibrium, the supplier optimizes his supplier's problem (1) using an
upper bound approximation to the revenue function obtained by omitting
in (7) many of the linear pieces defining it.
The supplier perceives the
lack of equilibrium because the last shadow price announced by the economic sector corresponds to a new linear piece which does not permit the
.
11
........ a.......
Sectoral
Problem
..................
Period t
)
T (rT,n
revenue
jt(r tn
nT,n+l/
)
supply
i
in
period
t
shadow
price
It,n+l
rT,n
V
Figure 2
supplier to receive as great an income as previously computed using the
approximation.
The situation is depicted in figure 3 where the dotted
line is the new linear piece added to the approximate revenue function
corresponding to the latest shadow price.
The iterative scheme con-
verges to an equilibrium because there are only a finite number of pieces
defining the revenue functions and at each iteration, a new piece is
added to at least one of them if equilibrium is not achieved.
The iterative scheme just described is a resource directive decomposition method applied to a mathematical programming problem embodying
both the sectoral and the supplier's problem.
This problem is
6Resource directive decomposition is a well known approach to the
solution of large scale linear and other mathematical programming
problems (see Shapiro (1978) for more details).
12
revenue
resource
supplied
Figure 3
13
1
L-I.
, ,(R) {t=l
St() - g ((
mx
r
t)
+
j=l
T T+l
aT T + l (R -
T
T
rt)
t=l
tt
s.t. px
t
-r
Alx
- s
Atxt
<
O
t = 1,2,...,T
<
0
t = 1,2,...,T
>
Ax
dt
(10)
t = 1,2,...,T
T
r
t
<
R
<
rt
t=l
0 < x_
t
~, ~0 <_ s t < Mt , 0
12
For a fixed supply schedule r
time.
,r
t = 1,2,...,T
T
, problem (12) is separable in
It breaks into T subproblems, the LP-sectoral problems (5) and
the resource levels rt are selected by the supplier's problem (1) under the
definition (8) for the supplier's revenue function.
Problem (10) can be
interpreted as a maximization of the discounted benefit, under the assumption of constant marginal utilities with 4t(0) taken as reference value.
14
The other interpretation is the minimization of the present value of
sectoral plus supplier's costs.
Moreover, when the unique economic sector considered here is an
aggregate of a competitive demand market and the unique supplier's
extraction cost function is an aggregate extraction cost function of a
competitive supply market, problem (10) is a maximization of the discounted sum of sectoral plus producers' surplus.
Note also that the
derived demand curve shown in figure 1 describes the sector's preferences for the depletable resource which in turn implies the sectoral
surplus.
By the definition (8) the unique supplier is assumed to capture
as revenue the full amount of this surplus.
Consequently the resource
allocation resulting from solving (10) through the iterative approach
described above will be efficient.
15
3.
Supplier's Problem
Thle supplier's problem (1) defined in the previous section has a
simple structure permitting it to be solved by dynamic programming.
To this end, we define the value functions
VT(S) = maximum present value of selling
decisions from period t through
T when S is the stock of the de-
(11)
pletable resource by the start of
period t.
The functions V
T
satisfy the recursions
V (S) = maximum {t(r) - gt(R - S,r) + aV
TO<r<S
(S - r)}
(12)
t = T, T-l,...,l
where
T+i
vT+ (s) =
T+1
T (s)
for all S.
The supplier's problem is solved by computing
VT(R)
V1 (R)
The computation of the return functions is facilitated by the fact that
they will be concave if the income and extraction cost functions are concave and convex, respectively.
The notation of the value functions V
makes explicit reference to
the assumed planning horizon of T periods after which we value the remainT
r according to the salvage function. To a certain
ing resource R t=l
ID
extent, the choice of planning horizon and salvage function are arbitrary,
but there are several instances in which the potential error is small.
Arbitrary accuracy can be obtained by increasing T until the factor a
discounting salvage reduces it to a negligible level.
T-l
Of course, the
computational burden increases rapidly with T which may make this approach
unrealistic.
