Quantity Measurement and Balanced Growth in Multi Sector Growth Models

Transcription

1 Quantity Measurement and Balanced Growth in Multi Sector Growth Models Georg Duernecker University of Munich, IZA, and CEPR) Berthold Herrendorf Arizona State University) Ákos Valentinyi University of Manchester, CERS HAS, and CEPR) December 5, 2017 Abstract Multi sector models typically rely on a numeraire to aggregate quantities whereas NIPA uses the chain index For three popular versions of the multi sector growth model, we provide analytical expressions for the growth of aggregate quantities under both measurement methods and establish that the compound differences are sizeable over long horizons We show that using the chain index captures more accurately the aggregate effects of secular changes 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 welfare 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 and suggestions Valentinyi thanks the Hungarian National Research, Development and Innovation Office Project KJS K ) All errors are our own

2 1 Things to add Quote Oulton 2007): What GHK viewed as investment specific technological change can be modeled/viewed as differences between sectoral TFP growth If quality improvements are properly measured, then the two views are observationally equivalent We follow Oulton s formulation Quote Moro 2015): Also pointed out that measuring GDP through chain indexes may make a difference 2 Introduction The one sector growth model has become a workhorse model of macroeconomics, which captures in a simple and tractable way the essence of modern economic growth The simplicity of the one sector version of the growth model necessarily implies that it abstracts from relevant features of reality like the secular movements in the relative prices and expenditure shares of subcategories of GDP To capture the aggregate implications of such movements, the profession has pursued several routes of disaggregation The resulting multi sector versions of the growth model present the researcher with the challenge of how to aggregate sectoral quantities to 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 calibrated to the postwar US economy We will start analyzing a two sector benchmark version with investment and consumption of the model of Uzawa 1963) We will then turn to two models that disaggregate this benchmark further into three sector versions of the models of Greenwood et al 1997) and Ngai and Pissarides 2007) The usual way of aggregating quantities within the context of these multi sector models is to define quantities in the unit of a numeraire from the same 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 result establishes that it matters quantitatively which measurement method we choose For the post war US economy they generate vastly different numbers for the growth of capital and GDP per hour On top of that, quantity growth in terms of the chain index is independent of the units in which the quantity levels are expressed, whereas quantity growth in terms of a numeraire depends on the choice of the numeraire Our second result establishes an additional advantage of using the chain index, namely, that it captures more accurately the economy wide effects of secular changes in relative prices We show for a version of the Ngai Pissarides model 1

3 of structural transformation that Baumol s disease reduces welfare growth and that quantity growth measured in terms of the chain index captures this whereas quantity growth measured in terms of a numeraire misses it Our third result offers correction factors for each model version that make it straightforward to move from quantity growth in terms of a numeraire to quantity growth in terms of the chain index These correction factors are helpful because it is typically easier to construct a balanced growth path when quantities are measured in terms of a numeraire Connecting the balanced growth path to the data therefore requires moving to quantity growth in terms of the chain index We show that the correction factors are constant along the balanced growth path if and only if the expenditure shares of sectoral output do not change when relative prices change Turning now to the details of the models, we first develop a two sector version of the growth model with investment and consumption that captures the implications of the secular decline in the price of investment relative to consumption While this model mainly serves as a point of departure for further disaggregation into three sector models, on its own it is already useful to illustrate an important difference between the two measurement methods: only when GDP growth 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 balanced growth path along which aggregate variables and sectoral outputs grow at the same rate while prices and sectoral expenditure shares remain constant But the growth rate of GDP per hour along the balanced growth path does depend on the choice of the numeraire Connecting the model with the postwar US economy, we find that the resulting differences in GDP growth are sizeable 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 of the units in which the corresponding levels are measured We derive correction factors that link GDP growth in terms of the different measurement methods to each other We show that given that along the balanced growth path the relative expenditures on consumption and investment are constant, these correction factors are constant as well Connecting the two sector model to the postwar US economy, we find that compounding the correction factors leads to sizeable differences in the level of GDP per hour Next, we analyze a version of the three sector version of the growth model developed by Greenwood et al 1997) This model version distinguishes between structures, equipment, and consumption and assesses the implications of the secular decline in the relative price of equipment Consistent with the data, we assume that structures experience the slowest and equipment experiences the fastest technological progress, with consumption in the middle We start by characterizing the balanced growth path of the model when quantities are measured in units of a numeraire We show that along the balanced growth path, the growth rates of capital and output per hour are the same and constant We then switch to measuring quantities through 2

