A CONSTANT LONG-RUN CAPITAL/OUTPUT RATIO IS TAUTOLOGOUS AND MEANINGLESS1 Julian L. Simon PART I: INTRODUCTION The ratio of the aggregate capital stock to total income is commonly to be one of the main parameters of an economy -- one of Klein's five "gr ratios" (1962). This concept appears frequently in economic theory. As stant, it is a fundamental stylized fact in growth theory, a benchmark ag which to test whether an economy is in equilibrium growth. I shall argue that even in principle, let alone in practice, no meas of the aggregate capital stock -- either in physical or money terms -- ca meaningful, and therefore no theory that rests upon it has economic meani stylized fact -- in Solow's words, that "the ratio of capital to output s systematic trend" (1970, p. 2) -- is inevitable because it is a tautology er a meaningful growth theory can be reconstructed without this concept i question not discussed here. There are two strands of the argument, one concerning the physical r the other concerning the monetized ratio. 1. Concerning the physical ratio: Language such as Solow's "the ca output ratio [is] a technological fact" (1988, p. 307) suggests that the is analogous to such physical laws as that which governs the relationship pressure, volume, and temperature. But economic progress consists of get output from a given amount of inputs. This contradicts the notion of con ratios in particular industries. And there is no way to add hammers and so as to construct the aggregate ratio that is necessary for macroeconomi 2. Concerning the monetized ratio: The observed monetized K/Y rati an appropriate proxy for an aggregate physical ratio, or for any analysis it is a tautology; the value of the capital stock depends on the value of rather than helping determine it. This argument is made in Part II, afte the main argument of the paper is complete. To give substance to the discussion, consider the case of East Germa 1990. Its industrial plant was much the same as it was in 1989, so there reason to believe that a "technological" K/Y ratio has changed. But afte with West Germany many of the factories were abandoned, implying that the value fell (assuming that there was earlier a market value for these fact The change in market value was not due to an accompanying change in physi condiion or in the replacement cost of the equipment or in any other acco measure of value. Rather, the fall in value was due to changes in the op ties for East Germans as workers and consumers. This implies that the va the capital is a function of the value of the services provided by the ca a measure of output rather than of input. This valuational concept is qu different than any standard conceptual or empirical work on the subject. Nor is this case an anomaly that could not occur within a closed eco While it is difficult to imagine a technical development that would affec entire industrial plant of a society, a new discovery can render much of in a given industry obsolete. For example, the discovery of new drugs fo illness eliminated many mental hospitals. If this can happen within spec sectors, the aggregate K/Y ratio is not immune to such change. The argument seems odd at first. Therefore I shall make the argumen steps. Notation i L = labor force; Y = total output in market i during period j; K j stock of capital; v = Y/K reciprocal of capital-output ratio; w = wage rate; r = rate of return on capital; c,a,B = constants. II. THE MAIN ARGUMENT In support of a stylized constant K/Y ratio, the original growth the adduced such data as Denison's for the U.S. and eight European countries 1962; Barger's for 1950-1964; Matthews' and Feinstein's U.K. data for 185 1900's (all cited by Solow, 1970, pp. 5-7); and the CED-Denison data for from 1909 to 1958 (U.S. Department of Commerce, p. 189). These data purp show that two independent entities -- K and Y -- remain in a constant rel ship. All the estimates of output are in monetized valuations from natio income accounts, and the estimates of capital come from surveys of assets in money.2 These modes of measurement are of the utmost importance for t ment. Later publications such as Goldsmith (1985) and Wolff (1991) do not with the "stylized fact" as well as did the earlier publications. For ex Goldsmith's key Table 18 the ratio for France varies from 1.28 to 2.98, a ratio for Yugoslavia drops from 3.31 in 1950 to 1.60 in 1965, and 1978 ra range from 1.31 for Mexico to 3.09 for Switzerland. In Wolff's Table 4b, ratio in Japan was 1.39, 2.33, 1.65, 1.80, and 2.47 in 1938, 1950, 1960, 1979 respectively, while in the United States it jumped from 2.87 to 3.13 1900 and 1913. There is no apparent explanation for these differences in ratios among countries, or for the differences without trend from period within countries. Such unexplained variations suggest to me that the mod measurement matters greatly, and that whatever the published figures repr little or no causal role in economic growth. Hence attempts to measure t tal stock empirically, even by master workmen, are seen to fail in terms validity. The central analytic point of the paper is this: In the long run th values of output and capital are not, and cannot be, in a very different than is observed because of market forces inherent in competitive markets is