FINAL REPORT TO NATIONAL COUNCIL FOR SOVIET AND EAST EUROPEAN RESEARCH

Transcription

1 FINAL REPORT TO NATIONAL COUNCIL FOR SOVIET AND EAST EUROPEAN RESEARCH TITLE: MEASURING RELATIVE USA AND USSR DEFENSE SPENDING USING TRANSLOG INFORMATION FUNCTIONS TO OBTAIN "TRUE" INDEXES AUTHOR: Paul R. Gregory James M. Griffin CONTRACTOR: Transecon, Incorporated PRINCIPAL INVESTIGATOR: Paul R. Gregory and James M. Griffin COUNCIL CONTRACT NUMBER: DATE: March 15, 1982 The work leading to this report was supported in whole or in part from funds provided by the National Council for Soviet and East European Research.

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3 Table of Contents Abstract Statement of the Problem A Two-dimensional Graphical Example Theoretical Implementation of Productions Possibilities Frontiers Empirical Estimation of the U.S. Defense Transformation Function Calculating "True" Indexes from the Translog Function Empirical Results Conceptual Problems and Interpretations Appendix A: An alternate Approach: Duality and the Translog Revenue Function Appendix B: TSP Programs for Estimating Translog Transformation

4 ABSTRACT Official estimates of relative military spending in the USA and USSR show that the USSR in recent years has spent some 50% more per annum than the USA when the defense outlays of both countries are valued in U.S. prices. When Soviet ruble prices are used, Soviet defense outlays are 30 percent above the U.S. These measures are designed to quantify the relative ability of one country to produce the defense output mix of the other country, but they contain an unknown degree of bias. The bias is introduced by the assumption implicit in these calculations that the one country is capable of transforming its defense output into the mix of the other country at fixed rates of transformation (defined by the price parameters of one or the other country). For this reason, it is presumed that established measures will overstate the ability of one country to produce the output mix of the other. The degree of bias can be estimated if the production possibilities frontier is known. Given the limitations of the Soviet data, we have worked only with estimating the U.S. defense transformation function with the objective of determining the value of U.S. defense output if the U.S. were constrained to produce the Soviet output mix. This figure, when compared with actual Soviet defense outlays valued in dollar prices, gives a "true" index of the relative ability of the U.S. to produce the Soviet output mix. We have estimated the U.S. defense transformation function from actual U.S. data. Our calculations show that, as expected, the average established estimates overstate the U.S. ability to produce the Soviet output mix. The percentage overestimate is between 1,6% and 3,7% depending upon whether the National Foreign Assessment Center or the Rosefielde estimates - 2 -

5 ABSTRACT of the Soviet mix are used. This adjustment is not very large, and it appears to confirm the Bergson-Moorsteen prediction that the bias will not be great because the Soviet output mix is being evaluated in terms of U.S. relative prices. The adjustment falls well below the maximum error interval (15%) that the CIA attaches to its own estimates for this period. These results should be regarded as preliminary and should be interpreted with caution. Our most important reservation is that our theoretical model is based upon the assumption of cost-minimizing behavior. If costs are minimized, the product transformation curve should have the expected quasiconvex curvature. Our tests of the properties of the estimated translog product transformation curve yielded some evidence of non-cost minimizing behavior. Econometrically speaking, the U.S. defense product transformation curve is not "well behaved" in all outputs. Second, we have worked at fairly high levels of aggregation (five outputs), primarily for econometric reasons. It remains to be seen what would happen if the model were estimated at lower levels of aggregation. Third, we are unable, due to data limitations, to perform the same calculation for the USSR; namely, to calculate the relative ability of the Soviet Union to produce the U.S. output mix, Presumably, established measures overstate the USSR's ability to produce the American output mix. Just as our measures increase the superiority index in dollar prices (from 40 to 42-45%) so would this reverse calculation reduce the Soviet superiority index in ruble prices. If one had two "true" indexes of the ability to produce the other country's mix, then defense planners could choose that one that they regarded as the most relevant for comparative analysis. have accepted the current and constant price U.S. data as correct. Fourth, we We have not - 3 -

