The Review of Regional Studies On The Economics Of Energy Allocation Merlin M. Hackbart, Charles W. Hultman, AND James R. Ramsey* The Arab petroleum embargo in late 1973, the gasoline shortage and rising fuel prices experienced by consumers in the last few years, and the difficulties encountered by some sectors in securing certain indus trial materials focused attention in recent years on the possibility of a serious energy shortage and consequent dislocations in the U.S. Econ omy. Furthermore, the "energy crisis" revealed the lack of coordinated state and federal policies for dealing with shortages and energy alloca tions. Market restrictions, such as the OPEC cartel which brought on the energy crisis, are possible in the future. Hence, policies and pro cedures for allocating fuel during energy cut-offs and shortfalls must be developed by national and state governments if impacts are to be minimized. During the 1974 petroleum shortfall, the federal government allocated a quantity of fuel to state energy offices for further distribu tion to industries within each state. In general, states did not have analytical procedures or formal policies for allocating the additional fuel. Purpose A difficulty in analyzing the impact of a fuel shortage on the state economy and establishing allocative procedures is the large amount of data which must he reduced to management proportions. One approach to this data management problem is the use of an input-output model. Such a model can he used to provide a substantial amount of informa tion about the economic impacts, both direct and indirect, of a fuel shortage on a state economy. In this study, the impact of a petroleum shortage on state industrial output and employment are estimated using such a model. Xhis paper includes a discussion of: (1) the procedures employed, and (2) the empirical results obtained from the application of the Ken tucky input-output model in combination with linear programming techniques to establish a petroleum allcication system which would minimize the impact of petroleum cutbacks for Kentucky. The pro cedures employed in the Kentucky study could be applied in other states and for other energy shortages, such as natural gas, if a suffi ciently disaggregated input-output model existed. *Associate Professor and Professor of Economics, University of Kentucky; and Assistant Pro fessor of Economics, Middle Tennesee State University, respectively. Volume 6, Number 1 99 Nature of the Modeh The Kentucky input-output table displays an array of interindustry flows of goods and services among sectors in the Kentucky economy. The flows depict industry or sector sales to all other industries or to final users (i.e., government, consumption, investment, and inventories). Consequently, all sector outputs are accounted for hy the I-O table. For this study, the Kentucky 1-0 table was converted to a model via the behavioral assumptions of production function stability, constant re turns to scale, and no substitution of production factors to carry out the petroleum supply cutback and allocation analysis.^ The 52-sector Kentucky model was aggregated to a model of 43 sectors for this analysis. The aggregation wa& necessitated due to the smallness of cer tain sectors and the lack of sector specific wage data required for certain calculations described in this paper. Formally, the aggregated dTsector Kentucky mode? can he defined as follows: Xi = Total production in industry i aij = Ratio of dollar of input used hy industry j and produced hy industry i Yi = Final demand in industry i Production in each Kentucky industry is either sold to final con sumers or bought hy other industries. Given this, an expression for industry i relating final demand to production is: X, - 2 aijXi =Y,. (1) j = l Collecting the XiS to form a vector X and Yis to give a vector Y and aijS to form a coefficient matrix A allows (1) to he written: X - AX = Y, (2) which can he further simplified to: (I - A)X = Y. (3) This expression gives all final demands Y, and the production necessary to deliver these Ys can be expressed: X = (I — A)-^Y. (4) Methodology The year 1974 was selected as a base year for analyzing the impact of a cutback in petroleum supphes on the Kentucky economy. By project- 100 The Review of Regional Studies ing an array of state final demands (Y) for 1974, total production (X) was computed by substitution into (4) above. Similarly, given (X), intersectoral flows were estimated by pre-multiplying the X vector times the coefficient matrix, A. The resulting meitrix represented Kentucky economy final demand and production transactions for 1974. In the sec tions which follow, the respective methodologies utihzed for estimating the employment and output impacts of a petroleum cutback as a guide for petroleum allocation policy utilizing linear programming techniques in combination with I-O are reviewed. Output impact. In the Kentucky Input-Output Model, row 22 of the A matrix represents the petroleum sector. (Consequently, each element of this vector represents a petroleum utilization coefficient as it repre sents the direct petroleum input per dollar output by sector. Multiplying these coefficients, aijS, times the respective sector output levels, the XjS, yielded a value C which can be represented as in (5): a22,iXi -|- 322,2X2 -|- ... 