Quantity Measurement and Balanced Growth inMulti–Sector Growth Models∗

Georg Duernecker (University of Munich, IZA, and CEPR)

Berthold Herrendorf (Arizona State University)

Akos Valentinyi (University of Manchester, CERS–HAS, and CEPR)

December 5, 2017

Abstract

Multi–sector models typically rely on a numeraire to aggregate quantities whereas NIPAuses the chain index. For three popular versions of the multi–sector growth model, we pro-vide analytical expressions for the growth of aggregate quantities under both measurementmethods and establish that the compound differences are sizeable over long horizons. Weshow that using the chain index captures more accurately the aggregate effects of secularchanges in relative prices. For example, in a standard model of structural transformation,measuring GDP growth with the chain index captures that Baumol’s disease reduces wel-fare growth, which using a numeraire misses.

Keywords: Balanced Growth; Baumol Disease; Chain Indexes; Structural Change.JEL classification: O41; O47.

∗We thank Fernando Martin, Richard Rogerson and participants of several presentations for comments andsuggestions. Valentinyi thanks the Hungarian National Research, Development and Innovation Office (ProjectKJS K 124808). All errors are our own.

1 Things to add

• Quote Oulton (2007): What GHK viewed as investment–specific technological changecan be modeled/viewed as differences between sectoral TFP growth. If quality improve-ments are properly measured, then the two views are observationally equivalent. Wefollow Oulton’s formulation.

• Quote Moro (2015): Also pointed out that measuring GDP through chain indexes maymake a difference.

2 Introduction

The one–sector growth model has become a workhorse model of macroeconomics, which cap-tures in a simple and tractable way the essence of modern economic growth. The simplicity ofthe one–sector version of the growth model necessarily implies that it abstracts from relevantfeatures of reality like the secular movements in the relative prices and expenditure shares ofsubcategories of GDP. To capture the aggregate implications of such movements, the profes-sion has pursued several routes of disaggregation. The resulting multi–sector versions of thegrowth model present the researcher with the challenge of how to aggregate sectoral quantitiesto economy–wide quantities, like the capital stock or GDP. This paper is about this challenge.

We will study the implications of two measurement methods for the behavior of economy–wide quantities in three popular multi–sector versions of the growth model that are calibratedto the postwar U.S. economy. We will start analyzing a two–sector benchmark version withinvestment and consumption of the model of Uzawa (1963). We will then turn to two modelsthat disaggregate this benchmark further into three–sector versions of the models of Greenwoodet al. (1997) and Ngai and Pissarides (2007). The usual way of aggregating quantities within thecontext of these multi–sector models is to define quantities in the unit of a numeraire from thesame period. We call this method “measuring quantities in terms of a numeraire”. In contrast,the national income and product accounts (“NIPA”) use chain indexes. We call this method“measuring quantities in terms of the chain index” or “using chained quantities”.

To put a synopsis of what is to come upfront, we derive three main results. Our first resultestablishes that it matters quantitatively which measurement method we choose. For the post–war U.S. economy they generate vastly different numbers for the growth of capital and GDPper hour. On top of that, quantity growth in terms of the chain index is independent of the unitsin which the quantity levels are expressed, whereas quantity growth in terms of a numerairedepends on the choice of the numeraire. Our second result establishes an additional advantageof using the chain index, namely, that it captures more accurately the economy–wide effectsof secular changes in relative prices. We show for a version of the Ngai–Pissarides model

1

of structural transformation that Baumol’s disease reduces welfare growth and that quantitygrowth measured in terms of the chain index captures this whereas quantity growth measuredin terms of a numeraire misses it. Our third result offers correction factors for each modelversion that make it straightforward to move from quantity growth in terms of a numeraire toquantity growth in terms of the chain index. These correction factors are helpful because itis typically easier to construct a balanced growth path when quantities are measured in termsof a numeraire. Connecting the balanced growth path to the data therefore requires moving toquantity growth in terms of the chain index. We show that the correction factors are constantalong the balanced growth path if and only if the expenditure shares of sectoral output do notchange when relative prices change.

Turning now to the details of the models, we first develop a two–sector version of the growthmodel with investment and consumption that captures the implications of the secular declinein the price of investment relative to consumption. While this model mainly serves as a pointof departure for further disaggregation into three–sector models, on its own it is already usefulto illustrate an important difference between the two measurement methods: only when GDPgrowth is measured with the chain index is it independent of the choice of units. To show this,we first measure quantities in terms of a numeraire. The two–sector model then has a balancedgrowth path along which aggregate variables and sectoral outputs grow at the same rate whileprices and sectoral expenditure shares remain constant. But the growth rate of GDP per houralong the balanced growth path does depend on the choice of the numeraire. Connecting themodel with the postwar U.S. economy, we find that the resulting differences in GDP growth aresizeable. Measuring quantities from the model in terms of the chain index avoids this problem,because the chain index was designed precisely so that chained growth rates are independent ofthe units in which the corresponding levels are measured. We derive correction factors that linkGDP growth in terms of the different measurement methods to each other. We show that giventhat along the balanced growth path the relative expenditures on consumption and investmentare constant, these correction factors are constant as well. Connecting the two–sector model tothe postwar U.S. economy, we find that compounding the correction factors leads to sizeabledifferences in the level of GDP per hour.

Next, we analyze a version of the three–sector version of the growth model developed byGreenwood et al. (1997). This model version distinguishes between structures, equipment,and consumption and assesses the implications of the secular decline in the relative price ofequipment. Consistent with the data, we assume that structures experience the slowest andequipment experiences the fastest technological progress, with consumption in the middle. Westart by characterizing the balanced growth path of the model when quantities are measured inunits of a numeraire. We show that along the balanced growth path, the growth rates of capitaland output per hour are the same and constant. We then switch to measuring quantities through

2

the chain index. We show that this brings about a key change along the same balanced growthpath from before: while the growth rate of capital per hour is still constant, it is now smaller thanthe growth rate of GDP per hour. We again derive the correction factors between the differentmeasurement methods and show that they are constant along the balanced growth path. Thereason is the same as in the two–sector model, namely, the relative expenditures on the differentinvestment goods are constant along the balanced growth path. Connecting this three–sectorgrowth model with the postwar U.S. economy, we find that compounding the correction factorsagain leads to sizeable differences in the level of GDP per hour.

One might think that our result that, measured in terms of the chain index, capital per hourgrows more slowly along the balanced growth path than GDP per hour must be at odds with theevidence. After all, Kaldor (1961) included in his growth facts the claim that the growth of thetwo is the same. When we look at postwar U.S. economy, however, we find that this is a myth:the average growth rate of chained capital was about half a percentage point smaller than theaverage growth rate of chained GDP. The intuitive reason, of course, is that the capital stockcontains a larger share of structures, and a smaller share of equipment, than investment andGDP. Since structures depreciate more slowly than equipment, the growth of capital is belowthe growth of investment and GDP. Using a chain index to aggregate the components of thecapital stock picks this effect up, whereas using a numeraire misses it.

