A Two-Sector Approach to Modeling U.S. NIPA Data

A Two-Sector Approach to Modeling U.S. NIPA Data Karl Whelan Division of Research and Statistics Federal Reserve Board(cid:3) April, 2001 Abstract The one-sector Solow-Ramsey model is the most popular model of long-run economic growth. This paper argues that a two-sector approach, in which technological progress in the production of durable goods exceeds that in the rest of the economy, provides a far better picture of the long-run behavior of the U.S. economy. The paper shows how to use the two-sector approach to model the real chain-aggregated variables currently featured in the U.S. National Income and Product Accounts. It is shown that each of the major chain-aggregates|output, consumption, investment, and capital stock|will tend in the long-run to grow at steady, but di(cid:11)erent, rates. Implications for empirical analysis based on these data are explored. (cid:3)MailStop89, 20thandCStreetsNW,Washington DC20551. Email: kwhelan@frb.gov. Iam gratefultoChadJones,MichaelKiley,SteveOliner,MichaelPalumbo,andDanSichelforcomments on an earlier draft, and to John Fernald, Jonas Fisher, and Kevin Stiroh for helpful exchanges on the issues covered in this paper. The views expressed are my own and do not necessarily reflect the views of the Board of Governors or thesta(cid:11) of theFederal ReserveSystem.

Since the 1950s, the Solow-Ramsey model of economic growth, which treats all output in the economy as deriving from a single aggregate production function, has been the canonical model of how the macroeconomy evolves in the long run. The model has also featured prominently in the analysis of economic fluctuations: Business cycles are commonly characterized as correlated deviations from the model’s long-run \balanced growth" path, which features the real aggregates for consumption, investment, output, and the capital stock, all growing together at an identical rate determined by the growth rate of the aggregate technology process. The main purpose of this paper is to make a simple point: Despite its central roleineconomicstextbooksandinbusinesscycleresearch,thetraditionalone-sector model of economic growth actually provides a poor description of the long-run behavior of the U.S. economy. A simple alternative two-sector model, in which technological progress in the production of durable goods exceeds that in the rest of the economy, explains a number of crucial long-run properties of U.S. macroeconomic datathatareinconsistent withtheone-sector growth model,andis farbettersuited for modeling these data as currently constructed. As such, this two-sector approach provides a better \baseline" model for macroeconomic analysis. Inarguingthecaseforatwo-sectorapproachtolong-runmodelingofthemacroeconomy, this paper re-enforces the principal message a recent paper by Jeremy Greenwood, Zvi Hercowitz, and Per Krusell (1997), which also employed a model in which technological progress in the production of durable goods is faster than in the rest of the economy. However, this paper di(cid:11)ers from the work of Greenwood, Hercowitz, and Krusell in using a two-sector model to (cid:12)t macroeconomic data as published in the U.S. National Income and Product Accounts (NIPAs). As a result, the approach taken heredi(cid:11)ers both in the evidence cited in favor of a twosector model, and in its treatment of aggregation. This approach produces some new results on the properties of the real NIPA aggregates that have far-reaching implications for empirical work in macroeconomics. Concerning the evidence for a two-sector approach, Greenwood, Hercowitz, and Krusell argued that the o(cid:14)cial price deflators for durable equipment in the NIPAs understate the relative price decline for these products, and consequently also underestimate the di(cid:11)erential in sectoral rates of technological progress. Thus, their work was not based on NIPA data, but instead featured the equipment price in- 1

dex developed by Robert Gordon (1990). More recently, much of the research on price mis-measurement has focused on non-manufacturing sectors, so this approach of adjusting only for price mis-measurement for durable equipment seems likely to overstate the true pace of relative price decline for equipment, and thus the case for a two-sector approach. However, I document that|once we include evidence from the 1990s|then a compelling case for the two-sector framework can be made using the o(cid:14)cial data from the NIPAs. The failure of the one-sector approach to (cid:12)t published NIPA data is important, because most researchers use these data for empirical work in macroeconomics, and much of this research relies on the assumption that the one-sector growth model provides an adequate approximation to the long-run behavior of these data. For example,researchersofteninvokethebalancedgrowthpropertytodemonstratethat theirempiricalspeci(cid:12)cationshavesoundlong-runproperties. However, thefactthat most investment spending is on durable goods, while most consumption outlays are on nondurable goods and services, implies that the balanced growth prediction is now (cid:12)rmly rejected, a (cid:12)nding that overturns an often-cited earlier result (based on a sample through 1988) of King, Plosser, Stock, and Watson (1991). The second new aspect of this paper, the approach to aggregation, requires a little more explanation. In a one-sector world, all output has the same price. In this case, aggregate real output, as measured by weighting quantities according to a (cid:12)xed set of base-year prices, should be independent of the choice of base year and should grow at a steady rate in the long-run. In reality, because the durable goods sector has faster growth in real output and a declining relative price, the growth rate of a (cid:12)xed-weight measure of real GDP will depend on the choice of base year: The further back we choose the base year, the higher the current growth rate will be, soreal aggregate outputmeasuredin this mannerwill always tend to accelerate. Thereareanumberofpossiblesolutionstothisproblemofbase-yeardependence in (cid:12)xed-weight measures of growth in aggregate real output. This paper follows the approach taken by the U.S. Commerce Department’s Bureau of Economic Analysis (BEA) in constructing the NIPAs. The BEA abandoned the (cid:12)xed-weight approach in 1996, switching to measuring all real aggregates using a \chain index" method, which uses continually updated relative price weights. I follow this procedure in modelingthemajorrealaggregates usingachain-weighting method. Thetwo-sector 2

model yields a number of important insights into the long-run properties of U.S. NIPAdata. Inparticular,Ishowhoweachofthemajorchain-aggregatedvariables| output, consumption, investment, and capital stock|will tend in the long-run to grow at steady, but di(cid:11)erent, rates. This result has important implications for empirical analysis based on these data. Anothersolutiontotheproblemofbase-yeardependentrealoutputgrowthisthe one chosen byGreenwood, Hercowitz, and Krusell, whichis to useonecategory as a numeraire, and de(cid:12)nerealoutputbydeflating allnominaloutputbythe priceindex for this category, which in their case was consumer nondurables and nonhousing services. Because the nominal output of the durable goods sector tends to grow at the same rate as other nominal output, this measure of aggregate real output can exhibit steady-state growth. Greenwood et al argue that this measure is superior to the measure of real output featured in the NIPAs. However, the position taken in this paper is that neither method produces an intrinsically \correct" measure of aggregate real output, but rather that the two approaches merely measure di(cid:11)erent concepts. The rest of the paper is organized as follows. Section 1 uses U.S. NIPA data to document the case against the one-sector model and in favor of a two-sector approach. Section 2 presents the two-sector model. Section 3 uses the model to (cid:12)t aggregate U.S. data constructed according to the current chain-aggregation procedures, and discusses the model’s implications for the analysis of these data. Section 4calculates thecontribution oftechnological progressinthedurablegoodssector to growth in aggregate output and welfare. Finally, Section 5 compares our approach with that of Greenwood, Hercowitz, and Krusell. 1 Evidence All data used in this section come from the U.S. National Income and Products Accounts. 1.1 A Look at the Data Thestandardneoclassicalgrowthmodelstartswithanaggregateresourceconstraint 1−(cid:12) (cid:12) of the form C +I = Y = A L K . Because all goods are produced using the t t t t t t 3

