Explaining Toal Facor Produciviy Ulrich Kohli Universiy of Geneva December 2015
Needed: A Theory of Toal Facor Produciviy Edward C. Presco (1998) 2
1. Inroducion Toal Facor Produciviy (TFP) has become he choice measure of produciviy TFP is ofen referred o as he Solow residual, and i is generally jus ha, namely a residual TFP is raher opaque as o he naure of he phenomena ha i perains o measure I is difficul o reconcile TFP wih various models of facor augmening echnological change Is echnological change neural or is i biased? If i is neural, is i neural in he sense of Hicks, Harrod, or Solow? 3
1. Inroducion, coninued Do increases in produciviy, as capured by TFP, necessarily imply increases in real wages? Wha abou he real reurn on capial, mus i necessarily increase oo? The purpose of his paper is o sor ou some of hese quesions and o show how TFP can be decomposed ino he conribuion of labor and he conribuion of capial As an illusraion, some esimaes for he Unied Saes are repored 4
2. Index Number Approach Toal facor produciviy can be defined as he par of oupu growh ha canno be explained by inpu growh Noaion: y, p quaniy and price of oupu x K,, w K, quaniy and price of capial services x L,, w L, quaniy and price of labor services 5
2. Index Number Approach, coninued A sae-of-he ar measure of TFP is given by he following index: (1) 1, 1, 1,!!! = X Y T where (2) 1 1,!! " y y Y (3)!! " # $ $ % & + + + ' ( ( ( ( ( 1,, 1,, 1,, 1,, 1, ln ) ( 2 1 )ln ( 2 1 exp L L L L K K K K x x s s x x s s X (4) j j j y p x w s,,,!, }, { L K j! 6
2. Index Number Approach, coninued Using he daa of Kohli (2010) for he Unied Saes, one finds ha TFP has averaged abou 1.09% per year over he period 1970 2001 While his is useful informaion, i ells us nohing abou he naure of echnological change, and wheher i benefied capial or labor, or boh 7
3. Producion funcion approach TFP can also be defined wih reference o a producion funcion This acually leads o for four inerpreaions of TFP 8
3. Producion funcion approach, coninued 9
3. Producion funcion approach, coninued Le µ! "ln y " be he insananeous rae of echnological change; we hen have: (8)! f (")! = µ y Following Diewer and Morrison (1986), we define he following index of TFP: (10) T,!1 " f (x K,!1, x L,!1,) f (x K,!1, x L,!1,!1) f (x K,, x L,,) f (x K,, x L,,!1) (inerpreaion 1) 10
3. Producion funcion approach, coninued Assume ha he producion funcion has he Translog form: (11) ln y " T = # + " +! 0 KT K (ln x ln x K, K, $ ln x + (1 $ " )ln x L, ) K 1 +! TT 2 2 L, + 1! 2 KK (ln x K, $ ln x L, ) 2 + The inverse inpu demand funcions: $ ln f (%) $ ln x (12) s = = " +! (ln x # ln x ) + K, K KK K, L,! KT K, $ ln f (%) $ ln x (13) s = = (1 # " ) #! (ln x # ln x ) #! L, L, K KK K, L, KT 11
3. Producion funcion approach, coninued 12
3. Producion funcion approach, coninued 13
3. Producion funcion approach, coninued TFP can hus be inerpreed in four differen ways: (1) i is he change in oupu made possible by he passage of ime, holding inpu quaniies consan (2) i is he average of he insananeous raes of echnological change of imes -1 and (3) i is he average rae of echnological change beween imes -1 and (4) i is he par of oupu growh ha canno be explained by inpu growh In he Translog case, all four inerpreaions are equivalen 14
3. Producion funcion approach, coninued Esimaes of he Translog producion funcion from Kohli (2010) are repored in Table 1, column 1 TFP compued according o (15) or equivalenly (16), (17), or (20) averaged 1.02% over he period 1970-2001 15
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4. Impac of TFP on facor renal prices Wih he index number approach, one does no need economeric esimaes of he parameers of he producion funcion o measure TFP; ha makes i very aracive On he oher hand, his approach ells us nohing abou he naure of echnological change, or abou is impac on income shares or on he wo facor renal prices The economeric approach is more revealing in his respec The sign of φ KT is essenial in deermining he impac of he passage of ime on facor shares 17
4. Impac of TFP on facor renal prices, coninued If φ KT > 0, as i urns ou in he U.S. case, one can say ha echnological is pro-capial and ani-labor biased, in he sense ha i increases he share of capial over ime and reduces he share of labor Capial is hus favored a he expense of labor Wha abou facor renal prices, hough? Clearly, if echnological change leads o an increase in oupu, for given facor endowmens, and o an increase in he share of capial, i mus increase he real reurn o capial Bu wha abou labor? 18