Lastly, we analyze a three–sector version of the multi–sector growth model developed byNgai and Pissarides (2007). This model has become the benchmark for studying how differ-ences in the pace of technological progress at the sector level lead to changes in the relativeprice of sectoral output, which in turn lead to the reallocation of economic activity from thegoods sectors to the service sectors (“structural transformation”).1 When we measure quanti-ties in units of a numeraire, we obtain the usual balanced growth path along which the growthof economy–wide quantities is constant while economic activity is reallocated underneath fromthe goods sector to the service sector.2 When we switch to measuring quantities through thechain index, we obtain the novel result that the growth of GDP per hour declines along the samebalanced growth path. This implies that the balanced growth path that we derived in terms ofthe a numeraire no longer is a balanced growth path in terms of the chain index, because thegrowth of GDP per hour no longer is constant. This also implies that now the correction factorbetween the two ways of measuring GDP per hour changes over time, which reflects that bothrelative prices and relative expenditures change along the balanced growth path. Lastly, weshow that when we connect the model to the postwar U.S. economy, then the correction factoragain has quantitatively sizeable implications.

The model implication that the chained growth of GDP per hour is slowing over time, in-

1See Herrendorf et al. (2014) for a recent review article of the literature on structural transformation.2Since the sectoral shares change, this is often called a “generalized” balanced growth path. A balanced growth

path in the strict sense would have constant shares.

3

stead of staying constant, is at odds with the first of the Kaldor growth facts that over longhorizons the growth of GDP per hour is constant. While at first sight this implication of themodel might appear to be a problem, there is actually some evidence that is consistent with it.We document for the postwar U.S. economy that there has been a secular decline of averagegrowth of GDP per hour. While this is often ignored by growth theorists who instinctively focuson a balanced growth path, this growth slowdown is widely documented; see for example therecent paper by Byrne et al. (2016). That there has been a growth slowdown is also consistentwith the observation of Baumol (1967) that structural transformation slows down GDP growthbecause it reallocates economic activity to the service industries which tend to have smallerthan average productivity growth (“Baumol’s disease”). Given that we have a tractable modelof structural transformation at hand, we can go further than Baumol did and show that wel-fare growth (where welfare is measured by the utility index) also slows down along the usualbalanced growth path constructed in terms of a numeraire. Since the growth of GDP per houris often used as a proxy for welfare improvements, one would hope that it picks up the slow-down in this welfare measure. Our results imply that measuring quantities with the chainedindex delivers this whereas measuring quantities with a numeraire misses it. This probably isthe strongest argument in favor of employing the chain index for measuring economy–widequantities in multi–sector models.

The rest of the paper is organized as follows. In the next section, we review several closelyrelated papers. In Section 4, we develop a two–sector benchmark model with consumptionand investment. In Section 5 and 6, we study the three–sector version that result when we dis-aggregate investment into structures and equipment and consumption into goods and services,respectively. Section 7 concludes and an Appendix contains the proofs of our results.

3 Related Literature

Our work is closely related to Whelan’s important studies on the use of chain indexes in macroe-conomics. Whelan (2002) provided an insightful discussion of the pitfalls associated with usingchain indexes in the context of the national accounts. Whelan (2003) discussed how to calibratethe steady state of a two–sector growth model in the spirit of Greenwood et al. (1997) usingdivisia approximations of the chained index. While Whelan (2003) shares with our paper theinterest in the issues that arise when one seeks to connect a multi–sector version of the growthmodel to NIPA, the focus of the two papers is different. He was interested in calibrating aversion of the model of Greenwood et al. (1997) to the national accounts in such a way that hemeasures model quantities through chain indexes (or approximations to them) in the same wayas it is done in the national accounts. In contrast, we are interested in understanding the differ-ences between measuring quantities in units of a numeraire versus in terms of the chain index

4

in the context of multi–sector models. This is relevant because the common practice in theliterature on structural change is to use a numeraire, instead of the chain index, when construct-ing aggregate quantities. The novelty of our work is that we provide analytical expressions forthe growth of economy–wide quantities that results under both measurement methods, deriveanalytical expressions for the correction factors between them, and assess how large they arequantitatively for different version of the multi–sector growth model. An important part of ourpaper analyzes these issues in the context of the structural change model of Ngai and Pissarides,which did not exist when Whelan (2003) wrote his study.

The working-paper version of Ngai and Pissarides (2007), Ngai and Pissarides (2004), alsorealized that structural change leads to a reduction in GDP growth when real GDP is calculatedin terms of constant prices instead of a numeraire.3 Our paper adds two important points to thisinsight. First, we prove that along the balanced growth path of a three–sector version of themodel of Ngai and Pissarides, both welfare and chained GDP growth slow down. Second, weestablish for empirically plausible parameter values that the resulting GDP growth slowdownhas quantitatively sizeable implications. We find that compounding it over the postwar period,the reduction in the level of real GDP per hour is too large to ignore.4

Our work is also related to a large literature about structural change that measures the growthof GDP per hour in units of a numeraire and proceeds to derive a balanced growth path; seeHerrendorf et al. (2014) and the references therein. To avoid misunderstandings, we do notsuggest to change this way of proceeding when the goal is to solve the model and characterizean equilibrium path in the mode convenient way. This statement is particularly true for modelsof structural change in which a balanced growth path exists only if quantities are measured inunits of a numeraire, like the model of Ngai and Pissarides. Instead, our point here is that oneneeds to be careful when connecting the model to the national accounts and when interpretingthe properties of that balanced growth path in the face of changing relative prices, and possiblyrelative expenditures. It is then preferable to aggregate quantities by using the chain index. Wehope that the correction factors we derive will help in this context, and make moving betweenusing a numeraire and using the chain index straightforward.

4 A Two–Sector Model with Investment and Consumption

We start with developing a two–sector version of the growth model with consumption andinvestment in the spirit of Uzawa (1963). Our goal throughout the paper is to keep things asstandard as possible. We therefore heavily lean on the exposition in Herrendorf et al. (2014).

3Subsequently, Boppart (2016) also picked up on this insight.4In a companion paper, Duernecker et al. (2017), we show numerically that this effect is even more sizeable

when one takes into account structural change within the service sector.

5

4.1 Environment

There is a measure one of identical households. Since we are interested in matching modelquantities to the data, we allow population growth and assume that each household has Nt = ηt

members where η ∈ (0,∞). Hence, the population is Nt and N0 = 1 is normalized. Eachhousehold member is endowed with the initial capital stock K0 > 0 and with one unit of time ineach period.

