F Loete

M Sorine

Electrical tagging devices for the removal of fault location ambiguities by reflectometry in complex electrical networks

Keywords: Time-domain reflectometry, complex network, distributed system, online wire diagnosis, wire fault detection ambiguity

In general, data from a single reflectometer are insufficient to locate defects in wires beyond a branching point. Knowing the distance to the defect, some additional information is required for selecting the faulty wire among possible candidates. This paper presents an innovative method of electrical tagging of wires to get uniqueness of fault location. It presents the tagging devices that may be used instead of additional reflectometers as in the expensive and complex distributed reflectometry techniques. Inexpensive electrical "tags" can be purposely designed with specific frequency signatures using only a few passive components. Preliminary experimental results show how their insertion on the branches of a network can give a proper identity to each one of them without affecting the useful signal frequency band. It is shown how each echo of the time domain impulse response of the line keeps a memory of the branches it has travelled along, thus cancelling its source location ambiguity

I. INTRODUCTION

Nowadays, since the electrical networks are getting bigger and more complex, their diagnosis has become a major concern. The most commonly used technique for diagnosing and locating faults is reflectometry. Time domain reflectometry consists in analyzing the impulse response of the electrical line under test from the considered injection point. A high frequency signal is sent down the electrical line, from a single injection point, where it is reflected at each impedance discontinuity. When diagnosing complex electrical networks, i.e. having at least one junction, the injected signal is forked into each branch. The echo pattern rapidly becomes extremely complex and difficult to interpret, especially since nothing can differentiate a signal that has travelled along a branch or another. Consequently, when a fault appears beyond a junction, and using a unique diagnosis system at a single injection point, it is impossible to identify on which branch the defect is placed [START_REF] Auzanneau | Wire troubleshooting and diagnosis: review and perspectives[END_REF]. Many time or frequency domain reflectometry methods have been developed [START_REF] Naik | Multicarrier reflectometry[END_REF]- [START_REF] Smith | Fault location on aircraft wiring using spread spectrum time domain reflectometry[END_REF] but they all suffer from this major ambiguity drawback. Very few works have been reported on methods aiming at solving this major problem. They are all based on the distributed reflectometry approach [START_REF] Lelong | Distributed reflectometry method for wire fault location using Selective Average[END_REF]- [START_REF] Ben Hassen | Diagnosis Sensor Fusion for Wire Fault Location in CAN Bus Systems[END_REF] which consists in duplicating the diagnosis systems at each but one extremities of the network. Although efficient, they are expensive and require complex signal conditioning and communication between the different modules.

In this work, we present an innovative, yet simple and very inexpensive method to locate, without ambiguity, the fault location in complex electrical harnesses from a single observation point. The method can also be considered in embedded applications since it doesn't perturb the useful signals conducted by the electrical line. For clarity, the principle will be illustrated on a simple Y star shaped network containing 3 lines of different length connected with a single junction (Fig. 1). However one has to note that the following demonstration can be easily adapted to any complex network topology.

II. ELECTRICAL TAGGING OF ELECTRICAL LINES METHODOLOGY

The impulse response of the entire healthy network, with lossless lines and frequency independent termination loads, can be simply expressed for the first reflections at the junction and each extremity, as:

. . . . +… (1)
Where and are respectively the length and the reflection coefficient on the loads at the end of the branches i.

One can notice in (1) that since the termination loads are frequency independent, the shape of the impulses responses associated to the reflections on the impedance mismatches contained on each branches are all Dirac pulses.

Consequently, as illustrated in Fig. 1, a defect created at the same distance after the junction on either one of the two branches would results on the same modification of the impulse response of the Y shaped harness. One can therefore not differentiate a pulse coming from branch 1 or 2 and state on the location of the defect with certainty.

The principle of the proposed method [START_REF] Loete | Electrical tagging method of complex electrical networks[END_REF] is presented in Fig. 2. It consists in functionalizing each branch with an electrical "Tag". These electrical devices can be inserted either at the junction, in the middle or at the end of a branch. In the frequency band used by the interconnected devices, each "Tag" is designed to have the same characteristic impedance as the line it is placed in, thus not affecting the useful conducted signals. Outside of this useful band, a specific frequency behavior of the impedance of the tag is purposely designed. Fig. 2. Each extremity branch of a complex electrical network Σ is equipped with an electrical "tag" designed with a specific frequency pattern. As an example, the specific frequency signatures of the two TAGs were designed as low pass filters with two different cutoff frequencies outside the useful band.

Consequently each tag has a specific impulse response that will affect the echoes coming back from each branch. Each echo is tagged with the identity of the branch it is coming from thus removing the location ambiguity of the defect.

III. EXPERIMENTAL TAGGING OF A STAR SHAPED NETWORK

A. Electrical tags characteristics

In the example presented in this work, two tags were used in order to give a unique signature to each extremity of the Y star shape network of Fig. 1. The network was made of 50 Ω RG58 coaxial cables. The lengths , , and were respectively 0.6 m, 1.6 m and 5 m. The two frequency dependent tags were designed with simple Pi Butterworth low pass filters of order 3 made of only two capacitances and one inductance (Fig. 3). The cutoff frequencies of the tags placed on the branches 1 and 2 were respectively 40MHz and 65MHz. Even if the Fig. 1 presented in section II clearly shows that the tags can be placed anywhere on the branches, since the tag are low pass filters and a limited frequency band can go through the filter, one could not expect to detect a defect with a good accuracy after the tag. Consequently the tags were placed at the extremities of the branches.

The experimentally measured scattering parameter of Tag1 and Tag2 are shown in Fig. 4. One can see that with such cutoff frequencies, a tag would not affect the transmission, or cause unwanted reflection in the network, on a signal composed of harmonics below the lowest cutoff frequency. As an example, the CAN protocol, which is widely used in automotive or aeronautical application, works at 500 kHz with buses made of 120 Ω twisted pairs. With minor modifications on the component values of the filters, the tags could be adapted still ensuring the CAN messages integrity.

B. Experimental results and identification of each branch of the network

The impulses responses and of the two tags presented in Fig. 4 are clearly different and can be distinguished without ambiguity. Once inserted on the Y star shaped network they dramatically affect the impulse response of the whole network and as expected each echo has a specific shape contrary to an untagged network as shown in Fig. 5.

Consequently the echoes coming back to the injection point have a unique shape that is the memory of their multiple travels through the branches. It follows that the location ambiguity of the defect can be removed easily. This can be conducted by analyzing the tag signatures present or absent in the impulse response of the line. If this healthy tagged network is affected by a defect such as an open circuit on a branch, as shown in Fig. 6, the signature of the tag associated to the defective branch disappears. By simply identifying the tags still present in the response, one can easily state on which branch the defect is located.

IV. CONCLUSION

In this paper we have proposed a new method allowing to efficiently solve the extremely important ambiguity location problem of the defects in complex electrical harness by reflectometry. The proposed electrical tagging method is powerful but yet very inexpensive and may give for embedded automotive or aeronautical systems a viable alternative to the costly existing methods. The analysis of the simple low pass filter tags present in the line allow to rapidly state without ambiguity on the location of a defect on a complex electrical network.

Fig. 1 .

 1 Fig. 1. Location ambiguity of a fault on a complex electrical network topology. Above: Time Domain reflectometer of a Y star-shaped network either adapted at each extremity (─) or affected by an open circuit on a branch (─ ─). Below: The only knowledge of the distance of the fault to the injection point doesn't allow to state on which branch the fault is located.

Fig. 3 .

 3 Fig. 3. Equivalent circuit of the electrical Tags inserted at extremities of the Y shaped network. The two tags are Pi Butterworth low pass filters with respective cutoff frequenies of 40 MHz and 65MHz.

Fig. 4 .

 4 Fig. 4. Impulse response, reflection and transmission scattering parameters of the Butterworth pi filters 1 (─) and 2 (─ ─).

Fig. 5 .

 5 Fig. 5. Comparison of the time domain response of an untagged network (─) and a tagged one (─ ─). Each echo has a shape which is the memory of its travel along the branches.

Fig. 6 .

 6 Fig. 6. Tagged Y star shaped network. Impulse response of the healthy tagged network (─) or the defective one affected by an open circuit on branch 1 at 1m after the junction. The impulse response of Tag1 has disappeared from the impulse response of the line.


Jorge Durán

Omar Licandro
email: omar.licandro@nottingham.ac.uk.

Timo Boppart

Fabrice Collard

Jeremy Greenwood

Jonathan Heathcote

Berthold Herrendorf

Boyan Jovanovic

Dirk Niepelt

Aurélien Poissonnier

Javier Ruiz-Castillo

Is the output growth rate in NIPA a welfare measure? *

Keywords: Growth measurement, Quantity indexes, Equivalent variation, NIPA, Fisher-Shell index, Divisia index, Embodied technical change. JEL classification numbers: C43, D91, O41, O47

Bridging modern macroeconomics and the economic theory of index numbers, this paper shows that real output growth as measured by National Income and Product Accounts (NIPA) is a welfare based measure. In a two-sector dynamic general equilibrium model of heterogeneous households, recursive preferences and quasiconcave technology, individual welfare depends on present and future consumption.

In this context, the Bellman equation provides a representation of preferences over current consumption and investment. Applying standard index number theory to this representation of preferences, it is shown that the Fisher-Shell true quantity index is equal to the Divisia index in turn well approximated by the Fisher ideal chain index used in NIPA.

