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P NK 0 40 80 120 160 200

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40 ° 1,665 2,310 2,777 3,066 3,178 3,lJ240 40 1,863 2,523 3,006 3,311 3,439 3,~40 80 1,968 2,644 3,143 3,464 3,607 3,57240 120 1,980 2,672 3,187 3,523 3,682 3,663

80 ° 1,729 2,374 2,841 3,130 3,242 3,17680 40 1,927 2,587 3,070 3,375 3,503 3,45280 80 2,032 2,708 3,207 3,528 3,671 3,63680 120 2,044 2,736 3,251 3,587 3,746 3,727

120 0 1,685 2,330 2,797 3,087 3,198 3,132120 40 1,883 2,544 3,027 3,332 3,459 3,409120 80 1,983 ' 2,665 3,163 3,484 3,627 3,592120 120 2,001 2,693 3,207 3,544 3,703 3,684

-72 -

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II ....,IIIIIIII .-IIIII

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iii 3.5 ~ K. P =~~~flS~~~iIii ' '-ilIlttrel;]' , {iHI"AAt1- ' 1iJj! K.P !f~iilr3f/Fm~*0

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-",' "I I' .... A.... :

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BrrE"I'1jl~~m1J (Marginal Productivity) ~;!f~~¢re~;ffiHiH};t·-¥{ft

fl'Jim1Jl§:l.ft.1JD$ffi1&~:gj;*mffi:!:S':JtzJ:,mi~mmi:ip%gtjjjHi1l (Marginal ValueProductivity) '1'iZ:'j!§-m;l$5!p;R!£.@fli: (Marginal physical productivity)Q~~~%~~~~~~~ffl.~~~~o~~~~'m~~~~~~~m~~

~1:m1Jtik~-~~1i:J3t-¥{ftfl'JiBrr1J~1i.1JDtt-Jm~!i&it0 riii:P}5tl.JU;Jt~ZP:l0tt-J

~~~ m1J (Average marginal productivity) fO~ Wtt-J ~p~ ~m7J(Exact marginal productivity) 0

.0~ 3.1 ~i7JJ ,1t; P -= 0 'rm N g 0 fWtJD~ 40 -0-JTfl'Ji ' fi'3~m:ii:§l1,013 1}JTJft~ l,528&-Fr '31ifl;f{~ N J'f:1ffl;lJD3J:H7j~ 40 1}JTrmID~~:Jl:tt-J:Lj

1JD$ffi~rJ;I$ 515 1}JT ' 1~~~M~2:.rt 515/40= 12.8 'E!Pfl;~ N§ 0~ 40 l'l.ij~-0JTJ'f:1~~1J!E}iHi:ZP:l0PJ.0:lWmHg~ 12.8 1}JT ' ~J7;t£ZP:lt:Jtt:J~~1::~jJ 0t1umi~~~~J{lJ~ AMP = 6.Yf6.X ,;rtq:l AMP ft~zp.:l0tt-J3:£~:1:mJJ}6.y fI;~m:m.tt-J~1l::' ,6.X ft~~gg~lW~:l:tt:J~1t' #iJffiJ::.~~15t* ' 1%

.A..PJ£J.re ,6.X m+m- 40 i}JTa:J!l&1[ocfI;ill!.' to 30, 20, 10, "M. 5~ , 'M~-&PJ.0-fR 6.X= 1 ffii*1\'J~*f~*:l~:J3t-1}JT~' m1Jl§~:VQ~:m~mzill: 0 ~rmi1J~re;f4l!-~tt-JPJ5t~iJ'f1: (Divisibility) 'RDPJ.05t~rJ~1~i5.!(.'J'~'4'I.{ft,flu 0.11}JT' 0.01 1}!T······~ ,~~v'§uHI':J~1l::ffl dX ft~ , mTI!:S':Jv-:iz+0Hl::ffl dYft=a£ ' JlUf~*~~p~~jg:fJ EMP fflr5:t'$~' EMP= dY/dX 0 ~@~~1'Et'J(~J:j£fiffl;~~5t ' *~~~*~~~~t~1JPJm~~TXiJ!ttfr:1'J.ne1

.fiiI~5} (3.1) ~'1iJJ2J.~$U.~t N, P =~~1Iilglj(f:j5i~~i£jJt

~3.5 N,P;.~-.t flfJ _.ti-it1!f.i..~ 11(~:M:) (11·&: *JTI*~)(A) N ~;«

~#:OOiJfllt 0 40 80 120 160 200

jJH~~2!:tJ 23.48 18.37 13.26 8.15 3.04 -2.08

(B) P ~~

~~;6'IIi}fHI: 10 20 30 40 50

il~~2!:tJ 8.02 3.45 1.42 0.21 -0.61

j]PJQUfl~~?t1J~~l:!:ltmr:

(1) N (t-jf!~~~:tJ : ~~ = 18.3449-0.1l11N+0.OO98K (3.8)

(2) P (t-j51~~~·:tJ: ~~ =5.6250--0.0641P (3.9)

(8) K (t-jil~~lf:tJ: ~i = 6.0995-0.0579K+0.OO98N (3.10)

EE~ (3.3) J:tt:j:l~1'fNxK ~~-1Ji ' iitttt NlQ:K1!iii~?t~?t}3IJ~ (3.8)». (3.10) =st' 81J:a-~T N 71-' ~*.fl K -:rJU&~ N (t-j~~~lf:tJ~:Yl:N ;;f;:Jr:&lIi,Ffl.!t:J~~:9~ , ~:Yl: K !t:J~" ' EiJ~ K (t-j~tt~iE' t&~ N (t-ji!~~~2!:tJ~K(t-j:L1i1JDffij:ti1JD u Il\i: N = 40*JT ' JtU N (t-jil~~lf:tJtE K= 0,4,080,120& 160*!THq' ?tJ.lU~13.90, 14.29, 14.69' 15.08&15.48, iiJJ!;ij"iflfii(t-j~~ 0 .¥~K (t-jil~~lf:tJtm (3.10) :rtm~ , W-::;1: N (t-j~' 0 P(t-j~r*~lf:t:r£{ (3.9) Jt~* '{I>1.:;tt;;f;:!rnlliJfltitt-Jf3lf1 0 .=:~~(t-jil~~lfj]£{ N ftj(' K *:2:' P ~;l' 0

~3.6 ~'*-.t~~1tif.1If.i...tJ

(~~) (A) N, K=~~(t-j:fi'Hi!Hl~':tlf:tJ* (¥'U'L : 1}!T)

KN

0 40 80 120 160 200

0 18.35 13.90 9.46 5.01 0.57 -3.886.10 6.49 6.89 7.28 7.67 8.07

40 18.74 14.29 9.85 5.40 0.96 3.483.78 4.18 4.57 4.97 5.36 5.75

80 19.13 14.69 10.24 5.80 1.35 -3.091.47 1.86 2.26 2.65 3.04 3.44

120 19.53 15.08 10.64 6.19 1.75 -2.70-0.84 -0.45 -0.06 0.34 0.73 1.12

160 19.92 15.48 11.03 6.59 2.14 -2.30-3.16 -2.77 -2.37 -1.98 -1.59 -1.19

-71-

x~~N~~~~~n'T.~K~~~~~h

(B) p~~~m~~~~~h

W~»Ilimil: 0 20 40 60 80 100

~~~~::fJ 5.63 4.28 2.94 1.60 0.26 -1.09

~M.-m:;kfS~~~:i:m?R.=:.~~pIfiJ1Ut'i'iJQJ,~(3.8) , (3.9) lZ (3.10)i:\;~M.- ~jj1J*ili '(3.9);rI:; .R%P, tIx. p PJ'QJ,Jf.!. jW3jt/:!:J ';ftf@:j0 83.79iHf 0

