Returns to Scale in U.S. Production: Estimates and Implications

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 546 March 1995 RETURNS TO SCALE IN U.S. PRODUCTION: ESTIMATES AND IMPLICATIONS Susanto Basu and John G. Femald N In F D P p m c s d c comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.

ABSTWCT A typical (roughly) two-digit industry inthe United States appears to have constant or slightly decreasing returns to scale. Three puzzles emerge, h F e at higher levels of aggregation. Second, estimates ofdecreasing returns inmany industries contradict evidence ofonly small economic profits. Third, estimates using value added differ substantially from those using gross output, a l r These puzzles inconsistent with a representative firm paradigm, but are consistent with simple stories of aggregation over heterogeneous units. We discuss implications of this heterogeneity for recent models of impetiect competition in macroeconomics.

RETURNS TO SCALE IN U.S. PRODUCTION: ESTIMATES Am IMPLICATIONS Susanto Basu and John G. Femald] Why isproductivity procyclical? That is,why do measures of labor productivity and total factor productivity rise in booms? The answer to this question sheds light on the relative merits of different models of business cycles. One recent class of explanations emphasizes the potential role of imperfect competition and increasing returns to scale. Measured total factor productivity then reflects not just technology shocks, butalso variations in input use. Robert Hall (1988, 1990),especially, has argued that relaxing the traditional assumptions of perfect competition and constant returns helps explain procyclical productivity. In addition, recent papers show that increasing returns and imperfect competition can modi~ and magnify the effects of various shocks in an otherwise standard dynamic general equilibrium model. In response to government demand shocks, for example, models with countercyclical markups can explain a r real wages while models with increasing returns can explain a rise in measured productivity. Perhaps most strikingly, if increasing returns are large enough, they can lead to multiple equilibria, in which sunspots or purely nominal shocks drive business cycles.z ‘The authors are respectively: Assistant Professor of Economics at the University of Michigan and Faculty Research Fellow of the National Bureau of Economic Research; and Staff Economist in the Division of international Finance, Board Governors of the Federal Reserve System, Please address correspondence to S. Basu. Department of Economics, University of Michigan, 611 Tappan St., Ann Arbor, MI 48109-1220; or J. Femald, Mail Stop 20, Federal Reserve Board, Washington, DC 20551.. This is a substantially revised version of a paper previously circulated as “Constant Returns and Small Markups in U.S. Manufacturing.” We thank Russ Cooper, Dale Jorgenson, Sam Kortum, Greg Mankiw, Stephanie Schmitt-Grohe, and an anonymous referee for helpful comments, and Barbara Fraumeni for help with the data. We particularly thank Mike Woodford for extensive written comments on an earlier version of this paper. Basu is grateful to the National Science Foundation for financial support and to the Hoover Institution for its hospitality. This paper represents the views of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or other members of its staff. 2 See, for example, Rotemberg and Woodford (1992), Farmer and Guo (1994), and Beaudry and Devereux (1994). For a survey of dynamic general equilibrium models with imperfect competition, see Rotemberg and Woodford (1995).

2 a m t m n e i i r i c p m o p d si f t m s m s i r a d i c I c r e s l i r ( F 1 t m m i c t p g i i m T p p e e i d c r p c U d i t c p b e i r t m i e at i c e d r s i s m p m c T f c w H ( D H P 9 e w e s l i r s h a s h d ma i s s e i r n i s s e d r M h t c im t n i a b s f p a p e t p d F at i a h si d r s a l p d r a f l i t f c p o b m c w o m e s S p e l a a t i s a d r t m t p e s a i r T v e d s g e a l r V a an m o g i s w p c i c v a af l s f o v b a v a s d a b t g o A p c w a b l v a j C L ( o d e r s d l a i t e p s a i a t a b p a b e

3 g o e m appears that aggregation biases can explain the lack robustness in empirical estimates using value-added data. Much ofthedifficul~r inproviding simple estimates ofreturnsto scale arises from the substantial heterogeneity d M t p r l g d he s m m a i f t h a ma m C t f m F s p s pr s i a p e m S h un the s e f p m m e d T t l s e p p i i h p d t u m p p w l c a h p e o e m e t p t p h p p i m p t i s l s i p m bu m a m c s u b c p e p c We m t p a s h s m c d he a d r s ( e a f r s a s s u c m s c s s d s e e d n i H general results appear unlikely: i h d p f m S [e e f t e r s r H s no m r c r s m st s g o v a S m c m a t i a m f i t p w our proposed solution in S 1 B a e n i a c m i f S c e a e m f our industries to various aggregates, demonstrating the importance of these effects.