4 the chain index We show that this brings about a key change along the same balanced growth path from before: while the growth rate of capital per hour is still constant, it is now smaller than the growth rate of GDP per hour We again derive the correction factors between the different measurement methods and show that they are constant along the balanced growth path The reason is the same as in the two sector model, namely, the relative expenditures on the different investment goods are constant along the balanced growth path Connecting this three sector growth model with the postwar US economy, we find that compounding the correction factors again 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 hour grows more slowly along the balanced growth path than GDP per hour must be at odds with the evidence After all, Kaldor 1961) included in his growth facts the claim that the growth of the two is the same When we look at postwar US economy, however, we find that this is a myth: the average growth rate of chained capital was about half a percentage point smaller than the average growth rate of chained GDP The intuitive reason, of course, is that the capital stock contains a larger share of structures, and a smaller share of equipment, than investment and GDP Since structures depreciate more slowly than equipment, the growth of capital is below the growth of investment and GDP Using a chain index to aggregate the components of the capital 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 by Ngai and Pissarides 2007) This model has become the benchmark for studying how differences in the pace of technological progress at the sector level lead to changes in the relative price of sectoral output, which in turn lead to the reallocation of economic activity from the goods sectors to the service sectors structural transformation ) 1 When we measure quantities in units of a numeraire, we obtain the usual balanced growth path along which the growth of economy wide quantities is constant while economic activity is reallocated underneath from the goods sector to the service sector 2 When we switch to measuring quantities through the chain index, we obtain the novel result that the growth of GDP per hour declines along the same balanced growth path This implies that the balanced growth path that we derived in terms of the a numeraire no longer is a balanced growth path in terms of the chain index, because the growth of GDP per hour no longer is constant This also implies that now the correction factor between the two ways of measuring GDP per hour changes over time, which reflects that both relative prices and relative expenditures change along the balanced growth path Lastly, we show that when we connect the model to the postwar US economy, then the correction factor again has quantitatively sizeable implications The model implication that the chained growth of GDP per hour is slowing over time, in- 1 See Herrendorf et al 2014) for a recent review article of the literature on structural transformation 2 Since 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

5 stead of staying constant, is at odds with the first of the Kaldor growth facts that over long horizons the growth of GDP per hour is constant While at first sight this implication of the model might appear to be a problem, there is actually some evidence that is consistent with it We document for the postwar US economy that there has been a secular decline of average growth of GDP per hour While this is often ignored by growth theorists who instinctively focus on a balanced growth path, this growth slowdown is widely documented; see for example the recent paper by Byrne et al 2016) That there has been a growth slowdown is also consistent with the observation of Baumol 1967) that structural transformation slows down GDP growth because it reallocates economic activity to the service industries which tend to have smaller than average productivity growth Baumol s disease ) Given that we have a tractable model of structural transformation at hand, we can go further than Baumol did and show that welfare growth where welfare is measured by the utility index) also slows down along the usual balanced growth path constructed in terms of a numeraire Since the growth of GDP per hour is often used as a proxy for welfare improvements, one would hope that it picks up the slowdown in this welfare measure Our results imply that measuring quantities with the chained index delivers this whereas measuring quantities with a numeraire misses it This probably is the strongest argument in favor of employing the chain index for measuring economy wide quantities in multi sector models The rest of the paper is organized as follows In the next section, we review several closely related papers In Section 4, we develop a two sector benchmark model with consumption and investment In Section 5 and 6, we study the three sector version that result when we disaggregate 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 macroeconomics Whelan 2002) provided an insightful discussion of the pitfalls associated with using chain indexes in the context of the national accounts Whelan 2003) discussed how to calibrate the steady state of a two sector growth model in the spirit of Greenwood et al 1997) using divisia approximations of the chained index While Whelan 2003) shares with our paper the interest in the issues that arise when one seeks to connect a multi sector version of the growth model to NIPA, the focus of the two papers is different He was interested in calibrating a version of the model of Greenwood et al 1997) to the national accounts in such a way that he measures model quantities through chain indexes or approximations to them) in the same way as it is done in the national accounts In contrast, we are interested in understanding the differences between measuring quantities in units of a numeraire versus in terms of the chain index 4

