On Certain Conditions for Generating Production Functions - II

Transcription

1 J o u r n a l o A c c o u n t i n g a n d M a n a g m n t J A M v o l 7, n o ( 0 7 ) On Crtain Conditions or Gnrating Production Functions - II Catalin Anglo Ioan, Gina Ioan Abstract: Th articl is th scond in a sris that will trat undrlying conditions to gnrat a production unction Th importanc o production unctions is undamntal to analyz and orcast th various indicators that highlights dirnt aspcts o th production procss How otn orgts that ths unctions start rom som prmiss, th articl coms just mting ths challngs, analyzing dirnt initial conditions On th othr hand, whr possibl, w hav shown th concrt way o dtrmining th paramtrs o th unction ywords: production unction; productivity; marginal rat o substitution o lasticity JE Classiication: D43 Introduction W din on R n th production spac or n ixd rsourcs as: SP(x,,x n)x i0, i, n whr xsp, x(x,,x n) is an ordrd st o inputs and a rstriction o th production spac to a subst D psp calld production domain It is now calld production unction (output) an application: which satisis th ollowing axioms: Q:D pr +, (x,,x n)q(x,,x n)r + (x,,x n)d p Th production domain D p is convx i x(x,,x n), y(y,,y n)d p [0,] ollows (-)x+y((-)x +y,,(-)x n+y n)d p Q(0,0,,0)0 3 Th production unction is continuous 4 Th production unction is o class C (D p) i admits nd ordr continous partial drivativs 5 Th production unction is monotonically incrasing in ach variabl 6 Th production unction is quasi-concav that is Q(x+(-)y)min(Q(x),Q(y)) [0,] x,yr p At th nd o this introduction, lt not that a unction is calld homognous i rr such that: Q(x,,x n) r Q(x,,x n) R* r is calld th dgr o homognity o th unction W will say that a production unction Q:D pr + is with constant rturn to scal i Q(x,,x n)q(x,,x n), with incrasing rturn to scal i Q(x,,x n)>q(x,,x n) and with dcrasing rturn to scal i Q(x,,x n)<q(x,,x n) (,) (x,,x n)d p Associat Prossor, PhD, Danubius Univrsity, Romania, Corrsponding author: catalin_anglo_ioan@univ-danubiusro Snior cturr, PhD, Danubius Univrsity, Romania, ginaioan@univ-danubiusro 58

2 J o u r n a l o A c c o u n t i n g a n d M a n a g m n t J A M v o l 7, n o ( 0 7 ) In what ollows w will considr only unctions o two variabls: capital and labor: QQ(,) and w will not W will call th marginal productivity rlativ to th capital : Q Q and with rspct to th labor: Also, w din w Q - calld th productivity o capital, and w Q - th productivity o labor bing th avrag productivity rlativ to th production actor, rspctivly From [5], w hav that in th gnral cas o th variation o all inputs, or 0 units o input and 0 units o input, and Q()0: Q( 0, 0) 0 (0t,0t)dt 0 (0t,0t) dt 0 W will call also th marginal rat o tchnical substitution o th actors and J th opposit chang in th amount o actor to substitut a variation o th quantity o actor in th situation o dx consrvation production lvl and not: RMS(,) in an arbitrary point x, and dx dx analogously: RMS(,) dx RMS(, ) Q It is calld lasticity o production in rlation to : - th rlativ variation o Q w production at th rlativ variation o and also th lasticity o production in rlation to : Q - th rlativ variation o production at th rlativ variation o Q w I th production unction is homognous o dgr r, atr Eulr s rlation: Q Q rq w obtain that r 0 RMS(, ) W inally din th marginal rat o substitution lasticity: RMS(, ) t suppos now that th unction is homognous o dgr r Bcaus Q(,) Q, r Q, w will not q Q, and w hav: Q(,) r q hav thror: 59 W

3 J o u r n a l o A c c o u n t i n g a n d M a n a g m n t J A M v o l 7, n o ( 0 7 ) Q r q' r q' Q Q w r q r r q q' r q q' Q w r q RMS(,) RMS(, ) q' q q' RMS(, ) RMS(, ) q( )q"( ) q'( ) q( ) q'( ) Conditions o Marginal Rat o Substitution Elasticity t suppos in what ollows that: (), Q bing homognous o dgr r W hav th ollowing dirntial quation: q'( ) q"( ) q'( ) or q( ) q( )q"( ) q'( ) q( ) q'( ) q"( ) q( ) q'( ) q( ) q'( ) q( ) that is: q'( ) t y( ) W hav: q( ) Th quation bcoms: y '( ) y ( ) y ( ) q"( )q( ) q'( ) y '( ) q( ) q"( ) q'( ) q( ) q( ) q"( ) y ( ) q( ) Again, i w not: x( ) w obtain by dividing at y (): y( ) y '( ) y ( ) y ( ) Th solution o homognous quation ln x( ) d x( ) x '( ) x ( ) - a linar quation o irst dgr d x '( C 60 ) x ( ) is: x( ) ln '

4 J o u r n a l o A c c o u n t i n g a n d M a n a g m n t J A M v o l 7, n o ( 0 7 ) Rturning at th nonhomognous quation, lt x ( ) C( ) d W ind that: C'( ) d C( ) d C( ) d thror d C '( ) and inally: C( ) W hav now: x ( ) d and thror: y( ) d d C d d C d d d d C Rturning to th dinition o th unction y: ln q( ) ' d d d C ln q( ) d d d C d D thror: q ( ) D d d d dc t F b an indinit intgral o that is: F d F ' W can writ: F d F dc q ( ) D But F F C d W hav inally: F d F C F F' d F d F 'd q ( ) D whr F d and, o cours: Q(,) r q F F F d d 6

5 J o u r n a l o A c c o u n t i n g a n d M a n a g m n t J A M v o l 7, n o ( 0 7 ) χ constant F d ln rom whr: I -: F d F d C C F d F C q ( ) D D C ln C ln C r and Q(,) D C For r w obtain th CES production unction I -: F d F d C C C ln ln C q ( ) D D C C r and Q(,) D C - th Cobb-Douglas production unction homognous o dgr r χ + F d d ln rom whr: F d F d C C C ' d C C C d C d C ln C C d C C ln C G whr G d C W hav thror: 6

6 J o u r n a l o A c c o u n t i n g a n d M a n a g m n t J A M v o l 7, n o ( 0 7 ) q( ) D F d F C D CG C r and, o cours: Q(,) D C CG In particular or, w hav, atr dvloping in sris: G d d C C C C C 3C 4C Now: q( ) D F d F C D 3 6C 8C C 4 and, o cours: Q(,) r D 3 6C 3 8C 4 C 4 3 Rrncs Ioan, CA & Ioan, G (06) On Crtain Conditions or Gnrating Production Functions I Acta Univrsitatis Danubius Œconomica, Vol, no, pp 9-50 Ioan, CA & Ioan, G (0) On th Gnral Thory o Production Functions Acta Univrsitatis Danubius Œconomica, Vol 8, no 5 Ioan, CA & Ioan, G (0) A gnralisation o a class o production unctions Applid conomics lttrs, 8, pp Ioan, CA & Ioan, G (0) Th Extrm o a Function Subjct to Rstraint Conditions Acta Univrsitatis Danubius Œconomica, Vol 7, no 3, pp Ioan, CA & Ioan, G (0) Mathconomics Galati: Zigotto Publishrs Mishra, S (007) A Bri History o Production Functions Working Papr Sris Social Scinc Rsarch Ntwork (SSRN), 63