The Measurement of Growth under Embodied Technical Change by Omar Licandro* Jorge Durán** Javier Ruiz-Castillo*** DOCUMENTO DE TRABAJO 2001-09

June 2001

* FEDEA.

** CEPREMAP

*** Universidad Carlos III de Madrid

Los Documentos de Trabajo se distribuyen gratuitamente a las Universidades e Instituciones de Investigación que lo solicitan. No obstante están disponibles en texto completo a través de Internet: http://www.fedea.es/hojas/publicaciones.html#Documentos de Trabajo

The Measurement of Growth under Embodied Technical Change*

Omar Licandro FEDEA

Jorge Durán CEPREMAP

Javier Ruiz-Castillo Universidad Carlos III de Madrid

June 2001

Abstract

The new U.S. data from NIPA contradict some of the well-known Kaldor stylized facts, and call for a reformulation of the modern theory of economic growth. Among these new facts, three must be stressed: A permanent decline in the relative price of durable goods, a permanent increase in the real equipments to real GDP ratio, and a large long run growth rate of equipments relative to the growth rate of non durable consumption. In order to be consistent with these new facts, growth models must include at least two sectors, and the problem of defining aggregate output arises. In this paper, the economic theory of index numbers is used to propose a definition of the output growth rate, which is consistent with a representative agent's preferences in a growth model with embodied technical change. The main findings are that (i) NIPA's methodology measures growth in accordance with the economic theory on index numbers, and (ii) when the growth rate is measured as in NIPA, the contribution of embodied technical change to per capital GDP growth in the U.S. is of 69%, much larger than the 58% found by Greenwood, Hercowitz and Krusell (1997).

*The authors have benefited from comments on a previous manuscript of Raouf Boucekkine, Fernando del Río, Juan Francisco Jimeno and Michael Reiter. We thanks also Robert Gordon for his helpful advice on NIPA's methodology. Finally, the first author acknowledges the financial support of the Spanish Ministry of Sciences and Technology (SEC2000-0260). Correspondence author: Omar Licandro, FEDEA, c/Jorge Juan 46, 28001 Madrid, SPAIN, e-mail: licandro@fedea.es.

1 Introduction

The well-known Kaldor stylized facts have guided from the sixties the research agenda on economic growth, giving empirical support to the neoclassical growth theory. Most of the developments of this theory are based on the one-sector Solow-Ramsey model, which predicts that the economy converges to a balanced growth path (BGP) where, among other things, relative prices are constant, all components of aggregate demand grow at the same rate, and the capital-output ratio stays unchanged.

However, new evidence from National Income Product Account (NIPA) for the U.S. published by the Bureau of Economic Analysis (BEA) contradicts some of the predictions of the neoclassical growth model. Over the last decades, the U.S. economy shows the following pattern:

1. A permanent decline in the price of equipment investment relative to the price of non durable consumption

2. A permanent increase in the ratio of real equipment investment to real GDP

3. A large long-run growth rate of equipment investment relative to the long-run growth rate of non durable consumption

These three facts call for a reformulation of modern growth theory in order to generate predictions consistent with the new evidence. As stated in Whelan (2001), such a reformulation requires a multisector dynamic general equilibrium model. A first step in this direction is in the two-sector version of the optimal growth model proposed in Greenwood, Hercowitz and Krusell (1997), hereafter GHK. The first sector produces one non-durable good, which is used both for consumption and as an input for the production of durable goods. This sector benefits from disembodied technical change. The second sector produces one durable good which is only used for investment, and it benefits form an additional source of technological progress, the so-called embodied technical change. This is the simplest way of accommodating the permanent decline in the relative price of equipments, and having predictions consistent with facts 2 and 3.

This paper rises the fundamental question of how to aggregate consumption and investment in a common measure of real output, in the framework of a two-sector optimal growth model. In a static framework, the problem of aggregating different final goods in an index of aggregate output has been addressed by the economic theory of index numbers, which is at least as old as consumer theory. Such an index is called a true quantity index. In order to apply index number theory in a two-sector optimal growth model, this paper argues that intertemporal preferences can be represented by an indirect utility function of current consumption and investment. Then, following Fisher and Shell (1971), a true index of real output growth can be defined.