same technology, a decentralized market equilibrium must feature consumption and investment goods having the same price. Thus, in the standard expression of the resource constraint, the variables C;I, and Y have all been deflated by an index for this common price to express them in quantity or real terms. Themodelusuallyassumesthatarepresentativeconsumermaximizesthepresent discounted value of utility from consumption, subject to a law of motion for capital and a process for aggregate technology. If the technology process is of the form logA t = a+logA t−1 +(cid:15) t , where (cid:15) t is a stationary zero-mean series, then it is well known that the model’s solution features C;I;K; and Y all growing at an average rate of a in the long-run. This means that ratios of any of these variables will be 1−(cid:12) stationary stochastic processes. This hypothesis of balanced growth and stationary \great ratios" has been held as a crucial stylized fact in macroeconomics at least as far back as the well-known contribution of Kaldor (1957). More recently, King, Plosser, Stock, and Watson (1991), using data through 1988, presented evidence that the real series for investment and consumption (and consequently output) share a common stochastic trend, implying stationarity of the ratio of real investment to real consumption. Figure 1 shows, however, that including data through 2000:Q2 (cid:12)rmly undermines the evidence for the balanced growth hypothesis. Since 1991, investment has risen dramatically in real terms relative to consumption, growing an average of 8.4 percent per year over the period 1992-99, 4.7 percentage points per year faster than consumption. Indeed, applying a simple cyclical adjustment by comparing peaks, there is some evidence that the ratio of real investment to real consumption has exhibited an upward trend since the late 1950s, with almost every subsequent cyclical peak setting a new high. It might be suspected that the strength of investment since 1991 has been the result of some special elements in the current expansion such as its unusual length, the surge in equity valuations, or perhaps the \crowding in" associated with the reduction of Federal budget de(cid:12)cits. If this were the case, then we would expect the ratio of real investment to real consumption to decline to its long-run average with the next recession. However, while the factors just cited have likely played some role in the remarkable strength in investment, a closer examination reveals that something more fundamental, and less likely to be reversed, has also been at 4

work. A standard disaggregation of the data shows that the remarkable increase in real investment in the 1990s was entirely due to outlays on producers’ durable equipment, which grew 9.1 percent per year; real spending on structures grew only 1 2.2 percent per year. Moreover, Figure 2a shows that this pattern of growth in real equipment spending outpacing growth in structures investment is a long-term one, dating back to about 1960. Importantly, this (cid:12)gure also documents that the samepatternisevidentwithintheconsumptionbundle: Realconsumerspendingon durable goods has consistently grown faster than real consumption of nondurables and services. These facts are summarized jointly by Figure 2b, which shows that, over the past 40 years, total real output of durable goods has consistently grown faster than total real business output (de(cid:12)ned as GDP excluding the output of government and nonpro(cid:12)t institutions). The faster growth in theU.S. economy’s real outputof durablegoods reflects an ongoing decline in their relative price. Remarkably, as can be seen from Figure 3a, the share of durable goods in nominal business output has exhibited no trend at all overthepast50years. Similarly,theincreaseinreal(cid:12)xedinvestmentrelativetoreal consumption has been a function of declining prices for durable goods and the fact the such goods are a more important component of investment than consumption: Figure3bshowsthat, onceexpressedinnominalterms,theratioof(cid:12)xedinvestment 2 to consumption has also been trendless throughout the postwar period. Figure 4 illustrates the relevant relative price trends. Relative prices for durable goods,ofbothconsumerandproducerkind,havetrendeddownsincethelate1950s. Thedecline for equipment did not show through to the deflator for (cid:12)xed investment until about 1980 because the relative price of structures, which show little trend over the sample as a whole, rose between 1960 and 1980 and fell thereafter. Viewed from this perspective, the post-1991 increase in real investment relative torealconsumptiondoesnotappearasaparticularlycyclicalphenomenon. Instead, it reflects long-running trends. Over the long run, nominal spending on investment 1Tevlin and Whelan (2001) provide a detailed discussion of the behavior of U.S. equipment investment in the1990s. 2Over 1960-1999, durable goods accounted for 13 percent of consumption expenditures and 47 percent of investment expenditures. 5

and consumption have tended to grow at the same rate. But the higher share of durable goods in investment, and the declining relative price of these goods, together imply that real investment tends to grow faster than real consumption. Two other factors|the increase in the relative price of structures from the early 1960s until 1980, and then the decline in the nominal share of investment in the 1980s| somewhat obscured this trend until the 1990s. However, over the entire postwar period, these two variables have been stationary. Unless those patterns change going forward, the ratio of real investment to real consumption should continue to trend up. Formal statistical tests con(cid:12)rm the intuition suggested by these graphs, that the real ratios are non-stationary and that the nominal ratios are stationary. An Augmented Dickey-Fuller (ADF) unitroot test for the ratio of real (cid:12)xed investment to real consumption gives a t-statistic of -0.78, meaning we cannot come close to 3 rejecting the hypothesis that this series has a unit root. The t-statistic for the corresponding nominal ratio is -2.99, which rejects the unit root hypothesis at the 5 percent level. Similarly, the ADF t-statistic for the ratio of real durables output to real business output is -0.05; the corresponding nominal ratio has a t-statistic of -4.03, rejecting the unit root hypothesis at the 1 percent level. 1.2 Some Price-Measurement Issues Before moving on to describe a model to (cid:12)t the patterns just documented, a discussion of the data is appropriate. In previous work aimed at replacing the onesector approach with a model containing multiple technology shocks, Greenwood, Hercowitz, and Krusell (1997) cited Robert Gordon’s (1990) evidence that price inflation for durable goods is overstated in the NIPAs because valuable quality improvements are often ignored. Their analysis replaced the o(cid:14)cial price deflator for equipment investment with the alternative index developed by Gordon, but used the NIPA price deflators for all other categories. The preceding discussion shows that it is not necessary to apply this type of adjustment to make the case against the one-sector approach (note this is partly 3This test is based on a regression of the variable on its lagged level, an intercept, and lagged (cid:12)rst-di(cid:11)erences. The number of lags used (two) was chosen according to the general-to-speci(cid:12)c proceduresuggestedbyCampbellandPerron(1991),buttheresultswerenotparticularlysensitive to this choice. 6

because we used a sample period that extends ten years beyond the sample of Greenwoodetal,whichendedin1990.) However,forsomeofthecalculationswewill performlater, suchasthecontributionoftechnological progressintheproductionof durable goods to aggregate output growth, the accuracy of the answers will depend on having the correct series for the relative price of durable goods, which raises the question of whether we should apply these price adjustments. The approach taken here will be to use the o(cid:14)cial data for all calculations. The reason for this is that much of the recent evidence on price mis-measurement, including some of the work of the Boskin Commission, has focused on prices for 4 non-manufacturing output. While durable goods are one of the areas where some of the adjustments for quality suggested by Gordon have actually been adopted by the NIPAs (most notably for computers), there are serious questions about the adequacy of the output deflators for sectors such as (cid:12)nancial services and construction. Measured productivity growth in these sectors has been extremely weak and seems likely to have been underestimated. Given that the nominal output series are generally considered to be relatively accurate, these considerations suggest that price 5 inflation in these non-manufacturing industries may be overstated. In the absence of convincing evidence pointing one way or the other for adjusting the relative price of durable goods, I have chosen not to apply any such adjustment. 2 The Two-Sector Model We have documented three important stylized facts about the evolution of the U.S. economy over the past 40 years: (cid:15) Therealoutputofthedurablegoodssectorhasgrownsigni(cid:12)cantlyfasterthan the real output of the rest of the economy. (cid:15) Because durable goods are a more important component of investment than consumption, real investment has grown faster than real consumption. (cid:15) These trends reflect relative price movements: The share of durable goods in nominal output has been stable, as have the shares of nominal consumption and investment. 4SeeBoskin et al (1998). 5SeeCorrado and Slifman (1999) and Gullickson and Harper(1999). 7