Introduction

The present paper bridges modern macroeconomics and the economic theory of index numbers to show that the class of chain indexes used by National Income Product Accounts (NIPA) properly reflects changes in welfare when applied to a two-sector dynamic general equilibrium economy with heterogenous households, recursive preferences and quasi-concave technology. 1 In doing so, it evaluates the suitability of NIPA methodology for measuring real changes in GDP in a general model economy with explicit preferences and technology. In this framework, preferences are defined over consumption streams, present and future, but NIPA is constrained to use observable information at a given point in time by aggregating the main components of current final demand, encompassing consumption and investment. The Bellman equation, indeed, provides a representation of preferences, such that all information about changes in welfare is contained in current (observable) changes in consumption and investment. In particular, we show that the Fisher-Shell true quantity index when applied to this representation of preferences is equal to the Divisia index, which is known to be well approximated by the Fisher ideal chain index used by NIPA. 2 This means that the real growth rate of output in National Accounts is a welfare based measure in the very precise sense of equivalent variation -see [START_REF] Mckenzie | Measuring Economic Welfare: New Methods[END_REF].

Until the 90's the Bureau of Economic Analysis (BEA) featured in its NIPA a Laspeyres fixed-base quantity index to measure real GDP growth. Traditional fixedbase quantity indexes yield a reasonable good measurement of real growth provided that relative prices remain stable. However, the situation radically changed in the mid-80s when following the seminal contribution of [START_REF] Gordon | The Measurement of Durable Goods Prices[END_REF] the BEA started deflating equipment investment by a constant quality price index, which from then it is observed to permanently decline relative to the price of non-durable consumption goods and services. It was then realized that, due to the permanent decline of the relative price of 1 For economic index number theory, see [START_REF] Diewert | Index numbers[END_REF], [START_REF] Triplett | Economic theory and BEA alternative quantity and price indexes[END_REF], [START_REF] Fisher | Economic Analysis of Production Price Indexes[END_REF] and IMF (2004, chapter 17), among many others. A renewal of interest in using money metric utility for price measurement is in [START_REF] Redding | Measuring aggregate price indexes with demand shocks: Theory and evidence for CES Preferences[END_REF].

equipment, investment tends to be overweighted in the fixed-base Laspeyres index due to the so-called substitution bias. As a consequence, GDP growth was overestimated when departing from the base year. All this lead the BEA to consider alternative measures of real GDP growth, switching in the early 90s to a Fisher ideal chain index which is known to well approximate a Divisia index. 3 This paper provides a rational for this methodological change based on the idea that real output growth as measured by index numbers reflects the underlying preferences of households in a well defined technological environment.

The recognition that the price of durables were declining relative to the price of non-durables had also consequences for macroeconomics. Growth theory has been reformulated in the late 90's in order to replicate this fact. Based on the hypothesis first formulated by [START_REF] Solow | Investment and technical progress[END_REF] that technical progress is embodied in capital goods, Greenwood et al (1997) proposed a simple two-sector optimal growth model with investment specific technical change where productivity grows faster in the investment than in the consumption sector. 4 Unlike the one-sector growth model, the aggregation issue needs to be raised in a two-sector economy. The general methodology suggested in this paper is then used to measure output growth in this family of models, by studing the two-sector AK model proposed by [START_REF] Rebelo | Long-run policy analysis and long-run growth[END_REF], which replicates the empirical regularities referred above in the simplest possible world -see [START_REF] Felbermayr | The under-estimated virtues of the twosector AK model[END_REF]. By doing so, it helps to understand the main issues behind the change in methodology referred in the previous paragraph. This theoretical framework also sheds light on an old debate in the growth and growth accounting literature. The so-called Solow-Jorgenson controversy was revived by the differing interpretations found in [START_REF] Hulten | Growth accounting when technical change is embodied in capital[END_REF] and Greenwood et al (1997). The controversy can be shown to boil down to the issue of the aggregation of consumption and investment when these are measured in different units and, more importantly, when its relative price has a trend. In our conceptual framework, it becomes clear that Greenwood et al (1997) take a path that is more consistent with the theory. However, implicitly, these authors-and others following like [START_REF] Oulton | Investment-specific technological change and growth accounting[END_REF]-also adhere to a modern version of the paradigm that consumption, and consequently its growth rate, is the relevant measure of real growth. 5 In this paper, we claim that investment growth, as reflected by the Divisia index, also matters for output growth. Notice that NIPA methodology stresses the fact that the growth rate of investment does contain information relevant to households welfare since it reflects utility gains associated with postponed consumption. This is particularly relevant in a world where technical change is embodied in durable goods, and hence where technical progress only materialize through the incorporation of new physical capital.

The issue of trends in relative prices and different sectorial growth rates is also critical for the recent literature on structural transformation, since agriculture, manufacturing and services grow at different rates during the development process. 6 In line with our findings, Duernecker et al (2017) claim that using chain indexes more accurately reflects the effects of secular changes in relative prices.

It is important to point out that the main result on this paper, that output growth in NIPA is a welfare based measure, does not require a representative household. The proof that a Fisher-Shell index is equal to the Divisia index holds true even when agents have different preferences and income, and consequently equilibrium may differ from the equilibrium of the corresponding representative agent economy. When a Fisher-Shell index is applied to a dynamic general equilibrium economy with heterogenous households, money is used as a common norm to evaluate welfare changes across individuals; money metric utility implicitly adopts an utilitarian approach weighting each households proportional to its own income. The approach leads to apply the Fisher-Shell index to individual households separately, and then aggregate their individual growth rates weighted by their shares on total income. Starting from the usual assumption that income levels may 5 Greenwood et al (1997), in fact, is not a normative paper. It does perform the positive exercise of measuring the contribution of embodied technical change to US growth. However, in doing so, they measure output and its growth rate in units of consumption, de facto identifying real output growth with consumption growth. [START_REF] Cummins | Investment-specific technical change in the United States (1947-2000): Measurement and macroeconomic consequences[END_REF] generalize the exercise and use standard NIPA methodology to the same objective, finding similar quantitative results. See also [START_REF] Greenwood | Accounting for growth[END_REF]. Sections 3 and 4 further discus these issues.

be used to make inter-households comparisons, inter-households differences in income growth aggregate to the average growth rate of the economy. Such an approach reduces to the analysis of the income side of NIPA and, if data were available, it may be used to study the social welfare implications of observed phenomena like job polarization, where different occupations face different growth rates in earnings depending on their position in the income distribution. 7 But, what do we mean when we say that output growth in national accounts is a welfare based measure? In order to answer this question, it is important to understand the implications of applying money metric utility in this context. Notice that when using national accounts to measure welfare gains, we face two fundamental problems.

We need first to understand the relation between current production and welfare, to then understand how by subtracting inflation from nominal output growth, real output growth measures gains in welfare.

How does current production relates to households' welfare? The answer somehow relates to the seminal contribution of [START_REF] Weitzman | On the welfare significance of national product in a dynamic economy[END_REF]. Let us decompose our main argument. First, since national accounts aggregate current consumption and current investment when measuring output growth, in order to apply index number theory to a two-sector dynamic general equilibrium economy, we use the Bellman equation to represent households preferences in the consumption-investment dimension. Second, in the Bellman equation, the value function represents the equilibrium value of assets, evaluated from the point of view of households preferences, i.e., the equilibrium discounted flow of consumption utility. The value function is then a measure of households welfare defined in the domain of households assets. Third, the Bellman equation states that the return to assets, evaluated using the subjective discount rate, has to be equal to the utility of current consumption plus the value of current net investment. In this sense, when we apply index number theory to the representation of preferences emerging from the Bellman equation, we are not directly measuring welfare gains, but gains in the return to assets, using indeed households preferences to do so. Fourth, at any time, the economy uses households assets to produce output, which actually is the return to these assets.

Consequently, in this context, output is nothing else than the sum of consumption utility plus the value of net investment as stated by the Bellman equation. Finally, let us stress that the return to assets, i.e. current production, is allocated not only to current con- 7 For job polarization, see [START_REF] Autor | The growth of low-skill service jobs and the polarization of the US labor market[END_REF] sumption, which generates current well-being, but also to current net investment, which will contribute through the creation of new assets to generate well-being in the future.

For this reason, as pointed out by [START_REF] Weitzman | On the welfare significance of national product in a dynamic economy[END_REF], from a welfare point of view when measuring current output we should add both consumption and net investment.

How inflation has to be subtracted from nominal income growth in order to real output growth measure gains in welfare? Notice, first, that a utility function is a particular representation of households preferences: monotonic transformations of it change the level of utility leaving the preference map intact; so the growth rate of a particular representations is meaningless. To overcome this problem, the traditional approach to welfare measurement is to create a money metric representation of the underlying preferences by the mean of the so-called equivalent variation. Suppose an agent with a given income optimally buy a particular consumption basket at given prices, but then she receives in exchange a new preferred basket. The equivalent variation to this problem is the hypothetical increase in income (at the given prices) that helps the consumer reach the same level of utility as with the new consumption basket -see chapters 1 and 2 in [START_REF] Mckenzie | Measuring Economic Welfare: New Methods[END_REF]. Since the same exercise may be repeated for all available consumption baskets, the equivalent variation can be seen as the change in a money metric utility represention of the underlying preferences because a basket is at least as preferred to another if the equivalent variation is non-negative. Second, in a dynamic economy like the one described in this paper, at any time we observe a current consumption-investment choice and current prices. The next period, since household income, the shadow value of capital and equilibrium prices are different, we observe a new combination of consumption and investment. To compare these two situations the Fisher-Shell true quantity index measures changes in welfare, as represented by the Bellman equation, by the mean of an equivalent variation using the initial prices and shadow value of capital as the benchmark. This paper shows that the equivalent variation in percentage of current income turns out to be the Divisia index. Hence, the chained index used in NIPA can be seen as a measurement of welfare change. In this sense, this paper extends Witzman (1976) result to the measurement of output growth in NIPA. This paper is organized as follows. Section 2 first describes a two-sector representative household general equilibrium economy with general recursive preferences and quasiconcave technology. It applies index number theory to the Bellman representation of these preferences and proves that the Fisher-Shell true quantity index is equal to the Divisia index. Then, it extends the result to an economy with household heterogeneity in preferences and income. Section 3 illustrates it in the interesting case of the twosector AK model, which replicates the observed permanent decline in the relative price of investment. In this framework, it discusses the issue of money metric utility and compare the Divisia index with a consumption equivalent measure of output growth.