Growth theorists analyzing multisector economies in a dynamic general equilibrium framework avoid dealing with this issue. In particular, in evaluating the quantitative properties of their model GHK argue that for growth accounting real output can be identified with the production in the non-durable sector. By doing so, their measure of real output growth differs from the one published by the BEA. This position has strong theoretical and quantitative implications. In particular, it affects the estimation of the contribution of embodied technical change to the growth rate of per-capita output, which is the main objective of their paper.

In this paper, a true quantity index is applied to the measurement of real output growth in a simplified version of the two-sector growth model proposed by GHK. Parameters are calibrated to the U.S. economy as close as possible to GHK' calibration, and the proposed true quantity index of output growth is computed. The main findings are the following two. First, NIPA's methodology leads to a very good approximation to the proposed true index of real output growth. Second, once real output growth is appropriately measured, it is found that the contribution of embodied technical change to per capita GDP growth in the U.S. is of 69%, much larger than the 58% found in GHK.

The paper is organized as follows. Section 2 presents the facts. In order to clarify the debate, the relation between quality improvements, embodied technical change and vintage capital is discussed in Section 3. A simplified version of the GHK's model is presented and solved in Section 4. In Section 5, the aggregation problem is analyzed, and a true index of real output growth is proposed based on the economic theory of index numbers. The quantitative implications are discussed in Section 6, where the contribution of embodied technical change to U.S. per capita growth is estimated. Finally, conclusions and extensions are in Section 7.

2 New Evidence

Concerning the first of the three facts referred to in the Introduction, Figure 1 shows the evolution of the relative price of durable consumption and equipment investment in NIPA relative to the price of non durable consumption.

Diewert (1981) is a good survey on the economic theory on index numbers.

Figure 1: Equipment investment and durable consumption prices relative to non-durable consumption prices. Source: BEA.

The observed decline in the relative price of equipments is a clear evidence of a permanent improvement in the efficiency of the durable goods sector relative to the non durable sector. This increase in efficiency is partially due to quality improvements, i.e., new durables have better and better quality. For this reason, this phenomenon has been called embodied technical change. New technologies are incorporated in new equipments, and new investments are required in order to profit from the progress in technology. Moreover, from an empirical perspective, the observation of embodied technical progress is closely related to the introduction of quality adjustments in the measurement of prices and quantities in NIPA. In particular, the introduction of hedonic prices in NIPA's methodology for computers is at the basis of the observed decline in the relative price of equipments. From 1969 to 1999, the relative price of computers has declined at the cumulative rate of per year, which explains most of the annual decline in the relative price of equipments.

Figure 2 shows the evolution of both the real (continuous line) and the nominal (dotted line) equipments to GDP ratios. Both lines coincide in 1959, which has been taken as the base year. The important observation, fact 2 in the Introduction, is that the real ratio is diverging with respect to the nominal ratio. In 1999, the nominal ratio is around 10%, but the real ratio is around 22%, more than twice as large. Under embodied technical change, even if the equipment investment share on nominal GDP is stable, the ratio of real equipments to real GDP is increasing over time. When equipments benefit from embodied technical change, real equipment grows faster than GDP implying that the real equipments to GDP ratio increases. According to new NIPA measurements, the output-capital ratio is non stationary but permanently decreasing, which contradicts one of the Kaldor stylized facts.

In his seminal work, Gordon (1990) measures the quality improvements in durable goods by mean of hedonic regressions and estimates the growth rate of embodied technical change in around 3% per year.

Figure 2: The ratio of equipment and software to GDP, in nominal and real terms

Finally, embodied technical change has also important consequences for the growth rates of GDP and its components, as stated in fact 3. As Table 1 shows, equipment investment is growing faster than GDP, and GDP is growing faster than non durable consumption.

