UNIVERSITY OF ILLINOIS LIRRARY AT UR3ANA Ci. .ivlPAIGN ENGINEERING NOTICE: Return or renew all Library Mali each Lost Book is $50.00. !,The Minimum Fee for . 2 7 im The person charging this material is responsible for its return to the library from which it was withdrawn on or before the Latest Date stamped. below. Theft, mutila nary action To renew calQTc UNIVERSITY nd underlining of books are reas ns for discipli- Kersity. ILLINOIS LIBRARY AT URBANA-CHAMPAIGN MM L161— O-1096 _3 Supersedes Doc. 146 CONFERENCE ROOM "Proceedings of the 1975 Summer Computer Simulation Conference" San Francisco, Cal. ENGINEERING LIBRARY UNIVERSITY OF ILLINOIS A MODEL FOR ANALYZING ENERGY IMPACT OF TECHNOLOGICAL CHANGE UKBANA, ILLINOIS Clark W. Bullard III and Anthony V. Sebald Center for Advanced Computation, University of Illinois, Urbana, Illinois ABSTRACT This paper describes the development of a linear model of the U. S. energy system. It is basically an input-output model that is tailored specifically for an- alysis of energy related problems. The most significant feature is the development of a set of ficticious "energy product" sectors which de- fine nonsubstitutable end-uses of energy. Thus instead of consuming fuels, the industrial sectors consume energy products (e.g. space heat, air conditioning, etc.). The advantage of this formulation is that it is no longer necessary to specify the production functions of each of the economy's many sectors to reflect fuel substitu- tion possibilities. Technological changes associated with fuel substitution are localized in a small sub- matrix of the model instead of the entire rows corres- ponding to the energy sectors. Next we examine the sensitivity of total energy demand to three types of technological change: changes in energy supply technologies, energy utilization efficiencies, and substitution among non-energy inter- mediate inputs. The coefficients describing these technologies fall in three distinct partitions of the matrix. Within each partition, the technological co- efficients are ranked according to their importance relative to arbitrarily specified policy variables. Finally, the accuracy of the model is analyzed by evaluating output tolerance for various values of the tolerance on its parameters. 1. INTRODUCTION Most applications of the large scale input-output model of the U. S. energy-economic system [l], have been to assess energy impacts of changes in demands for final system outputs (e.g. as would result from car-bus sub- stitution) [2-6]. Because of the necessary 1-0 assump- tion that the model's parameters* are constant, all early applications of that model were of an assessment, rather than predictive nature. Recently, however, the model has been expanded and modified to accommodate projected future values of certain parameters important for energy policy analysis [?]• Some of these parameters, technical coefficients that would change as a result of fuel substitutions, are essentially the design parameters for the U. S. energy production system. Using the results of other models specifically designed to determine future values of these parameters [8], the 1-0 model may be partially updated to reduce the uncertainty of predictive results. Updating the parameters specifying energy supply technologies is necessary in order to use a 1967 (the latest base year for which data are available) 1-0 model for predicting energy demand. However, it is not at all clear that updating only these parameters would be sufficient, for each of the model's 135,000 parameters affects the results directly or indirectly. Since each parameter specifies part of a production technology, each technological change (no cvvtter how obscure) has an energy impact . The purpose of this paper is twofold. First we review the rationale and development of the energy input-output model, with emphasis on the structural features that facilitate its applicability to problems of a predictive nature. Second we will discuss methods for quantifying the energy impact of techno- logical change and demonstrate that these techniques can be employed to identify a subset of the model's parameters whose accurate updating would be sufficient to reduce the uncertainty of the model outputs. Results of three calculations are presented in Section U . The examples were selected to demonstrate the use of the model for quantifying impacts of techni- cal change, identifying parameters to which model out- puts are most sensitive, and tightening output error tolerances . 