Any difficulties due to terminal conditions will disappear naturally
if there are activities in the LP-sectoral problem (5) in later periods
which are efficient substitutes for the depletable resource.
This will
be the case, for example, if there is a T* such that for all t > T*,
do (O)
dr
<
dg (0,0)
dr
In words, this condition states that the marginal income at a zero level
of supply in period t is less than the marginal cost of extraction at a
zero level of supply even if no extraction before period t has taken place.
In this case, we could take the planning horizon to be T* and the salvage
function
T*+(y)
0 for all y > O.
Another approach to defining the salvage function is to formulate and
solve a stationary, infinite horizon approximation to (12).
The result
would be a function V(S) which is the value to the supplier of S units of
the depletable resource at the start of an infinite, stationary planning stage
horizon.
The function V could serve as the salvage function after a long,
finite planning horizon. 7
Generalizations of the revenue function are possible describing
alternative behavioral assumptions about the section's willingness to pay
to the supplier the full amount of its cost savings.
7
For example, the
See Modiano (1978) for a definition and solution of the infinite
horLzon problem.
17
revenue function could be defined as
t(r), r]
gt(r)
If t
is concave and non-decreasing in its first argument, then it can
be shown that B
is concave.
In this case, the iterative approach of the
previous section remains entirely valid and operational.
revenue functions
t
The original
are computed as before from the sectoral model, but
it is the revised revenue functions Bt which are used in the supplier's
problem.
This extension permits several institutional phenomena to be
modeled:
i)
price regulation
Bt(r) =
fr min {pt
lt ()}
dE
is an exogenously given upper bound on the unit price
where p
that can be paid to the supplier.
In the derived demand curve
in figure 1, this corresponds to cutting the curve by a straight
-t
line at a price p parallel to the quantity axis.
ii)
cost of externalities
Bt(r)
-
t(r)
where qt is convex and increasing reflecting, for example, the
social costs of pollution.
iii)
tax depletion allowance
B(t) = [1 +
_L3 B t (r)
where y is the percentage of depletion allowance and T is the
tax rate.
For 0 < y < 1, the before-tax revenue function 8t
is higher than St.
The it functions implicitly defined by these three examples satisfy the
conditions ensuring concavity of
t.
18
The supplier's problem as originally stated in (1) has no restrictions besides the depletable resource constraint.
There are several
ways to elaborate upon this formulation in order to make it more realistic.
The elaborations are'not necessarily mutually exclusive.
However,
the addition of other constraints to (1) is likely to make the supplier's
problem too complex to be solved by dynamic programming methods, and
other mathematical programming methods would be required.
few illustrative'extensions are discussed in this context.
A
Once again,
the iterative scheme described in the previous section remains valid and
operational with these extensions.
The-supplier's problem can be extended to include extraction capacity constraints and investment decisions to enlarge those limits.
0
we let y
denote the initial extraction capacity available and
t
If
the
amount of capacity installed'in period t, we further require that
t
r
<
t
Z
j
y
0
t = 1,2,...,T
+ y
(13)
j=1
Since capacity installment cost includes in general a fixed charge and a
variable component we need introduce the zero-one variables 6t satisfying
0
6
if yt = 0
t = 1,2,... ,T
=
1
oifif
(14)
>
The present value of capacity build-up expenditures is then given by
T
E at-l{Qtst + qtytl
t-l
(15)
19
where Qt is the fixed charge and qt is the cost per unit of capacity
installed in period t.
An enriched supplier's problem would result from the simultaneous
incorporation of (13) and (14) as constraints and (15) in the objective
function of (1).
The addition of the integer zero-one variables, how-
ever, further complicates the solution of the supplier's problem.
The
state-space for dynamic programming recursions is two-dimensional;
namely, the pair (Y,S) where Y is the total capacity available and S, as
previously defined, is the stock of depletable resource bv the start of
Deriod t.
The supplier's problem would be even more complex if we were to take
into account some of the intricacies of the capacity build-up process
such as the lags existing in capacity expansion.
While we have enough
flexibility in the specification of the extraction cost-functions in (1),
by allowing it to vary over time, it is more plausible to assume that
transition to cheaper extraction technology can only be made at a fixed
charge.
