規模報酬不變的生產函數
生產要素用量X1,X2,...,Xn, 生產函數 Y=F(X1,X2,...,Xn)
規模報酬不變則有 F(tX1,tX2,...,tXn)= t F(X1,X2,...,Xn)
d F(X1,X2,...,Xn)= Fx1(X1,...,Xn) dX1 + Fx2(X1,...,Xn) dX2 +...+ Fxn(X1,...,Xn) dXn
d F(tX1,...,tXn)= Fx1(tX1,...,tXn) d(tX1) + Fx2(tX1,...,tXn) d(tX2) +...+ Fxn(tX1,...,tXn) d(tXn)
= t [Fx1(tX1,...,tXn) dX1 + Fx2(tX1,...,tXn) dX2 +...+ Fxn(tX1,...,tXn) dXn]
又 d F(tX1,...,tXn)= d[t F(X1,X2,...,Xn)]= t [Fx1(X1,...,Xn) dX1 + Fx2(X1,...,Xn) dX2 +...+ Fxn(X1,...,Xn) dXn]
因對任意(dX1,...,dXn)兩式皆須成立,可得多維相似形效果
Fx1(tX1,...,tXn)=Fx1(X1,...,Xn),
Fx2(tX1,...,tXn)=Fx2(X1,...,Xn),
Fxn(tX1,...,tXn)=Fxn(X1,...,Xn),
今欲知空間中一點(A1,A2,....,An)產量F(A1,A2,....,An)
連接該點與原點線上之點可表為(tA1,tA2,....,tAn)產量F(tA1,tA2,....,tAn)
該線上點與點間(t變,A1...An不變)之dF
dF(tA1,tA2,....,tAn)=Fx1(tA1,...,tAn) d(tA1) + ...+ Fxn(tA1,...,tAn) d(tAn)
=Fx1(A1,...,An)A1 dt + ...+ Fxn(A1,...,An)An dt
又dF(tA1,tA2,....,tAn)=d[t F(A1,A2,....,An)]= F(A1,A2,....,An) dt
則兩式又可得F(A1,A2,....,An)=Fx1(A1,...,An)A1+...+Fxn(A1,...,An)An
寫回Y=F(X1,X2,....,Xn)=Fx1(X1,...,Xn)X1+...+Fxn(X1,...,Xn)Xn
1=(∂F/F)/(∂X1/X1)+...+(∂F/F)/(∂Xn/Xn)= ε1+...+ εn
若單位要素成本為P1,P2,...,Pn,商品價格P
固定總成本預算下最高產量發生於Fx1/P1=Fx2/P2=...=Fxn/Pn=1/t時
總成本=P1X1+P2X2+...+PnXn=t(Fx1 X1 + Fx2 X2 +...+ Fxn Xn)=tY
總產值=PY=P(Fx1 X1 + Fx2 X2 +...+ Fxn Xn)
利潤為(P-t)Y = (P-t)(Fx1 X1 + Fx2 X2 +...+ Fxn Xn)
(若完全競爭使利潤為0,則t=P,Fx1=P1/P,...,Fxn=Pn/P)