(3.8) "& (3.10) =i:\;1t-:tr=OO*nJ1J&' i'&.~ffl!&P:V:1jW;r\;~m~~ili , ~P N=177.121}Fr"& K= 135.51 -0Fr 0 reJ:ITii.=:.OOJ1tz1iN: f\;;\. (3.3) :rt' 15J1,~HI:7~tlt8~:i%mf&~ilfflw*1i'{3,767*JT0

.=.~~ N !£~i¥J£U~~~hslHUi1lmM.-~ 3.6 , ffiilPJ'nJ ' N~~i¥JfS~.~~~h~~#~ffl.~~'*QJ,R#~.~,m~*~'.~.~om-oo

!Wftf~~T-:ttWi-l\1lii¥JiElii'M':!: ' ~PfmJJE:~~J:f!&\\~±J.~~~~hPv.iEJ:tiJIJ ' ~~jJ~~~ , ;ft~JiI,\t~\7'( 0

re~'4'=:'W~ft:JfS~~~~~1JBfzrtJ:mt ' -fuPJ'J2J.m*~I

N (P,,=5.00) * P (Pp=2.22) * K (PII:=2.00) *i!iC1¥:i: 3!~1:~1:1 i!iCi!.'Ht ~~~~jJ i"iCi!.'!l: 3ip~~~jJ

~ ;It 80 5.89 27 6.09 10 -**J't **

100 10.70 36 0.65 10 -**#t til 90 8.35 30 3.61 15 5.23

*fl*~:~~~~i!.'.(*ITI*m)~~~~~+=$~~~~~

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*W~51~Il:J~'-~~~M~~~ffl.~X::~ll:Jwm~~ElHmofB•••Y ~FJP:*~1·':::.~~ N, P R K ;UtlUfj:m.ru~l't:Jr;,%-';;&P1:PLIm~PJJ?ltr.:1T;R~ :

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J1t:it-ft-R;~ Y ~ N Il:JIBJ·f*'PJ£1)';'jzp.I!ffili~ (EZ=5G*~) ~~

o:ff-zp.I!ff~~J::m~~~Y~NZ

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) 1fJ;rr~7.i1j ~-~~:ff{1h~~mjfmlilWjlltiJE~

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III 4.4"&111 4.5 ?}}.JIj~~:f* N, P =~~tt-Ji3~1!~I!ll~ 0 f!I1IiIJ:RJ~ ,

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N ~*flfS~miitt:J~lHJtk' ffij p ~*tt:J~m!lllt*tE p= 10 Qt]§fJ~iMU4

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N'" 183.77±7.8261V25.2506VP-l.9418P+843.4238-0.255556Y··-(4.2)

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P NdN dP

P NdN dP

dP dN dP- dN

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30 70.93 -0.10 -10.14 30 103.31 -0.14 -7.22

40 70.39 - -0.01 -68.27 40 102.57 -0.02 -48.90

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14.24 $)~lf)i!lt}i.\: (~~m (~~o)

(8) P!lf! K jiijfl~~(fJm~~~.I!l3~ (~ N =0)P = 83. 79±14.8966v'0.8189K-OJl03885K2 +232.1903-0.134259Y

•..... (4.5)

J-.iii.=:i'tPJQI,filfYUmlil 4.22 , iii 4.23&111 4.24 0 ~1IiII 4.22 PJQI,;ffl:fjfEfl~~!i:EI!I~rg. P f4t;~ZjS~ , =;£~ Nm~tt~(fJ P~* 0 fil4.23 ED (4.4)Jt*~~, tmQl, Y=2,700 l'J3~ll:lltJf!J~~~J' N ~~I'J3~iIJ~~~:i0 50, KJlU 110 ,~~ N ~~ttfl;:JJ!~1'J3 K 0

~t£ Y=3,100 If!j, N 1'J3~1!iJwltJ~-t

~ 70 , K JlU 100 ,~~ N mi§igM;:g:I¥J K if\Q Y = 2, 700 1f!j;~.rJ' 0

1il4.24~ (4.5) Jtl'J3lilM' tEY= 1,600EU, 7001f!j , ~I.JJffl P ,K Ji:l1f-~~1:~' (g;ff y= 1,8001l"J' JlIJ~~-\'tm p ~~~.i1ff ' itt~ K 9if8~&:!I1'J3ff.m~ P =ifi~* 0 ~Iil4.24Ji:l Y=1,800"& 1,900I'f:JK, P ~~~1'f:J

*Jl.1r;&!gJ¥J:i:71Jtil:i0~ 4.3' 10m 1,800~JTf8~1f!j , mm K!1l! P I'f:Jtlli?;ntiffl:i:, ~Jtm 40 $I! 22.25, 50~ 14.31, 60~ 8.31, 11Z 70~ 3.83 1}R~ 0

1:TIif1'f*~ iiiI:fjml!l~~t£-A::~ rID f3 PJQI,:*.i' jff~nI!~fPJ I'J3 *3.1::Yilim ffiffl0.1:~f§j-~:i:I'f:Jf8*'i ' .fl~-~~~~~;tm~ , ¥')-~~JlIJ~Ilfr~j;' , jiJ/'!!;-fJ.~~~fj1Jo--\'t1'tL1f!j , 3}-~~~j;'~W·1'tL ?~t!Zt51~Tm~~~rm(p.[:IJa N JJFtP ~71J) 1'f:J3!~f~*

J2L (4.1) ~~{91JPJJ2L:1t~~J!~'Ht~$:tlIl""f ;dP 8 (1.9028-0.0146N)

MRS = -erN = - vl.9028N-0.OO7292N2+187.1634-0.125000Y;(E~r:p:5tJlIjf-t7,-:t:fifllXf[(~ y ~1 N 'JlIJPJ)jt1~:?~'.WHlUB~~f\tJf$" 7f}:. 4.1 ¥~ 4.3lfl ' {~JU1Jmfl~~FJJft-Jj'1~~ft:W~ , tinJ2L7f}:. 4.3 , Y =1,800~{91J ' ~K= 40~ , -'¥ill.A'JK ~~PJJ.2J.ft~ P~~ O.92~ill. ' ~ K= 50 , 60 , ~70~, JllJftZjzRnt:1"t'!'!r 0.69,0.52,"& 0.38 .ill.~ P ~~ 'PJJt1'!:ii~tt~~~~"~~:tt(ih~7f}:."%;Hfc(J-Jf!.* ' ~fI~~ft~~l&~B1m~-&~W!¥a~~j::j£

1Ifitz~-;kfHl c

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P (KfJ /h~).Ilbilt (.&.-t: -I!4,;lO )

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if:m.g~ 2,000 -0JTlF!iftrJ~::ffIJ ' t&~;f.f~:i: 210 :Jc!¥Jll!t8~~~~l&*!&1dt ' ~~~2,OOO-0rr!¥J18~ , ffiifYr5Jf N , P =~#!¥Jtt:!t~CJC2JJij~.If:l:IlB~z-!IJ~T 2 l¥J~~' VjJ!t T 2 ~~RDtt~~.gJEttIDt*rJi 0

fPj}.!l!.1jJJ2U1(~ , ;f.f 150 j(;~~!¥J"~ , 1iJ£1.~~¥tl~ C1 C1 1'§-!lJt¥J~'@:i:

llB~m~~(t-]~it 1.8OO1}JT' 1tfYr5i: N' P !¥Jfdi~-!IJ:\'li T 1 I'!J~~ 0 {t>:J!t1Ji:j:f£ , JWf£* T 3'" , )J!ti!J!t~-!lJrfJifYr~~J3M *$ (tE*~J~-Ilrn$) 'RPtt~w:.ggf:i*1-PJ~~, B.lJ*~ (Least cost line) WZfjlH~J.i£l:!I~ (Expansion line , RP~7f