to say, the observed K/Y ratio in monetary terms does not indicate a c physical relationship between capital and output. (And please keep in mi as commonly used in the literature, a production function is defined as a cal-technical construct; see Solow, 1988.) In a competitive market, the output is theoretically determined only by the total cost of inputs; if t of bulldozers falls, the price of bulldozing will fall. Hence if the mon of capital or labor input falls due to technical change, the money value of output in that market must fall, also. The rest of the argument is an elaboration of this simple observatio straightforward proof that, on quite realistic assumptions, the observed and output measures should change by the same proportions from period to This is the reason for the observed constant K/Y ratio. (Deviations from across countries can easily be the results of differences in measurement tions; the movement toward convergence that Wolff perceives can be the re convergence in accounting techniques.) A word about the valuation and measurement of capital3: Estimates o capital shares of a given country made by governmental organizations and al scholars are, of course, subject to a large number of difficulties. I one would measure the current market value of each asset, thereby reflect current technology and capital condition. But the actual published estim derive largely from (a) estimates of depreciation, which are not likely t very close to market facts, together with (b) original purchase prices, w current meaning is much affected by technical change and inflation. Thes culties are among the many which muddy the meaning of a capital-stock est But in the long run these errors of estimation should wash out and leave rough correspondence between the book value and the market value of a cou capital, as is assumed by those who estimate countries' capital stocks; t all that is required for discussion here. Furthermore, to the extent tha is indeed relative constancy of the observed K/Y ratio, it is itself evid the book-value estimates bear some stable relationship to market value. There are also the daunting conceptual difficulties concerning the n an aggregate production function which are associated with the famous Cam controversy of decades past. But these issues need not detain us here be aim is not to elucidate the meaning of the capital term in the K/Y ratio aggregate production functions, but rather only to explain why the moneti ratio is likely to remain unchanged with time. Before stating the argument more precisely, it is illuminating to lo related situation where capital and output also remain in a constant rati agriculture, the market value of a unit of land consistently is roughly 3 times the market value of the gross output of that unit of land (Clark's countries ranging from China to Australia to ancient Greece, in all the y which data are available.4 The explanation of this constancy is that whe value of agricultural output changes, the price that is bid for land chan response. Therefore, the monetary ratio of land value to output value te nothing whatever about the physical facts of production, but only about t ior of capital markets and portfolio behavior. In other words, farm output value determines capital (land) value; h two quantities clearly are not independent and could not meaningfully sta physical K/Y ratio. This observation about agricultural land is akin to central point of the paper about industrial capital. (For plant and equi capital, however, the price of [new] plant and equipment influences the p industrial output, whereas the price of land has less influence on the pr agricultural production; the weather has a large and unpredictable influe the latter but not on the former. In an emergency short-run crunch in wh supply of industrial plant and capital were largely fixed -- say, the sup machinery to make perfume during wartime -- the price of output would pro influence the price of plant and equipment more than vice versa, and the would then be just the same as for farmland.) Land may not seem perfectly analogous to industrial capital because all quantity in acres varies less from year to year than physical measure capital in various industries. But as with industrial capital, land's ma price is influenced both by the demand for it as well as by the supply-si of clearing new land and of improving old farmland with irrigation and dr Perhaps the main difference between land and industrial capital is that t in farmland is mostly sales of old farmland into which the costs have bee long ago, whereas a much larger proportion of industrial capital transact in new capital. Now to the industrial K/Y ratio. Assume that labor's share of outpu (Strict constancy is not necessary for the argument, but only to make the neat, a point I shall amplify below.) We can also take as a stylized fac the rate of return on capital has no trend. We can now write (l) Y = wL + rK and (2) wL = bY. Hence (3) rK = aY, since Y = aY + bY. Assume that these relationships hold in representative market 1, in p Let us further suppose that (where market is the superscript, and period 1 subscript) in market 1 the amount of capital needed to produce output Y 1 sharply, because either labor or capital becomes more efficient. To fix imagination, consider a computation that costs a dollar on an IBM compute that costs much less than a penny now (constant dollars). Assume for con that demand is completely inelastic. If so, the value of the capital in fall sharply to some cK , where c is a constant less than unity, and the output will also fall until the new price of computing services equals th cost. We can figure the new value of output as follows (writing every st intuition in this counter-intuitive matter): By assumption 1 1 (2a) wL = bY and 2 2 1 1 1 (la) Y = WL + rK . 