6 ABSTRACT sought to make adjustments in this data set. We cannot rule out the possibility that our results are due to problems in these data. Despite our reservations, this technique suggests a new approach to making comparisons of US and USSR military spending. We suggest that future research in this area focus on the following issues: 1. The introduction of non cost minimization behavior into theoretical modelling. 2. Investigation of the behavior of the bias measure at lower levels of aggregation. To what extent to our findings reflect a high level of aggregation? We assume that disaggregated data are available; the problem would be in developing econometric procedures that handle a much larger number of outputs. 3. Estimation of the Soviet product transformation curve to determine the Soviet ability to produce the U.S. defense mix. We do believe that our research does yield one strong conclusion: we believe that we have ruled out the possibility that corrections for diminishing marginal rates of transformation are large with respect to the U.S. ability to produce the Soviet mix. The correction we obtain is well below the suggested margin of error applied to the CIA's calculations that assume constant transformation rates. This does not appear to be a correction we have to worry about in such comparisons

7 defense A STATEMENT OF THE PROBLEM The conceptual problems of assessing the relative magnitudes of Soviet efforts and the effectiveness of U.S. and American forces are discussed in research papers by the National Foreign Assessment Center: Estimated Soviet Defense Spending: Trends and Prospects and A Dollar Cost Comparison of Soviet and U.S. Defense Activities, (June 1978 and January 1981 ). The basic conceptual problem is the well-known phenomenon of index number relativity discussed by Abram Bergson, Richard Moorsteen, G.W. Nutter, and Herbert Levine. 1 Discussions of these problems in the context of Soviet-American comparisons are also available in the literature. According to the National Foreign Assessment Center's calculation (of January 1981), Soviet defense outlays in 1980 were 30 percent above those of the U.S. in Soviet ruble prices but were 50 percent 2 above the U.S. in U.S. prices. Franklyn Holzman has argued that the Foreign Assessment Center has understimated the effect of index number relativity- an 3 assertion challenged by Steven Rosefielde. The Holzman-Rosefielde debate focuses on computation problems and on the conceptual problem of index number relativity. relativity will The important issue of aggregation and its effect on index number not be dealt in this report. As Bergson and Moorsteen have shown, 4 substantial biases could be involved in such measures, for they employ the unrealistic assumption that the USA, for example, is capable of transforming its current mix of physical defense outputs at fixed rates of transformation, into that mix currently being produced in the Soviet Union, where the transformation rates are those implied by U.S. relative prices.* The reverse is true for comparisons in ruble prices: the * Bergson and Moorsteen demonstrate that this method (as opposed to using Soviet prices as transformation rates) reduces the bias in measured index numbers

8 established calculations assume that the Soviet mix of physical defense outputs can be transformed into the United States "composite commodity" at the constant rates of transformation dictated by relative ruble prices. In the language of production frontiers, established measures assume straight-line production possibilities frontiers for both the US and USSR. As students of index number relativity have demonstrated, serious biases can be introduced by assumptions of linear product transformation curves. The transformation of one country's output mix into the composite commodity of the other's actually proceeds at diminishing marginal rates of transformation; the use of fixed transformation rates (dictated by one country's price parameters) to assess the country's ability to produce the defense output mix of another country (the relevant standard) will yield biased measures. The only way to obtain a "true" measure of the relative ability of country A to produce the composite defense commodity of country B is to know how marginal transformation rates behave as the defense output mix changes. In our research we develop the analytical and econometric techniques for such an appraisal. The potential for bias is greatest when there are substantial differences in relative prices between the two countries and when the defense output mix is quite different. Both conditions appear to be met in the case of USSR-USA comparisons according to the National Foreign Assessment Center's calculations. Generally speaking, the relative prices of high technology products are lower in the U.S., and the relative prices of manpower and low technology products are low in the USSR. Stated in different terms, valuing Soviet defense expenditure in U.S. prices assumes that the Soviet Union would be capable of transforming its output mix from its relative emphasis on conventional forces, etc. into the - 6 -

