A dynamic optimization model of depletable resources

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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.

his IioldIings over a finite planning horizon. A number of extensions

are discussed briefly. An implemented version of the model to the dynamic allocation of coal supply to the U.S. energy sector is also presented.

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 sec-tor is derived from the cost savings realized by the secsec-tor 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}

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 con-sumers. 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 systeMathe-matically 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 sub-stantial 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

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 re-sulting 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 ob-tained by solving the supplier's problem:

T t t t-1l t T VT(R) = max t-lt(rt) gt( r,rt)} + aT (R - rt) t=l j=l t=l T S.t. r < R (1) t=l t rt > t = 1,2,...,T

where (<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 () = O.

The convex, increasing shape of the curve reflects increasing marginal costs resulting from increasing effort necessary to extract deeper deposits [e.g. (Zimmerman, 1977)].

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. If t 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 drt drt T i t+l i-t dgi _ 1-t + T-t+l dBt+ a -r < 0 i=t+l drt drt -t = 1,2,...,T drt drt (2.a) T i-t 4 lt T-t+l d ) rt = O a AC -X + )r =0 it+l drt drt t = 1,2,...,T T t Z r < R, t=l ^ T ^ ( r - R) t=l ^t r > 0 = O0 >0 t = 1,2,...,T (2.b) (2.c) (2.d) (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-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 t T i -t 1- T+1

d - = i-t d + t T-t+ d(3)

drt drt i=t+l drt 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 Hot-elling's classic model (1931). These models are different, for example, in that they do not incorporate a sectoral model using the depletable resource.

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 avail-able 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

~t(r) = min ctx + fs (5.a) s.t pt t < r (5.b) t t Axt - s < 0 (5.c)

A

x d

(',.d)

O < xt, O< st <Mt (5.e)

mission 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 alter-native 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 Finally, we assume that

lim t(r) > - I, a condition that is ensured, for example, if ct > 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

end-use goods in period t when r is the quantity (6) 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).

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 _ t,kr} ,

k=l,...,Kt

where the scalars at,k and tsk are derived from the extreme points of the linear programming dual to problem (5). The non-negative scalars

t,k

k ' 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 {(4t(o) _ at,k) + nt,kr} .(7) k=l...,K _t

The derived demand curve of the sector for the depletable resource is shown in figure 1 where the shadow prices t,K are decreasing because %t is a convex function. The final shadow price HtK = o because we have

assumed lim t(r) > - a. r-eo

With the above assumptions on the basic model, we proceed to deter-mine 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, t(0) - t(r). Any quantity paid for a

supply of r units less than t(0) - t(r) lowers the sector's cost of meet-ing demand. 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),

(8)

price

quantity rt

"For any non-negative resource

12 T

levels r ,r ,...,r satisfying T

Z rt < R we say that the sup-t=l

plier's and the sectoral problems (9)

are in equilibrium if these re-source 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

re-12 T

source, r ,r ,...,r ; 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 eco-nomic sector corresponds to a new linear piece which does not permit the

Sectoral ... a... Problem ... Period t revenue jt(r tn ₎ shadow price It,n+l

V

T (rT,n) nT,n+l/ supply in period t rT,n 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 decom-position 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).

revenue

resource supplied

1 L-I. , ,(R) {- mx St() - g - ( ( r t) + t=l j=l T T+l T T aT T +l(R - rt) t=l tt t t s.t. px -r Alx - s Atxt Ax T rt t=l 0 < x_ t

~ ~

, 0 < _ st < Mt, 00 < O < 0 > dt t = 1,2,...,T t = 1,2,...,T t = 1,2,...,T < R t < r t = 1,2,...,T 12 T

For a fixed supply schedule r ,r , problem (12) is separable in time. 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 assump-tion of constant marginal utilities with 4t(0) taken as reference value.

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 dis-counted sum of sectoral plus producers' surplus. Note also that the derived demand curve shown in figure 1 describes the sector's preferen-ces 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.

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 satisfy the recursions T V (S) = maximum {t(r) - gt(R - S,r) + aV (S - r)} (12) T O<r<S t = T, T-l,...,l where T+i T+1 vT+ (s) = 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 con-cave 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

remain-T

ing resource R - r according to the salvage function. To a certain t=l

extent, the choice of planning horizon and salvage function are arbitrary, but there are several instances in which the potential error is small.