.8~Rutm~·~:±it:1~y ';ft1:i~H~~JJ!':~ll{iiJ~~t¥JIlB~) 0

LOO(t-]W~~~T~~~*~~~~.IlB~mw(t-]~~~.g~ft~*~'~

t~J!t~Wll'.6JlU1iJ~,~H~l*llBt~ mlt~~-!IJ:rJJ(t-]*1$:f§~ , ttOz.mf.$~j~H~fI;tt$t.~iso-clines) 0 4-:j3}iiftry;j~.!Kl!IJ~!fI!{~tmM:1

ili:dN _MPPp __Pp~dP =MPPn= Pn (5.3)dP -MPPn - P, -, dN MPPp Pp

(5.3) J:tf;0=:flf~~rJJltt,gJlEf'lpx:*t,ill.--&~fF,f.ft-z~, J)-~ft-~~1Wlk&:'~tEJ:mJ§fmWJW.~ (Border line) Zrq , J[P;tt~I;~Htt2r$+lft~ 0 *J:~ti7~'Jf:it~ltt~)ffii':'tJlX*Ut1**E!.JZif{fftE!B~PJ0ED (3.4) ».. (3.5) J:t>JtiliPnT :

dP 15.2222-0.1167N r,dN = 7.7824-0.0625P = Pp = a

J:tlfl a= ~; '1t1ml:J:tPJ~~if{1WE!Bi;ljPnT:N= 130.48- a (66.706-0.5357P) ...... (5.4)

J:J:t0 Pn!Pp= 1.80 ,2.25 ;&.3.oof-tf,. ,J{lri1J>Jtf1j.!fmg~~ffltE!Bi;lj 'PnIllS.2

, liIJ:1¥.J:tJJi*%f\;~W~ ,·~~rq~~II1::t~:fl~~:5HlT1¥.Jil1~ 0 'ffi":fImlt+~ ,~~E!Il~~ttJ:~' ffiifN N .mxlft'li'~r'i'iJzl5 0 J[P~&7.f- N~~1¥.J{J{m;f§!fitf!,g1l'Jj, 0:H~m N ~~~~:fl;fJJ ; RZ 'tlll* P ~jt~1Wmt§!fitf!1J:ill'!j , J{rJ~~E!B~{j::E~ , **~1Mffj P ~%~~;ff;fiJ 0 ""ire Pn/PP = 1.80 ;&. 2.25 IFIf ' if{~Ht!i;ljtg.~~lEitI!!l~t¥.JY~~7Ulft*5.1 0

1~5 .1 N ,P z:"*:tif..::r- Fl{jt ~Ht.T ji1~ne.;H-~~~tJl.~~m~

y

1,8001,9002,CXX>2,1002,200

N .Pfflm.Lt(Pn/Pp)= i.so N.pm:mltCPn/Pp)= 2.25N P N P

~.~ M.OO 37.49 47.37OO.~ ~.~ ".~ ~.~

57.45 48.78 52.11 59.5065.33 56.95 60.55 66.51U.~ 0024 ro.V U.G

ED .i:;tti1J~ , ~~ll:(t-]*1Jo ' f'fr~1¥.J N, P =~~t¥.JmaJfHitf!.1~tJO 0 ~{fl#lIt (Pn/Pp) ED 1.80 ~m 2.25 ~,~7f,N~t~~(t-]~m;j:g~itf!~1Jo 'E!.JZ P~~1¥.J{flm;f§flitf!~j/ 0 ~~~rm~~:l:Pn y= 1,8OOlt.Y , m1\'Uf:JN~~ED ~.83 ~:¥37.49*R' ~ffi]P~~Jl.IJEf:IM.66~~47.37*R0

[115.2J:~if{~I!lI~~1&~1ft-ati?J~ItJ.li ' ~J[P~*1;.~IZ~~&mw N,P=~~!&:i:1¥.JJ1].k.R:fl~fj, ~~tIr!J~~ffi~~iXif&(t-]*¥f4:z~ 0 -%t1±D1aP.~J:m~1!J(fj~JE:lt~ , PnN = 2P~1i!£JfHt~1l'Jj , Jl.U~ OA~ , J!tIFlfPn*~mH:Pn!Pp=3.00 IF!j , -MY! =~:a:=~~D1aJfHI:1¥.J milk J:tf~~*1¥.J~~' ~~JlQf4U~mftl5ffi].R~~~&noo1¥.J~M~~o~~)ffi~(fj~frlt~~~~~~m'~~~

JlE.pt*~ J&,g 0 ~ ill:if{~ E!B i;lj-tf!"iiJ01l/.f11&:li1~ (f:j JlE~re k H~~ I!!l.~ (Qptimumfertilizer ratio curve) 0

- 92-

o

60

~ Jl:f*z N,P =~~~it.gJfEfl.P.Xt~~~> 1 *~I!b¥RPJJ2J.rn (3.6) 1Z (3.7) 5Vjt

II I±J~T:: dP _ 23.4815-0.1278N: dN - -7.5985+49.4033 PiI PnJ =-Pp= a

---:~ {tfml:itPJ~I N= 183.77+59.466a-386.6344 a

---I':;"*~ 1___I y' P ...... (5.5)

:='k; l:itPJJ2J.JilMmGJil5.3 J&.gJfE*1-P.X*~---~"3

I ~P.X~~'~~~~~~T~J&.gJfE~

----~;~~ ~*~m~~m.Z~~~*oI

,0 "0 ~o #t~z N, P, K ~~~~~1!j;JfE

*5.3 :%~~~~(;¥i~ *,tmG*~~~PJJ2J.rn (3.8) , (3.9)..+\;~:f!~ (~,jf.) J;7. (3.10) it5tJ.lU;1tili!mT:

(1) P. N =~~l¥J.l&1!j;~E*1P.X*Jfu¥Rf.5: (K"-'O,Pn/Pp= al)N= 165.10-a. (50.625-0.6042P) ...... (5.6)

(2) K. N =~~t't-JJ&1!j;JfE*1-mG*I!IJ~~: (P=O,Pn/PIr:= a2)N= 18.3449-6.0995a2+ (O.0579a2+0.0098) K ...... (5 7)

O.0098az+O.l111 .(3) P, K =~~t'fJ.l&1!j;JfEnmG*~~~: (N=O,Pp/Pk= as)

P=83.79- as (90.862-0.8621K) (5.8)

l:ffii~;rtPJJ2J.5tfJIjIlilMf.51f515.4 'Ill 5.5 &.lil5.6 0 rntn N t!i! K =*I~~filJ*~J,@iWll{f ' tOclll 5.5 l:t't-JW~{Mxlt~~~o 83111 5.4 PJ0.tfl±J 'IE Pn=

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1Be>

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140

100

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* P,K =--$~~it-JMH{E!ll*,~, rtr~=~~~~~~~~~£~~*'MePp

=1.00Pk-~ , B~jliDJ1::U 0 jffi')\t P~~it-Jffi:";ffr;j:g~{jf,jI\1J 'fm Pp > l.50Pk~'M.~.~~~.~K=~'~~

, ~P~~(t:]{fi~ts~1lffi1!r; 0

Yo P p=O.50Pk 'JlIJt!tJKf!M~~ll.P):!J;fJ.~ P=38o

\000

1200

~-mie ~111t!Hl!!mm$~~~~I£~ , jt~H{ I!!l~~J:(f:j~~f\~i&1!&re:¥1~*lY:mi1r ,j)tiHl*(f:j:fIJlf!,!J~ (f!Pffimn.flWJrJe1Jn(f:jm~~~~~*liiJiJJ!':IJX:* , W-'f!Pamre§f~:fIj~) ?J1'&:':(£MH{I!!lt~~ 0 Jl;*~rC19~fi0"trilM*1±ijjtdrz*~;f:ljll!!l!!lli '~iW~~1Wlr,,~~&:'~51~m~~{J;ffr 0