2. MODEL DEVELOPMENT The model of ref. [l] defines and solves a system of N energy balance equations for each of the N sectors of the economy. Published data from the U. S. Depart- ment of Commerce allows implementation at a very detail- ed level, exceeding 360 sectors [9]. The model's deri- vation is described in ref s . [l, 10] where detailed results are presented. The model's parameters, elements o'f the matrix A of technological coefficients, are defined as follows: ij amount of output from sector i sold to sector .1 (2-1) unit output from sector J Our values differ from the published values of ref. [9] for the base year 1967 in two ways. First, outputs of energy sectors are expressed in physical units (Btu) rather than current dollar values to account for the fact that energy is sold to different sectors at mostly different prices. Physical data are preferable for all sectors, but are not available. Second, sector outputs are domestic outputs only; this facilitates use of the results for analyzing energy impacts of foreign trade policies. The energy costs of goods and services are given by the first five rows (corresponding to the five energy sectors) of the solution matrix (I-A)~l. They are designated by the 5xN matrix e expressed in units of Btu's of energy type k J required directly and indirectly to produce a unit of output from sector J for final consumption. The matrix (l-A)"l has the same units as A, shown in fig. 1 on the following page. * A matrix of coefficients fixing the production technologies of all sectors of the economy. The Library of the SEP 8 1977 University ot Illinois at Urbana-ChamDaign SEP 3 1977 The person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN E in EiiOIO SfcERCDVCTiW A L161 — O-1096 Supplies Industries Supplies Industries Btu Btu Btu T" Btu + Figure 1 . Matrix of Technical Coefficients consumption goods, to produce energy products. Also note that A^ p = by definition. It will he seen that none or these definitions are essential to the model; they may be relaxed later to accommodate special cases. They are described here only to highlight the relationships between the newly defined energy product sectors and the existing energy supply and industrial sectors. To add these new sectors, it was necessary to derive the ba9e year coefficients Apg and Apj. The method described in reference [ll] was based on overall energy product control figures from refer- ence [12] reconciled in each sector with actual fuel use data from reference [13]. As indicated in Table 1, energy supply sectors were expanded from 5 to 8 to distinguish technologies for electricity generation and to accommodate a coal gasification sector. 2.1 Fuel Substitution Parameters The most obvious drawback of this model was its inability to account for fuel substitution, and for the rapid changes in the technology of direct energy use brought about by recent substantial increases in fuel prices. Recognizing that fuels are highly substi- tutable for many purposes, we address that problem first by identifying a set of end uses: space heating, water heat, process heat, feedstocks, etc. To retain the advantages of the input-output formulation, we assume that while fuels may be substitutable, the end uses (called "energy products") are not. We accommodate this by adding a set of new sectors to this model, one corresponding to each energy product. The non-substi- tution assumption is reflected in the constant techno- logical coefficients (Btu's of energy product/unit output from any sector) . Supplies Products Industries Supplies Products Industries Btu Btu Btu Btu J_ Btu Btu T" FiKure 2 . Expanded Matrix of Tech nical Coefficients The A matrix corresponding to this new model is shown in Figure 2, where energy supply sectors (S) sell directly to the energy product sectors (P) which in turn distribute their outputs to the rest of the industries (I). Thus, by definition, there are no inputs of energy supplies to the industrial sectors; H = 0, and i instead. Since A represents " only current account transactions, cooperates with capital IP = because energy equipment rather than Supply Sectors Product Sectors Coal Ore Reduction Feedstocks Crude Oil & Gas Other Feedstocks High Btu Coal Gas Motive Power Refined Oil Miscellaneous Thermal Natural Gas Water Heat Fossil Elec. Space Heat Nuclear Elec. Air Conditioning Hydro Elec. j> Miscellaneous Electric Table I . Energy Supply and Product Sectors Notes 1. Electric supplies converted at 3*»13 Btu/kwh. 2. Motive power defined as energy at the drive shaft to allow for fuel and electric substitution within the model. 3. Miscellaneous thermal energy is that heat available for industrial processes or other uses. U. Water heat is that transmitted to the water. 5. Space heat and air conditioning measure heat transferred to or from the building. 