Finally, Section 4 discusses the main implications of our results and Section 5 concludes and suggests some possible extensions.

Measuring output growth

Consider a two-sector non-stochastic perfectly competitive dynamic general equilibrium economy with two goods, consumption and investment, and a general quasi-concave technology transforming capital and labor into these two goods. Firms hire capital and labor to produce them, and under the usual intertemporal budget constraint, a representative household chooses continuously consumption and savings in order to maximize intertemporal utility. All along this paper, we assume that preferences and technology are such that an equilibrium path exists and is unique.

In this framework, a National Statistical Office (NSO) would like to measure output growth consistently with individual preferences and technology. Indeed, at equilibrium the NSO only observes current nominal consumption and investment, and the corresponding prices, but has no information about individual preferences, technology and future consumption. Let us finally assume that the NSO uses this information to measure the growth rate of both consumption and investment and, then, uses index number theory to compute the growth rate of output. The general problem in national accounts is to find an index built out of observables at t, current consumption and investment, and the associated prices, that measures changes in real output. For the fictitious economy, the NSO aggregates equilibrium consumption and investment by the mean of a Fisher-Shell true quantity index -controlling for changes in equilibrium prices. Section 2.2 shows how to construct a Fisher-Shell index in this context and proves that the Fisher-Shell index is equal to the Divisia index, which in continuous time is equal to the Fisher ideal chain index -the one used in NIPA to measure GDP growth. 8 The resulting rate of output growth is then welfare based. Section 2.3 generalizes the result to heterogeneous households.

Bellman equation under recursive preferences

The economy evolves in continuous time. For any date t ≥ 0 and any consumption path

C : [0, ∞) → R + , t C
d dt U ( t C) = -f (c t , U ( t C)). (1)
The generating function f is assumed to be differentiable, with f 1 > 0 and f 2 < 0.

Note that f 1 is the marginal utility from current consumption, lost when we move an infinitesimal period of time ahead, and so the negative sign in [START_REF] Acemoglu | Introduction to Modern Economic Growth[END_REF]. In turn, f 2 < 0 is related to the implicit subjective discount rate. 9 For instance, the classical additively separable utility function is an important particular case of the general specification above in which

U ( t C) = ∞ t
e -ρ(s-t) u(c s )ds with u (c) > 0, u (c) < 0 and ρ > 0. Differentiate with respect to time t to write

d dt U ( t C) = -u(c t ) + ρU ( t C).
Hence, in this case, f (c, U ) = u(c) -ρU and indeed f 1 (c, U ) = u (c) > 0 while f 2 (c, U ) = -ρ < 0. Indeed, this clarify the interpretation given above that f 1 is the marginal utility from current consumption, lost when we move an infinitesimal period of time ahead, and f 2 is the return to household assets, which value is represented by U ( t C) and the discount rate is ρ.

Each
f (c, v(k t , Θ t )) + v 1 (k t , Θ t )x + v 2 (k t , Θ t ) Θt . (2)
The intuition behind this equation becomes clear if one notes that along an optimal path

v(k t , Θ t ) = U ( t C) so dv(k t , Θ t )/dt = v 1 (k t , Θ t ) kt +v 2 (k t , Θ t ) Θt = -f c t , v(k t , Θ t ). Note
as well that, in a sense, with all past actions summarized in k t , the objective function in

(2) is giving us the preference relation over consumption and investment at instant t. 10

Fisher-Shell true quantity index

In this section, we show that in the dynamic general equilibrium framework developed in the previous section, the Divisia index is a true quantity index. In regard of the Bellman equation ( 2), preferences of the representative consumer over consumption and

w t (c, x) . = f (c, v(k t , Θ t )) + v 1 (k t , Θ t )x + v 2 (k t , Θ t ) Θt .
To save notation, we write w t (c, x), but time enters this function only through the stock of capital k t and the exogenous states Θ t , both given at time t.

For a given state of the system, as represented by k t and Θ t , the function w t (c, x) can then be seen as a representation of individual preferences over consumption and net investment, the last summarizing postponed consumption. To the extent that the exogenous states and the stock of capital will change along an equilibrium path, these preferences are time-dependent. This is precisely the building block of the true quantity index introduced by Fisher and Shell (1968). Since welfare comparisons must be done within the same preference map, the Fisher-Shell true quantity index proposes to fix not only prices but also preferences. In particular, it compares income today with the hypothetical level of income that would be necessary to attain the level of utility associated with tomorrow's income and prices with today's prices and today's preferences -as evaluated by w t (c, x). The remain of this section elaborates this idea.

In the following, the NSO adopts the consumption good as numeraire. Of course, the choice of the numeraire is inconsequential. Let us define equilibrium nominal net income = c t + p t x t . Under standard assumptions, optimal choices will lie in the boundary of Γ(k t , Θ t ) so that there is a welldefined equilibrium price of net investment p t > 0 expressed in units of consumption (see Figure 1). The constraint (c, x) ∈ Γ(k t , Θ t ) can be replaced by the linear constraint c+p t x ≤ m t in the problem of the Bellman equation (see Figure 1). Hence, the associated indirect utility function of the representative household problem is defined as

u t (m t , p t ) . = max c+ptx≤mt w t (c, x)
while the expenditure function is

e t (u t , p t ) . = min wt(c,x)≥ut c + p t x.
The fundamental idea behind money metric utility is to use the expenditure function to make welfare comparisons, by associating utility u t to observed income m t = c t + p t x t .

Notice that at the equilibrium of the fictitious economy, the NSO observes {c t , x t , p t } at both the current period t and the future period t + h, h > 0. 11 Since comparisons must be done within the same preference map, the Fisher-Shell true quantity index fixes both prices and preferences. In particular, it compares income today m t with the hypothetical 11 Of course, the NSO produces these measures after period t + h.

level of income mt+h that would be necessary to attain the level of utility u t (m t+h , p t+h ) associated with tomorrow's income and prices m t+h , p t+h with today's prices p t and today's preferences as represented by functions e t (u, p) and u t (m, p). This artificial level of income is given by mt+h = e t u t (m t+h , p t+h ), p t .

The idea is illustrated in Figure 2 in a situation where nominal income increases and the price of investment declines. The preference map corresponds to instant t preferences as represented by w t . Point A is the observed situation at instant t. Point B is the hypothetical choice using instant t preferences when facing observed prices p t+h and income m t+h . Point C represents the choice that maintains such level of utility but with prices p t . In the end, it does compare two levels of income that correspond to the same price vector so it is clear that the index is extracting price changes. In this particular case, the true quantity index is just reflecting the fact that the true output deflator is dropping with the price of investment, that is to say that income in real terms is growing more than nominal income m t+h /m t . 12 Note that the difference mt+h -m t is the equivalent variation which can be seen as the composition of nominal growth m t+h -m t and the compensating variation mt+h -m t+h . The equivalent variation measures by how much income would have to increase beyond its nominal increase to compensate for not having the price of investment dropping.

In continuous time, the reasoning is the same but the time gap h tends to zero. The instantaneous Fisher-Shell index is defined as 13

g FS t . = d dh mt+h m t h=0 = 1 m t d mt+h dh h=0 ,
that is, the instantaneous growth rate of the factor mt+h mt when h gets arbitrarily small. To compute this index note that

d mt+h dh h=0 = e 1,t u t (m t , p t ), p t u 1,t (m t , p t ) ṁt + u 2,t (m t , p t ) ṗt
12 If alternatively, the investment good were the numeraire, nominal income would be growing faster than the hypothetical income m and consumption price changes should be subtracted from nominal income growth to get real income growth. Indeed, the real growth rate will remain unchanged, since it does not depend on the choice of the numeraire. 13 See Appendix A1 for a rationale of this definition.

where subscripts denote the partial derivatives with respect to the corresponding arguments. To obtain an expression for all these derivatives let us go back to the dual and primal problems discussed above. Let µ be the Lagrange multiplier of the maximization problem in the definition of the indirect utility function, measuring the marginal contribution of income m to welfare w. We have, from the the primal problem

u 1,t (m t , p t ) = µ u 2,t (m t , p t ) = -µx t ,
and, since the expenditure function is the inverse of the indirect utility function,

e 1,t (u t , p t ) = 1 µ .
As expected, the marginal contribution of income to welfare, ∂u/∂m = µ, is equal to the inverse of the marginal contribution of utility u to total expenditure, ∂e/∂u = 1/µ.