4 S r i r c m m S c Estimating Firm-Level Returns to Scaled 1nt s q r g H ( m e r s t r e r s g o t v d e p b f u v a a m p M E R S a f l c m p r g o p i c l w p i i m e L T d s t w f p f Y = L . d s c i J t r s t c a t f p t f m m i g o e r s ym co g i g a p g T g r L =y + + - c ] + = y + This equation generalizes Hall (1990, equation 5.29). Hall’sequation, in t g e d S r a b n r p e p C m i t r s y e r a m c Increasing returns can take different forms, e.g., no fixed costs but diminishing marginal cost; or fixed c w f u s m c e c a co m p o m c An identity links returns to scale and the markup: c a p s f s r r c w ~ A 1d e s g d

5 construct c s ( e y i e e a p r m 3 p H g relatively small profits, equation (3) shows that p approximately equals y; large markups. for example, require large increasing returns. Given lowestimated profits,equation (3)also showsthat strongly diminishing returns imply firms consistently price output below marginal cost. S i m e s c t f r s m e c i n m p l g p c o e m m i e H s a i r v a p i c l a d r m t c l s D d v a ~ - { ~ ( (4) (1-sM) M V a l ap S r s i g o g w s i i r s e s m t r c t v a g g o i co m p i g d a to e ( +C + . T e e ( o inequation( e t i a s p t d r b p y b s R W (1995) a p av e s p r c zero. 6 B 1 U N I P A m s r v a c F i C F i d ap co D d h

6 V A a M Production Why do macroeconomists often use value-added data? A compelling reason is that macroeconomists are ~~pically interested in understanding value-added aggregates, especially GDP. Summed across firms or industries, real value added has the desirable property of equalling total national expenditure. Thus, aggregate value added is clearly appropriate for focusing on the uses of output. But as we now discuss, value added isnot generally appropriate for studying productivity growth, attempting to understand the s o c s g a v p m G r s y and markup p are the primitives of technology and behavior. Thus.they areconceptually the natural parameters for, say,calibrating multi-sector models with imperfect competition. On theother hand,aswenow demonstrate, value-added data generally yield biased estimates of returns to scale p i c t r e & = Y(~-cM)&v + + T u d [ d f c c that yc~= ys~, we f ‘ v = [y ‘7) T e g m T o v e e u v a o m T v i c t p co t p e m c e s b m o i z t e S p ~ c m ( w p s om d c b p o i ( J i p p S m t z e s p e e y( M f a a t p e i t q S p f t f s f Y =G Q . F l u d e w g p v a

‘7 t c g p i d t s ( g n e p v a r t = v + e e w r c l c g s a m t p f s a t r s a p o c l r G h d H, H h d o e w r i w - H re b y {1) Y’ = Y 1-y CM“ Thus, p corresponds to y“, the parameter ofien of interest to macroeconomists. For example, if G L t are m e r s e a p f G ma r a h o d t u v d p u e c w a a ma r e a g o c i f e ( H L “ To p c our results, we briefly discuss how large deviations from constant returns and perfect competition n g r f d 7 R W ( B (1995a) explores some ofthe implications ofdropping the assumption that G is Leontief. Note that the data do not support the assumption of a zero elasticity of substitution. Bruno (1984) reviews a number of studies and concludes that the elasticity is between 0.3 and 0.4. Rotemberg and Woodford (1993, Appendix III) conclude that a reasonable value for this elasticity is 0.7.

8 co v m Co m g m e i c i r a m w implicit collusion and countercyclical markups, for example, Rotemberg and Woodford (1992) f t a s m s e a q r R w r w g p r s f m s l d s T r s d c r p c s c m p m r m i h s r e s l m r s S ( e c p m in f m v t s s ne g m i a s r a s “ s m a ra e M ac m F ( t r m e 1 am c m r m e 1 (Schmitt-Grohe emphasizes, h t m u a an i l s e a l s 0 T q “ l l a a s m H w e m p a b i e r r p i i d “ m r m e i f in w t r s i r e m m g i w r s i r e b s f do s c c l a f s i f f p q i 8 M existing m a e r p w p f s c e ( w e s b V S p s c t r s m r e t in 9B F ( P (

II. Data and Puzzling Results We now a t f S I e r s p v l a f s m p a f t r a d w m e i a t S a r t f s c f e n s U e E m w r d a s d f i s d i e e Butapplying firm-level theory to aggregate data produces three puzzling results: Returns-to-scale estimates differ at different levels of aggregation; estimated returns to scale are sometimes strongly diminishing; and r s g o v v a d t p h a r t f s p A. D d p D J i t c p b e 1 T d p c s p d r a m e e c p t H t c ob p i i g o J G F ( d d t A d g d d b c s v f d F e i c o i growth rates as logchanges. S c p c e c c ( T c D a o i g l n d t ma w t p b e F c v e i a u d p r b i u ( a d s in u c w p g d s Ap v t p a d q f r c m w t d p d r