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10 U.S. output mix emphasisizing tactical air and naval forces at transformation rates prevailing in the United States. Commonsense would suggest that substantial changes in output mixes would occur at diminishing marginal rates of substitution. A TWO-DIMENSIONAL GRAPHICAL EXAMPLE We supply a graphical example (figure 1 ) to illustrate the sources of bias in existing estimates of relative defense expenditures. The example is not intended to depict reality-- simply to indicate the importance of adjusting for variable marginal transformation rates. The example is two-dimensional and assumes that manpower is relatively cheap in the USSR relative to military hardware. R denotes the observed Soviet physical output combination.* U denotes the observed U.S. physical output combination. The curvature Of the U.S. product transformation curve (uu) suggests diminishing marginal rates of transformation as the U.S. alters its output to the existing Soviet mix (from U to A). The same is true of the USSR product transformation curve (rr), which indicates changing transformation rates as the USSR shifts its output mix to that of U.S. (from R to F). The location of each product transformation curve is determined by the supply of factors (X) available to each country and by the technology (T) at hand to transform the factors of production into outputs. While R and U can be * We must be clear about the meaning of "outputs" in this context. The ultimate "output" is "military power" and these two "outputs" are really inputs into the production of military power. "Military power", however, cannot be measured unless the social indifference curve is known; so we must deal in this study not with military power, but with those products that "produce" military power.

11 observed, X and T are not directly observable in this example. The various measures of relative defense outlay can be depicted using Figure 1. If various index number formulae are employed, the indexes will represent: (1) a) Soviet ability to produce the U.S. output mix using fixed U.S. transformation (U.S. prices): OE/OU b) Soviet ability to produce the U.S. output mix using fixed Soviet transformation rates (Soviet prices): OD/OU c) The "true index" of the Soviet ability to produce the U.S. output mix (allowing for diminishing rates of transformation): OF/OU (2) a) U.S. ability to produce the Soviet output mix using fixed Soviet transformation rates (Soviet prices): OR/OC b) U.S. ability to produce the Soviet output mix using fixed U.S. transformation rates (U.S. prices): OR/OB c) The "true index" of the U.S. ability to produce the Soviet output mix (allowing for diminishing rates of transformation): OR/OA As Bergson and Moorsteen have demonstrated, paradoxically, the index that measures the ability of A to produce B's output mix is closer to the "true" index when the prices of country A are used. Both OD/OU and OR/OB are less biased than OE/OU and OR/OC

12 As this example illustrates, the degree of curvature in the respective product transformation curves determines the degree of bias contained in the conventional index number measures. The bias can be substantial even if the indexes preferred by Bergson and Moorsteen (lb and 2b) are employed. Of course, the basic index number ambiguity remains. The two "true" indexes will yield different results (OF/OU will not equal OR/OA), but the biases have been removed. Both OF/OU and OR/OA are accurate measures of the ability of one country to produce the defense output mix of the other. Which output mix to accept as the standard for comparison is a judgment to be made by the users of the respective "true" indexes. THEORETICAL IMPLEMENTATION OF PRODUCTION POSSIBILITY FRONTIERS In order to determine the ability of a given country to produce the output mix of the other country, it is necessary to identify the production possibility frontier of at least one country. To calculate OR/OA, we must know the U.S. production possibilities frontier. To calculate OF/OU, we must know the Soviet production possibilities frontier (PPF). To compare the two "true" indexes, the PPFs of both countries must be known. For purposes of exposition (and later empirical implementation), let us consider the problem of deriving the PPF for the U.S. THEORETICAL BASIS Traditional economic analysis begins by positing the existence of a neoclassical production function (with given technology) relating the single output, Q, to the input levels of the n inputs, X i : 0) Q = f(x 1...X n )