6. Miscellaneous electric measured at the wall socket. Includes all nonsubstitutable uses of electricity (motors, lighting, etc.). In this new framework total direct and indirect requirements for energy supplies and products are given by the same equations derived in reference [1] for any given vector of final demands for goods and services Y. X = e Y (2-2) where e represents the supply and product rows of the matrix (i-A) -1 . These energy intensities are functions of A alone; the chief result of our redefinition of A is that parameters specifying fuel substitution techno- logies are now isolated in the small partitions A gs and A<^ . These coefficients account for fuel substi- tution and may be computed exogenously as a function of relative prices, import quotas, and other factors, using models such as that of ref. [8J. Digitized by the Internet Archive in 2012 with funding from University of Illinois Urbana-Champaign http://archive.org/details/modelforanalyzin146bull One might expect, in addition, that these fuel substitution parameters would be among the most impor- tant with respect to the energy cost of goods and services. This expectation follows from the fact that A is well-conditioned and its eigenvalues are all less than unity, so the energy intensities can be expressed as a series expansion (i-A)" =I+A+A +A" (2-3) In this equation the actual Btu content of the fuel appears in the first term, and the second term is the direct energy used in the last stage of the production process for each good or service. Non-energy techno- logies (such as substituting fiberglas for steel in cars) don't appear until the higher order terms. From this convergent series one might expect direct energy use technologies to be most critical, and that change in most non-energy technologies would be less important. This hypothesis will be tested in sec. U. 3. ERROR BOUNDS AJTD PARAMETRIC STABILITY As mentioned earlier, conventional 1-0 analyses are based on the assumption that all parameters, the matrix A of technical coefficients, are relatively stable over time. Since 1-0 data are typically six or seven years old when first published, and are only infrequently (once every five years) updated, some account must be taken of parametric uncertainty re- sulting both from measurement errors and from techni- cal change over time. The longer the planning horizon for the predictive applications of the model, the greater the uncertainty. In this section we first discuss methods for estimating maximum error bounds on the results given uncertainty in A. Then we describe a technique for identifying a subset of A where technical changes or parametric uncertainty could have the greatest energy impact. 3.1 Maximum Error Bounds 2.2 Direct Energy Use Parameters The parameters in the technological coefficient matrix that one would expect to be the next most important are A__ and Ap T , the energy product rows. These reflect the efficiency of direct energy use in economic production processes. Since stochastic error analyses are unwieldy for problems of this type and scale, we restrict our attention to estimates of maximum error bounds. A simple approach utilizing matrix norm analysis bounds the uncertainty on the inverse of a matrix due to an uncertainty in its elements as follows Direct energy conservation measures implemented at the point of use, building insulation for example, can reduce the requirements for space conditioning represented by the coefficients A_ T in the production functions of each industry. Similar options exist for each of the other energy products. These coefficients could change as a result of changes in the price of energy relative to capital or labor, or from a variety of conservation policies such as investment tax credits, subsidized loans for insulation programs, etc. Evaluation of the impact of such changes on total energy demand or on energy inten- sities of particular products is straightforward, using an efficient, inexpensive updating technique based on a square root-free Givens method for solving the system of equations X = (I-A) -1 Y as described in reference [lU], (2-»*) l«C| C" 1 -(C+6C)" 1 I C 1-M provided «C < 1 JicLL (3-1) where M = the condition number of C = | |c| | ■ | |C~ | | . Eq. (3-1) says in effect that given a percentage (in the norm sense) perturbation on C, the resulting percentage perturbation on C"-^ will be less than or equal to M times as large provided that ||c|| • ||6c|| is small. As demonstrated in section h, this technique gives a very loose upper bound. An example will suffice here to show how the triangle inequality on which (3-1) is based could produce overly conservative results. 