Clearly, if we are to explain the trend in the relative price of durable goods, we need a two-sector approach that distinguishes these goods separately from other output. Thissection develops, andempiricallycalibrates, asimpletwo-sector model that(cid:12)tsthefactsjustdescribedbyallowingforafasterpaceoftechnologicalprogress 6 in the sector producing durable goods. The model presented here shares a number of common features with that of Greenwood, Hercowitz, and Krusell (1997) in that it focuses on a two-sector economy with one sector having faster technological progress than the other. One difference worth noting is that I focus on the durable goods sector as a whole as the sector with faster technological progress, rather than limiting the focus to the production of producer durables. This is necessary if we wish to (cid:12)t the facts about real expenditures on consumer durables, and this aspect of the model will be important for the welfare calculations presented later in the paper. Also, as noted above, the empirical calibration and treatment of aggregation in this paper are very di(cid:11)erent from that in the work of Greenwood et al. 2.1 Technology and Preferences The model economy has two sectors. Sector 1 produces durable equipment used by both consumers and producers, while Sector 2 produces for consumption in the form of nondurables and services and for investment in the form of structures. The notation to describe this is as follows: Sector i supplies C units of its consumption i good to households, I units of its capital good for purchase by sector j, and keeps ij I unitsof its capital good for itself. Theproductiontechnologies in the two sectors ii are identical apart from the fact that technological progress advances at a di(cid:11)erent pace in each sector: Y1 = C1+I11+I12 = A1L 1 1 −(cid:12)1 −(cid:12)2K 1 (cid:12) 1 1K 2 (cid:12) 1 2 (1) Y2 = C2+I21+I22 = A2L 1 2 −(cid:12)1 −(cid:12)2K 1 (cid:12) 2 1K 2 (cid:12) 2 2 (2) (cid:1)logA = a +(cid:15) (3) it i it 6Thereisplentyofavailableevidencethatpointstowardsafasterrateoftechnologicalprogressin the durable goods sector. For example, the detailed Multifactor Productivity (MFP) calculations published by the Bureau of Labor Statistics reveal durable manufacturing as the sector of the economy with thefastest MFP growth. Seehttp://stats.bls.gov/mprhome.htm. 8

Equipment and structures depreciate at di(cid:11)erent rates, so capital of type i used in production in sector j accumulates according to (cid:1)K ij;t = I ij;t −(cid:14) i K ij;t−1 (4) To keep things as simple as possible, we will not explicitly model the labor-leisure allocation decision, instead assuming a (cid:12)xed labor supply normalized to one L1+L2 = 1 (5) and that households maximize the expected present discounted value of utility " (cid:18) (cid:19) # X1 1 k−t E t ((cid:11)1logD t +(cid:11)2logC2t ) (6) 1+(cid:26) k=t Here, D is the stock of consumer durables, which evolves according to (cid:1)D t = C1t −(cid:14)1D t−1 (7) 2.2 Steady-State Growth Rates Wewillfocusonthesteady-stateequilibriumgrowthpathimpliedbyadeterministic version of the model in which A1 and A2 grow at constant, but di(cid:11)erent rates; in a full stochastic solution, all deviations from this path will be stationary. In this steady-state equilibrium, the real output of sector i grows at rate g . We will also i show below that the fraction of each sector’s output sold to households, to Sector 1, and to Sector 2, are all constant, and that L1 and L2 are both (cid:12)xed. Along a steady-state growth path, if investment in a capital good grows at rate g then the capital stock for that good will also grow at rate g. Consequently, taking log-di(cid:11)erences of (1) and (2), we get g1 = a1+(cid:12)1g1+(cid:12)2g2 (8) g2 = a2+(cid:12)1g1+(cid:12)2g2 (9) The solutions to these equations are (1−(cid:12)2)a1+(cid:12)2a2 g1 = 1−(cid:12)1 −(cid:12)2 (10) (cid:12)1a1+(1−(cid:12)1)a2 g2 = 1−(cid:12)1 −(cid:12)2 (11) so the growth rate of real output in sector 1 exceeds that in sector 2 by a1 −a2. 9

2.3 Competitive Equilibrium We could solve for the model’s competitive equilibrium allocation by formulating a central planning problem, but our focus on (cid:12)tting the facts about nominal and real ratios makes it useful to solve for the prices that generate a decentralized equilibrium. In doing so I assume the following structure. Firms in both sectors are perfectly competitive, so they take prices as given and make zero pro(cid:12)ts. Householdscantradeo(cid:11)consumptiontodayforconsumptiontomorrowbyearninginterest income on savings. This takes the form of passing savings to arbitraging intermediaries who purchase capital goods, rent them out to (cid:12)rms, and then pass the return on these transactions back to households. Relative Prices: Because relative prices, rather than the absolute price level, are what matter for the model’s equilibrium, we will set sector 2’s price equal to one in all periods, and denote sector 1’s price by p . Given a wage rate, w, and rental t rates for capital, c1 and c2, the cost function for (cid:12)rms in Sector i is (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) Y i w 1−(cid:12)1 −(cid:12)2 c1 (cid:12)1 c2 (cid:12)2 TC i (w;c1;c2;Y i ;A i ) = A i 1−(cid:12)1 −(cid:12)2 (cid:12)1 (cid:12)2 Prices equal marginal cost, so the relative price of sector 1’s output is A2 p = (12) A1 Combinedwith(10)and(11), thistellsusthattheratioofsector1’snominaloutput to sector 2’s nominal output is constant along the steady-state growth path: The decline in the relative price of durable goods exactly o(cid:11)sets the faster increase in quantity. We can also show that the fraction of each sector’s output devoted to consumption and investment goods is constant. Together, these results imply that the ratios of all nominal series are constant along the steady-state growth path. Clearly, this result is a direct consequence of our restrictive choice of log-linear preferences and technology, but it (cid:12)ts with the evidence just presented. So,ourmodeliscapableof(cid:12)ttingthestylizedfactsaboutthestabilityofnominal ratios, and di(cid:11)erential growth rates for the real output of the durable goods sector and the rest of the economy. To calibrate the model empirically to match the behavior of aggregate real NIPA series, we will also need to derive the steady-state 10

values ofthenominalratios of consumption,capital, investment, andoutputof type 1 relative to their type-2 counterparts, as functions of the model’s parameters. Consumption: Denominating household income and (cid:12)nancial wealth in terms of the price of sector 2’s output, the dynamic programming problem for the representative household is (cid:20) (cid:21) Max 1 V t (Z t ;D t−1) = C1t;C2t (cid:11)1logD t +(cid:11)2logC2t + E t V t+1(Z t+1;D t ) 1+(cid:26) subject to the accumulation equation for the stock of durables and the condition governing the evolution of (cid:12)nancial wealth: Z t+1 = (1+r t+1)(Z t +w t −p t C1t −C2t ) Here r is the real interest rate de(cid:12)ned relative to the price of sector 2’s output. The (cid:12)rst-order condition for type-2 consumption and the envelope condition for (cid:12)nancial wealth combine to give a standard Euler equation: 1 1 1 = E t [1+r t+1] (13) C2t C2;t+11+(cid:26) So, using a log-linear approximation, the steady-state real interest rate is r = g2+(cid:26) (14) Some additional algebra (detailed in an appendix) produces the following equation p(g1+(cid:14)1+(cid:26))D (cid:11)1 = (15) C2 (cid:11)2 Also, as noted earlier, the steady-state solution involves the stock of durables growingatrateg1,sothatC1 = (g1+(cid:14)1)D byequation(7). Substitutingthisexpression for D into (15) implies the following condition for the ratio of nominal consumption expenditures for the two sectors: (cid:18) (cid:19) pC1 (cid:11)1 g1+(cid:14)1 = (16) C2 (cid:11)2 g1+(cid:14)1+(cid:26) Investment: Capital goods are purchased by arbitraging intermediaries who rent them out to (cid:12)rms at Jorgensonian rental rates: (cid:18) (cid:19) (cid:1)p c1 = p r+(cid:14)1 − = p(g1+(cid:14)1+(cid:26)) p c2 = r+(cid:14)2 = g2+(cid:14)2+(cid:26) 11