13 The production function is, however, a special case of the transformation function which allows for both multiple inputs and outputs: (2) G(Q 1...Q m, X 1...X n ) = 0 In the case in which the inputs and outputs form weakly separable groups, equation (2) can be rewritten to express the output aggregate ( ) as a function of the input aggregate g( ): (3) (Q 1 Q m ) = g(x 1 x n ) A special case of (3) is that with two outputs (military hardware (H) and military manpower (M)) and one composite input, X: (4a) X = (H,M) or more generally: (4b) X = (Q 1..., Q m ) Equation (4a) can be used to describe the production possibility frontier shown in Figure 1. For a given level of input X, one can solve the various combinations of military hardware (H) and manpower (M) producible with resources X available. For different levels of input X, the production possibility frontier can be shifted inward or outwards. The product transformation function in (4) is in many ways analogous to the standard production function except the inputs and outputs are reversed. Instead of maximizing output for a given combination of inputs, as in the production function, the product transformation function elicits the minimum input necessary to produce a given output combination. The Translog Transformation Function The translog function offers a local approximation to any transformation function. For this reason, it is suited to the problem at hand because it places few restrictions on the properties of the transformation function to be

14 estimated. While other generalized functional forms offer similar properties, we have determined to use the translog because of its simplicity and widespread application to an array of problems. The translog transformation function for equation (4b) is written as Following previous researchers, we do not attempt to estimate equation (5) directly; rather, use is made of marginal productivity conditions. Partial differentiation of (5) with respect to the logarithms of outputs yields the following set of equations: Assuming competitive markets, the marginal cost of an additional unit of Q i. supplied at price P. is: (7) We must emphasize an important point; namely, this formulation is based upon the assumption of competitive profit-maximizing behavior. Its empirical usefullness is limited if this condition is not approximated in the real world. Substituting (7) into (6), we obtain the condition that the revenue shares (s i ) are linear functions of the logarithms of output. Thus given data on the revenue shares (S i ), or equivalently the expenditure shares for military products and their output levels (Q i ), it is possible to estimate empirically all of the parameters in equation (5) except α 0 Equation (8) sets

15 up a system of m equations. Each expenditure share is a function of the logarithms of its own output and of the outputs of the m-1 other products. Only m-1 of these equations are independent. The mth equation is therefore derived as a residual from the parameters of the other equations. A TWO-DIMENSIONAL ILLUSTRATION To illustrate the various conditions to be met by the translog transformation function, we use the simple two output model with H and M. The system of equations in (8) is: A variety of theoretical conditions must be met. First, the expenditure shares of hardware (S H ) and manpower (S M ) predicted by applying observed outputs to the α and β parameters must be positive in order for the marginal product of X to be positive in equation (7). Second, the symmetry conditions, which state that follow by assumption from the original translog formulation in equation (5). Third, the equality conditions follow from the accounting nature of the data whereby the cost of inputs must equal expenditures on H and M. Consequently, the intercept terms in (9) sum to unity and the column sums of the sum to zero. Combining the equality and symmetry conditions, we obtain: We know that S H + S M = 1; therefore:

16 Fourthly, it is particularly important that the transformation function enjoy the proper curvature property; namely, the marginal rate of transformation in Figure 1 be declining. Mathematically, this requires that equation (4) be convex in the two military outputs: manpower and hardware. Consequently, this condition requires that the Hessian matrix of equation (5) be positive semi-definite. EMPIRICAL ESTIMATION OF THE U.S. DEFENSE TRANSFORMATION FUNCTION The foregoing theoretical analysis demonstrates the applicability of the translog transformation function to estimating the U.S. defense product transformation curve. Once the parameters of this transformation functions are known, the "true' index of the ability of the U.S. to produce the Soviet output mix in a given year can be calculated. How this is done is elaborated in the next section. DATA We have estimated the U.S. defense product transformation curve from annual data on U.S. defense spending in current and constant dollars for the years 1949 to The data are drawn from official U.S. sources. The expenditures shares are disaggregated into five outputs: 1) MP, military personnel (without retirement pay), 2) 0 & M, ordinance and maintenance, 3) P, procurement (with and without military construction) 4) RDT&E, research and development, and 5) MC, military construction. Following equation (8), the annual expenditure shares -- of military personnel, procurement, RDTE, etc. in prices of the current year serve as the dependent variable in our model. Real expenditures ( in constant 1972 dollars)on each output serve as the independent variables according to the format of equation (8)