Moreover, the negative marginal contribution of prices to welfare is ∂u/∂p = -µx, since an increase in prices reduces income by x units. These properties are critical for the result below and they are directly related to the money metric utility nature of the Fisher-Shell index, which defines the hypothetical income m using the expenditure function to valuate changes in utility after controlling for changes in prices.

Using the three conditions above in the definition of the Fisher-Shell index, we conclude that

g FS t = ṁt -x t ṗt m t = ṁt m t - p t x t m t ṗt p t .
Notice that the marginal terms e

g FS t = (1 -s t ) ċt c t + s t ẋt x t . = g D t
where g D t denotes the Divisia index. We have then shown that, for all t, the Fisher-Shell index g FS t is equal to the Divisia index g D t . In this framework, by definition, the Divisia index is the average of the growth rates of consumption and net investment, weighted by their corresponding equilibrium shares in total net income.

We have then shown that in this framework the Divisia index is a true quantity index, and as such it is a welfare measure. The interpretation is straightforward. It is clear that g FS t is a measure of real growth since it is constructed as the growth rate of nominal income subtracting pure price changes, in this case the change of the relative price of investment p t . The index only keeps changes in quantities. It is also clear that it is a true index because it is constructed from the representative household's preferences using standard theory. 14 The beauty of the result is that the NSO does not need to know people preferences and production technology, but current quantities and prices.

On household heterogeneity

The argument above was built under the assumption of a representative household. In this section, we show that the same reasoning applies to a heterogeneous agents economy where households have both heterogeneous preferences and heterogeneous income. Critical in the result is the fact that the utility representation of preferences derived from the Bellman equation is quasilinear, belonging to the Gorman family. 15First, let us assume that there is a continuum of heterogeneous households of unit mass with recursive preferences represented by the utility U i generated by the differential

equation 1 dt U i ( t C i ) = -f i (c i,t , U i ( t C i )),
where t C i represents the consumption path of household i and function f i has the same 14 This equivalence would come as no surprise to index number theorists. The Fisher ideal chain index is known to approximate in general some sort of true quantity index because both are bounded from above and below by the Laspeyres and Paasche indexes respectively. In continuous time, these indexes tend to each other as the time interval h tends to zero. Further, in general, the Divisia index coincides with the Fisher ideal chain index if the growth rates of consumption and investment are constant.

properties as above. Second, at equilibrium capital is distributed across households according to ϕ t , which maps any individual i at any instant t into a quantity of capital.

Let us finally assume that, for this economy, an equilibrium exists and is unique. Notice that equilibrium may likely be different from the corresponding equilibrium with a representative household, implying that the distribution of preferences and capital across individuals matters.

In the recursive competitive equilibrium representation of this economy, with exogenous state Θ t and an equilibrium distribution of capital ϕ t , the problem of a household i with capital k i,t can be written as

0 = max f i (c i , v i (k i,t , Θ t , ϕ t )) + v i,1 (k i,t , Θ t , ϕ t )x i + π i,t s.t. c i + p t x i = m i,t
where c i and x i are household's current consumption and net investment, respectively, p t is the equilibrium price, common to all households, and m i,t is the equilibrium net income of individual i. π i,t represents the differential terms of v i (k i,t , Θ t , ϕ t ) with respect to time that are exogenous to the problem of the consumer, i.e., those corresponding to Θ t and ϕ t .

As in section 2.2, the optimization problem of household i is associated with the instantaneous utility function over consumption and net investment

w i,t (c i , x i ) . = f i,t (c i ) + x i ,
where Notice that, in general, the average hypothetical income mt+h at the equilibrium of the heterogeneous household economy, may be different from the hypothetical income mt+h at equilibrium of the representative household economy, since these two economies may likely have different equilibrium paths.

f i,t (c i ) . = f i (c i , v i (k i,t , Θ t , ϕ t ))/v i,
As in section 2.2, let us define the Fisher-Shell index for the economy with heterogeneous households as

gFS t . = 1 m t d mt+h dh h=0 .
Operating on the definition of mit+h above

d mt+h dh h=0 = ṁt + p t Ā t (p t ) -m t /p t ṗt . Notice that Ā t (p t ) = i A i,t (p t )di = i f i,t c i,t -1/p t c i,t =0, since f it =1/pt + c i,t /p 2 t di = c t /p 2 t ,
where Of course, at equilibrium, consumption and investment may be growing at different rates than in the corresponding representative household model, and the saving rate may also be different. Consequently, even when the growth rate, as measured by the Divisia index is a welfare measure in both economies, these two economies may be growing at different rates.

c t = i c it di
Two assumptions are critical for the main result in this section, i.e., that the Fisher-Shell index is equal to the Divisia index under heterogeneous households. First, as in the case of homogeneous households, under money metric utility, nominal income is the metric used to measure households' utility, implying that gains in welfare are measured as gains in nominal income minus inflation; the main principle used by national accounts.

The second critical assumption is the use of the quasilinear representation of preferences that emerges from the Bellman equation representation of household preferences in the space of current consumption and current net investment. This assumption is not critical at all in the case of a representative household; in facts, in Section 2.2, we show that the Fisher-Shell index is equal to the Divisia index for a general function w(c, x). Indeed, it is critical in this section, since we profit from the quasi linearity representation of preferences to show that aggregate utility gains, as measured by the Fisher-Shell index, are equal to gains in nominal per capita income minus inflation.

This result comes at non surprise. By adopting aggregate nominal income as a norm for measuring aggregate output, the Fisher-Shell index implicitly assumes that the aggregate welfare function is utilitarian, giving to each household a weight proportional to its income. This clearly reflects in the definition of the artificial income measure mt . is the simplest endogenous growth model that replicates the observed permanent decline in the relative price of equipment and the permanent increase in the investment to output ratio. 16 We have preferred to use it instead of the original Greenwood et al (1997) model, since the AK model has the advantage of jumping to the balanced growth path from the initial time, which allows for an explicit solution of the value function. This is very useful to understand the role of money metric utility in the main statement of this paper that the growth rate in NIPA is a welfare measure.

Embodied technical progress

The two-sector AK model

The model in this section is based on [START_REF] Rebelo | Long-run policy analysis and long-run growth[END_REF]

c t = h α t ,
where α ∈ (0, 1). The remaining stock k t -h t ≥ 0 is employed in the production of new capital goods with a linear technology

kt = A(k t -h t )
where A > 0 is the marginal product of capital in the investment sector net of depreciation. There is a given initial stock of capital k 0 > 0. Again, we will write x t = kt for net investment.

The representative household has preferences over consumption paths represented by17

∞ 0 c 1-σ t 1 -σ e -ρt dt, (3)
that is, the additive case mentioned above, where ρ > 0 is the subjective discount rate and σ ≥ 0 the inverse of the intertemporal elasticity of substitution.

In the absence of market failures, equilibrium allocations are solutions to the problem of a planner aiming at maximizing household's utility subject to the technological constraints. The Bellman equation associated with the planner's problem is

ρv(k t ) = max x=A(kt-c 1/α ) c 1-σ 1 -σ + v (k t )x. (4)
The value function v(k t ), which is (3) evaluated at equilibrium, represents the value of capital measured as the discounted flow of consumption utility. The return to assets, as measured by the subjective discount rate ρ, is equal to the utility of current consumption plus the value of net investment, representing the utility of postposed consumption -the future consumption that this additional capital will produce in the future.

The equilibrium growth rate of capital is

γ = A -ρ 1 -α(1 -σ) . (5)
From the feasibility constraints, it is clear that the growth rate of investment is also γ, and that αγ is the growth rate of consumption. Competitive equilibrium allocations are balanced growth paths from t ≥ 0.

Returns to scale differ between sectors. Since α < 1, as the stock of capital grows the investment sector becomes more productive with respect to the consumption goods sector. Differences in productivity causes the decline in investment prices relative to consumption goods prices. This difference in returns to scale can be interpreted, as put forward by [START_REF] Boucekkine | Embodied technological change, learning-by-doing and the productivity slowdown[END_REF], as a consequence of strong spillovers in the production of investment goods. 18 From the feasibility constraints, we can obtain the competitive equilibrium price of investment in terms of consumption units as the marginal rate of transformation:

p t = - dc t dx t = - dc t dh t dh t dx t = αh α-1 t A .
Since the stock of machines used in the consumption goods sector grows at the constant rate γ, the price of investment relative to consumption decreases at rate (α -1)γ < 0.

The competitive equilibrium allocation displays the regularities observed in actual data. Investment grows faster than consumption since γ > αγ. The relative price of investment decreases at rate (α -1)γ < 0. Indeed, the nominal share of investment in income remains constant. To see this, let us take the consumption good as numeraire and define nominal income as in the general case as m t = c t + p t x t . From the equilibrium equations, one can show after some simple algebra that the investment share

s t = p t x t m t = p t x t c t + p t x t = α(A -ρ) ρ(1 -α) + ασA . = s
for all t ≥ 0. To be precise, s is the equilibrium share of investment in total income. 19 18 Cummins and Violante (2002) observe that their measure of investment-specific technical change occurs first in information technology and then accelerates in other industries. They conclude that information technology is a "general purpose" technology, an interpretation that matches well with the spillovers' interpretation. See also [START_REF] Boucekkine | Obsolescence and modernization in the growth process[END_REF]. 19 It is interesting to notice that in the two-sector AK model, as well as in the balanced growth path of the Greenwood et al (1997) model, the investment share is constant. Consequently, measuring output growth by the mean of a fixed-base or a chained index does not make any difference.