4. Impac of TFP on facor renal prices, coninued 19
4. Impac of TFP on facor renal prices, coninued 20
4. Impac of TFP on facor renal prices, coninued As long as he echnology is progressing, he firs erm on he righ hand side is posiive If φ KT is posiive, echnological change is ani-labor biased I migh even be ha φ KT /s L, > µ, in which case echnological change would be ulra ani-labor biased: echnological change would hen lead o an acual fall in he wage rae even hough echnological progress would unambiguously increase average labor produciviy 21
4. Impac of TFP on facor renal prices, coninued As i urns ou for he U.S. case, φ KT /s L, < µ ; echnological case is hus ani-labor biased, bu no ulra ani-labor biased Noneheless, he rae of increase in real wages is less han he rae of growh of TFP and of average labor produciviy Over he enire sample period, real wages increased by abou 46%, wih 27% explained by echnological change, he res being explained by capial deepening Alhough he economeric approach yields much richer resuls han he index number approach, he fac remains ha i sill does no each us much abou he naure of he echnological change process, or as o why echnological change is ani-labor biased 22
5. Disembodied facor augmening echnological change 23
5. Disembodied facor augmening echnological change, coninued 24
5. Disembodied facor augmening echnological change, coninued 25
5. Disembodied facor augmening echnological change, coninued 26
5. Disembodied facor augmening echnological change, coninued 27
5. Disembodied facor augmening echnological change, coninued Esimaes of (31), based on Kohli (2010), are shown in Table 1 Technological change in he Unied Saes comes close o being Harrod neural TFP, compued on he basis of (36) or equivalenly (37), (38), or (41) averaged 1.02% per year beween 1970 and 2001 28
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6. The decomposiion of TFP beween labor and capial 30
6. The decomposiion of TFP beween labor and capial, coninued 31
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7. Facor augmening echnological change and TP flexibiliy 33
7. Facor augmening echnological change and TP flexibiliy, coninued 34
7. Facor augmening echnological change and TP flexibiliy, coninued 35
7. Facor augmening echnological change and TP flexibiliy, coninued 36
7. Facor augmening echnological change and TP flexibiliy, coninued 37
7. Facor augmening echnological change and TP flexibiliy, coninued 38
7. Facor augmening echnological change and TP flexibiliy, coninued 39
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8. A parsimonious and ye flexible model 42
8. A parsimonious and ye flexible model, coninued 43
8. A parsimonious and ye flexible model, coninued 44
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9. The impac of echnological change on facor renal prices reexamined, coninued We can now explain why echnological change is ani-labor biased in he case of he Unied Saes As shown by (22), echnological progress mus increase he real reurn of a leas one facor, bu no necessarily of boh Take he exreme case of Harrod-neural echnological progress, which is a reasonable approximaion for he Unied Saes; in ha case, echnological progress leads o an increase in he endowmen of labor measured in efficiency unis Oupu necessarily increases, and so does oupu per uni of labor (average labor produciviy) The reurn o capial mus increase as well since in he woinpu case, he wo inpus are necessarily Hicksian complemens for each oher 47
9. The impac of echnological change on facor renal prices reexamined, coninued The reurn o labor per efficiency uni mus necessarily decrease because of diminishing marginal reurns; by how much depends on he size of he elasiciy of complemenariy If capial and labor are srong Hicksian complemens, he reurn o labor per efficiency uni will fall by a large amoun, so ha he reurn o labor per observed uni may decline, even hough each uni of labor has become more efficien! 48
9. The impac of echnological change on facor renal prices reexamined, coninued 49