§~~nm-~~(N)~~~m~(Y) ~~~~ili~,~m~~~I!!l~

Y=f (N) PJ'£LliliWYolil5.7 oG;~mnJtl N~~H;1 '~~:I:~~ , Q1J&~ N fi3iiillJ' Y iB:lffiJil , &¥. N =130 1tIj, Y ~f}:l1i ' jfm&l~llJi1&~:!IjI;ii N t¥J:lifil[]~I(~L jffii'Wf~ 0 am~~!Jl¥J1:itJI~ , :g:A~

1400 ~ , ft*tIJlf!,!J~f;M!~~1&;f!t~1J{'f~Ht~~

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P r (L',Y) =Pn (L',N)

~:rtPJ'&~:L',Y_ PnL',N- Pr

ifi]G;L',y ~ L',N.w!.+IrIfRpmTt1:R* :-dY _ P,

dN - Pr~1WlM~:rt~PJ' rtr~!J 1511~ iln£L illtI¥.Jz

30 60 qo I~O 150N (K9/ha) 0 ~1l'ft.~tIJlM ' JtIJ :

'is.''' J)...~H;,~~~-iJ~ IT =PrY-PnN-94 -

~~ Y ;!1$ N l'fJimtk ' Ml:1:~iiJfl&A :tt =Pyf (N) -PuN

11$1~:1I!i*(tj'fIJlM (rr) 'JtH1:l1:~~ N v.&5HE1i/!;G~M-~Jl.PIiJ ' Jl.pdrr df (N) dY

d1r= PYd N- - Pn=O' ~IlP r. dN -Pn=OE81:~1iJ1~~~ = ~; ,RP N ~~(f:j~~~~j]~M-JlEl;ftm~{!;f§.lt (Pn/Py

) Il'If ' raf.HfflJ 0 ~1-tE~~riIJ~1:~*1S-!JiA9~~ , ffii-J&*;fIJIM:fJil'fJ~~~MJJe#fB~ii'U1Ht '"l&:t?iAIiJBJt;31ttl:J~4~~M-JJe*'lfS~1'R;f§.ltziil~ , ~~2jSIT

~l'JZ ' ffii1~~~~EIB~m1WZiJi;!1$W:*;flJlMJ.li ' 4f.t ~; = 5,Jl.PJlE~m~Am

~m;f§.l'fJliffi'~17fj , Jlfj1EliI 5.7 1:iiJ1-!J.1\i*fIJ~J.li~ A lrJi ' ttll'lf N=88, Y =

870 0 ~-~-;- :i'1fto ' Jl.PN~%(tjtrl:fti:t§t-J:ljJJo , ~m~{I~;j:g!fAA::P' ' ~ ~;

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"l&:%1Jl[':;J4fi3

N

6.00 79.055~5O 83.34

5.00 87.624.50 91.91

4.0096.19

P

3.89 62.28

3.33 71.24

2.78 80.242.22 89.00

1.67 97.80

§ Mft-JnE*'tm'€k~#HtRJEE~JE*,t{l#} (I!PQl.fS~~7Fft-J{I#}) 1Rtt·JWf3~

~~~s~~~~~~~~fS~{I~~(*~~Yre~fS~m~~~7F) 'M-*Rft-J~~~~*~m~-*rr'~~f3~N~~~W~~~~%'~RJ*iliN

~~-*rrft-J~~f.Hu~1l.*rr0 ¥.a §1Wft-JJJB*t~~1t$ (I!PJi;fL~~-*rrY:~~Bflt-1~rr 'M\i&ll~ji§ (f3P18%) x~mf,l: 0.5 *rr 'j\'{[:il¥ (~K5O%) X~fS~ O.~rrtrr·~.=:~~m~ffll#}1tJtU~.5JU~ Pn/p,. = 5.00 , Pp/p,. = 2.78,

&PkjP,. = 1.60 '1EJlt~{fHg·1tT ' RJY*iliJ'&:k7.'lj~1.JiP.fr~Ji N"&. P =~~:$(f:j;rtJiflUl:j887 .62-0rr~80.24*Fr 0

re:tE~r~ft-JJJe*'l-m~{I~ltT.zlZjJ&·@;(f:j~~bii!fflflttA~~i?1ilt (3.1)

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:ffE7l!Uft-Jm~l.ut;!02.361*rr, EE;rtJifflJE*4j]ijJ:i1JrJft-J~:i:~1.348*rr ' D'l!iJlEJR*~6080JT(I!P 4 x96.19+2.78x80.24) , i'&:;rtJiJlEJW~;fIJ~:ffl740(I!Pl.348-608)

*JTt¥Jm~ 0 :Jt~;rtJiJJBQt*~1'iJjEH*Jrft-J1R.*7I- ' li1f.~-t2.1~.Hi5;1t{ih~;rtJiJre.\it(f:j~

~~~~(f:j~ili'ffi~~m~8~~.'i'&:~*~~o

(~;ft) (f!l-fft : -0JTj*~D

Pn/P" Pp/P" N P 1fElflIJtf:JfS~~Jl: ;rtJiJlEP.frii ;fIJ iJi:

6.00 2.78 79.05 80.24 2.275 565

6.00 2.22 79.05 89.00 2.297 6125.00 2.78 87.62 80.24 2.322 6485.00 2.22 87.62 89.00 2.345 9654.00 2.78 96.19 80.24 2.361 740

4.00 2.22 96.19 89.00 2.382 787

re~~JJB*,tm~.fi'llta-1tT N, P =~~~1&~~;rtJim:i:p.-~~lflU~f£3'€k~:li:Jf!

ili' JtIH~*-5.4 (A) 'EE~~RJY~m,~Pnjp,.=5.00~,1&;$i1&(f:jN!£#»IJifflii:~ 87.62 1}rr' 11; PpjP,.=2.781!'Jj' Ji\'d~·.%~ P ~~;rtJifflit~80.24

*rr ' ffij:tj:c!'IF.lflU~fS~~:i:M~2.322*rro~~:1t;rtJiJlEP.fr1i;fIJ~&:M!im::~*Jm711

-96-

~Jltl~ (B) 0 -DJm-iEJ:~m• .IE*lmf.ll:iJU~.Itr, N ~:~U~~'&"ft-J:8fljjIHi:z

~*£Umft¥fi'L~.A 4381}f? 'ffii P ~~ft-J.5X*Jltl~ 2231}rr ,t&:lt~~*

AOO11}f?'ffii~~1}f?ft-J:8flj./E~~om*J:~~~~~'M./E~~~~~.It

ft-J~~ii~~ft-J~*:8fljfflmij1JlJ' Jt~~l't:JfEf.J::a:i:~J'*1JlJ ':tm~filtl* (A) ~

J:fljft-JfE~m¥.llt2. 2151}JTt'0100 , J!IJtlrjFjl't:Jm*~~:m 2. 3991}JTA:ltl08.31 0

1

;tt~:jt l:i:l ~*Z~:*;fIJ lfiJlM~-~"it3ml'" :N= 183.77-7.8261 an, (an= PulPy) ..•... (5.11)

49.4033P= ( 7.5985+ pa)2, (ap=Pp/P,.) ...... (5.12)

~A~~~an&ap~'~~~~~~~~~~~m~~m~l'"~~~~~m

:I: '3m~ 5.5 0 rn~ft:f* N ~~tnm&~ii~~~3ii/f~~~t$f~* ,ioOlitZtfpj-JlE*,I-m~;1~m~l'" ' iW$f!&~~tt-J~m:i:-W*~1&~ , 3m Pn/P:r = 5.00 E;J ,~~15 144.63 1J:-JT ' ~~tJtU.R 87.62 1J:-JT 0 fiZz , 1{t P ~**~. , ~;l;:jd&~'ii~~re:ltJFJ~)jHi;lt~3O% 0

