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ANALYSIS OF QUALITY ADJUSTED PRICE INDICES AND GROWTH ACCOUNTING: FOCUS ON SOLOW VINTAGE MODEL.

Introduction

There has been a revival of interest in the problem of how to handle embodied technical progress with the information revolution, and the development of quality adjusted computer price indices. One set of such estimates has been incorporated into the U.S. national income accounts. (Dean and Kunze (1) describe the changes. Jorgenson and Landau (2), Gordon (3), and Foss, Manser, and Young (4) supply quality adjusted computers price indices. There has been much controversy about whether and how such quality adjusted price or quantity estimates should be used in growth accounting (see Denision (5,6) for instance). In 1966 and 1967 Jorgenson and Griliches (7,8) pointed out how errors in price indices for investment goods could lead to errors in productivity estimates, and suggested that high rates of reported productivity growth could be explained by measurement errors. The purpose here is not to criticize their 'by now somewhat dated' estimates, but to point out a difficulty so that others may avoid it. Understanding the treatment of capital embodied technical change is important because their methodology has continued in use, and others have echoed their call for quality adjusted price indices : quality adjusted price indices for computers have been adopted in the U.S.A

Since investment data is in money terms, obtaining physical quantities of investment requires dividing by a suitable price index. In the method used by Jorgenson and Griliches, capital stocks are derived by a perpetual inventory method in which net investment in each year (net investment is gross investment minus what they called replacement) is cumulated to give a capital stock, and then capital services deduced from this (by assuming services to be proportional to stock for each type of capital and then aggregating to form indices of total capital services). They then derived a formula which showed how errors in the price index would lead to errors in the productivity growth rate. They were not very specific about what they would regard as an error free index, but many readers assumed that an error free index would be one that fully adjusted for all quality changes. This interpretation was made more plausible because their treatment assumed that the price indices useful for measuring the output of the capital goods industry were the same as those required for computing the quantity of capital input. Since it is plausible that higher quality goods should be regarded as equivalent to a greater quantity of goods, and it was widely recognized that the quality adjustment methods used in deriving the producer goods price indices did not adjust for all quality improvements, their argument was widely read as a call for using indices that were fully corrected for all changes in the quality of capital goods.

The purpose of this paper is not to start a controversy about what type of quality improvements Jorgenson and Griliches thought should be included in the indices (for they never clearly said) or about what items were included (since the details of their original methods were substantially modified in their later studies), but to discuss whether an adjustment for all quality changes (including what is often referred to in the literature as costless ones) should be made in the price indices used for growth accounting.

Definitions of the quality of capital goods and capital embodied technical change may be useful to readers who have not yet come across these concepts (further discussions can be found in Solow (9,10) Triplett (11), and Early and Sinclair (12)). Capital embodied technical changes are those which require changing the design of the capital goods used. For instance, introducing diesel technology to railroads required replacing steam locomotives with diesel ones. However, not all improvements in capital good quality come from capital embodied progress. Any capital good's quality may be improved by simply spending more to produce a higher quality good. This is costly quality improvement. There is general agreement that quality changes of this type should be reflected in the price indices. The quality improvements due to capital embodied technical progress are usually referred to as costless, since they do not necessarily involve raising the cost of producing the goods, Gordon (13) refers to these as non-proportional quality changes since the quality improvement is more (than proportional to any associated cost increases.

Prelude

Several other strands in the literature make the handling of capital embodied progress an interesting question. Jorgenson (14) has published a paper on the role of capital embodied technical progress in productivity growth measurement, that derives the same relationship between errors in prices and the growth of productivity that appears in the two papers with Griliches. Here the emphasis is on the effect of different rates of embodied technical progress on the estimation of what he (like others) referred to as disembodied productivity growth. It is shown how changing the embodied technical progress built into the price indices changes the estimated total factor productivity growth. Indeed, for every estimated rate of capital embodied technical progress there is a corresponding estimate of the productivity growth rate. His paper also points out that because the observed output series could be matched by a wide range of pairs of time series for embodied and disembodied technical change, disembodied and embodied technical progress cannot be separated using only time series data - - Griliches paper emphaises that:
one can never distinguish a given rate of growth embodied technical
change from the corresponding rate of growth in disembodied technical
change.


This statement should be read as applying to econometric methods. A conceptual distinction remains (possibly requiring clearly defining capital quantity) and someone familiar with a specific technical change (say automation) could easily classify it as being embodied or disembodied on the basis of whether capital goods of new design are required.

It is also pointed out that the amount of investment required to attain a given amount of economic growth should not depend on the index of capital embodied technical progress used, as long as the index of total factor productivity is correctly paired with that index of capital embodied technical change used. There is no criteria for choosing which of these pairs of possible time series would be more meaningful. or which should be used for predicting the effects of changing levels of investment on growth. Unfortunately, what type of index would be regarded as error free is not stated, although the implications seems to be that all quality changes should be reflected in the price index for capital goods. This omission is somewhat surprising since the original derivation is in terms of errors in the price of investment goods. The term error would normally be used only if the writer would regard a particular index of investment goods prices as accurate. (*)

The question of whether to adjust the quantity of capital goods for capital embodied technical progress and whether to adjust the capital good price indices for all (including costless) quality changes are the same question. This question is the subject of this paper. As Jorgenson (14) put it:
We may interpret the reciprocal of the relative error in the price of
investment goods, I/Q, as an index of the quality of investment goods.
The rate of growth of this index may he interpreted as the rate of
embodied technical change. In this interpretation, the erroneously
measured quantities of investment goods output, I (*), and capital
input, K (*), are surrogate investment and surrogate capital, that is,
investment and capital corrected for quality change.


This quote is very strong evidence for a particular interpretation of the argument in the two papers coauthored with Griliches, namely that errors in quality adjusting price indices would lead to errors m total factor productivity. This interpretation is that failure to adjust investment series for captial embodied technical progress related quality improvements would create errors in total factor productivity estimates.

This paper will argue that adjustment the investment for all quality changes will worsen, rather than improve, productivity growth estimates. This holds especially if the quantity of capital represented by older goods has been reduced for obsolescence (as Griliches and Jorgenson did). However, even if double counting is avoided, using price indices adjusted for changes in utility to the purchaser, as Griliches (15) proposes, would still be incorrect.

Solow's Vintage Model

Solow introduced the distinction between embodied and disembodied technical progress. In a series of papers (9,16,17) he first drew a distinction between technical progress which had the effect of raising the efficiency of production without requiring a change in the design of capital goods, and progress which required capital goods of new designs. He called the former disembodied progress, and the latter embodied. (*) He introduced a new growth model in which a more modem vintage of capital goods had the same effect on output as 1 + x old goods. He showed how such capital could be aggregated, and how U. S, output growth might be described with such a model. A surrogate capital stock could be constructed by a version of the usual perpetual inventory procedure in which new capital goods represented more surrogate capital because of their higher quality. If it could be estimated how much better goods of the new vintage were, capital stock series suitable for growth accounting could be constructed. This would be done by weighting the vintages by how much capital embodied technical progress they had benefited from.