The pro(cid:12)t functions for the two sectors are (cid:25)1 = pA1L 1 1 −(cid:12)1 −(cid:12)2K 1 (cid:12) 1 1K 2 (cid:12) 1 2 −wL1 −c1K11 −c2K21 (cid:25)2 = A2L 1 2 −(cid:12)1 −(cid:12)2K 1 (cid:12) 2 1K 2 (cid:12) 2 2 −wL2 −c1K12 −c2K22 The (cid:12)rst-order conditions for inputs can be expressed as p(1−(cid:12)1 −(cid:12)2)Y1 (cid:12)1Y1 p(cid:12)2Y1 L1 = K11 = K21 = (17) w g1+(cid:14)1+(cid:26) g2+(cid:14)2+(cid:26) (1−(cid:12)1 −(cid:12)2)Y2 1 (cid:12)1Y2 (cid:12)2Y2 L2 = K12 = K22 = (18) w pg1+(cid:14)1+(cid:26) g2+(cid:14)2+(cid:26) These conditions imply that, for each factor, the ratio of inputused in sector 1 relative to that used in sector 2 equals the ratio of nominal outputs, which is constant. Oneimplication is thatbothL1 andL2 are(cid:12)xedinsteady-state, as assumedearlier. From these equations, we can also derive the steady-state ratios for nominal investment in capital of type 1 relative to nominal investment in capital of type 2, as well as the corresponding ratio for the capital stocks. Letting (cid:12) (g +(cid:14) ) i i i (cid:22) = (19) i g +(cid:14) +(cid:26) i i these ratios are p(I11+I12) (cid:22)1 = (20) I21+I22 (cid:22)2 p(K11+K12) (cid:12)1g2+(cid:14)2+(cid:26) = (21) K21+K22 (cid:12)2g1+(cid:14)1+(cid:26) where equation (20) uses the fact that I = (g +(cid:14) )K in steady-state. ij i i ij Output: Combining the resource constraints, (1) and (2), with the (cid:12)rst-order conditions for capital accumulation, (17) and (18), and the fact that I = (g +(cid:14) )K ij i i ij in steady-state, we obtain the following expressions for real output: (cid:22)1 Y1 = C1+(cid:22)1Y1+ Y2 p Y2 = C2+p(cid:22)2Y1+(cid:22)2Y2 Combined with (16) these formulae can be re-arranged to obtain the share of consumption goods in the output in each sector: C1 1−(cid:22)1 −(cid:22)2 = Y1 1−(cid:22)2+ (cid:11)2(cid:12)1 (cid:11)1 12

C2 1−(cid:22)1 −(cid:22)2 = Y2 1−(cid:22)1+ (cid:11)1(cid:12)2 (cid:11)2 Note that these equations, combined with the constancy of the ratios of capital of type i used in sector 1 to capital of type i used in sector 2, implied by (17) and (18), con(cid:12)rm our earlier assumption (made when deriving the steady-state growth rates) that the fraction of each sector’s output sold to households, to Sector 1, and to Sector 2, are all constant. Finally, we can combine these expressions with equation (16) to obtain the ratio of nominal output in sector 1 to nominal output in sector 2: 0 1 pY1 = 1−(cid:22)2+ (cid:11) (cid:11) 2(cid:12) 1 1 pC1 = @ 1−(cid:22)2+ (cid:11) (cid:11) 2(cid:12) 1 1 A (cid:22)1(cid:11)1 = (cid:18) (22) Y2 1−(cid:22)1+ (cid:11)1(cid:12)2 C2 1−(cid:22)1+ (cid:11)1(cid:12)2 (cid:12)1 (cid:11)2 (cid:11)2 (cid:11)2 2.4 Calibration To use the model for empirical applications, we will need values for the following eight parameters: a1, a2, (cid:12)1, (cid:12)2, (cid:14)1, (cid:14)2, (cid:26), and (cid:11)1=(cid:11)2 (only the ratio of these two parameters matters). The depreciation rates are (cid:12)xed a priori at (cid:14)1 = 0:13 and (cid:14)2 = 0:03, on the basis of typical values used to construct the NIPA capital stocks 7 for durable equipment and structures. The values of the other parameters are set so that the model’s steady-state growth path features important variables matching theiraverage valuesover theperiod1957-99, wherethissamplewaschosen tomatch the start of the pattern of declining relative prices for durable goods: (cid:15) A ratio of nominal consumption of durables to nominal consumption of nondurables and services of 0.15. (cid:15) Aratio of nominal outputof durables to other nominal business sector output of 0.27. (cid:15) A growth rate of real durables output per hour worked of 3.9 percent. (cid:15) A growth rate of real other business sector output per hour worked of 1.8 8 percent. 7SeeKatz and Herman (1997) for a description of thesedepreciation rates. 8Allper-hourcalculations in this paper refer to total business-sector hours. 13

(cid:15) An average share of income earned by capital of 0.31. (cid:15) A rate of return on capital of 6.5 percent, as suggested by King and Rebelo (2000). In terms of the model’s parameters these conditions imply the following six equations: 0 1 (cid:11)1 @ (1− 1 (cid:12) − 2 (cid:12) )a 1 1 − + (cid:12) (cid:12) 2 2a2 +(cid:14)1 A = 0:15 (23) (cid:11)2 (1− 1 (cid:12) − 2 (cid:12) )a 1 1 − + (cid:12) (cid:12) 2 2a2 +(cid:14)1+(cid:26) (cid:18) = 0:27 (24) (1−(cid:12)2)a1+(cid:12)2a2 = 0:039 (25) 1−(cid:12)1 −(cid:12)2 (cid:12)1a1+(1−(cid:12)1)a2 = 0:018 (26) 1−(cid:12)1 −(cid:12)2 (cid:12)1+(cid:12)2 = 0:31 (27) (cid:12)1a1+(1−(cid:12)1)a2 +(cid:26) = 0:065 (28) 1−(cid:12)1 −(cid:12)2 These solve to give parameter values a1 = 0:03, a2 = 0:009, (cid:12)1 = 0:145, (cid:12)2 = 0:165, (cid:26) = 0:047, and (cid:11)1 = 0:19. We will now use these parameter values to discuss some (cid:11)2 empirical applications of the model. 3 Fitting Aggregate NIPA Data We have described the long-run growth path for each of our sectors, how they price their output, and how they split this output between consumption and investment goods. In this section, we consider the combined behavior of the two sectors: How do aggregate real variables behave? In particular, we explore the properties of the aggregate real series published in the U.S. NIPAs. 3.1 Unbalanced, Unsteady Growth? Because the two sectors grow at steady but di(cid:11)erent rates, we have labeled our solution a steady-state growth path. However, it is clearly not a balanced growth path in the traditional sense. Indeed, according to the usual theoretical de(cid:12)nition of aggregate real output as the sum of the real output in the two sectors, aggregate 14

growth also appears to be highly unsteady. The growth rate will tend to increase each period with sector 1 tending to become \almost all" of real output. Similarly, we should expect to see the growth rates for the aggregates for consumption and investment also increasing over time, as the durablegoods components become ever larger relative to the rest. To illustrate this pattern, consider what we usually mean when we say that aggregate real output is the sum of the real output of the two sectors. If our economy produces four apples and three oranges, we don’t mean that real GDP is seven! Rather, we start with a set of prices from a base year and use these prices to weight the quantities. Recalling our assumption that the price of output in sector 2 always equals one, we can construct such a measure by choosing a base year, b, from which to use a price for sector 1’s output: Y t fw = p (cid:16) b Y1t +Y2t (cid:17)(cid:16) (cid:17) = p0e (g2 −g1 )b Y10eg1t +Y20eg2t = p0Y10eg2beg1 (t−b) +Y20eg2t (29) The growth rate of this so-called \(cid:12)xed-weight" measure of real output will tend to increase every period, asymptoting towards g1. And the time path of this acceleration will depend on the choice of base year: The further back in time we choose b, the faster Yfw will grow. Intuitively, this occurs because the (cid:12)xed-weight series t has the interpretation \how much period t’s output would have cost had all prices remained at their year-b level" and the percentage change in this measure must depend on the choice of base year, b. Because the fastest growing parts of the output bundle (durable goods) were relatively more expensive in 1960 than in 1990, the cost of the this bundle in 1960 prices must grow faster than the cost in 1990 prices. This problem of unsteady and base-year-dependent growth is not just a prediction of our model; it is an important pattern to which national income accountants have had to pay a lot of attention. In the past, the BEA dealt with this problem by moving the base year forward every (cid:12)ve years. This re-basing meant that currentperiodmeasuresofrealGDPgrowthcouldalwaysbeinterpretedintermsofarecent set of relative price weights. However, this approach also had problems. For example, while1996 priceweights may beusefulfor interpreting 1999’s growth rate, they are hardly a relevant set of weights for interpreting 1950’s growth rate, given the 15