As Whelan (2000) pointed out, this ratio must be carefully interpreted, because chained quantity indexes employed actually in NIPA are non additive. In particular, the real investment ratio is no longer a share, and it can become larger than one if the relative price of equipments continues to decline.

Table 1: Annual growth rates (%) for 1969/1999

3 Quality Improvements, Embodied Technical Change and Vintage Capital

Quality improvements in the durable goods sector, embodied technical change, and vintage capital are very related concepts, but they have slightly different meanings. Quality improvements refer to an economic process in which new goods have higher and higher quality, in the sense that they have new or improved attributes or characteristics that allow them to perform better than before. Personal computers are a good example of quality improvements: in particular, new processors are permanently developed and they perform better and better. The importance of quality improvements in the growth process has been recently stressed by Aghion and Howitt (1992), among others.

Embodied technical change means that new investments are required in order to diffuse the changes in technology. In the case of PCs, the diffusion of the advantages of new processors requires the production of new PCs, since old PCs have old processors and cannot benefit from the advantages of recent innovations. Only if new investments are made, new technologies can have an impact on the performance of the economy. As referred to in the Introduction, GHK introduce embodied technical change in an optimal growth model.

Finally, vintage capital models assume that equipments of different vintages have different performance, implying that the capital age structure is relevant for the determination of the equilibrium. In this framework, the diffusion of new technologies is a process of creative destruction, where new and more efficient equipments permanently replace old ones. This process is very short in the computer sector, where PCs live for some few years. Benhabib and Rustichini (1991), and Boucekkine, Germain and Licandro (1997) are recent contributions to this area.

When technical progress in the durable sector adopts the form of quality improvements, new technologies are incorporated in new equipments, which makes the idea of embodied technical change more precise. However, embodied technical change and quality improvements are different concepts, since new technologies can also be more efficient by producing exactly the same durable good with a smaller amount of resources. Even in this case where machines of different vintages are identical, new investments are needed to profit from technological progress, implying that technical change is embodied. In GHK, technological change in the durable sector must be understood as embodied, in the sense that new investments are required to benefit from the progress in technology. However, this model does not require quality improvements, since the increased efficiency in the durable sector can also be interpreted as process innovation, with the same good produced more efficiently.

A final precision in terminology: Vintage capital and embodied technical change are also related but different concepts. In a vintage model, capital is an heterogeneous good, and the equilibrium depends on the age distribution of equipments. Moreover, vintages of capital are the natural way of modeling embodied technical change, in the sense that new machines are more efficient than old machines, because with the same amount of resources they produce more or better quality goods. Finally, embodied technical change does not require capital heterogeneity. GHK is a good example of embodied technical change without an explicit vintage structure of capital.

From an empirical point of view, it is very difficult to distinguish product from process innovation on aggregate data, and consequently it is difficult to quantify the importance of quality improvements. The BEA collects nominal data for different goods and uses price deflators to measure quantities. The price evolution of any item in NIPA depends on product and process innovation. Process innovation affects directly the observed price, but the BEA also adjusts at least partially for quality improvements. Even for computers, for which the BEA uses hedonic regressions to adjust for quality improvements, the contribution of quality improvements to the decline in the corresponding price deflator is not published. In this context, the permanent decline in the relative price of equipments must be understood as resulting from embodied technical change. It might be mainly due to quality improvements, but its relative importance with respect to process innovation cannot be quantified.

However, there are some interesting examples of vintage capital models without embodied technical change. For example, Benhabib and Rustichini (1991) analyzes the so-called “one hoss-shay” depreciation assumption, under which equally productive machines depreciate suddenly after some period. At any moment in time, machines of different vintages have different production horizons, and the equilibrium depends on their distribution across vintages.

Solow (1960) shows that Cobb-Douglas vintage technologies can be aggregated in a Cobb-Douglas aggregate technology, with a well defined capital aggregator.