T-l Arbitrary accuracy can be obtained by increasing T until the factor a

discounting salvage reduces it to a negligible level. 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) < dg (0,0)

dr 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. For example, the

7

See Modiano (1978) for a definition and solution of the infinite horLzon problem.

gt(r) t(r), 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. The original revenue functions t 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

where p is an exogenously given upper bound on the unit price 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 Bt(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

The supplier's problem as originally stated in (1) has no restric-tions besides the depletable resource constraint. There are several ways to elaborate upon this formulation in order to make it more

realis-tic. 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. A

few illustrative'extensions are discussed in this context. 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 capa-city constraints and investment decisions to enlarge those limits. If

0 t

we let y denote the initial extraction capacity available and the amount of capacity installed'in period t, we further require that

t t j 0

r < Z y + y t = 1,2,...,T (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 if yt = 0

6 = t = 1,2,... ,T (14)

1 oifif >

The present value of capacity build-up expenditures is then given by

T

E at-l{Qtst + qtytl (15)

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 re-source 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 mathe-matical programming methods.

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 incor-porate 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 in

problem (10) to correspond to a new activity with p = 0 for all t. Suppose further that there is a fixed investment cost F > 0 to be incurred if

varia-t t

ble x is ever used at a positive level, in which case x can be used at any level up to K > 0. The fixed cost is modeled by the introduction of zero-one variables y for all t and the constraints

T z Yl < 1 (16) t=l and t t s x₁ - K Z Y₁ < 0 t = 1,...,T. (17) s=1 8

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.

Similarly, we add

T

F Z ay (18)

t=l

to the objective function. The constraints (16) ensure that the new activity can be introduced at most once, and the constraints (17) ensure that yl = 1 and the fixed cost (18) is incurred in the first period t such that x > 0.

The decomposition scheme given in figure 2 for the basic model can be extended to permit for the case ust described a double decomposition as shown in figure 4. 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 are fixed. 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).

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.

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

represen-tative 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 con-straints. The objective function is to minimize the supply and conversion

1 1

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).

demands for the reference year 1985. The costs are dollars per 106BTU's delivered to intermediate nodes in the energy network optimization prob-lem; specifically, these are the nodes immediately preceding the conver-sion of BTU's to electric energy and end-use devices. No exogenous value is assigned to the demand for peak load electricity. 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 elec-tricity 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 BTU:

15

the amount of coal used for process heat not to exceed 3.4 x 10 BTU and turbine generators to be able to handle the peak demands assigned endo-genously 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.

a) 0 rl H 0 CD cn r-I .4 N 4 ¥ 4 H H .0 H-N LI N H Ln N-N '0 H/ laodsul ,_ alVATId r-V0 Co Cr- M LI -n Cl Eodsuz N C o L Ot Ina l V 1 i 00 ,_ It n -t r-C C H N 4 Cl n ·· 0 ) co'.0 _HD~4 _ -e . .. I..

4

_ 0 .. _J 0 CU N L I 0 1.0 a -oxla - -r4 H ...c3 n tln 01H~ .0 - Cl O H 0 4.J H N C .4 I ' . 4o4 H4 Cl '. - in a peo° aBtq 1 o It ) Ln H Ln 00 w cr 'o H '.0 m 0 P0 VCU H~ N CNI- 04 c; pe°1 asoa C 00 00 r - < 4) I I I I I 4n 1A I -I Ln c uCn O C C r di oo LI L I '.0 eO O N o -f L '4-4H N C n I _U)~r .- V C) c U)H aLdsI L 0 I LI Q 0 C Nc 0 0% 0 N- H N C w l W l u H N N O l I LI cl o 0 0 .0 0 Ca to -W o 0 O uO 4-W_co 4i-C U H U ti o H~ N 'IO 0 U, a u 4 -4 C)A -0 nH r-

The nonlinear supply function we used was

1 2

f(s₁, s2 ) = a s1 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 2) al-l 2 P as = a asl 1 s2 (19) as1 and af(sl,s2) a1 a2 -P = - a o2s1 s2 (20)

Taking logarithms and inverting, we obtain the supply function in explicit form as

in s1\ 1 /a 2 l -a 2 \ kn(P1/aal)

Kn s2 Ci-a1 a2

k-a

a1 1 -1

k

n(2/aa2)/

The constant parameters and 2 can be estimated from their relation-ships to constant own and cross-price elasticities

13Manne (1976) used this type of function to model endogenous demand for electric and non-electric energy.

