A contribution to estimate a benchmark capital stock. An optimal consistency method

Abstract

There are alternative methods of estimating capital stock for a benchmark year. However, these methods are costly and time‐consuming, requiring the gathering of much basic information as well as the use of some convenient assumptions and guesses. In addition, a way is needed of checking whether the estimated benchmark is at the correct level. This paper proposes an optimal consistency method (OCM), which enables a capital stock to be estimated for a benchmark year, and which can also be used in checking the consistency of alternative estimates. This method, in contrast to most current approaches, pays due regards both to potential output and to the productivity of capital. It is applied to 45 cases for nine OECD countries and six Latin American ones. It works reasonably well, and it requires only small amounts of data, which are readily available. It appears to exhibit similar accuracy to alternative methods, but it is virtually inexpensive in both time and funding.

Keywords:

• benchmark capital
• perpetual inventory method (PIM)
• potential output
• capital productivity
• optimal consistency method (OCM)

JEL Classification:

• O4
• B4

Notes

1. There are other convenient methods that can give an idea about a benchmark capital stock, but they are rough and unreliable, as no provision for productivity and optimal output are made, e.g. Hall and Jones (1999).

2. There are various possible assumptions for calculating depreciation, as can be seen in OECD (2001). For example, there could be a variable depreciation over time, diminishing at the beginning and accelerating after a trough or just a variable one according to other economic factors. Or there could be a straight‐line depreciation pattern that assumes that efficiency declines linearly over the lifetime of capital, therefore, the same depreciation rate applies over the capital‐service life. This means that the rate of depreciation for a given piece of capital stock is equal to the inverse of this capital life, i.e. if the capital stock life was 40 years, then the depreciation rate would be 1/40 a year. When applied over the initial capital stock this should take into account the retrospective accumulation to that benchmark year. It can be seen that the results between a straight line‐depreciation and a geometrical depreciation, as applied here, do not differ significantly. So the latter would normally be a good proxy for the former.

3. The benchmark for a GFCS is simply the accumulation of gross fixed capital formation (GFCF) over its working life, without allowing for depreciation. That is, if the working capital life is, say, 40 years, and the benchmark year is, say, 1950, then the GFCS at the benchmark year will be the sum of GFCF from 1911 to 1950. After that, this capital will undergo the retirement of any GFCF in their 41st year of age. Notice that this formulation, contrary to that of the Net Fixed Capital Stock, assumes that capital does not depreciate over its service life, but its switches off its service once its working life finishes, and therefore should be discounted (retired) from the capital stock. For example, a light bulb has about the same efficiency until it bursts. However, most types of capital actually require a good deal of maintenance and repair over their service lives, so as to delay efficiency losses, e.g. rolling stock or roads. The problem with using this capital for growth studies is that the expenditure in maintenance and repairs, which clearly is there to counteract economic and technical depreciation, is not counted as (replacement/restoration) investment. Therefore, there is a massive amount of investment‐like outlay that is simply ignored. This might make sense for taxation or accounting reasons, but it does not make much economic sense. In addition, most investments actually lose efficiency over their working lives, even allowing for maintenance and repairs, like dams, housing, and most machinery and equipment, which may make NFCS a more appropriate measure for economic studies.

4. This is a generic definition that does not tell us anything about the state of the economy, e.g. capital idleness or demand levels. We, however, estimate this at optimal levels, i.e. assuming under given constraints that capital is fully used and that it maximises its productivity. In other words, under given assumptions, we optimise the incremental and the average productivity of capital as well as the level of output. We call the later optimal or potential output (see especially Section 5).

5. For all our sub‐periods in OECD countries, the standard deviation as a percentage of the mean (i.e. the variation coefficient) ranges from 1.2% (Belgium 1982–92) to 6.3% (France 1972–82), with an average of 2.8%. In turn, the correlation coefficients between capital and GDP vary from 0.89 (Finland 1982–92) to 1.00 (Australia 1972–82) with an average of 0.98. For Latin American countries, the variation coefficient ranges from 1.7% (Colombia 1967–77) to 8.3% (Venezuela 1982–92) with an average of 4.3%. The correlation coefficients here vary from 0.84 (Venezuela 1982–92) to 1.00 (Mexico 1967–77), with an average of 0.95. These high coefficients respond partly to the fact that GFCF is a component of both the capital stock and GDP. But they also respond to the possibility that capital may be an all‐dominant factor or a good proxy for other productive factors that contribute to GDP, which are then dragged by capital and move in similar proportions and directions, within a range. This is especially true when capital is corrected for idleness (Thirwall 2003), which is what the optimisation in this paper implicitly sorts out.

6. Notice that these ratios do not represent the contribution of capital to output alone, as output is also made up of other contributions, such as those coming from labour, technical change and the institutional setup. Hence, the inverse of the capital–output ratio more properly represents the amount of output that can be ‘sustained’ by a unit of capital, allowing for the existence of other contributions. However, following normal practice, we will refer to the inverse of the capital–output ratio as the average productivity of capital.

7. Given that parameters (5) and (6) above are assumed to be fixed, then if capital depreciates by a given percentage, output will depreciate (be reduced) exactly by the same percentage, i.e. 100% of investment in year ‘0’ (I ₀=ΔK ₀) will produce 100% of the variation of output in year ‘1’ (ΔY ₁), as according to our one‐year lag assumption there can be no depreciation yet (of λ annual rate). Now, in year 1, assuming an annual capital depreciation of 10% (λ=0.1), the stock of capital accumulated up to year ‘−1’ (K ₋₁) will then contribute with only 90% (1 − λ =0.9) of the output of year ‘0’ (Y ₀), corresponding to Y ₀(1 − λ). This is the reason why equation (4) can be written in this way.