Finally, NIPA provides very good information on investment (creation), but there is no information on capital stock destruction, which makes difficult to measure the quantitative importance of vintage capital.

4 Growth under Embodied Technical Change

The new evidence on the U.S. growth patterns during the last decades of the 20th century calls for a reformulation of the modern theory of growth. A first and important attempt in this direction is in GHK. Based on Solow (1960), these authors propose a two sector version of the optimal growth model with embodied technical change. As stated in Section 3, the GHK's model avoids all the difficulties of vintage capital models and does not refer explicitly to quality improvements in the durable goods sector. A simplified version of this model can be represented by the following planner's problem:

\[\max \sum_ {t = 0} ^ {\infty} \frac {c _ {t} ^ {1 - \sigma}}{1 - \sigma} \beta^ {t},\tag{1}\]

st.

\[z _ {t} k _ {t} ^ {\alpha} = c _ {t} + x _ {t}\tag{2}\]

\[i _ {t} = q _ {t} x _ {t}\tag{3}\]

\[k _ {t + 1} = (1 - \delta) k _ {t} + i _ {t}\tag{4}\]

\[q _ {t + 1} = \left(1 + \gamma_ {q}\right) q _ {t}\]

\[z _ {t + 1} = (1 + \gamma_ {z}) z _ {t},\]

given , , and . The endogenous variables , , and are in per capita terms. Equation (2) is the feasibility constraint in the non-durable sector: technology is Cobb-Douglas, and non-durable production is allocated to consumption and to the production of equipments, in quantity . is total factor productivity in the non-durable sector, and is the rate of disembodied technical change. From (3), technology in the investment goods sector is linear, with productivity . Technological progress also affects the investment goods sector, with being the rate of embodied technical change. Equation (4) is the law of motion for capital. Investment and capital are measured in durable units, and consumption is measured in non durable units. Concerning parameters, is the capital share, is the depreciation rate of capital, is the time preference parameter, and is the inverse of the intertemporal elasticity of substitution.

In addition, GHK distinguish two types of investment goods, structures and equipments, assume that preferences are also defined on leisure and introduce taxes. The version presented in this paper takes the minimum assumptions required to reproduce the evidence in Section 2.

The Euler equation associated to this problem is:

\[\left(\frac {c _ {t + 1}}{c _ {t}}\right) ^ {\sigma} = \frac {\beta}{1 + \gamma_ {q}} \left(1 - \delta + \alpha z _ {t + 1} q _ {t + 1} k _ {t + 1} ^ {\alpha - 1}\right).\tag{5}\]

Given the exogenous process of technological progress, the equilibrium of this economy is thus characterized by equations (2) to (5) and the initial condition . This model is consistent with fact 1 in the Introduction, since the relative price of equipments declines permanently at the rate . It is very important to notice that, if is calibrated as the decline rate of equipment prices in NIPA relative to non-durable consumption, then and should be measured in the same units as real investment and the capital stock in NIPA.

Along the BGP, non durable consumption grows at the rate

\[1 + g _ {c} = (1 + \gamma_ {z}) ^ {\frac {1}{1 - \alpha}} \left(1 + \gamma_ {q}\right) ^ {\frac {\alpha}{1 - \alpha}},\]

and the growth rate of investment is

\[1 + g _ {k} = (1 + \gamma_ {z}) ^ {\frac {1}{1 - \alpha}} (1 + \gamma_ {q}) ^ {\frac {1}{1 - \alpha}} > 1 + g _ {c}.\]

GHK's model predicts that the long-run growth rate of equipments is larger than the long-run growth rate of non durable consumption, consistently with fact 3 in the Introduction.

For simplicity, the non durable good is taken as numeraire. Then, and are nominal consumption and investment, respectively. Nominal output is just . However, real consumption and real investment are measured in different units. The measurement of real output is undertaken in the next section.