The vision of economic growth provided by Solow vintage model (16) was quite different from his earlier original model (18) which was a model with a single type of constantly improving Capital in which technology merely multiplied output. In the vintage model investment was necessary to actually bring technology progress into use. This corresponded to the common observation at most inventions (diesel locomotives, jet planes, computers, etc.) required investment in new equipment before they could actually be used. Production functions containing a vintage capital stock had the property that the effect of a given investment was much higher than in similar disembodied models (19). While the return to additional investment in the Solow vintage model is indeed greater than if technical progress is disembodied, it will be argued that this is because his model contains only a single type of capital. With multiple types of capital goods, recognizing embodiment does not strengthen the case for additional investment. The Solow vintage models attracted much comment. Probably the most important was a Denison (20) comment which argued that embodiment was unimportant. The Jorgenson (14) embodiment paper discussed above was a response to Solow's calculations with his vintage models. (*)

Many of the issues raised in the earlier debates among Jorgenson, Solow, and Denison appear to have never been resolved. The major players continued to use their original methodologies (see the publications by Denison and by Jorgenson and coauthors in the references). The question of whether (and how) to incorporate capital embodied technical progress (also called costless quality improvements) into capital stocks estimates continued (see Panel to Review Productivity Statistics (21), rdon (13) Griliches (15) Triplett (11) and Demson (5) for more recent contributions). The description of the procedures behind the official U,S. Bureau of Labour Statistics (22) total factor productivity indices contains signs of the thoughts of both Denison and Jorgenson. Perhaps because of this attempt to incorporate both strands of thought, what is to be measured when a capital stock is constructed for growth accounting is not specified. This paper has insights to add to this already voluminous discussion. The basic procedure used will be to set out the logic of different ways of handling capital embodied technical progress, pointing out that adjusting investment for costless quality improvements, and reducing old capital for obsolescence, and both mechanisms for handling the higher quality of new goods arising from capital embodied progress. It will be shown that adjusting for costless quality change leads to a fundamental inconsistency.

Role of Obsolescence: Although several writers (see Boddy and Gort (23), Brown (24) Tice (25), and Solow (10)) have pointed out that obsolescence and capital embodied technical change are closely related concepts, little has been written about how to incorporate both obsolescence and the higher quality of capital into growth accounting. All estimates of the rate of decline in capital services incorporate obsolescence. The basic reason for this is that there is no way to determine a deterioration only rate of decline and hence estimation of one is seldom even attempted. Obsolescence is included if estimates of capital service decline come from rates of decline in rents or asset prices (see Hulten and Wykoff, (26) and the many studies they refer to).

Logic of Growth Accounting: A restatement of the theory behind growth accounting may be useful here: (This summarizes Miller (27)). Any output increase that cannot be attributed to increased capital input, increased labor input, or increases in other inputs is the residual, and is attributed to increased productivity. This requires knowing how much output was increased by new capital goods added through net investment. If the number of units of capital services provided is known, the price of a unit of capital services can be derived by dividing the total income from capital by the number of units supplied. (Note that the logic of deriving a per unit price depends on all units of capita services selling for the same price). Since under the assumed competitive conditions the marginal product of a factor equals the price of its service, this price also equals capital's marginal product. The net investment multiplied by this marginal product gives the growth attributed to the net investment (i.e., the increase in the capital stock).

Total income is Y and capital's fraction of national income is a, making capital's earnings a Y. Thus, the price the services of a unit of capital is a Y/K where K is the total number of units of capital input. With net investment being [DELTA]K, the total income growth due to the new investment is [DELTA]Ka Y/K. As a fraction of income it is [DELTA]KaY/KY or [DELTA]Ka/K. This is the standard growth accounting method. This can be seen by rearranging the terms so that the contribution of capital growth to output growth is ([DELTA]K/K)a, or the growth rate of capital input multiplied by capital's share of total output. In words, the procedure of multiplying the number of units of added capital by a marginal product estimated by dividing total capital income by the number of units of capital services used is equivalent to multiplying the growth rate of capital input by capital's income share. This alternative formulation makes explicit the role of the price of capital services and how it is calculated. It thus provides a critical insight into how capital services must be measured for the formula to work. The insight is that the measurement must be such that all units of capital rent for the same price.

The above is essentially Solow's growth accounting argument. However, expressing it this way makes clear a necessary condition for the argument to hold. The quantity of a factor's services must be defined in such a way that the services of all items sell for an amount proportional to the number of units of services they are considered to provide. For instance, the services of a unit of capital must be priced the same (and hence have the same marginal product) regardless of whether these services come from a new building or a new machine. It follows that Solow was wrong to dismiss in a footnote the question of how capital was defined or aggregated. Only if the aggregation is such as to permit regarding all capital as having the same price and marginal product does his argument hold. As will be shown, there are plausible sounding measures of the stock of capital or the quantity of capital services which do not meet the equal marginal product condition. While such series may have their uses, they are not suitable for growth accounting. The growth accounting methodology is usually derived by assuming that a production function exists, and then manipulating it. This derivation logically depends on the existence of an aggregate production function. If the existence of an aggregate production function is considered only an approximation, one cannot argue that a formula derived by differentiating the functions must hold. A benefit to the direct derivation above is that no assumption about an aggregate production function's existence was required.

Measuring Capital Inputs: Consider the simplest case that illustrates the points to be made. Machines last forever, thus eliminating the need to consider deterioration. Technical progress has occurred once (and is not expected to occur again). There are now [n.sub.1] machines of an earlier design and [n.sub.2] machines of a later design, with the cost of the machines unchanged between the periods. After construction of the earlier models, capital embodied technical progress has occurred such that the new models have a marginal product p times that of the old models. This implies that they rent for times the rent of the first machines. This progress did not change the price to the machine's purchaser. In the current period there is investment in m machines. Naturally, these will be of the improved type. It is desired to use growth theory to discover how capital growth contributes to the growth of the economy. The problem is made simple so that the implications of using growth accounting with different capital definitions can be explored. The theory is stated in terms of capital input and the change in capital input. Thus, it is necessary to convert our data on the number of machines of different types in existence and the number being added into capital input measures. (This is the problem of how to aggregate the old and new machines). To avoid being embroiled in stock and flow considerations (which are not the issue here), the customary assumption will be made that the number of units of capital service is proportional to the stock of capital.

One possibility is to adjust the new machines for their improved quality. To do this, each new machine must be regarded as equivalent to p old machines. The numeraire is the number of old machines. In this case the total stock is [n.sub.1] + p[n.sub.2], and the capital input is [n.sub.1] + p[n.sub.2] The total income to capital is a Y, as above. This makes the income per unit of capital services aY/([n.sub.1] + p[n.sub.2]). By Solow's insight, this is also the marginal product of a unit of capital, with capital measured in units of the old machines. There are m new machines added in the current period. Since the unit of capital is the old machines, this is equivalent to pm new units of capital, and pm new units of capital services. The added output is pma Y/[n.sub.1] + p[n.sub.2]). The rate of output growth is obtained by dividing this by Y, giving pma/[n.sub.1] + p[n.sub.2]). This is the contribution to the growth in output of the growth in capital input.