10 p p p c t i a e l s h e u r r F i c e u d t e v a e s t a c i i i p m a v i g o w t c e t l p a w f t g i a S t a e a i d t e c ag s S i r w c i i E i s p ( t p N S ( S S ( a o i b r w p c d s b t O R T i 2 r e e T 1r s e u d v l a T 2 r w a r s r i t c t a t f r e y f g d s e c t g e e y t r e p ( e y u c m v a p s t l s b p s O T e f c w a r i i w s i a f w t n g o s t r w t n v a N t s T 2 a s t e s f F a a W i a q r u o as e q r c w t i r s b w i

c e i b a B w i r T f s g o t p e w n m s s e a i r W m d s s e i r s m i r s e p e e 1 v e l e j e m m e E e d m 1 c j m e s m G q a i b p s r u point estimates are, for the most part, Iitile changed; the point estimates are (surprisingly) lower uninstrumented fortheentire economy, aswell as fordurables. Pointestimates generally risefordurables, The major effect is on standard errors. Uninstrumented, we can now reject constant returns at the 1 percent level t m g o C T 1 2 d f p E r s s l a t in industry-level data. Table 2, only durable-goods manufacturing industries show any evidence of increasing returns to scale. t p e 1 a t e e e w l e i ma r w g m e m m o t i d r g o w t i r c r v a p e e r s a s d p u v d n r s pa d m T f c s p R e o s We convert each e y u s a c i a S e c l e u s v f l e w a s e c a e i u s f v a r v E s e m u ( d p e i w a n c e m a s d s e w a h o 0

12 one. Returns to scale are larger inthe instrumented regressions (opposite what we expect ifthe probiem ispositive feedback technology’s to input use), but even there, w a v e p e m e s 1 H e l t o (S f i e s 5p l t si g t o n d f e s d r i s e profits, which are not observed in U.S. data. Furthermore, the possibility of replication suggests that m s e a c t b up t r s s l c c t p c f c g v r T p e u v d c s i g e s e l e n r c r in r most striking estimate is for manufacturing non-durables, w p e s r s a z s e l h r l un T 2 s a s p E s t v e s ap o b b e d b g v e s d r c o v ( s i i c p a a a w c p S a p m g e e i t v e s b u T 1 h d v a e s i g o e T f t p d b t e e d a f r T 1 2t l t p F a r d W he J i i s y s e t l s e ( c T h h p g o v a e d

si f s r t i r S w a s e T 2o s as s d d r s T r s W t e g o v a this paper attempts to r t p C \ e l Consistent with f p literature r a r r s e d t d l a e m F s w c p f s i i r m ( e r l i r u d t m v a w T 2 r d r w s d T d a p b e e r r i g o g t i r c d r r a a s s s s i r w r r t h l s o ri r n c d o I v n e r b r F l s p s t b l i r s c e e v o e b d b B ( p M C e s r s r r s S instruments are not only weak, but are bad (correlated with the error term), then the resulting bias is likely to be larger in the r r d c b d f v e e v w i f v output necessarily covaries with the disturbance term. In S a t i P o r r t p v s p u f e p e i f r f n t s f e p D H P (1 f g o r g

I 14 m a 1 f t r i l b t i w t m t p r s c P i t l co d i o F h i p e a r l s v a o a o N 1 s l h more cyclical than capital stock, measured productivity b c p S s o r w i l f p d r e B ( e r i r a s i l s p U H m v 1 r a we r s a B E R ( s s q t m d e r e b 0 s t sp r s t T T s a a t p p a i r r f c v i i r e l h p t a c v c u t d s v e S v c h e t p F e a i r a d r f i d S c e d b i a e s c u e a a i r e a i f i r T v i c l a v a g o r same d c a d r d d p a c c u u h s l e r s e e B ( t e c v c u e we g r s r 0 F m p e s e c r s e B H C ( p a c r e s b 1t a G R ( a e c S e B ( B E R ( (

returns are needed to rationalize the o l d e s w a g i F Ca ( f n p r s h I ag T i t e e p s a i H f t e c F i s e s s a m p S F ( C L sp w i i a i i r s a ex v d g d t p s e s p b s d T c t s d c t e III. Aggregating Over Firms Ma m a p s f ma a t r p c d s i w f t s f a a m t e i c t f a g o v a D i T a g output g v g a Z @ = ‘i”@i i (11) dv = ~ Wvi”dv.i i w s f r t i r s f n v a ( m in c t i v a We a t t e p t e t r f F (

m p c T a i g i i e w f i & = ~ ‘ (12) S i e ( g rates f D ~ we r s ~ ~ r ~~ + R + w e w a t s A g o g ( d t s r s “ f m a i g a o g d re w c p g f r s h a g i D s “ p r c d b c r r ~ H b p p c v a s i e e r w a v g as r d c i m “ v r s t a t h = fvW&v+ Rv+ I . (16) d v w a t s d A a f b f r p r d a s T e d e e e e t i c s t a f t f a h H c f r th S B F ( a c d