5 The Aggregation Problem

The problem of aggregating consumption and investment must be addressed in the GHK model. A simple measurement of real output growth consistent with NIPA's methodology is:

\[\tilde {g} \simeq (1 - s) g _ {c} + s g _ {k} = g _ {c} + s \gamma_ {q},\tag{6}\]

where is the nominal saving rate, which is constant along the BGP. In accordance with fact 2, .

However, GHK claim that the appropriate aggregation procedure for ‘growth accounting’ is to identify real output with the production in the non durable sector. Remember that the main objective in GHK is to measure both sources of technological progress, and in particular disembodied technical change in the non durable sector. Some precisions on the calibration procedure adopted in GHK could be very informative at this stage. They use the series of durable prices estimated by Gordon as an appropriate measure of . They take nominal measures of consumption and investment, and an index price of non durable consumption from the NIPA. They deflate nominal consumption and nominal investment by the non durable prices, and obtain measurements of and . Given the obtained measures of and , they use (3) and (4) to compute and . Disembodied technical change is then derived from (2), using information on and . In this sense, is the appropriate aggregation procedure for growth accounting, since it allows for a consistent estimation of the growth rate of disembodied technical change. However, as it is shown in the next, this is not an appropriate measurement of real output. Index number theory provides the appropriate framework for the measurement of the growth rate of real output in this economy.

5.1 Index Number Theory

The economic theory of index numbers was developed to provide theoretical foundations for the construction of price and quantity indexes. It assumes that individuals have well defined preferences on a set of goods, and that they optimally allocate a given amount of income to the consumption of these goods at given prices. The problem is purely static, in the sense that current income cannot be transferred to the future. All prices are nominal and change over time.

This result from the computation of a chained-type Laspeyres quantity index for real output. In NIPA, a chained-type Fisher index is computed. However, as the example in the next section shows, differences between these two indexes are not really significant in this context.

Suppose for simplicity that a representative agent has access to two different goods, which he consumes in quantities in period t. Preferences are represented by the utility function , increasing in both arguments and concave. At each period t, the solution of the agent's problem is

\[u \left(p _ {t}, Y _ {t}\right) = \max _ {\{x _ {t} \}} U \left(x _ {t}\right)\]

st.

\[p _ {t} x _ {t} = Y _ {t},\]

given nominal prices and nominal income . The optimal utility level depends on nominal income and prices. The dual associated to this problem is

\[\kappa (p _ {t}, \upsilon) = \min _ {\{x _ {t} \}} p _ {t} x _ {t}\]

st.

\[U \left(x _ {t}\right) = v,\]

where is the so-called cost function, and represents the minimum cost required to achieve a given level of utility at prices .

Suppose that prices and nominal income are observed for two adjacent periods, t-1 and t. In order to make comparisons in real terms, a reference price vector must be selected. For example, let us take current prices as reference prices and compute the minimum cost to obtain past utility at those prices, , where . It measures the income required at reference prices to achieve the same utility level that the agent has achieved at past prices and income. The ratio

\[\frac {Y _ {t}}{Y _ {t - 1} ^ {*}} = \frac {\kappa (p _ {t} , v _ {t})}{\kappa (p _ {t} , v _ {t - 1})}\]

gives a measure of real income change.

In many economic problems, as it is in our growth context, the relevant preference map is time dependent. Fisher and Shell (1971) have extended the definition of price and quantity indexes to a situation where preferences change over time. In this case, the utility function is time dependent, say , implying that the optimal utility level and the cost function are both time dependent. They postulate that comparisons must be done in terms of current preferences, so that the Fisher-Shell real income index is defined as

Pollak (1975) is an exception. He extends the standard theory of the cost of living index to a multiperiod setting. More recently, Reiter (1999) proposes true quantity indexes for real wealth and real savings in a similar framework.

\[\mathcal {F S} _ {t} = \frac {Y _ {t}}{Y _ {t - 1} ^ {\mathcal {F S}}} = \frac {\kappa_ {t} (p _ {t} , v _ {t})}{\kappa_ {t} (p _ {t} , \hat {v} _ {t - 1})},\]

where and . measures “How much income is required ‘today’ to make me indifferent between facing yesterday’s budget constraint and facing ... today’s prices and the income in question.” In this definition, taken from Fisher and Shell, page 19, ‘today’ means evaluated at current preferences.