There is another way to do the calculations. One can work in units of the new capital. This requires treating the services of the old capital as less because of its obsolescence. Each old machine is then equal to 1/p new machines. Last period's stock is [n.sub.1]/p + [n.sub.2], and the original capital input is [n.sub.1]/p + [n.sub.2]. The total income to capital is again aY. This makes the income per unit of capital services a Y /([n.sub.1]/p + [n.sub.2]). By Solow's insight, this is also the marginal product of a unit of capital, with capital measured in units of the new machines. There are m new machines added in the current period. The added output is ma Y/([n.sub.1]/p + [n.sub.2]). The growth rate in output is obtained by dividing this by Y, giving ma/([n.sub.1]/p + [n.sub.2]) This is the contribution to output growth of the growth in capital input.

The two procedures are equivalent. This can be seen by multiplying the numerator and denominator of the latter expression by p. This completes the demonstration that it makes no difference whether the new capital is regarded as embodying technical progress, or the old capital goods are regarded as having become obsolete. It appears purely a matter of taste and ease of getting the required data whether to use price indices adjusted for the new capital goods' improved quality including the costless quality improvements) or to lower the quantity of capital represented by the old capital goods because of their obsolescence. However, it is easy to slip into a subtle trap. The new capital may be regarded as embodying technical progress, while the old capital is also reduced for its obsolescence. The reduction in old capital services for obsolescence is likely to occur because the only efficiency decline series available incorporates both obsolescence and deterioration.

To adjust the new investment and new capital for its improved quality while also reducing the quantity of old capital for its obsolescence, gives a capital grow rate of pm([n.sub.2] +[n.sub.1]/p). This third method would clearly be wrong. The numerator is in the old machine equivalent of the machines, while the denominator is in new style machines (with the old machines expressed in terms of old style machines.) The argument has been made for a stable society, where the only disturbance is a new machine's introduction. There could be continuing technical progress, but this progress must be sufficiently well anticipated that the marginal products remain proportional to the prices of each machine's services. (If the price proportional to marginal product assumption is violated, the new capital goods' marginal product cannot be estimated from the total income to capital and the capital services input.) Also, the argument has been made for only two models. It could readily be extended to more periods and more models.

The above discussion is not in the form of a continuous growth model. It is not necessarily to have one to do growth accounting, although historically the idea of growth accounting (and Solow's formula) emerged from consideration of continuous growth models. The practical problem is what was the contribution to output growth from capital growth in historical periods, and what will be the contribution to future income of proposed investments. Investment decisions must be made at regulation intervals, and little is accomplished by posing, the problem as a continuous growth problem. When embodied in a national income accounting framework with deductions for capital consumption, current growth accounting does not result in a situation quite as bad as in the third method discussed above, even when price indices for new capital goods are adjusted for quality change, and depreciation rates reflect obsolescence. The reason is that a partial offset exists. Quality adjusted investment grows more rapidly than if new capital were not considered to be of higher quality. However, the quality adjusted capital stock also grows more rapidly. This implies that the depreciation deductions also grow more rapidly. The rapidly growing depreciation deduction provides an offset to the more rapidly growing gross capital formation. In a steady state, the investment, the capital stock, and the depreciation allowances are all growing at the same rate. This growth rate is higher than if new capital goods were not considered to embody a better technology, but not quite as high as the above simple example (where there is no capital consumption and hence no capital consumption allowances) might be considered to imply. However, the above effect provides only a partial offset.

Of course, when the error is made, it is not in such an obvious form. If authors had stated clearly the units they were working in (as is customary in the physical sciences), the error would be less likely to be made. However, for some reason this is not customary in economics. Writers typically discuss the problem of expressing new machines in terms of old machines as a problem of finding the appropriate price index to use in measuring investment. They also discuss the fact that old machines lose efficiency. Efficiency is a phrase that is not defined (or at least not defined in terms of other terms which are defined), and efficiency decline sometimes appears to refer to a decline in capacity, and sometimes to a decline in rent or marginal product (see Miller (28) for a detailed discussion of alternative definitions). Often, it appears to refer to physical deterioration, with obsolescence excluded. However, the word obsolescence is seldom used, occasionally, efficiency will be defined as the decline in "services" provided by a capital good, but without "services" being defined. Unfortunately, without units being defined, it is relatively easy to slip into the error of using inconsistent units in a long and complex argument.

Documentation that the Error has Actually Been Made

Since there may be some question as to whether the possible error discussed above is actually made, lets us look at some cases where it has been made. Probably the most important example is provided by the current United States Government productivity indices. They recently started using computer price indices that reflected the costless quality improvements being made in computers'. However, there is no indication that the rates of decline in capital services that had previously been used were altered to reflect only the decline in capital services due to deterioration, and to exclude those due to obsolescence. Indeed, given the available data, it is hard to see how the effects of obsolescence could have been excluded. Used asset prices, rental rates, depreciation rates used by firms, and depreciation rates allowed by the U.S. government all reflect the effects of both obsolescence and deterioration.

Jorgenson and Griliches (8) argued for adjusting capital stocks for embodied technical change, showing mathematically how using incorrect numbers could lead to errors in productivity estimates. However, in applying their ideas they appear to have used capital decline rates with obsolescence included. Because obsolescence is not explicitly discussed it is hard to be certain what they are doing. The term used in the discussion is replacement. This is not explicitly defined, but appears to correspond to the concept of depreciation, but applied to services rather than to stock. This would normally include the effect of obsolescence. Jorgenson and Griliches used a double declining balance with the same length of life as was allowed for income tax purposes (obsolescence is deductible for tax purposes). In using lengths of life taken from estimates for tax purposes, Jorgenson and Griliches incorporated capital embodied technical progress and the related changes in capital quality. Indeed, the underlying length of life data come from 1942 U.S. Treasury (29) tables titled Income Tax Depreciation and Obsolescence: Estimated Useful Lives and Depreciation Rates. The very title shows that obsolescence is incorporated.

Probably the strongest evidence that their capital series included the effects of obsolescence is that there is no obvious way they could have excluded the effects of obsolescence in order to obtain a deterioration only series. If they had actually calculated such a series, they would have explained how it was done. If they had been aware of the need for such a series, but had not been able to calculate it, they would have commented on the difficulty. Thus, it appears that they failed to realize that their decision to adjust investment for capital embodied technology improvements implied a need for a series on the decline in capital services that reflected only deterioration, not obsolescence.