6 in the context of multi sector models This is relevant because the common practice in the literature on structural change is to use a numeraire, instead of the chain index, when constructing aggregate quantities The novelty of our work is that we provide analytical expressions for the growth of economy wide quantities that results under both measurement methods, derive analytical expressions for the correction factors between them, and assess how large they are quantitatively for different version of the multi sector growth model An important part of our paper 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), also realized that structural change leads to a reduction in GDP growth when real GDP is calculated in terms of constant prices instead of a numeraire 3 Our paper adds two important points to this insight First, we prove that along the balanced growth path of a three sector version of the model of Ngai and Pissarides, both welfare and chained GDP growth slow down Second, we establish for empirically plausible parameter values that the resulting GDP growth slowdown has 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 growth of GDP per hour in units of a numeraire and proceeds to derive a balanced growth path; see Herrendorf et al 2014) and the references therein To avoid misunderstandings, we do not suggest to change this way of proceeding when the goal is to solve the model and characterize an equilibrium path in the mode convenient way This statement is particularly true for models of structural change in which a balanced growth path exists only if quantities are measured in units of a numeraire, like the model of Ngai and Pissarides Instead, our point here is that one needs to be careful when connecting the model to the national accounts and when interpreting the properties of that balanced growth path in the face of changing relative prices, and possibly relative expenditures It is then preferable to aggregate quantities by using the chain index We hope that the correction factors we derive will help in this context, and make moving between using 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 and investment in the spirit of Uzawa 1963) Our goal throughout the paper is to keep things as standard as possible We therefore heavily lean on the exposition in Herrendorf et al 2014) 3 Subsequently, Boppart 2016) also picked up on this insight 4 In 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

7 41 Environment There is a measure one of identical households Since we are interested in matching model quantities to the data, we allow population growth and assume that each household has N t = η t members where η 0, ) Hence, the population is N t and N 0 = 1 is normalized Each household member is endowed with the initial capital stock K 0 > 0 and with one unit of time in each period The period utility is of the log form This is an analytically convenient special case of the constant relative risk aversion functional form, which is required for balanced growth The intertemporal utility function is then given by: t=0 ) β t Ct N t log = N t β t log c t ) where β βη 0, 1) is a modified discount factor If η > 1, then β must be sufficiently smaller than one to ensure that β < 1 We use lower case letters to denote per capita variables, so c t C t /N t 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 be equal to per worker and per hour quantities We therefore do not distinguish among them and refer to all model variables that are deflated by N t as per capita variables In per capita terms, capital accumulates according to: t=0 ηk t+1 = 1 δ)k t + x t 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 total income In per capita terms, the production functions are: c t = k θ ctγ t cn ct ) 1 θ, x t = k θ xtγ t xn xt ) 1 θ where θ 0, 1) is the capital share parameter; k it and n it are sectoral capital and labor per capita eg, n it N it /N t ); γ i 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 the production side aggregates Note too that sectoral TFP at time 0 equals one: γ 01 θ) i = 1 This corresponds to a choice of units and is without loss of generality 6

8 Figure 1: The Relative Prices of Investment 16 Price of investment relative to consumption P t = t Source: NIPA, Fixed Asset Tables, Bureau of Economic Analysis, own calculations Capital and labor are freely mobile between the two sectors The feasibility constraints are: k ct + k xt k t, n ct + n xt 1 The welfare theorems will hold in the versions of the growth model we employ Since the usual notion of balanced growth and the Kaldor Facts involve prices, we will nonetheless solve for competitive equilibrium 42 Competitive equilibrium The firm problem in each sector is to minimize costs subject to producing a given quantity of sectoral output We choose investment as the numeraire The first order conditions then are: ) θ 1 kct r t = p t θ γ t1 θ) kxt c = θ n ct n xt ) θ kct w t = p t 1 θ) γc t1 θ) = 1 θ) n ct ) θ 1 γ t1 θ) x, kxt n xt ) θ γ t1 θ) x where r t and w t are the rental prices for capital and labor and p t is the relative price of consumption to investment These first order conditions imply that the relative price of consumption is inversely related to the sectoral TFPs: p t = γx γ c ) t1 θ) 2) Figure 1 suggests that the empirically relevant case is γ x > γ c This case is often referred to as investment biased technological progress The production side of this two sector environment aggregates To see this, we divide the 7

9 first order conditions by each other to obtain the usual result that the capital labor ratios are equalized: This implies that for i {c, x} Combining 2) and 3), we get: k t = k ct n ct = k xt n xt y it = k θ t γ t1 θ) i n it 3) y t = x t + p t c t = k θ t γ t1 θ) x, 4) r t = θkt θ 1 γ t1 θ) x, 5) w t = 1 θ)k θ t γ t1 θ) x 6) Note that if one wants to think of 4) as a technology in the strict sense of the concept, then it is preferable to eliminate the relative price and write it as: x t + γx γ c ) t1 θ) c t = k θ t γ t1 θ) x The household problem is to maximize utility subject to the budget constraint and several constraints on the choice sets Since leisure does not generate utility here, the household allocates its entire time endowment to working Imposing this, the household problem can be written as: max {c t,k t+1 } t=0 β t log c t st p t c t + ηk t+1 1 δ + r t )k t + w t, c t, k t+1 0, k 0 = k 0 > 0 t=0 Solving this problem gives the following necessary conditions for optimality: p t c t + ηk t+1 = r t k t + w t + 1 δ)k t, 7) p t+1 c t+1 = β p t c t η 1 δ + r t+1), 8) β t k t+1 lim = 0 t p t c t 9) The first condition restates the budget constraint, which has to hold with equality if the household optimizes The second condition is the standard Euler equation that governs the optimal allocation of consumption over time Note that the left hand side contains the relative prices of consumption because quantities are expressed in units of the numeraire investment The last condition is the transversality condition which says that in the limit the discounted marginal 8