The supplier's problem can be extended to include the decision
on a transition period again by the addition of zero-one variables.
We
cannot neglect also the possibility of investments for depletable resource reserves expansion.
However, since the outcome of a reserve
expansion project is uncertain as the size of the new discoveries are
not known a priori, this would lead us away from deterministic mathematical programming methods.
20
4.
Sectoral Problems
We have proposed the linear programming problem (5) to describe the
economic sector in which the depletable resource competes with other
primary supplies in a variety of conversion activities to meet end use
demands.
Linear programming problems have simple data requirements, but
other mathematical programming representations of the sector are clearly
possible.
The use of convex nonlinear supply functions ft(st), for exam-
ple, would smooth out the resulting cost and income functions, and the
derived demand curve shown in figure 1.8
Another important extension to the LP sectoral problem is to incorporate constraints which cut across time to model dynamic effects such as
the introduction of new technologies which substitute for the depletable
resource.
As a specific example, suppose we consider variable x
problem (10) to correspond to a new activity with p
in
= 0 for all t.
Suppose
further that there is a fixed investment cost F > 0 to be incurred if variable x
t
is ever used at a positive level, in which case x
any level up to K > 0.
zero-one variables y
t
can be used at
The fixed cost is modeled by the introduction of
for all t and the constraints
T
z Yl < 1
(16)
t=l
and
t
x1 - K
8
t
s
Z Y1 < 0
s=1
t
=
1,...,T.
The decomposition scheme shown in figure 2 would not converge finitely
if the sectoral problems were nonlinear. This is a theoretical rather
than a practical difficulty since computation would be terminated when
the calculated error in optimality was small enough.
(17)
21
Similarly, we add
F
T
Z ay
t=l
(18)
The constraints (16) ensure that the new
to the objective function.
activity can be introduced at most once, and the constraints (17) ensure
that yl
=
that x
> 0.
1 and the fixed cost (18) is incurred in the first period t such
The decomposition scheme given in figure 2 for the basic model can
be extended to permit for the case
as shown in figure 4.
ust described a double decomposition
Problem (10) with the additional constraints
separates into T LP sectoral problems as before if, for all t, both the
depletable resource variables rt and the investment variables
fixed.
are
The shadow prices in each period from the LP sectoral problems
are now used to update the investment plan as well as the supply schedule
of the depletable resource.
These are shadow prices on constraints
other than the depletable resource constraint (5.b) in the LP sectoral
problems for which the activity corresponding to x
has positive coeffi-
cients.
There are two other extensions of the LP sectoral problem which we
briefly mention.
First, uncertainty in supply of the depletable resource,
or uncertainty in demand, in each period can be modeled by splitting the
period into two stages.
The economic sector makes its main decisions
in the first stage and then some compensating decisions in the second
stage after the uncertainties have been resolved.
A second extension is
We refer here to the well known two-stage linear programming model
under uncertainty; see Wagner (1975).
22
Figure 4
23
to allow endogenous end-use demand dt in every period in problem (10).
This
would permit the end-use consumer to respond to prices paid to the sector
for consumer goods using the depletable resource.
The objective function
in (10) in this case requires the inclusion of a consumer surplus function
of the satisfied end-use demand. 1 0
10Shapiro (1977) discusses the relationship of endogenous demand and
consumer surplus to mathematical programming constructs.
24
5.
Application of the Model to U.S. Coal Supply
In this section, we discuss the application of the model given in
section two to the optimal depletion of U.S. coal supplies assuming that
the U.S. energy sector is the sole user of the coal.
Our main purpose
is to illustrate how the model can be implemented, and to give additional
modeling and methodological details arising from the implementation.
The
results should not be viewed as definitive but rather representative of the type of results that can be obtained by our approach.
However, we attempted to use valid data throughout the model, and the
results about the dynamic pattern of U.S. coal supply were reasonable.
* LP model of U.S. energy sector.
For this study, we used a linear
programming representation analogous to (5) derived in large part from
11
the Brookhaven Energy System Optimization Model (BESOM).
