Explanng Movements of the abor Share n the Korean Economy: Factor Substtuton, Markups and Barganng ower Bae-Geun, Km January 2, 26 Appendx A. Dervaton of the dervatve of et us start from eq. (). For notatonal convenence, the subscrpt t s suppressed. Notng that Y Af(k), @Y@ A [f(k) kf (k)] and @Y@K Bf (k), the elastcty of output wth respect to each nput can be expressed as a functon of k. kf (k) f(k) ; K kf (k) f(k) : (A) The margnal rate of techncal substtuton (MRT S) s equal to the rato of margnal product of labor to margnal product of captal. So we have and dmrt Sdk captal and labor s Modfyng eq. (A2), we have MRT S @Y@ @Y@K A f(k) B f (k) k (A2) (AB)f(k)f (k) [f (k)] 2 >. The elastcty of substtuton between d ln(k) d ln(mrt S) MRT S @Y K @ @Y @K Y K Y K (A3)
By takng logs on both sdes and d erentatng, d ln(mrt S) d ln(k) d ln Because d ln(k) d ln(mrt S), the above equaton s mod ed to ( )d ln(mrt S) d ln : (A4) Now snce both MRT S and are functons of k, eq. (A4) s further mod ed to ( ) MRT S dmrt S dk dk d ( dk d ) dk dk: ( ) 2 By arrangng terms, we nally arrve at d dk ( ) ( ) MRT S dmrt S : (A5) dk B. E cency of Nash barganng To see that the Nash barganng s e cent, we can start from dervng workers nd erence curves and the ndustry s so-pro t curves that can be plotted n the (! ; ) space where! W s the real wage pad to workers employed n ndustry. The slope of the nd erence curve s gven by d! U (! ) U (! r ) d U (! ) where! r W r. The so-pro t curve s the combnaton of the real wage and the employment level that yeld the same sze of real pro t. functon (B) D erentatng the followng pro t Y W R K (F ( ; K ))F ( ; K ) W R K ; we obtan the slope of the so-pro t curve. d! @Y d @! (B2) 2
The nd erence curves are downward-slopng when! >! r whereas they are at when!! r. As usual, the so-pro t curves have an nverted U-shape. The schedule represented by ( )( )(@Y @ ) s the labor demand curve when workers barganng power s zero. The slope of the so-pro t curves s zero along ths labor demand curve. E cent contracts occur at ponts where the nd erence curves are tangent to the so-pro t curves. So the condton for e cent contracts s U (! ) U (! r ) U (! ) @Y @! : (B3) Ths concdes wth the condton derved from the Nash barganng, that s, eq. () n Subsecton 3.2. Therefore, the Nash barganng s e cent. The contract curve s the set of ponts that sats es eq. (B3). The slope of the contract curve can be derved n the followng way. Frst, modfy eq. (B3) as below:! U (! ) U (! r ) U (! ) (Y )F ( ; K ) : Next, d erentate both sdes of the equaton above to have d! d (F ) 2 F (B4) where [U(! ) U(! r )] U (! ) [U (! )] 2. Snce < and F <, the slope of the contract curve s postve as long as! >! r and U (! ) <. C. Returns to scale and ncome shares Suppose that the producton functon of ndustry s homogeneous of degree, whch s the ndustry-spec c parameter. s Y F (sa ; sb K ) (C) By Euler s theorem, we have @Y @ @Y @K K Y Ths can be mod ed n terms of the elastcty of output wth respect to each nput. (C2) @Y @ Y @Y @K K Y, or K (C3) 3
et s (A ) n eq. (C). It leads to A Y F ; B K A Now let us de ne k to be the captal-labor rato measured n e cency unts (k B K (A )), and f(k ) F (; k ). Thus we obtan Y (A ) f(k ) Snce @Y @ (A ) A [ f(k ) k f (k )] and @Y @K (A ) B f (k ), the elastcty of output wth respect to each nput can be expressed as a functon of k. k f (k ) f(k ) ; K k f (k ) f(k ) The margnal rate of techncal substtuton (MRT S ) s MRT S @Y @ A f(k ) @Y @K B f (k ) k (C4) (C5) In order to have convex soquant curves, t should be that dmrt S dk >. As n eq. (A3), we have MRT S K @Y @ Y @Y K @K Y By takng logs on both sdes and d erentatng, d ln(mrt S ) K d ln(k ) d ln (C6) Because d ln(k ) d ln(mrt S ), the above equaton s mod ed to ( )d ln(mrt S ) d ln Both MRT S and are functons of k. So by d erentatng and arrangng terms, we obtan d ( ) ( ) dk MRT S dmrt S dk (C7) Now let us start from eq. (2), whch s wrtten below once more, to derve ncome shares 4
when the producton functon exhbts non-constant returns to scale. W ( ) @Y Y @ R K Modfy (7) to ( )( )(@Y @K ) R. lug ths nto the above equaton. Then by arrangng terms, we have W @Y Y @ @Y Y @ @Y @ @Y K @K @Y @ @Y @K K Recall that (@Y @ ) (@Y @K )K Y, and put ths relatonshp nto the above equaton. Then we can express t n terms of the product wage. W @Y @ Y (C8) Thus the labor share n ndustry (S ) s W Y @Y @ Y (C9) Moreover, by modfyng eq. (7), the captal share n ndustry (S K ) s RK Y @Y @K K Y K ( ) (C) Fnally, usng the de nton of the pro t share (S S S K ), we have S ( ) (C) 5
Fgure A- Trend Component of Income Shares (a) Nonfarm busness: abor share (b) Manufacturng: abor share -.5 -.5 -.6 -.6 -.7 -.7 -.8 -.8 -.9 -.9 -. 97 975 98 985 99 995 2 25 2 og of labor share near trend H trend -. 97 975 98 985 99 995 2 25 2 og of labor share near trend H trend -.5 (c) Nonfarm busness: Captal share -.5 (d) Manufacturng: Captal share -. -. -.5 -.5-2. -2. -2.5 97 975 98 985 99 995 2 25 2 og of captal share near trend H trend -2.5 97 975 98 985 99 995 2 25 2 og of captal share near trend H trend 6
Fgure A-2 Impulse Response Functons for NFB Sector (a) IRFs to a rse n rms market power 6 4 2 Market concentraton 5 - -2 abor share -3 5 Employment -.5 - -.5-2 -2.5 5 (b) IRFs to a rse n barganng power of workers Market concentraton abor share Employment.5 3.5 -.5-5 2 5.5 5 7