10 utility of capital is zero 5 The definition of competitive equilibrium is standard: given prices, the allocation solves the household problem; the allocation solves the firm problems; markets clear 43 Balanced growth It is well known that given this choice of units, the two sector model has a balanced growth path along which the Kaldor 1961) facts hold 6 To see why, note that 5) implies that the rental price for capital is constant if and only if k t grows at factor γ x So if we want a constant r t, then we need to impose that k t grows at factor γ x Then, 4) implies that y t grows at the same factor at k t, that is, γ x Using 1), constant growth of k t implies constant growth of x t 4) implies that p t c t also grows at factor γ x The Euler equation 8) then pins down r and k 0 : ηγ x = β 1 δ + r) = β ) 1 δ + θk θ 1 0 Given k 0, the resource constraint implies a unique value for c 0 The transversality condition holds because: β t k t+1 lim = γ xk 0 lim β t = 0 t p t c t p 0 c 0 t Hence, we have constructed a path along which the real interest is constant and all equilibrium conditions are satisfied Along this equilibrium path, the Kaldor facts hold Specifically, r t is constant Fact 5); y t and k t grow at the same constant factor Facts 1 and 2); Hence k t /y t and r t k t /y t 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 of sectoral capital and sectoral employment in total capital and total employment are constant Second, k t and c t grow at different rates: k t grows at rate γ x ; c t grows at rate 1 + γ x ) θ 1 + γ c ) 1 θ Third, x t, k t and y t grow at the same rate and the relative prices of investment and capital are equal by construction 5 The precise statement of the transversality condition is that lim t β t k t+1 )/p t c t ) 0 The equality sign follows because capital and marginal utility are non negative 6 These are: the trend growth rates of GDP per worker and capital per worker are constant and the same; there is no trend growth in the share of the payments to capital in GDP, the capital output ratio, and the gross return on capital 9

11 44 Aggregation Even in this simple two sector version of the growth model, the problem arises that quantity growth measured in terms of a numeraire depends on the choice of the numeraire When we derived the equilibrium conditions and the balanced growth path above, we defined quantities in terms of the numeraire x t For this choice of units, GDP growth is: γ num,x y,t+1 = p t+1c t+1 + x t+1 p t c t + x t = γ x Since the choice of numeraire is arbitrary, we may also define quantities in terms of the numeraire c t Combining 2) and 4), we then get: p 1 t y t = c t + p 1 t x t = p 1 t kt θ γ t1 θ) x = kt θ γc t1 θ) Going through the same steps as before, it is straightforward to show that GDP growth for this different choice of units is: γ num,c y,t+1 = c t+1 + p 1 t+1 x t+1 c t + p 1 t x t = γ c In other words, the growth rate depends on the choice of the numeraire Changing the numeraire implies that the growth factor of real GDP changes from γ x to γ c This is unsatisfactory, as the choice 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; see Diewert 2004) for further discussion The simplest indexes are fixed weight indexes There are two natural choices for the fixed weights: either we evaluate quantities at the current period prices or the base period prices, where the base period is an arbitrary past period The former is called the Laspeyres index and the latter is called Paasche index Unfortunately, these indexes also generate different numbers for quantity growth It is easy to see that if price changes are 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 the Paasche index This is the Gerschenkron effect: early weighted quantity indexes are biased upwards whereas late weighted indexes are biased downward The faster the relative price changes are the more severe this problem becomes The chain index solves the problem that the choice of units matters for quantity growth It is based on the Laspeyres and the Paasche quantity indexes for the current and the previous period: γ paas y,t+1 = p t+1c t+1 + x t+1 p t+1 c t + x t, γ lasp y,t+1 = p tc t+1 + x t+1 p t c t + x t 10