2 - 1 -a2

1- a - a₂ 1 a ₁ - 02

n21 = n22 =

1 2 1 2

14

Erickson and Spann estimate the own-price elasticities

nil

= '83 and n2 2 .69

which imply that

a .726 and

a

2 = .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,s2) 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.

natural gas. For an average conversion factor of 5.8 x 106 BTU per barrel of oil,

a = (7/5.8) = 1.66 .726

However, table 2 shows that radically different values of a are implied by observations from alternative models. For our study, we took a = .75

consistent with the values in Table 2. Clearly, more statistical analysis of this parameter and others would be required to ensure greater confi-dence in the numerical results.

Table 2

Comparison of Parameter Estimates from Alternative Models

F.E.A. - Project Independence @ $7 oil

F.E.A. - Project Independence @ $11 oil

M.I.T. - Econometric Model @ $11 oil

M.I.T. - Judgmental Model @ $11 oil Oil Domestic Production (10 BTU) 23.1 31.3 31.33 28.79 Gas Domestic Production (10 5 BTU) 23.9 24.6 35.78 24.35

Source: Federal Energy Administration (1974), Tables F-8, F-9 and I-10). Parameter a .48 .79 .61 .77

I

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 co-efficient 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 k a a

¢(r) = min cx + a Z X (si) (s2) + qy (19a)

k=l s.t. s.t. (coal): Px < r (19b) K kk (oil): K1x- X s - y < 0 (19c) k=l kk (natural gas): K2x < (19d) k=l K k

(convexity constraint): = 1 (19e)

k=l

(alternative primary Alx < s (19f)

supplies):

(end-use demands): A₂x = d (19g)

(side constraints): B x < b (19h)

Table 3

Grid Linearization of Oil and Natural Gas Supply

Grid Oil Natural Gas 109 $

1015 BTU 1015 BTU sk k k .7 2 6 k .6 7 0 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

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

end use demands. All costs, including those in table 1, are computed in constant 1972 dollars. We used a discount rate a = .9.

Table 4 gives the data we used on nuclear energy and oil import prices. The escalation of nuclear energy is significant and assumes there is no serious moratorium imposed in the 1980's. The increases

16

in oil import prices are due to Hnyilicza and Pindyck. 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

exhaus-17

tible resources. We assume in our supplier's problem (1) an extrac-tion cost funcextrac-tion of the form

t-1 t t-l

gt ( r , ) = h( ) - h( r) t = 1,2,...,T

j=l j=l 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 extrac-tion takes place.

16

See Hnyilicza and Pindyck (1976). 17

Hotelling (1931), Gordon (1976), Herfindahl (1967) and Weinstein and Zeckhauser (1972) have all considered the possibility of increasing marginal costs in depletable resources extractions.

Equilibrium Model II

Year Escalation of Nuclear Energy (1015 BTU) 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 6.10 7.20 8.30 9.40 10.50 11.60 12.70 13.80 14.90 16.00 16.50 17.00 17.00 17.00 17.00 17.00 17.00 17.00 17.00 17.00 17.00 17.00

Oil Import Prices

$1975 $1972/ 106 BTU 9.82 1.15 9.88 1.16 10.07 1.18 10.26 1.20 10.45 1.23 10.64 1.25 10.84 1.27 11.07 1.30 11.30 1.32 11.53 1.35 11.76 1.38 11.98 1.40 12.22 1.43 12.46 1.46 12.70 1.49 12.94 1.52 13.18 1.55 13.44 1.58 13.70 1.61 13.96 1.64 14.21 1.67 14.46 1.70 __ __

Table 5

Intertemporal Growth of End-Use Demands: Summary

Demand Year Geometric

15 Growth (10 BTU) 1979 1985 1991 2000 Rate 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% Peak Load Process Heat 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%

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).

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. However, the

1 8

:3

co CI, 0 *-H I, n a o a3 u '-Ia C V_o Pro_p '-4 0 0 ud

a

*ts

ea S

2

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 loca-ted 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 selec-ted to give a basis of comparison between the productive regions. Also, the BTU content per ton of coal is the measure of concern to us. 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 respec-tive 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. This was accomplished by approximating the mar-ginal costs in Table 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

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o

C 08_o 0 C

o

0

o) 5 4 V- \0

go

t 4 4 0 0 t_ a H _eq '0₀ 0 v H S_ 00 *s ₄ 0 U _H 0 _HU0 ro 4 u o * S 0 0 *3 0 o _{3 0\} 0 _r4 CQi 0 Qr% \0 C- V CQCA u t 4 C\ C El A CD C; r( C *4 Cv a C o co t H C * 0 * * N C N \0 : H -4 H Hq + o Cc S ^ CQ \0 0 N -f o n cUn 0t Cq m C m m C N N * 0 * *~ 0 0\ N 4.4.