322,43X43 = C. (5) C, then, represents the total amount of petroleum (in dollars) directly required to produce the 1974 sector output totals. By changing the equality sign of (5) to a less-than-or-equal-to sign, the sum of oil usage was constrained to a level no greater than the total amount available. For analyzing the impact of a petroleum availability cutback on output, C was then assumed to be reduced to reflect the petroleum cutback. Notationally, the reduced petroleum availability can be represented as C and (5) becomes (6) reflecting the petroleum cutback. 322,1X1 -j- 322,2X2 -|- ... 322,43X43 = C' (6) A constraint was then imposed on each se<;tor independently so that constraint (6) was expanded into a series of one-element relations written as follows: 322,1X1 = Ci' 322,2X2 = C2' 322,43X43 = C43' From (3) it can be shown that given a final demand Y there is only one X which would solve the system. The hnear programming problem was initially written as foUows: Volume 6, Number 1 maximize Z = S Xi i = 1 subject ̂to: (I — A) X = Y (i) BX = C (ii) and X = 0, (iii) (8) where B is row 22 of A as described in (6). Here total production in the state (sum of the Xi) is maximized subject to the constraint of the input-output relations and the added constraint on oil supplies. In order to achieve a linear programming solution, it was also necessary to specify an upper limit on projected final demands. FDu represented the vector of 43 sectoral final demands. The new problem was to: maximize Z — 2 Xi i -=1 subject to: (I — A)X — Y = 0 (i) Y = FD„ (ii) BX C (iii) X 0 (iv) (9) where Y = the amount of final demand actually delivered. Solving the linear programming system (9), total production for the state, for each industry, and final demand delivered could have been estimated. However, a possible consequence of the absence of an alloca tion scheme for distributing oil, given a petroleum cutback and that total state production was to he maximized, was that entire industries could have been eliminated in the solution as far as supplying final demand. This situation was economically untenable hut was possible with the final demand constraint stated simply as: Y = FDa (10) Therefore, an alternative was to place a lower limit on the above con straint, such as: The Review of Regional Studies FDl = Y = FDo (11) where FDl was an assumed lower limit on final demands and FD„ repre sents the previously specified upper limits. The final step for estimating output impacts involved the development of dual activity variables or shadow prices. A dual variable shows the amount by which the objective function will change for a unit change in the appropriate resource constraint. The original solution variables were production figures, X, and final demand figures, Y. These numbers generally refer to the "columns" of the linear programming arrangement whereas the dual variables indicate the importance of the rows or con straints. In the construction of: maximize Z = 2 Xi i =1 subject to: (I — A)X — Y = 0 Y ̂ FDu (ii) BX ̂ C and X, Y ̂ 0 (iv) (12) the dual variables are associated with the 43 rows of line i, 43 rows of > line ii, and 1 row of line hi of (11). Specifying X, Y = 0 simply states that only zero or positive values are allowed in the solution. The concern of this study was with impacts of changes in oil supplies, therefore, of concern was the dual variables of line iii. It is important to note the difference between the system (12) with a constraint such as that in line iii and a similar system with the follow ing constraints: a22,iXi = Ci' 822,2X2 = C2' 822,43X43 = C43'. Volume 6, Number 1 103 Each component of the summation of line iii in (12) is now a separate constraint. Given the values for Ch, Cz', ... Cis', a modified system using all 43 rows of (13) was simultaneously solved. If the Ci's are binding then the dual variables define the expected change in each sector's output attributable to a marginal increase in the quantity of oil allocated to each sector (i.e., the