12 Taking the geometric average gives chained growth of real GDP: y,t+1 = pt+1 c t+1 + x t+1 p t+1 c t + x t p t c t+1 + x t+1 p t c t + x t Chain linking these indexes across many periods chaining ) implies the quantity growth between any two periods The resulting growth rates are independent from the choice of the numeraire 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 results that link GDP growth in terms of a numeraire to GDP growth in terms of the chain index along the 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 c t or x t equals the growth of chained GDP times a constant correction factor: y,t+1 = γnum,x y,t+1 γ c /γ x ) 1 θ p 0 c 0 + x 0 = γ num,c c 0 + γ x /γ c ) 1 θ p 1 γ x /γ c ) 1 θ y,t+1 p 0 c 0 + x 0 0 x 0 c 0 + γ c /γ x ) 1 θ p 1 0 x 0 If γ x > γ c, then γ c /γ x ) 1 θ + x 0 /p 0 c 0 ) γ x /γ c ) 1 θ + x 0 /p 0 c 0 ) < 1 < γ num,c y,t+1 c 0 + γ x /γ c ) 1 θ p 1 0 x 0 c 0 + γ c /γ x ) 1 θ p 1 0 x, 0 < γchain y,t+1 < γnum,x y,t+1 Proof: See Appendix Note that the correction factor and chained GDP growth are both constant along the balanced growth path of the two sector model The intuitive reason is that along the balanced growth path, p t c t and x t grow at the same rate and so the relative expenditures do not change: p t c t /x t = p 0 c 0 /x 0 As a result, the two sector model has a balanced growth path for both ways of measuring the growth of aggregate quantities, because in both cases the growth of GDP and capital is constant and the ratios in current prices are also constant because they are the same in both cases) To get a sense of the difference that the correction factors make, Table 1 reports the average annual growth rates of GDP per hour in different units The period is again We can 7 Note that it is equivalent whether we used the chain quantity index or the chain price index The reason for this is that is y p,t+1 = γchain y,t+1 γchain p,t+1 11

13 Table 1: Annual growth rates of GDP per hour in different units in %) Units growth rate chained prices 210 investment 238 consumption 196 equipment 358 structures 134 see that there are sizeable differences, suggesting that when we want to connect GDP growth from the model with the data, it is crucial to measure GDP growth in the same way in the model and in the data The natural approach to this is to measure GDP growth in the model through the use of chain indexes This has the advantage that chained growth rates are independent of the choice of units In contrast, as we argued above, growth rates in units of a numeraire are not In sum, we have shown that disaggregating GDP into two components matters for calculating GDP growth if the relative price of the components changes considerably The next section explores the same logic for the growth of the capital stock The left panel of Figure 2 shows that the relative prices of structures and equipment have changed a lot We therefore disaggregate investment and capital into structures and equipment and study how to calculate growth rates in a three sector version of the growth model akin to that of Greenwood et al 1997) 5 Disaggregating Investment into Equipment and Structures 51 Model There are three sectors that produce consumption and the two capital goods structures and equipment Structures and equipment accumulate according to: ηk bt+1 = 1 δ b )k bt + x bt, ηk et+1 = 1 δ e )k et + x et, where δ b, δ e [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

14 Figure 2: The Relative Prices and Shares of Structures and Equipment 16 Price of equipment and structures relative to consumption Share of equipment in total investment and capital at current prices P t = t P t = t Equipment Structures Source: NIPA, Fixed Asset Tables, Bureau of Economic Analysis, own calculations Share in investment Share in capital Source: NIPA, Bureau of Economic Analysis, own calculation straints are: k b jt k bt, j {b,e,c} j {c,b,e} j {c,b,e} k e jt k et, n jt 1 The sectors have Cobb Douglas production functions that use structures, equipment, and labor as inputs, have constant returns to scale, and have the same share parameters θ b and θ e Figure 2 suggests the Cobb Douglas functional form is a good approximation, as the income shares of structures and equipment remain almost constant although the relative prices change considerably Labor augmenting technological progress is again denoted by γ j and is exogenous and sector specific Solving the firm problems in each sector and following the same steps as before, we can establish 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 are given by: p it = γe γ i ) t1 θb θ e ) which, of course, is a generalization of equation 2) to three sectors Figure 2 suggests that the empirically relevant case is γ b < γ c < γ e This case is often referred to as equipment biased technological progress 13

15 Again, the production side aggregates: y t = p ct c t + p bt x bt + x et = k θ b bt kθ e et γ t1 θ b θ e ) e, 10) y t r bt = θ b, 11) p bt k bt r et = θ e y t k et, 12) w t = 1 θ b θ e )k θ b bt kθ e et γ t1 θ b θ e ) e, 13) To avoid confusion, we should mention an important difference between our environment and that of Greenwood et al 1997) Using our notation, their production function was y t = c t + x bt + x et = k θ b bt kθ e et γ t1 θ b θ e ) e This production function does not include in GDP the effects of equipment biased technological change on relative prices To capture this, Greenwood et al 1997) modified the accumulation equation of equipment: k et+1 = 1 δ e )k et + q t x et, where q t is inversely related to the relative price of equipment In other words, the quality improvement of equipment was not measured in GDP but was treated as embodied technological change that leads to more quality adjusted) equipment capital given the same equipment investment This view of the world made sense when Greenwood et al 1997) wrote their study, because the quality improvements of equipment production were not well captured by NIPA They therefore took the NIPA data at face value and used additional information from Gordon 1990) to calibrate q t Since their study was written, the BEA has spent a great deal of effort to capture the implications of the quality improvements of equipment in the national account Most observers agree that now it is preferable to write the production function and the capital accumulation equation as we have done above; see for example the discussion in Whelan 2003) The intertemporal utility function is also as before, and so the household problem is: max {c t,k bt+1,k et+1 } t=0 β t log c t st p t c t + ηp bt k bt+1 + k et+1 ) = 1 δ b + r bt )p bt k bt + 1 δ e + r et )k et + w t, t=0 c t, k bt+1, k et+1 0, k b0 = k b0 > 0, k e0 = k e0 > 0 14