2.3 Changes in Non-Energy Technologies Certain other changes in non-energy technologies, elements of A and A TT , may also substantially affect energy demands or intensities. As an example, observe that substitution of fiberglas for steel in auto manufacturing actually causes a shift in energy demand from coal to oil. This is because steel is coal-intensive and fiberglas is oil-intensive. Technological changes of this type are usually accompanied by others in the same industry, or column of A. For example, the reduction of steel use would call for less welding and more epoxy bonding, which would alter other elements of the auto sector column. 5 Let A = 200 1 B = .1 ab| 1 < ||A|| ■ ||B| 20 < 200 • 1 This is clearly a very conservative bound. Whether these technological changes, which affect only the higher order terms in eq. (2-3), are negli- gible depends on how they enter into the problem- specific importance functions defined in the next section. A much tighter bound on the inverse uncertainty may be obtained using the following procedure, which involves creation of two perturbed A matrices, one which is the perturbation causing the greatest possible increase (i.e. positive tolerance) on all elements of (l-A)~l and the other which causes the greatest negative tolerance on (i-A) . The inverses of these two matrices give the worst case plus and minus element-by-element tolerances on (l-A)~ . It can be proved [15] that for a wide spectrum of 1-0 models, these perturbed A matrices are easily created by per- mitting all elements of A to assume their maximum (minimum) values simultaneously. This gives a much tighter worst case bound than the norm bound, but it must be remembered that tightness of the bound will depend on the likelihood that all elements might be in error in the same direction simultaneously. 3.2 Identification of Important Parameters To tighten the error bounds on results of any application of the model, one could update some or all of the model's parameters to reduce uncertainty. Using a method developed in ref. [15], we will define importance with respect to an 'importance function' of the general form: J = f [A(I-A)" 1 , Y] where J may be a scalar, vector, or matrix expression of the problem solution, and is of order less than or equal to that of (l-A) -1 . The notation A(l-A)- 1 repre- sents a perturbed inverse matrix, resulting from perturbations of the parameters A. After specifying an uncertainty level on elements of A, one can evaluate the resulting AJ using the Sherman and Morrison relation [l6]. A parameter k^^ is said to be important if its uncertainty causes some element of AJ to exceed a prescribed threshold. Its importance with respect to the entire J is quanti- fied by JI v where x mn mn it | v AJ where t is the =7 applicable importance thres- \ hold and v >_ 1. otherwise. Note that a brute force application of the Sherman- Morrison relation will not suffice for large 1-0 models. If m, n, i, J e{l,2. . . ,370} , 1.9 x 10 10 tests would be required. Ref. [15] presents an efficient method for reducing the number of these tests needed to identify all i, J, m, n for which v >. 1. !». EXAMPLES The model presented here is intended for predicting future demands for energy supplies and products as a function of magnitude of the GNP, the market basket of goods and services comprising it, and technology. While the overall energy demand may be predicted using simpler models, demand for specific energy supplies may not. Shifts among requirements for various types of energy supplies are highly sensitive to changes in technologies related to fuel substitution: changes that might be induced by supply constraints on certain resources, environmental regulations, taxes, or subsidies. We shall first present a calculation showing the magnitude of such changes over a relatively short period of time. Then we focus on demand for a parti- cular form of energy - electricity - and identify the parameters in the model to which electric demand is most sensitive. Finally, we use the model to predict demands for energy products and discuss the implications of accuracy in parameter estimation and the effects of technological change on the levels of these demands. All results presented here were obtained using a version of the model aggregated to 101 sectors , because more detailed data on energy products are only preliminary. However, all algorithms have been verified on similar models at the full 360-sector detail. l».l Requirements for Energy Supplies Estimates of the technology for converting energy supplies to energy products in 1969 and 1985 (projected) were available from ref. [17]. These were derived from actual and expected values of supply, demand, and technological constraints acting on the U.S. energy supply system. These estimates were used to update the A<,„ and A partitions of the matrix of technologi- cal coefficients of our model. The total energy supplies required directly and indirectly to produce the actual 1967 GNP were then computed. This standard 1967 bill of goods (the latest available) was chosen so the results in Table 2 would reflect only evolution in the technology of producing energy products from energy supplies. ENERGY SUPPLY 1969 1985 1969-1985 INCREASE Coal 12,250 12,620 3% Crude Oil & Gas 1*9,080 1*5,1*60 -1% Hi Btu Coal Gas 1*90 — Refined Oil 2** , 390 20,760 -15? Natural Gas 19,630 20,070 -2% Fossil Electric 3,6lU 2,781 -23* Nuclear Electric 39 1,395 3,1*77? Renewable Elect. 