The model is
essentially a network optimization problem consisting of source nodes
corresponding to primary energy supplies and sink nodes corresponding
to energy end-use demands.
The arcs in the network connect source nodes
to sink nodes and they correspond to activities converting and transmitting
supply BTU's to BTU's satisfying demand.
Losses in arc flows are exper-
ienced due to inefficiencies in conversion and transmission devices.
Unit costs are associated with each arc.
Demands are given exogenously,
and supplies are variable with associated unit costs per BTU supplied up
to fixed upper bound levels.
In addition, there are side constraints
on the BTU flows in the arcs due to capacity restrictions and policy constraints.
11
The objective function is to minimize the supply and conversion
Hoffman and Jorgenson (1977) give a recent discussion of BESOM. The
specific representation we used was a smaller version of the model
suggested by Beltrami and Frauenthal (1977).
25
costs of meeting exogenous energy end-use demands in a given year.
Table 1 gives the unit costs, upper bounds on supply and exogenous
demands for the reference year 1985.
The costs are dollars per 106 BTU's
delivered to intermediate nodes in the energy network optimization problem; specifically, these are the nodes immediately preceding the conversion of BTU's to electric energy and end-use devices.
is assigned to the demand for peak load electricity.
No exogenous value
This demand is assumed
to be endogenous and formed by summing 20% of the demand for electricity
air conditioning, 5% of the demand for base load electricity, 5% of the
demand for electric process heat and 10% of electric public transport.
In
addition, no supply limit is specified for turbine generators because they
are oil fired.
Side constraints included in the model for the year 1985 require:
the
cooling demand not to exceed 4/5 of the heating demand; the amount of electricity used for space heat not to exceed 2.8 x 1015 BTU; the oil-to-gas
usage ratio in the petrochemical industry to be exactly 5.6; the amount of
electricity used for ground transportation to be .17 x 1015 BTU; the amount
of coal used by the petrochemical industry not to exceed .35 x 10
the amount of coal used for process heat not to exceed 3.4 x 10
15
BTU:
BTU and
turbine generators to be able to handle the peak demands assigned endogenously to them.
ENonlinear oil and natural gas supply model.
Previous experience with
BESOM has indicated that linear pricing of primary supplies can produce
highly elastic derived demand curves for those supplies. 2
For this reason,
we expanded the model to include a nonlinear supply function for oil and
natural gas in each period.
Thus, we used an empirical model of oil and
gas supply in the process of deriving a normative model of coal supply.
Shapiro, White and Wood (1976) and Shapiro and White (1977) report on
these demand curves and experiments with nonlinear supply functions.
26
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27
The nonlinear supply function we used was
f(s1 , s2 ) = a s1
1
2
s2
= cost of supplying s
BTU's
of crude oil and s2 BTU's
of natural gas.13
This function can be related to an econometrically estimated supply and
demand function by noting that the long-run marginal costs of supply or
supply prices should satisfy
af(lS
as
P
=
a
2
al-l
asl
s2
1
(19)
-
a
a1
a 2o2s 1 s2
(20)
2)
as1
and
af(sl,s2 )
P
=
Taking logarithms and inverting, we obtain the supply function in explicit
form as
in s1 \
Kn s2
The constant parameters
1
/a 2
k-a
-a
l
2
kn(P 1 /aal)
\
k
Ci-a
1
a2
and
2 can be estimated from their relation-
a11
-1
n( 2 /aa2 )/
ships to constant own and cross-price elasticities
1 3 Manne (1976) used this type of function to model endogenous demand for
electric and non-electric energy.
28
2
1-
-a 2
- 1
a
- a2
1
a1
- 02
n22 =
n21 =
1
14
Erickson and Spann
2
1
2
estimate the own-price elasticities
nil
=
'83
and
n2 2
.726 and
a2
.69
which imply that
a
=
.670
These are the values used in our study.
A major difficulty we encountered in the complete specification of
the supply cost function f(sl,s 2) stems from the determination of the
parameter a. Note that in order to completely specify it, only one
observation is required from either of the supply curves (19) or (20).