16 Figure 3: Chained Growth of GDP and Capital per Hour 008 Growth rate of real GDP per hour 008 Growth rate of real capital stock per hour Source: NIPA, Bureau of Economic Analysis, own calculations Source: NIPA, Bureau of Economic Analysis, own calculations The necessary conditions are: p t c t + ηp bt k bt+1 + k et+1 ) = 1 δ b + r bt )p bt k bt + 1 δ e + r et )k et + w t, 14) p ct+1 c t+1 p ct c t β t p bt k bt+1 lim t p ct c t = β 1 δ + r bt+1 )p bt+1 = β η p bt η 1 δ + r et+1), 15) = lim t β t k et+1 p ct c t = 0 16) 52 Chain indexes and balanced growth Again, we start by deriving a balanced growth path Following similar steps as in the previous section, 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 Proof: See Appendix γ γ θ b b γ1 θ b e 17) In this model version, the growth of GDP in terms of the numeraire y et and in terms of the chain index are: γ num,e y,t+1 y,t+1 = = j {b,e,c} p jt+1 y jt+1 = γ, j {b,e,c} p jt y jt j {b,e,c} p jt+1 y jt+1 j {b,e,c} p jt+1 y jt j {b,e,c} p jt y jt+1 j {b,e,c} p jt y jt 15

17 Following the same steps as for the proof of Proposition 2, it is straightforward to show an analogous result to Proposition 2 Since capital has two components now, we state the result for 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: y,t+1 = γnum,e y,t+1 k,t+1 = γnum,e k,t+1 γ c /γ e ) 1 θ b θ e + γ b /γ e ) 1 θ b θ e p b0 x b0 )/p c0 c 0 ) + x e0 /p c0 c 0 ) γ e /γ c ) 1 θ b θ e + γ e /γ b ) 1 θ b θ e p b0 x b0 )/p c0 c 0 ) + x e0 /p c0 c 0 ), 18) γ b /γ e ) 1 θ b θ e + k e0 /p b0 k b0 ) γ e /γ b ) 1 θ b θ e + k e0 /p b0 k b0 ) 19) Comparing the growth rates of GDP and capital in terms of the chain index from the previous proposition, it turns out that for plausible parameter values GDP grows more strongly than capital along the balanced growth path: 8 k,t+1 < γchain y,t+1 To see this, we set θ e +θ b = 1/3 We start with 18) p ct+1 /p ct = γ e /γ b = 10159, x bt /p ct c t = 013 and x et /p ct c t = 009 on average over the period This implies that y,t+1 /γnum,e y,t+1 = Turning to 19), p bt+1 /p bt = γ e /γ b = and k et /p bt k bt = 021 on average over the period This implies that k,t+1 /γnum,e k,t+1 = Since γ num,e y,t+1 = γ num,e k,t+1, it follows that k,t+1 < γchain y,t+1 At first sight it is worrying that the three sector model contradicts the Kaldor facts by implying that for plausible parameter values that GDP grows faster than the capital stock However, looking at the evidence from the postwar US, we find that this is actually borne out by the data: indeed the average growth rate of GDP per hour was half a percentage point larger than the average growth rate of capital per hour; see Figure 3 For the above numbers, Proposition 4 implies a difference between capital and GDP per hour of 02 percentage points So our model accounts for 40% of the observed difference in the growth rates of capital per hour and GDP per hour In order to account for the rest, one would have to disaggregate capital further than just into structures and equipment Figure 3 also shows that the growth rates of GDP per hour and capital per hour both declined In the next section, we will show that the growth rate of GDP per hour naturally declines along the balanced growth path of a structural change version of our benchmark model, which 8 For a different version of the model of Greenwood et al 1997), Whelan 2003) obtained a similar result 16