(17) As with gross output,thevalue-addedreallocatioenffectRY’reflects the “ b r s c i g 1r v i i input use within f t i c The aggregation equations (14) and (16) have several implications. First, aggregate demand instruments need u w r e h i p t d shocks need not lead to equiproportionate changes in input use for all firms within the a e g d s u f w i some firms may m s t o c p S a p e p d S h c t e d l a l d A e p c e e r f d l a H a e f p h e ina i s p a i r s industries. F a e likely to d g o v a p explaining t y d r ” T p e a s c co s p b r r s T n a s r b i o d p r d e s Hence, e s s p Fourth, it b u w p w c r s A c o m t f e a v r s if e d d l a s a r t p S We g e w g b e f p t v d v v T t g p i b e m w p

c n t m s a f u c w f “ v a w f p f p c l w t f c b f U t a t F ( w a e a p f F t a s h s f n a e m o S d f d d m p (c d d r s f ad v m p d u T r a f a c a o IV. Aggregation Results T s e e i a e i p s a m r s t e m M t r p y m a e f p t T i d e ( u s e r s c e e “ c g v g s t a t a g a g o v t r r a i a e o o a i w i a a e s p u e a p c e r t s s c t r T 2 i a d o b s e e o h i t c s a e t e a c i a a c a f f i T o b e f l p f a d t pa d r i i a p r p u r b g p a e i d i . v c e i p r e a t

e in ( n h i v c gr v t u i i e T t r m d i u e c w p o a a r n p e aw e c c price s n 0 l v a g o c l c s n l v - 0 l g o G d s p a r ma d o i p p p T r t v e i t p i o v i a l no reason to expect that the instruments are any more valid at an industry level. InTable 3,we present aggregation-corrected aggregate results, u u i e b w g o F o e t p e c s c r S c i v v c d v e t r r S s t r e i e d r b g o v a A t i a c r m s e i r T p in durables, where all of the estimates show st s i r n e s d r t t s d w e t n T 3 p s s d r O n t e d r t r st d f o c r s d t shown T s s e u r h w n t s e f d s r i s T r a h s s d t T A p t r r d

O a e a p l e s i e r s U v m ap n t r c s e s c a For durables, corrections to value added make less of a difference. s v re t p i n r u d r T 3 s t c a r i a f a d n i r T 3 r e a f p economy? Without detailed firm-level data we cannot be certain, but can suggest the factors that matter. Suppose equation (14) g o a a o f l of an individual industry i. We then aggregate over industries to the level of the entire economy. Omitting the technology term, this gives: (18) u wi r t ~ p c s aggregate input g e i s r p e pr f h l e of returnsto scale; systemic shocksto dx may change factor p l lo f e h r e a a s l r a i c l m p ( w at c increase their inputs relatively more. Thus, industry-level reallocations associated with idiosyncratic s s n h s s S & r a( s w = + e co r c o a e o i e ~ S ~ e R g With s r e ( ag g a o e p n s h a f p un s h r d t w w s e e t f w c s

21 - R = ( - ~ Wi(uiqi-fi)”(di-h) . i When we e e ( v r p d s magnitudes of the parameters. As one case, suppose the ~i equal zero. Then equation e ~ + i. These are s quantities estimated in Table 2, with the same biases. In c ag e s r T — a s c t s t 5 ~ e m o s t c z t c aggregate estimate from equation (20) correctly estimates ~, without bias. Then estimated R correctly controls not just a i a a f f i T e T 3p c w a f r s w e T 2 (though relying only on di d n d c e s w c t s o Ne p t c c t h a c e p b e a p F T 2 s ~ n w a m l - ( a p e a m e g 1 S r s s a r R pr r r a s s w a i p T w a 5 w z i n co b unity.F e l c s c i t b f o f e p s r t s t s a a a c r s s t a e e t p p in S 1 Ne r p c a p t c s t a s c g ~i d d h i a v i as m p b a u i a s b e r s c f - w d e - E q l ( s s