~5.6 ~.~~~••m~~$~~•••~

~.JIe:.~k~~~j,i (~;jYt)

(A) ~'@;~~*:8!liltUi&:ttf,!jl€!uft:1m~~:i: c.& :1})1" 11}~)~-J.ltr/P:r 6.00 5.50 5.00 4.50 4.00

~~:i:...~p 136.80 140.71 144.63 148.54 152.45Pr/P:r :8!liffllt3.89 18.49 3,44-4 3,466 3,487 3,505 3,5223.33 20.43 3,451 3,473 7,494 3,512 3,5292.78 22.66 3,458 3,480 3,501 3,519 3,5362.22 25.30 3,464 3,487 3,507 3,526 3,5421.67 28.41 3,470 3,493 3,513 3,532 3,548

~-- Pn/Py 6.00 5.50 5.00 4.50 4.00------------~m-"g:~

p ·-!Jj!~ftt*~ffjEr'~ 821 774 723 668 610

Pp/Py .!U.JJX*3.89 72 1,408 1,478 1,549 1,622 1,6983.33 68 1,419 1,489 1,560 1,633 1,7082.78 63 1,431 1,500 1,572 1,645 1,7202.22 56 1,444 1,514 1,585 1,658 1,7341.67 47 1,459 1,529 1,600 1,673 1,748

3?:Ii~;:t~.fil!m:*HE'€k{Jl ffr l:tl'" ' 1S-~*~:l~dr~fa;~ffl!lt ' JlAiE7~~~m'i&~m:Ri1tiJlEETr~fiJ1.;1:frn't£ 5.6 0 rn::&J:~~ri.lii Pn/Py = 5.00 Hz Pp/P y = 2. 78 rf~ , :fa:~'~~~~~m:lt~ N= 144.63 R P=22.66 *R' 1t~:m:~~ 3,EQ1 1}JT'ffij:8!liwm{!j.fUj~~ 1,572 1}JT ' ;OOi~*QUE~15.{{rft;tffi:!0 786 'l}JT ' fliJ;flf:!01&~~llfr'f 0

~~*~~~~~!&*~~~~~~~~~:8!lim.o

-98-

~5. 7 ~*t-A~ Fl~ lie.;ft~:t1:{f ~t~T:J:ii.'t~~m~ (#U~)(A) P~~ ('1ii1ft : %JT/%l¥{)

~:M-.lt (;;) 2.00 2.20 2.50 2.80 3.00

ift~&"(rj1lfl!ffliit 54.00 51.02 46.55 42.08 39.10

(B) N,K'='~~

Pk/Py == 1.60 Pk/Py == 2.00

e.t», ~:~'@jNJtHi: B:~~~KJtBi: Pn/P:t B:~&"Nffl:m: B:~N;Kffllt3.20 145.38 102.47 4.00 137.45 94.21

4.00 138.07 101.23 5.00 128.31 92.65

4.80 130.76 99.98 6.00 119.17 91.10

5.60 123.45 98.74 7.00 110.03 89.55

6.40 116.14 97.50 8.00 100.90 87.99

p== 83.79-14.8965 ap (ap==Pp/Py) (5.13)

0.1l11N-0.OO98K==18.3449-an, (an""P"'/PY) (5.14)

0.0098N-0.0579K=6.0995+ all: (all:=P"/Py) (5.15)

J:TIii-=::;:trp, (5.13) :rtRm P I~ ap(rjlUj.f* ,it( P PJ£L'1ii~rn a p3jt1:i:l

, tfili:*1%:R 5.7 !t:JIM~ (A) 0 ~~ N,K =~~F!Ur~lllI.trm~' ;ltl&~·~!t:Jma

m:l:mrn (5.14) "& (5.15) Z~:lL1ft¥:;:t3jt1Jj.'ll.P :

{K,= 135 .51- 1. 5534 a n- 17 .5441 all: ;.. (5.16)

N=0.0885K+165.10-9 an (5.17)

rnJ:Uff::::APJ"?n N ,K !t:J~~1IfI!JtHft::;t an"& a" !t:J~1f '4-:j'If~ ak= tan'

~ll.P t ==~= PplI: / Ppn = _pPIr 'IlP t f1:;~ K.N =~~!t:JfJ{:m-.lt ,..JO,J:illi;IUj1¥an n Y n

ft,A.. (5 .16) ~t ' PJ1i-K==135.51- (1.5534+17.5441t)all: (5.18)

it(~ N, K !t:JJ'&~'&1IfI!fflit-~iffi:;t an & t == Pk/Pn !t:J~1'J d!{).rt1J~.Lti~5.7!¥JlilU:R (B) '$:{7!J*iift,·~ Pn/Pr:=4.00 ,ffij Pk/P:t = 1.60 ,ll.P t =Pk/PQ:= 0.40 ~ 'JR~a!t:J~~~Ymm:i:~E!:I (5.18) ;?t:;:f~ K,= 101.23 'ffij

rn (E.17) ~PJ"~~ N:= 138.07. 1.E!.~ Pk/P:t:= 2.00 , ll.P Pk/Pu == 0.501t!j , -l&~~ft:J·~~~tlJimi1~~ N=137.45"& K=94.21, PJ~K~~fJ!l;fttflflitB~'

it(1IfI!;tHi~:P ' ffij N ~~t~'i!imfi!;fJ'Jmrm:fjij~ 0

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-99-

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4.80 2.80 1.60 130.76 42.08 99.98 3,569 1,1704.00 2.50 1.60 138.07 46.55 101.23 3,615 1,2913.20 2.20 1.60 145.38 51.02 102.47 3,654 1,419

3.00 2.20 1.00 148.14 51.02 113.31 3,677 1,5132.50 1.80 1.00 152.71 56.98 114.09 3,702 1,6102.00 1.40 1.00 157.27 62.94 114.86 3,722 1,711

~:i:"iiJ~ 3,722 1H=r, D!Ii)f-l'W~5flJ~~ 1,711 1}Tf' 5HJIJmJ::f9Ij£f:j 106.28 %& 171.96 % ' mN.D!Ii~~;fIJ~~w':

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Y n = 15.2222N-0.O58333N1

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-107-

An Economic Analysis of .Rice Production and the

Application of Fertilizer Nutrients in Taiwan

Hung-Teh Chen

I. Introduction

Fert i lezer use in Taiwan is enormous. 659.017 metric tons of

-74.70%. of the chemical fertilizers-and several rni l ion tons of

compost were dumped on paddy fields in 1957.

How fertilizer use is related to the soaring rice yields can

be illustrated by a few figures. In 1948. only 128,770 metric tons

of chemical fertilizers were used. but in 1957. the consumption

jumped to 659,017 metric tons. Rice output in the same period

rose from 1,068,400 to 1,839,000 metric tons.

Indeed, farmers in Taiwan know the value of chemical fer-

tilizers. But how to use them effectively, economj caljy , and

safely poses a big problem.

So far, there are many stud.ies on the optimum application of

fertilizer nutrients, but their interests are largely concentrated

on the problem of the physical quantity. In fact. the economic

profit only is thoroughly considered by all the farmers.

Rice yield is known to be a function of innumerable produ-

ction factors. such as land, capital, labor, and so forth. This

statement can be expressed in a function form as fol low s:

Y =f(XI. X" x., ,Xu)where Y stands for rice yield and X1.X2 ....... ,X" for each produc-

tion factor. But for the convenience of study as well as in the

short run period as one year. many production factors except

the chemical fertilizers may be taken as constant factors. There-

fore. rice yield becomes a function of chemical fertilizers.

which are hereafter symbolized by N, P, and K s tandirig for

fertilizer nutrrents of nitrogen. phosphate. and potasium respec

t lv el y . Such a function appears as follows:

Y = f (N, P, K,)

The above function is generally called the fertilizer respons

curve or the crop production curve.

