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The devil is in the details: Capital stock estimation and aggregate productivity growth—An application to the Spanish economy

A Correction to this article was published on 29 October 2021

This article has been updated


The variables that contribute to explaining the major puzzles and paradoxes in macroeconomics and economic growth literature always appear related, directly or indirectly, to capital stock and depreciation. Depreciation defined in a narrow sense refers only to physical wear and tear, but in a broader sense, it also includes economic deterioration and obsolescence. In this study, we explore the link between these two depreciation concepts, the capital deepening and total factor productivity (TFP) growth. We propose a double growth accounting framework that allows us to establish a relationship between variables in statistical terms and variables in economic terms. Then, with Spanish data for 1964–2015, we first analyze the role played by capital intensity and TFP in explaining the evolution of labor productivity. The results are substantially different depending on whether we use statistical or economic measures of capital and depreciation. Second, we focus on the paradox of productivity, concluding that the apparent absence of a positive correlation between investment in information and communication technology and the TFP growth rate may be due to the delay effect associated with such investment combined with the statistical under-estimation of true economic depreciation.

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Source: Authors’ elaboration

Change history


  1. 1.

    Solow’s original work also included some theoretical assumptions that, if violated, would contribute to the mismeasurement of TFP growth. These assumptions are that all inputs adjust instantly to their optimal levels, there are constant returns to scale, there are no external factors, and the economy is perfectly competitive. Several authors have since questioned their validity, considering the effects of quasi-fixed and external inputs, non-constant returns to scale, a variable degree of capacity utilization, or imperfect competition, allowing them to propose alternative measures of TFP growth using the primal approach (Basu et al. 2006) as well as the dual approach (Morrison and Schwartz 1996).

  2. 2.

    Baily (1981) considered the US productivity slowdown a consequence of poorly measuring the services of capital, and suggested using the economic value of capital stock computed as Tobin’s average q.

  3. 3.

    See the seminal contributions of Solow (1957) and Denison (1962).

  4. 4.

    The non-financial business sector is defined as total activities in the economy excluding the financial intermediation sector, real estate, and non-market services.

  5. 5.

    The Spanish regional database BD.MORES (see De Bustos et al. 2008) is compiled by the Budget General Directorate of the Spanish Ministry of Finance and is available at:

  6. 6.

    From 2014 onward, a new period of economic recovery began. During these years, the Spanish economy experienced a strong increase in both production (3.65%) and employment (2.50%), but not in labor productivity (1.16%). In this section, however, we do not provide a detailed analysis of the most recent years due to the provisional nature of some data and the unavailability of other data.

  7. 7.

    For the crisis period 2008–2012, Hospido and Moreno-Galbis (2015) use balance sheet information from a sample of Spanish manufacturing and services companies to point out that labor productivity is also affected by the behavior of TFP. The authors find a positive link between TFP and certain composition effects associated with the proportion of temporary workers and the weight of exporting firms facing international competition, which contribute significantly to the recent improvement in labor productivity.

  8. 8.

    According to Solow (1987): “the fact that what everyone feels to have been a technological revolution ... has been accompanied everywhere ... by a slowing down of productivity growth, not by a step up. You can see the computer age everywhere but in the productivity statistics.“

  9. 9.

    The idea that the measured consequences of investment in ICT need time to become visible in the macroeconomic aggregates initially put forward by Basu et al (2001, 2003). In the Spanish case it was pointed out by Mas and Quesada (2006) and Martínez et al. (2008).

  10. 10.

    Given that this period of strong economic growth represents an expansive phase of the business cycle, a higher rate of productive capacity utilization is also expected. Therefore, greater depreciation due to economic deterioration appears in our records combined with the greater depreciation caused by obsolescence.


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We are very grateful to the editor and especially to an anonymous referee for their helpful suggestions and constructive comments. The authors acknowledge the financial support from the Spanish Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación and FEDER project PGC2018-095821-B-I00, the Spanish Ministerio de Ciencia, Innovación y Universidades projects ECO2016-76818-C3-3-P and PID2019-107161 GB-C32, the Belgian research programs ARC on Sustainability, as well as the Generalitat Valenciana PROMETEO/ 2020/083.

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Correspondence to María-José Murgui-García.

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The original version of this article was revised. The citations for the equations (8), (9) and (10) are incorrect.



Two measures of capital stock and the depreciation rate

Here we provide a quick revision of the traditional statistical and the new economic measures of the depreciation rate and thus of capital stock. The availability and quality of data series for production inputs determine the results concerning the residual variable TFP in growth accounting exercises. Although there is no widespread debate regarding labor input and a consensus dictates how it should be measured, there have been significant discrepancies and contradictory approaches as regards the way in which the series of physical capital stock should be obtained. Even so, a certain agreement on the measurement of capital has been reached among empirical and theoretical economists. It involves the use of both Jorgenson’s theorem of proportionality, which states that the depreciation of capital goods is proportional to the capital stock, and the mechanics of the PIM, which is widely used to estimate capital stock series and studies according to the accumulation equation

$${K}_{t}={I}_{t}^{G}+(1-{\delta }_{t}){K}_{t-1}$$

The dynamics of capital stock depends on investment and depreciation flows. There are no fundamental problems involved in the measurement of the gross investment flow, \({I}_{t}^{G}\), since it represents the acquisition of new capital goods according to explicit transactions that are market-observable transactions. However, the case of the depreciation flow is different because economic transactions related to capital depreciation activities are basically unobservable.