22 a re f w i r the e q r W e a c e f b c s e q V. Implications for Calibrating Macro Models W i h p a r p o D m m p e u a p f G t s f p s w a s he t a p s c a a p f p e a i he a t q u a s s e c i he m a m w t s c u o a d w c a o m a c p m In a p d n i e T e s t t g a q p a d p m e m B ( w s T a c c i by ic[O.1]. E m e u c c [ s s b c = E(ac + c s x + w(T-x) where / is an individual’sendowment of time, x is p w w We o l s / = a V’>0, v < a v > I s p e c a s e w A l d 1 , f p f [o Y = . , w y y t f m p a m a m c c

A o f p g t a t a l n p e e c c e p i p m e z p f e f m ~ e r s wage in this economy is thus ~ALY-l . A~Y-l . P E t e s t u e e f c u m v =a . F C J ( e n d m s Pa e s t s e a d < d (24) l =y dlnL C ( h y m s l I r a s (d m c l u l d c h rise with aggregate labor supply. t e s s i more than o d s a l G a a v a s h n l e h e P e t c p d w t m e d r s r f w e usinga d An E w H now consider e w p e c f e p w c r s H f a h l 2 noted a i r d m c ar f m w t i r a ac m ( F 1 c e s i r s m average cost exceeds marginal cost, and iscompatible with any slope of the marginal costcurve. One can calibrate the slope of the marginal cost curve from estimates of the degree of returns s a that there are no fixed costs.

pr t o T = A w a B f c p m c H u 1s > e b h p h w f e u c ( e p H h r T 1 w w f 2 Wl=A1. r t h a s a l h d w r g L L T a w + (l-s)AZ. F s d a o s r m a s z s e “ s e e (2) or(6) u a f t e B ( dI.nY AI-AZ 1‘1 Y ‘ ~ = 1-se , w z “ [ S positive, ~ > 1. Let us suppose that c,, Al, and Az are c ~ e re f y e ( p s e w ( g d w e m e E r S f c i l s w m “ r U t s e w a af n h 1 t f w m h w e H i w b c b c < x + + w N t m l H l a w u t e f a p f c i c l m a a s S s b h a i d S ( a t e e f s l r a e T ( c t as T a d c h s s s e m d w s p ac e e c r p

25 S c Y ( h s c p p w c a w i is not reflected in a change in the marginal compensation for labor. Then we can see that equation (24) is not satisfied, and this e a u e T a t e p l p p l co h e s d o e m e t c a a r c p e f a d d a d economics at work. R ! s t i r h h e a w o w h f a l w e l s a n h w w r w e T w m t l s d w k t i w k h p d w S p i ( a u f c b V = aE =a +( l c m e t d < d dlnz dw = M E ( s t t c t e h m e u j s c o e ~e a a s c v b c im T e s t c g l a c o m a d p s h N s f e

s h g i F n t s s s u l d r s e f a d } c c c e t p c c p b a l p a l co ( a t f s Ao m c w a y ( 1 m e would be unable to replicate this behavior. Second. this example shows that the aggregate parameter also can fail to capture the relevant e m e although italways does so inthe s e r t m e p a p a r p “ m p l u c t a p “ m p l H l r e d t c l s T l s i re i b m a m f p l t H d would occur, since there is no completely-accepted theory of imperfect competition that explains howthe same input can have different marginal products in different uses. VI. Conclusions both empirical and theoretical grounds, heterogeneity p a i ma A e r s w across r d i a i a p c d r T l a w d r c e e s e p T i r a c s a r l m p b e w s l i r A r f m r m m c e e t f A p e p pr u c f u c e t f s v u e a i i r r a d r e t f n i a b f

particular. aggregating across imperfectly”-competitiveproducers can explain the puzzles we identif}. since similar factors employed in different industries can then have different marginal products. e d~ i a h l r s a i output of durable-goods industries isalso more procyclical than the average. Thus, the additional factors employed a b h m p t h the average products of factors in use, leading aggregate productivity to be procyclical. If long-run returns to scale cannot be lower than one. this story r i h i r H a e a m d i r ec m m as evidence for large increasing returns at a representative fire. On the other hand, reallocation effects can p o d m pr co s i e p a i d r H a i e d inresultsatdifferentievcisof ag d t d o d s p c e r e f avaiiabie data. T r s t a e i d s a t p w ap c r s a c r a m d i r T r c t p s B H C ( M ma i d p p w e H r p w a returns to scaie over ail firms e C re s a b e r scale; c o v t s i m m i t p as m w m h f producing c r s T e d m P e a r r i i r p t p a s f F c p d r s t ec w e u a w c As e S Vs t t i c f m

re T m e d a e m e h and w d no f e T i r h ma m n c m d g m i c h c a i f p U t f t w s b d a c r l n a r m p w i r a r f p ac r e economic fluctuations. Ascertaining which paradigm provides better macroeconomic insights is an important task for future research.