Hence, to estimate a, we may solve equation (19) or (20) by setting a
"reference price" and a "reference level" of supply.
It is our firm
belief, however, that this model can be highly sensitive to an arbitrary
selection of the reference point.
In a first attempt we follow Manne and arbitrarily set a price of
$7 per barrel of oil at unitary (1015 BTU) supply levels of oil and
1 lErickson
and Spann (1971)
15Manne (1976) in his model of constant elasticity of substitution for
the demands for electric and non-electric energy sources has arbitrarily
set a low price at low levels of demand.
29
natural gas.
For an average conversion factor of 5.8 x 106 BTU per barrel
of oil,
a = (7/5.8)
.726
=
1.66
However, table 2 shows that radically different values of a are implied
For our study, we took a
by observations from alternative models.
consistent with the values in Table 2.
=
.75
Clearly, more statistical analysis
of this parameter and others would be required to ensure greater confidence in the numerical results.
Table 2
Comparison of Parameter Estimates from Alternative Models
Oil Domestic
Production
(10
BTU)
Parameter
Gas Domestic
Production
(10
5
BTU)
I
a
F.E.A. - Project Independence
@ $7 oil
23.1
23.9
.48
F.E.A. - Project Independence
@ $11 oil
31.3
24.6
.79
M.I.T. - Econometric Model
@ $11 oil
31.33
35.78
.61
M.I.T. - Judgmental Model
@ $11 oil
28.79
24.35
.77
Source:
Federal Energy Administration (1974), Tables F-8, F-9 and I-10).
30
A second major difficulty stems from the elimination of the
extraction costs for oil and natural gas from Table 1.
Since the original
data were not available to allow us to recompute these cost coefficients,
we had to rely on an arbitrary criterion.
We chose to deduct the smallest
entry (.64) from all the cost coefficients in the natural gas cost coefficient row in Table 1.
In an analogous fashion for oil, the smallest
cost coefficient entry on the rows for oil, oil-fired electricity and
oil-fired turbine generators (1.21) was deducted from all entries on these
rows.
In order to retain the linear programming format of the sectoral
model, the nonlinear oil and natural gas supply function was incorporated
using a grid linearization.
Table 3 gives the linearization we used.
The
LP sectoral problem incorporating the linearization is
K
¢(r) = min
cx + a
Z
k
a
X (si)
a
(s2)
+ qy
(19a)
k=l
s.t.
s.t.
(coal):
Px
(oil):
K 1 x-
K
< r
(19b)
< 0
(19c)
<
(19d)
= 1
(19e)
kk
X
s
-
y
k=l
kk
(natural gas):
K2x
k=l
K
k
(convexity constraint):
k=l
(alternative primary
supplies):
Alx
<
s
(19f)
(end-use demands):
A2 x
= d
(19g)
(side constraints):
B x
< b
(19h)
0 < x, 0 < y, and 0 <
k, k = 1,2,...,K.
31
Table 3
Grid Linearization of Oil and Natural Gas Supply
Grid
Oil
Natural Gas
1015 BTU
1015 BTU
109 $
k .7 2 6
k .6 7 0
sk
k
1
20
15
54.017
2
20
25
76.062
3
20
40
104.214
4
40
15
89.346
5
40
25
125.810
6
40
40
172.376
7
60
15
119.928
8
60
25
168.872
9
60
40
231.376
10
80
15
147.783
11
80
25
208.096
12
80
40
285.117
jZ
The variable y in this formulation corresponds to oil imports at the
price q.
*
Dynamic aspects of LP model.
Until now, we have omitted dynamic
or time dependent details about the LP model of the U.S. energy sector.
The dynamic features we introduced were (1) escalation of nuclear
energy supply, (2) increases in oil import prices and (3) growth of
All costs, including those in table 1, are computed
end use demands.
We used a discount rate a =
in constant 1972 dollars.
.9.
Table 4 gives the data we used on nuclear energy and oil import
prices.