The above, with a simple example, has shown that it is unnecessary to adjust new investment for the embodied technology that causes it to be of higher quality. Yet, it is common to see arguments for incorporating capital embodied technical progress into the model by using a price index for capital goods that incorporates quality changes. A Panel to Review Productivity Statistics (21) appointed by the National Academy of Sciences (U.S.) concluded that use of a resource-cost quality adjustment was not adequate. A recent such call for incorporating quality Change into price indices and into measures of capital stock is by Gordon (13), who has recently published such a set of indices (3). Triplett (11) has argued for constructing input indices using a user value criterion for input indices, reporting the idea has widespread acceptance, based on what he calls an informal poll. Jorgenson and Griliches (7,8) and Jorgenson (14) appear to argue for this. Unfortunately, the treatment of obsolescence is not discussed by these authors. However, their very failure to discuss the treatment of obsolescence strongly suggests that they have not considered the possibility that capital embodied technical progress is already being included. Their argument is not limited to cases where obsolescence is not also being deducted.

Some of the many criteria for capital input indices conflict with the need for all goods to have equal marginal products per unit of capital services. In particular, the intuitively plausible goal of having a particular item always represent the same quantity of capital of physically unchanged is inconsistent with recognizing obsolescence, since a physically unchanged good may be made partially obsolete by an improved good. While some goods are actually replaced because of obsolescence, there is a problem with the goods that remain in service, but whose rent has been reduced. (It may be thought that a solution is to use the older good as a numeraire and let the improved good represent more capital. However, as shown later in this paper, this does not solve the problem.) The individual choosing or designing a capital input index for productivity studies must keep clearly in mind that the services from all goods must have the same price (rent) per unit of service. Otherwise, the choice of the index may be determined by some other criteria: leading to the use of an inappropriate index.

Treatment of Obsolescence in Capital

Aggregation: Ideally, all discussions of quality adjusting investment would include an explicit discussion of how to handle obsolescence. Griliches (30) has provided elsewhere an explicit discussion of obsolescence in the context of alternative capital concepts for explaining investment. His indices include ones where the stock is adjusted for the number of machines remaining, for deterioration only, and for both deterioration and obsolescence. At one point an explicit distinction is drawn between deterioration (excluding obsolescence effects) and devaluation (including obsolescence). Unfortunately, similar conceptual distinctions were not made in his work with Jorgenson. The reader is left with no indication of which of several possible capital concepts were employed in the theory. Instead, the role of obsolescence was either not noticed, or assumed away without discussion. (However, in a receipt note, Griliches (31) mentions his early recognition that
the observed depreciation rates in second hand markets contain a large
obsolescence component that is induced by the rising quality of new
machines.


The growth accounting theory of Jorgenson and associates is derived as if depreciation is caused only by deterioration (32 to 37). Capital service decline is described as due only to the need for "replacement". An explicit discussion of obsolescence is not to be found. Of course, subsequent to the early jorgenson - Griliches work, Denison (38,39) successfully argued that the surrogates used for quality adjusted capital price indices were inappropriate, and their new estimates (40) did not quality adjust the investment goods price index. (The specific indices substituted were not adjusted for cost less quality change, leaving open the question of whether their goal was to adjust for such changes.) The changes made were because of problems with the specific indices used, rather than doubts about the conceptual desirability of quality adjustments. While their pioneering attempt to find price indices that better incorporated quality improvements failed, the arguments presented were influential, and calls for better indices have continued. No one asks whether the improved quality may already be included via obsolescence. Over the year separate literatures have developed on the problems of growth accounting constructing price indices, capital aggregation, capital embodied technical change, and depreciation without a full recognition of the need for a system approach in which all the components are consistent.

How Should Capital Embodied Technical Change be Handled?

Let us consider how capital embodied technical change should be handled. First of all, it is clear that it does exist and should be allowed for. If capital embodied technical change is not allowed for somehow, goods with equal costs but of different vintages and hence of different marginal products will be treated as representing the same quantity of capital. This would violate the growth accounting requirement that marginal products be proportional to quantities of capital services goods are considered to provide. Having decided that capital embodied technical progress should be incorporated, there remains the question of how to do so. Should introduction of new goods be viewed as causing current investment to be more capital, or as lowering the capital services supplied by old capital? In one sense this is just a question of whether to use a Laspeyres or a Paasche index. The big debate on capital embodied technical progress is not usually viewed as just a question of choice of index number, but on close analysis it is. Unfortunately, labeling it an index number problem moves us no closer to a solution.

In practice, there appear to be compelling reasons for choosing to reduce old capital for obsolescence. The simplest reason is that the costless quality change adjusted price indices required to revalue new investment are generally not available (computers are an exception) and difficult to produce. Thus, capital embodied progress can only be incorporated by reducing old capital for obsolescence. A less obvious factor is that estimates of the physical deterioration of old capital are required in a model adjusting each new vintage for technical change, and these do not exist. Estimates that have been used (from length of life data and from market value data) have really been estimates of rent decline with the passage of time. They have incorporated both deterioration and obsolescence in an unknown proportion. To separate deterioration from obsolescence would be almost impossible. In contrast, if embodied progress is to be incorporated by reducing service from old capital for obsolescence, implementation requires only estimating the combined effects of obsolescence and deterioration from observed lengths of life, or decline in rents with time.

Multiple Sectors Case : If there were only one kind of capital (as in Solow's models), it would be immaterial whether the actual procedure were to recognize obsolescence in old capital or to adjust new goods for embodied technical progress. Rates of growth for capital input would not depend on the procedure chosen. However, this is no longer true if there are several types of capital goods, each of whose quality improves at a different rate, and over several periods. For simplicity, consider a case with two types of goods, say trucks (representative of goods not undergoing capital embodied technical change) and computers. Both last forever.

Computers are frequently used as an example of a good whose quality has vastly improved. Also it is easy to imagine technical progress taking the relatively simple form of making one new computer able to do the work of two old computers, a situation in which it would seem logical to view one new computer as equivalent to two old computers. If adjusting for quality is incorrect in this simple case, it probably would not be correct in any case. Initially one truck costs as much as one computer and the value of their annual services are equal. However, computers rapidly improve while trucks undergo no progress.

Consider a future time when accumulated technical progress has caused new computers to be twice as productive as old computers. Now the procedure adopted becomes important. Suppose there are equal quantities of both types of good initially, and the same number of dollars of new goods is added for each (without any retirements).

If obsolescence is recognized in the accounting system, a unit of new trucks is equated to a unit of new computers. There would then be one unit of new trucks, and one of old trucks, for a total of two units of capital. Since old computers are only half as productive as new computers, old computers represent half a unit of capital.

Capital has increased 33 per cent in going from 1.5 units (1 of trucks and .5 of computers) to 3.5 tits (2 of trucks and 1.5 of computers).

The other option is to equate computers and trucks by their cost in the original year. Table 2 shows the accounts.

Originally there were 2 units of capital (1 of trucks and 1 of computers). The added capital is 1 of trucks and 2 of computers (since the price of computers is considered to have been halved. The result is that there are now 5 units of capital, an increase of 150 per cent. Which should be used ? Are new goods more capital because of embodied technical progress, or are old goods less because of obsolescence ?

In might be argued that there is no problem because with sufficient disaggregated data on computers and trucks, and their income shares, one could compute an income share weighted average growth rate. This is correct, if one has the data. However, one seldom has data at the fine level of detail needed to calculate separate growth rates and income shares for such detailed categories as trucks and computers. This still leaves the theoretical question of whether to reduce old capital goods for obsolescence or to increase new capital goods for improved quality.