18 generates Baumol disease as described in the introduction 9 To establish this, we disaggregate consumption into goods and services and capture the fact that the relative price and the relative expenditures 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 and Pissarides 2007), which highlights the role of relative prices behind structural change, but abstracts from income effects While income effects play an important role in the context of structural change [Boppart 2016)], they are not crucial for the key points we want to make here Focusing on relative price effects then has the advantage that it permits simple analytical derivations of all results 61 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} j {g,s,x} k jt k t, n jt 1 The sectors again have Cobb Douglas production functions with an equal capital share parameter Following the same steps as before, we can show that the production side aggregates: y jt = k θ t γ t1 θ) j n jt, 20) y t = x t + p gt c gt + p st c st = k θ t γ t1 θ) x, 21) r t = θkt θ 1 γ t1 θ) x, 22) w t = 1 θ)k θ t γ t1 θ) x, 23) where γ x is exogenous, labor augmenting technological progress in the investment sector and p gt and p st are the prices of consumption goods and services relative to investment Again, the 9 See Baumol 1967) for the initial contribution and Oulton 2001), Nordhaus 2008) and Baumol 2013) for restatements of his observation 17

19 Figure 4: Structural Change Facts 50 Relative nominal value added and hours worked 25 Prices of services relative to goods, 1947= Value added of services to goods Hours worked of services to goods Source: WORLD KLEMS, April 2013 Release, own calculations Source: WORLD KLEMS, April 2013 Release, own calculations relative prices satisfy: p it = γx The evidence presented in Figure 4 suggests that γ s < γ g γ i ) t1 θ) 24) 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 the period utility: c t = ω 1 εg c ε 1 ε gt ) + ω 1 ε s c ε 1 ε ε 1 ε st, where ε 0, 1) is the elasticity of substitution and ω i are relative weights that are non negative and 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 that this 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 subproblems The intertemporal problem is as before, that is, allocate total income among the composite consumption good and savings The static problem is new: allocate the period t consumption expenditure p t c t among goods and services This representation separates growth from structural change From the perspective of balanced growth in the aggregates k t and c t, the representation looks like the two sector growth model from Section 4 From the perspective of structural change, the representation implies that we can focus on the solution to a static problem that allocates each period s consumption expenditure between goods and services The intertemporal problem is: max {c t,k t+1 } t=0 β t log c t st p t c t + ηk t+1 = 1 δ + r t )k t + w t, c t, k t+1 0, k 0 = k 0 > 0 t=0 Herrendorf et al 2014) showed that the relative price of consumption is a weighted average of 18

20 the relative prices of consumed goods and consumed services: The necessary conditions are: p t ω g p 1 ε gt ) + ω s p 1 ε 1 1 ε st The static problem is: p t c t + ηk t+1 = r t k t + w t + 1 δ)k t, 25) p t+1 c t+1 = β p t c t η 1 δ + r t+1), 26) β t k t+1 lim = 0 t p t c t 27) max ω 1 ε g c ε 1 ε gt c gt,c st The first order conditions are: ) + ω 1 ε s c ε 1 ε ε 1 ε st st p gt c gt + p st c st = p t c t p gt c gt p st c st = ω g ω s pgt 20), 24), and 28) imply the result of Ngai and Pissarides 2007): p st ) 1 ε 28) p gt c gt p st c st = n gt n st = ω g ω s γs γ g ) t1 θ)1 ε) 29) Since sectoral TFP grows faster in goods than services, γ g > γ s, and since goods and services are assumed to be complements, ε < 1, structural transformation happens and expenditure and labor are reallocated from goods to services as the economy grows These patterns are consistent with the evidence presented in Figure 4 62 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 10 Kongsamut et al 2001) suggested to call this equilibrium path a generalized balanced growth path, because a balanced growth path in the strict sense would have constant expenditure shares We are somewhat sloppy here and do not distinguish between the two 19

21 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 44 above that measuring GDP growth through chain indexes is preferable to a numeraire because only the growth rates of chained quantities are independent of the choice of base year This point applies generally, and so it is relevant in the three sector models as well Measuring GDP growth through chain indexes has the additional advantage in the context of structural change that it captures the behavior of the growth rate of welfare as measured by the consumption index To establish this, we first characterize how welfare growth behaves along the balanced growth path from before: Proposition 6 The growth rate of the consumption index c t declines along the generalized balanced growth path Proof: See Appendix The intuition for this result is that as consumption expenditure grow at a constant rate, the growth of c t slows if and only if the growth of p t as given by 61) accelerates This happens because the share of services in total consumption expenditure increases at the same time as which the relative price of services increases The reason for this, of course, is that goods and services 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 consumption index along the balanced growth path unfortunately, measuring growth in the units of a numeraire does not deliver this because it leads to constant GDP growth along the balanced growth path Fortunately, measuring growth in terms of the chain index is more successful in this regard To build some intuition for why this is the case, note that chained GDP growth is now defined as: y,t+1 = which is straightforward to rewrite to: y,t+1 = γ x j {g,s,x} p jt+1 y jt+1 j {g,s,x} p jt y jt j {g,s,x} p jt y jt+1 j {g,s,x} p jt+1 y jt, i {g,s} γ i /γ x ) 1 θ p it+1 c it+1 /p t+1 c t+1 ) + x t+1 /p t+1 c t+1 ) i {g,s} γ x /γ i ) 1 θ p it c it /p t c t ) + x t /p t c t ) This equation shows why chained GDP growth picks up the growth slowdown of the consumption index All terms on the right hand side are constant along the balanced growth path except for p it c it )/p t c t ) and p it+1 c it+1 )/p t+1 c t+1 ) They both change over time because structural transformation increases the share of service consumption and decreases the share of good 20