The escalation of nuclear energy is significant and assumes
The increases
there is no serious moratorium imposed in the 1980's.
in oil import prices are due to Hnyilicza and Pindyck.
16
Table 5
gives a summary of growth in end-use demands.
I
Coal supplier's model.
The essence of depletion is the movement
from cheaper to more costly deposits.
The effect of cumulative pro-
duction in elevating extraction costs in the mineral industry has been
widely recognized since the origins of the economic theory of exhaus17
tible resources.
We assume in our supplier's problem (1) an extrac-
tion cost function of the form
t
t-1
gt (
r ,
j=l
) = h(
t-l
r)
) - h(
j=l
t = 1,2,...,T
j=l
where h gives the total cost of extracting a total of E units of the
depletable resource independently of the time periods in which extraction takes place.
16
See Hnyilicza and
17
Hotelling (1931),
Zeckhauser (1972)
marginal costs in
Pindyck (1976).
Gordon (1976), Herfindahl (1967) and Weinstein and
have all considered the possibility of increasing
depletable resources extractions.
J
Table 4
Escalation of Nuclear Energy and Oil Import PriLces for
Equilibrium Model II
Year
Escalation of Nuclear Energy
Oil Import Prices
(1015 BTU)
__
$1975
__
$1972/
106 BTU
1979
6.10
9.82
1.15
1980
7.20
9.88
1.16
1981
8.30
10.07
1.18
1982
9.40
10.26
1.20
1983
10.50
10.45
1.23
1984
11.60
10.64
1.25
1985
12.70
10.84
1.27
1986
13.80
11.07
1.30
1987
14.90
11.30
1.32
1988
16.00
11.53
1.35
1989
16.50
11.76
1.38
1990
17.00
11.98
1.40
1991
17.00
12.22
1.43
1992
17.00
12.46
1.46
1993
17.00
12.70
1.49
1994
17.00
12.94
1.52
1995
17.00
13.18
1.55
1996
17.00
13.44
1.58
1997
17.00
13.70
1.61
1998
17.00
13.96
1.64
1999
17.00
14.21
1.67
2000
17.00
14.46
1.70
34
Table 5
Intertemporal Growth of End-Use Demands: Summary
Demand
15
BTU)
(10
Year
Geometric
Growth
Rate
1979
1985
1991
2000
Space Heat
9.14
10.60
12.29
15.35
2.5%
Air Conditioning
1.90
2.20
2.55
3.19
2.5%
Base Load
5.43
7.20
9.65
14.97
4.5%
17.07
19.00
21.15
24.83
1.8%
Petrochemical
7.19
8.00
8.90
10.64
1.8%
Air Transport
2.93
3.20
3.50
4.00
1.5%
Public Transport
1.46
1.60
1.75
2.00
1.5%
Private Transport
2.19
2.40
2.63
3.01
1.5%
Peak Load
Process Heat
Sources:
The geometric growth rates for all end-use demands except
base load electricity were obtained from F.E.A. - Project
Independence Report (1974). The demand growth rate for
load electricity was suggested by Manne (1976).
35
Figure 4 depicts a convex, increasing cumulative cost function h
and a piecewise linear approximation to this function.
h
E
Figure 4
The Shape of the Coal Cumulative Extraction Cost Function
It is in the derivation of such an approximation that we concentrate our
efforts in this section.
For this purpose, Zimmerman's model, based upon
the geology of remaining coal deposits, was extensively used.18
The current version of Zimmerman's model considers six distinct
coal production regions, depicted in Figure 5. The model generates
marginal extraction costs for three coal production levels, for each
region and different categories of sulfur content.
The predictions for
one, five and twenty times the 1975 production levels are presented in
Table 6.
This would give us three observations of the cumulative ex-
traction cost function of each region by sulfur content.
18
See Zimmerman (1977).
However, the
36
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co
CI,
0
*-H
I,
n
a
o
u
a3
CV
o
'-Ia
Pro p
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0
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ud
a
*ts
ea
S
2
37
pure horizontal summation of the marginal extraction costs in Table6 ,
trrespective of its region of production would be meaningless.