The growth accounting formula gives no guidance as to whether to recognize obsolescence (treating old computers as half of new computers) or to adjust the price indices for costless technical progress (treating new computers as twice old computers). Both procedures appear reasonable.

Paradoxical Result of Adjusting for Costless Quality Change

If price indices are adjusted for costless quality improvements, the new computers are viewed as equivalent to two old computers, and hence as equivalent to two old trucks (since old computers have been equated to old trucks). This would require that a new computer's marginal product be twice that of a new truck (since in the absence of technical progress in trucks, new trucks still equal the older trucks they are identical to). However, economic theory tells us that this cannot be. In equilibrium, the marginal products of all capital goods whose services have the same production costs must be equal, a new computer and a new truck still have the same costs. The new computer and the new truck must have the same marginal product. Thus, use of quality adjusted price indices here involves a logical contradiction, in that one cannot maintain equality of current marginal products for both the earlier model and the later model

The problem arises from the fact that capital is a produced means of production (as Rymes (41) emphasized), and hence will be produced up to the point where its value (capitalized rent) equals its costs of production. In equilibrium, its marginal product is determined by its costs of production, not by how much capital embodied technical progress it has benefited from. This problem does not occur if old capital goods are adjusted for obsolescence. A new truck and a new computer costing the same are regarded as providing the same quantity of services. An old truck that is identical to a new truck provides the same quality of capital services as either a new truck or a new computer. An old computer is considered to provide half the services of a new computer or a new truck. In practice, the question of whether to reduce the services of old capital for obsolescence, or to increase that of new capital for embodied technical progress, is likely to be determined on pragmatic grounds. Even if a way were found to adjust price indices for costless quality improvements (as has been done in the United states for computers), use of these indices requires that the rates of deterioration in capital goods must be known. These rates are not known. At best, estimates exist or can be made for the rates of decline in rent (=marginal product) due to the combined effects of obsolescence and deterioration.

Sectoral Investment Policy

The reason for doing growth accounting is to find out how much of the economic growth arising from adding new machines is due to the growth in capital represented by the new machines. Presumably, this will be a contribution to discussions about whether it is worthwhile to add new machines. Adjusting for obsolescence is the preferred approach for such economic planning. It is commonly desired to know the increase in national income that can be expected from investing in a particular type of capital good or a particular sector. This can be calculated by multiplying the services to be expected from the good by the price for these services (which in theory equals the value of the marginal product). In turn, the price for these services can be calculated by dividing the total income from capital in the last period by the total capital services used in that period. As pointed out, this will work only if the weights used for the services of the different capital goods are proportional to their current marginal products. This will hold if embodied technical progress has been reflected by adjusting the services of older capital goods for obsolescence, but not if the new good are considered to provide a greater quantity of capital services due to embodied technical progress. To see this, consider the case discussed earlier, where a new computer is considered to provide twice the services of an old computer that costs the same amount.

As an illustration, suppose the situation is the one described earlier in which there are five units of capital services, measured in old truck units, being provided (an old truck = 1 unit, an new truck = 1, an old computer = 1, a new computer = 2 units). These five units were calculated with a new computer as equivalent to two old computers. Suppose total income is $.3.50. The apparent average price for a unit of capital services is $.70. This would imply that a new truck which rented for $.70 would provide services worth $.70, and a new computer (whose rent was also $.70) would provide services worth $.1.40. An economic model that included computers and trucks as separate economic inputs would predict that any given level of investment would add more to output if made in computers than in trucks (R-member that the stock and flows are considered by as proportional to one another.) For instance, in this example suppose each unit of capital services cost $3.50 to produce (perhaps $35 of capital good was required, and investors required a 10 per cent return). The economic model would predict greater income gains if the investment were made in computers than in trucks. This conclusion is wrong, as can been from elementary economic theory. The standard neoclassical theory used here holds that all potential profitable investments are made. This requires that the value of the marginal product equal the price of the good's services. This condition is met for trucks. Their marginal product is worth $.70, which is the same as the price of their services, however, it is not met for computers, remember that the price of providing capital services via computers is the same $ .70 as for services via trucks. Thus, all opportunities for employing computers that yield more than S.70 would have already been exploited, leaving the marginal product of computers at $.70, the same as trucks.

In the above example, the error is easy to detect, but in an elaborate economic model with multiple sectors and many different investment goods, the inconsistency might not be detected and might lead to error. Even if there were no errors in sectoral allocation, there could be errors in estimating the average return from new investment. The income increase from new investment would depend on how it was allocated among the sector, and would be $.1.40X the increase in service from computers plus $.70 X the increase in services from trucks. If most of the new investment were in computers rather than trucks, the return to new investment would be overestimated. Now consider the procedure of reducing the capital services of old goods for obsolescence. To compute the prices for current services, it is most useful to again treat the services of a current truck as one unit. Since a current truck and a new computer rent for the same amount, they must each provide one unit of capital services. The old computer provides half as much services. Thus the total services are 1 from the old truck, 1 from the new truck, 1 from the new computer, and .5 from the old computer. The total is 3.5. The price per unit of services is $3.50, or $1 per unit. The marginal products of both new computer services and truck services would be estimated to be equal. The model would not say that additional investment in computers would do more for national income than investment in trucks. Likewise, if the only use for the calculations is to tell planners by how much income should go up if the services from capital can be increased by a small amount, the calculations using the obsolescence procedure reach the correct answer.

This example shows how, for purposes of policy making, the procedure of reducing old goods for obsolescence will give the correct answers, while the procedure of treating the new model goods as representing more capital will not. The example is obviously on the simple side. The obsolescence of the old computer design is treated as unanticipated, and firms do not anticipate any further obsolescence. More elaborate examples could be constructed with obsolescence anticipated and planned for, but doing so would not change the conclusion that models that treated capital goods benefiting from costless quality improvements as more capital are likely to lead to incorrect predictions. Incidentally, directly discussing the marginal product of investment goods may be preferable to using the vocabulary of growth accounting to discuss the appropriate level of investment. Using growth accounting, the growth contribution is determined from the last increment of capital. It is presumed that further additions of capital will lead to a similar amount of growth. The key assumption is that the marginal product of the machines whose addition is being considered will be the same as that of the machines added last period, a simpler way to understand the problem is to directly calculate the marginal product of new machines. One method of estimation is to use the historical data on the income from capital, and a measure of the capital input when that income was used, thus deriving a price for capital services which is interpreted as capital's marginal product. This latter approach (while using the same data) has the virtue of making the reasoning and any approximations involved explicit.

The simple example discussed above, along with other possible procedures, is summarized in the table below. Capital services are measured in truck equivalents The capital stock consists of one old truck, one new truck, one old computer, and one new computer. The total income from capital is $3.50. The calculations of the increase in income from adding one new truck and one new computer are exhibited in detail for various methods. The technology is such that trucks have not changed in design or in cost. However, one new computer does the work of two old computers, but costs the same as the old computer. No equipment wears out, and no further technical progress is expected. The industry being considered produces a consumption good whose quality has not changed. Output is measured in terms of dollar's worth of this product.