9. The impac of echnological change on facor renal prices reexamined, coninued 50
9. The impac of echnological change on facor renal prices reexamined, coninued 51
9. The impac of echnological change on facor renal prices reexamined, coninued Looking a he resuls for he Unied Saes, i is clear ha echnological progress leads o an increase in he reurn o capial since all hree righ-hand-side erms in (80) are posiive For labor he firs wo erms of (82) are posiive, alhough hey are close o zero given ha echnological change urns ou o be almos Harrod-neural and ha λ is numerically small; he hird erm is posiive as long as ψ KL < 1/s K, which indeed urns ou o be he case So we can conclude ha echnological progress also increases he reurn o labor in he U.S. case; noe, however, ha because he share of labor declines, he increase in real wages is less ha he increase in average labor produciviy, or of TFP for ha maer 52
9. The impac of echnological change on facor renal prices reexamined, coninued Technological change in he Unied Saes is ani-labor biased because i is mosly labor augmening, and because he Hicksian elasiciy of complemenariy beween capial and labor is greaer han one; hese wo findings ogeher explain why echnological change has a negaive impac on he share of labor This could no have been inferred from he mere finding ha φ KT is posiive: echnological change would also be ani-labor biased if i were Solow neural and if he elasiciy of complemenariy were less han one 53
10. Generalizaion o an arbirary number of inpus 54
10. Generalizaion o an arbirary number of inpus, coninued 55
10. Generalizaion o an arbirary number of inpus, coninued 56
11. Conclusions In his paper we aemped o explain TFP in erms of disembodied, facor augmening echnological change This led us o come up wih five differen inerpreaions of TFP: (1) i is he par of oupu growh ha canno be explained by inpu growh (2) i is he change in oupu made possible by he passage of ime, holding inpu quaniies consan (3) i is he average of he insananeous raes of echnological change of imes -1 and (4) i is he average rae of echnological change beween imes -1 and (5) i is a moving geomeric mean of he raes of facor efficiency augmenaion In he Translog case, all five inerpreaions are equivalen 57
11. Conclusions, coninued We have shown ha in he case of a TP-flexible Translog producion funcion TFP can always be inerpreed as he oucome of disembodied, facor augmening echnological change Indeed, we have proposed a convenien way o derive he facor-augmening raes of echnological change from he esimaes of such a Translog producion funcion We have found ha echnological change is almos Harrodneural in he case of he Unied Saes, so ha TFP is overwhelmingly explained by labor 58
11. Conclusions, coninued Furhermore, echnological change is ani-labor biased, in he sense ha i ends o decrease he income share of labor; his is due o he relaively large Hicksian elasiciy of complemenariy beween capial and labor Noneheless, echnological change has a posiive effec on he reurn of boh capial and labor, alhough he benefi o labor is less han wha TFP or average labor produciviy would sugges 59
Thank you for your aenion! 60
Growh facors 1970-2001 Quaniy of capial services: XK 2.25706 Quaniy of labor services: XL 1.70513 Price of oupu: P 3.76623 Quaniy of oupu: Y 2.52563 Price of labor services: WL 5.50201 Toal facor produciviy: T 1.37071 Capial componen of TFP: T K 1.01850 Labor componen of TFP: T L 1.34581 Capial efficiency: ΓK 1.06789 Labor efficiency: ΓL 1.50832 Labor share: SL 0.98628 Oupu per uni of labor: A Y/X! 1.48119 Real wage rae: M W! /P 1.46088 Relaive capial inensiy: X X! /X! 1.32369 Relaive efficiency facor: Γ Γ! /Γ! 0.70800 Capial inensiy in efficiency erms: K X Γ 0.93718 Approximae mean levels: Share of capial: sk 0.28 Inverse price elasiciy: εlk 0.49 * * * Decomposiion of produciviy growh 1970-2001: Average labor produciviy A = T K T L X s K = 1.48119 Capial produciviy effec (4.67%) T! = 1.01850 Labor produciviy effec (75.60%) T! = 1.34581 Capial deepening effec (19.73%) X!! 1.32369 0.28 1.08060 Marginal labor produciviy M = X ε LK Γ ε LK Γ L = 1.46088 Capial deepening effec (36.43%) X!!" 1.32369 0.49 1.14809 Relaive efficiency effec (-44.86%) Γ!!" 0.70800 0.49 0.84362 Facor augmenaion effec (108.43%) Γ L = 1.50832