17 The data set therefore consists of 33 annual observations of expenditure shares and real outputs. This data set allows us to estimate the and parameters of the U.S. defense transformation function up to the constant In the course of this project, the U.S. defense transformation function was estimated in different variants. First, variants with and without retirement pay for military manpower were estimated with generally similar results, but we concluded that it is more appropriate to exclude retirement pay on the grounds that retirement pay did not contribute to the current output of "military power". In the four sector variants, military construction was added to procurement expenditures, In the five sector variants, military construction entered as an independent sector. In all cases, the last sector was estimated from the parameters of the other 3 or 4 equations because the final equation is not independent of the other equations (see equation 10). The terms were constrained so that = to enforce the symmetry assumption. The basic model (equation 5) does not allow for technological change or for scale effects, other than those that would be captured econometrically by generally rising real expenditures. Therefore, in addition to the basic model (equation 9), two additional models were estimated -- one that entered time (T) as a time trend reflecting technological change, and the other entering the scale of total real defense output (S) to capture scale effects, These models were all programmed using a modified version of TSP. (Appendix B). * For example, equation (5) becomes: when T (technological change ) is introduced

18 CALCULATING "TRUE" INDEXES FROM THE TRANSLOG FUNCTION We have on hand the National Foreign Assessment Center's estimates of the dollar value of Soviet defense expenditures. These estimates have been the object of considerable controversy -- a controversy that must remain outside the realm of this research project. We shall use the National Foreign Assessment Center's calculations in our illustrations: researchers who wish to substitute their own estimates of the dollar value of soviet defense spending may do so. Our task is to estimate the dollar value of U.S. defense outlays if the United States were to produce the Soviet output mix with its available resources. Thus, the theoretical standard towards which we are aiming is the OR/OA measure of relative defense spending. If the National Foreign Assessment Center's estimates of the dollar value of Soviet defense spending are accepted as accurate, we have already on hand the OR/OB measure of relative defense spending. Insofar as OR/OB is the measure with the least bias, we are interested in determining the amount of bias introduced by diminishing marginal rates of transformation. This bias is, in affect, OB/OA. THEORETICAL FORMULATION Once the American product transformation function is known ( the and parameters), the estimation of the OA vector is relatively straightforward: Returning to (5), we have: where the denote estimated parameters; the * denote actual real output in a given year, and equals Substituting actual Q*. values into (11) yields an estimate of the inputs (X) required to produce the vector of real outputs up to an arbitrary

19 constant This value is denoted as By definition, a product transformation function defines the alternate mixes of outputs that are producible from a fixed amount of inputs The question therefore is the following: If the U.S. were constrained to produce the defense output mix of another country -- the USSR -- without an increase in inputs, what are the maximum quantities of Q, Qm that could be produced? The answer is obtained by inserting the actual Soviet output mix of a given year into equation (11), holding inputs constant at and then solving for the Q.: where the denote the ratio of product i to product 1 in the actual Soviet output mix. Because is known and the actual Soviet output mix is also known (the ), equation (12) reduces to a simple quadratic equation with one unknown, Q,. Equation (12) can therefore be solved for Q,. Once Q, is known, the other Q. are known by multiplying Q, by the respective μ i. In this manner, we can solve for all the Q. in (12). What will these Q. represent? They represent the vector of real outputs that the U.S. would be capable of producing with input levels fixed if the U.S. were constrained to produce the Soviet output mix. In terms of Figure 1, these Q. are the outputs depicted by A in two-dimensional space. When multiplied by U.S, prices and summed, the total yields the value of U.S. defense spending in U.S. prices if the U.S. were constrained to produce the Soviet output mix. This value can then be compared to the actual value of U.S. defense outlays to obtain the comparison OB/OA and with estimates of Soviet defense outlays in U.S. prices

20 to obtain the "true" index OR/OA. A Two Dimensional Example To insure an appropriate understanding of our methodology, let us show how it works in two-dimensions (with outputs H and M). Using actual data on H and M, let us assume that we have estimated the necessary parameters of equation (11 ). Therefore we have: where the * represent actual U.S. values for a given year and the " represent estimated parameters of the transformation function. Equation (13) yields a value for -- the level of U.S. inputs in the given year. The actual Soviet output mix is then employed (M = μh) in equation (13): Equation (14) can be solved for H and then for M The H and M represent the physical outputs the U.S. could produce with given inputs if it were constrained to produce the Soviet mix The biases could be estimated directly: OB/OA = H*/H and OR/OA = H R /H, where H denotes the actual Soviet output of H and H* denotes the actual U. S. output of H. An Alternative View There is an indirect approach to this problem; namely, to estimate the changes in input levels required to produce different output mixes. Stated differently, for the U.S. to produce output vector OB would require a greater level X than - 18-