22 consumption Since services have a slower productivity growth than goods, the chain index captures that this leads to Baumol disease and slows down GDP growth A different way of putting this is that the correction factor between GDP growth in terms of a numeraire and in terms of the chain index now changes, because both the relative price and the relative expenditure of goods and service change In contrast, in the previous two model version, the correction factor stayed constant because only the relative price changed Proposition 7 The chained growth rate of GDP per capita is given by: γg ) 1 θ [ γg ) 1 θ y,t+1 = γ x γ x γs γnum,e y,t+1 ) 1 θ ] γ x γx ) 1 θ [ γ g + γx ) 1 θ γ s γx ) 1 θ ] γ g ω s ω g γ s /γ g ) t+1)1 θ)1 ε) +ω s + x 0 p 0 c 0 ω s ω g γ s /γ g ) t1 θ)1 ε) +ω s + x 0 p 0 c 0 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 indexes when one wants to seriously think about the implications of structural transformation for welfare and growth, it also means that if one measures GDP growth with the chain index, then the balanced growth path that we derived above no longer is a balanced growth path This seems to fly in the face of Kaldor s observation that GDP per capita has grown at a constant trend rate To resolve the resulting tension between this implication of the model and the Kaldor facts more careful, it is useful to return to Figure 3 While evidence like the one in the Figure is often taken to imply that there is no change in the trend growth of GDP per hour, a more accurate representation is that the growth of GDP per hour slowed down in the second half of the sample This is reflected by the fact that clearly the average growth rate was higher in the first part of the sample than in the second one 11 In other words, the implications of Proposition 7 are entirely consistent with the evidence To get a sense of the difference between GDP growth calculated through a numeraire versus GDP growth calculated through chain indexes, we provide a back of the envelope calculation of the correction factor 30) To do this, we take the final expenditure perspective The averages of the relevant variables during were p gt+1 /p gt = γ x /γ g = 09922, p st+1 /p st = γ x /γ s = and x t /p t c t ) = 022 Using θ = 1/3 together with the expenditure share of service in ) of ), respectively, the correction factor equals in 1947 and in 2015 If the ) correction factor is compounded over 68 years, then that implies 5% 16%) lower GDP per capita than with the model way Since the correction factor 11 The work of Antolin-Diaz et al 2017) conducts a serious statistical analysis and confirms that the trend of postwar US GDP underwent a structural break in the middle of the sample 21

23 is declining over time, the 1947 factor implies the smallest and the 2015 implies the largest correction Hence, the true correction factor implies a GDP per capita that is between 5% and 16% 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 the model The table shows that the average growth rates of GDP per hour in units of the numeraire investment was and in chained prices was Hence, in the data, the correction factor was 09978, which implies a 14% lower GDP per hour over 68 years It is reassuring that the 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 accumulate to 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 and are the same for all practical purposes The previous calculation shows that this would be the wrong conclusion to draw The results of this section suggest that it might be necessary to rethink whether the first Kaldor Fact really holds for developed countries that massively reallocate their economic activity to the service sector which has slower than average productivity growth This point is related to a tension that exists in the literature On the one hand, one of the striking features of the basic, one sector growth model is that it captures the long run growth experience of the US reasonable well, because prolonged periods of below average growth like the Great Depression 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 average GDP growth rate of the US economy has slowly declined and there is no indication until now that it is about to return to the balanced growth path Our analysis implies that this is also what a standard multi sector growth model implies This raises the question about what such a model has to say about the future growth slowdown A particular worry is that the slowest growing 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 sector models through the chain index We have shown that when relative prices and expenditure shares change only chained quantity growth is independent of the chosen units and captures how equilibrium quantities and equilibrium welfare are affected We have established a general principle for moving between quantity aggregation in terms of a numeraire versus in terms of the chain index If the relative price of the components of GDP 22