very di(cid:11)erent relative price structure prevailing then. So, re-basing improves recent measures of growth at the expense of worsening the measures for earlier periods. In addition, periodic re-basing leads to a pattern of predictable downward revisions to recent estimates of real GDP growth. 3.2 Chain-Aggregates Because of the problems with (cid:12)xed-weight measures, the BEA abandoned this approach in 1996. Instead, it now uses a so-called chain index method to construct all real aggregates in the U.S. NIPAs, including real GDP. Rather than using a (cid:12)xed set of price weights, chain indexes continually update the relative prices used to calculate the growth rate of the aggregate. Since the growth rate for each period is calculated using relative price weights prevailing close to that period, chain-weight measures of real GDP growth do not su(cid:11)er from the interpretational problems of (cid:12)xed-weight series, and do not need to re-calculated every few years. The \base year" for chain-aggregates is simply the year chosen to equate the real and nominal series, with the level of the series obtained by \chaining" forward and backward from there using the index; the choice of base year does not a(cid:11)ect the growth rate 9 for any period. The speci(cid:12)c method used by the NIPAs is the \ideal chain index" pioneered by Irving Fisher (1922). Technically, the Fisher index is constructed by taking a geometricaverageofthegrossgrowthratesoftwoseparate(cid:12)xed-weightindexes,one a Paasche index (using period t prices as weights) and the other a Laspeyres index (using period t−1 prices as weights). In practice, however, the Fisher approach is well approximated by a Divisia index, which weights the growth rate of each category by its current share in the corresponding nominal aggregate. The stability ofnominalshares,documentedearlier, impliesthatunlike(cid:12)xed-weightseries,chainaggregates do not place ever-higher weights on the faster growing components, and will grow at a steady rate over the long run, even if one of the component series continually has a higher growth rate than the other. 9Chain-aggregated levelsneedtointerpretedcarefully. Forexample, becausethelevelofchainaggregated real GDP is not the arithmetic sum of real consumption, real investment, and so on, one cannot interpret the ratio of real investment to real GDP as \the share of investment in real GDP"becausethisratioisnotashare:Thesumoftheseratiosforallcategories ofGDPdoesnot equal one. SeeWhelan (2000) for more discussion of theproperties of chain-aggregated data. 16

The switch to a chain-aggregation methodology has important implications for the analysis of U.S. macroeconomic data. Table 1 illustrates that these series can behave very di(cid:11)erently from their (cid:12)xed-weight counterparts. It shows the growth rates of the chain-weight and 1992-based (cid:12)xed-weight aggregates for GDP, invest- 10 ment, and consumption over the period 1992-98. For every year after 1992, each of the (cid:12)xed-weight series grows faster than the corresponding chained series, with these di(cid:11)erences increasingover time. Thisis mostapparentfor investment because durable equipment is a larger component of that series, so the relative price shift between equipment and structures is more important. For 1998, six years after the base year, the chain-aggregate for investment grows 11.4 percent while the 1992based (cid:12)xed-weight series grows 22.5 percent. The di(cid:11)erence between (cid:12)xed- and chain-weight measures of real GDP, as we move away from the base year for the (cid:12)xed-weight calculation, are also quite notable. For 1997 and 1998, the (cid:12)xed-weight measure of GDP grows 5.2 percent and 6.6 percent, while the corresponding chain series grows at a steady 3.9 percent pace. Our two-sector model is well-suited to replicating the behavior of the NIPA chain-aggregates. From(16), (20), (21), and(22), wecanderivetheshareofdurable goods in each of the nominal aggregates as functions of our model’s parameters. Using the Divisia approximation, our model predicts the steady-state growth rates for the chain-aggregates for output, consumption, investment, and the capital stock (all on a per hour basis) should be as follows: (cid:18) 1 g Y = g1+ g2 (30) 1+(cid:18) 1+(cid:18) (cid:16) (cid:17) g C = (cid:11)1 (cid:11) (cid:16) 1 g1 g + 1 g + (cid:14) 1 g 1 (cid:14) + 1 + 1 + (cid:14) (cid:26) 1 (cid:14) (cid:17) + 1 (cid:26) +(cid:11)2 g1+ (cid:11)1 (cid:16) g1 g + 1 + (cid:14) (cid:11) 1 (cid:14) + 2 1 (cid:26) (cid:17) +(cid:11)2 g2 (31) (cid:12)1 (g1 +(cid:14)1 ) (cid:12)2 (g2 +(cid:14)2 ) g I = (cid:12)1 g1 + + (cid:14)1 +(cid:26) (cid:12)2 g1+ (cid:12)1 g2 + + (cid:14)2 +(cid:26) (cid:12)2 g2 (32) g1 +(cid:14)1 +(cid:26) g2 +(cid:14)2 +(cid:26) g1 +(cid:14)1 +(cid:26) g2 +(cid:14)2 +(cid:26) 10The (cid:12)xed-weight (cid:12)gures shown in this table are unpublished estimates obtained from the Department of Commerce’s STAT-USA website. Earlier estimates, going through 1997, were published as Table 8.27 of Department of Commerce (1998). In this table, I have used 1992-based data, which pre-date the 1999 comprehensive revision to the NIPAs, to better illustrate how the chain- and (cid:12)xed-weight series di(cid:11)er as we move away from the base year. However, all other data used in the paperare current as of March 2001 and usea base yearof 1996. 17

(cid:12)1 (cid:12)2 g K = (cid:12)1 g1 + + (cid:14)1 +(cid:26) (cid:12)2 g1+ (cid:12)1 g2 + + (cid:14)2 +(cid:26) (cid:12)2 g2 (33) g1 +(cid:14)1 +(cid:26) g2 +(cid:14)2 +(cid:26) g1 +(cid:14)1 +(cid:26) g2 +(cid:14)2 +(cid:26) The model has been calibrated to match the average share of durable goods in nominal consumption and output, and also the average real growth rates of our two sectors. Thus, essentially by design, its predicted steady-state values for growth in chain-aggregated output, consumption, and investment match their empirical averages closely (see Table 2). The model’s steady-state growth rate for output per hour exactly matches the publishedaverage of 2.3 percent, and the predicted values for consumption and investment are only one-tenth di(cid:11)erent from the empirical averages of 2.2 percent and 3.1 percent respectively. The steady-state growth rate for the chain-aggregated capital stock (per hour) is 2.3 percent, a good bit higher than the empirical average value of 1.7 percent. Given that capital stocks adjusts slowly, empirical averages for this variable may be less likely to correspond to a long-run growth path. Using a slightly longer sample starting in 1948, this average value increases to 2.0 percent, closer to our predicted steady-state value. Note that the model’s calibrated steady-state values for g and g are both K Y 2.3 percent. This prediction of a constant long-run capital-output ratio may seem familiar from the Solow-Ramsey model but, in this case, it is simply due to a coincidence of coe(cid:14)cients taking the right values rather than being a fundamental feature of the model’s steady-state growth path. 3.3 Implications for Analysis of NIPA Data Ourmodel’spredictionsaboutthesteady-statebehavioroftheNIPAchain-aggregated series have far-reaching implications for how these series should be treated in empirical analysis. In particular, much empirical analysis in macroeconomics relies implicitly on the assumption that the one-sector growth model provides a good description of the long-run properties of these data, and this assumption can lead to misleading conclusions. I will point out three simple examples. 1. Calibration of Business Cycle Models: The long-run growth paths for most general equilibrium business cycle models have the balanced growth feature of the Ramsey model. Indeed, the usual approach in empirical applications of 18