Data Analysis

The first column shows the price indices for new computers. Where price indices are adjusted for costless quality change, the indices for new computers is .5, reflecting the fact that a new computer docs twice the work of an old computer. Where the price indices are based on the resources used in production, the computer price index has not changed and is hence equal to one. If obsolescence is to be adjusted for the old computer is considered to be half of the new computer. The are four possible combinations of the two rules for handling the price indices for new computers and old computers. Column I shows the procedure argued for here Obsolescence is allowed for, but cost less quality improvements are not adjusted for. The second column adjusts for costless quality improvement, but not for obsolescence. The third column adjusts for costless quality improvements and obsolescence. This appears to be the Jorgenson-Griliches procedure. The fourth solution allows for neither costless quality improvements nor for obsolescence. This fourth solution appears to be the procedure used by Denison during the sixties, when he argued that costless quality improvements could not be estimated, and that capital inputs should be based on the cost of producing the good. Obsolescence should be deducted only when the good was abandoned. Thus old computers still in use would not be considered to provide less capital services just because never computers with the same cost provided more services (see Millert (42,43) for a detailed critique of Denison).

The table shows what the flow of capital services estimated by each of these methods would be, as well as what the price of a unit of capital services would be and tile estimated marginal products and additional output from investing in one more truck or in one more computer (which would naturally be new). For the policy analyst, the last two rows show the output to be expected from investing in either trucks or in computers. For the economist merely trying the estimate the contribution to growth made by investment in various goods, these numbers indicate the amount of output growth that should be attributed to adding the designated goods to the capital stock. As can be seen, the different accounting systems give quite different numbers, with the increase in national income from adding capital services priced at $1.00 ranging from $.70 to $1.56. Only the proposed system actually estimates the marginal product of capital services from all goods to be equal to the price of capital services from those goods, as is required by economic theory.

Consumption Foregone Approach

The proposed system is equivalent to measuring the quantity of capital services provided by a good as being proportional to the minimum consumption that must be forgone to obtain the services of the good. (Notice that if new capital goods are measured in terms of the consumption goods that must be sacrificed to produce them, the resulting measure is a stock measure. However, the stock represented by a new good is also the present value of all its anticipated future services. Since it is desirable to use the concept of capital services, this stock measure must be multiplied by an appropriate conversion factor to give the services provided for any particular year of its life.) Also notice the use of the phrase minimum consumption. If technical progress has changed the cost in consumption good forgone of obtaining the services of capital goods (as in the computer example), the older capital goods still existing have undergone obsolescence. Then the old goods must have a weight equivalent to the minimum amount of consumption goods that must be forgone to obtain the equivalent of their services. The most economical way to obtain the equivalent of the old goods' services is not to produce goods of the old design, but to produce goods of the new design that give equivalent services (*). Notice that this quantity of capital services provided by newly produced goods will be the same regardless of whether technical progress has reduced the cost of producing a good of the old design, or whether a new design has been introduced that can be produced at a lower cost. The latter case might be described as embodied technical progress.

If one tries to draw a distinction between decreased cost of producing capital goods, and new goods that produce more output, the distinction will often be difficult. For instance, what if there is a minor design improvement, which permits a capital good to be produced at lower cost? A realistic (and important) case occurs if several semiconductor chips in a computer are replaced by a single chip. A new computer model emerges that, from the consumer's viewpoint, is exactly like the order model, but which costs less to produce. Should is be treated as a price reduction, or as the introduction of a new product? If the product is unchanged, this should be price index. If a new product has been introduced, the price index for computers should not be reduced. Quite different values for the computer price index, and for the stock of computers are obtained depending on which interpretation is adopted.

After the new models are introduced, what is the contribution to economic output that is to be attributed to them? With embodied technical progress, the answer is critically dependent on whether the new chips are considered as reducing the price of the old model, or as creating a new model embodying technical progress. If it is the same old model, the new computers go into the stock at their old value. If it is a new model, they go into the stock at a multiple of their old value. The latter result arises because the computer price index is considered to have declined. The contribution to output that is considered to arise from adding to the stock functionally identical new computers is critically dependent on whether they are considered to be a new product or not. This is an odd result. In the proposed system, in which the new models are valued according to the consumption goods that must be forgone to obtain them, the contribution to output growth from adding the new models is the same regardless of which system is used. This is obviously correct, since the economic outcome cannot depend on {whether the technical development that permits more circuits on a chip is regarded as reducing computer prices, or as permitting the introduction of better computers at the old prices. The investment in the new model computers accounts for a quantity of output equivalent to the marginal product of capital times the quantity of capital added (valued in consumption forgone). The attribution of output increase to growth in the capital stock does not depend on whether the new computers are considered to be a new model or the old model. This is as it should be. Any output increase accompanying the addition of the new computers that exceeds the above amount is attributed to technical progress.

Denison (5) has discussed the magnitude of the contribution to U.S. growth accounted for by capital accumulation when the consumption forgone approach is used, versus the contribution made when alternative methods are used. He shows that the new U. S. methods for handling the improved quality of computers increases the rate of growth of the capital stock after 1982. Thus, in the computer-using industries, the rate of growth of total factor productivity is reduced. (This is offset by increased total factor productivity growth in the computer manufacturing industry.) If measurement is in terms of consumption forgone, total factor productivity regards the new method of measuring input of computers to be undesirable.

Basic Aggregation Problem : The reader probably now sees the basic problem. When technical progress improves a capital good, it normally lowers the prices of its services, with a lower price, firms use it in more applications, lowering its marginal product. Arguments that depend on holding the relative marginal product constant while technical progress lowers the price wont work. For instance, if computing power is very scarce in 1950, the marginal product of computing power (and hence the marginal product of computers) will be high. If computing power is abundant in the 1990's, the marginal product will be much less. There is no way the appropriate relative weights for computers and trucks in 1950 will be the same as the appropriate weights in 1993. The usual procedure of deriving growth accounting from an aggregate production function leaves he impression that the correct construction of a time series for capital input is a key aggregation issue. This paper has shown that the only necessary condition is that the marginal product of capital services from net investment can be estimated by dividing total income from capital by the nub mer of units of capital services input. This requires merely an aggregation system such that all units of capital have the same marginal product per unit of capital at the time of the calculation. Unfortunately, many apparently desirable attributes for time series are inconsistent with this apparently simple requirement. For instance, requiring that a physically unchanged good still in service always be considered as providing the same number of units of capital service is inconsistent with equal marginal product condition, since it requires excluding obsolescence. If this is the meaning attached to physical capital, time series of physical capital are unsuited for growth accounting. It is plausible that a new machine that can do as much as two old machines should always be treated as equal to two old machines. Unfortunately, this plausible condition is inconsistent with growth accounting.