Three important types of soil are considered in this analy-

sis: (0 sandstone and shale alluvial soils. (2) slate alluvial soils.

and (3) red soils. Taipei City, Yuanlin Township, and Taoyuan

- 35-

Township, respectively. are chosen to represent the above three

types of soils.

The original data used in this analysis are fertilizer experi-

ments completed on the above three places for the first crop of

paddy rice in 1953.

Then the same procedure of analytical methods is employed

separately on each type of soils. And the analysis of the ferti-

lizer experiment in Taipei City is presented in this summary.

only the important finding of the other two, Yuanl in Township

and T'aoy uan Township. appear in the appendixes attached to

this summary.

The fertilizer experiment completed in Taipei City is shown

in Table 1.

Table 1. Fertilizer Experiment of Rice

in Taipei. Ist Crop of 1953(Unit: Kg/Ha)

NP K

0 60 120----_._----. ~--- - -- -----------

0 0 1,190 1,410 1.765

0 60 860 1.880 1.910

0 120 1,150 1.645 2,375

60 0 1,160 2.480 2,330

60 60 1.120 2,050 2.535

60 120 1,640 1.940 2.120

120 0 1.215 1.970 2,600

120 60 1.720 1,900 2,190

120 120 1.575 2.685 2,685

Notes: (I) This experiment was done by Taiwan ProvincialAgricultural Experiment Station in Taipei, 3x3x3designed in one repl lcat ian .

(2) N, P, arid K stand for nitrogen, phosphate andpotasium nutrients respectively.

(3) The result of analysis of variance shows that N issignificant at p == 1%, and P at p = 1096.

The result of fertilizer experiment shows us the practical

relationship of rice yield and the amount of fertilizers used. and

accordingly. the production function may be derived according

-36-

to the statistical method. of curve fitting.

Several models are commonly employed in deriving production

functions. The important ones are: (1) Spillman function; it is

very commonly used in fertilizer response curv e as) Cobb-Douglas

function; it is also very widely used in economic analysis, (8)

'Quadratic function; it can reprsent the increasing and de-

creasing of the product with respect to fertilizer applied, (4)

square-root function; its characteristcs is the same as the

quadratic function, and (6) other functions.

The choice of model is very important in economic analysis.

Each of the above models has its own d ef ect.s of its qualities.

In this article the quadratic and square-root, forms are selected

for the analytical models. These two models are recently

widely adopted by research ers in the United States and are

first applied in Taiwan.

The result of analysis of variance-- to the test effect of

each factor, involving effects of separate nutrient and interactions

between them, upon the rice y ield-« shows that only the effects

of Nand P are significant. Therefore, only N an dP are considered

in the deriving of the prod u ct ion function as:

Y=f(N, P).

After the comparisons of the multiple correlation coeff ic il ents

and standard errors among all the various functions derived,

the most representative one is chosen as:

Y =1012.78+ 15. 2222N+7'. 7824P-0.058333N2-0.031250ps •..•..••.•..... (1).R=0.8904 (R=multiple correlation coeff ic ient.)

II. Predicted Rice Yield and Production SurfaceRice yields Can be predicted from equation (1). If we give

any arbitrary numbers of Nand P in the right hand side, then

the rice yield Y Can be predicted as Table 2.

Table 2 Predicted Rice Yield, Taipei (Unit; Kg/Ha)

P N

0 40 80 120 1600 1,013 1,528 1,857 1,999 1,955

4') 1,274 1,790 2,119 2,261 2,21680 1,435 1,951 2,28) 2,422 2,378

120 1,497 2,012 2,341 2,483 2,439160 1,458 1,974 2,302 2,445 2,400

- 37-

The fourth row P=40 which in function form is Y=f(N.40).

shows that the rice yield varies as N changes. Meanw hl le ; the

second column shows the result of Y = f(O. P). From the above

table we find that the effect of N is greater than that of P.

Each column or row shows that nlce yields change at first in-

creasingly and then decreasingly.

The diagram of the production function is shown in three

dimensions and forms the production surface as Figure 1.

Fig. 1 Predicted productionsurface of rice yieldin Taipei

IIIIII

........... t',!.

"'~">,",.... I

... ~>:P

''lP\,,"'tf'¥~

From the figure we get the clearer view of the relationship

between the rice yield and two fertilizer nutrients applied.

The prod uct ion surface shows

"""" first increasing and the succes-

sive decreasing along' the N-and

P-direction respectively, as if a

"""" surface of a part cut from an

apple. We can also imagine that

there is the highest point. which

represents the maximum rice yield

with respect to certain amounts

of Nand P applied. exi st i ng at

somewhere on the production sur-

face. The fact of the steeper slope

along N-direction than. that along

P-direction tells us that N is mone

effective than P in the contribu-

tion to the rice yield.

Marginal physical product i-

v it y of each nutrients. which means

the increment of rice yield resulting from adding a unit of

f ertd l i aer nutrient. can also be derived from the production

function by finding the partial derivatives as follows:

The marginal physical productivity of N is

aY IaN = lS.2222 - O.1l67N·········(Z)The marginal physical productivity of P is

aYlap =7.7824 - O.0625P •..••. (3)

The above two equations represent the linear variation of

the marginal physical prod uct.iv it ies with respect to Nand P

separately. because equation (2) and (3) are two downward

-38-

straight lines and the marginal physical p roduct iv it ies decrease,

of course, as the inputs increase. The marginal physical pro-

ductivities at selected -Ievels of Nand P are shown in the tablebelow.

Table 3 Marginal Physical Prod.uct iv it i es of Nand PTaipei (Unit: kg.)

"

If a vertical plane QRST of the base plane cuts into the pr.o-

duction surface at K==40. the Intersect.lng parts forming a curve

as LL in the figure is known as the production curve with

respect to N at K==40. With the same procedure. two figures

represent the production curves of Nand P are shown in Figures

3 and 4.

The vertical dot lines represent the limits of the original

fertilizer experiment. The forms of the production curve of each

single nutrient are parabolic because the original production

function is in quadratic forrn.

Three production curves of N at P == 0,40 and 80 are shown in

Fig. 3. Each curve increases at first. reaches its highest at

about N =130. then decreases gradually. The distance betweenthe lower two is wider than that of upper two. This fact also

shows that the cont rIb ut ion of the f trst 40 Kg/Ha of P is greater

than the successive input of 40 Kg/Ha. The curves in Firgur'e 3

are always steeper than, that in Figure 4 because the effect of

y(f5/, r----......--.-- y(""""),.....----.---.--~

-

--N~40

----·-N·~eo

.. _ ..···...=·0

l.,,;·-----_·_--t ."--

oo

__ ..-.t----_~",.- :

- /'/ I

-·,.-...;_....... .•••,I·II

... ---- P-o-- P-Ao_._._p-..

•oo

",.--.;.-...../'

//-:

---

• L----L.---'----'--_-t-_..Jo 4D .., 120 .-

P (~/ ... )

Fig.3 Rice Production curvesof N in Taipei

Fig.4 Rice production curvesof P in Taipei

N is greater than that of P.

(II) The production curves of two fertilizer nutrients applied

under constant ratios can also be derived from the crd g lnal

product ion function. If we let N = aP. i. e. the amount of N used

-40-

the above procedure, several production curves

Nand P applied with different ratios are

shown in Figures 6 and 7.

After the comparisons of the

three production curves in Fig. 6.

we conclude that the combination

.... ratio of P=1.25N is the besb. But ,

in fact. a farmer does not like to

adopt this combination ratio be-

__ cause not only the fact that was

mentioned in the previous section

that the effect of N is twice grea-

low ter than that of P but also the

is~ times of P, the production function of two fertilizer nutri-

ents as Y = fCN. P) may be transformed into Y = fCa P, P) or Y =

fCP). Finally, the problem is transformed into the relationship

between the rice yield and phosphate nutrient applied.