At this point it may be worth stressing that the choice of the consumption good as numeraire is inconsequential. The argument above follows equally if we choose to measure income in units of investment, p -1 t c t + x t , or, for that matter, in any other arbitrary monetary unit provided that relative prices are respected -that is, that the price of investment relative to consumption is p t . This is important because identifying real growth with growth of nominal income is as arbitrary as the choice of the numeraire in which nominal income is expressed.

Measuring output real growth

In this section, we apply the general theory proposed in Section 2 to the two-sector AK model. As in the general case, in regard of the Bellman equation ( 4), the function

w t (c, x) = c 1-σ 1 -σ + v (k t )x
can be seen as representing preferences over current consumption and current net investment. Again, the constraint in the Bellman equation ( 4) can be replaced by the budget constraint c + p t x ≤ m t because the budget line is tangent to the production possibilities frontier locally at the optimum. Notice that in this example the utility representation w t (c, x) changes over time only because the marginal value of capital does. In this sense, applying the Fisher-Shell index in this context is equivalent to compute a quantity index that compares income today with the hypothetical level of income that would be necessary to attain the level of utility associated with tomorrow's income and prices, with today's prices and today's shadow value of capital.

Let us define the indirect utility u t (m t , p t ) and the expenditure function e t (u t , p t ) as in Section 2. Recall that the Fisher-Shell true quantity index compares income today m t with the hypothetical level of income mt+h that would be necessary to attain the level of utility associated with tomorrow's income and prices m t+h , p t+h with today's prices p t and today's preferences as evaluated by e t , u t . It is important to notice that, since v (k t )

is the shadow price of capital, applying the Fisher-Shell index in this context reduces to compare current income with an hypothetical income using today's market and shadow prices of investment. Nothing else changes with t in the preference representation w t (c, x)

than v (k t ).
Denote again this artificial level of income as mt+h = e t u t (m t+h , p t+h ), p t .

From the definition of g FS t in Section 2, we conclude that, for all t ≥ 0,

g FS t = (1 -s)αγ + sγ = αA(A -ρ) ρ(1 -α) + ασA
As already said, the Fisher-Shell quantity index is equal to the Divisia index. As in the general case, the interpretation is straightforward: g FS is a measure of real growth because it is constructed as the growth rate of nominal income substracting pure price changes, in this case the change of the relative price of investment p t . The index only keeps changes in quantities. It is also clear that it is a true index because it is constructed from the representative household's preferences.

On money metric utility

What are the implications of money metric utility for the measurement of output growth in the case of the two-sector AK model? This section argues, that money metric utility, implicit in the Fisher-Shell index, selects a particular representation of preferences that makes welfare to grow at the rate g FS . This particular representation depends crucially on preferences and technology. Let us develop this argument.

Since the two-sector AK model jumps to the balanced growth path at the initial time, a constant fraction of total capital will be permanently allocated to the production of consumption goods and capital will be permanently growing at the endogenous rate γ.

After substituting the optima consumption path in (3), the value function reads

v(k t ) = Bk α(1-σ) t , with B = (A -γ) α(1-σ) (1 -σ) ρ -αγ(1 -σ) . ( 6
)
Notice that both the exponent of k t and B depend on preferences and technology.

The argument is the following. The utility function in (3) is one among many representations of the same preference order -constant intertemporal elasticity of substitution preferences. The Fisher-Shell index arbitrarily choses another, the one that at equilibrium grows at rate g FS and adopts nominal income at some reference time as its benchmark. We build the argument in two steps.

First, let us denote by vt the equilibrium welfare of the representative agent at time t measured on an arbitrary unit. Let us then make two assumptions concerning vt , consistently with the main implicit assumptions of the Fisher-Shell index. We assume first that at the initial time, t = 0, v0 = (c 0 + p 0 x 0 )/ρ. This is the money metric utility assumption that the return to assets is equal to nominal income at the reference time, here t = 0. The second assumption is that vt grows at the rate g = g FS , meaning that vt = gv t . Then, for all t ≥ 0, vt = v0 e gt .

Consequently, if a utility representation of household preferences consistent with the Fisher-Shell index exists, it has to be that at equilibrium welfare is a potential function of k t with exponent g/γ. Let us now show that such a representation exists.

Second, we adopt the following alternative representation of the constant intertemporal elasticity of substitution preferences in ( 3)

v(k t ) = max C ∞ t c 1-σ s 1 -σ e -ρ(s-t) dt β ,
which is maximized subject to the technological constraints in Section 3.1; with βC > 0.

Since this new utility function represents the same preferences as those of the original two-sector AK model, the equilibrium path is the same. Consequently, we can easily

show that at equilibrium v(k t ) = C v(k t ) β = v0 e gt ,
where

C = v0 B -β k -α(1-σ)β 0 and β = g αγ(1-σ)
, both depending on preferences and technology parameters, and C additionally dependign on both the initial capital stock and the initial nominal income. 20 We have then proved that the growth rate as measured by the Fisher-Shell index is a welfare measure in the sense that it is equal to the growth rate of a particular representation of household preferences. The choice of this representation directly results from the key assumptions in money metric utility that welfare is measured in units of nominal income at some reference time.

On consumption equivalence

When measuring welfare gains, consumption equivalence is a usual way of dealing with the previous referred problem of the representation of preferences. Notice that consumption equivalence is a compensating variation measure, in terms of the entire consumption path instead of current income. In our context, the problem reduces to measure the hypothetical increase in the consumption path that makes an individual evaluating her welfare at time t indifferent between staying at t or jumping directly to time to t + h.

Let us formulate the problem formally in the case of the two-sector AK model developed in this secction.

Let us assume that preferences are separable and constant intertemporal elasticity of substitution, like in [START_REF] Asheim | Does NNP growth indicate welfare improvement?[END_REF]. The hypothetical increase in the consumption path λ h that makes an individual indifferent between staying at t or jumping to t + h has then to verify the condition

λ 1-σ h v(k t ) = v(k t+h ),
where v(k t ) and v(k t+h ) refer to the equilibrium welfare of the representative household at times t and t + h as measured by the value function v(k). Indeed, the compensating variation λ h directly depends on the length of the interval h.

From ( 6), the previous condition becomes

λ h = v(k t+h ) v(k t ) 1 1-σ = e αγh .
The growth rate of welfare consistent with consumption equivalence is then the derivate of λ h with respect to h evaluated at h = 0, which in the case of the two-sector AK model reads

g ce = 1 λ h dλ h dh h=0 = αγ.
The consumption equivalence measure of the growth rate of the the two-sector AK economy is equal to the growth rate of consumption. The result comes at no surprise, since the consumption path implicit in v(k t+h ) is the same as the consumption path in v(k t ) multiplied by the factor e αγt . We have then to conclude that in the case of the two-sector AK model, measuring growth by the mean of the growth rate of consumption is also welfare based. The simplicity of the results relies on the fact that the two-sector AK economy is always at its balanced growth path. In the general case of a concave technology, like in Greenwood et al (1997), if the economy is not at its balanced growth path, the calculation of the consumption equivalent growth rate is not straightforward and this results cannot be easily extended.

Discussion

In the framework of two-sector dynamic general equilibrium models, Section 2 shows that the Divisia index is, in fact, a true quantity index. This is of substantive interest since the Fisher ideal chain index used in actual national accounts approximates well the Divisia index, implying that the growth rate of output in NIPA is welfare based.

This section discusses the implications of this result. To make our main point clear, this section refers to representation of preferences like in equation ( 4).

More on money metric utility. Notice that at equilibrium the welfare of the representative household, v(k) in the Bellman equation ( 4), measures the value of assets, represented here by the capital stock. Then, ρ v(k) is the return to these assets as evaluated using the subjective discount rate ρ. From (4), at equilibrium the return to assets is equal to the utility of current consumption plus the value of current investment, the latter being assessed at the marginal value of capital v (k). Of course, welfare as measured by v(k) is defined in an arbitrary unit: monotonic transformations of preferences will change the level of utility leaving the preference map intact; consequently, the growth rate of different representations will not be necessarily the same. To overcome this problem, as discussed in the introduction, the literature on welfare measurement adopts current income as a sensible norm to measure changes in welfare; it does by using an equivalent variation measure (i.e., money metric utility). Since income as measured by national accounts represents the return to the stock of assets, the Fisher-Shell quantity index and then the Divisia index are equivalent variation measures quantifying changes in the return to capital. Since the subjective discount rate in (4) is time independent, the Divisia index also measures changes in welfare.

Investment matters. The following example makes it more clear why investment matters in the definition of output growth. Consider a world with embodied technical progress -as the one in Greenwood and Yorukoglu (1997), for example. Let the consumption path in this economy be depicted as in Figure 3. In period T there is an unexpected permanent technology shock to the investment sector: embodied technical progress accelerates. New machines, if produced and added to the capital stock, can make the productivity in the consumption goods sector grow faster indefinitely. In our example, hence, after observing the unexpected acceleration of investment specific technical change in T , the consumer finds optimal to initially reduce consumption in order to increase investment and, then, profit from technical progress. In this world, at time T households welfare increases: the drop in consumption reflects the interest of the consumer in benefiting from faster growth thereon; if this move would have not increased her welfare, she would have chosen not to increase investment and remain in the original Figure 3: Investment matters path with lower growth. Then, the consumption growth rate at time T does not measure welfare correctly. In fact, it has the opposite sign! However, the growth rate of output as measured by the Divisia index does, since it captures well the gains in welfare coming from the acceleration of technical progress and the associated optimal increase in investment. Remind that technical progress is assumed to be investment specific. Then, gains in productivity require new investments. The discussion above helps to illustrate why the growth rate of investment matters for output growth measurement. Faster growing investment today represents our best proxy for the preference for faster consumption growth tomorrow.