The period utility is of the log form. This is an analytically convenient special case of theconstant–relative–risk–aversion functional form, which is required for balanced growth. Theintertemporal utility function is then given by:

∞∑t=0

βtNt log(Ct

Nt

)=

∞∑t=0

βt log (ct) .

where β ≡ βη ∈ (0, 1) is a modified discount factor. If η > 1, then β must be sufficiently smallerthan one to ensure that β < 1. We use lower–case letters to denote per capita variables, soct ≡ Ct/Nt denotes consumption per capita. As usual for basic versions of the growth model,there is no intensive margin for hours worked. This implies that per capita quantities will beequal to per worker and per hour quantities. We therefore do not distinguish among them andrefer to all model variables that are deflated by Nt as per capita variables. In per capita terms,capital accumulates according to:

ηkt+1 = (1 − δ)kt + xt. (1)

where δ ∈ [0, 1] is the depreciation rate.There are two sectors that produce consumption and investment. We use the Cobb–Douglas

production function because it naturally implies constant shares of capital and labor in totalincome. In per capita terms, the production functions are:

ct = kθct(γtcnct)1−θ,

xt = kθxt(γtxnxt)1−θ.

where θ ∈ (0, 1) is the capital–share parameter; kit and nit are sectoral capital and labor per capita(e.g., nit ≡ Nit/Nt); γi ≥ 1 is exogenous, labor–augmenting sectoral technological progress.Note that we restrict the capital–share parameter to be the same in both sectors so that theproduction side aggregates. Note too that sectoral TFP at time 0 equals one: γ0(1−θ)

i = 1. Thiscorresponds to a choice of units and is without loss of generality.

6

Figure 1: The Relative Prices of Investment

Pt=10.81−0.005⋅t

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1950 1960 1970 1980 1990 2000 2010Source: NIPA, Fixed Asset Tables, Bureau of Economic Analysis, own calculations

Price of investment relative to consumption

Capital and labor are freely mobile between the two sectors. The feasibility constraints are:

kct + kxt ≤ kt,

nct + nxt ≤ 1.

The welfare theorems will hold in the versions of the growth model we employ. Since theusual notion of balanced growth and the Kaldor Facts involve prices, we will nonetheless solvefor competitive equilibrium.

4.2 Competitive equilibrium

The firm problem in each sector is to minimize costs subject to producing a given quantity ofsectoral output. We choose investment as the numeraire. The first–order conditions then are:

rt = ptθ

(kct

nct

)θ−1

γt(1−θ)c = θ

(kxt

nxt

)θ−1

γt(1−θ)x ,

wt = pt(1 − θ)(

kct

nct

)θγt(1−θ)

c = (1 − θ)(

kxt

nxt

)θγt(1−θ)

x .

where rt and wt are the rental prices for capital and labor and pt is the relative price of consump-tion to investment. These first–order conditions imply that the relative price of consumption isinversely related to the sectoral TFPs:

pt =

(γx

γc

)t(1−θ)

. (2)

Figure 1 suggests that the empirically relevant case is γx > γc. This case is often referred to asinvestment–biased technological progress.

The production side of this two–sector environment aggregates. To see this, we divide the

7

first–order conditions by each other to obtain the usual result that the capital–labor ratios areequalized:

kt =kct

nct=

kxt

nxt.

This implies that for i ∈ {c, x}

yit = kθt γt(1−θ)i nit. (3)

Combining (2) and (3), we get:

yt = xt + ptct = kθt γt(1−θ)x , (4)

rt = θkθ−1t γt(1−θ)

x , (5)

wt = (1 − θ)kθt γt(1−θ)x . (6)

Note that if one wants to think of (4) as a technology in the strict sense of the concept, then itis preferable to eliminate the relative price and write it as:

xt +

(γx

γc

)t(1−θ)

ct = kθt γt(1−θ)x .

The household problem is to maximize utility subject to the budget constraint and severalconstraints on the choice sets. Since leisure does not generate utility here, the household al-locates its entire time endowment to working. Imposing this, the household problem can bewritten as:

max{ct ,kt+1}

∞t=0

∞∑t=0

βt log ct s.t. ptct + ηkt+1 ≤ (1 − δ + rt)kt + wt, ct, kt+1 ≥ 0, k0 = k0 > 0.

Solving this problem gives the following necessary conditions for optimality:

ptct + ηkt+1 = rtkt + wt + (1 − δ)kt, (7)pt+1ct+1

ptct=β

η(1 − δ + rt+1), (8)

limt→∞

βtkt+1

ptct= 0. (9)

The first condition restates the budget constraint, which has to hold with equality if the house-hold optimizes. The second condition is the standard Euler equation that governs the optimalallocation of consumption over time. Note that the left–hand side contains the relative pricesof consumption because quantities are expressed in units of the numeraire investment. The lastcondition is the transversality condition which says that in the limit the discounted marginal

8

utility of capital is zero.5

The definition of competitive equilibrium is standard: given prices, the allocation solves thehousehold problem; the allocation solves the firm problems; markets clear.

4.3 Balanced growth

It is well known that given this choice of units, the two–sector model has a balanced growthpath along which the Kaldor (1961) facts hold.6 To see why, note that (5) implies that the rentalprice for capital is constant if and only if kt grows at factor γx. So if we want a constant rt, thenwe need to impose that kt grows at factor γx. Then, (4) implies that yt grows at the same factorat kt, that is, γx. Using (1), constant growth of kt implies constant growth of xt. (4) implies thatptct also grows at factor γx. The Euler equation (8) then pins down r and k0:

ηγx = β (1 − δ + r) = β(1 − δ + θkθ−1

0

).

Given k0, the resource constraint implies a unique value for c0. The transversality conditionholds because:

limt→∞

βtkt+1

ptct=γxk0

p0c0limt→∞

βt = 0.

Hence, we have constructed a path along which the real interest is constant and all equilibriumconditions are satisfied. Along this equilibrium path, the Kaldor facts hold. Specifically, rt isconstant (Fact 5); yt and kt grow at the same constant factor (Facts 1 and 2); Hence kt/yt andrtkt/yt are constant (Facts 3 and 4).

In sum, we have just shown that:

Proposition 1 There is a unique balanced growth path of the two–sector model. The Kaldor

facts hold along the balanced growth path.

There are several noteworthy features of the balanced growth path. First, the shares ofsectoral capital and sectoral employment in total capital and total employment are constant.Second, kt and ct grow at different rates: kt grows at rate γx; ct grows at rate (1 +γx)θ (1 +γc)1−θ.Third, xt, kt and yt grow at the same rate and the relative prices of investment and capital areequal by construction.

5The precise statement of the transversality condition is that limt→∞(βtkt+1)/(ptct) ≤ 0. The equality signfollows because capital and marginal utility are non–negative.

6These are: the trend growth rates of GDP per worker and capital per worker are constant and the same; thereis no trend growth in the share of the payments to capital in GDP, the capital–output ratio, and the gross return oncapital.