21 necessary to produce OA. The output vector OB lies outside the product transformation curve that goes through U. OB could not be produced by available inputs. We illustrate this indirect procedure in terms of two dimensions, H and M. The derivation of the vector OB is quite simple. Let us measure the output of the Soviet mix (Z) as a fixed-proportions relationship: Where and are fixed parameters reflecting the relative proportions of H to M for any given total output Z. The point in Figure 1 satisfies the condition that the output combination at OB can be produced at U.S. prices at actual U.S. expenditures (C ). Because we can solve for in equation (16) and then for. The values and indicate the quantities of M and H the U.S. could produce if it could transform its output mix to the Soviet mix at fixed (U.S.) transformation rates. The indirect procedure calls for calculating the ratio of inputs necessary to produce vector OB relative to vector OA. First, using calculate: This expression gives the logarithm of inputs necessary less a constant, to produce the output mix at point Note that by the argument above, a similar expression is obtained for vector OA which is obtained by inserting the actual U.S. values in the translog transformation function. The bias can be

22 obtained as follows: The appropriate interpretation of (18) is as follows: measures the inputs required to produce A. Point B lies outside the original PPF; therefore more inputs are required to produce B; namely, The bias is therefore the difference between the two input levels. Note that α 0,the unknown parameter, subtracts out. The direct and indirect procedures should yield identical results. The direct procedure estimates A and compares A with U. The indirect procedure determines the resources necessary to produce B and compares this input level with the resources required to produce U. In the first case, the magnitude of the shift in iso-expenditure curves is being measured. In the second case, the magnitude of the shift in the product transformation curve is being measured. Both are equal upward shifts from A to B. EMPIRICAL RESULTS Regression Results Our calculations of bias are based upon the five sector defense model. We have estimated this model econometrically from U.S. time series data for the year 1949 to 1981 with the fifth sector coefficients (military construction) calculated as residuals from the properties of the translog function. The estimated coefficients of the U.S. translog product transformation curve are given in Table 1:

23 Table 1: Estimated Coefficients of the U.S. Translog Product Transformation Function Coefficient T-Value Coefficient T-Value Tests of the properties of the estimated translog transformation curve include tests for nonnegativity and tests for quasi-convexity. The first test rules out negatives marginal products, and the second test insures that the transformation function has the proper curvature properties. Our tests for non-negativity show that there are no cases of negative marginal products, but the estimated translog does not pass the test for quasi-convexity. An intuitive test for curvature properties is to check the signs of the own-price supply elasticities, which should be uniformly positive. We have found, however, numerous data points in which the own-price elasticities of procurement and research and development are negative, on the other hand, the own-price elasticities of the other three sectors are uniformy positive. However, the existence of perverse own- price elasticities is sufficient

24 evidence to call into question the curvature properties of the estimated translog production transformation funtion and to question the underlying assumption of cost minimization in all sectors. The estimated function also failed to pass the formal positive-semidefinite test. The product transformation function is not "well behaved" over the full sample range. Bias Estimates The bias estimates that are calculated from the estimated U.S. product transformation curve must be treated with caution because of the questionable curvature properties of the estimated translog function. We have estimated the bias in existing estimates of the U.S. ability to produce the Soviet output mix using the direct and indirect procedures outlined above. Conceptually, the two methods should yield identical results and they do in reality. The first method calculates in step 1 the resources required to produce actual U.S defense outputs (in constant dollars) by plugging these actual values into the estimated translog function. Then holding inputs constant, we calculate in the second step (by inserting the Soviet output mix) into the estimated translog function, the hypothetical outputs the U.S. could have produced if constrained to produce the Soviet output mix. The indirect procedure inserts the hypothetical U.S. outputs that would have been produced had the U.S. been able to convert to the Soviet output mix at fixed U.S. transformation rates into the estimated translog function This procedure yields the amount of inputs that would have been required to produce the hypothetical constant-transformation outputs. This amount is then