The same concluston may be got from the production surface

in Figure 5. If a vertical plane QRST runs from the origin in thedirection of N on the base plane a production curve LL is got rep-

resenting rice yields with respect to K as well asN =K is applied.According to

with respect to

relative prices of fertilizer nutri-

ents should be considered. Since

the production curvs in Figure 7

are more practical. The following

Fig. 5 Illustration of rice findings are recommended. Whenp rod uct lon curve oftwo nutr.ients applied a farmer has a small amount ofwith a constant ratio P nutrfent , i. e. P is less than

48 Kg!Ha, the ratio N = 3P is most recommended; when the

amount of P nutrient is between 48 and 72 Kg{Ha, the ratio

N = 2P is recommended; when the amount of P nutrient is rich

enough. the ratio of N = P is recommended.

The above ratios are profitable when they are considered

only from the physical stdevtf the economic profit. in addition,

is taken into consideration the ratios will .change ,

(8) An analysis of the problem. Min what range and how ferti-

lizer nutrients can substitute one another,?" is introduced below:

An interesting question arises when we imagine a plane

-- 41-

y r----.------~---.-...,(~)

----- p- "-- P-o.T6,.-·--P·'~S"

-//._. ". ",,'". ..."", ...."""/ r > ,

I /'~ ,,'. ,

I,'. ,----- tJ- P

--IoI:ilP

-----1oI-3P

Fig. 6 Rice production cur- Fig. 7 Rice production cur-ves of P with constant ves of N with constantratio of N applied in ratio of P applied inTaipei Taipei

parallel to the base plane cut i ng the production surface, What

does the part of their intersection mean? Figure 8 is devised to

solve this question.

The plane QRST, paralleling to the base plane at a height

of Y:=2.800 KgjHa , cuts the production surface at LL which is a

curve. The height of each point on the curve LL is constant at

Y=2.800. Then. LL is projected to the base plane as L'L'. Since

the curve L'L' represents the d lfferent combinations of Nand K

to produce the rice yield fixed at 2.800 Kg!Ha, the curve L'L' is

thus called the iso-product curve. Now. four iso-product curves

are shown in Figure 9 to discuss their characteristics: All the

i so-product curves are convex to the origin, and the larger the

yield the farther they shift upward, and accordingly, there

exists only one combination of Nand P, 130.78 KgjHa and 124.52

KgjHa respectively, to produce the highest yield of 2.490 Kg!Ha

shown as M in the figure. Meanwhile, none iso-product curves

intersect one another. And after a view from technical side two

border lines. AM and BM, are plotted to separate the profitable

and unprofitable parts of the application of fertilizer nutri-

ents. Hence. the area enveloped by OAMB is taken into consi-

-42 -

M(~901

and P to produce Y=2,OOO and 2,200 and their marginal rates of

substitution are shown in

Fig. 10 Iso-product curves of. Nand P in Taipei

Table 4. The combinations show

that P becomes smaller while

N is greater. and vice versa.

This Is shown clearer in columns

3 and 4 as the marginal rate of

substitution of N for P. or of P

for N respectively.

The tangent lines at each point

upon the iso-product curve are

called the marginal rate of

substitution (MRS) between two

nutrients, Le.

MRS=dP/dN. or MRS=dN/dP.

To take column 3 for example, the

MRS of N or P is -1.64, -1.20,' ••

and -0.31. This is the so-called

decreasing marginal rate of subs-

titution.

,,- 1t3

".... k'"""'k3

1'300 '"19lx>":j

Table 4 Fertilizer NutrientsCertain Rice YieldSubst ltt.ut ion , Taipei

Combinations for producing aand Their Marginal Rates of

(Unit: Kg/Ha)

Rice Yield Y = 2.000 Rice Yield Y =2.200

N P dP/dN dN/dP N P dP/dN dN/dP

60

70

8090

100

110

44.39

30.3619.94

12.10

6.382.41

-1.64

-1.20

-0.90

-0.63

-0.48

-0.31

-0.61

-0.83

-1.11

-1.49

-2.08

-3.18

60

70

80

90

100

110

120.00

74.87

57.18

45.56

37.58

32.28

-29.05

-2.27

-1.40

-0.96

-0.65-0.42

-0.03

-0.44

-0.71

-1.05

-1.53

-2.40

IV. Economic Optima of Fertilizer NutrientsSince all combinations of Nand P on an iso-product curve

may be applied to produce the same rice yield. there arises the

question, "which combination is the best?" In order to answer

this question we take for granted that the combination with

the least cost is the best.

-44-

The least cost of fertilizer combination, according to the

principles of production economics. is the point where marginal

rate of substitution of two fertilizer: nutrients equal to their

inverse price ratio as:

MRS= dN/dP= Pp/Pn. (Pp and Pn stand for the prices of P and

N respect tvely ,')

Then, these least cost points are connected to form the least

cost curve (a line in this analysis) or the expansion line which

means a farmer adjusts the scale of production with regard to

his capital.

Marginal rate of substitution between two nutrients can

also be derived from the producbton function according to the

following procedure as:

MRS= dN/dP=(dY/dP)/(dY/dN)

This procedure shows that MRS of P for N is the ratio of the

marginal productivity of P, Le. dY/dP, to the marginal produ-

ctivity of N. I .e. dY/dN.

Hence. the least cost combinations of two fertilizer nutrients

are derived as:

dP 15.2222 - O.1167N PndN= 7.7824-0.0625P=pp=a

Then, the above equation is simplified as:

N = 130.48-aC66.706 - O. 5357P)

When a is given, there exists a linear relationship between Nand P, therefore, it is called the least cost line or the expansion

line.

When a= Pn/Pp = 1.80,2.25 and 3.00 three expansion lines are

plotted in Figure 11. If.!! is smaller , the expansion line tends tomove to the left hand side. In common parlance, the cheaper

the price of the more it is applied.

A line OA representing N =2P considered only from technical

side but not from the least cost principle also suggests many

combinations of Nand P in rice production. But. such a dect-

sion. very clear in the figure, is not profitably adopted by the

farmers.

The intersections of each expansion line, Pn/Pp = 1.80 and

. 2.25 separately, and the iso-product curves are shown in Table

5.

-45 -

l.l(Io/"')r-~-~-...--..-----.....,

Fig. 11 Iso-product curves andleast cost lines of Nand P in Taipei

When Y increases, the nee-

ded Nand P also increase. And

when PnJPp = 1.80 changes to

2.25, meaning a relatively high

price of N or low price of P,

the needed N decreases but

needed P increaes in order to

produce the same rice Yield.

Any point on the expansion

line is the least cost combina-

tion of two fertilizer nutrients,

but, which is the most profita-

ble one?

Here. the price of rice must

be taken into considerasion.

Let 7t stand for profit; Py ,

Pn , and Pp for p rices of Y, N.

and P respectively. Then 7t can

be written as:

7t=PyY - PnN - PpP

In order to get the largest 7t (the maximum profit), the first

partial derivatives of above equation should be equal to zero.

Thus.

fj7tIaN = Py(fjYfj/N) - Pn= 0, or fjYlaN;= PnJPy

fj7tJap=py(aY/fJP) - Pp=O, or fjY/fjP=Pp/Py

Also. aNJaP = Pp/Pn can be derived from above two equat ions ,

This shows that the maximum profit point always exists on the

least cost line.