2 2 2 By definition and previous assumption 1 1 1 (4) rK = aY = rcK 2 2 1 and 1 1 (3a) rK = aY . 1 1 Divide (3a) by (4) 1 1 rK aY 1 1 (5) ---- = --- 1 1 rcK aY 1 2 cancel and rearrange 1 1 (6) Y = cY . 2 1 By definition 1 1 Y Y 1 1 1 2 (7) v = -- , and v = -- 1 1 2 1 K K 1 2 and hence by definition and cancellation 1 cY 1 1 1 (8) v = --- = v . 2 1 1 cK 1 Hence the K/Y ratio in market-value terms (current or constant dolla remains unchanged when there is an increase in productivity.6 The analys easily be extended to the economy as a whole.7 An example: Assume that an inventor creates a new device that incre bagel machine's speed by 10%, thereby decreasing the cost of the capital by 1 cent per bagel. The cost of bagel-machine capital services will fal market competition, and the price of bagels will fall--but the K/Y ratio up the same as before. Physical output probably has changed (due to the reduction) and the capital input clearly has changed, so the physical inp relationship between them has not remained the same. But the market-valu relationship will have remained the same. Another illustration (from Steinmann, 1981, 1989) is Ricardo's analy seeds, in which the capital good and the output good are the same (seed). that someone finds a new variety of seed that yields 10:1 instead of 3:1. monetized K/Y ratio obviously stays the same (because a seed unit of capi seed unit of output are identical). Yet there has clearly been a physica in the production relationships (though one might not wish to say that th cal K/Y relationship has changed). Steinmann draws another conclusion from the seed example, by way of in capital's share of output as measured by the rate of return on capital small changes in the seed yield ratio, one can imagine a compensating cha in the algebra above. But for big yield changes this is most implausible more important, there is no reason to suppose that r would change at all, alone magically change to the required size. So again the monetized K/Y remains the same while the physical relationship changes. For labor's share the argument is similar to that for the return to Assume for now that labor's share is increasing. But the largest possibl crease is from, say, 60% to lOO% of all income. Even if this change were by some change in physical capital efficiency (physical output per dollar capital cost, in a case where labor remains the same), it is far too smal offset the big observed changes in capital efficiency, e.g., in computers small proportional increase in labor's share relative to the increase in tivity over the years also makes clear that the constancy of the K/Y rati be explained by a substitution of capital for [efficiency] labor). And t no reason to consider labor's share change as induced, anyway. Hence the cy of labor's share is not crucial to the argument, though it makes possi above neat and simple algebraic explanation of the argument. In short, a constant monetized K/Y ratio is simply an inevitable out the pricing process rather than a structural element of the production pr CAN THE CAPITAL STOCK BE ESTIMATED INDEPENDENTLY? The fact that the value of the capital stock is a function of the va output, and hence is tautologous, is reason enough to consider the K/Y ra without meaning. But if it is the case that the capital stock cannot eve measured meaningfully, the main argument of the paper should seem even mo pelling. Furthermore, the fact that the measured monetary ratio is more-or-le stant, even though the physical ratio -- for which the monetary ratio is to be a proxy -- must be declining, should be reason enough to conclude t monetary ratio is not a meaningful proxy for the theoretical concept unde sion. But there are also reasons in the monetary measurement alone which the measurement without meaning in itself, let alone as a proxy. The monetary value of the stock of agricultural land can be measured fully because frequent sales indicate the market value of all parcels wit able accuracy (though the K/Y ratio nevertheless is not meaningful for th given above). The same is true of the housing stock. But a market value be estimated readily or at all for the stock of equpment. This is becaus difficulty in transferring equipment from seller to buyer -- the costs of ing, transporting, and re-installing are often very large relative to the the equipment; b) the absence of active markets, and c) the great variety and conditions of machinery. Hence estimating the aggregate value by mar assessment is not done in practice, which is why other measures such as i depreciated original cost and reproduction cost are used. Solow (1989) suggests that reproduction cost is the appropriate meth long-run estimation. But this is not likely to be an