9

4.4 Aggregation

Even in this simple two–sector version of the growth model, the problem arises that quantitygrowth measured in terms of a numeraire depends on the choice of the numeraire. When wederived the equilibrium conditions and the balanced growth path above, we defined quantitiesin terms of the numeraire xt. For this choice of units, GDP growth is:

γnum,xy,t+1 =

pt+1ct+1 + xt+1

ptct + xt= γx.

Since the choice of numeraire is arbitrary, we may also define quantities in terms of the nu-meraire ct. Combining (2) and (4), we then get:

p−1t yt = ct + p−1

t xt = p−1t kθt γ

t(1−θ)x = kθt γ

t(1−θ)c .

Going through the same steps as before, it is straightforward to show that GDP growth for thisdifferent choice of units is:

γnum,cy,t+1 =

ct+1 + p−1t+1xt+1

ct + p−1t xt

= γc.

In other words, the growth rate depends on the choice of the numeraire. Changing the numeraireimplies that the growth factor of real GDP changes from γx to γc. This is unsatisfactory, as thechoice of units ought to be irrelevant for the behavior of quantity growth.

The alternative to a numeraire is to measure aggregate quantities with index numbers; seeDiewert (2004) for further discussion. The simplest indexes are fixed–weight indexes. Thereare two natural choices for the fixed weights: either we evaluate quantities at the current–periodprices or the base–period prices, where the base period is an arbitrary past period. The former iscalled the Laspeyres index and the latter is called Paasche index. Unfortunately, these indexesalso generate different numbers for quantity growth. It is easy to see that if price changesare negatively correlated with quantity changes of the components, which usually the case,then quantity growth according to the Laspeyres index will be smaller than according to thePaasche index. This is the Gerschenkron effect: “early–weighted” quantity indexes are “biasedupwards” whereas “late–weighted” indexes are “biased downward”. The faster the relativeprice changes are the more severe this problem becomes.

The chain index solves the problem that the choice of units matters for quantity growth. Itis based on the Laspeyres and the Paasche quantity indexes for the current and the previousperiod:

γpaasy,t+1 =

pt+1ct+1 + xt+1

pt+1ct + xt,

γlaspy,t+1 =

ptct+1 + xt+1

ptct + xt.

10

Taking the geometric average gives chained growth of real GDP:

γchainy,t+1 =

√pt+1ct+1 + xt+1

pt+1ct + xt

ptct+1 + xt+1

ptct + xt.

Chain linking these indexes across many periods (“chaining”) implies the quantity growth be-tween any two periods. The resulting growth rates are independent from the choice of thenumeraire or the year in whose units we express the values of the aggregates in other periods(reference year).

Given the analytical tractability of our two–sector model, we can provide analytical resultsthat link GDP growth in terms of a numeraire to GDP growth in terms of the chain index alongthe balanced growth path:7

Proposition 2 Along the balanced growth path of the two–sector model, the growth of GDP in

terms of the numeraires ct or xt equals the growth of chained GDP times a constant correction

factor:

γchainy,t+1 = γnum,x

y,t+1

√(γc/γx)1−θp0c0 + x0

(γx/γc)1−θp0c0 + x0= γnum,c

y,t+1

√c0 + (γx/γc)1−θp−1

0 x0

c0 + (γc/γx)1−θp−10 x0

.

If γx > γc, then √(γc/γx)1−θ + x0/(p0c0)(γx/γc)1−θ + x0/(p0c0)

< 1 <

√c0 + (γx/γc)1−θp−1

0 x0

c0 + (γc/γx)1−θp−10 x0

,

γnum,cy,t+1 < γchain

y,t+1 < γnum,xy,t+1 .

Proof: See Appendix.

Note that the correction factor and chained GDP growth are both constant along the bal-anced growth path of the two–sector model. The intuitive reason is that along the balancedgrowth path, ptct and xt grow at the same rate and so the relative expenditures do not change:ptct/xt = p0c0/x0. As a result, the two–sector model has a balanced growth path for both waysof measuring the growth of aggregate quantities, because in both cases the growth of GDP andcapital is constant (and the ratios in current prices are also constant because they are the samein both cases).

To get a sense of the difference that the correction factors make, Table 1 reports the averageannual growth rates of GDP per hour in different units. The period is again 1947–2015. We can

7Note that it is equivalent whether we used the chain quantity index or the chain price index. The reason forthis is that is

γchainy·p,t+1 = γchain

y,t+1 · γchainp,t+1.

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Table 1: Annual growth rates of GDP per hour in different units (in %)

Units growth rate

chained prices 2.10investment 2.38consumption 1.96equipment 3.58structures 1.34

see that there are sizeable differences, suggesting that when we want to connect GDP growthfrom the model with the data, it is crucial to measure GDP growth in the same way in the modeland in the data. The natural approach to this is to measure GDP growth in the model throughthe use of chain indexes. This has the advantage that chained growth rates are independent ofthe choice of units. In contrast, as we argued above, growth rates in units of a numeraire arenot.

In sum, we have shown that disaggregating GDP into two components matters for calculat-ing GDP growth if the relative price of the components changes considerably. The next sectionexplores the same logic for the growth of the capital stock. The left panel of Figure 2 shows thatthe relative prices of structures and equipment have changed a lot. We therefore disaggregateinvestment and capital into structures and equipment and study how to calculate growth ratesin a three–sector version of the growth model akin to that of Greenwood et al. (1997).

5 Disaggregating Investment into Equipment and Structures

5.1 Model

There are three sectors that produce consumption and the two capital goods structures andequipment. Structures and equipment accumulate according to:

ηkbt+1 = (1 − δb)kbt + xbt, ηket+1 = (1 − δe)ket + xet,

where δb, δe ∈ [0, 1] are the two depreciation rates. Note that we use the index b for structures(“buildings”), because we reserve s for services later.

Structures, equipment, and labor are freely mobile among the sectors. The feasibility con-

12

Figure 2: The Relative Prices and Shares of Structures and Equipment

Pt=−12.04+0.007⋅t

Pt=27.75−00.014⋅t

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1950 1960 1970 1980 1990 2000 2010

Equipment Structures

Source: NIPA, Fixed Asset Tables, Bureau of Economic Analysis, own calculations

Price of equipment and structures relative to consumption

0.173

0.423

0.0

0.1

0.2

0.3

0.4

0.5

0.6

1950 1960 1970 1980 1990 2000 2010

Share in investment Share in capital

Source: NIPA, Bureau of Economic Analysis, own calculation

Share of equipment in total investment and capital at current prices

straints are: ∑j∈{b,e,c}

kb jt ≤ kbt,∑j∈{c,b,e}

ke jt ≤ ket,∑j∈{c,b,e}

n jt ≤ 1.

