Cobb-Douglas Production Function PDF

WlUl JJIV\,,=.I~ UIIU U JJOlL W1Ul 1-'lU\,~:::,::, '-'. ' "- I Linear Homogeneous Production Function Production functio~ can take several forms but a p~rticular form of production function enjoys wide popularity among the economists.· This is a linear homogeneous...

WlUl JJIV\,,=.I~ UIIU U JJOlL W1Ul 1-'lU\,~:::,::, '-'. ' "- I Linear Homogeneous Production Function Production functio~ can take several forms but a p~rticular form of production function enjoys wide popularity among the economists.· This is a linear homogeneous production function, that is, production function which is homogeneous of the first degree. Homogeneous production ~ctio~ of the first degree implies that if all factors of production are increased in · a given proportion; output also increases in the same proportion. Hence linear homogeneous production function represents the case of constant returns' to scale·. If there are two factors X and Y, then homogeneous production function of the first degree can be mathematically expressed as : · mQ.=: f(mX, mY) ' , ' where Q stands for the total production and m is any real number. The above functio~ m~s that if factors X and Y are increased by m-times, total production Q also increases by m-times. It is because of this that homogeneous function of the first degr,ee yields constant returns to scale..... --~ -. ' s More generally, a homogeneous production function can be expressed as.. -...! k. -. ·~ ·--.. ·-··. Qm = (mX, mY) where m is any real number and k is co~stan~. This·fun~tio~ is homoge~e~us function of the kth degree. If k is equal to one, then the above homogeneous function becomes homogeneous of the first degree. If k is equal to two, the function becomes homogeneous of the 2nd degree. If k is greater, fu?ln one, the production ,fµnction will yield increasing return~ to scale. If on the 1 _. other hand, k is less than 1,· it will yield decreasing retur~s to scale.... \ ~·. · · Linear homogeneous production function is extensively used in e~pirical studies by economists. This is because in view of the limited analytical tools at the dis~sal of the economists, it can be easily handled and used in empirical studies. Further, because of its. possessing highly useful economic features and properties, (for instance, constant returns·to scale is a very important p~operty of.homogeneous production function o( the firs~ ~egree)t it is easily used in calculations by cc,mputer~ and on accollilt of this it is extensively employed in linear programming and input-output analysis, Mor~over, because ~f its simplicity and close approximation to reality; it i~ widely used in model analy~is.regarding procl~cti~:>n, distribution , and economic growth.... As we shall prove in the next chapter, the expansion path of the homogeneous production function of the first degree is always a straight line· through the origin. This implies·that  27 8 h.co ns ~~ re la ti ~ n qf th~ f~r st. de gr~ e~ ~t ucti?n ~c tio will alwaYs ~ hl case_.of.ho~og~n~~~ nsprod ee n th e fac tor s th at wi ll ~ used for production prices, optimal proportio be tw be pr od uc ed. Be ca us e of the simple nature of th nt of output to the same whatever the amou th~ first de gr ee , th e task of the entrepr~neur is qlJi~ nc tion of r proportions cllld homogeneous production fu es on ly to fin d ou t ju st one op ti~ um facto req uir decisio simple and convenient; he m ain co ns tan t, he has.no t to make any fresh ice s re oduction. Moreover' thne so Jong as relative factor pr he ex pa nd s his lev el of pr to be used as regarding factor proportions tor pn ces at different l ) (w ith co ns tan t rel ati ve ,ac tor proportions ~e is also very USefuI in ' use of the same optimum fac n fu nc tio n of th e first de gr eous productio first degree, which, as said ' levels of output in homogen