雙資本三生產函數資本均衡配置不存在?

 

雙資本三生產函數資本均衡配置不存在?

no capital equilibrium to triple production function and double capital ?

K1+K2+K3=K
L1+L2+L3=L
產品Y1,Y2,Y3單價P1,P2,P3
欲最大化產值須使
P1δY1/δK1=P2δY2/δK2=P3δY3/δK3......(1)
P1δY1/δL1=P2δY2/δL2=P3δY3/δL3......(2)
若皆Cobb-Douglas函數
Y1=c1 K1^a1 L1^b1
Y2=c2 K2^a2 L2^b2
Y3=c3 K3^a3 L3^b3
兩式相除(1)/(2)
a1/b1 L1/K1=a2/b2 L2/K2=a3/b3 L3/K3=1/t
則K1=a1/b1 L1 t, K2=a2/b2 L2 t, K3=a3/b3 L3 t
Y1= c1 (a1/b1 t)^a1 L1^(a1+b1)
Y2= c2 (a2/b2 t)^a2 L2^(a2+b2)
Y3= c3 (a3/b3 t)^a3 L3^(a3+b3)
代回(2)
P1 b1 c1 (a1/b1 t)^a1 L1^(a1+b1-1)
=P2 b2 c2 (a2/b2 t)^a2 L2^(a2+b2-1)
=P3 b3 c3 (a3/b3 t)^a3 L3^(a3+b3-1)
若三者皆為規模報酬不變a1+b1=a2+b2=a3+b3=1
P1 b1 c1 (a1/b1 t)^a1 =P2 b2 c2 (a2/b2 t)^a2 =P3 b3 c3 (a3/b3 t)^a3
P1 c1 a1^a1 b1^b1 t^a1 =P2 c2 a2^a2 b2^b2 t^a2 =P3 c3 a3^a3 b3^b3 t^a3
由P1 c1 a1^a1 b1^b1 t^a1 =P2 c2 a2^a2 b2^b2 t^a2 得t=t1,
由P2 c2 a2^a2 b2^b2 t^a2 =P3 c3 a3^a3 b3^b3 t^a3 得t=t2,
t1不一定會等於t2, 無均衡配置?

動態配置過程迴圈

若a1+b1=a2+b2=a3+b3=1
給定L1,L2,L3套用(1)式而令=1/t
P1a1c1 K1^(a1-1) L1^b1=P2a2c2 K2^(a2-1) L2^b2=P3a3c3 K3^(a3-1) L1^b3=1/t
K1= L1 (P1a1c1 t)^(1/(1-a1))
K2= L2 (P2a2c2 t)^(1/(1-a2))
K3= L3 (P3a3c3 t)^(1/(1-a3))
由K=K1+K2+K3可解出t代回而得K1,K2,K3
再給定此組K套用(2)式而令=1/s
L1= K1 (P1b1c1 s)^(1/(1-b1))
L2= K2 (P2b2c2 s)^(1/(1-b2))
L3= K2 (P3b3c3 s)^(1/(1-b3))
又可由L=L1+L2+L3解出s代回而得L1,L2,L3
然後再套用(1)形成迴圈


實例:
若a1=0.2,a2=0.5,a3=0.8,P1c1=P2c2=P3c3=1,K=3,L=6
K1= L1 (a1 t)^(1.25)
K2= L2 (a2 t)^(2)
K3= L3 (a3 t)^(5)
由K=K1+K2+K3可解出t代回而得K1,K2,K3
再給定此組K套用(2)式
L1= K1 (b1 s)^(5)
L2= K2 (b2 s)^(2)
L3= K3 (b3 s)^(1.25)

由(L1,L2,L3)=(2,2,2)開始
t=1.23568, (K1,K2,K3)=(0.34850,0.76345,1.88803)
s=2.08816, (L1,L2,L3)=(4.53387,0.83224,0.63387)
t=1.49219, (K1,K2,K3)=(1.00008,0.46327,1.53664)
s=1.74075, (L1,L2,L3)=(5.23804,0.35095,0.41094)
t=1.61777, (K1,K2,K3)=(1.27821,0.22963,1.49216)
s=1.67123, (L1,L2,L3)=(5.46051,0.16034,0.37923)
t=1.65093, (K1,K2,K3)=(1.33250,0.10491,1.37701)
s=1.66452, (L1,L2,L3)=(5.69243,0.07325,0.34996)
t=1.67250, (K1,K2,K3)=(1.44808,0.05123,1.50072)
s=1.63792, (L1,L2,L3)=(5.59379,0.03436,0.37192)
t=1.66530, (K1,K2,K3)=(1.41533,0.02382,1.56087)
s=1.64549, (L1,L2,L3)=(5.59481,0.01612,0.38906)
t=1.65523, (K1,K2,K3)=(1.40490,0.01104,1.58403)
s=1.64805, (L1,L2,L3)=(5.59690,0.00750,0.39561)
(K2,L2)趨近於(0,0),Y2最後停止生產,
(K3,L3)趨近於(1.6,0.4),
(K1,L1)趨近於(1.4,5.6)

1/t為K邊際產值,1/s為L邊際產值