The sectors have Cobb–Douglas production functions that use structures, equipment, andlabor as inputs, have constant returns to scale, and have the same share parameters θb and θe.Figure 2 suggests the Cobb–Douglas functional form is a good approximation, as the incomeshares of structures and equipment remain almost constant although the relative prices changeconsiderably. Labor–augmenting technological progress is again denoted by γ j and is exoge-nous and sector specific.

Solving the firm problems in each sector and following the same steps as before, we canestablish that for each subcategory of capital the capital–labor ratios are the same in all sectors.Choosing equipment as the numeraire, the relative prices of consumption and buildings aregiven by:

pit =

(γe

γi

)t(1−θb−θe)

.

which, of course, is a generalization of equation (2) to three sectors. Figure 2 suggests that theempirically relevant case is γb < γc < γe. This case is often referred to as equipment–biasedtechnological progress.

13

Again, the production side aggregates:

yt = pctct + pbtxbt + xet = kθbbt k

θeetγ

t(1−θb−θe)e , (10)

rbt = θbyt

pbtkbt, (11)

ret = θeyt

ket, (12)

wt = (1 − θb − θe)kθbbt k

θeetγ

t(1−θb−θe)e , (13)

To avoid confusion, we should mention an important difference between our environmentand that of Greenwood et al. (1997). Using our notation, their production function was

yt = ct + xbt + xet = kθbbt k

θeetγ

t(1−θb−θe)e .

This production function does not include in GDP the effects of equipment–biased technologi-cal change on relative prices. To capture this, Greenwood et al. (1997) modified the accumula-tion equation of equipment:

ket+1 = (1 − δe)ket + qtxet,

where qt is inversely related to the relative price of equipment. In other words, the qualityimprovement of equipment was not measured in GDP but was treated as embodied technolog-ical change that leads to more (quality–adjusted) equipment capital given the same equipmentinvestment. This view of the world made sense when Greenwood et al. (1997) wrote theirstudy, because the quality improvements of equipment production were not well captured byNIPA. They therefore took the NIPA data at face value and used additional information fromGordon (1990) to calibrate qt. Since their study was written, the BEA has spent a great dealof effort to capture the implications of the quality improvements of equipment in the nationalaccount. Most observers agree that now it is preferable to write the production function andthe capital–accumulation equation as we have done above; see for example the discussion inWhelan (2003).

The intertemporal utility function is also as before, and so the household problem is:

max{ct ,kbt+1,ket+1}

∞t=0

∞∑t=0

βt log ct s.t. ptct + η(pbtkbt+1 + ket+1) = (1 − δb + rbt)pbtkbt + (1 − δe + ret)ket + wt,

ct, kbt+1, ket+1 ≥ 0, kb0 = kb0 > 0, ke0 = ke0 > 0.

14

Figure 3: Chained Growth of GDP and Capital per Hour

0.02

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

1950 1960 1970 1980 1990 2000 2010Source: NIPA, Bureau of Economic Analysis, own calculations

Growth rate of real GDP per hour

0.015

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

1950 1960 1970 1980 1990 2000 2010Source: NIPA, Bureau of Economic Analysis, own calculations

Growth rate of real capital stock per hour

The necessary conditions are:

ptct + η(pbtkbt+1 + ket+1) = (1 − δb + rbt)pbtkbt + (1 − δe + ret)ket + wt, (14)pct+1ct+1

pctct=β

η

(1 − δ + rbt+1)pbt+1

pbt=β

η(1 − δ + ret+1), (15)

limt→∞

βt pbtkbt+1

pctct= lim

t→∞

βtket+1

pctct= 0. (16)

5.2 Chain indexes and balanced growth

Again, we start by deriving a balanced growth path. Following similar steps as in the previoussection, we can show the following result:

Proposition 3 If the growth of real GDP per capita is measured in terms of the numeraire

equipment, then there is a unique balanced growth path of the three–sector model. The Kaldor

facts hold along the balanced growth path. Moreover, investment per capita, GDP per capita,

and capital per capita all grow at the same factor.

γ ≡ γθbb γ

1−θbe . (17)

Proof: See Appendix.

In this model version, the growth of GDP in terms of the numeraire yet and in terms of thechain index are:

γnum,ey,t+1 =

∑j∈{b,e,c} p jt+1y jt+1∑

j∈{b,e,c} p jty jt= γ,

γchainy,t+1 =

√∑j∈{b,e,c} p jt+1y jt+1∑

j∈{b,e,c} p jt+1y jt

∑j∈{b,e,c} p jty jt+1∑

j∈{b,e,c} p jty jt.

15

Following the same steps as for the proof of Proposition 2, it is straightforward to show ananalogous result to Proposition 2. Since capital has two components now, we state the resultfor the growth of both GDP and the capital stock:

Proposition 4 Along the balanced growth path of the three–sector model, growth of real GDP

in a numeraire equals chained growth of real GDP times a constant correction factor:

γchainy,t+1 = γnum,e

y,t+1

√(γc/γe)1−θb−θe + (γb/γe)1−θb−θe (pb0xb0)/(pc0c0) + xe0/(pc0c0)(γe/γc)1−θb−θe + (γe/γb)1−θb−θe (pb0xb0)/(pc0c0) + xe0/(pc0c0)

, (18)

γchaink,t+1 = γnum,e

k,t+1

√(γb/γe)1−θb−θe + ke0/(pb0kb0)(γe/γb)1−θb−θe + ke0/(pb0kb0)

. (19)

Comparing the growth rates of GDP and capital in terms of the chain index from the previ-ous proposition, it turns out that for plausible parameter values GDP grows more strongly thancapital along the balanced growth path:8

γchaink,t+1 < γ

chainy,t+1 .

To see this, we set θe+θb = 1/3. We start with (18). pct+1/pct = γe/γb = 1.0159, xbt/pctct = 0.13and xet/pctct = 0.09 on average over the period 1947–2015. This implies that γchain

y,t+1/γnum,ey,t+1 =

0.9899. Turning to (19), pbt+1/pbt = γe/γb = 1.0221 and ket/pbtkbt = 0.21 on average over theperiod 1947–2015. This implies that γchain

k,t+1/γnum,ek,t+1 = 0.9880. Since γnum,e

y,t+1 = γnum,ek,t+1 , it follows

that γchaink,t+1 < γ

chainy,t+1 .

At first sight it is worrying that the three–sector model contradicts the Kaldor facts by imply-ing that for plausible parameter values that GDP grows faster than the capital stock. However,looking at the evidence from the postwar U.S., we find that this is actually borne out by thedata: indeed the average growth rate of GDP per hour was half a percentage point larger thanthe average growth rate of capital per hour; see Figure 3. For the above numbers, Proposition 4implies a difference between capital and GDP per hour of 0.2 percentage points. So our modelaccounts for 40% of the observed difference in the growth rates of capital per hour and GDPper hour. In order to account for the rest, one would have to disaggregate capital further thanjust into structures and equipment.