5.2 Value Function and Indirect Utility

In GHK's growth model, the representative agent owns an initial stock of capital and an endowment of labor. At each time , he produces consumption and investment goods in order to consume today and accumulate capital for future production. His preferences in (1), however, are defined on his intertemporal consumption flow. In order to apply index number theory, intertemporal preferences must be represented by an indirect utility function defined on current consumption and investment. Consider the Bellman representation of the problem:

\[v (q _ {t}, z _ {t}, k _ {t}) = \max _ {\{c _ {t}, i _ {t} \}} \frac {c _ {t} ^ {1 - \sigma}}{1 - \sigma} + \beta v (q _ {t + 1}, z _ {t + 1}, k _ {t + 1})\]

st.

\[z _ {t} k _ {t} ^ {\alpha} = c _ {t} + \frac {i _ {t}}{q _ {t}}\tag{7}\]

\[k _ {t + 1} = (1 - \delta) k _ {t} + i _ {t},\tag{8}\]

where is the value function. The right hand side of the Bellman equation can be interpreted as the maximization of an indirect utility function on current consumption and investment:

Comparisons could also be done in terms of past preferences. However, Fisher and Shell found it ‘natural’ to use current preferences, since evaluations are made today, not yesterday.

\[U _ {t} \left(c _ {t}, i _ {t}\right) = \frac {c _ {t} ^ {1 - \sigma}}{1 - \sigma} + \beta v \left(q _ {t} \left(1 + \gamma_ {q}\right), z _ {t} \left(1 + \gamma_ {z}\right), (1 - \delta) k _ {t} + i _ {t}\right).\tag{9}\]

The indirect utility function is time dependent, since it depends on current states , , and . Along the BGP the value function takes the following form

\[v (q _ {t}, z _ {t}, k _ {t}) = v ^ {S} (q _ {t}, z _ {t}, k _ {t}) \equiv \Lambda \frac {\left(z _ {t} k _ {t} ^ {\alpha} - (\delta + g _ {k}) \frac {k _ {t}}{q _ {t}}\right) ^ {1 - \sigma}}{1 - \sigma},\tag{10}\]

where

\[\Lambda = \left(1 - \beta \left(1 + g _ {c}\right) ^ {1 - \sigma}\right) ^ {- 1}.\]

Index number theory can now be applied to compute the growth rate of real output.

5.3 The Index of Real Output Growth

The indirect indifference map is time dependent. In order to compute a true quantity index, according to Fisher and Shell, comparisons must be done in terms of today's preferences.

Let the economy be in its BGP at least from time t-1, and let the non durable good be the numeraire. Prices at time t are given by the vector , and represents nominal income in per capita terms. In order to compute a true quantity index, prices are taken as reference prices. Denote by an allocation at time t. A Fisher-Shell quantity index for output growth in the GHK economy is

\[\mathcal {F S} _ {t} = \frac {Y _ {t}}{Y _ {t - 1} ^ {\mathcal {F S}}} = \frac {\kappa_ {t} (p _ {t} , v _ {t})}{\kappa_ {t} (p _ {t} , \hat {v} _ {t - 1})}.\]

To compute , the utility function has been evaluated near to the BGP. For that, in equation (9), has been substituted for his expression in (10). Given this parametric function , the cost function , and the utility levels and have been computed.

It can be easily proved that , which implies that the FS index is larger than . Notice that production measured in non-durable units is growing at the rate . However, the reduction in investment goods prices due to embodied technical change makes individuals richer, in the sense that the production required yesterday to obtain the same level of utility at today prices is larger than actual production. For this reason, the index gives a larger measure of output growth than the growth rate of consumption. This result has important consequences for the measurement of growth. In particular, it shows that the GHK growth rate permanently underestimates the growth rate of real output, and consequently it underestimates the contribution of embodied technical change to the U.S. growth.