these models is to calibrate the preference and technology parameters to ensure that, over the long-run, the models feature the traditional \great ratios," such as the real consumption-output ratio or the real capital-output ratio, conforming to 11 their sample averages. Our results imply that this approach is flawed. Because the real NIPA aggregates for output, consumption, investment, and the capital stock each tend to grow at di(cid:11)erent rates in the long run|reflecting their di(cid:11)erent mixes of durable goods and other output|ratios of these variables are unlikely to be stationary. Consequently, model parameters calibrated to match these empirical averages (andhencetheempirical resultsfrom thesemodels)willdependarbitrarily on the sample used. 2. CointegrationofConsumptionandIncome: Thatconsumptionandincome should move together in the long-run is one of the most basic ideas in macroeconomics. However, our model implies that, when using real NIPA series, one needs to be careful about exactly what de(cid:12)nition of real income is being used. In general, it should not be presumed that commonly-used aggregate measures of real consumption and real income will grow at the same rate in the long run. For example, real aggregate consumption will tend to grow slower than real GDP as currently constructed. Empirical work on consumption tends to focus on real outlays on nondurables and services, and this series will exhibit slower long-run growth than both 12 aggregate real consumption and real GDP. Our model does predict that the ratio of nominal consumption to nominal income should be stable in the long run. Note though that the model ignores governmentspendingandtaxes, soitismoreprecisetointerpretthispredictionasrelating to disposable income as opposed to total income. So, under the assumption of a stationary tax rate, if one deflates disposable income by the aggregate consumption 11SeeCooleyandPrescott(1995)andKingandRebelo(2000)fortwostandardexamplesofhow these models are calibrated. 12Thisargument appearstocontradict theresultsof Cochrane (1994), whofound that theratio ofrealconsumptionofnondurablesandservicestorealGDPwasstationary,andusedthis(cid:12)nding to explore the \error-correction" properties of this ratio for forecasting output growth. However, Cochrane’s(cid:12)ndingofstationarityderivedfromhisuseoftotalrealGDP,whichincludesgovernment purchases, a category that has grown slower in real terms than private output. Once private real GDP is used,theresults are as predicted by ourmodel, with a downward drift in theratio of real consumpion of nondurablesand services to real private GDPevident from thelate 1950s onward. 19

deflator, then this de(cid:12)nition of real income will grow at the same rate as aggregate 13 real consumption. 3. Aggregate Investment Regressions: Most macroeconomic regression speci- (cid:12)cations for investment are derived from the assumption that there is one type of capital, which depreciates at a constant rate. In this case, the ratio of aggregate real investment to the lagged aggregate real capital stock, It , summarizes the Kt−1 growth rate of the capital stock. For this reason, the aggregate It ratio has been Kt−1 the most commonly used dependent variable in macroeconomic investment regres- 14 sions. Our model implies that, if applied to current NIPA data, such regressions would be mis-speci(cid:12)ed. The problem, of course, is that not all types of capital are identical. In particular, our model captures two crucial di(cid:11)erences between equipment and structures: The stock of equipment grows faster than the stock of structures, and equipment also depreciates signi(cid:12)cantly faster. From equations (19), (20) and (21), our model predicts that, because g1 +(cid:14)1 is greater than g2+(cid:14)2, equipment should make up a higher share of total nominal investment than of the total nominal capital stock. This is con(cid:12)rmed by the data: Over 1957-99, equipment accounted for 47 percent of nominal investment, and 19 percent of the nominal capital stock. Because it places a higher weight on the fastgrowing asset (equipment), the growth rate of chain-aggregated investment will be higher than the growth rate of the chain-aggregated capital stock. Thus, the aggregate series for It will tend to grow without bound, making it a very poor Kt−1 proxy for g (which is constant along the steady-state growth path). Remarkably, K over the period 1948-98, the correlation between It and g is -0.05. Kt−1 K Forthisreason,investmentregressionsusingthisdependentvariablewillbemisspeci(cid:12)ed. Incontrast,though,onecaneasilyre-arrangethe(cid:12)rst-orderconditionsfor pro(cid:12)t maximization to obtain separate speci(cid:12)cations for equipment and structures, in which It is a function of chain-aggregated output growth and the growth rate Kt−1 of the asset-speci(cid:12)c real cost of capital. 13Note that this is exactly how the BEA de(cid:12)nes real disposable income. See, for example, standard NIPATable 2.1. 14See, for example, Blanchard, Rhee, and Summers (1994), Hayashi (1982), and Oliner, Rudebusch, and Sichel (1995). 20

4 Technological Progress, Growth, and Welfare Our calibrated parameters can be used to calculate the contributions of the two typesoftechnologicalprogresstolong-runrealaggregateoutputgrowthasmeasured in the NIPAs. Inserting (10) and (11) into (30) we get (cid:20) (cid:21) (cid:20) (cid:21) (cid:18) (1−(cid:12)2)a1+(cid:12)2a2 1 (cid:12)1a1+(1−(cid:12)1)a2 g = + Y 1+(cid:18) 1−(cid:12)1 −(cid:12)2 1+(cid:18) 1−(cid:12)1 −(cid:12)2 (cid:20) (cid:21) 1 (cid:12)1+(1−(cid:12)2)(cid:18) (cid:12)2(cid:18)+(1−(cid:12)1) = 1−(cid:12)1 −(cid:12)2 1+(cid:18) a1+ 1+(cid:18) a2 (34) This equation has a simple interpretation along the lines of the equilibrium growth rate of theone-sector model, which is a , whereais thegrowth rateof themodel’s 1−(cid:12) 1 unique aggregate technology process and is a scaling factor representing the 1−(cid:12) additional output growth caused by induced capital accumulation. In the twosector case, the picture is a little more complex|with two types of technological progress, therearetwo separate directe(cid:11)ects on output,and withtwo capital goods used in two sectors, there are four separate induced accumulation e(cid:11)ects|but the logic is the same. Plugginginourparametervalues,we(cid:12)ndthat58percentofthelong-rungrowth in business sector output per hour as currently measured can be attributed to technologicalprogressintheproductionofdurablegoods(a1),with35percentagepoints of this coming from the directe(cid:11)ect on the productionof durables,and the other 23 points representing the induced e(cid:11)ect of this technological progress on the production of other goods and services. In contrast, of the 42 percent of long-run growth due to technological progress in the production of nondurables, services, and structures (a2), only 2 percentage points represents induced e(cid:11)ects on the production of durable goods. Recalling the representative agent’s utility function from equation (6), we can also calculate the contribution of each type of technological progress to the growth inwelfare. Speci(cid:12)cally, alongthesteady-state growth path,thechangeintheutility of the representative consumer is (cid:1)U(D t ;C2t ) = (cid:11)1g (cid:20) 1+(cid:11)2g2 (cid:21) (cid:20) (cid:21) (1−(cid:12)2)a1+(cid:12)2a2 (cid:12)1a1+(1−(cid:12)1)a2 = (cid:11)1 1−(cid:12)1 −(cid:12)2 +(cid:11)2 1−(cid:12)1 −(cid:12)2 (cid:11)1(1−(cid:12)2)+(cid:11)2(cid:12)1 (cid:11)1(cid:12)2+(cid:11)2(1−(cid:12)1) = 1−(cid:12)1 −(cid:12)2 a1+ 1−(cid:12)1 −(cid:12)2 a2 (35) 21