marginal value of an additional unit of oil to each sector). More specifically, each element of the resulting vector of dual variables, 0, represents the expected change in each sector's output given a $1.00 reduction in the petroleum available to that sector or in dustry. Vector 0 is of dimensions 1 X 43. The dual variables were used in the empirical results which are summarized in a following section. Employment impact. In addition to output impacts as indicated by the dual variables, the input-output model was used to estimate the em ployment impacts of reductions in petroleum supplies. Using the pre viously defined notation, now let V = A value-added coefficient vector for Kentucky industries V' = A labor coefficient row vector V" = Residual value-added coefficient matrix. The value-added coefficients in the V vector represented the total value added (labor, capital, land, and entrepreneurship) by industry per $1.00 output. The vector was disaggregated into a (labor) vector and a V" (residual) vector. To disaggregate V, it was necessary to determine labor's share of value added.^ To find the appropriate labor share of value added in Kentucky, average factor shares for each aggregate sector (manufacturing, non- manufacturing, and utilities) were utilized.' Factor shares for these major sectors had been estimated in an earlier study and are summarized in Table 1. In disaggregating the V vector into Vh the appropriate labor share presented in Table 1 was multiplied by the V vector of value-added coefficients. The columns of the (I — A) ' matrix contain coefficients showing the direct and indirect requirements by industry for an expansion of final demand by $1.00 for each sector. TABLE 1 Allocation of Factor Shares By Industry Capital Manufacturing Non-Manufacturing Utilities Source: Harold K. Charlesworth and William G. Herzel, Kentucky Gross State Product, 1969, (Lexington, Kentucky: Office of Business Development and Government Services, 1972), p. 13. 104 The Review of Regional Studies Pre-multiplying (I — A)-^ by the vectoi- will yield a row vector of labor value-added multipliers; or (I - A)-^ where: L = Direct and indirect labor requiremeiits per dollar of output by sector Finally, the multiplication of each element in 6, the expected changes in each sector's output, given a reduction in petroleum supplies, times the corresponding sector element in the vector L, the direct and indirect labor requirements, will yield the change in each sector's output resulting from an oil cutback which can he attributed to labor. That is. ©jlj = Q (j = 1 . ■ • n) where: 6j = jth elements of vector 6 Ij = jth elements of the L vector Q = change in each sector's labor requirement resulting from a petroleum cutback Q can he transformed into an employment impact estimate in terms of numbers of workers by division by the mean wages for each sector for the given period or time. Or, N =- where: W = Average wages for each sector N = Employment impact Results The above-outlined procedures, equations 12 through 16, were used to estimate both the output and employment impacts of a given p^roleum cutback. To derive these estimates, an array of final demands (Y) were projected for the state for 1974. The projected final demand was $14,322 billion and total production was computed (by [4]) to be $18,081 billion. As previously indicated, these figures describe the situation in the state economy given no constraints. It should also be noted that the original cutback in oil supplies was arbitrarily assum(jd to be 12 percent. Given this constraint, total state production declines to $17,454 billion and final demand declines to $13,880 billion. Finally, note that the calculated Volume 6, Number 1 105 dual variables show the cost to the state in terms of output of a $1.00 reduction in petroleum to each industry. To avoid infinitesimal employ ment impact figures, the dual variable was arbitrarily converted to show the impacts of $1,000.00 reduction in petroleum supplies. The estimated output and employment impacts are presented in Tables 2 and 3, respectively. For example, as seen in Table 2, if the petroleum available to the cigarette and cigar industry (SICs 211 and 212) is re duced by $1,000.00 then the total output for the state will be reduced by $10,105,090. A rational allocative technique could be developed based upon the im pact the reduction in petroleum has on industrial output. Hence, given a reduction in petroleum supplies allocative efforts should be made to guarantee fuel supplies for those industries near and at the top of Table 2. It is these industries for which the loss in state output is the greatest for fuel cutbacks. At