ov

H C H C 0' Hi C 0 C 0 0 CQ C,,\ \0 0 N 4. % \0 -OD _{C' (s-} O\ _(W A C_H Ol O C 0 ao q _{4 N} 0 cr 0 0 cV2 CQ C> a, ' 0 ' O' N

g a:)

-~~~~~~~~~(-H NQ Ce- . .1-₀ 0 : 4J m

g

ur Q

EqX

0 C-.. c Cd o E-4 4 x

'0 ₀ '0 ₀ r%0 CVl. i 01 0 0 NI o O 0 CQ * * 0 N .* ('Q .* _so *- _l% -O N 0 0 0 s0

*

*

-,- CN 1-1 -a 0) W) _rl .,I_I 'A 4I c 0 E o E- o 0' --I o_d .. as Wn r rq

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) 1 2 3 23.6 23.6 22.0 4 5 6 17.0 21.8 20.0 Source: Zimmerman (1977), (1978). ---

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-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 pre-sented 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 consump-tion, 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. Natural gas consumption is unchanged in years 1979, 1985 and 2000. 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.

Equilibrium Coal Consumption Coal Year Consumption (1015 BTU) 1990 16.67 1991 18.00 1992 19.67 1993 21.44 1994 23.16 1995 25.05 1996 26.82 1997 28.71 1998 30.74 1999 35.04 2000 -37.26 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 15.25 15.05 14.90 14.59 14.51 14.49 14.51 14.58 14.81 15.00 15.81

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References

Beltrami, E., and Frauenthal, J. (1977), "An Energy Supply and Demand Model," undergraduate teaching module, State University of New York, Stony Brook.

Benders, J.F. (1962), "Partitioning Procedures for Solving Mixed

Variables Programming Problems," Numerische Mathematik, 4, pp. 238-252. DasGupta, P., and Heal, G. (1974), "The Optimal Depletion of Exhaustible Resources," Review of Economic Studies (Symposium), pp. 3-28.

Erickson, E.W., and Spann, R.M. (1971), "Supply Response in a Regulated Industry: the Case of Natural Gas," The Bell Journal of Economics, 2, No. 1, pp. 94-121.

F.E.A. (1974), Project Independence Report, Federal Energy Adminis-tration (November).

Gordon, R.L. (1976), "A Reinterpretation of the Pure Theory of Exhaustion," Journal of Political Economy, 75, pp. 274-286.

Grinold, R.C. (1972), "Steepest Ascent for Large Scale Linear Programs," SIAM Review, 17, pp. 323-338.

Herfindahl, 0. (1967), "Depletion and Economic Theory," in Extractive Resources and Taxation, M. Gaffney (ed.), Madison, Wisconsin: University

of Wisconsin Press.

Hnyilicza, E., and Pindyck, R.S. (1976), "Pricing Policies for a Two-Part Exhaustible Resource Cartel," European Economic Review, 8, pp. 139-154. Hoffman, K.C., and D.W. Jorgenson (1977), "Economic and Technological Models for Evaluation of Energy Policy," Bell Journal of Ecomomics, 8,

No. 2, pp. 444-466.

Hotelling, M. (1931), "The Economics of Exhaustible Resources," Journal of Political Economy, 39, pp. 137-175.

Manne, A.S. (1976), "ETA: Model for Energy Technology Assessment," The Bell Journal of Economics, 7, No. 2, pp. 379-406.

Modiano, E.M. (1978), "Normative Models of Depletable Resoulrces," Doctoral Dissertation, M.l.T., Technical Rport No. 51, M. 1.1'. O()perltIo.lu; Rst-:carhl

Cent:cr (Apr1).

Shapiro, J.F., D.E. White, anid I).O. Wood (1977), "Sensitivity Antal sis

of the Brookhaven Energy System Optimization Model," OR 060-77, M.I.T. Operations Research Center.

Shapiro, J.F. (1977), "Decomposition Methods for Mathematical Programming/ Economic Equilibrium Energy Planning Models," OR 063-77, M.I.T. Operations Research Center.

Supply Models and the Brookhaven Energy System Optimization Model," OR 071-78, M.I.T. Operations Research Center.

Shapiro, J.F. (1978), Fundamentals of Mathematical Programming- John Wiley Publishing Co., New York, in press.

Wagner, H.M. (1975), Principles of Operations Research, 2nd Edition, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.

Zimmerman, MI.B. (1977), "Modeling Depletion in a Mineral Industry: The Case of Coal," The Bell Journal of Economics, 8, No. 1, pp. 41-66.