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F R G J “ B C A S H Econ. T ( 1 4 F F A C M p 1 G R E s and the Form OJt)le P F A No 1 H R “ R b P M I J ( 1 9 .“ p S P R Gr e P D C M P 1 H R J D “ P I B E ( 1 J D G F F B Productivity and U.S. Economic Gro}\’th. C M H U P 1 K L L S “ R E I B P E A fM 1( 2 N C S R “ F R E S S P In V E E ( 9 N S “ R b P and Marginal Cost in U.S. IndustV’:A Contradiction.” J ( 1 1 1 P R “I H P B C A C A M Y Y U A 1 R J W M “ P E A D E A ” J ( 1 1 and . “ C E E P I E A M C M 1 . “ G E M I C P M F B C R e T C P N P U P 1 Sc S “ F M F S E M W B G F R S 1 S D S J “ V R I T W P 1 C M N J 1 T R “ B P E Activi~~IMicroeconomicS). no. 1(1989).

Tablel Aggregate Estimates Private Manufact Manufact. Manufact. Econom}s uring Durables Nondurable PARAMETER Two-Stage Least Squares Gross O 1 1 Y ( ( ( ( ] ] Implied Value- Added y’” ( ( (0.21) ( Direct Value- 1 1.10 Added (0.38) (0.33) ( ( Estimate Ordina~~ Least Squares 1 ] I Gross Output Y ( (0.04) (0.03) ( Implied Value- 1 1.41 1.40 Added y’” ( ( ( ( 1 ] D Value- Added ( ( ( ( Estimate S P 1 b p f p s e e w are converted in s u e ( t row e e I p o government defense spending, and the political part>’ofthe president. with one lag of each.

Table 2 Weighted Average of Sectoral Estimates P Manufact Manufact. M E uring Durables d PAWMETER T L S Gross Output 0.97 0.92 0.73 Y ( (0.1 Implied Value- 1 Added y“ ( ( Direct Value- 0.94 0.87 Added (0.22) (0.15) ( Estimate I Ordinary Least Squares I I G Output 0.83 0.93 1.07 0.77 Y (0.04) I (0.03) (0.02) I (0.05) Implied Value- 0 1 1.28 0.66 Added y“ (0.07) (0.05) (0.06) (0.06) D V 0.54 0.66 Added (0.09) (0.08) (0.07) ( Estimate S P 1 b p first row presents gross-output-weighted averages s equation industry estimates of equation (2). The second row converts each industry estimate u e ( t a w v w t p v e e ( I p o g d spending, p p p w e

Table3 A Estimates Corrected for Reallocations Private Manufact Manufact. Manufact. Econom~7 uring Durables durable Gross Output 1 1 0.96 Y (0.05) (0.04) (0.03) (0.08) Implied Value- ].o~ 1.26 1.33 0.87 Added y“ ( (0.16) (0.11) ( Direct Value- 1.03 1.19 1.36 0.81 Added (O.18) (0.18) (0.15) (0.27) Estimate Sample Period is 1959-89. Estimated aggregation terms are subtracted from gross output grolvth and ~’alue added growth before regressions orI input gro~vth. For further descr~ption. see text.

Al Appendix I Detailed Derivations of Equations in Section I T a d e S 1 p detail. We begin with the following production function for a firm: Y = . Y is gross output. K and L are primary inputs of capital and labor, while M is intermediate inputs of e m T is i s t a p f h d y c l i g Di p f (Al), we can e g o (A2) L l r l t u c so q v di g r r d F r a co n u e o r t E s t o g d t s a w i g w w depend on the output elasticities. The o e e d r s ,=[~)+(y)+[y). (A3) S t f h s d m p g m p t f m T e f c m i ?

K,L,M, w a L m with the interpretation of marginal cost, and PJis the shadow value of the Jth input as perceived by the fim~. (As discussed below, this shadow value ma}’or may not be observable as the input price.) definition. the markup p of the o p marginal cost is Hence. we can rewrite the above equation as: ‘[:1[$1=[$ $) Y [p Py (A5) : J =K, L, M. = p PYI. = ‘s” T the e of output with respect to an} factor J equals a markup p multiplied b~’the share of t i in total revenue, s). N that the price of capital, PK,must be defined as the rental cost of capital. It does not include possible profits, which generall}’are also payments to capital. With perfect competition, where p = equation ( j s e o r toan} input e i s r U i c e o exceeds the r s U f l e ( r e (1 1 xJp~ . AC (A6) Me Y Thus. cost minimization implies that returns to s ye ratio ofaverage to marginal cost. Increasing returns can take different forms. e.g.. f costs but diminishing marginal cost: or fixed costs ~“ithflat or upward sloping marginal cost. Under cost minimization, we can use equation (A6) to write an identity linking returns to scale and the markup:

(A7) * ‘ = 3 w Sx s p t revenue. Equation (A7) then implies that the output elasticities, which equal psj. also equal ye,,where CJis the share of payments to input J t c H write the t d o @ = Y“lLcLdl + + - C ] + (A8) = y l & * dt. a cost-weighted g r v i w f t s r t r s s H ( s markup pricing the measured Solow residual is procyclical. even with constant returns and no change t n a d r p m d s p f m i S t f f v t q f o m p L c s r c f H J ( ( a c s however. capital isquasi-fixed ( s m p c e st r r r c s r theusercost c Ha f s m p i g w u p s v c m s r i e c g T p s e r s h r F a quasi-fixity affects only the period-by-period computation of the input shares. not g r c o q i S t s c a f T

A4 ap e c f t m s l s S mi r r c strongesteffecton capital’sshare.Butsince g r c a u b c e in measuring capital’s share u c s b a s c p C L (1 p s i t m b q l o 3V0 e co c d a c p i c l S “v a c d ‘v=[.:~.Ll&+[c:~cLld (A9) also rewrite (A8) as: l @ = Y [(l-cM)*v ‘ cM~~+]df o (A1O) m tr w p u b s y fromboths r w p e (All) I g o m r v a where measures of real value added attempt to subtract from gross output the productive contribution of intermediate goods. Hence, g o s w v a “ l l without power” (Domar 1961, p D u n r v a least two reasons. First, s s v a s n o a n simplification for macroeconomists f even at a sectoral level, on value added and primary inputs of capital and labor. Second, b n

a d W d d s v a b k gr d u c t p w m o o i i D i g r D i o v a d * ( ~ @ ~ ( . - M) V a l ap S r s i g o g w s i i r s equa]ityshows that m r c t v a g g o i S f e ) f (A13) S y e p t e a w t c r s m g v a e g r p i p t p

i e t c (1 n u c c r p r - c a D i v a un w v t n c p s c p s a c w e r s s t profits. Inthe presence of markups. h e s i i i ( d a v g I } a c s g o p c i g a e o r i i e r s markups. thiselasticitey r s H s c m e a v a ar v pr r c v c i g u r w c t T H a u c r c S r s i g d g t w v d c m equal 1.or i i a u f p o p m e a e p w c i e f a g v a n t p of interest may not be d r s g o C f s (Al): Y = V . ( Ma u m s p f a s ( D e T t p s i r i r i r i r V H c p E t d s i o l c o i i

ad d m c S a r s m h t e r s t v e h o e y w r M (A17) Y = y + ~ H“ G h d V H h d e (l f orderconditionstellusthattheoutputelasticitw}i~threspectto m y H r t e y = - + (A18) re b y t 1 (A19) = Y — 1-y CM9 W i r v r s e gross-outputreturns s s y a w a o A k o e a mu d j e s e i u c c c “econom}.-wide” returns to scale. as small increasing r p l t l increasing returns forthe e o S e f o p i number s e s a r r s y output ofthe pre~’iouss i i p i c l the percent change in n i o ( s

As = - +YcMdyn.l (A20) = - + y - + (Yc~)2dYn-2“ s intot e d j i S i s i a T g i w t c w g r a p i T plausibly the appropriate concept for calibrating d r s p f ao m m N t c c i s d c r “ g f g o v T i s r l r s g o t s o p g satisf~~final demand t p p m g i i p o g N t d s t t a l e c b E w t v r s s g r i i r t o V d i o g o in i N t e h a w m g p v a w d i d f (A21) C t e d s v a s o el w r m v w (A22)

T c i p e a e ( ( s t e r d b g r ofthe standard measure ofvalue added and a measure w a a m p o F s d equation( e ( f y‘CM y ‘ + ‘M 1 1 -CM [ t z p t a w [ = y + (yV-l) & +~ . (A24) 1 M T e S p d a e

AIO Appendix II Data Sources and Methods We use unpublished data provided b}’Dale Jorgenson and Barbara Fraumeni on i g output and inputs of capital, labor, energ), and materials. T i m industries and 13non-manufacturing industries.z~T sectoral accounts seek to provide accounts that a extentpossible.consistentwiththeeconomic theog ofproduction.These dataarea~’ailable b w w a i q q a essentiali}rinvolves taking a changes over time in input composition. Computers. e a h s d t f s t d f S e andjanitors make a different m c o i r f p a di c ~ n f J d e d c t c a d d instantaneous shares with averages in periods t and t-1. This Tornquist approximation is exact if the u p f t o t p a f s ap a f e r p c f J (1 ( and Caballero and Lyons (1992). and compute a series forthe usercost ofcapital r. The required PaJ”ment for any type of capital. PKK.isthen rz~K, where z“K is the current-dollar value of the stock of this t}’pe of capital. In each sector. we use data on the current value of the 50 types of capital. plus 24 The manufacturing industries match the ~vo-digit classification, except that transport equipment (SIC 37) is divided into “motor vehicles” (SIC 371) and “other transport equipment.” The nonmanufacturing industries comprise agriculture. metal mining, coal mining, oil and gas extraction. m m co tr c i i e ies,gasu t f i s ns