Ourparametervaluesimplythattechnological progressintheproductionofdurable goods accounts for 53 percent of the change in the utility of the representative agent. This is lower than the 58 percent calculated for g because durable goods Y receiveahigherweightinginsteady-statenominaloutputthanintheutilityfunction ((cid:18) > (cid:11)1). (cid:11)2 5 Comparison with Greenwood et al (1997) As noted earlier, the model presented in this paper shares some important similarities with that of Greenwood, Hercowitz, and Krusell (1997). This section compares the approach and results of this paper with their work. 5.1 An Alternative Representation There are two main di(cid:11)erences between the model presented in this paper, and that of Greenwood, Hercowitz, and Krusell (henceforth, GHK). The (cid:12)rst is that the model in this paper acknowledges that technological progress has been faster for all durable goods, including consumer durables. GHK’s model did not include a separate treatment of consumer durables, instead featuring two production technologies, one for consumption goods and structures, and the other for producers’ durable equipment. I will return to this issue below. The second di(cid:11)erence is a presentational one. GHK did not emphasize their model’s two-sector interpretation. Instead of working with a model in the form Y1 = A1F (K;L) (36) Y2 = A2F (K;L) (37) they summarized these two production technologies in a single equation of the form Y1 +Y2 = zF (K;L) (38) q Note that this equation is consistent with the following two-sector representation: Y1 = 0:5(cid:3)zqF (K;L) (39) Y2 = 0:5(cid:3)zF (K;L) (40) 22

Here, the 0:5 is merely a normalization and can be ignored. Within our framework, z can be thought of as A2, and q can be thought of as A1. Clearly, though, the A2 two systems|equations (36) and (37) on the one hand, and (39) and (40) on the other|represent two di(cid:11)erent ways of summarizing the same model. GHK constructed an empirical q series as the price of consumer nondurables and services relative to the price of producers durable equipment, as in equation (12) of this paper. Note now that, because q is the deflator for sector 2’s output (consumer nondurables and services) divided by the deflator for sector 1’s output (durableequipment),thenzF (K;L)inequation(38)canbecalculated asaggregate nominal output divided by sector 2’s price. Thus z can be calculated as a Solow residual for this measure of real output. GHK used their empirical series for z and q to calculate that 58 percent of the growth in zF (K;L) was due to the q variable, with the rest due to the z series. This 58 percent (cid:12)gure should not be confused with our calculation in the previous section|based on equation (34)|that 58 percent of the growth in real chainaggregated businessoutput(g )wasduetotechnological progressintheproduction Y of durable goods. In our model, the direct equivalent to GHK’s zF (K;L) series (total nominal output deflated by sector 2’s price) can be calculated as pY1+Y2 = (1+(cid:18))Y2 (41) (Recall that we normalized sector 2’s price to equal 1). Obviously, this series will have a steady-state growth rate of g2. Thus, the comparable calculation is not our previous (cid:12)gure of 58 percent, but rather the boost to g2 coming from a1 growing faster than a2, which is only 25 percent. 15 Our calculation of the e(cid:11)ect of A1 (or q) A2 on g2 is lower primarily because we used a di(cid:11)erent measure of the relative price of durablegoods. Asnotedabove,GHK’sprocedure|usingGordon’salternativeprice deflatorfordurableequipmentandtheNIPAdeflatorforconsumernondurablesand services|likely overstates the relative price decline for durable goods, and thus the impact of technological progress in this sector. 15This (cid:12)gurewas calculated as (cid:12)1(a1−a2) . (cid:12)1a1+(1−(cid:12)1)a2 23

5.2 Which Measure of Real Output is Correct? Beyond the presentation of the two-sector model and its calibration, a more substantive di(cid:11)erence between GHK’s approach and that in this paper concerns the measurement of aggregate real output. Speci(cid:12)cally, GHK use zF(K;L) as de(cid:12)ned in equation (38)|nominal outputdivided by the deflator for consumer nondurables and services|as their measure of aggregate real output, and argue that this is superior to the real output measure in the NIPAs. It turns out, though, that their characterization of the NIPA measure is somewhat misleading, for two reasons. First, GHKinterpretthemeasureofaggregate realoutputintheNIPAsasbeing 16 of the form Y1+Y2, summing the real output of the two sectors. However, while this was true for the real NIPA data used by GHK in their study, this is no longer an accurate description of how U.S. real GDP is measured. As we described above, since 1996, a chain-weighting method has been employed. Second, GHK argue that the NIPA measure of real output is inconsistent with thefactthatdurablegoodspriceshave beenfallingrelative topricesforothergoods and services. Speci(cid:12)cally, they associate the NIPA measure of real output with an aggregate production function of the form Y1+Y2 = zF (K;L) (42) and point out that if type-1 and type-2 outputare perfect substitutes in production 17 (as this equation implies) then, counterfactually, they must have the same price. This argument, however, appears to confuse measurement with theory, speci(cid:12)cally one-sector theory. Even if aggregate real output was measured as in the left-handside of equation (42), the use of such a measure wouldn’t necessarily imply any judgement on the properties of the \aggregate production function" or on the economy’s ability to substitute production of type-1 output for type-2 output. Indeed, if one adopts the multisector approach used in this paper, then there is no aggregate production function. In this case, the choice between di(cid:11)erent measures of real output depends only on what exactly it is that one wishes to use the measures for. These considerations suggest that neither the NIPA measure of real output nor GHK’s measure are intrinsically the \correct" one. Rather, they measure di(cid:11)erent 16This can beseen from equation 24 of their paper. 17Hercowitz (1998) makes this argument in more detail. 24

concepts. However, as a summary statistic for aggregate real output, the NIPA measure|which weights the growth rates of the two sectors according to their shares in nominal output|is more representative of the growth occurring in the economy as a whole. As can be seen from (41), in the long run, GHK’s measure will only reflect the growth in production of consumer nondurables, services, and structures. One argument for a consumption-unit measure of real output is that it may be a better proxy for welfare. However, GHK’s measure does not account for the welfare gains due to rising consumption of durable goods. In fact, for this reason, our model suggests that the NIPA measure of real output is the superior statistic for welfare. In steady-state, the change in utility of our representative consumer is (cid:11)1g1 +(cid:11)2g2. While the weights on g1 and g2 determining the steady-state growth rate of real chain-aggregated output are not exactly the same as that for the change in utility|g Y places 21 percent of its weight on g1, compared with 16 percent for (cid:1)U(D t ;C2t )|this series comes a lot closer than GHK’s measure, which grows at rate g2 in the long-run. 6 Conclusions This paper has argued that, despite its central role in economics textbooks and in business cycle research, the one-sector model of economic growth provides a poor description of the long-run evolution of the U.S. economy. In particular, the model’s central prediction, that the real aggregates for consumption, investment, output, and the capital stock, should all grow at the same rate over the long run, is (cid:12)rmly rejected by postwar U.S. NIPA data. The reason for the failure of the balanced growth hypothesis is the model’s inability to distinguish the behavior of the durable goods sector from that of the rest of the economy. The real output of the durable goods sector has consistently grown faster than the rest of the economy, and most investment spending is on durable goods while most consumption spending is on nondurables and services. As a result, real investment has tended to grow faster than real consumption, a pattern that has been particularly evident since 1991. These patterns are easily modelled using the simple two-sector approach of this paper. 25