772 588 -2U* Table 2. Energy Demands: 19 69 vs . 1985 Energy Technologies 12 (Units: 10 Btu) It can be seen from the table that if the 1985 optimal energy supply technology were in place in 1969, consumption of various 'energy resources would have been substantially different. To accomplish a shift of this magnitude in the intervening sixteen years will evidently require that the nuclear power industry expand much more rapidly than the rest of the sectors, dominating an overall 10? increase in size of the electricity share. If capital shortages, safety or environmental problems , or other factors should act to retard expansion of the electric industry, conser- vation policies may be needed to close the gap between supply and demand. The effects of such electricity- conserving technical changes on electric demand are evaluated next . !*.2 Technical Change to Conserve Electricity Here we select electricity demand as our importance >-l function, J_: L J„ = f [ A(T-A) -1 , Y] J r = u-(l-A) _1 Y = u-X 1 i = electric utilities elsewhere (l*-l) where u is a row vector which extracts and sums the total requirements for outputs from the electric utility sectors. Using the methods of sec. 3.2, the parameters to which this scalar function is most sensitive were determined. Table 3 ranks the most important technolo- gical coefficients in each of the major partitions of the matrix. Some of the non-energy technical coeffi- cients are more important 'by this criterion than many direct energy use technologies, contrary to intuitive expectations (e.g. livestock + food is more important than air conditioning -* wholesale and retail trade). Table 3 identifies those technical coefficients where relatively small percentage changes result in the largest payoffs in electricity conservation. The tech- nical feasibility of such changes and the likelihood of implementing them through various policies must be evaluated independently. The results here simply identify those areas where the feasibility of such policies should be examined. Energy Supply Parameters fossil electric hydro electric fossil electric fossil electric fossil electric fossil electric fossil electric refined oil + misc . electric ■* misc . electric -» air conditioning ■+ fossil electric including transmission losses) + water heat ■* cooking and refrigeration + space heat ■* motive power Energy Use Parameters misc. electric ■» chemicals misc. electric ■*■ primary nonferrous metals misc . electric -► primary iron and steel misc. electric -*■ wholesale and retail trade air conditioning -► wholesale and retail trade misc. electric ■* medical, educational services misc. electric -► food misc. electric ■* paper Non-Energy Parameters livestock ■+ food chemicals ->■ plastics stone and clay products -► new construction fabrics * apparel heating equip. -*■ new construction chemicals ■♦ grain agriculture nonferrous metals -*■ new construction printing * business services Table 3- Parameters (Of Various Types) To Which Total Electric Demand is Most Sensitive planning horizon. It would be prohibitively expensive to accurately update and project future values of the more than 10,000 parameters of our 101-sector model; efforts must be concentrated on a few of the most important parameters. To help identify the point of diminishing returns (e.g. how many updated parameters is enough?), expected errors in the absence of para- metric updating must be compared with those afterward. Below we describe a method for doing this based on analysis of maximum error tolerances. Let us assume the nominal values of the technical coefficients over the entire prediction period are given by the base year values , and let the uncertainty on all parameters be ± 10? over the time period of interest. We compute the Leontief inverses of the perturbed matrices and postmultiply by the actual base year final demand vector. We find that the perturbed matrices indicate only that the exact electric demand lies within an interval of +30. U? and -23. ** 96 about the nominal value. While this tells us much more than the condition number criterion, such