od uc tio n fu nc tio n of th e eneous pr agriculture as well as in input-output analysis. Homog s be en ac tua Jly fo un d in r~s to scale, ha ve been made for van0Us above, implies constant retu m m an ag ~m en t stu di es ha es. In India, far uts. Analysing the data many manufacturing industri ric ult ur al in pu ts an d ou tp collected for ag hed the conclusion that States and data have been s, Dr. A. M. Kh us ro re ac agement studie rical studies conducted collected in these farm man ag ric ult ur e!. Li ke wi se , e~ pi ailed in Indian industries are·characterised constant returns to scale prev th at m an y m an uf ac tur ing in have found again implies constant in the United States and Brita ge ·co st (L AC ) cu rv e wh ich by a long phase of constan t long-run avera en eo us prod uc tio n fu nc tio n of th~ first degree. re ~_to scale and homog.,.► ~.,:;,_ ~._;'flf~~~J'."'"... y ,-9~ ,r.~·u·t yr, '"'/....,.......... "t':""'7: l';' ';. t ION FUNCTIONi 'f',,y ,.-,.,: ,~t,:( " ~ ,. r-1< t~~, 7 "' ~,..,~., , ft"'r: 'Gr" 'l,..._._.. !"f. -DOUGLAS PRODUC T t COBB ed statistical tion functions an d have us d actual produc Many economists have studie n changes in physical inputs ' and physical outputs. A s betwee bb- methods to find out relation n fun cti ~n fo un d ou t by statistical methods is the Co uctio applied _most familiar empirical prod igi na lly , Co bb -D ou gla s production function was n. Or ing Douglas production functio an ind ivi du al firm bu t to the whole of the manufactur s of b!r not to the prcxluction proces thu s ori gin all y ~h ol e m an ufacturing production. In Co n wa s on indusby. Output in this functio e are two inputs, labour and capital, Cobb-Douglas producti n, th er Douglas prcxluction functio ing ma the matical _form:. functio n tak es th e follow I Q = AL al( l3 - ;, t L th e qu an tity of Ja bo ur ~mployed, K is the quantity. where _Q ~ the ~u fa ct ur in g output, is s ar e po sit ive co ns tan ts. Ro ughly speaking, Cobb-Dougla of ~p t~ em pl o~ ~' and A ,and a in manufacturing productio n was at ab ou t 75 % of th e incr ea se production functi~n found ilithe remaining 25% was due to the cap1·ta1 m put.. e labour input, an d _.. f c bb- _du , e to..th ally it wa s fo un d th at th f o o · ·. It is importa nt to note that originth. - e sum ~ exponents yer~ ~o~ ~ e r_research and lt t Douglas production was equa L arialysis 'itWas ·geh~mifsed more than one arid Jess than · · equal to one constant ret ur ~~ ett ( at w~ ~ arid't ~. s ;a ~.-·, '.-~ l~.. o:e~ - pr ~I ~ ov ~ of = ex ~o lin~{ ~omogeneous ·produc ~e ual' to on~, ponents (a + 13} could be eq (a + ~) 15 nts e later when sum of expone ction) occur, tion fun t~ ~c ale (1.e s when a +:13,>l;w e get i ns -te tur n~ to sc ale ,_an ,d ,w he n~ + 13 _ 1. Cobb-Douglas production Junction ~xh~bt~ , its When a + p < 1, say '· t·t ts. equal to 0.8, then.in this. production. function new output. Q' = ga+f3ALa Kl3.= 'gO.BQ ' ' _That is, increasing each i~put by constant factor g will cause output to increase by g0.a that 1s, le~ then g. Returns to scale in this case are decreasing. ' It therefore follows from above iliat the ~um of th~ exponents of C~bb-Douglas productio function, that is, a+ J3 measures returns to scale. · : ' n If a + J3 = 1, returns to scale are constant. If ex + J3 > 1, returns t~ scale a~e ln~reasing. ' If ex+ J3 < 1; returns to scale are decreasing. Cobb-Douglas Production Function and Output Elasticities of Factors Thirdly, the exponents of labour and capital in Cobb-Douglas production function measure output elasticities of labour and capital. Output elasticity of a factor refers to a percentage change in output caused by a given per~harge change in. a variable factor, other factors and inputs_remaining constant. Thus,. ' - '. _...__ ao aL ·- ao -L Output el~ticity _o f labour = Q -. _; L ~