In GHK's model, only non durable consumption provides utility, which calls for a natural normalization: Measuring real output in non durable goods. In the BGP, this measurement of real output grows at the same rate than non durable consumption. However, the true index of real output growth shows that measuring real output in non durable units misses the point, because it does not take into account the increase in efficiency generated by the embodied nature of technical progress.

6 Calibration

In order to measure real growth, the model is calibrated as close as possible to GHK' calibration. The following parameter values are taken directly from GHK: , , , , and Along the BGP, the growth rate of consumption is and the growth rate of investment is . The growth rate of consumption is here slightly smaller than in GHK, because structures are excluded. Nominal output shares are and , where is nominal production (i.e., production in the non durable sector).

Using NIPA's methodology, the chained-type Fischer index gives an annual growth rate in real terms of along the BGP. The Fisher-Shell quantity index of output growth is per year, very close to the NIPA measurement. In addition, both the and the NIPA growth rates have been computed for different values of in [.01, 10]. The growth rate ranges in the [1.366, 1.558] interval, in percentage points, but it is in all cases very close to the corresponding NIPA measurement. It must be concluded that NIPA's methodology provides an appropriate measurement of real output growth.

In GHK, consumption preferences are represented by a logarithmic function.

The chained-type Laspeyres quantity index of output growth is . The chained-type Paasche quantity index is and the chained-type Fisher quantity index is a geometric mean of both: . Observe that all chained-type indexes are almost equal.

6.1 The Contribution of Embodied Technical Change

The main result in GHK is the estimation of the contribution of embodied technical change to per capita growth. By assuming that real GDP must be measured in non durable units, they actually measure the contribution of embodied technical change to the growth of per capita non durable output. However, as it has been argued in this paper, NIPA gives a more accurate measurement of the growth rate of per capita GDP, which calls for a new estimation of the contribution of embodied technical change.

First, the contribution of embodied technical change to the growth rate of per capita non durable output is estimated as 58%. This simplified version of the GHK's model gives the same contribution of embodied technical change as the one reported in GHK. Second, when NIPA's methodology is used to measure real output growth, the contribution of embodied technical change to the growth rate of per capita output is 69%. Equation (6) helps to understand this result. In NIPA's methodology, the positive difference between the growth rate of output and the growth rate of consumption depends directly on . The additional term is a pure contribution of embodied technical change. By measuring real output in non durable units, GHK do not take this contribution into account.

As GHK pointed out, for growth accounting the production in the non durable sector must be used to compute the rate of disembodied technical progress. However, NIPA's methodology leads to the appropriate measurement of real output. In this sense, this paper claims that GHK underestimate the contribution of embodied technical progress to the growth rate of per-capita GDP in the U.S..

7 Conclusions and Extensions

The new U.S. data from NIPA contradict some of the well-known Kaldor stylized facts, and call for a reformulation of the modern theory of economic growth. Among these new facts, three must be stressed: A permanent decline in the relative price of durable goods, a permanent increase in the real equipments to real GDP ratio, and a large long run growth rate of equipments relative to the growth rate of non durable consumption. In order to be consistent with these new facts, growth models must contain at least two sectors, so that the problem of defining aggregate output must be addressed. The definition of output growth proposed in this paper is in accordance with the economic theory of index numbers, and it follows closely Fisher and Shell (1971).

A simplified version of GHK model is calibrated on U.S. data and the Fisher and Shell index of output growth is computed. The first finding is that NIPA's methodology measures growth consistently with a Fisher-Shell true quantity index. This contradicts GHK claim that output must be measured in non durable units for growth accounting. Secondly, when the growth rate is measured as in NIPA the contribution of embodied technical change to per capita GDP growth in the U.S. is of around 69%, much larger than the 58% found by GHK.