Net National Product. In connection with these considerations, the use of the Bellman equation makes it clear why production in national accounts is measured as final demand. Since present and future consumption is all that matter for welfare, and net investment measures the value of the future consumption it will produce, a welfare measure of output growth has to weight the growth rate of both final demand components, consumption and net investment. This interpretation is consistent with Weitzman (1976)'s claim that "net national product is a proxy for the present discounted value of future consumption." 21 In fact, his equation ( 10) is in spirit equivalent to the Bellman equations ( 2) and ( 4), which rationalize our choice of taking current net income as the proper norm in the Fisher-Shell true quantity index. It is important to point out that Weitzman (1976) is not about output growth and its relation to welfare gains in the growth process, but about the level of output and its relation to the level of welfare. In this sense, the non trivial question of the appropriate measurement of output growth has remained open until our days. The best result in this direction is in a subsequent paper by [START_REF] Asheim | Does NNP growth indicate welfare improvement?[END_REF]. That paper builds a measure of the level of output and shows that output growth is a necessary and sufficient condition for welfare growth, but without providing any specific insight on how output growth should be measured. This papers gives a fundamental step ahead in this direction: by applying standard index number theory, we show that the precise way NIPA measures growth is welfare based.

At this point it may be worth clarifying that, as pointed out by [START_REF] Weitzman | On the welfare significance of national product in a dynamic economy[END_REF], it is not GDP but Net National Product (NNP) what matters for welfare. 22 Depreciated capital is a lost resource that does not contribute to welfare. It is in this sense that some authors claim that NNP is relevant for welfare and GDP for productivity -see the discussion in [START_REF] Oulton | Productivity versus welfare; or GDP versus Weitzman's NNP[END_REF]. If the depreciation rate is constant, however, net and gross investment grow at the same rate. Indeed, when investment growth faster than consumption, NNP grows slower than GDP since the share of net investment on net income is smaller than the corresponding share of gross investment.

Paradox of endowment vs production economies. It is very important to understand that a true quantity index of output growth is a welfare based measure conditional on both preferences and technology. In other words, it does not reflect changes in welfare independently of the possibilities allowed by technology. The example below shows the interplay between technology and preferences in the definition of output growth emerging from index number theory applied to this family of problems.

Consider the following example that clarifies further the meaning of measuring welfare changes. For the two-sector AK model in Section 3, take any configuration of parameters such that, for example, the growth rate of investment at equilibrium is 6% and the 21 Weitzman's argument is developed in a simple optimal growth model with linear utility and the proof is based on the assumption that current income remains constant over time. In its own words, he gets "the right answer, although for the wrong reason." 22 Since the model economy in Section 2 is closed, GDP and GNP are equal, as well as NDP and NNP.

investment share is 20%. Let α be equal to 1/3. The Divisia index tells us that this economy will be growing at 2.8%, since consumption represents 80% of output and will grow at 2%. Alternatively, consider an endowment economy with exactly the same preferences and the same equilibrium consumption flow. In this economy, consumption is mana from haven. Indeed, a household would be indifferent between living in the AK or in the endowment economy, since she will get the same consumption path, that she will evaluate using the same preference map. In the endowment economy, indeed, index number theory will associate income with current consumption; the Divisia index will then measure output grow as consumption growth; 2% in our example. Why is it the case that two economies where people have identical preferences and face exactly the same consumption path do not grow at the same rate? The reason is that a true quantity index takes current income as a norm and current income is defined differently; at any time, both economies share the same consumption utility, but investment goods are produced only in the production economy. These seemingly paradoxical example illustrate well the intimate relation between preferences (what we want to do) and technology (what we can do) when measuring output growth. Indeed, in this particular example, both measures of output growth are welfare based and consistent with NIPA methodology.

The example makes also clear the implications of measuring production as final demand:

since there is no investment in the endowment economy, output growth becomes identical to consumption growth.

Growth accounting. To end this discussion, let us review the implications for growth accounting. In terms of model representations of actual economies, the introduction of more than one sector with different growth rates raises the practical and conceptual issue of how output growth has to be measured. The choice of the appropriate output growth rate affects every quantitative exercise based on the measurement of growth. This is the case in the literature on growth accounting under embodied technical change, the socalled Solow-Jorgenson controversy. 23 To measure the contribution of investment specific technical change to growth, Hulten (1992) measures growth (his equation ( 7)) following [START_REF] Jorgenson | The embodiment hypothesis[END_REF]. He suggests a raw addition of consumption and investment units, calling the outcome quality-adjusted output. Using our notation, this strategy amounts to c t + x t . Greenwood et al (1997) note that, in their setting, adding consumption and effective investment turns the economy into a standard [START_REF] Solow | Investment and technical progress[END_REF] growth model 23 See Hercowitz (1998) for a review of the Solow-Jorgenson controversy.

with no embodied technical change. Greenwood et al (1997) correctly state that any aggregation requires the different quantities to be expressed in a common unit and they adopt the consumption good as their standard. For this purpose, investment has to be multiplied by its relative price, in our notation their choice of output level would be y t = c t + p t x t . 24 Oulton (2007) generalizes the argument and suggests that output components have to be deflated by the consumption price index in order to measure growth. But this is indeed what Greenwood et al (1997) suggest when they identify non-durable production with real output and the real growth rate with the growth rate of consumption. What the present paper clarifies is that the issue is not the units used to measure real output levels but the choice of the right index of real output growth. In this sense, we follow [START_REF] Licandro | The measurement of growth under embodied technical change[END_REF] and conclude that the "true" contribution of ETC to output growth, reflecting welfare changes, has to be measured using NIPA methodology as in [START_REF] Cummins | Investment-specific technical change in the United States (1947-2000): Measurement and macroeconomic consequences[END_REF]. 25A word of caution. We have to be careful in the way we interpret the output growth rate in this framework. Since raising the growth performance of an economy is costly, it is well-known in endogenous growth theory that there exists an optimal growth rate. In the case of the two-sector AK model above, the optimal growth rate of capital is γ as shown in equation [START_REF] Becker | Capital Theory, Equilibrium Analysis and Recursive Utility[END_REF]. Let us then assume, for example, that the two-sector AK model is at equilibrium growing at its optimal growth rate, but at time t = 0 an uninformed government decides to introduce some incentives to promote growth, for example by subsidizing capital production and then distorting the private return to capital. At the time of the reform, t = 0, the economy starts growing faster at the cost of a reduction in welfare. From this time on ahead, the growth rate of output in the distorted economy, like in Section 3.3, will measure welfare gains, which will be larger than in the efficient economy. Unfortunatelly, the initial welfare losses will not be captured by national accounts, since changes in the value of assets are in general not registered.

Let us formalize the previous statement by following the same steps as in Section 3.3.

The value function of the distorted economy reads, for t ≥ 0,

v d (k t ) = B d k α(1-σ) d,t , with B d = (A -γ d ) α(1-σ) (1 -σ) ρ -αγ d (1 -σ)
.

where

γ d = τ A -ρ 1 -α(1 -σ) and k d,t = k 0 e γ d .
The distortion introduced by the subsidy is represented by the wedge τ > 1. It is easy to see that |B d | < |B| and decreasing with τ > 1, meaning that at t = 0 the policy generates welfare losses, which are larger the larger is the distortion. Paradoxically, welfare in the distorted economy is growing faster; reflecting the fact that there exists a finite time

t d > 0 from which v d (k d,t
) becomes larger than v(k t ).

Conclusions and extensions

This paper shows that a Fisher-Shell true quantity index is equal to the Divisia index when applied to a two-sector dynamic general equilibrium economy with heterogeneous households, general recursive preferences and general technology transforming production factors (capital and labor) into consumption and investment. Indeed, it turns out that the chained-type index used by national accounts to compute real output growth is well approximated by the Divisia index. Consequently, real GDP growth in NIPA is welfare based in the precise sense of equivalent variation. This result is illustrated in the framework of the two-sector AK model, which replicates the well-know stylized facts that investment grows faster than consumption and that the relative price of investment permanently declines. More important, changes in the growth rate of investment induced by changes in embodied technical progress turn out to be a relevant part of welfare increases along an equilibrium path. Investment then matters in the measurement of output growth. In general, this paper can be seen as a recall that index number theory has an important role to play clarifying the criteria with which we construct our indexes.

In particular, this approach may be of great relevance for the debate on the use on index number theory to rationalize the use of the Penn World Tables (see [START_REF] Neary | Rationalizing the Penn World Table: True multilateral indices for international comparison of real income[END_REF] and van Veelen and van der Weide (2008)).

Let us finally comment on those dimensions in which this approach could be extended and those in which it will be hard to do. Broaden it to many durable and non-durable goods seems straightforward. The approach could also be applied to many forms of nonoptimal equilibria. Notice that, in this case, the production possibility frontier will not be tangent to an indifference curve at equilibrium, and hence the generalization will not be straightforward. However, if the representative household is price taker in all markets, irrespective of the fact that prices are distorted, at equilibrium the budget constraint will be tangent to an indifference curve. Under theses circumstances, index number theory could be applied to compare different points in the equilibrium path in a similar way we did in Section 2. In particular, for a stationary economy moving from a distorted to a non distorted equilibrium, the Divisia index could be measuring the welfare gains period by period.