25 compared to the actual amount to determine the increase in inputs necessary to produce the constant transformation outputs. Both procedures require the actual Soviet output mix. In this study, we have used the output mix reported by the National Foreign Assessment Center in its January 1981 publication (p,11) ). As an alternate, we have used Rosefielde's output mix which increases the weight of Soviet procurement. Our results are shown in Table 2. Table 2: Percentage Overestimate of U.S. ability to produce Soviet Output Mix, implicit in existing Estimates National Center's Mix Foreign Assessment Estimates of Soviet Rosefielde Estimate of Soviet mix Direct Method Indirect Method CONCEPTUAL PROBLEMS AND INTERPRETATIONS Table 2 shows that the assumption of constant transformation rates results in a overstatement of the U.S. ability to produce the Soviet output mix of 1.6 to 3.7 percent for the period This would raise the Soviet superiority index from 40 percent to 42 percent (National Foreign Assessment mix) or to 45 percent (Rosefielde mix). Our method of calculating the index number bias appears to offer a new approach to correcting for biases to obtain a "true" index of the U.S. a b i l i t y to produce the Soviet output mix. It yields answers that are directionally consistent with theoretical considerations. While the estimates are not large, they nevertheless do make a substantive difference in reported estimates of Soviet defense -23-

26 expenditure superiority. They appear to support the Bergson-Moorsteen prediction that the bias will be smaller when the output mix of the USSR is used as a standard for the two countries while U.S. relative prices are applied to both countries. We would caution that these estimates be used with extreme caution and be regarded only as a first experiment with this type of econometric tool. Our reasons for urging caution and further research are: 1. The estimated U.S. product transformation curve does not appear to be well behaved. It shows signs that cost minimizing behavior is lacking in certain defense sectors. Yet the theoretical model assumes cost minimization. The implications of non-costminimization should be thoroughly, investigated. 2. We have dealt with this problem using quite high levels of aggregation. Yet index number theory suggests that the degree of bias will vary positively with the level of aggregation. The model should be estimated at lower levels of aggregation. 3. The proper interpretation of our results is that we are estimating the short run ability of the U.S. economy to transform its mix into the Soviet output mix. This is the appropriate interpretation of translog functions estimated from annual time series data. Our calculations therefore likely overstate the long run difficulties that the U.S. would encounter in shifting to the Soviet mix. Whether short run or long run measures are more appropriate in such calculations remains to be resolved. 4. We have not experimented with breaking the sample up into shorter periods, primarily in order to maximize the degrees of freedom. This is a long period of time, and there may be discontinuities in our sample. Future research should experiment with different samples to establish the effect of the sample selection on the bias estimates

27 4. A parallel study should be conducted on the Soviet data to establish the Soviets ability to produce the U.S. mix. Such a study could not be conducted by these researchers because of data limitations

28 REFERENCES 1 Abram Bergson, The real National Income of Soviet Russia Since 1928 (Cambridge: Harvard University Press, 1961) Chap.3; Richard Moorsteen, "On Measuring Productive Potential and Relative Efficiency," Quarterly Journal of Economics, vol.75, no.3 (August 1951); G.W. Nutter, "On Economic Size and Growth," Journal of Law and Economics, vol.9, no.2 (October 1966); Herbert Levine, "On Measuring Economic Growth," Journal of Political Economy, August National Foreign Assessment Center, Soviet and U.S. Defense Activities, , SR , January Franklyn Holzman, "Are the Soviets Really Outspending the U.S. on Defense?" International Security (Spring 1980); Steven Rosenfielde, "Are the CIA's Estimates of Total Soviet Defense Activities Really Overvalued? A Reply to Franklyn Holzman," typescript, October Bergson, chap Steven Rosefielde, False Science:Underestimating the Soviet Arms Buildup (in press), chap. 13. The Rosefielde mix and the CIA mix (from National Foreign Assessment Center 1979, pp ) are: Personnel 0 & M Procurement Construction R & D Rosefielde CIA