Figure 3 also shows that the growth rates of GDP per hour and capital per hour both de-clined. In the next section, we will show that the growth rate of GDP per hour naturally declinesalong the balanced growth path of a structural change version of our benchmark model, which

8For a different version of the model of Greenwood et al. (1997), Whelan (2003) obtained a similar result.

16

generates Baumol disease as described in the introduction.9 To establish this, we disaggregateconsumption into goods and services and capture the fact that the relative price and the relativeexpenditures of the two change along the balanced growth path.

6 Disaggregating Consumption into Goods and Services

In this section, we study a three–sector version of the structural change model of Ngai andPissarides (2007), which highlights the role of relative prices behind structural change, butabstracts from income effects. While income effects play an important role in the context ofstructural change [Boppart (2016)], they are not crucial for the key points we want to makehere. Focusing on relative price effects then has the advantage that it permits simple analyticalderivations of all results.

6.1 Model

There are three sectors that produce consumption goods, consumption services, and investment.Capital and labor are freely mobile between the sectors. The feasibility constraints are:∑

j∈{g,s,x}

k jt ≤ kt,∑j∈{g,s,x}

n jt ≤ 1.

The sectors again have Cobb–Douglas production functions with an equal capital–share param-eter.

Following the same steps as before, we can show that the production side aggregates:

y jt = kθt γt(1−θ)j n jt, (20)

yt = xt + pgtcgt + pstcst = kθt γt(1−θ)x , (21)

rt = θkθ−1t γt(1−θ)

x , (22)

wt = (1 − θ)kθt γt(1−θ)x , (23)

where γx is exogenous, labor–augmenting technological progress in the investment sector andpgt and pst are the prices of consumption goods and services relative to investment. Again, the

9See Baumol (1967) for the initial contribution and Oulton (2001), Nordhaus (2008) and Baumol (2013) forrestatements of his observation.

17

Figure 4: Structural Change Facts

0.0

1.0

2.0

3.0

4.0

5.0

1950 1960 1970 1980 1990 2000 2010

Value added of services to goods Hours worked of services to goods

Source: WORLD KLEMS, April 2013 Release, own calculations

Relative nominal value added and hours worked

0.5

1.0

1.5

2.0

2.5

1950 1960 1970 1980 1990 2000 2010Source: WORLD KLEMS, April 2013 Release, own calculations

Prices of services relative to goods, 1947=1

relative prices satisfy:

pit =

(γx

γi

)t(1−θ)

. (24)

The evidence presented in Figure 4 suggests that γs < γg.Different from the previous section, Figure 4 suggests that now the expenditure share change

markedly when the relative prices change. To capture this, we use a CES aggregator for theperiod utility:

ct =

(ω

1εg c

ε−1ε

gt + ω1εs c

ε−1ε

st

) εε−1,

where ε ∈ (0, 1) is the elasticity of substitution and ωi are relative weights that are non–negativeand add up to one. Note that we restrict the elasticity of substitution to be between zero and one,which implies that goods and services are complements. Ngai and Pissarides (2007) argued thatthis is the empirically relevant case and Herrendorf et al. (2013) provided supporting evidence.

As shown by Herrendorf et al. (2014), the household problem can be split into two sub-problems. The intertemporal problem is as before, that is, allocate total income among thecomposite consumption good and savings. The static problem is new: allocate the period t

consumption expenditure ptct among goods and services. This representation separates growthfrom structural change. From the perspective of balanced growth in the aggregates kt and ct, therepresentation looks like the two–sector growth model from Section 4. From the perspectiveof structural change, the representation implies that we can focus on the solution to a staticproblem that allocates each period’s consumption expenditure between goods and services.

The intertemporal problem is:

max{ct ,kt+1}

∞t=0

∞∑t=0

βt log ct s.t. ptct + ηkt+1 = (1 − δ + rt)kt + wt, ct, kt+1 ≥ 0, k0 = k0 > 0.

Herrendorf et al. (2014) showed that the relative price of consumption is a weighted average of

18

the relative prices of consumed goods and consumed services:

pt ≡(ωg p1−ε

gt + ωs p1−εst

) 11−ε.

The necessary conditions are:

ptct + ηkt+1 = rtkt + wt + (1 − δ)kt, (25)pt+1ct+1

ptct=β

η(1 − δ + rt+1), (26)

limt→∞

βtkt+1

ptct= 0. (27)

The static problem is:

maxcgt ,cst

(ω

1εg c

ε−1ε

gt + ω1εs c

ε−1ε

st

) εε−1

s.t. pgtcgt + pstcst = ptct.

The first–order conditions are:

pgtcgt

pstcst=ωg

ωs

(pgt

pst

)1−ε

. (28)

(20), (24), and (28) imply the result of Ngai and Pissarides (2007):

pgtcgt

pstcst=

ngt

nst=ωg

ωs

(γs

γg

)t(1−θ)(1−ε)

. (29)

Since sectoral TFP grows faster in goods than services, γg > γs, and since goods and servicesare assumed to be complements, ε < 1, structural transformation happens and expenditureand labor are reallocated from goods to services as the economy grows. These patterns areconsistent with the evidence presented in Figure 4.

6.2 Chain indexes and the growth slowdown

We again start by deriving a balanced growth path along which the Kaldor facts hold:10

Proposition 5 If the growth of quantities is measured in terms of a numeraire, then there is

a unique balanced growth path of the three–sector model. The Kaldor facts hold along the

balanced growth path.

10Kongsamut et al. (2001) suggested to call this equilibrium path a “generalized” balanced growth path, becausea balanced growth path in the strict sense would have constant expenditure shares. We are somewhat sloppy hereand do not distinguish between the two.

19

Proof: The proof is the same as in the two–sector model. The key step is to split the three–sector model into the intertemporal and the static part. QED.

We argued in Subsection 4.4 above that measuring GDP growth through chain indexes ispreferable to a numeraire because only the growth rates of chained quantities are independentof the choice of base year. This point applies generally, and so it is relevant in the three–sectormodels as well. Measuring GDP growth through chain indexes has the additional advantage inthe context of structural change that it captures the behavior of the growth rate of welfare asmeasured by the consumption index. To establish this, we first characterize how welfare growthbehaves along the balanced growth path from before:

Proposition 6 The growth rate of the consumption index ct declines along the generalized bal-

anced growth path.

Proof: See Appendix.

The intuition for this result is that as consumption expenditure grow at a constant rate, thegrowth of ct slows if and only if the growth of pt as given by (6.1) accelerates. This happensbecause the share of services in total consumption expenditure increases at the same time aswhich the relative price of services increases. The reason for this, of course, is that goods andservices are complements.