the same time, however, it must be remembered that the figures in Table 2 provide only a first general guide to the cost in terms of output of reducing the availability of petroleum to each of the state's industries. The possibility of other offsetting factors must be recognized. For ex ample, some industries might be able to absorb a small reduction in petroleum availability by using existing supplies more efficiently. Second, some industries initially affected adversely by the indirect impact of petroleum shortages might be able to secure essential input from outside the state. As shown in Table 3, the reduction in state employment resulting from a $1,000.00 cutback in petroleum supplies to the cigarette and cigar in dustry would be 234.74 workers. It can be concluded that the industry rankings presented in Table 3 offer a logical distribution procedure for allocating oil cutbacks. It would follow that an effort should be made to guarantee fuel supplies for those industries near or at the top of Table 3 since it is for these industries that the loss in emplojnnent is greatest if petroleum supplies are reduced. Conclusions The industry rankings presented in Tables 2 and 3 are, as would be expected, very close. In fact, the computation of a rank-order correlation coefficient shows a +0.91 relationship between the two different rank ings of impacts from the reduction in petroleum supplies.® A perfect correlation between the rankings would be indicated by +1.00. The results presented in Tables 2 and 3 must be considered tentative and interpreted with caution. First, the findings are subject to the data limitations of input-output analysis.^ Second, due to the openness of a state economy, it must be assumed that the interregional trade coeffi cients are stable.® Any change in trade patterns to supplement energy resources would, of course, mitigate projected impacts. The Review of Regional Studies Changes in State Output Given a 3)1,000 Change in Petroleum to Each Industry Output Change Per YearIndustry Cigarette and Cigar Manufacturing Tobacco Stemming and Redrying Amusements Communications Medical and Educational Services Business Services Insurance Lumber and Wood Printing and Publishing Apparel Finance Retail Trade Primary Metals Real Estate Hotels and Lodging Textile Mill Products Government Enterprises Electric, Gas, Water, and Sanitation Fabricated Metals Personal Services Wholesale Trade Stone, Clay, Glass Coal Mining Other Transportation Equipment Mining, Other Electrical Machinery Construction Paper and Allied Products Machinery Except Electrical Automotive Repairs Tobacco Motor Vehicle and Equipment Agriculture, Other Food and Kindred Products Professional and Scientific Instruments Motor Freight Transportation Paints and Allied Products Miscellaneous Manufacturing Plastics Furniture and Fixtures Chemicals Petroleum Products Leather Products $10,105,090.00 7,580,350.00 1,559,960.00 1,309,310.00 1,092,260.00 863,250.00 862,050.00 844,050.00 694,860.00 670,710.00 648,030.00 615,970.00 592,660.00 587,380.00 493,750.00 483,440.00 423,420.00 443,040.00 319,150.00 234,670.00 220,540.00 196,130.00 183,140.00 180,900.00 145,410.00 130,310.00 130,270.00 122,720.00 117,990.00 116,980.00 116,050.00 108,490.00 103,780.00 100,670.00 97,200.00 87,950.00 63,830.00 59,200.00 56,910.00 53,176.96 39,690.00 29,460.00 27,510.00 211, 212 214 79 48 80, 82 73 63 24 27 23 60-62 52-59 33 65 70 22 91-97 49 34 72 50, 51 32 12 372-379 10,13,14 36 15-17 26 35 75 0132 371 01,02 20 38 42 285 39 30 25 28 29 31 Charles Hultman, Tom Patrick, and James Watts, "Energy in the Ken tucky Economy: A Preliminary Report," 1974, (an unpublished paper), p. 46. Source: Volume 6, Number 1 TABLE 3 Change in State Employment Resulting from a $1,000 Change in Petroleum to Each Industry Industry Change in Employment Per Year 211,212 214 79 65 73 60-62 52-59 80, 82 48 63 70 91-97 24 23 27 50,51 22 0132 33 75 12 36 ^9 10,13,14 42 15-17 38 26 35 285 20 372-379 25 30 39 371 28 29 31 Cirgarette and Cigar Manufacturing Tobacco Stemming and Redrying Amusements Real Estate Business Services Finance Retail Trade Medical and Educational Services Communications Insurance Hotels and Lodging Government Enterprises Lumber and Wood Apparel and Other Printing and Publishing Wholesale Textile Mill Products Personal Services Fabricated Metals Electric, Gas, Water, and Sanitation Agriculture, Other Tobacco Primary Metals Auto Repairs Coal Mining Electrical Machinery Stone, Clay, Glass Mining, Other Motor Freight and Warehousing Construction Professional and Scientific Instruments Paper and Allied Products Machinery Except Electrical Paints and Allied Products Food and