Appendix: Quantity indexes in continuous time

A1. Quantity indexes in continuous time

In continuous time, let us define a growth factor Γ t t+h as the gross rate of growth of an arbitrary variable between a base time t and a current time t + h, h ≥ 0. When h ≤ 0, Γ t t+h measures the gross rate of growth between the base time t + h and the current time t. In the jargon of national accounts, Γ t t+h is referred as a volume index. Let us then define the instantaneous growth rate of the underline variable at time t + h when the base time is t as

g t t+h = dΓ t t+h dh 1 Γ t t+h . (7)
Notice that in continuous time, h ≥ 0, the derivate of a growth factor at any time t + h is equal to the growth rate of the variable itself at t + h. Let z t be a continuous-time variable and Γ t t+h = z t+h /z t the growth factor. Apply [START_REF] Boucekkine | Obsolescence and modernization in the growth process[END_REF] to get

dΓ t t+h dh 1 Γ t t+h = żt+h z t+h .
This way of defining the instantaneous growth rate may look odd but it may be useful in those cases in which we have an index like Γ t t+h but no explicit variable giving rise to it like z t in this example.

Using the notation introduced in Section 2, the starting point in index number theory is some nominal aggregate income c t + p t x t . Remind that we have adopted consumption as the numeraire so that its price is normalized to one while the price of investment in The Fisher ideal growth factor with time base t and current time t + h, h ≥ 0, is defined as

F t t+h = L t t+h P t+h t 1 2 . ( 8
)
The definition in equation ( 7) is also applied in Section 2.2 to the Fisher-Shell quantity index since we have a well-defined factor mt+h /m t . Notice that in the definition of g FS t , we use the property that lim h→0 mt+h = m t .

A2. Fixed-base quantity indexes in continuous time

Traditional measures of real growth stem from fixed-base quantity indexes. The most common among them are the Laspeyres and Paasche indexes referred in Appendix A1.

From Appendix A1, the Laspeyres factor of change between t and t + h is L t t+h = c t+h + p t x t+h c t + p t x t , for h ≥ 0, where t represents the base time and t+h the current time. In continuous time, the Laspeyres index g L t t+h is the instantaneous growth rate of factor L t t+h as a function of h -see equation [START_REF] Boucekkine | Obsolescence and modernization in the growth process[END_REF]. That is,

g L t t+h = dL t t+h dh 1 L t t+h =
ċt+h + p t ẋt+h c t+h + p t x t+h , which measures the instantaneous real growth rate at t + h for the given base time t.

The Laspeyres index is popular because it is conceptually simple.

However, if the relative price of investment permanently declines and substitution makes real investment permanently grow faster than real consumption, as observed in the data, the Laspeyres index tends to give too much weight to investment as we depart from the base time t. In order to illustrate it, let us assume the economy is at a balanced growth path with constant investment and consumption shares, s and 1 -s respectively, s ∈ (0, 1), the relative price of investment goods p t declining at a constant rate, and investment and consumption growing at the constant rates g x and g c , respectively, g x > g c > 0.

Note, indeed, that the Laspeyres fixed-base index reads g L t t+h = c t+h c t+h + p t x t+h g c + p t x t+h c t+h + p t x t+h g x .

Since x t+h grows relative to c t+h , it is easy to see that along a balanced growth path the weight of consumption in the Laspeyres fixed-base index decreases and the weight of investment increases with h. This effect is known in the index numbers literature as the substitution bias. Fast growing items when weighted using past (relatively high) prices are overweighted, overstating the real growth rate of output. The effect is larger the farther we are from the base time, converging to the growth rate of investment as h goes to infinity.

The Paasche index uses current prices as a base, instead of past prices, and hence tends to understate real growth as we go back in time.

under the assumption that the growth rates of both consumption and investment are constant. As h grows, so t -h decreases, the weight of consumption increases because

x t-h /c t-h decreases, converging to the growth rate of consumption as h goes to infinity.

For the arguments developed above, both Laspeyres and Paasche fixed-base indexes yield poor measures of real growth when output components grow at different rates because of changing relative prices. The farther we are from the base time, the more the Laspeyres index overstates growth, and the more the Paasche index underesates it. 26Indeed, it is easy to see that in continuous time both Laspeyres and Paasche quantity indexes are equal to the Divisia index when evaluated at t: xt the growth rate of investment. 27 Given that in continuous time, both Laspeyres and Paasche quantity indexes are equal to the Divisia index at t, it is easy to show that the Fisher ideal index is equal too. It is trivial to see that this property also applies to the Fisher ideal index.

dL t

A3. Chained-type quantity indexes in continuous time

In this appendix, we use our simple framework to review the BEA methodology. 28 The introduction by the BEA of quality corrections in equipment prices in the mid-eighties revealed a persistent declining pattern in the price of equipment relative to the price of non-durable consumption goods. Since then, real investment appears to be growing much faster than real non-durable consumption. In this new scenario, fixed-base quantity indexes face the severe substitution bias problem explained in Appendix A2 above. For this reason, the BEA moved to a chained-type index based on a Fisher ideal index computed for contiguous periods. 29 Let us first define a Fisher ideal index to them define a Fisher ideal chain index both in continuous time.

Let us now define a Fisher ideal chain (factor) index for the time interval (0, T ), where t = 0 represents now the reference time (in contraposition to the base time). The key assumption of chained indexes is that the base time moves with t, by taking t as the base time when computing the growth rate at time t. From Appendix A2, for any time 27 In discrete time, the weights of consumption and investment growth rates in the Laspeyres and Paasche indexes are different from current income shares. 28 Young (1992) is a non-technical presentation of the methodological changes introduced in NIPA. [START_REF] Whelan | A guide to the use of chain aggregated NIPA data[END_REF][START_REF] Whelan | A two-sector approach to modeling U.S. NIPA data[END_REF] provides a more detailed guide into the new methods in use at BEA to measure real growth. For economic index number theory see [START_REF] Diewert | Index numbers[END_REF], [START_REF] Triplett | Economic theory and BEA alternative quantity and price indexes[END_REF] A chained factor index for a time interval t ∈ (0, T ) is build in two stages. First, at any time t ∈ (0, T ) a growth rate is computed using t as the base time. Second, the time t growth rates computed at the first stage are chained in order to build growth factors in an interval of time t ∈ (0, T ). Notice that fixed-base factor indexes are equal to one at the base time. In the case of chained indexes base times are changing. For this reason, the time at which the factor index is set equal to one is now called the reference time.

  denoting the restriction of C to the interval [t, ∞), preferences of the representative household are represented by some recursive utility function U generated by the differential equation

Figure 1 :

 1 Figure 1: The production possibilities frontier and competitive prices

Figure 2 :

 2 Figure 2: The Fisher-Shell true quantity index

  As referred in the Introduction, following[START_REF] Gordon | The Measurement of Durable Goods Prices[END_REF]'s observation that quality adjusted equipment investment prices were permanently declining relative to the price of non-durable consumption goods and services, the Bureau of Economic Analysis (BEA) moved first to control for quality improvements in the measurement of investment prices, and second to a Fisher ideal chain index to measure output growth. The first change made investment to grow faster than non-durable consumption. As an undesirable consequence of trends in relative prices, the fixed-base quantity index used to measure GDP growth became obsolete fast enough to provided appropriate growth figures. In facts, in this case, fixed-base quantity indexes suffer from the well known substitution bias problem that tends to overestimate the weight of the fast growing items (see Appendix 2).The second change addresses this last problem by making the NIPA measure of output growth to be approximately equal to the Divisia index.Almost contemporaneously, a new literature developed in macroeconomics aimed to accommodate growth theory to this new evidence.Greenwood et al (1997), in their seminal paper, extend the Ramsey model to a two sector (consumption and investment) growth model with two sources of technical progress, consumption and investment specific technical change (disembodied and embodied in capital goods, respectively). This model is able to replicate the permanent decline in the relative price of equipment investment, as well as the fact that investment grows faster than consumption (implying that the investment to output ratio is permanently growing). In this context, it is particularly clear that the aggregation issue is far from trivial since consumption and investment grow at different rates.This section describes a simple version of the two-sector AK model proposed by[START_REF] Rebelo | Long-run policy analysis and long-run growth[END_REF] and apply to it the Fisher-Shell index proposed in Section 2.2 to show that the BEA had good fundamental reasons to use a Fisher ideal chain index to measure output growth. As shown in[START_REF] Felbermayr | The under-estimated virtues of the twosector AK model[END_REF], the two-sector AK model

  instant t, a social planer chooses individual consumption c t and per capita net investment kt such that (c t , kt ) ∈ Γ(k t , Θ t ) is feasible, where k t is capital and Θ t

	utility is U ( t C) and the current change in welfare as measured by U ( t C) is simply given
	by (1).
	A NSO aiming to build a true quantity index measuring output growth in this econ-
	omy faces, in addition to the well known problem that preferences are not univocally
	represented by a utility function, the additional problem from an accounting point of
	view that neither preferences and technology nor foreseen consumption are observable.
	In this context, a NSO wishes to build a quantity index that reflects changes in welfare
	using only current consumption c t and current net investment x t = kt , both observables
	at instant t; and all that matters of the level of k t is summarized in the price of investment
	p t as we will argue below. To this end, however, a NSO shall need to express preferences
	as a function of variables observed at t. Since preferences are recursive, this amounts
	to express changes in welfare as a function of current consumption c t and current net
	investment x t .
	In other words, a NSO needs a representation of preferences over current consumption
	and current net investment, and this is what the Bellman equation gives us. The original
	problem is to maximize U ( t C) subject to (c s , ks ) ∈ Γ(k s , Θ s ) for all s ≥ t, k t > 0 given,
	where Θ s is a vector of exogenous states that directly affect technology. The associated
	Bellman equation is
	0 =	max
	(c,x)∈Γ(kt,Θt)
	represents a vector of exogenous non-stochastic states. In the following, we assume that,
	for a given k t > 0, there exists a unique consumption and investment path equilibrium
	(c s , ks ) s≥t that maximizes U ( t C) subject to the technological constraint. Then, total
	shown in Appendix A3.