operational concept sider, for example, a Kaypro computer now in use, though no longer manufa which is suddenly knocked to smithereens. What is the appropriate reprod cost? A new IBM-compatible is surely more valuable capital from any poin view. And a second-hand Kaypro will be priced at some market value which nothing to do with the cost of production. Once more the valuation of th depends upon the value of its output, among other things. DISCUSSION 1. Nothing written here is intended to suggest that the current rat investment (or savings) relative to current income has no meaning. It ma meaningful to note, for example, that because there is a great deal of un in an economy at a given time, perhaps for fear of war or political chang is little investment, and this may reduce income. On the other hand, Sol that "the capital-output ratio is not of much short-run use because the d utilization of capital may fluctuate quite a lot from year to year" (1989 this issue is outside the scope of this article. 2. Having seen that there is no empirical reason in the observed mo K/Y ratios to believe that the physical K/Y ratio is constant, we are fre theorize about the course of the physical K/Y ratio as economic developme ceeds. A falling physical capital-output ratio would seem an obvious techni But to show this rigorously we must proceed in step-by-step fashion, maki those comparisons that can legitimately be made. This means that we cann pare entire capital stocks, largely because new consumer products are int with the passage of time. Unfortunately, space is lacking to pursue the here, but a longer version is available upon request. SUMMARY AND CONCLUSIONS The apparent constancy of the observed capital-output ratio is misle Capital and output measured in value terms maintain a constant relationsh cause the price of output depends only on the cost of inputs in competiti kets. The appropriate capital and output concepts for a growth model are logical rather than value measures, and substitution of the latter for th (due to difficulty of measuring the former) results in confusion, wrong-h theory, and unsound conclusions. \article0 capout3 1-234 June 11, 1992 REFERENCES Hans Brems. Labor, Capital and Growth. (Lexington, Mass.: Lexington Books, 1973). Colin Clark. Conditions of Economic Progress. 3rd ed. (New York: Macmillan, 1957). Edward F. Denison. Why Growth Rates Differ. (Washington: The Brookings Institution, 1967). __________. "The Puzzling Drop in Productivity," The Brookings Bulletin, Vol. 15, No. 2, 1978. Raymond W. Goldsmith. Comparative National Balance Sheets (Chicago: C Press, 1985). Lawrence R. Klein. Introduction to Econometrics (Englewood Cliffs: Prentice Hall, 1962). Julian L. Simon. The Theory of Population and Economic Growth (Oxfor Basil Blackwell, 1986). Robert Solow. Growth Theory: An Exposition. (New York: Oxford Univer Press, 1970). __________. "Growth Theory and After," American Economic Review, 78 June, 1988, 307-317. __________. Correspondence, April 4, 1989. Gunter Steinmann, private correspondence, 1981 and 1989. U.S. Department of Commerce. Long Term Economic Growth, 1960-1965. (Washington: GPO, 1966). Edward N. Wolff, "Capital Formation and Productivity Convergence Ove the Long Run", American Economic Review, vol. 81, June, 1991, 579. FOOTNOTES 1I appreciate a helpful reading from Hans Brems, conversations on th with Richard Kihlstrom, James Kurish, Leonard Mirman, and Walter Primeaux correspondence with Paul Davidson and Robert Solow. This article draws u (1986, Appendix A). 2All such studies have many methodological problems, of course, espe those that use the Cobb-Douglas production function, which constrains the utive shares to be constant and thereby biases the K/Y ratio estimate. M important here, however, is not whether or not the ratio is constant, but theorists assume it to be constant and build models in consonance with th sumption. 3See Goldsmith (1985) for detailed discussion of measurement issues. 4See Clark, 1957. In China in 1921-25, the K/Y ratio in Chinese doll the average farm acre (excluding livestock and supplies) was $1736/$376 = land alone, it was $1374/$376 = 3.7. In Orissa, India, 1958-59, the avera vestment per acre was Rs. 535, and in land alone it was Rs. 474, whereas there was Rs. 125 per acre, a K/Y of 4.3. In the Punjab, 1955-56, the va land averaged Rs. 840 per acre, where the value of output per acre was Rs giving a K/Y ratio of about 4.3. In Andhra Pradesh in 1959-60, the K/Y ra plus all other capital) was 6.4. In the years 1910-16, the ratio of land output ranged from 3.05 to 3.56 in Australia, Canada, France, Russia, Swi and the United States countries in which land was surely a smaller propor total capital than in India or China. (Paragraph largely drawn from Simo p. 257, with references given therein.) The U.S. Midwest was for several the only major exception; land prices confounded all analysts until 1982, they went into a sharp decline, as this ratio predicted must happen. 