Since the growth of GDP per capita is often used as a crude proxy for welfare improvements,one would hope that our measure of GDP growth picks up the slowdown in the consumptionindex along the balanced growth path. unfortunately, measuring growth in the units of a nu-meraire does not deliver this because it leads to constant GDP growth along the balanced growthpath. Fortunately, measuring growth in terms of the chain index is more successful in this re-gard. To build some intuition for why this is the case, note that chained GDP growth is nowdefined as:

γchainy,t+1 =

√∑j∈{g,s,x} p jt+1y jt+1∑

j∈{g,s,x} p jty jt

∑j∈{g,s,x} p jty jt+1∑j∈{g,s,x} p jt+1y jt

,

which is straightforward to rewrite to:

γchainy,t+1 = γx

√√∑i∈{g,s} (γi/γx)1−θ pit+1cit+1/(pt+1ct+1) + xt+1/(pt+1ct+1)∑

i∈{g,s} (γx/γi)1−θ pitcit/(ptct) + xt/(ptct).

This equation shows why chained GDP growth picks up the growth slowdown of the con-sumption index. All terms on the right–hand side are constant along the balanced growth pathexcept for (pitcit)/(ptct) and (pit+1cit+1)/(pt+1ct+1). They both change over time because struc-tural transformation increases the share of service consumption and decreases the share of good

20

consumption. Since services have a slower productivity growth than goods, the chain index cap-tures that this leads to Baumol disease and slows down GDP growth. A different way of puttingthis is that the correction factor between GDP growth in terms of a numeraire and in terms ofthe chain index now changes, because both the relative price and the relative expenditure ofgoods and service change. In contrast, in the previous two model version, the correction factorstayed constant because only the relative price changed.

Proposition 7 The chained growth rate of GDP per capita is given by:

γchainy,t+1 = γnum,e

y,t+1

√√√√√√√√(γg

γx

)1−θ−

[(γg

γx

)1−θ−

(γsγx

)1−θ]

ωsωg(γs/γg)(t+1)(1−θ)(1−ε)+ωs

+ x0p0c0(

γxγg

)1−θ+

[(γxγs

)1−θ−

(γxγg

)1−θ]

ωsωg(γs/γg)t(1−θ)(1−ε)+ωs

+ x0p0c0

. (30)

Chained GDP growth declines over time along the balanced growth path, implying that Kaldor

Fact 1 no longer holds. All other Kaldor facts still hold.

Proof: See Appendix.

While the result of Proposition 7 is an additional reason in favor of employing chain indexeswhen one wants to seriously think about the implications of structural transformation for wel-fare and growth, it also means that if one measures GDP growth with the chain index, then thebalanced growth path that we derived above no longer is a balanced growth path. This seemsto fly in the face of Kaldor’s observation that GDP per capita has grown at a constant trendrate. To resolve the resulting tension between this implication of the model and the Kaldorfacts more careful, it is useful to return to Figure 3. While evidence like the one in the Figureis often taken to imply that there is no change in the trend growth of GDP per hour, a moreaccurate representation is that the growth of GDP per hour slowed down in the second half ofthe sample. This is reflected by the fact that clearly the average growth rate was higher in thefirst part of the sample than in the second one.11 In other words, the implications of Proposition7 are entirely consistent with the evidence.

To get a sense of the difference between GDP growth calculated through a numeraire versusGDP growth calculated through chain indexes, we provide a back–of–the–envelope calculationof the correction factor (30). To do this, we take the final expenditure perspective. The averagesof the relevant variables during 1947–2015 were pgt+1/pgt = γx/γg = 0.9922, pst+1/pst =

γx/γs = 1.0093 and xt/(ptct) = 0.22. Using θ = 1/3 together with the expenditure share ofservice in 1947 (2015) of 0.49 (0.73), respectively, the correction factor equals 0.9997 in 1947and 0.9975 in 2015. If the 1947 (2015) correction factor is compounded over 68 years, then thatimplies 5% (16%) lower GDP per capita than with the model way. Since the correction factor

11The work of Antolin-Diaz et al. (2017) conducts a serious statistical analysis and confirms that the trend ofpostwar U.S. GDP underwent a structural break in the middle of the sample.

21

is declining over time, the 1947 factor implies the smallest and the 2015 implies the largestcorrection. Hence, the true correction factor implies a GDP per capita that is between 5% and16% lower than with the model way. To check for the plausibility of this model implication,we return to Table 1, remembering that GDP per capita and GDP per hour are the same in themodel. The table shows that the average growth rates of GDP per hour in units of the numeraireinvestment was 1.0224 and in chained prices was 1.0202. Hence, in the data, the correctionfactor was 0.9978, which implies a 14% lower GDP per hour over 68 years. It is reassuring thatthe range implied by our model includes the data number. Note that the previous back–of–the–envelope calculation nicely illustrates how tiny differences in the annual growth rate accumulateto sizeable differences in the levels of GDP over 68 years. Although this point is well known,it is easy to overlook and to think that correction factors of 0.9997 and 0.9975 are the same forall practical purposes. The previous calculation shows that this would be the wrong conclusionto draw.

The results of this section suggest that it might be necessary to rethink whether the firstKaldor Fact really holds for developed countries that massively reallocate their economic ac-tivity to the service sector which has slower than average productivity growth. This point isrelated to a tension that exists in the literature. On the one hand, one of the striking featuresof the basic, one–sector growth model is that it captures the long–run growth experience ofthe U.S. reasonable well, because prolonged periods of below average growth like the GreatDepression ultimately proved to be just deviations from the unchanged balanced growth path.On the other hand, it is impossible to overlook the fact that since World War II the averageGDP growth rate of the U.S. economy has slowly declined and there is no indication until nowthat it is about to return to the balanced growth path. Our analysis implies that this is alsowhat a standard multi–sector growth model implies. This raises the question about what sucha model has to say about the future growth slowdown. A particular worry is that the slowestgrowing service industries might take over the economy and thereby lead to permanently low,or even zero, growth of GDP per hour. We study this important question in a companion paper,Duernecker et al. (2017).

7 Conclusion

We have argued that it is generally preferable to measure aggregate quantities of multi–sectormodels through the chain index. We have shown that when relative prices and expenditureshares change only chained quantity growth is independent of the chosen units and captureshow equilibrium quantities and equilibrium welfare are affected.

We have established a general principle for moving between quantity aggregation in termsof a numeraire versus in terms of the chain index. If the relative price of the components of GDP

22

changes over time but the expenditure share stays roughly constant, then the correction factorbetween GDP growth in terms of a numeraire and the chain index differs from one and staysconstant. Since GDP growth in terms of a numeraire is constant along the balanced growthpath, this means that chained GDP growth is constant as well. In contrast, if the expenditureshares of the components of GDP change along with their relative prices, then the correctionfactor, and therefore chained GDP growth, changes over time. In our example of structuralchange, chained GDP growth therefore picks up that there was a growth slowdown which GDPgrowth in terms of a numeraire misses.