The GHK model constitutes an important step for the reconciliation of the modern growth theory with the new evidence. However, these new facts call for a more general framework. First, new NIPA's data indicate that embodied technical change also affects durable consumption. Second, the observed increase in the real equipments to real GDP ratio reflects the rise in the efficiency of the durable goods sector. However, from the point of view of individual saving behavior, it is difficult to understand why the U.S. economy would converge to a situation where almost all real income were allocated to investment and very few resources were allocated to consumption. Therefore, in line with Whelan (2001) the following step is to extend the growth model with embodied technical change in order to include durable consumption and services. This would allow for a permanent substitution of durable services for non durable consumption. In the far future we will still eat potatoes but robots will coke it for us.

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12. [12] Whelan, K. (2001), “A two sector approach to modelling U.S. NIPA data,” Federal Reserve Board DP 2001-4.

COLECCION RESUMENES

98-01: “Negociación colectiva, rentabilidad bursátil y estructura de capital en España”, Alejandro Inurrieta.

TEXTOS EXPRESS

2001-01: “La reforma de las pensiones en el contexto internacional”, José A. Herce y Juan F. Jimeno.

2000-03: “Efectos sobre la inflación del redondeo en el paso a euros”, Mario Izquierdo y Simón Sosvilla-Rivero.

2000-02: “El tipo de cambio Euro/Dolar. Encuesta de FEDEA sobre la evolución del Euro”, Simón Sosvilla-Rivero y José A. Herce.

2000-01: “Recomendaciones para controlar el gasto sanitario. Otra perspectiva sobre los problemas de salud”, José A. Herce.

DOCUMENTOS DE TRABAJO

2001-09: “The Measurement of Growth under Embodied Technical Change”, Omar Licandro, Jorge Durán y Javier Ruiz-Castillo.

2001-08*: “Análisis económico de los comportamientos adictivos no saludables: Principales propuestas teóricas”, Fabiola Portillo y Fernando Antoñanzas.

2001-07: “Las migraciones interiores en España”, Samuel Bentolila.

2001-06: “Is the Deficit under Control?. A generational Accounting Perspective on Fiscal Policy and Labour Market Trends in Spain”, Gemma Abío, Eduard Bernguer, Holger Bonin, Joan Gil y Concepció Patxot.

2001-05: “Duración de los regímenes del SME”, Simón Sosvilla-Rivero y Reyes Maroto.

2001-04: “Assessing the Credibility of a Target Zone: Evidence from the EMS”, Francisco Ledesma-Rodríguez, Manuel Navarro-Ibáñez, Jorge Pérez-Rodríguez y Simón Sosvilla-Rivero.

2001-03: “La política macroeconómica en economías interdependientes”, Simón Sosvilla-Rivero.

2001-02: “Smoking in Spain: Analysis of Initiation and Cessation”, Namkee Ahn y José Alberto Molina.

2001-01: “La privatización de las pensiones en España”, José A. Herce.

2000-28: “Multinational Enterprises and New Trade Theory: Evidence for the Convergence Hypothesis”, Salvador Barrios, Holger Görg y Eric Strobl.

2000-27: “Obsolescence Vs modernization in a Schumpeterian vintage capital model”, Raouf Boucekkine, Fernando del Río y Omar Licandro.

2000-26: “Provisión de servicios públicos y localización industrial”, Luis Lanaspa, Fernando Pueyo y Fernando Sanz.

2000-25: “Labor Force Participation and Retirement of Spanish Older Men: Trends and Prospects”, Namkee Ahn y Pedro Mira.

2000-24: “Paridad del poder adquisitivo y provincias españolas, 1940-1992”, Irene Olloqui y Simón Sosvilla-Rivero.

2000-23: “Optimal Growth under Endogenous Depreciation, Capital Utilization and Maintenance Costs”, Omar Licandro, Luis A. Puch y J. Ramón Ruiz-Tamarit.

* Este Documento solamente está accesible en pdf en nuestra página web: http://fedea.es/hojas/publicaciones.html