In order to measure depreciation and provide useful data series, the most common practice has for years consisted of assigning accounting values based on assumptions about the mathematical functional form of the survival (retirement) profile, the efficiency profile according to age, and the age-price profile of an asset or cohort of assets. Hence the measure of capital stock is a statistical measure because depreciation is estimated under arbitrary assumptions about the parameters that characterize the previous functions. The statistical measurement of depreciation implies that the only loss in the value of the assets taken into account is the loss experienced as they age. Consequently, because it is ignored the role of utilization, maintenance and embodied technical progress, depreciation is treated more as a technical necessity than as a result of economic decisions.

In such a context, accuracy in implementing the perpetual inventory method depends on the choice of the asset retirement distribution. A survival profile is required in order for the retirement process to be modeled, and a key parameter in this process is the average service life. Although a subject of debate, it is usual to assume fixed service lives and one ad hoc pattern for retirements (one-hoss-shay, linear, or a bell-shaped function like Winfrey, Weibull and lognormal distributions). The age-efficiency function is also assumed to be fixed, with several possible shapes (hyperbolic, linear or geometric profiles). In coherence with the above, the age-price function is also taken as fixed and can be of the straight-line type, with prices falling by a constant amount each period, or of the geometric type, with prices falling by a constant rate each period.

In the search for a measure of the depreciation flow, different combinations of retirement patterns with age-efficiency patterns or with age-price patterns are admitted for the purposes of achieving the goal. The functional form of these interconnected statistical functions is important when it comes to determining the functional form of the depreciation pattern, the parameters of which are taken mainly from empirical studies (company accounts, statistical surveys and second-hand asset price records exploited using econometric methods). However, the more recent OECD recommendations (OECD 2009) imply the explicit recognition that the geometric depreciation pattern is the most suitable approximation to the loss in value of assets as they age. Now the usual method employed to estimate the depreciation rate is the well-known double-declining balance method, as summarized in the expression \({\delta}_{i}=2/\overline{T}_{i}\), where \(\overline{T}_{i}\) is the average service life for assets of type i. According to this method, the measurement of depreciation is directly associated with the fixed average service life of the different assets.

In short, the measurement of capital according to the perpetual inventory method depends on a statistical measure of depreciation, which implies that the variability observed in the implicit depreciation rate \({\delta }_{t}\) mainly reflects changes in the composition of the capital stock.

Alternatively, we have the economic measure of capital stock. This is different in nature from the statistical measure above and also provides different results. In a recent paper Escribá-Pérez et al. (2018) revisits the intertemporal behavior of firms in a perfectly competitive environment. It represents a generalization of Hayashi’s (1982) paper because the investment-related adjustment costs function is complemented with a function incorporating maintenance and repair expenditures. The latter allows the depreciation rate to be considered an endogenous control variable together with investment. The control problem consists in choosing the optimal investment and depreciation that maximize the present discounted value of cash-flow. This problem has a single state variable, so it is subject to one dynamic constraint that expresses the accumulation process of capital stock,

$${K}_{t}^{*}={I}_{t}^{G}+\left(1-{\delta }_{t}^{*}\right){K}_{t-1}^{*}.$$

In fact, the economic measurement of capital \({(K}_{t}^{*})\) and depreciation \(\left({\delta }_{t}^{*}\right)\) translates to the empirics the fundamental assumptions of the purest neoclassical theory of capital, which suggests measuring aggregate capital at equilibrium in terms of value. Following Hayashi’s work, we can associate the economic value of the capital stock along the optimal equilibrium path, to the market value of the firm, \({V}_{t}^{*}\), by introducing the financial market measure of Tobin’s q ratio. Moreover, under the usual assumptions of competitive markets and static expectations we get

$${q}_{t}=\frac{{V}_{t}^{*}}{{{p}_{t}^{K}K}_{t}^{*}} =\frac{{B}_{t}^{*}}{{{r}_{t}{p}_{t}^{K}K}_{t}^{*}}=\frac{{B}_{t}^{G}-{\delta }_{t}^{*}{p}_{t}^{K}{K}_{t-1}^{*}}{{{r}_{t}{p}_{t}^{K}K}_{t}^{*}} .$$

This contribution integrates market prices like the real interest rate \(\left({r}_{t}\right)\) and the price of investment goods \(\left({p}_{t}^{K}\right)\), and profitability indicators like distributed profits (net profits \({B}_{t}^{*}\) or gross profits \({B}_{t}^{G}\)) and the observable Tobin’s \({q}_{t}\), into the evaluation process (Escribá-Pérez et al. 2019). The only data needed from official sources are price variables and the value of flows such as gross investment and gross distributed profits. Finally, from a known initial value of capital stock, equations (9) and (10) can be used sequentially to obtain the series of depreciation rate \({\delta }_{t}^{*}\) and capital stock \({K}_{t}^{*}\). According to this algorithmic system of equations we can identify a set of correlations between the independent variables mentioned above and the two endogenous ones. First, intensive investment expenditures will increase the capital stock but reduce the depreciation rate. Second, higher levels of distributed profits are associated with higher values of the depreciation rate and lower values of the capital stock. Finally, smaller values of Tobin’s q ratio and real interest rates are correlated with higher depreciation rates and a smaller capital stock.

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Escribá-Pérez, FJ., Murgui-García, MJ. & Ruiz-Tamarit, JR. The devil is in the details: Capital stock estimation and aggregate productivity growth—An application to the Spanish economy. Port Econ J (2020).

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  • Capital
  • Depreciation
  • ICT
  • Slowdown
  • TFP

JEL classification

  • E22
  • O33
  • O47