Our work is related to recent work that challenges some of the Kaldor facts. To beginwith, many students of economic growth suggested reasons for why the growth of GDP andproductivity may have slowed down since the 1980s; see Fernald and Jones (2014) for furtherdiscussion and Antolin-Diaz et al. (2017) for a statistical analysis. Our contribution to thisliterature is to show that Baumol disease is a force behind the growth slowdown. There is alsoa related literature that notes that, contrary to another Kaldor fact, the share of income paidto capital in total income seems to have increased considerably since around 2000. Examplesinclude Elsby et al. (2013), Karabarbounis and Neiman (2014), Koh et al. (2016) and Bridgman(2017). Since the income share of capital equals the payments to capital in current pricesdivided by total income in current prices, calculating it does not require an additional choice ofunits. Therefore, our analysis about the choice of units has no bearing on the important questionwhy the capital share might have increased.

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Appendix: Proofs

Proof of Proposition 2

Rewriting the definition of the chain index, we have:

γchainy,t+1 =

√pt+1ct+1 + xt+1

ptct + xt

ptct+1 + xt+1

pt+1ct + xt=

√pt+1ct+1 + xt+1

ptct + xt

(pt/pt+1)pt+1ct+1 + xt+1

(pt+1/pt)ptct + xt.

Using that both ptct and xt growth at factor γx, we get:

γchainy,t+1 = γx

√(pt/pt+1)ptct + xt

(pt+1/pt)ptct + xt.

Using (2) and again that both ptct and xt growth at factor γx gives:

γchainy,t+1 = γx

√(γc/γx)1−θp0c0 + x0

(γx/γc)1−θp0c0 + x0.

25

QED.

Proof of Proposition 3

(11) and (12) imply that the rental rates for capital are constant if and only if yt grows at thesame factor as ket and pbtkbt. The capital accumulation equations imply that the growth factorsof two investments are the same as those of the capital stocks. Hence, both xet and pbtxbt growat the same factor as yt. (10) implies that this is possible if and only if pctct grows at the samefactor as yt, xet and pbtxbt, which we call γ.

Using that pbtxbt grows at factor γ and that

pbt =

(γe

γb

)t(1−θb−θe)

it follows that kbt and xbt grow at factor γ(γb/γe)1−θb−θe . To determine γ, we substitute theseresults into (10) and take the ratio of two adjacent periods to find:

γ = γθb

(γb

γe

)(1−θb−θe)θb

γθeγ1−θb−θee .

Solving for γ gives (17).All other steps are as in the proof of Proposition 1 and are omitted. QED.

Proof of Proposition 6

We know that along the balanced growth path ptct grows at factor γx. To show the claim, itsuffices to show that the growth of pt accelerates.

Recalling the definition of the price index, we have

pt =(ωg p1−ε

gt + ωs p1−εst

) 11−ε . (31)

This implies:(pt+1

pt

)1−ε

=ωg p1−ε

gt+1 + ωs p1−εst+1

ωg p1−εgt + ωs p1−ε

st=

ωg p1−εgt

ωg p1−εgt + ωs p1−ε

st

(pgt+1

pgt

)1−ε

+ωs p1−ε

st

ωg p1−εgt + ωs p1−ε

st

(pst+1

pst

)1−ε

.

The relative weights on the relative price changes add up to one, implying that

ωg p1−εg

ωg p1−εg + ωs p1−ε

s= 1 −

ωs p1−εs

ωg p1−εg + ωs p1−ε

s.

26

Using this, we get:(pt+1

pt

)1−ε

=

(pgt+1

pgt

)1−ε

+

( pst+1

pst

)1−ε

−

(pgt+1

pgt

)1−ε ωs p1−εst

ωg p1−εgt + ωs p1−ε

st.

Using (24), this can be rewritten as:(pt+1

pt

)1−ε

=

(γx

γg

)(1−θ)(1−ε)

+

(γx

γs

)(1−θ)(1−ε)

−

(γx

γg

)(1−θ)(1−ε) ωs

ωg(γs/γg)t(1−θ)(1−ε) + ωs.

Since ε ∈ (0, 1) and γg > γs the right–hand side is increasing over time. QED.

Proof of Proposition 7

We first prove that the other Kaldor Facts still hold. xt grows at factor γx along the generalizedbalanced growth path from above. Since investment is the numeraire, that fact is not affectedby what price index we use. kt/yt = θ/rt and rtkt/yt = θ constant along the generalized balancedgrowth path from above. The ratios are calculated from quantities in current prices, so chainingdoes not matter. rt is constant along the generalized balanced growth path from above. rt is unitfree, so chaining does not matter.

The proof that fact (1) no longer holds is as follows. Chained GDP growth can be writtenas:

γchainy,t+1 =

√pt+1ct+1 + xt+1

ptct + xt

∑i∈{g,s}(pit/pit+1)pit+1cit+1 + xt+1∑

i∈{g,s}(pit+1/pit)pitcit + xt.

Using (24), this can be rewritten as:

γchainy,t+1 =

√√γx

pt+1ct+1

ptct

∑i∈{g,s} (γi/γx)1−θ pit+1cit+1/(pt+1ct+1) + xt+1/(pt+1ct+1)∑

i∈{g,s} (γx/γi)1−θ pitcit/(ptct) + xt/(ptct)

= γx

√√∑i∈{g,s} (γi/γx)1−θ pit+1cit+1/(pt+1ct+1) + xt+1/(pt+1ct+1)∑

i∈{g,s} (γx/γi)1−θ pitcit/(ptct) + xt/(ptct)

= γx

√√√√ (γg/γx)1−θ +[(γs/γx)1−θ − (γg/γx)1−θ

]pst+1cst+1/(pt+1ct+1) + x0/(p0c0)

(γx/γg)1−θ +[(γx/γs)1−θ − (γx/γg)1−θ

]pstcst/(ptct) + x0/(p0c0)

.

To rewrite this further, note that (29) implies that:

pstcst

ptct=

ωsγt(1−θ)(ε−1)s

ωgγt(θ−1)(ε−1)g + ωsγ

t(1−θ)(ε−1)s

=ωs

ωg(γs/γg)t(1−θ)(1−ε) + ωs.

27

Combining the last two equations gives:

γchainy,t+1 = γx

√√√√√√√√(γg

γx

)1−θ−

[(γg

γx

)1−θ−

(γsγx

)1−θ]

ωsωg(γs/γg)(t+1)(1−θ)(1−ε)+ωs

+ x0p0c0(

γxγg

)1−θ+

[(γxγs

)1−θ−

(γxγg

)1−θ]

ωsωg(γs/γg)t(1−θ)(1−ε)+ωs

+ x0p0c0

.

Given that γs < γg < γx, it is clear that γchainy,t+1 < γx for all t. Moreover, given that empirically

ε ∈ (0, 1), γchainy,t+1 is declining. In the limit,

limt→∞

γchainy,t+1 = γx

√(γs/γx)1−θ + x0/(p0c0)(γx/γs)1−θ + x0/(p0c0)

< γx.

QED.

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