The reader should not presume that the concepts of aggregate capital required for growth accounting are suitable for other purposes. In particular, weighting different vintages by their marginal products is not the most useful procedure in studies of factor inputs (including investment studies), although there is not space to fully develop the argument here. (It has been made elsewhere (44)) In essence, weighting different vintages by marginal product is equivalent to weighting them by rent, orby quasi-rents. From the definition of quasi-rents as the value of output minus the costs of non-factor input, it follows that isoquants will have linear sections an production functions will be nonconcave. This is logically inconsistent with the usual assumption in factor input studies of strictly concave production functions and unique factor inputs determined by factor prices. The author has discussed these points in this Journal and elsewhere (45,46,47).

Conclusions

A vintage capital model may calculate different capital stocks for different types of capital. With a base year several years in the past, a dollar of investment in computers will appear to increase income much more than a dollar's worth of investment in trucks, or other machinery. A social planner, seeing these results, may conclude that national income would be increased by a program of subsidizing computer purchase by a tax on purchases of other machines. Except in the case of certain market imperfections, this would be unwise. National income would be maximized by investing up to where the value of the marginal product equaled marginal cost of all types of capital. Economic planners using a detailed vintage model with different types of capital would not arrive at a solution that met this condition. With a base year five years in the past, and computer prices dropping by 20 per cent year (about the correct rate since Gordon (3) found that between 1951 and 1984 quality adjusted computer prices declined an average of 21.8 per cent year) the marginal product of a dollar invested in computers could be only a third of the marginal product of a dollar invested in other machinery of the same life and distribution of services over the machine's life. Thus, there is a real potential for misallocation of capital from use of a vintage capital model in a planned economy.

Adjusting the price indices for new investment for its improved quality and reducing the quantity of capital represented by old capital goods for obsolescence are alternative means for incorporating capital embodied technical progress. One of these techniques should be used to insure that the proper ratio between the quality of new investment goods and the existing capital stock is maintained. Deflating investment with price indices adjusted for costless quality change violates the condition that good of equal cost must have equal marginal products in equilibrium. When there are several types of capital goods, each undergoing a different rate of improvement, the correct procedure is to reduce the quantity of capital services provided by older goods for their obsolescence.

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(6.) Denison, E. F, and Robert J., Gordon's concept of capital, Review of Income and Wealth (March 1993)

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(12.) Early. J. F. and Sinclair, J. H., Quality adjustment in the producer price indexes, in Foss. M. F., (edited), The U. S. National Income and Product Accounts : Selected Topics (Chicago, 1983)

(13.) Gordon, R., Energy efficiency, user cost change and the measurement of durable goods prices, in Foss, M., (edited), The U. S. National Income and Product Accounts: Selected Topics, (Chicago, 1983)

(14.) Jorgenson, D.W., The embodiment hypothesis, Journal of Political Economy (February 1966)

(15.) Griliches. Z., Comment, in Foss, M. F., (edited), The U.S. National Income and Product Accounts : Selected Topics (Chicago, 1983)

(16.) Solow, R.M., Technical progress, capital formation, and economic growth, American Economic Review (May 1962.)

(17.) Solow, R. M., Capital, labor and income in manufacturing., in The Behavior of Income Shares (Princeton, 1964)

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(19.) Abramovitz, M., Economic growth in the United States, American Economic Review (September 1962)

(20.) Denison, E. F, The unimportance of the embodiment question, American Economic Review (March 1964)

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(22.) U. S. Bureau of Labor Statistics, Trends in Multifactor Productivity: 1948-81, Bulletin 2178 (Washington. D.C., 1983).

(23.) Boddy, R. and Gort, M., Obsolescence, embodiment, and the explanation of productivity change, Southern Economic Journal (April 1974)

(24.) Brown, M., On the Theory and Measurement of Technical Change, (Cambridge, 1966)

(25.) Tice. H. S., Depreciation, obsolescence, and the measurement of the aggregate capital stock of the United States 1900-1962, Review of Income and Wealth (June 1967)

(26.) Hulten. C. R. and Wykoff, F. C, The measurement of economic depreciation, in Hulten, C. R., (edited), Depreciation, Inflation, and the Taxation of Income from Capital (Washington, D.C., 1981)

(27.) Miller. E. M., The definition of capital for Solow's growth accounting formula, Southern Economic Journal (July 1989)

(28.) Miller. E. M., Is efficiency decline rent decline or capacity decline ? Southern Economic Journal (January 1992)

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(30.) Griliches, Z., Capital stock in investment functions : some problems of concept and measurement, in Christ, C. F., Measurement in Economics (Stanford 1963)

(31.) Griliches, Z., Hedonic price indexes and the measurement of capital and productivity : some historical reflections, in Berndt, E. R. and Triplett. J. E., (edited), Fifty Years of Economic Measurement, (Chicago, 1990).

(32.) Jorgenson, D. W., The economic theory of replacement and depreciation, in Sellekaerts, W. (edited), Econometrics and Economic Theory (White Plains, 1974)

(33.) Jorgenson, D. W., Accounting for capital, in von Furstenberg, G. (edited), Capital Efficiency and Growth (Cambridge, 1980)

(34.) Cristensen, L. R., and Jorgensen, D. W., The measurement of U. S. real capital input 1929-1967, Review of Income and Wealth (December 1969)

(35.) Gollop, F. W., and Jorgenson, D. W., U. S. productivity growth by industry, 1947-1973. in Kendrick, J. W. and Vaccara, B. N., (edited), New Developments in Productivity Measurement and Analysis (Chicago, 1980)

(36.) Christensen, L. R., Cummings, D. and Jorgensoen, D. W., Economic growth, 1947-1973, An international comparison, in Kendrick, J. W. and Vaccara, B. N., (edited), New Developments in Productivity Measurement and Analysis (Chicago, 1980)

(37.) Fraurneni, B. F. and Jorgenson, D. W., The role of capital in U. S. economic growth : 1948- 1976 in von Furstenberg G, Capital, Efficiency, and Growth (Cambridge, 1980)

(38.) Denison, E. P., Some major issues in productivity analysis: An examination of estimates by Jorgenson and Griliches, Survey of Current Business (May 1972).

(39.) Denison E. F., Final comment, Survey of Current Business (May 1972)

(40.) Jorgenson. D. W., and Griliches, Z., Issues in growth accounting: a reply to Edward F. Denison, Survey of Current Business (May 1972)

(41.) Rymes, T. K., On Concepts of Capital and Technical Change (Cambridge 1971)

(42.) Miller, E. M., The importance of the embodiment effect: a comment on Denison's growth accounting methodology, Journal of Macroeconomics (Winter 1985)

(43.) Miller, E. M., Growth and embodiment: a reply to Denison, Journal of Macroeconomics (Winter, 1988)

(44.) Miller, E. M., The unique 'rent defined' capital input assumption in investment theory, Southern Economic Journal (April 1989)

(45.) Miller, E. M., Is aggregation of capital by its rent reasonable ? Implications for growth accounting, Journal of Financial Management and Analysis (January-June 1992)

(46.) Miller, E. M., Used capital : implications for isoquants, production functions, and Shephard's lemma, Eastern Economic Journal (April-June 1988)

(47.) Miller, W. M., What units is capital he measured? in Joan Robinson's classic question revisited, in Rime, I., (edited), The Joan Robinson Legacy (Armonk, 1991)

Professor EDWARD M. MILLER, Ph.D.