This
is because in the derivation of the aggregate U.S. coal cumulative
cost function, a low marginal extraction cost mine would be given priority
over higher marginal extraction cost mines, even if the former is located at a much greater distance from the demand centers.
In order to avoid
these anomalies a reference location in the U.S. map (Chicago) was selected to give a basis of comparison between the productive regions.
the BTU content per ton of coal is the measure of concern to us.
Also,
A
comparison of mined coal being produced in different regions can only be
carried out in these units.
The addition of the transportation costs to
the point of reference (Table 7) and the subsequent conversion to BTU
(Table 8) yields Table 9.
At this point we proceed with the horizontal summation of the marginal
costs for coal irrespective of sulfur content.
For this purpose we assumed
that the marginal costs in Table 9 were constant to the left of the respective production volumes.
For example, for region 1 and low sulfur coal,
the marginal cost curve would be given by $1.80/106 BTU between 0 and
.18 x 1015 BTU, $1.85/106 BTU between .18 and .91 x 1015 BTU, $1.97/106 BTU
between .91 and 3.62 x 10
BTU and so on.
A further approximation was
required to reduce the number of linear segments in the resulting aggregated
cumulative cost function.
ginal costs in Table
This was accomplished by approximating the mar-
9 to a single decimal digit.
Table 10 gives, then,
a piecewise-linear approximation to the U.S. coal cumulative cost function,
h.
This is the extraction cost function that we consider for the supplier's
38
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40
Table
7
Transportation Costs to Chicago
Region
Costs
($/ton)
1
6.94
2
7.74
3
4.26
4
14.40
5
13.49
6
16.89
Table 8
Conversion Factors
Region
Contents
(106 BTU/ton)
Source:
1
23.6
2
23.6
3
22.0
4
17.0
5
21.8
6
20.0
Zimmerman (1977), (1978).
-----
--
41
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44
problem (1).
For that problem, it was further assumed that the amount
of coal reserves R are the Federal Energy Administration estimates of
the total U.S. coal reserves: 9.371 x 10
BTU.
Finally, we assume no
salvage value for coal remaining after the year 2000.
*
Results.
The equilibrium coal allocation obtained by the model
is presented in Table 11.
The optimal consumption of coal by the
energy sector as predicted by the model should slightly decrease from
1979-1984, when it starts to rise again up to 37.26 x 1015 BTU in year
2000.
This may result from the progressive escalation of nuclear energy
and the mild variations in oil import prices.
The allocation of the sectoral alternative primary supplies is presented in Tables 12 through 15 for the years 1979, 1985, 1991 and 2000.
A comparison of these tables shows U.S. imports of oil starting at a
high level in 1979, approximately 64% of the total oil domestic consumption, vanishing in the years 1985 and 1991 and picking up again in year
2000.
Domestic oil production varies sharply in the periods 1979/85 and
1991 to year 2000.
1985 and 2000.
Natural gas consumption is unchanged in years 1979,
It goes above the level of 15 x 1015 BTU in year 1991
when it reaches 38 x 1015 BTU.
The major change in those years occurs
from year 1991 to 2000 where the oil utilization for space heat approaches
the gas levels and the allocation of coal-fired electricity to base load
electricity is sharply increased.
45
Table 11
Equilibrium Coal Consumption
Year
Coal
Consumption
(1015 BTU)
1979
15.25
1990
16.67
1980
15.05
1991
18.00
1981
14.90
1992
19.67
1982
14.59
1993
21.44
1983
14.51
1994
23.16
1984
14.49
1995
25.05
1985
14.51
1996
26.82
1986
14.58
1997
28.71
1987
14.81
1998
30.74
1988
15.00
1999
35.04
1989
15.81
2000
-37.26
46
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References
Beltrami, E., and Frauenthal, J. (1977), "An Energy Supply and Demand
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Benders, J.F. (1962), "Partitioning Procedures for Solving Mixed
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the Case of Natural Gas," The Bell Journal of Economics, 2,
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Grinold, R.C. (1972), "Steepest Ascent for Large Scale Linear Programs,"
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