error bounds are totally unsatisfactory for policy purposes. If, however, we focused our attention on updating the most important 2% of the model's parameters, and were able to predict them 'exactly', our uncertainty would be reduced. To quantify the extent of that reduction, we set these parameters to their nominal values while perturbing the other 98? to their maximum upper (lower) bounds, and computed a new solution. The resulting 'predictions' of electric demand tightened the interval to only *-U.J% and -U.2? about the nominal. This seems to be a quite satisfactory range of uncer- tainty in view of the rather conservative assumptions implied about the distribution (maximum upper bound) and additive nature of the assumed perturbations. SS Those important parameters lying within the A, and A partitions of the A matrix could be esti- mated with considerable certainty using specialized models such as the Brookhaven model, and recognizing that the long lead times associated with construction of energy supply facilities make predic- tions over a 10-year planning horizon relatively straightforward. Certain conservation options may be implemented between now and 1985 and could signifi- cantly affect total energy demand at that time. These should be explicitly recognized as these key parameters are updated. The example described above was quantitatively unrealistic to the extent that the updated parameters cannot be known with absolute certainty. It shows, however, that confidence in the results can be sub- stantially increased by updating only a very small fraction of the model's parameters. A budget to obtain a 'best estimate' could be most effectively spent in the manner dictated by these results. U .3 Demand for Energy Products The model may also be used to estimate future demands for electricity, given the fact that the model's parameters are subject to some uncertainty over the Let J be a vector importance function repre- senting the eight elements of the total require- ments vector X = (T-A) - -'- Y representing demand for each of the energy products. The 1-0 model may be used to estimate future values of this importance function. * Standard Monte Carlo-type statistical analyses of these problems are infeasible due to the size of the matrices and the nonlinear! ty of the matrix inversion step. The condition number of (I - A) is !*9. This merely assured us that our 10? parametric errors might be magnified up to 1*9 times! which in turn are input to the energy system optimi- zation model of ref. [8] to determine optimal values of the parameters defining energy production technologies. Solution of the combined 1-0 and LP models proceeds in an iterative fashion, with energy supply system parameters and demand constraints (respectively) being updated at each step [7]. Here we shall examine the sensitivity of the importance function J p to changes in key technological coefficients. As in the previous example, the base year final demands are used to estimate the eight elements of the X vector corresponding to energy product demands. With all technological coefficients perturbed to their maximum upper or lower bounds (± 10? as in all examples discussed here), the intervals for the results averaged about + 18?. After identifying the most important 2% of the technical coefficients and holding them to their nominal values, the average was reduced to about + V. Results for each of the eight energy service demands are shown in Table U. Upper limits of interval for the +10? perturbation are shown; lower limits are smaller in all cases) . Coke Other Feedstocks. Motive Power Process Heat Water Heat Space Heat Air Conditioning. Misc. Electric. . . Uncorrected Corrected 32 27 8 30 5 9 12 20 Table U . Energy Service Requirements : Maximum Upper Bound6 (Percent) Before and After 'Correction' of Most Important Parameters The most important parameters were determined using the methods of section 2, and the importance function J p implicitly weighted all eight elements equally. The eight elements could have been differentially weighted if the analyst wanted to identify a different set of parameters that would further tighten the intervals on certain elements. For this specific application, specifying constraints for a linear programming model, a weighting scheme based on the LP shadow prices would perhaps be appro- priate. The shadow prices finally resulting from the complete iterative solution of the 1-0 and LP models with updated coefficients would certainly be different from the roughly estimated weights initially employed, but these could be altered a posteriori if the resulting error bounds on the combined solution were unacceptable. 