While the case for a multi-sector approach to long-run modeling of the macroeconomy has also recently been made by Greenwood, Hercowitz, and Krusell (1997), this paper makes two new and important contributions. First, I show that the one-sector model can be strongly rejected based on published NIPA data. This is important because most empirical work in macroeconomics uses these data, and often relies on the one-sector model’s balanced growth predictions. The message here for practitioners is that consideration of a two-sector approach to long-run modelling should not be limited to datasets that have adjusted the NIPA equipment prices for measurement error. In any case, given the recent research on price measurement error in the (cid:12)nance and construction industries, one could argue that the evidence presented here for the two-sector approach as a model of the underlying reality is more convincing than that cited by Greenwood et al, because it does not relyonadjustments topublisheddatathatignorepotential pricemis-measurements outside the durable goods sector. The paper’s second main contribution is the use of the two-sector framework to establish the long-run properties of the real NIPA aggregates constructed according to the current chain index method. Speci(cid:12)cally, it has been shown that each of the major chain-aggregated variables|output, consumption, investment, and capital stock|will tend in the long-run to grow at steady, but di(cid:11)erent, rates. This result has far-reaching implications for empirical practice in macroeconomics. In particular, our examples have shown how treating these data as if they are (cid:12)xed-weight aggregates generated byaone-sector modelcanoftenresultinincorrectconclusions. Theresultsinthispapersuggestthattheremaybelargegainstofutureempirical research aimed at understanding the apparently large gap between technological progress in the durable goods sector and in the rest of the economy. In particular, anexploration ofwhetherthispatterncan beexplainedbydi(cid:11)erential rates ofR&D activity, or other \spillovers" stressed in the endogenous growth models of Romer (1990) and Jones and Williams (1998), would appear to be particularly worthwhile, and could yield important policy implications. 26

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[12] Hercowitz, Zvi (1998). \The ‘Embodiment’ Controversy: A Review Essay." Journal of Monetary Economics, 41, 217-224. [13] Jones, Charles I. and John Williams (1998). \Measuring the Social Return to R&D." Quarterly Journal of Economics, 113, 1119-1135. [14] Kaldor, Nicholas (1957). \A Model of Economic Growth." Economic Journal, 67, 591-624. [15] Katz, ArnoldandShelbyHerman(1997).\ImprovedEstimatesofFixedReproducible Tangible Wealth, 1929-95." Survey of Current Business, May, 69-92. [16] King, Robert, Charles Plosser, James Stock, and Mark Watson (1991). \Stochastic Trends and Economic Fluctuations." American Economic Review, 81, No. 4, 819-840. [17] King, Robert and Sergio Rebelo (2000). \Resuscitating Real Business Cycles," in John Taylor and Michael Woodford (eds.), The Handbook of Macroeconomics, North-Holland. [18] Oliner, Stephen, Glenn Rudebusch, and Daniel Sichel (1995). \New and Old Models of Business Investment: A Comparison of Forecasting Performance." Journal of Money, Credit, and Banking, 27, 806-826. [19] Romer, Paul (1990). \Endogenous Technological Change." Journal of Political Economy, 98(5), S71-102. [20] Tevlin, Stacey and Karl Whelan (2001). \Explaining the Investment Boom of the 1990s." forthcoming, Journal of Money, Credit, and Banking. [21] U.S. Dept. of Commerce, Bureau of Economic Analysis (1998). \National Income and Product Accounts Tables." Survey of Current Business, August, 36-118. [22] Whelan, Karl (2000). A Guide to the Use of Chain Aggregated NIPA Data, Federal Reserve Board, Finance and Economics Discussion Series Paper No. 2000-35. 28

A Derivation of Equation (15) The (cid:12)rst-order conditions for consumption expenditures are (cid:20) (cid:21) (cid:20) (cid:21) (cid:11)1 1 @V t+1 1 @V t+1 + E t = E t p t (1+r t+1) (43) D t 1+(cid:26) @D t 1+(cid:26) (cid:20) @Z t+ (cid:21) 1 (cid:11)2 1 @V t+1 = E t (1+r t+1) (44) C2t 1+(cid:26) @Z t+1 while the envelope conditions for the value function are (cid:20) (cid:21) @V t = (1−(cid:14)1) (cid:11)1 + 1−(cid:14)1 E t @V t+1 (45) @D t−1 D (cid:20) t 1+(cid:26) @ (cid:21) D t @V t 1 @V t+1 = E t (1+r t+1) (46) @Z t 1+(cid:26) @Z t+1 From (43) and (44), we know that (cid:20) (cid:21) (cid:11)1 1 @V t+1 p t (cid:11)2 + E = (47) t D t 1+(cid:26) @D t C2t Thus, (45) can be re-expressed as (cid:18) (cid:19) @V t = (1−(cid:14)1) p t (cid:11)2 @D t−1 C2t Shifting this equation forward one period and taking the expectation we get (cid:20) (cid:21) ! E t @V t+1 = (1−(cid:14)1)E t p t+1(cid:11)2 @D t C2;t+1 Inserting this into (47) and re-arranging we get ! (cid:11)1 = p t (cid:11)2 − 1−(cid:14)1 E p t+1(cid:11)2 t D t C2t 1+(cid:26) C2;t+1 which becomes ! (cid:11)1 C2t = 1− 1−(cid:14)1 E C2t p t+1 t (cid:11)2p t D t 1+(cid:26) C2;t+1 p t Usingourformulaeforsteady-state growthratesofconsumptionandrelative prices, this becomes (cid:11)1 C2t = 1− (1−(cid:14)1)(1+g2 −g1) (cid:25) g1+(cid:14)1+(cid:26) (cid:11)2p t D t (1+(cid:26))(1+g2) as required. 29

GDP Consumption Investment Chain- Fixed- Chain- Fixed- Chain- Fixed- 1992 2.7 2.7 2.8 2.7 5.7 5.5 1993 2.3 2.4 2.9 3.0 7.6 7.7 1994 3.5 3.6 3.3 3.4 8.6 9.0 1995 2.3 2.7 2.7 2.9 5.5 7.2 1996 3.4 4.1 3.2 3.6 8.8 11.2 1997 3.9 5.2 3.4 4.1 8.3 12.7 1998 3.9 6.6 4.9 6.1 11.4 22.5 Table 1: 1992-Dollar Chain- and Fixed-Weight Growth Rates for Major Aggregates Source: Department of Commerce STAT-USA Website (www.stat-usa.gov) Steady-State Values Average, 1957-99 Output Per Hour 2.3 2.3 Consumption Per Hour 2.1 2.2 Investment Per Hour 3.0 3.1 Capital Stock Per Hour 2.3 1.7 Table 2: Growth Rates for Chain-Aggregates 30

Figure 1 Ratio of Real Private Fixed Investment to Real Consumption 0.30 0.28 0.26 0.24 0.22 0.20 0.18 0.16 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 31

Figure 2a Ratios of Major Components of Investment and Consumption Real Equipment Investment / Real Structures Investment Real Durable Goods Consumption / Real Nondurables and Services Consumption 0.20 1.8 0.18 1.6 0.16 1.4 0.14 1.2 0.12 1.0 0.10 0.8 0.08 0.6 0.06 0.4 0.04 0.2 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 Figure 2b Ratio of Total Real Durables Output to Real Business Output 0.26 0.24 0.22 0.20 0.18 0.16 0.14 0.12 0.10 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 32

Figure 3a Ratio of Total Nominal Durables Output to Nominal Business Output 0.26 0.25 0.24 0.23 0.22 0.21 0.20 0.19 0.18 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 Figure 3b Ratio of Nominal Private Fixed Investment to Nominal Consumption 0.32 0.30 0.28 0.26 0.24 0.22 0.20 0.18 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 33

Figure 4 Relative Prices of Durables and Fixed Investment (In Logs, All Prices Relative to Price of Consumer Nondurables and Services) All Durable Goods Producers’ Durable Equipment 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 -0.2 -0.2 -0.4 1950 1960 1970 1980 1990 1950 1960 1970 1980 1990 Fixed Investment Structures 0.40 0.20 0.35 0.15 0.30 0.25 0.10 0.20 0.05 0.15 0.00 0.10 0.05 -0.05 0.00 -0.10 -0.05 -0.10 -0.15 1950 1960 1970 1980 1990 1950 1960 1970 1980 1990 34