29 APPENDIX A: THEORETICAL AND EMPIRICAL IMPLEMENTATION OF THE REVENUE FUNCTION APPROACH The aim of this appendix is to employ duality theory to outline an altenative method of describing the technology in Figure 1. The analogue to the product transformation curve in equation (4) is shown in the following section to be a revenue function. The translog approximation to the revenue function is subsequently described. Finally, it is shown how to calculate the "true index" of the U.S. ability to produce the Soviet output mix. This "true index", derived from the revenue function, differs from the "true index" derived in the previous section from the product transformation curve. Since there is no necessity that the two measures yield identical results, the two approaches serve as useful reliability checks. Theoretical Basis for Revenue Function Our purpose here is not a rigorous derivation and proof of duality relationships, but rather an intuitive explanation. Let us first review the duality relationship as it relates production and cost functions. Consider the simple production function (19) Q = f(x 1 X 2 ). Shepherd, and more recently, McFadden and Lau show that the dual to the production technology is the following cost function: (20) C = f(p X1, P x2, Q) Either equation (19) or (20) is sufficient to describe the underlying technology even though equation (19) utilizes input quantities while equation (20) utilizes

30 input prices. Thus there is a fundamental duality between input prices and quantities. For purposes of this problem, the above framework must, be modified to account for two outputs and only one input. As stated in equation (4): The dual to equation (19)) is the revenue function (R) showing for output prices P H and P M and a given level of inputs (X), the maximum revenue obtainable: (22) R = g(p H, P M, X) Thus output prices have replaced input prices, and the input quantity enters in the revenue function while output enters in the cost function. Nevertheless, the same duality theory is at work. Just as minimizing the cost of producing a given output requires using the same input as the production plan that maximizes output for a given level of costs, it is likewise true that the strategy which maximizes revenue for a given level of input necessitates choosing the same outputs which would minimize the input needed to produce a given revenue. The Translog Revenue Function We utilize a translog function as an approximation to the revenue function in equation (20). Partial differentiation of equation (23) yields the following equations: (24) Just as the partial derivative of the cost function with respect to the input price yields the input quantity, partial differentiation of the revenue function with respect to output prices yields the output quantities:

31 Substituting these relationships into equation (24) shows that the left hand side of equation (24) can be written as revenue shares (S): Equation (25) serves as the basis for econometric estimation. The data requirements are the prices of military outputs and their respective quantites. The parameters of the translog revenue function must meet a set of conditions. First, monotonicity requires positive revenue shares. Second, symmetry requires that Third, the additivity or accounting conditions requires that at every data point: Fourth, since the revenue function is quasi-concave in output prices, the Hessian matrix must be negative semidefinite. Use of Translog Revenue Function To Obtain A Measure of Relative Superiority The purpose of this section is to show that identification of the and terms in the translog revenue function are sufficient to derive a "true index" of the U.S. ability to produce the Soviet output mix given by OR/OA in Figure 1. Empirically, we propose a procedure which will identify point OA directly, then by simply dividing OR by OA, the "true index" is calculated. Initially, we begin by calculating the value of the translog function at point U for the actual U.S. prices (23)

32 Next, we raise the price of P M and lower that of P., gradually, causing the desired output mix to rotate in the direction of OR, the Soviet mix. Finally, we reach a relative price vector (P H /P M *) that yields the Soviet output mix Substituting these relative prices into (26), we obtain, the revenues associated with the new mix less a constant. Solving for R*, we obtain: Given revenues at point A and the output mix we have the following linear system sufficient to solve for quantities of H and M at point H: two equation Given the exact values of H A and M A at point A, one can obtain the ratio OR/OA by simply dividing the quantity of M or H at R (e.g. M R or H R ) by the corresponding quantity at point A: Thus the measure of the U.S. ability to produce the Soviet output mix is obtained

33 APPENDIX B NOTE A program of translog regressions and bias calculations used in the preparation of this Final Report is on file and available at the National Council for Soviet and East European Research.

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