  , differentiate the definition of nominal income m t = c t + p t x t with respect to time and define the equilibrium share of net investment to net income as s t . = p t x t /m t

	which implies that
	Finallyto write
	ṁt m t	= (1 -s t )	ċt c t	+ s t	ẋt x t	+ s t	ṗt p t	,

[START_REF] Acemoglu | Introduction to Modern Economic Growth[END_REF]

, u 1 and u 2 in the definition of the Fisher-Shell index simplify as a direct consequence of the properties discussed in the paragraph above; all three are related to the marginal value of income µ. It is in this sense that money metric utility operates in the Fisher-Shell index. Since gains in welfare are measured as a compensating variation by comparing the artificial level of income mt+h with the nominal income m t , and prices enter linearly in the budget constraint, gains in welfare are equal to the change in nominal income (arbitrarily measured here in units of the consumption good) minus the contribution of prices to it (which comes only from the change of investment prices, weighted by the equilibrium (net) investment share).

  (m i,t , p t ) = A i,t (p t ) + m i,t /p t e i,t (u i,t , p t ) = p t u i,t -A i,t (p t ) , where A i,t (p t ) is defined below. In fact, from the household problem, optimal consump-By denoting the implicit solution for c i as c i,t (p t ), it is then easy to show thatA i,t (p t ) = f i,t c i,t (p t ) -c i,t (p t )/p t .Let us define the artificial level of household i income as in section 2.2, i.e., mi,t+h = e i,t u i,t (m i,t+h , p t+h ), p t = p t A i,t (p t+h ) -A i,t (p t ) + p t /p t+h m i,t+h , which is linear on income due to the fact that preferences are quasilinear. Consistently with national accounts, aggregate income is defined as m t = i m i,t di, which also measures per capita income since population has been normalized to unity. Let us now define the aggregate hypothetical income consistently with the definition of per capita income as mt = i mi,t di. Using the results just above, mt+h = p t Āt (p t+h ) -Āt (p t ) + p t /p t+h m t+h ,

	where
	Āt (p t ) =

1 (k i,t , Θ t , ϕ t ). Notice that we have subtracted π i,t from the right hand side of the Bellman equation and then divided it by v i,1 (k i,t , Θ t , ϕ t ).

Since non of these two terms depend on c or x, such a transformation has no effect on the households program. Function w i,t (c i , x i ) is maximized under the budget constraint c i + p t x i = m i,t , where, as said above, p t is the equilibrium price and m i,t , for all i, represents equilibrium household net income. Since this utility representation is quasilinear, it belongs to the Gorman family. It is easy to show that the indirect utility and expenditure functions become u i,t tion c i solves f i,t (c i ) = 1/p t . i A i,t (p t )di.

  . = p t x t /m t as before and x t = i x it di is per capita net investment. The Fisher-Shell index is, indeed, equal to the Divisia index, meaning that the growth rate in NIPA is a welfare measure irrespective of households being either homogeneous or heterogeneous.

	where s t
	is per capita consumption. Then
	gFS t =	ṁt m t	-s t	ṗt p t	= (1 -s t )	ċt c t	+ s t	ẋt x t	,

  , follows[START_REF] Felbermayr | The under-estimated virtues of the twosector AK model[END_REF] closely, and entails all the characteristics that are relevant to the present discussion in the simplest possible framework. The stock of capital at each instant t is k t , from which a quantity h t ≤ k t is devoted to the production of the consumption good. Consumption

	goods technology is

  consumption units is p t . Laspeyres quantity indexes use time t (the base time) prices as weights based on the following growth factor L t t+h = c t+h + p t x t+h c t + p t x t . It does allow to compute the growth rate of output by putting all nominal values at base time prices. Notice that in this framework the real unit in which quantities are measured is nominal income c t + p t x t at the base time. Paasche indexes take current prices as

	weights by defining the growth factor as
	P t t-h =	c t + p t x t c t-h + p t x t-h	,

h ≥ 0. Real output growth is measured at current t prices.

  The Passche factor isP t t-h = c t + p t x t c t-h + p t x t-hfor all h ≥ 0 and the growth rate + p t x t-h g c + p t x t-h c t-h + p t x t-h g x ,

	g P t t-h =	dP t t-h dh	1 P t t-h	=	c t-h	c t-h

  -s t )g ct + s t g xt , where s t = ptxt ct+ptxt is the investment share, g ct = ċt ct the growth rate of consumption and g xt = ẋt

	t+h dh	1 L t t+h h=0	=	dP t t-h dh	1 t-h h=0 P t	= (1

  , Fisher and Shell (1998) and IMF (2004, chapter 17).t ∈ (0, T ), the instantaneous growth rate of the Fisher ideal index is -s t )g ct + s t g xt .Even if there is a trend in relative prices, inducing the substitution of one good for another, the chained-type index allows weights to change continuously to avoid the emergence of any substitution bias.Let us assume that s t , g ct and g xt are continuous function of t, then the Fisher ideal index g F t is continuous too. A Fisher ideal chain (factor) index C F

	g F t =	dF t t+h dh	1 F t t+h h=0	= (1 t is defined by the
	differential equation
			ĊF t = g F t C F t ,
	C F 0 = 1, which solution is
			C F t = e	t 0 g F

s ds .

See Fisher and Shell (1968) for a definition of a Fisher-Shell index and for a discussion about the conditions of its applicability. See[START_REF] Triplett | Economic theory and BEA alternative quantity and price indexes[END_REF] for a discussion on the properties of the Fisher ideal index. National accounts in Europe measure real growth by the mean of chained-type Laspeyres index following the Commission Decision 98/715/EC.

In the conceptual framework developed in this paper, Appendix A.2 formally explains the substitution bias inherent to the Laspeyres fixed-base index. Appendix A.3 shows in the same framework that a Fisher ideal chain index is equal to the Divisia index.

[START_REF] Autor | The growth of low-skill service jobs and the polarization of the US labor market[END_REF] Many other papers have followed. See, for example,[START_REF] Krusell | Investment-specific R and D and the decline in the relative price of capital[END_REF], Gort et al (1999), Greenwood et al (2000), Cummins and Violante (2002), Whelan (2003), Boucekkine et al (2003, 2005) and Fisher (2006).

For structural transformation, see Acemoglu and ) (2008), Duarte and Restuccia (2010), Herrendorf et al (2013) and Ngai and Pissarides (2007), among many others.

The property that in continuous time the Fisher ideal chain index is equal to the Divisia index is

[START_REF] Epstein | The global stability of efficient intertemporal allocations[END_REF] explores conditions under which a generating function f represents a recursive utility function U . Becker and Boyd (1997, chapter 1) motivates the study of general recursive preferences.

The planner solves a standard recursive program in which the state variable summarizes at each instant t all past information that could be relevant for today's decisions. For a brief exposition of recursive techniques in continuous time see[START_REF] Obstfeld | Dynamic optimization in continuous-time economic models (A guide for the perplexed)[END_REF].

See[START_REF] Gorman | Community preference fields[END_REF][START_REF] Gorman | On a class of preference fields[END_REF].

See also[START_REF] Acemoglu | Introduction to Modern Economic Growth[END_REF], chapter 11.3.

This is a particular case of the general preferences in Section 2.1. Here the correspondence Γ is defined for every k ≥ 0 as the set Γ(k) of pairs (c, k) such that there exists h with 0 ≤ h ≤ k, c ≤ h α , and k ≤ A(k -h).

Notice that β and C may be both positive or negative depending on σ being smaller or larger than one, respectively. The constraint βC > 0 then holds for any σ > 0.

In their setting, this choice looks somewhat natural because the investment sector uses as input the consumption good. In their notation y t = c t + p t x t is total output in the non-durable sector, even if only c t is consumed and the remaining production p t x t is allocated to the investment sector.

Indeed, as suggested in section 3.4, applying consumption equivalence to the balanced growth path of the Greenwood et al (1997) economy makes the growth rate of consumption an alternative welfare based measure of output growth.

Updating regularly the base is not a solution because it would imply a permanent revision of past growth performance. It posses the additional problem of multiple real growth measures for each period, each of them affected differently for the substitution bias depending on the associated base period.

[START_REF] Diewert | Index numbers[END_REF] provides a clear explanation of the index suggested by[START_REF] Fisher | The Making of Index Numbers[END_REF].

Licandro acknowledges financial support from the Spanish Ministry of Sciences, Innovation and Universities under contracts ECO2010-17943 and ECO2013-48884-C3-1-P, and IMERA-AMSE that he was visiting the 2018 Autumn term.