Professor of Economics and Finance

University of New Orleans

Louisiana, U.S.A.

(*) The Choice of Base Year for Measuring Prices: The above argument can he extended to the question of what base year to use for measuring capital goods prices for a growth accounting exercise. In one sense this is again just a question of whether to use a Laspeyres or a Paasche index (or something else). However, just because a problem can be expressed as one of choosing an index doesn't mean it lacks a solution, or that the choice of index is not important. Since the argument presumes that the services of all items of capital sell for the same price (rent divided by number ofunits of capital services provided), the hase year for measuring prices should he chosen to make this assumption possible. This implies the use of current price ratios for different types of capital goods. Only if the prices of goods considered to provide the same quantity of capital services are equal can their marginal products be considered equal (which makes possihle assuming a single marginal product of capital). This requires use of current price ratios for ditferent capital services (which will usually he proportional to current prices for the goods themselves). If in a previous period one computer rented for the same amount as one truck, hut now two computers rent for as much as one truck, one truck should he considered equal to two computers for the growth accounting purpose of calculating what the effects on output of adding capital will be.

The relationship between the previous argument concerning capital embodied technical progress and price changes for other reasons can be seen by again considering the case of a computer model that has heen so improved that it now does twice as much as an older model. It was argued above that the new computer should be equated to a truck whose cost of production of services is the same, since in equilibrium both will have the same marginal product. Now consider the case in which the ability of the computer to do calculations is unchanged, hut a new manufacturing procedure has made the cost of producing the computer only half as high. (One might imagine this was done by putting more circuits on each chip, but this would involve introducing a new model, which was the earlier case.)

From the social viewpoint the two innovations are equivalent. Either one halves the cost of purchasing enough computing power to do a specified number of calculations. Both should be treated the same way in growth accounting, as in the proposed procedure. If originally the computers needed to do a certain number of calculations cost as much as a truck, but technical progress (or price changes) have caused equivalent computers to sell for as much as half a truck, the new computers are considered to he equivalent to halt" a truck. (Since rational firms use inputs up to where their marginal products equal their prices, the marginal product of this amount of computing power is half that of a truck).

The argument for using current price ratios holds regardless of why computer prices have dropped. They may he down because technical progress has reduced the cost of producing them, (which in this case has impacts similar to an improvement which makes them twice as productive), or because the producers reduced the price they charged, or because a key resource used in producing them has become less scarce. Regardless of the reason for the price change, when trying to discover the effect on output of more investment, the different items should he aggregated by their current prices (which equals their marginal products or their cost of production).

(*) However, if the Solow Vintage model is to be applied to the real world, it is necessary to have a specific definition for the quantity of capital, including the base year. The arguments of this paper about the most useful definition of capital apply to Solow's vintage model with embodied capital, as well as to his model with non, embodied capital. Indeed, with the proposed definition of the quantity of capital services, the capital stocks in the two models should not differ, and they should not differ greatly from a model where capital is measured by consumption forgone. However, in practice the capital definition used may not be the one suggested here. The most likely reason is that the statistical system uses a base year other than the current year. For instance, the United States national income statistics use Laspeyres indices for pries, with the base year being the year of the last economic census. When this happens, which of several prices indices are used to deflate investment forthe construction of capital stocks will affect the growth rate of the capital input.

(*) Examples of embodied technical progress would include the developinent of new methods of machine aided computation, which required the production of computers, the introduction of jet travel, which required jet planes, and the introduction of atomic power, which required the, production of atomic reactors. In all of these cases, benefitting from the technical progress required not merely the development of the new techniques, but the replacement of old capital goods with new capital goods incorporating the technology. Investment in these new goods is necessary to realize benefits from-the new technology. One implication of this fact is that it will often be a number of years between when an invention is made and when it is finally in full use, during this period the old machines are being replaced with new ones.

In contrast, disembodied technical progress consists of new techniques which can be introduced by merely changing the process of manufacure. For instance, more output might be obtained by someone devising a more efficient arrangement of machines on the shop floor, also, in machines that are controlled by computers one could conceive of a new, more efficient program being written. Here no physical investment is required.

(*) A dollar or new investment will raise output more than it would have earlier. Since today's computer is more powerful than yesterday's computer (correct), at first it seems correct to treat a dollar's worth of today's computers is making a larger contribution to output progress than a dollar's worth of yesterday's computers, where the dollar measures the production cost of yesterday's computers. If we decided to make our unit of capital a dollar's worth of capital in an earlier base period, today's computer would represent more capital than a dollar's worth of capital. If one uses the simplified formulas (including Solow's vintage capital mode) and examine the effect of investing in different types of capital goods, one gets the result that a dollar invested in computers raises output more than a dollar invested in trucks, (even when the two types of capital have the same lives and services distributed in the same pattern over their lives).
TABLE 1
UNITS OF ACCOUNTS ARE NEW TRUCK EQUIVALENTS

TRUCKS   COMPUTERS   TOTAL   CAPITAL

Old      1           5       1.5
New      1           1       2.0
Total    2           1.5     3.5

TABLE 2
UNITS OF ACCOUNT ARE IN OLD TRUCK EQUIVALENTS

TRUCKS   COMPUTERS   TOTAL   CAPITAL

Old      1           1       2
New      1           2       3
Total    2           3       5

TABLE 3
TABULAR SUMMARY

                              Adjusting for
                              obsolescence,      Adjusting
                              but not for        for capital embodied
Indicators                    capital embodied   progress but not for
                              progress           obsolescence
                              (The preferred
                              solution)

Obsolescence rate for           5                0
computers
Price index fix new             1                5
computers
New computers in terms          1                2
of trucks
Old computers in term           5                1
of trucks
Total capital services in      35                5
truck equivalents
Income from capital ($)       350                3.50
Price of capital ($)            1                 .7
Marginal product of             1                 .7
capital services ($)
Income increase from            1                 .7
investing in a truck ($)
Income increase from            1                1.40
investing in a computer ($)

                              Adjusting for both     Adjusting for
                              obsolescence,          neither
                              and capital            obsolescence and
Indicators                    embodied               capital embodied
                              technical progress     technical
                              (Jogenson-Griliches)   progress
                                                     (Denison)

Obsolescence rate for          5                      0
computers
Price index fix new            5                      1
computers
New computers in terms         2                      1
of trucks
Old computers in term          5                      1
of trucks
Total capital services in     45                      4
truck equivalents
Income from capital ($)        3.50                   3.50
Price of capital ($)            .778                   .875
Marginal product of             .778                   .875
capital services ($)
Income increase from            .778                   .875
investing in a truck ($)
Income increase from           1.56                    .875
investing in a computer ($)
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Author:Miller, Edward M.
Publication:Journal of Financial Management & Analysis
Article Type:Report
Date:Jan 1, 2018
Words:12559
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