According to this procedure the maximum profit point of

equation (1) is obtained as follows:

15.2222 - 0.1l67N = Pn/Py = an7.7824 - 0.0625P=Pp/Py=ap

or simplified as:

N = 130.48 - 8.571400P = 124.52 - 16.0000ap

-46-

Table 5 Least Cost Combinations of Nand P under DifferentPrice Ratios. Taipei (Unit: KgfHa)

Y PnfPn=1.80PnfPp=2.25

N P N P1,800 43.83 34.66 37.49 47.371,900 50.35 41.42 44.49 53.182,000 57.45 48.78 52.11 59.502,1000 65.33 56.95 60.55 66.512,200 74.29 66.24 70.17 74.49

When different values of an· and ap are given Table 6 is

tabulated. It also shows that the amounts of fertilizer nutrients

applied to get the maximum profit. in other words. the economic

optimum amounts of fertilizer nutrients. are subject to change

with the fertilizer-rice price ratio; the higher the ratio. the

smaller the amount of fertilizer nutrients are used and vice versa.

Table 6 Economic Optima of Fertilizer Nutrients under Differ-ent Fertilizer Rice Price Ratios. Taipei

PnfPy

6.005.505.004.504.00

NEconomic Optima

of N (KgfHa)

79.0583.3487.6291.9196.19

PpfPy

3.893.332.782.221.67

P

Economic Optimaof P (KgfHa)

62.2871.2480.2489.0097.80

Economic Optima of Fertilizer Nutrients and the Pre-dicted Rice Yield, Taipei (Unit: KgfHa)

Substitute the economic optimum amounts of fertilizer nutri-

ents in the production, function (1), the respective rice yield is

predicted and the gain, from the fertilizer applied is calculated

by reducing the fertilizer cost from rice yield. Such result is

shown in Table 7.

Table 7

PnfPy

6.006.005.005.004.004.00

PpfPy

2.782.222.782.222.782.22

N

79.0579.0587.6287.6296.1996.19

P

80.2489.0080.2489.0080.2489.00

PredictedRice Yield

2,2752.2972,3222,3452.3612,382

Gain from Fer-t.i ltzer Applied

565

612648695740787

-47-

Since most of the chemical fer.tilizer is sold to the farmer at

a fixed price and quantity by the government and a farmer

must pay the price with rice, Table 8 is compiled, for the farmer

to decide the scale of production.

Table 8 Economic Optima of Fertilizer Nutrients, the PredictedRice Yield and the Gain from Fertilizers Applied.Taipei

CA) Economic Optima of Fertilizer Nutrients and the PredictedRice Yield (Unit-Kg/Ha)

~- PN/PY

fertilizers applied. 1,013 Kg .• and the fertilizer cost. 661 Kg.

V. Comparrisons between the Results ofthis Analysisand the Status Quo

To conclude this article, the comparisons are made between

the optima of this analysis and the actual s i t uat iorn . This com-

parison may be interesting and meaningful to the workers who

concentrate their interest in the fertilizer problem in Taiwan.

Before making the comparisons. It is worth d'iscussirig the

characteristics of fert.ilizer problem in Ta i w ant u) All the chemical

fertilizers are sold to the farmers by the Food Bureau of Tai wan

Provincial Government. Hence. this function may be called a

rationing system. (2) The farmers must pay rice instead, of cash

to the Food Bureau for the fertilizers they purchase. (8) The

price of fertilizer, commonly called the exchange ratio of rice

for fertilizer. is fixed by the Food Bureau, and is not subject to

change in a pnoduct ion period or a year. (4) The rationing amount

of fertilizer is fixed in each county, except a small feasibility

allowed. (5) Different fertilizers are sold to the farmers in a

constant combination ratio in quantity. (6) The fertilizer loan

helps farmers expand their scale of production. (7) The scale of

rice production will not change when the price of rice changes

but exchange ratio of rice 'fertilizer remains unchanged.

Three separate Comparisons are made

(l) The comparison between economic optima of fertilizer

nutrients and the r:ationing amounts of ]954's. It is seen from

Table 9 the amounts of nitrogen get closer, but the amounts of P

and K differ greatly.

Table 9 The Comparison of Economic Opirna and the Practical

Rationed Amounts ef Fertilizer Nutrients (Unit: KgjHa)

N P KEconomic Rationed Economic Rationed Economic RationedOptima Amcunt Optima Amount Optima Amount

Taipei 88 80 80 27 -* 10Yuanlin ]45 100 23 36 -* 10Taoyuan 128 90 42 29 93 15

Source: Data of rat ioned amounts from the Bulletin of TaiwanProvincial Government

.. Data not available

-49 -

These difference, specially distinct in P and K. may be

caused by two factors: (a) The fertilizer inventory of the Food

Bureau is so small that it can't allow more rationing amounts,

and (b) The chemical fertilizer may be used' in many lines of

crop production. Thus, the rationing amount should be conside-

red upon different productivities occuririg from different crop

productions.

(2) The rationing amounts of fertilizer varies in different

counties and years. Here, the rationing amounts of nitrogen in

three' places in 1954 and 1959 are compared.

Table 10 The Comparison of Rationed Amounts of NitrogenNutrient in 1954 and 1959 (Unit: KgjHa)

PracticalRiceYield·

1954 1959R' d Mrginal Practical R t i ed Marginal

at ione Practical. Rice a Ion Physical.Amount Productivity Yield. Amount Productivity

Taipei 80 5.89 844 60 8.22 703

Yuaulin 100 10.70 1.709 120 8.15 1,898

Taoyuan 90 8.35 1,201 90·· 8.35 1.218Total 270 3,754 270 3,819

Source: Same as Table 9• Marginal physical productivity and predicted nice yield are

calculated from this analysis .•• The real rationed amount is 92 Kg/Ha (with MPP =8.12). but

shown in the table with 90 KgjHa for the convenience forcomparison that the two rationed amounts are the Same.

The same amount of 270 Kg. of nitrogen nutrient is rationed

in 1954 and 1959 in three places. while the rationed amount in

Taipei decreaes by 20 KgjHa; but that in Yuanlin increases by

20 KgjHa; and Taoyuan remains unchanged. Their respective

marginal physical product iv it.i es calculated according to this

analysis are Widely different in 1954. but closer in 1959. The

rlce yields are also predicted from the production functions.

The total -out puts of these two years are seen in the last row,

3,754 Kg. in 1954 and 3.819 kg with the same input of 270 Kg. of

nitrogen nutrient. Therefore, the conclusion is that the rationed

amounts of 1959's is better than that of 1954's.

(8) The rationed ratios among three fertilizer nutrients are

-50-

shown in Table 11.

Table 11 Rationed Amounts of Ferttliaer Nutrients and TheirRatios (Unit: Kg/Ha)

Year N P K Katio

1954 80 27 10 N=3.0PTaipei

1959 60 29 12 N=2,lP

1954 100 36 10 N =2.8PYnanlin

1959 120 43 30 N=2.8P

1954 90 29 15 N=3.1P=6KTaoyuan

1959 92 43 30 N =2.1P =3.1K

Source: Same as Table 9

The thorough analysis of production curves of two fertilizer

nutrients applied under constant ratios is completed in the pre-

vious section. Then. the conclusion is that the rationed ratios

among three fertilizer nutrients in three places is also better in

1959 than that in 1954.

But we know already that the above ratios are profitable

considered from physical side only, i. c. a greater rice output is'

got from the application of fertilizer under a certain ratio. But

this greater rice output does not necessarily represent the max-

imum economic profit to the farmers.

VI Conclusion

Because the experimental design is originally for the soil

scientists. the levels of experiment (3 levels for each fertilizer

nutrient) are not sufficient for fitting a good production function.

One interesting fact is that there is a coincidence of the

results of this analysis and that of works done by soil scientists

as shown in section V. (Though many assumptions are made for

that comparison.)

Further, the result of this analysis can be used for designing

a fertilizer 'experiment for economic analysis in the future. (This

will be treated in another article)

However, this article will serve as an introduction of a new

research method to the agrfcultral economists in Taiwan.

- 51-

150107-2-1150107-2-2