8. This is an exercise in constrained optimisation. But contrary to classic optimisation, the constraints (i) are many and simultaneous, and (ii) are expressed as inequalities, implying that there is no requirement that all constraints are exactly satisfied. Classical optimisation cannot deal with this, except in the most simple of exercises, in which the Lagrange method might still be usable. This is the main reason why this type of constrained optimisation has to be tackled via Linear Programming (LP). In the context of our equation (10), the LP says: given available data on actual output Y and on the depreciation λ (and therefore βs), find the parameters α_aK_by and α_b so that they minimise the difference between actual output Y and an unobservable optimal output Y*, provided that the optimal output is always larger or equal to actual output and the parameters are non‐negative. Then the linear programme, by means of an iterative process called the simplex method (which contrary to the Lagrange method is discrete rather than continuous), will first find a feasible solution and then improve it until its reaches an optimal solution, hence satisfying the task in hand (Vaserstein 2003). For another application of a similar LP programme, see Albala‐Bertrand (2007).

9. Following Note 3, the Gross Capital Stock (GFCS) level for a benchmark year can be defined as GFCS_b = ΣGFCF_i , where b is the benchmark year, and the sum subindex i ranges from the year of the initial investment (GFCF) up to the benchmark year, matching its service life. For example, if b = 1950 and capital life d = 50 years, then the sum index would accumulate all investment from 1901 to 1950. Thereafter, this capital will undergo retirements, corresponding to the cessation of the working life of the GFCF, in the 51st year. Therefore, the GFCS for any year after the benchmark year would be GFCS_{b +t} = GFCS_{b +t−1} − KR_{b +t−d} + GFCF_{b +t} . Where KR is the value of capital retired due to service life exhaustion. Accordingly, the subindex t corresponds to additional years after the benchmark year and the sub‐index d to the service life of GFCF at the moment of its inception. For example, if we want to know the residential GFCS for 1955, assuming that d=50 years and b=1950, then the equation above will look as GFCS ₁₉₅₅=GFCS ₁₉₅₄−RK ₁₉₀₅+GFCF ₁₉₅₅. Following a similar iterative procedure to that for the NFCS, the equivalent equation would take the form of GFCS_{b +t}=GFCS_b −ΣRK_i +ΣGFCF_i . Where the range of the first sum over RK is from b−d to b−d+t and that of the second sum over GFCF is from b to b+t. This can be also approximated via a geometrical depreciation rate, using equations (2) to (9) as in the NFCS. The retirement (RK) would now represent an average percentage of GFCS. So our method (OCM) can also be applied to the gross fixed capital stock, if need be. Lastly, we used GAMS (General Algebraic Modelling System) as a solver for the linear programme, but any other alternative would do.

10. To clarify, following equation (9), when the chosen final benchmark year is 1962, the initial benchmark year K_by will be 1950. And the initial years for the required GDP and GFCF series should be 1952 and 1951, respectively.

11. For example, the OECD reference series for France shows that the idleness‐uncorrected productivity of capital was decreasing, implying that our optimal productivity for the accumulated capital should have been larger than 0.37. So if we had some knowledge about this trend, then α _b>α_a , making the uncorrected departure significantly smaller than the one above. From Hofman (2000a) and OECD (1997) it can be seen that in periods in which the range of capital–output ratios is large, the overestimation of the base capital might also be large. This is especially the case for France 1970 and 1975, Belgium 1970, Finland 1970, Mexico 1950 and Brazil 1950 and Argentina 1965. It can also be seen that in most cases there was a strong upward trend in idleness‐uncorrected capital–output ratios or, its inverse, a strong downward trend in average capital productivity. So if the base capital should be used as benchmark capital, then we should devise some reliable test to correct the initial optimal α_b in relation to α_a . This should be mostly based upon the 10‐year series for GDP and GFCF in relation to the rule of capital accumulation. The asymptotic property of a PIM from any initial capital stock towards a reference PIM is likely to provide the answer, but this escapes the present paper.

12. This estimate of a benchmark capital might be more accurate than the alternatives as, in contrast to them, it pays due regard to capital productivity associated with potential output, and implicitly to capital idleness. But if the aim is to produce benchmark values closer to our references, then estimates about productivity trend associated with our series of GDP and GFCF can be entertained. An acceptable convergence from an arbitrary initial capital value may take many decades of PIM accumulation, but there will be no independent way to judge how long that should be. The method proposed here, however, requires only 10‐year series for GDP and GFCF, as our starting optimal base‐year capital stock would normally be close enough to the reference or actual capital stock.

13. Given that the sample was selected by a systematic rule (given years), independently from any characteristic of the countries in question, then the sample can be considered as random. This means that the frequency of the results in any of the intervals in the table, i.e. the percentage column, can be considered as the probability of the result within that interval. Then, by characterising each interval by its middle point, we can calculate the mathematical expectation of the departure (or ‘error’) associated with our method as compared to the reference ones. Notice that by construction these values would also correspond to the absolute average departure (in percent), coming from Table 1. This shows that the gap between the OCM‐PIM benchmark capital stock and the reference capital stock narrows to one‐third of the one associated with the OCM initial or base‐year capital stock. The speed of gap narrowing for individual countries would be associated with both the growth of capital via GFCF and the magnitude of its initial capital departure. In equality of conditions, we would expect that the faster that growth, the faster the narrowing of the gap and vice versa. Likewise, stable growth of GFCF and GDP would reduce the size of the departure. So the expectation of departure (or ‘error’) can be improved if we have some knowledge about the characteristics of the country over the period considered.