5Some would question whether land should even be considered capital i context, on the grounds that land is a natural endowment and the returns a rent to the descendants (natural or financial) of the person who simply a mark to it. But this argument will not suffice. First, if the land we used and used, it would soon be useless for agriculture. Land must const maintained, and this maintenance is gross investment. Second, when it wa nally claimed, the land had to be cleared and made useful for agriculture we know from contemporary landclearing projects in Siberia, Brazil, Afric elsewhere, the cost of land clearance is not far from the market value of ble land presently in use -- as one may expect. Costs per acre (2.471 ac one hectare) among a sample of land-development projects were: Guatemala, $91; Nigeria, $118; Sudan, $218; Ceylon, $307; Morocco, $307; United Stat Kenya, $973. The weighted average of a world sample of projects in settl was $400 per acre. More recently, a considerable lower estimate has been the FAO: "to add 5-7 million hectares to food production would cost betwe and $312 per hectare". For irrigation alone, on already cultivated lands India, the estimate is $250-$300 per acre. The weighted average cost for ty of world projects was $325 per acre, and omitting one large project it per acre. And though some land may be cheaper to clear than others (e.g. U.S. Midwest was cheaper than the Brazilian jungle) even the cheapest-to- land was far from the "free gift" that some historians speak of. The cos disease, danger from Indians, isolation, lack of markets, and so forth, w substantial. Proof is found in eighteenth-century New England where peop mained even though population grew to the point at which it was considere crowded" and land prices rose, because the perceived cost of opening the was greater. Additional evidence is the relationship of the cost of purc under-developed land to the cost of readying it for farming. In the 18th a representative 40-acre U.S. field cost $50, but fencing, clearing and c building cost $250. (Footnote largely from Simon, 1977, p. 240. Referenc therein.) 6Brems (1973, p. 171) also noticed the discrepancy between the money physical concepts of the capital coefficient, and he suggests a reconcili way of the rise in the prices of producer goods relative to consumer good explanation is not at all incompatible with the explanation suggested abo my inspection of the data suggests that the prices of producer and consum are not likely to explain much of the effect under discussion here. 7The only qualification is that the drop in the (present value of th duction cost of the amount of capital required to produce a given amount (assume it is a case where the labor input remains the same) must be grea the cost of the R&D that goes into the improvement. If the inventor lice invention at the rate of .9999.... cent per bagel, then the price of outp fall only by an epsilon and nothing else would change. (And if [the pres of] the inventor's take comes to equal the market value of his or her tim the invention, then we might even assume no increase in the value of the stock. But this is not at all necessary for the argument.) A CONSTANT LONG-RUN CAPITAL/OUTPUT RATIO IS TAUTOLOGOUS AND MEANINGLESS ABSTRACT Standard growth models -- those in the Harrod-Solow tradition with t ical progress exogenous, as well as models following Kaldor and Arrow tha technical progress endogenous -- are validated by their results being in with the stylized fact of a roughly-constant capital-output ratio. This argues that the concept of a constant K/Y ratio is (a) fundamentally unso physical-technical sense, because individual industry ratios must fall if technical progress, and (b) meaningless in an economic sense, because a r constant K/Y ratio is a tautology; it is a necessary outcome of an econom roughly-constant aggregate distributive shares. A well-working market wo the ratio constant in money terms no matter how it changes in physical te Hence the observed constant ratio in market-valuation terms tells nothing physical relationships. References taken out of reference list on 5-25-92. Gustav Cassel. The Theory of Social Economy. 5th ed., trans. (New York: Harcourt Brace, 1932). A. K. Dixit. The Theory of Equilibrium Growth. (New York: Oxford University Press, 1976. Walter A. Eltis. Growth and Distribution. (London: Macmillan, 1973). Nicholas Kaldor and James A. Mirrlees. "A New Model of Economic Growth," Review of Economic Studies, 29, June, 1962, pp. 174- 192. Reprinted in M. Mueller (ed.), Readings in Macroeconomics, 2nd ed. (Hinsdale: Dryden, 1970), pp. 306- 322. __________ and Richard F. Kosobud. "Some Econometrics of Growth: Great Ratios in Economics," Quarterly Journal of Economics, Vol. LXXV, May, 1961. Simon Kuznets. Population, Capital, and Growth. (New York: W. W. Norton, 1973). __________. "Population Change and Aggregate Output," in Universities-National Bureau of Economic Research, Demographic and Economic Change in Developed Countries. (Princeton: Princeton University Press, 1960) . William Petty. Another Essay in Political Arithmetic (1682) in The Economic Writings of Sir William Petty, Charles H. Hull, editor,(Cambridge: CUP, 1899). Julian L. Simon. The Economics of Population Growth. (Princeton: PUP, 1977). Robert Solow. "Technical Change and the Aggregate Production Func- tion, The Review of Economics and Statistics, 39: 1957, 312- 20. E. C. West. Canada-United States Price and Productivity Differences in Manufacturing Industries, 1963. (Ottawa: Economic Council of Canada, 1971).