Kindred Products Other Transportation Equipment Furniture and Fixtures Plastics Miscellaneous Manufacturing Motor Vehicle and Equipment Chemicals Petroleum Products Leather Products 234.74 229.39 136.40 110.55 109.45 99.98 98.52 93.40 80.31 69.47 44.44 40.38 39.86 33.51 24.48 20.45 21.42 19.44 14.25 14.19 14.10 13.40 12.84 10.46 9.10 8.20 6.96 6.38 5.93 5.06 4.40 3.62 3.57 3.53 3.12 2.73 2.15 2.01 2.00 1.64 1.58 .84 .09 Given these cautions, the results presented offer two strategies for the allocation of petroleum: (1) allocation so as to maximize state industrial 108 The Review of Regional Studies output, and (2) allocation so as to maximize state employment. Alloca tion based on either strategy has much meirit. Yet while it may seem initially that theire could be no objections to allocations based on maximizing either employment or output, in fact, other priorities could be established. For example, an allocative priority system might be based on the nature of the output of the respective in dustries. Thus, it could probably be argued that a higher priority should be given to medical and educational services (SIC 80 and 82) than to the amusement industry (SIC 79) regardhjss of the impact on employ ment or output. The allocative scheme described, then, does not offer value judgments regarding which objective should be achie\'ed in coping with an energy shcmtage. It simply provides a rational pidority system to achieve an objective which has already been specified. In the end, the planning agencies must combine their perception of social needs and desires with such tools of analysis as input-output tables and linear programming in order to implement an acceptable allocative scheme. FOOTNOTES iSince the cost of constructing an input-out- put table employing primary data is generally prohibitive for a state, the present model is based on secondary data. The 1963 U.S. Input- Output table is the source of the basic data. The data have been modified by use of survey and interviews to reflect the methods of pro duction for Kentucky industries which vary considerably from those used in the U.S. Input- Output Model. =Slee W. Leontief, Input-Output Economics New York: Oxford University Press, 1966); and William H. Miernyk, The Elements of Input-Output Analysis (New York: Random House, 1965), for further discussion of these behavioral assumptions of input-output model ling. 3The Kentucky model uses 1969 data for gross outlays (obtained from state corporate income tax returns, the Internal Revenue Ser vice, and the U.S. Bureau of Census). For the non-manufacturing sectors, two assumptions were made: (1) production data are not sig nificantly affected by inventory variation ex cept in the trade sectors, and (2) receipts were considered to be production. Finally, it is noted that»the model has been adjusted to account for jDrice changes from 1963 to 1969. ■i.Previously, economic research has estab lished different weights for labor's share of value added in the production process. For example, Kendrick and Nelson in separate works assign 0.25 as the weight for capital's share and 0.75 as the weight for labor's share. However, Solow's research reveals a weight of 0.51 for capital and 0.49 for labor. See John Kendrick, Productivity Trends in the United States (Princeton: Princeton University Press, 1961); Richard R. Nelson, "Aggregate Func tions and Medium-Range Growth Projections," America n Economic Review, LIV, No. 4, (Sep tember 1964), p. 578; and Robert Solow, "Tech nical Change and the Aggregate Production Function," Review of Economics and Statistics, XXXIX., No. 3, (August 1957), pp. 312-320. (The Solow figures of 0.51 and 0.49 are taken from the Cheneny, Hollis, Arrow, Minhas, and Solow Jirticle in same journal, August 1961. However, this figure is based upon Solow's work wliich we are citing.) ■■•Harcld K. Charlesworth and William G. Herzel, Kentucky Gross State Product, 1969 (Lexington, Kentucky: Office of Business De velopment and Government Services, 1972), pp. 12-13. "Let, H!o: There is no relationship between the ranking of results presented in Tables 2 and 3. H!j: Reject the null hypothesis. > For this test of significance (n = 10), the rank correlation coefficient can be tested using the t distribution. For a 5 percent level of sig nificance, the critical t is approximately 1.684 and the computed t is 14.05; hence, the null hypothesis is rejected. "See note 1 above. "For a more detailed discussion of the stability of trade coefficients, see M. Jarvin Emerson, F. Charles Lamphear, and Leonard D. Atencio, "Toward a Dynamic Regional Ex port Model," The Annals of Regional Science (Deceml)er 1969), pp. 127-138.