5. SUMMARY An energy input-output model has been modified to facilitate its utilization for predictive applications. This was accomplished by defining a set of ficticious 'energy product' sectors corresponding to nonsubstitu- table end uses of energy. One advantage of this new formulation is that it is no longer necessary to specify the production functions of the economy's many sectors to reflect fuel substitution possibilities. Parameters relating to fuel production and substitution technolo- gies are now localized in a small submatrix in a form compatible with the outputs of other models specifically designed to project their future values. Methods were presented for identifying these and other parameters to which model outputs were most sensitive. Energy impacts of technological changes were quantified and example calculations demonstrated that prediction uncertainty could be reduced by as much as a factor of five through selective updating of a small (2?) subset of the model's parameters important to particular problems. ACKNOWLEDGEMENT This work was supported by the Energy Research and Development Administration (ERDA) . We thank Dr. Lee Abramson of ERDA for his helpful comments. REFERENCES [l] C. W. Bullard and R. A. Herendeen, "Energy Cost of Consumption Decisions", Proc. IEEE, March, 1975. Also available as Document 135, Center for Advanced Computation, University of Illinois at Urbana-Champaign, Urbana, Illinois 6l801. [2] B. Hannon and F. Puleo, "Transferring from Urban Cars to Buses: The Energy and Employment Impacts," paper presented at the International Symposium on the Effects of the Energy Shortage on Transporta- tion Balance, Pennsylvania State University, May, 197 1 *. Also CAC Document No. 98. [3] R. Herendeen, "Affluence and Energy Demand" presented at the 9^th Winter Annual Meeting of the American Society of Mechanical Engineers, Detroit, November, 1973 (paper 73-WA/ENEP-8) . To be published in Mechanical Engineering . Also available as Document No. 102, Center for Advanced Computation, University of Illinois at Urbana- Champaign, Urbana 6l801, July, 1973. [h] R. Bezdek and B. Hannon, "Energy, Manpower, and the Highway Trust Fund," Science , Vol. 185 , p. 669, August, 197 1 *. [5] C. Bullard and R. Herendeen, "Energy Use in the Commercial and Industrial Sectors of the U.S. Economy, 1963", Center for Advanced Computation, University of Illinois at Urbana-Champaign, Urbana 618OI, Document No. 105, November, 1973. [6] C. Bullard, "Energy Conservation Through Taxation", Center for Advanced Computation, University of Illinois at Urbana-Champaign, Urbana, 6l801, Document No. 95, February, 197 1 *. [7] C. W. Bullard, "An Input-Output Model for Energy Demand Analysis", Center for Advanced Computation, University of Illinois at Urbana-Champaign, Urbana 6l801, Document No. ll»6, December, 197 1 *. [8] E. Cherniavsky, "Topical Report: Brookhaven Energy System Optimization Model", BNL-19569, Department of Applied Science, Brookhaven National Laboratory, December, 197**. [9] "Input-Output Structure of the U.S. Economy V. 1-3", U.S. Dept. of Commerce, Bureau of Economic Analysis, Washington, D.C., 197 1 *. Available from U.S. Government Printing Office. [10] C. W. Bullard and R. Herendeen, "Energy Cost of Consumer Goods 1963/67", Center for Advanced Computation, University of Illinois at Urbana- Champaign, Urbana, 618OI , Document No. l^O, November, 197 1 *. til] R. L. Knecht and C. W. Bullard, "End Uses of Energy in the U.S. Economy, 1967", Center for Advanced Computation, University of Illinois at Urbana-Champaign, Urbana, 6l801, Document No. 1^5, January, 1975. tl2] "Patterns of Energy Consumption in the U.S." a report to the U.S. Office of Science and Technology by Stanford Research Institute, Menlo Park, California, 1972. [13] D. Simpson and D. Smith, "Direct Energy Use in the U.S. Economy, 1967", Center for Advanced Compu- tation, University of Illinois at Urbana- Champaign, Urbana, 618OI, Technical Memorandum No. 39, December, 197**. [ll»] K. Noh and A. Sameh, "Computational Techniques for Input-Output Econometric Models", Center for Advanced Computation, University of Illinois at Urbana-Champaign, Urbana, 6l801, Document No. 13**, September, 197*1. [15] A. V. Sebald, "An Analysis of the Sensitivity of Large Scale Input-Output Models to Parametric Uncertainties", Center for Advanced Computation, University of Illinois at Urbana-Champaign, Urbana, 618OI, Document No. 122, November, 197 1 *. [l6] J. Sherman and W. J. Morrison, "Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix", Annals of Mathematical Statistics , vol. 21, pp. 12U-127, 1950. [17] M. Swift, "Reference Energy System Supply Technologies: 1969 and 1985", Brookhaven National Laboratory, private communication, January, 1975.