Al I inventories. distinguished bj the F3EAin c national product accounts. H t a c u c c - - ~ = (p + b,) = 1 to 52. (A25) p r r r c a d rate, i c p v d a ~ c r assume that the required return p equals the dividend y o u d I ~f D J e t t i a p a 3 p G un a w d y a c f experimented with several alternative measures of the capital cost, as discussed F ( E a z p t r s e l e results. This is unsurprising, since economic profits do not appear large m c a D i u i a g o s s d c e n a i p s e r s e a e a s i d c

A]? Appendix 111 Are the Hall-Ramey’Instruments Valid? This appendix suggests that they are not. because they are correlated with aggregation effects. H w d s i I v a l a v h I a T a p r c S e l a c g a R v - t ( d S 1 a d t t c r d t r t a t H i t b p m s l ( T 5 i v in u w d t n t u w a t c s d t i i r o t a b M i a r a i d t e s s e l ) c a t i l e r s i v w i e i r s i v c t t i e i ( t f f t s i g b t r e l c r t c i c t t b w T s r u UNINSTRUMENTED estimates of industry paramers. w T s r u I e i c o v g of the world p ( ( g r g d s d d ( ( p p p p (

c w p u r i are not a w a e T p e as a w example. the coefficient on the cument oil price is significantly negative at the 0.00! level; the coefficient on the lagged change is significantly negative at the 0.01 level for value-added and the 0.05 level g o G d s p a reallocations for manufacturing durables, but is otherwise insignificant. as is the political party of the president. These results thus v e importance of the theoretical point that the instruments we (and many others) use are not valid as instruments at an aggregate level.

A14 T Al Dependent Variable:EstimatedGross Output Aggregation Terms(Uninstrumented E Entries are marginal significance levels for regression on each of the instruments Private Non - Manufact. Durables Non-Dur. Economy Manufact. Manufact. Manufact. oil 0.034* 0.012* 0.113 0.455 0.019* o.o~]* oil(-1) 0.015* 0.181 0.600 0.056 gdef 0,382 0.587 0.189 0.001** 0.945 g 1) 0 0 0 0 0 0 0 0 0 0 p 0 0 0 0 0 Dependent Variable: Estimated V Added Aggregation Terms (Uninstrumented Estimate) Entriesare marginalsignificance levels forregression i P - M D N E M M M 0 0 0 0 0 o 1) 0 * 0 0 0 0 g 0 0 0 0 * 0 g ) 0 0 0 0 0 0 0 0 0 0 p 0 0 0 0 0 * Significant at the 5 percent level. ** Significant at the 1percent level

T Dependent Variable: Estimated G O A Terms ( E _ E m s l r i P - M D N E M M M O.000** O.000** 0 0 0 o 1) 0 0.007** 0 0 0 g 0 0 0 0 0.400 g ) 0 0.577 0.779 0.334 0.300 0 0.439 0.370 p ) 0 0 0.794 0 Dependent Variable: Estimated V A A T ( E E m s l r i P - M D N E M M M 0.001** 0.014* 0.015* 0.247 0.168 oil(-1) 0.005** 0.002** 0.250 0.777 0.005** g 0 0 0 0 * 0 g ) 0 0 0 0 0 0 0 0 0 0 p 0 0 0 0 0 * S 5 p l l* S 1p l

1 I F D P I N m t B Returns to Scale in U.S. Production: Estimates S and Implications F Mexico’s Balance-of-Payments Crisis: A Chronicle G C of Death Foretold E M The T C C B G K Ba P C R H R I R A G K Di i a L C L L Pr P B a L C B P M H U O P I P A P W M An Application to Oil Prices During C C T M P E E S K B St C M R C W C I D M U A T S P L M I E E J G f I F M U M E I A B E a M P S R S D R b F F R i N R R I E F E T S L f M C C E M J F E T A ‘ P a r c I F D P D In F S B G F R S W D 2

2 I F D P I u & R P P C E R G C R 95 A P P S A F A C T E C J J M U S M I A G B E Hy S C M M R Z 528 I C M Guy V.G. Stevens P A 526 U I C U H N E M Zhu Ma E T I 1 R C R F J W E D In M I E M H C A F M l G A R E R R A B T M E A D S M T M M M C A W W M P E A A B C B I I P L G T E N S